Sheepish Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⣠⠴⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⠀⣸⠁⠈⣧⢄⣰⠟⠁⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⡋⠁⠀⠀⠀⠀⠁⠀⠐⠲⠶⠆⠀⠷⢦⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀⢀⠀⠀⣠⠴⠚⠛⠳⢤⡄⠀⠀⠀⠉⠈⢷⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠒⠢⢄⡀⠀⠀⠀⠉⢻⠀⡼⠁⠀⠀⠀⠀⠀⠹⡄⠀⠻⢆⠀⠈⠉⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈⠑⠒⠤⠄⢀⢘⡇⠀⠀⠀⠀⠀⣶⣀⡥⠒⠒⠢⣄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠂⠀⠀⠀⠒⠺⣷⡀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⠀⠀⠀⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡙⠦⠄⠠⠤⠚⠛⢁⠀⠀⣠⡟⠀⣰⠷⢬⣯⣄⣤⠤⠤⠤⠒⠒⠚⠋⠉⠙⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠀⠀⠀⠀⠀⠀⠀⢹⡏⠁⠀⠀⣸⠳⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠴⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠉⠉⠛⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠈⠀⢰⠃⠀⠀⠀⠀⠀⠀⢴⡃
⠀⠀⠀⢀⠴⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⢀⠉⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠀⠀⣰⠃⠀⠀⡠⠓⠠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢄⡠⠃⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠘⡯⠹⡆
⠀⠀⣏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠠⢀⠔⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠑⠀⠀⡠⠉⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡀
⠀⠀⠀⠈⠙⠲⠤⠤⠄⠀⣀⡀⠀⠤⠤⠐⠚⠋⠁⠀⠀⠀⠀⠀⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠠⡤⠒⡶⠂⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠠⡤⠤⠤⠄⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠟⠒⠓⠒⢦⡀⠐⢏⠀⠀⠀⣠⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡀⠀⠀⠒⠚⢫⡀⠀⣼⠀⣠⠔⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠊⠁⠀⠀⠹⡖⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡰⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠞⠂⠁⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠓⠖⠶⠤⠦⣄⣀⣀⣠⣴⣶⠚⠓⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣾⠀⢹⡀⠀⠀⠀⠀⠈⠉⠉⢡⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⡥⠤⠀⠀⠈⠉⠓⠀⢳⡀⢀⡤⠄⠒⠒⠶⠸⢇⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠦⠤⣤⣤⡤⠤⠤⠖⠚⠁⢼⣛⠉⠉⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠃⠀⠂⠁⠀⠘⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇
⠀⠀⠀⠀⣴⣿⣿⠟⠛⠛⠿⠟⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣿⣿⣿⠁⠀⠀⢀⣀⣀
⠀⠀⠀⢸⣿⣿⡁⠀⠀⣀⣠⣤⣤⣤⣿⠿⢿⣿⣿⣿⠿⡿⣇⣤⣶⣿⠿⣿⣷⣤⣿⣿⣷⣶⣾⣿⣿⣿⣿⡿⢦⡀
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⠀⣿⣿⠁⠀⠘⣿⣿⡏⠀⠀⢈⣿⣿⣿⣿⡿⣿⢛⡛⡛⠉⠁⠒⠀⢻⡀
⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⢸⣿⣿⡇⣿⣿⠀⠀⠀⠘⣿⣿⣦⣶⣿⠟⢹⣿⣿⡗⣿⠈⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄⣀⣀⣤⣾⣿⠟⢠⣿⣿⣶⡄⠀⢸⣿⢿⣿⡏⠀⠀⢸⣿⣿⡇⠈⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠿⠛⠁⠀⢸⣿⣿⣿⡇⠀⠘⣿⠈⢿⣿⡄⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⣧⣤⣴⡿⠀⠈⠉⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⡀⠀⠀⢀⠀⡀⣀⡀⠀⠀⢀⢀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠖⠋⠉⠙⠓⢦⡀⠀⠀⠀⠀⢀⡀⢀⠀⢀⢀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠉⠓⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣄⠹⠄⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠒⠂⠤⢤⣄⡀⠀⠀⠀⠰⡇⠀⠀⠀⠀⢀⡠⠞⠋⠀⣠⠖⢙⣾⠳⣄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣞⣀⡀⠀⠀⢉⣠⣴⢎⡔⠁⠀⠘⠛⠀⢸⡃⣀⣀⣀⣀⣀⣀⣤⣤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⣤⣄⣀⣀⣀⣀⣀⣀⣨⠇⠀⡿⠉⠀⠉⢸⡀⠀⠀⠀⠀⠀⣸⠥⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⣳⣄⣀⣀⣸⣄⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡟⠈⠩⠋⠁⠈⢓⣲⣶⣪⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⢀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠊⢇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⡴⠋⠁⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣅⣀⣸⠀⠀⡰⠋⠀⢢⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⣁⣀⣀⡤⢧⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⡤⠞⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠐⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⡤⠤⠴⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⣶⣶⣶⣶⠖⠒⠲⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠤⠠⡇⢸⠀⠼⠄⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣶⡖⠒⠒⠂⠐⣷⠀⠂⠐⢀⣶⣿⡷⠆⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠆⠒⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⢀⣠⣤⣤⣾⣦⡀⠀⠀⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⡀⠀⠹⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⣠⣴⣾⣿⣦⠀⠀⢠⣿⣿⣿⣶⡄⠀⣰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣾⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⢿⣿⣿⡧⠀⣿⣿⣿⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⣠⣦⣤⣤⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣦⠀⠀⠀⣼⣿⣿⡿⢁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢿⣿⣿⣷⡀⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠸⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣯⣀⠀⠀⠀⠀⢈⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⡠⠾⢿⣏⣧⡾⠷⠾⣷⡾⠀⠀⠈⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣤⣒⣊⠀⣀⢀⡘⢁⡖⢡⣤⠁⠀⠀⠀⠀⠀⠀⢻⣧⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⣿⡅⠀⠀⠀⠈⠁⠀⠀⠀⢿⣷⣦⣀⣀⠀⠀⠀⣰⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠷⠈⠀⠀⠀⢠⡦⠀⣾⣿⠘⠋⠿⠛⠷⢦⣿⠻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⡞⠛⠁⠀⠀⠀⠀⠀⠀⠹⣷⡿⣷⣿⣿⣿⣿⣿⣿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⠷⣤⣿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣼⣧⠀⠹⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠛⢲⣦⠀⢸⣷⠀⠀⢀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⣀⣀⣀⡀⠀⠀⠐⠀⣾⣾⡷⠀⢽⣿⡶⠃⠈⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣩⣿⣿⠃⠀⠀⠀⠀⠺⠻⠋⠀⠘⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣾⣿⣿⣿⡇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⣀⣴⣿⡿⣿⣿⡟⣻⡿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢡⣦⡶⢶⣾⣿⣾⣿⡇⢹⣿⣇⣬⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⠀⠀⠀⠠⠀⢀⡀⠤⠈⠉⢸⣧⣾⣿⣿⡟⠉⠁⢸⣿⢿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣾⡿⠿⠟⠙⠀⠀⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣆⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⢂⣶⠋⠒⠚⠋⠉⠁⢦⣽⡦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⣴⠃⠁⠈⠃⠀⠀⠀⠀⠀⢀⠛⠻⡿⢦⡝⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⠉⠓⠘⢷⡀⠀⠀⠀⢛⡀⠂⠀⠸⣷⡄⠀⠀⠘⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⢠⡾⠁⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣀⣤⡴⠾⠃⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠉⠁⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠁⣠⣴⣾⠁⣶⣶⣶⣶⣦⢼⣿⠀⠀⠀⠀⠀⣀⣤⡟⠁⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣴⣾⣿⠟⠋⠀⠀⢛⡿⠊⠛⠂⠀⠹⣦⠐⠀⠘⠻⣯⡦⠀⠀⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠿⠟⠋⠁⠀⠀⠀⠀⠸⣿⣦⠀⠀⢀⡴⢾⣿⣦⣒⠁⠈⠱⡘⢦⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠙⠛⠛⡟⠋⢣⠀⠀⠙⣆⠉⠙⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⣾⠆⠀⠀⠀⠀⠀⠀⢀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠈⢻⠳⢶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⢠⡄⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠈⠦⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢼⡆⡌⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣄⡀⠀⠀⠀⠀⠒⢲⣤⣀⠀⠀⠈⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⣀⠀⠀⠠⡙⠿⣿⣶⡤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠈⠙⠶⣤⣈⠁⠙⠋⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡆⠀⠈⠀⠉⠉⠐⠋
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡀
⠀⠀⠀⠀⠀⠀⣀⣾⠏⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢁⡼⠏⢯⣄
⠀⠀⠀⠀⣠⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡿⠖⣄⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⢿⣿⣿⣇⠀⠀⠈⠰⣏⡐⡀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣆
⠀⠀⣰⠟⠀⠀⠤⠀⡠⠃⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠆⠐⠂⠀⢫⠽⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⣰⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⡇
⢀⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡦⠀⡉⠀⠀⢈⠴⠀⠘⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢻⣤⡀⠀⠀⠀⠴⠛⠀⠀⠀⠈⠛⣈⣶⣤⣀⠀⠀⠀⠀⠈⠃⠋⠀⠀⠀⠀⣲⡆⠀⠀⢀⣄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⣈⣿⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠠⣵⠂⠀⠀⠀⠀⡠⠾⠃⠀⣻⣧⣀⡀⠼⠇⢀⣤⣶⣿⣿⣿⣿⣿⡟⢿⣿⣿⣶⣿⣿⣿⣦⡀
⠐⠿⡉⠛⣃⠀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⢛⣴⣄⡀⠀⠀⠀⠀⢀⣄⠛⠽⢷⠶⠃⢸⣿⣿⣿⣿⣿⡿⣿⠃⠈⠛⢋⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⢧⣿⠟⢷⣤⣠⣿⣿⣄⣠⠞⠋⠉⠛⠿⣿⣿⣷⡄⠀⠀⠀⢼⡋⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⢻⣇⠀⠀⠸⡄⠹⢿⣿⠟
⠀⣠⣶⡶⣦⡀⠀⣘⢿⣶⠻⠋⠙⠲⠏⢁⠀⢠⠀⠀⠀⠈⠻⣿⠛⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣦⠀⠘⣿⡄⠀⢠⣿⠀⠀⣽
⡼⣟⠉⠛⢦⢇⣘⠋⠀⠈⠀⠀⠀⠀⠀⠐⠒⠊⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣍⣤⣤⣤⣤⣶⣶⣶⣷⣿⣷⣾⣿⣧⢤⣿⡗⠒⠚⠛⠒⠒⠿⠆⢀
⠀⢿⣧⠀⠈⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣲⣶⣒⣻⣿⣿⣿⣿⣿⣿⢿⣿⡟⣿⣿⢻⣿⣿⣿⣿⣿⣷⣿⠀⡇⠀⡈⠉⠉⠉⡿⠉⢉⡇⢰⡎
⠀⠀⠿⠂⠀⠀⢀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⣿⣿⣿⣿⣾⣿⣿⣿⣿⡧⠀⡇⠸⡇⠀⠀⡼⠀⠀⢨⠃⣼
⠀⠀⣤⠟⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⣠⠀⠀⠀⡇⠀⠀⡈⢀⣿
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⣿⡇⠀⡇⠛⠀⠀⠀⡇⠀⠀⠇⢸⡏
⢸⣇⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⣸⠀⠀⠀⠀⡇⠀⢰⠀⣼⠇
⠀⣻⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⠀⠀⡟⠀⠀⠀⣰⠀⠀⡆⢀⣿
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⢠⡇⠀⠀⢠⡆⢠⠎⠀⢸⡇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠷⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⠁⠀⠘⠇⠀⠀⠘⠁⠸⠀⠀⣾⠁
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠻⡿⣿⣿⣿⣿⣿⣿⢿⡧⠂⣿⣿⣿⣿⣿⣿⣿⡿⠀⠋⠀⠈⠀⠃⠀⠀⠀⠀⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠙⣿⣿⣧⣿⢿⢿⡇⢠⣿⣿⣯⣹⡟⣿⣿⡇⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⠈⠈⠇⠘⣿⣿⢫⡿⠁⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠋⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀⠀⣠⠒⢢⡀⡴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠑⠢⢄⣠⠖⠚⠀⠀⠀⠈⠉⠈⠀⢀⣙⠁⢹⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⡴⠑⠲⠞⠧⠀⠀⠀⠀⠀⡤⠤⠤⠜⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠫⠀⠀⡴⠤⣀⠀⠀⠀⠠⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠣⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠎⠒⠂⠀⠘⠲⡄⠸⠀⠀⠀⣱⡈⢙⡤⣞⡠⠤⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠒⠒⠂⢒⣷⠆⠁⠄⠀⢀⣷⡒⠭⠄⠀⠠⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠄⠒⠉⠁⠀⠀⢁⡁⠀⠘⠁⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠙⠀⠀⠀⠀⠀⠀⢙⣄⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠷⠾⠃⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠆⠀⠀⣠⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣶⢄⠐⠂⠐⡏⠀⠀⠀⠀⠠⢄⣀⠀⣀⣴⠋⠀⡄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣋⡴⠀⠀⠈⠑⠇⢠⠁⠀⠀⠀⠀⠀⠀⠈⠋⠁⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⣠⠄⢀⡄⠠⠼⢤⣀⣊⣀⠤⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠛⠀⢀⠀⠈⣀⡀⡄⣤⢘⣿⠁⠀⠀⠀⠀⠀⠁
⠀⢀⣀⣀⠀⠀⢰⣯⡀⠐⠀⣤⡾⢃⣇⠙⢹⣇⢉⣟⡇
⠀⠛⢤⠀⠐⣠⠾⠹⢅⡠⡾⠀⠂⢘⣠⠆⠸⣧⣾⣿⠃
⠀⠀⠑⠰⠤⣄⠀⠀⠀⠀⣇⡀⠟⠈⢹⢻⣇⡁⣿⠞
⠀⠀⠀⠀⠀⠈⠐⣀⣠⡀⠀⣠⠋⠁⠀⠀⣩⢿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⡿⠫⡟⣁⣤⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠴⡡⠧⣄⣩⠋⠯⠤⠀⠠⠤⣀
⠀⠀⠀⠀⠀⠺⢄⡀⠀⠀⠀⠐⠢⣽⡆⠀⠀⠀⢒⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠚⠀⠉⠉⠉

⠀⠠⠄⠀⠀⠤⠀⠄⠠⠤⠀⠀⠤⠀⠤⠄⠀⠀⠀⠀⢀⡀⢀⡤⣄⡴⠊⠉⠉⠑⢦⠀⠀⠀⠀⠠⠄⠀⠠⠤⠄⠠⠤⠠⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠉⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⣄⣀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢀⡴⠗⠀⢀⠔⣿⢦⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⠓⠂⠤⣼⣀⡀⠀⠐⣉⣤⢖⡔⠁⠀⠛⠀⢹⢂⣀⣀⣀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⠤⠤⣤⡄⠀⣤⠬⠇⠀⠏⠀⠈⢸⠀⠀⠀⠀⠀⡼⠤⠄⣀⣀⡀⠀⠀⠀⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡷⠤⢤⠼⢦⣀⡀⣀⢼⡇⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡗⠀⠀⢀⠁⠀⠁⠀⠀⢸⠛⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢈⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣁⢰⠃⠀⡠⠋⢠⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠁⢼⣀⣀⡠⢧⠀⠀⠀⠀⠀⠈⠑⠦⣤⠔⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣶⣶⠿⡖⠚⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣉⢉⠉⠃⢨⡀⠉⠉⠉⣱⣦⡄⠀⠀⠀⡄⢀⠀⡄

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⣶⣌⠛⢷⣄⠀⠙⠃⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⢷⣶⣤⣄⡀⠀⠀⣤⡶⠀⠀⣠⣴⠟⣫⣵⡶
⠀⠀⠀⠻⢶⣝⠻⢦⡌⠁⠀⠀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠛⠻⣶⣄⡀⢀⡴⠞⣉⡴⠞⠉⢀⣠⣶
⠀⣴⣤⣀⠀⠉⠻⠂⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠛⢿⣷⢦⣄⠀⠙⢿⣿⣶⡞⠉⢀⣠⠶⠛⢉⣴
⠀⠀⠉⠙⠳⠦⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢮⣙⠲⢤⡉⢷⣌⠳⣄⡀⠘⢿⣷⡞⠉⢁⣠⠾⠋⠁
⢀⣤⣀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠠⠄⠠⡤⠀⠀⠀⠀⣠⢀⡄⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣝⠲⠄⠲⣌⢿⣦⡀⢻⣿⡞⠋⠁
⠀⠉⠙⠛⠂⠀⣼⡿⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⣰⢿⣿⡿⠀⠀⠀⠙⢷⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⢰⣹⡎⣿⣧⠀⣿⣇⠀⠀⣤⠾⠃
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠉⠁⠀⠘⠃⠀⠀⣸⠇⣾⢹⡇⠀⠀⠀⠀⢸⣧⠀⢸⡇⢰⡄⠀⢠⠀⠀⠀⢀⣿⡈⠉⡈⣧⡀⠈⢻⡆⣿⣿
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⠈⠁⠀⠀⠀⠀⠀⠉⠀⢸⡟⢸⡇⣿⢸⡇⣷⠀⣿⣿⣷⢸⡇⠉⠀⠀⠸⣇⢻⣿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢸⢣⡏⣼⢡⣿⣾⣿⣿⣿⡾⠀⠀⠀⠀⠀⢻⡼⡿⠀⠀⠀⠛⢛
⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⢶⣄⡀⠀⠀⠀⢿⠃⠹⣷⡄⠀⠀⠀⠀⠀⠀⣰⠏⠀⡏⡞⣰⢃⣾⣿⣿⣿⢯⡾⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠛⣋
⠀⠀⠀⠀⢸⣟⠀⠀⠀⢀⡀⠘⣿⠃⡀⠀⠀⠐⢷⡄⠀⠀⠀⠀⠀⠀⠀⠸⠿⠆⠘⠛⠛⢃⣾⣏⣼⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠚⠛⠉
⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠳⡄⠹⡆⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣿⡆⠀⠀⣀⣀⣠
⠀⠀⠀⠀⢸⣿⡌⠀⠀⢧⡀⣾⠆⠹⣌⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣸⣿⠇⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠸⣿⣳⡰⣆⠠⣽⣌⢦⡀⠉⠈⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⢿⣿⣅⠈⠑⢬⡛⢷⣅⡀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠈⣭⣥⣭
⠀⠀⠀⠀⠀⠈⠻⣿⡳⠶⣄⠙⣂⠙⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⣠⣶⣶⣄⠀⣾⡿⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠛⠿⠿⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣂⣤⣤⡄⢀⡀⠀⠀⠻⠟⠀⢸⣿⠿⠀⠉⠀⢻⡇⠀⠀⠀⠀⠉⠉⢉
⣶⣦⣄⡀⠀⠀⣿⠃⠀⣠⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠄⠀⠀⢸⣷⠀⠀⠀⠐⢦⡀⠠⠶⠖⠀⠀⠀⢀⡀⠀⠀⠈⠛⠛⠛
⠈⠉⠉⠀⠀⠸⣿⠀⢰⡍⠀⠈⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠉⢛⣉⣤⣾⠇⠀⠁⠀⣠⠀⠀⠀⢀⣷⠐⠂⠀⠡⠄⠂⢿⡄
⢰⡶⠞⠂⠀⠀⢿⣆⠀⠀⠀⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡍⠉⣀⡀⠀⠀⠀⠀⣤⣤⠀⠛⠋⢠⣀⠀⠀⠁⠀⢸⣧⠀⠀⠒⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣥⠀⢀⣤⠀⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⠀⠀⣾⡇⠀⠀⠲⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠉⠉⠀⣰⡿⠁⠀⠰⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⡄⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⢷⣤⣤⣤⡤⠶⠶⠞⠛⠋⠁⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⣤⡀⢈
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⣴⣶⡾⠟⠃⠀⢀⣴⣿⣿⣷⣦⠀⠀⠀⠀⠰⣦⠘
⠀⠀⠀⠀⣀⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⠙⠿⣷⣦⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠇⢀⣾⠀⠀⠀⠙⣷
⠀⠀⠀⠸⣿⣷⣄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠉⠛⠛⠿⠿⠟⠻⠿⢷⣶⣤⣤⣀⣀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⡟⢀⣾⡟
⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠘⠿⠗⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠿⠿⠋⠀⠺⠿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢀⠔⠀⠀⠀⠀⢀⠀⠢⡀⠀⠀⠀⡠⠂⠀⠁⠀⠀⣉⠐⢄
⠀⠀⠀⠀⠠⢊⣽⠀⠨⣌⠱⠀⠀⠀⠸⣀⠂⠁⠀⠀⠀⠀⠈⠁⢾⡆⠀⢰⠁⠐⠀⣹⡄⠀⢠⣿⠀⠈⡄⠀⡜⠀⢰⢒⡆⠀⠠⠄⡄⠀⢣
⠀⠀⡀⠀⠀⠈⠁⠀⠀⠑⠐⠀⢠⠀⡇⠀⣴⠒⠂⠠⠤⠤⠐⣲⠄⡇⠀⠀⠀⠀⠄⠟⠀⠀⠐⠋⠀⠀⡇⢠⡇⠀⠰⠼⠃⠀⠠⠼⠃⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠠⠄⠀⢀⠆⠀⢱⡀⢛⢉⠒⠒⠒⠒⢏⡙⠀⠂⠀⢀⠀⠀⢀⣀⣀⣤⣄⡀⠀⠀⡇⠀⢧⠀⠀⠠⠔⠲⢄⡀⠀⠀⢠⠇
⠀⠀⠀⠙⠀⡀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠳⢄⠙⠁⠄⠀⠘⠋⡠⠂⠀⠀⠀⠣⡀⠘⠅⠉⠀⠠⠃⢀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⠤⠄⠊⠁⠀⠀⠀⠀⠀⠉⠓⠲⠶⠤⠾⠋⠁
⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⡠⠒⠉⠁⠉⠉⠁⠢⠀⠀⠀⠀⠀⢀⡀⠒⠘⠈⠓⠒⢄⠀⠀⠀⠀⠀⢀⠤⠒⠀⠀⠀⠒⠄⡀
⠀⢠⠃⠀⣾⣭⡂⠀⠀⣃⣹⡆⢰⡀⢀⠎⠀⠀⡒⢄⠀⠠⡒⢄⠈⢂⠀⠀⡰⠃⠀⠀⠀⠀⠁⢀⣠⠙⣄⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡀⠈⢦
⠀⣼⠀⠀⠨⠉⠇⠀⠀⠋⠙⠀⠘⡇⢸⠀⠀⢌⣊⡸⠀⠈⠉⡸⠀⠈⠀⢰⠃⠴⣶⣶⡄⠀⠐⠉⠀⠀⢸⡄⢰⠁⠀⡶⣭⣆⠀⢲⣮⢶⠀⢸
⠀⠸⡀⠀⠐⡒⠠⠤⠤⢔⠎⠀⠀⠅⢸⠀⠀⠀⢀⣄⠀⠦⣄⠀⠀⢠⠁⠘⡆⠐⠚⠉⠀⠀⢀⡠⣴⠁⢸⠇⢸⡄⠀⠘⠔⠏⠀⠘⠏⠏⠀⢸
⠀⠀⠳⡄⠀⠘⢏⡉⢉⡟⠀⢀⠞⠀⠈⠣⡀⠀⠸⠥⠤⣀⠬⠃⢀⠎⠀⠀⠹⣄⠀⠒⢦⣮⠭⣾⠋⢀⠞⠀⠀⢧⡀⠀⣤⠬⠤⠬⠿⣦⢀⡞
⠀⠀⠀⠈⠒⠦⠤⠴⠏⠤⠚⠁⠀⠀⠀⠀⠈⠒⠦⠤⣀⣠⡤⠔⠁⠀⠀⠀⠀⠈⠳⠤⣄⣀⣉⡠⠔⠁⠀⠀⠀⠀⠙⢦⣉⡉⠉⣉⣀⡤⠊
⠀⠀⠀⠀⡠⠀⠀⠐⠀⠒⠠⣀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠉⠁⡀
⠀⠀⡠⠃⠀⣠⣀⠀⠀⣀⢬⡀⠱⡀⠀⢀⠜⠁⠀⠀⠀⠀⢀⠀⠀⠑⠄⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠃⠀⢸⢰⣮⠇⠀⡇⣲⡾⠀⢠⠀⠸⠀⣠⠤⣶⡄⢤⣾⢾⢹⠀⠘⠀⠀⡜⠀⡊⠓⠀⡦⢠⠀⠒⣳⢢⠀⢰⠀⠈⠉⠁⠀⠀⠉⠉⠁⠘⡀
⠀⠈⡄⠀⠀⠉⠉⠀⠀⠀⠈⢀⠀⢸⠀⢀⠀⠸⡀⠋⠙⠀⠈⠫⠊⠀⢸⠀⠀⡇⠀⠙⠦⠔⠁⠀⠣⠔⠁⢸⠀⢻⠀⠠⢄⣀⣀⣀⣀⠠⠂⠀⡇
⠀⠀⢱⡀⠙⢤⣐⣒⣒⣠⣭⠏⢀⡎⠀⠘⣆⠀⠀⠀⠄⠠⠤⠤⠀⠀⡌⠀⠀⠠⡀⠀⡖⠲⠦⠤⢶⡄⢀⡎⠀⠸⣄⠀⠀⠀⠒⠒⠀⠀⠀⢠⠃
⠀⠀⠀⠱⢄⠈⠷⣒⣒⣲⠎⡠⠎⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣊⡠⠊⠀⠀⠀⠑⢴⣄⡀⠀⠀⣠⣤⡶⠃
⠀⠀⠀⠀⠀⠉⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠁

⠀⠀⠀⠀⢀⠀⣠⡶⠶⠶⠶⠶⠶⠒⠒⠒⠚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣒⠒⠒⠶⠶⠶⣤⡀⢀⡀
⠀⠀⠀⠀⠎⣼⠃⠀⣴⠶⠒⠛⠛⠛⣛⣛⠉⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⡉⢉⣩⣍⠉⣉⣉⡙⠛⣷⠀⠀⢹⡆⠓
⠀⠀⠀⠀⠀⣿⠀⠀⣿⢰⣟⣻⡇⢼⣏⣻⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠒⠚⠀⠻⠶⠇⠀⣿⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠹⠟⠁⠈⠙⠿⠁⠘⠷⠶⢤⣤⣤⣤⣤⣤⠾⠃⠸⠭⠝⠀⠯⠭⠇⢰⡏⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣓⣚⣒⡒⠲⠶⠶⠶⠶⠶⠶⠶⠛⠛⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠞⣛⣩⠭⠭⠭⠭⠭⠭⢭⣛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⠾⢋⠴⠒⠋⢁⣀⣠⣤⣤⣤⣤⣤⣄⡈⠙⠲⢬⡙⣿⡉⠉⠛⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃⡴⠁⢠⡴⢛⣩⠭⠒⠚⠛⠛⠛⠒⠮⣝⢷⣄⠀⠙⣎⢻⡝⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⡼⠀⢠⡟⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡹⣆⠀⠸⡀⢿⡀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠘⡇⢀⡶⢾⡇⡇⠀⢸⡅⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢻⡀⠀⣇⢸⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣷⠘⣇⠸⡇⣇⠀⠸⣇⢧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⢸⠸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⢻⡸⡄⠀⢻⡌⢮⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⠁⠀⢸⢰⡇⠀⢠⡏
⠀⠀⠀⠀⠀⢠⢸⡇⠘⠳⠾⢿⡜⣆⠀⠙⣦⡙⠦⢌⣀⣀⣀⣀⣀⣀⡤⢞⣱⠟⠀⢠⢏⡾⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠘⠈⣇⠀⠀⠀⠀⢻⣌⠳⣄⡈⠙⠳⠶⣤⣭⣭⣭⣭⣤⠶⠛⠁⣀⠔⢁⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠳⣦⣙⠲⠤⣀⣀⣀⠀⠀⢀⣀⡤⠖⢊⣡⡴⠋⠀⠀⠀⢀⡟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠐⠦⠉⠛⠶⠶⠤⣬⣭⣭⣥⠴⠶⠞⠋⠉⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠒⠶⠶⠶⠶⠧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟
⠀⡀
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⢋⣩⡿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢿⡀⠀⠈⠉⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠙⣇⠑⡄⠀⠀⢀⣠⡤⠄⠀⠀⢸⠁⠀⠈⠳⣄⠀⠀⠀⠲⢶⣶⠦⢽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢿⠀⠀⢀⣈⡭⠴⠶⣄⠀⢀⡾⠀⠀⣠⠔⠋⠀⠀⠀⠀⡀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⣠⠖⠋⠁⠀⠀⢠⣿⠴⠋⠁⠀⣸⠷⠀⣀⡀⠀⠠⣤⣝⣯⡝⠛⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣄⣰⠞⠁⠀⠀⠀⠀⢠⡟⠁⠀⠀⣠⢾⡥⠶⠒⠛⠉⠉⠓⠾⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠿⠾⣿⠃⣬⠇⠀⠀⢀⡴⠋⠀⠀⢀⣾⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠠⠔⢒⡶⠀⢸⣤⠤⠤⠴⠶⣯⠀⠀⠀⢀⣾⠾⣛⣉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⡀⠀⢀⠀⡀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠚⠛⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠤⣤⡤⠴⢶⡞⠋⠁⠀⠀⠀⠀⠀⣀⣬⡷⠦⣄⠀⠀⠀⠀⣰⠋⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⣲⢤⡴⠋⠀⠀⠀⢤⠖⠲⡟⠋⢯⠀⠳⣄⡼⠧⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡓⠻⣾⠁⠀⠀⠀⠀⠸⡆⠀⢹⡄⢈⣧⠴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡈⣳⠞⠛⣦⠀⠀⢻⣄⡤⠗⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⡿⠃⠀⠀⢸⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⠀⢡⠀⠀⠀⢸⠛⢦⡀⠀⠀⠀⠀⠀⠀⣰⠏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠻⠏⠀⠀⣀⣙⡟⠀⠀⠀⢸⡀⠀⠙⠶⣄⣀⣀⢀⡜⢻⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣴⠛⠉⠁⠈⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⠻⡄⠀⣠⠾⡏⢉⡿⢦⣸⡄⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⡀⠀⢸⡀⢻⡔⢋⡴⢧⣼⣀⣀⣯⣤⣶⣾⠓
⠀⠀⠀⠀⠀⠀⣼⠁⠈⠳⣤⡀⠀⠀⠀⠀⠀⢨⣿⣿⡟⠛⠁⠀⠀⡾⠀⣿⣩⡤⢬⣙⣣⣻⠿⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⣸⠁⣼⠀⠀⠀⠀⢀⣿⠖⠁⠧⠤⠴⠾⢿⣜⣇⢳
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡇⠀⠛⠒⠒⠒⠒⠚⠛⠀⣼⠛⠓⢶⣤⡴⣋⣁⣸⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⢉⡟⠉⠉⠛⠶⣬⡽⣟⣉⣉⡿⠁⣈⣡⡤⠶⠖⠲⣦⠢⡀⠈⢦
⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣋⣀⣀⣠⣤⣤⣼⣧⣀⣉⡿⠛⠉⠁⠀⠀⠀⠀⠀⢹⠀⢱⠀⠜
⠀⠀⠀⢀⣠⠴⠻⡘⢆⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⣡⣤⡴⣁⠞⠁⠀⠀⠀⠀⢸⠀⠋
⠀⠀⠀⠸⡇⠀⠀⠑⠌⠁⠀⠀⠒⠚⠉⠁⢹⠀⠀⠀⠀⠙⣯⠉⣉⣠⣤⠴⠾⠁⠀⠈⢹⠇⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢨⠟⣹⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⣴⠋⢠⡇⠀⠀⠀⠀⠀⠀⠀⠷⠶⠒⠛⣏⠈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣯⠿⢷⣖⠛⠉⠀⠀⠀⠀⠀⠀⠘⣇⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠋⠉⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⢦⣤⣠⣾⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣦⣄⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠛⠛⠛⠶⣶⣿⣿⣿⡿⠟⠛⠛⠛⢷⣦⣄⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠻⢿⣿⣿⣿⣿⠋⠀⢠⠶⣤⠀⢸⣿⣿⣿⣷⣶⣄⣉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣤⣀⡀⠀⠀⠀⠸⢿⣿⣿⣿⢿⣮⣽⣛⡶⣦⣌⣉⠛⠿⢿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣤⣀⡀⠈⠉⠁⠀⠈⠉⠛⠿⢶⣭⣝⣛⠶⢦⣌⣉⡙⠛⠛⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣤⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣬⣽⡿⠛⠷⢶⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠙⠛⢦⣤⡤⠚⠋⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣄⡈⠻⠿⣿⣿⣿⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠉⠁⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⡟⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠉⠉⠛⠳⠶⣤⣤⣠⡞⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⣿⡁⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠋⣿⣦⣤⣤⣤⣤⡀⠀⠀⠀⠉⠀⠀⠀⣿⣿⠷⣦⣤⣤⣤⡶⠛⠁
⠀⠀⠀⠀⣰⡟⣡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⣠⣿⠿⠗⠲⠶⠤⣤⣄⡀⠀⣿
⠀⠀⠀⣰⠟⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢀⠀⠈⠙⢶⣿
⠀⠀⣰⡏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀⠀⣻
⠀⢠⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣶⠏
⠀⠸⣧⠘⢷⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⡾⠋⠀⠉⠛⠉⠙⣇⠀⠀⢀⣤⣠⠾⠋⡀
⠀⢠⡿⢧⣄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⣦
⠀⢸⡇⠀⠙⠳⢶⣤⣀⡴⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⣾⠿⣤⡀⠀⠀⠀⢹⡇⠀⢀⠿⠶⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟⠉⢻⣦⣤⣤⣤⣤⣤⣠⣤⣠⣿⡀⣀⣀⣀⣀⣀⣀⡀
⢸⡏⠀⠈⠛⠳⠶⣤⣼⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⢸⣧⣤⣤⣤⣤⡤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠙⣿
⣾⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⢶⣤⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⣦⡀
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠈⢻
⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⣿⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⣿⣋⠀⠀⠀⠓⠠⠄⣀⡀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠞⠛⠉⠁⠀⠀⠀⢈⣿⣿⣷⣦⣤⣀⡀⠀⠀⠙⠒⠠⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡶⠶⢿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠸⠷⢶⣶⠶⠟⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠐⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣤⣀⡀⠀⠈⠐⠢⠤⣀⡀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠈⠁⣲⡾⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⣤⣤⣠⡟⠁⡴⠃⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠋⠁
⠀⠀⢀⣀⠀⠀⠓⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢶⣤⣀
⠀⠀⣏⠀⠴⢤⠦⣾⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠙⠿⠆⠀⢰⣿⢐⡇⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣸⠀⠀⢀⣤⣤⠞⠋
⠀⠀⣀⣌⣤⠆⣼⠃⢸⣷⠀⢸⡇⠀⢰⡀⠀⠀⠀⠀⣠⠄⠀⢠⣿⠀⠀⣀⡈⠁⢀⣴⠟⠉
⠀⢰⡏⠀⠁⠀⢹⡄⢹⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⣼⡿⠀⠀⢸⡏⢀⡀⣨⠟⠀⣼⡟
⠀⠀⣿⠂⣀⣀⡀⢻⡌⢻⡀⢸⡇⠀⢸⡟⠀⠀⣰⡏⣴⠀⠀⣸⡟⠈⣹⠁⠀⣼⣿⡁
⠀⠀⠈⢻⡋⠀⠁⢀⣽⠶⢿⠈⣧⠀⣿⡁⠀⠀⣿⠃⢿⠀⠀⣿⣁⡿⠋⢀⣾⠟⢉⣷
⠀⠀⠀⣾⣧⡾⠀⣾⠃⠀⡘⣧⣹⡄⢸⣇⠀⠀⢿⡀⠘⢧⣀⡟⠈⣠⡴⠛⠁⠀⠈⢿⡀
⠀⠀⠀⢿⠀⠀⠀⣿⢀⡾⢁⠀⠉⢳⣜⣿⡀⠀⢸⡇⠀⠀⢉⣴⠾⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡷⠂⠀⢻⡾⣱⠏⠀⣴⡿⠉⠛⢻⡄⠈⣿⢀⣴⡾⠋⠀⠀⠀⠀⠀⢀⣷⠏
⠀⠀⠀⣼⠋⠀⠀⠸⢷⡟⢀⣸⡟⠁⢀⡖⠀⢻⣦⠟⠋⠁⠀⠀⡄⠀⠀⢀⣷⠾⠃
⠀⠀⠀⠘⣧⠀⠀⠀⠘⠛⠛⠋⠀⠀⠻⠶⠶⠛⡉⠀⠀⠐⢦⡖⠃⠀⣀⡾⠁
⠀⠀⠀⠸⣧⡀⠀⡀⠀⠀⠀⢧⣠⠀⠀⢀⣀⣠⠷⣄⣀⣠⡾⠃⠀⠉⣿
⠀⠀⠀⠀⠸⣧⣴⠏⠀⠀⠀⠀⢙⣻⣷⡚⠉⠁⠀⠸⣏⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠛⠋⣩⢙⠻⢷⣤⣀⠀⠈⠛⠒⠚⠋
⠀⠀⠀⢰⣞⣷⢸⣿⣄⡀⠀⠘⠶⠟⣠⠤⢤⡈⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠁⢸⡇⠈⠛⢶⣄⡀⠀⠙⠚⣯⣤⣶⣶⠀⠙⠻⢶⣦⣄⠀⠀⠀⠀⣏⣹⠆
⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠋⡵⢶⣻⠆⠀⠈⠙⢻⣶⡄⠀⠈⠉
⠀⠀⠀⠀⣴⣋⠙⢷⣄⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⣀⣤⣤⡶⠞⠋⠉⡇⢀⡠⢤⡀⠀⠀⠀⡶⠛⡆
⠀⠀⣀⣸⣇⠙⢷⣾⣉⠳⣤⡀⠀⠀⠀⠀⠀⠈⠙⣿⠟⠋⠉⠀⠀⠀⠀⠀⣇⠈⣀⣨⡿⠀⠀⠀⠀⠉⠁
⠀⠈⢯⡽⠙⠛⣿⠀⢈⣷⠈⠙⢶⣄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠁⣰⠛⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠈⠻⣦⡀⠀⠀⢺⠀⠀⠀⢀⣠⣴⠶⠛⠉⠀⠀⠀⠀⠻⠤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠚⠛⠛⠛⠛⠓⠲⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣴⠏⠀⠀⢀⣠⠶⠛⠛⣛⣛⣛⣲⣦⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣚⣛⣛⡛⠛⠶⣤⡀⠀⠀⠹⣦
⠀⠀⢀⡾⠁⠀⠀⣠⣿⠵⠖⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠳⠶⣿⣦⠀⠀⠈⢷⡀
⠀⢀⡾⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢷⡀
⠀⣼⠃⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠘⣧
⢠⡏⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⢹
⣸⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈
⣿⠀⠀⠀⢀⡏⠀⠀⠀⠀⣴⣿⣿⣿⣦⡀⠀⠀⠀⢿⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣾⣿⣿⣷⡄⠀⠀⠀⢹⡄
⣿⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇
⢹⡀⠀⠀⠀⢿⡀⠀⠀⠀⠈⠛⠿⠛⠋⠀⠀⠀⢠⡟⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⡾⠀⠀⠀⢀
⠘⡇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸
⠀⢻⡄⠀⠀⠀⠀⠈⠓⠦⣤⣄⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣀⣤⡴⠞⠋⠀⠀⠀⠀⢠⡟
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠦⠤⣤⣤⣤⣤⡤⠴⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣻⣯⠤⠴⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣍⡀
⠀⠀⠀⠀⠀⢀⡴⠋⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣦⡀
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⢷⡀
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢷⡀
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣧
⠀⠀⢿⠀⠀⢀⠀⠀⠀⠀⣴⠏⠀⠀⣤⣄⣀⡀⠉⣷⣄⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿
⠀⢀⣸⡆⣀⣼⡄⠀⢀⡾⢷⣠⠴⠋⠉⠈⠉⠛⠻⠇⠈⢷⡀⠀⢀⡞⢹⡄⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠋⠉⢹⠃⠘⣧⠀⣼⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⡾⠁⠀⠿⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿
⢿⣄⣀⣸⡆⠀⠘⢧⣿⠀⠀⢀⣤⣶⠶⠄⠀⠀⠀⠀⠀⠀⠘⠋⠀⠐⠛⠿⣶⣄⠹⡆⠀⢤⣀⠀⠀⠀⠀⢸⠷⠷⢶
⠀⠙⠛⠛⣇⠀⠀⠀⠁⠀⠀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠉⠉⠉⠉⠳⣄⠀⠀⣿⠀⠀⢀
⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⢀⣤⣶⣶⣶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⢷⣸⣿⡶⠾⠛
⠀⠀⠀⡼⠙⣧⠀⠀⠀⠀⠀⢠⡾⢻⣿⣿⡏⠉⢳⡄⠀⢀⣾⣿⣿⣿⢻⣀⣸⣿⡄⠀⣠⣶⠶⠆⠀⢈⡿
⠀⠀⣸⠁⠀⠘⣧⠀⠀⠀⠀⣾⠀⠘⢿⣿⣾⣶⣿⣿⠀⢸⡟⠿⣿⣿⣿⣿⠟⢁⡇⠀⠁⢸⡆⠀⢠⣾⡁
⠀⠀⡿⠀⣸⠀⠘⣧⡀⠀⠀⢹⡄⠀⠀⠙⠛⠛⢋⡟⠀⠈⣧⡀⠀⠈⠀⠀⠀⣼⠇⠀⠀⢸⠇⣠⠟⠀⢻⡀
⠀⠀⣇⡀⣻⠀⠀⠈⠳⣄⠀⠀⠙⢶⣤⣤⣤⠶⠋⠀⠀⠀⠈⠻⢦⣤⣤⡴⠞⠁⠀⠀⠀⣼⡴⢋⣀⣀⣨⡇
⠀⠀⠘⡏⠻⣇⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⣼⠉⢩⡏
⠀⠀⠀⠱⡄⠹⣆⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⠆⠀⠀⠀⠀⠀⢹⣷⣤⣤⠶⠾⠛⠉⠀⠀⣰⠇⣠⠟
⠀⠀⠀⠀⠙⠳⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠉⠛⠶⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡶⠟⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢤⣀⡀⠀⠉⠉⠙⣻⣿⠛⠛⠛⢿⣏⣉⠉⠉⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠉⠙⠛⠛⠛⠛⢻⡇⠀⠀⠀⣾⠀⠉⠉⠛⠛⠉⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣇⠀⠀⠀⠀⢀⣴⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠶⠂⠀⠀⠲⠶⠆⣀⣀⣈⣳⠆⠠⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⠛⠋⠁⠀⠀⠀⠀⠀⠉⠉⠛⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⢀⡠⠄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢀⣀⠐⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣤⠞⠚⡋⠀⠈⠄⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡿⠿⣆⣨⣄⣀⣧⣀⠠⣶⠦⠾⠟⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⢀⠖⠈⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⡘⠃⢀⡤⠋⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠜⠋⢱⡄
⣄⠈⠳⠆⠉⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠳⠖⠲⠶⠂⠉⠉⠁⠀⠀⠀⠀⣤
⠉⠓⠀⠀⠀⠀⠀⢀⡄⠚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄
⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⠈⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣷
⠀⠀⠀⠀⠀⣾⠀⢀⡴⠋⠀⠀⢣⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠖⠉⠐⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠛⠓⠈⠀⠀⠀⠀⠈⢳⡄⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠒⠂⠈⠀⡤⠊⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠰⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠙⢦⡀⠀⠠⠤⠀⠉⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⣴
⠀⠀⠀⠰⣦⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠉
⠀⠀⢰⡀⢹⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶
⠀⠀⢸⡇⠈⠇⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉
⠀⠀⠘⢃⠀⣧⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠘⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠠⠊⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠋⠙⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣠⣀⣀⣠⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⡤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣀⣠⣤⢀⣀⣤⠤⣄⣀⣠⣤⣶⣦⣼⣿⣿⣿⣤⣀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡼⡶⣿⣿⣿⠋⠁⠀⠀⠀⠁⠀⣼⣿⣿⣿⡟⠁⠀⠀⠈⠳⣄⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢻⣿⣿⣶⠆⡄⡐⢀⠄⠀⢠⠀⠈⡌⠉⠙⣼⠀⠀⠀⠀⠀⠀⠘⣆⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⠿⢻⠃⠰⠀⠇⡾⠀⠀⠘⣇⠀⢰⢠⠀⠸⣧⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⡨⠇⢳⠀⠀⠀⢛⣅⣘⡮⢦⠀⡇⠱⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠠⠋⡼⣿⢾⢆⠀⠰⠛⡿⠹⣿⡤⠀⡷⢤⡟⢦⢀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢿⡄⠸⠴⡇⠁⣬⠀⠀⠂⠀⠨⠄⠈⠹⣿⠉⢸⠇⠈⣧⡆⢀⣀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⣸⠳⣾⡤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⣽⠔⠋⠀⠀⣿⡇⢸⡿⢹
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⠸⣿⠀⢰⣏⢤⣄⡀⠀⠀⠀⠀⢀⣤⣚⠻⣿⠀⠀⠀⠼⢋⡧⠞⠁⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠓⢿⠓⢈⣧⣑⣴⣛⣛⢿⣦⡀⢨⣿⣿⣷⡄⠀⠀⠀⠀⠈⠁⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠣⢀⣀⣀⡀⣰⡻⢥⠊⠙⢑⡐⣿⣿⣷⣿⣿⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠢⢄⡀⠀⠀⠀⣀⠴⠋⠉⠀⠀⡛⠋⠠⠔⢲⡄⣴⠭⠤⠤⠉⠛⢿⣛⣽⠾⠛⠉⠉⠉⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡰⢷⣿⣅⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⠉⢹⣦⡀⠀⠀⣨⣷⣄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣦⠴⠚⠛⠉⠉⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠳⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⠒⣦⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠙⠒⠒⠖⠚⠉⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⠤⠤⠶⠚⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣤⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢻⣿⡿⠋⠘⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⢿⣿⠏⠀⠛⠁⠐⠒⠲⠶⠄⠀⠀⠀⠀⠀⠴⠖⠒⠒⠋⠀⠉⠛⢿⠏⢉⠙⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⣇⠈⠏⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠠⣶⣶⡂⠀⠀⠀⠀⠘⢀⢼⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⠈⣡⣤⠀⠀⠀⠀⡀⡈⠀⠀⠀⠀⢀⡽⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣶⣦⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⣀⡴⠀⠀⠀⠁⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠒⠒⠛⣿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠺⣿⣿⣿⣿⣿⣿⣿⣿⠟⢸⡇⠀⠀⠀⡿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢹⠟⡿⢿⣿⣟⣁⣀⣨⡇⠀⠀⠀⣯⣀⡀⣠⣝⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢒⣶⣶⣇⣼⠁⠀⠀⡸⠁⠀⠙⠮⠷⠶⠯⠞⠁⠈⢳⠀⠀⠀⣇⢸⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠢⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠔⠚⣿⣿⣿⣿⣿⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠓⠒⠦⠤⠶⠒⠒⠒⠒⠒⠲⠶⠦⠤⣄⣀⡤⣼⣿⣿⣿⣿⣿⣈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡿⠛⠋⠉⠉⠙⠓⠒⠒⠒⠛⠉⠉⠉⠉⠙⠛⠓⠒⠶⠶⠶⣿⣿⣿⣿⣿⡟⠒⠒⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡅⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉⠉⠓⠒⠶⠶⠖⠒⠶⠶⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣠⡤⠤⠶⠶⠬⠭⠭⠭⠤⠤⠤⠤⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠶⠦⠤⠶⠶⠶⠶⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠛⠛⠻⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠿⡏⠉⠩⡈⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠀⠀⢀⠀⠁⠀⠀⠀⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⣶⢠⡀⠑⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⢠⡱⡆⠑⢤⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠙⡅⠀⠈⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢣⠀⠀⠀⡇⠄⠀⠀⡸⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⢀⢷⠀⡠⢿⠂⠀⠂⠀⠀⡀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠍⠉⠁⠈⠻⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⡀⠑⠀⠀⢀⠀⠄⣀
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠀⠸⡄⠙⠒⠁⠀⠀⠈⠙⠐⠄⠀⢀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠀⠀⠘⠆⠈⠄⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠂⠠⢄⡀
⠀⠀⠀⠀⠀⣸⣿⠀⡀⢀⠠⠀⠀⠀⠁⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡀⠀⡆
⠀⠀⠀⠀⠀⠿⠘⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠤⠀⠀⠠⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣷⠀⠁⠀⢃⠀⠀⠈⠀⠁⠀⠀⠈⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠧⠲⣄⠶⢏⣤⠐⠛⠁
⠀⠀⠀⠀⠀⡿⠀⠀⠀⢘⡷⠦⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠶⠒⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢃⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⠛
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠋⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢀⠈⠁⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠶⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⢀⣴⠿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠓⠗⣦⢶⡿⠟⠀⠀⠀⠀⡠
⠀⠀⠀⠸⡇⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡎⠀⠀⡶⠋⠙
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠳⡀⠰⣅
⠀⠀⠀⠀⠀⠀⠀⣯⣦⣀⠀⠀⠀⣀⡤⠐⣲⠖⠻⠀⣰⡆⢠⣿⣷⢄⣠⡀⠀⠀⠀⠀⠙⠍⣓⢤
⠀⠀⠀⠀⠀⠀⠀⠉⠿⠻⡿⣾⠞⠁⡠⠐⡁⠀⡠⣴⣿⠀⠈⠙⢿⣦⡙⢻⣿⣶⣦⣤⣀⣤⠟⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⣴⠋⠀⣿⣿⠀⠀⠀⠀⠙⢿⣦⠈⠓⠦⣤⣅⣠⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢿⣤⣤⣟⡋⠁⠢⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⠀⠘⡀⠀⠀⠀⠀⣽⡟⣠⠄⢠⠀⠀⠠⠀⠈⣿⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣄⠤⣤⣤⣾⡫⠘⠁⠀⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠀⠰⠋⠀⠀⠀⠀⢀⠀⠀⠈⠛⠻⠋⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠔⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣿⣦⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠿⡏⠙⠋⠙⠋⣿⡿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠖⠂⠀⠀⠀⠀⠋⠁⠠⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣦⡘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⡀
⠀⠀⠀⠀⠀⣴⠚⠉⠉⠉⠉⠉⠉⣹⠆⠀⠀⠀⠀⠀⠀⠸⣧⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⢟⠛⠳⣄
⠀⠀⠀⠀⣀⣹⡶⠤⠴⠖⣶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣧⣆⣤⡴⠚⢷⣠⢞⠕⠁⠀⠀⠈⢷⡄
⠀⠀⠀⢰⢱⠛⠓⠒⠒⠒⠛⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠑⢁⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⢸⡘⢦⡤⠤⠤⢤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⣴⠏⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⣷⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⡅⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⠀⣠⡾⢁⣴⠋⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⢠⢎⣠⠼⠁⠀⠀⢀⣴⠟⠛⣦⡞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢻⡀
⡔⠒⢤⡀⠀⢀⣀⡀⠀⠀⢸⠀⠀⣇⣀⣉⠉⠀⠀⠀⣀⣀⣿⣀⣰⠟⠻⣦⡀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠙⠒⠤⢵⡄⠸⣄⡙⡄⠀⣼⠀⢀⣯⣍⠉⠙⠛⠋⠉⠉⠉⠈⣿⠁⠀⠀⠀⠙⢦⡀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣶⡏⠀⠻⣍⣉⠛⠳⠶⠶⠶⠶⠒⠛⠋⢳⣄⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠶⠖⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⡀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠠⣏⣉⡩⠶⠀⢻⣿⠿⣛⣋⠀⠀⠀⠀⣠⡾⠛⠛⠛⠉⠉⠉⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣠⠀⠀⠀⠀⠀⡈⣿⣟⣭⣵⠞⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡔⠋⣠⠏⠀⠀⠀⡴⢚⡇⠀⠉⠈⠛⠋⠁⠀⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠾
⠈⠋⠁⠀⠀⠀⠸⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠁⠀⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⡀⡤⠀⠀⠀⠐⠂⠀⠉⠁⠀⠀⠀⠀⢣⡀⢀⡀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠄⡀
⠀⠀⠀⠀⡏⠔⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠢⣀⠑⢄⡀⠀⠀⡀⠄⠐⢈⡁⠤⠀⠐⠐⠉⢔⠂⠀⠀⢀⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠂⠠⢄⡀⠀⠀⠀⠀⠁⠢⣀⣰⡆⠀⠈⠈⠁⠀⡒⠋⠀⠀⠀⣀⠠⠀⠐⠈⠀⠀⠀⠀⠀⠙⠒
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠁⠀⠀⠀⠀⠀⢀⣀⡠⠤⠀⠀⠈
⠀⠀⠀⠀⡀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⡄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠒⠀⠉⠀⢸⠀⠘
⠀⠀⠀⠀⠂⠸⠈⠐⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⠠⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣯⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢄⡀⠀⠀⣀⣀⢰⠀⠔⠐⠺⠟⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠛⠿⠁⠀⠀⠀⠀⠀⢀⠈⠀⠘⠋⠿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⠃⠀⠀⠀⡀⠀⢠⣶⠙⠀⢰⠆⠀⠀⡟⠁
⠀⠀⠀⣨⡤⠶⢤⣤⡤⠤⠤⠶⠶⡶⠒⠒⠒⠒⠶⠶⠾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠾⠃⠀⠀⣧⡀⠀⠀⠀⡀
⠀⠀⣾⠁⣇⣠⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢰⠁⢹⡄⠀⠀⡇
⠀⠀⠹⣄⠀⣀⡷⠋⠉⠉⠉⠙⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠦⠤⠞⠀⡴⠋⢹⠚⢳⡀
⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠀⣀⠀⠀⠀⠀⣀⣤⣞⣁⣠⣤⡤⠾⠃⢐⠊⠀⠀⡇
⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡠⠄⠂⠐⡖⠉⠀⠈⢋⠉⢉⠍⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⠐⠊⠀⠀⠀⠀⢠⠀⠀⠀⠘⡤⠶⠤⠤⠤⠴⠶⠖⠚⠋⠉
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣿⠑⠒⢲⠖⢣⡀⠀⠀⢀⠀⠀⠈⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⣿⠀⠀⠋⠀⠀⠙⢤⡴⣭⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠘⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣠⠞⠁⣸⡄⠀⠀⢿⠀⠁⢸⡇⡀⠠⠀⠀⠂⢉⡀⠄⠀⠐⠈⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠤⠶⢾⣥⠴⠟⠁⣇⠀⠀⠿⠀⠓⠈⣁⣠⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⢸⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠃⠀⠀⠙⠅⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠀⠶⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠚⠛⠋⠉⠙⠛⠓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠦⣄⡀⠀⠀⠀⠀⠀⠈⠓⢬⡲⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⠖⠒⠒⢤⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⡶⠚⠛⠉⠙⢦⡀⢢⡀⠀⠀⠀⠀⠙⢮⡳⣄⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢳⡀⠀⠘⣇⠀⠀⢀⣴⠟⠁⠀⢀⡀⠀⠀⠀⠉⠳⢌⡒⣤⢬⣱⣤⡀⠙⢮⠂⠀⠀⠀⢰⣿⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀
⢤⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣽⡄⢠⡾⠁⠀⢠⠞⠛⠛⠃⠀⠀⠀⠀⠀⠉⠁⠘⠛⠶⡝⢦⡀⠳⡀⠀⠀⢸⡷⣤⠞⠉⠁⠀⠈⠉⠙⠒⠲⠶⢚⣿
⠀⠉⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⢹⣀⣹⣿⡁⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣴⠦⣄⠀⠀⠻⣄⠀⠀⢠⣿⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣤⣤⣴⡿⠋⠉⠈⣧⠀⠀⠀⢰⠋⣿⣏⣉⣦⡀⠀⢀⣾⣿⣷⡶⢻⠀⠀⠀⢹⣆⣴⠟⠉⠉⠳⣦⣀⡀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢈⣧⡀⠀⠸⣄⠈⠙⠛⢋⡇⠀⠘⡄⠉⠉⢀⡾⠀⠀⢠⡾⡏⠁⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⢦⡀⠈⠓⠲⠖⠋⠀⠀⠀⠉⠓⠛⠉⠀⣠⡶⢻⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣧⠀⠉⠓⠦⣤⣤⣤⠀⠀⠀⣷⠤⠶⠖⠋⠁⡠⢺⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⣧⡀⠀⠀⠀⠀⠈⠳⠶⠚⠁⠀⠀⢀⡤⢊⢕⡟⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠙⣍⠲⢤⣀⡀⠀⠀⣀⣀⡤⠴⢚⣥⠞⢳⡞⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡈⠳⣦⣬⣍⣉⣉⡭⠤⠖⠋⠁⢟⡣⠊⣠⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠙⣄⠈⠳⣔⣂⠄⣒⡆⠠⠤⠖⢋⡤⡞⠁⡧⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢣⣀⡼⣳⢦⣀⣱⣋⣠⠖⡊⣩⠝⠁⢳⡈⠉⠁⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⢀⡴⠃⠀⠀⠉⠣⣤⣀⡴⠟⠀⠀⠀⠙⠲⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⣠⣤⣀⣴⠀⠀⠀⠀⠀⠀⠠⣤⣶⡖⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠙⠉⠉⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⢰⡶⠄⠀⠀⠀⠾⠀⠀⠀⠀⠺⠛⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣀⣠⡴⠂⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠶⣤⣀⡀⠁⠀⠀⣀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠛⠛⠛⠛⠛⠶⠦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠀⠒⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⠴⠖⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣤⣤⣀⡀⠀⠿⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⢶⣶⠶⠚⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢿⣛⣛⣶⣚⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⠉⠉⠉⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣀⣤⣄⣸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⣯⣀⠀⠉⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣦⣌⣁⡀⠀⠀⠀⣸⣧⡤⠴⣶⠶⢾⡟⠋⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⢤⣤⣤⠤⠞⠁⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣅⣼⣧⠤⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣼⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⣤⣤⣤⣤⣤⣤⡤⢶⡶⠛⠋⠛⠻⣏⣧⠈⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣤⣀⣀⣀⣀⣀⣀⣀⣘⣶⣾⣙⣧⣤⡤⠶⢿⣏⣉⣭⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢤⣬⣉⣉⣉⣉⣉⣉⣩⣭⣤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⠀⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠒⠦⣴⠚⠳⠜⠉⢳⣠⠊⠙⡶⠋⠙⡦⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠃⠈⠀⢀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⣁⠀⠈⠁⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠞⠉⠀⢀⣰⠋⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⠁⠀⠀⠰⠁⠘⠃⠀⠀⢠⠖⠢⠀⠀⠀⠀⠀⠀⠐⢯⠽⠳⣆⠀⠀⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡴⠋⠁⢀⡔⠚⠂⠀⠀⣴⠋⠙⠂⠀⠀⠀⡰⠀⠐⠲⢤⡀⠀⠀⠸⠓⢦⠀⠀⠀⣾⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡷⠀⡔⢹⣳⠀⠀⢠⠊⠻⠂⠀⠀⢀⣤⠞⣇⠀⠀⡀⣴⣷⢦⡀⠀⠀⠈⢷⡀⠀⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿⠁⠘⠀⠀⠀⢀⠴⠲⠄⠀⠀⠀⠀⢀⡟⠀⠙⠶⢾⡇⠈⠁⠀⣽⠶⣄⠀⠸⠏⠳⡄⢀⣾⣤⡀
⠀⠀⠀⠀⠘⢿⣶⣤⣀⢸⣰⡇⠀⠀⠀⢀⢴⡷⡄⠀⠀⠀⠀⣰⠖⠋⠀⠀⠀⠀⠈⠷⡤⣶⠀⠀⡀⠈⢷⡀⠀⠀⣿⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⡿⠧⠀⠀⠀⠁⠈⡛⠁⢀⡠⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢺⡇⠀⠈⠋⠹⡄⠉⠈⢳⣴⣇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⣴⠒⠦⣄⣠⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠶⣶⠀⠿⠳⣆⠘⡇⠈⡇⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢠⠖⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢿⡀⣠⡄⠘⡄⠃⣸⣷⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠉⠹⣧⣀⡄⠈⢻⡿⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⣸⣷⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢸⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⣹
⠀⠀⠀⠀⠀⠀⣼⠁⢹⣾⡗⠘⣇⠀⠀⠀⢰⣤⣶⣶⣴⣶⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣷⡄⠀⠀⠀⣸⡟⠙⢦⠉⢷⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠸⣏⣠⠀⢿⡀⠀⠀⠘⠛⠛⠛⠛⠛⠋⠀⠀⠸⠧⠤⠀⠀⠀⠀⠀⠀⠻⠿⠛⠛⠛⠋⠀⠀⢀⣿⢃⣄⣸⠀⣸⠇
⠀⠀⠀⠀⠀⠀⡸⠧⠀⢸⠃⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠤⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠸⣾⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠑⡆⢀⡘⠷⣃⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠛⡧⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠛⠛⣧⡴⠛⢦⡼⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢀⣠⠞⠃⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⢻⡤⢶⣠⠷⠴⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⠇⡴⢀⡄⣠⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣾⣧⣾⣤⣇⣞⣰⠋⡼⢡⡶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⡋⠉⠉⠉⠙⠛⠻⢷⣿⣠⡏⢰⠇⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢿⣟⠦⠙⠦⣄⢀⡀⠀⠀⠀⠀⠉⠻⣧⣼⢋⠜⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠉⠓⠦⠀⠀⠀⠀⠈⠙⢯⣰⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣤⣬⣍⣻⣄⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⢾⣿⣿⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣏⣿⣿⣟⣿⣿⣆⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⢈⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣿⣿⣿⣿⣿⢿⣿⠉⢉⣼⣿⣿⠏⠀⠀⡼⣿⣿⣥⣌⡻⢦⡀⠀⢨⣼⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣿⣿⣿⡿⠃⠀⠈⠛⣩⣿⣿⠃⠀⠀⢴⠁⠸⡿⣿⣿⣿⣾⣿⠖⣤⡏
⠀⠀⠀⢀⣀⣠⣼⣿⣿⣿⣿⣿⣿⡶⠂⠀⠀⢠⣾⡿⠁⠀⢀⠀⡟⠀⠀⠻⣌⣴⡟⠉⠁⣠⣿⣠⣄
⢠⣴⠾⡿⠉⣿⠃⢠⡿⠋⣿⣿⣷⣿⣁⠀⢀⣾⠟⠀⠀⣠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡏⣩⣿
⢸⡏⢠⠃⣸⠇⠀⡿⠁⣸⣿⣿⣿⣏⣰⣾⣿⠋⠀⠀⠀⣉⣙⣿⣂⢀⣤⣶⣦⣤⣤⣴⣿⣯⣾⡿⠃
⢸⣤⣾⣴⣿⢀⣼⣃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣴⣾⣿⣿⣿⣿⣿⠃⣰⠟⢁⡼⠉⢩⡿⠻⣶⣶⣶⣶⣤⣤⣤⣤⣤⡤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣝⢿⣷⣍⠻⣝⢿⡻⣿⣟⣿⡿⠁⣰⠃⢀⣾⠁⣠⡟⢀⣼⣿⣯⡻⣮⡻⣿⣿⣝⠻⣝
⢸⣿⡿⣿⡻⣿⣝⢯⡻⣟⢿⣝⢝⢮⡳⣍⠳⣕⣮⡳⣝⢮⡻⣝⢿⣾⣿⣴⣿⣧⣴⣿⣶⣾⣝⠻⣟⢿⣮⡻⣮⠻⣿⢷⣌
⣸⣿⣝⢮⡛⢮⡻⢷⣝⢮⡳⣝⠳⣝⡮⡪⡻⣮⡻⣯⡪⡳⣝⠢⣕⢝⢿⣿⡷⣍⠻⣿⢯⡻⣯⡳⣮⡳⣌⠻⣮⡳⣌⠳⡝
⣿⣗⢍⠳⣝⢶⣝⢦⡙⢷⣌⠪⡑⢌⠻⣮⡢⡈⠻⣮⡙⢮⡪⡓⢬⡳⣝⢮⡻⣮⡳⣮⡳⣝⢮⡻⣮⡻⣮⡳⣌⠻⣌⢳⣌
⣿⣝⢷⡕⢌⠳⣝⢷⣝⢦⡙⣷⣌⠢⣑⢮⠻⣮⡢⡨⡻⣦⡱⣌⢦⡙⣮⡢⡙⢮⡻⣮⡻⣮⡳⣝⢮⡻⣮⡻⣮⡳⣌⠳⣝
⢿⢿⠳⡝⠢⠑⠌⠳⡙⢷⡝⢮⣿⣷⣜⢦⡑⣌⠻⣾⣮⣯⣿⣾⣷⣽⣮⣿⣮⢪⡻⣮⠻⣮⠻⡎⢣⠙⣿⣿⣮⡻⣜⢷⣌

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⡀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⢒
⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⠤⠄⠰⠊⠁⡇⠀⠈⠉⠓⢲⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⢐⡀⣆⠈⢳⠀⣇⠀⠀⠀⢀⡾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠙⠢⡀⠈⣢⡭⢹⣗⠚⢢⡜⢝⠲⣠⠏⢰⣀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⠏⠀⠳⠾⢥⠠⢃⠄⠈⢑⣾⠿⢆⣠⢗⣊⠭⣤⠘⠂⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄
⠀⢀⡠⠞⠃⠀⠀⠀⠀⠘⠶⣧⡞⠁⣉⠀⠀⠀⢙⠀⠀⢀⠽⢏⣳⠀⠦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡠⠀⠈⠀⢁
⠀⠘⠒⠒⠉⠉⢹⠓⠢⠤⣴⡻⠁⠀⠈⠀⠀⠃⠛⠒⢂⠤⠒⠈⠁⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣵⣾⣿⠮
⠀⠀⠀⠀⠀⢠⣁⠤⣒⡴⢮⣝⡓⣤⠀⠐⠂⡀⠾⠂⠀⡀⠠⠶⠶⢖⣲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⢀⠜⡡⠚⢁⠖⢹⠀⠈⠉⠉⡘⢋⠀⣠⡔⠉⡴⠃⣀⣠⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡛⡅
⠀⠀⠀⠀⠸⢴⠐⢃⠾⢴⣸⣆⠀⠀⢰⣃⣈⣉⣀⣇⡞⡠⠊⠁⡉⢀⡼⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣱⡀
⠀⠀⠀⠀⠀⢨⠋⠴⠶⡜⠁⠈⠳⠖⡟⡟⢆⠀⠉⠙⠿⣧⠶⠋⡰⢏⡀⠀⠘⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆
⠀⠀⠀⠀⠀⠈⠳⠗⣾⠁⠀⠀⠀⠀⠇⡇⠘⣆⠀⠀⠀⢰⢀⠞⠁⢀⣿⡄⠖⢀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢓⡀
⠀⠀⠀⠀⠀⠀⣠⢴⡇⠀⠀⠀⠀⢸⠀⡇⠀⠹⡀⠀⠀⢸⡏⠀⣠⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡇⠀⠁
⠀⠀⠀⢀⡠⠞⣡⢿⡇⠀⠀⠀⠀⡘⠀⡇⠀⠀⢳⡀⠀⢸⡧⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡛⠃
⠀⢀⠔⠉⣠⢚⡇⣿⠀⠀⠀⠀⢀⡇⢸⡇⠀⠀⠀⢧⠀⢨⣿⣿⣿⣿⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠈
⣰⠃⢀⠜⠁⠈⡇⢿⠀⠀⠀⠀⢸⠃⢸⡇⠀⠀⠀⠘⡆⢸⢿⣿⣿⠏⠙⠁⠀⠀⠈⢻⡠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃
⢇⠀⣏⠀⠀⠀⠙⣾⠀⠀⠀⠀⡼⠀⠈⡇⠀⠀⠀⠀⢣⡾⣿⣿⡇⢰⠀⠀⠀⠀⢰⡌⣳⡤⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠤⠄⠠⠇
⠀⠉⠉⠀⠀⠀⠀⠸⡇⠀⠀⢀⡇⠀⠀⡇⠀⠀⠀⠀⢸⢧⣿⣿⣿⡎⣆⠀⢱⣄⢲⡵⢶⢾⡏⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠄⣷⡀⢀⣸⣅⣠⡴⢛⢷⡶⠒⠛⢻⢸⣿⣿⣡⣟⠻⡶⠋⠈⠉⣿⠃⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠂⠁⣎⢻⡏⠀⠀⢸⢂⢢⢿⡇⠀⠀⣸⢶⣿⠟⣡⠎⣩⠆⠀⠦⢰⠷⠦⢤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠂⣾⠻⠯⢱⡎⡀⠣⢘⣯⠄⢶⡏⣾⠏⡐⠁⠀⠁⠀⠀⡴⠁⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⡁⢻⠀⠀⠁⠘⣿⣅⣈⡅⠛⢶⣀⣠⣆⡀⠀⣸⠇⢣⠀⠀⠐⡆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠙⣋⡟⠃⠀⠀⠀⠠⢷⠶⠤⠇⠠⡝⡼⡟⠀⠉⠙⣿⡆⠯⣀⡀⣠⠟⠹⢿⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠤⠄⠚⠋⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⠃⡇⠀⠀⠀⠈⠹⡟⠙⢯⡀⠀⠀⠀⠈⣿⢅⠈⠗
⠄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⠀⡇⠀⠀⠀⠀⠀⠹⠀⠀⠘⣦⣀⡀⣀⣾⣎⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢽⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣧⠀⠳⣄⠀⠀⠀⠀⠀⣀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠶⠶⠶⠶⠖⠒⠺⠿⠿⠖⠚⠛⠓⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠈⠀⠀⠀⠀⠀⠉⠀⠀⠠⠖⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⡠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡞⣠⠖⠉⠀⢠⡞⠁⢀⡾⠁⢀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣠⠗⠁⠀⢀⡠⣾⠁⠀⣾⠀⠀⡼⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣩⣤⡤⠴⠒⠉⠀⣿⠀⣸⠇⢀⡾⠁⢠⠞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢿⣰⣯⡶⣋⠤⠚⠛⠳⠆⠹⣄⠀⡆⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠀⡀⠀⠀⠀⠈⢦⠀⠀⠾⠋⠀⠀⠉⠉⠉⠀⠀⡠⠀⠐⢤⠀⠘⣶⡇⢪⡖⠀⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁⠂⣤⠠⠙⠒⠀⢠⢩⡆⠀⣠⣬⣌⠑⡄⠀⠀⠀⢰⣿⣿⣶⡀⠁⠀⠛⠁⠈⡛⠠⠞
⣀⡀⠤⠄⣠⠂⠀⠀⠀⠁⠀⠀⠙⢄⡀⠄⠐⠈⠀⠀⠑⠻⠀⠹⢿⣿⡇⠄⡠⠖⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⡜⢀⠀⠄
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⠀⠀⠀⠀⠙⠒⣀⢰⡄⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠉⠀⠀⠀⠀⣀⡤⢲⠙⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⡀⠸⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢲⠶⡲⠖⠒⡉⠍⠀⠃⠀⠀⠀⢡⡀
⠀⠀⠀⡆⠀⠀⡀⠀⠀⠀⡇⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⡀
⠀⡇⠀⡇⠀⠀⠧⠆⠀⠀⡇⠀⣿⣀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠈⠄⠀⠀⠀⠀⠀⣀⣠⡤
⠀⣷⠀⣧⠀⠀⠀⠐⠀⠐⠓⠂⠉⠀⠀⠀⠀⡀⠀⠠⠤⢤⣀⣀⡠⠔⠁⠀⢑⣄⡄⠀⠀⠀⠀⣀⠜⠐⠂⠠⡤⠀⠀⠀⠈⢄⣤⠴⠒⠋⠙⠢⢄
⠀⢹⠀⢸⡀⠤⠄⠀⠀⣀⠠⠄⠂⢀⡀⢀⣀⠀⠤⠀⠠⠗⣴⠂⠀⢀⠠⠖⠋⠡⢑⠦⠂⠐⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⣦⡀
⠠⠘⠉⠀⠀⠀⡠⠂⠁⡀⠀⠀⠐⠃⠀⠀⠀⢙⠁⠀⠹⢤⡙⡢⠈⠉⠒⠀⢴⡒⠈⡀⠀⢀⡀⠄⠐⠉⠀⠀⠀⠀⢀⠠⢖⠉⠄⣈⡱⠶⢖⡊⠉
⠀⠀⠀⠀⠀⠀⣄⠀⠣⡀⠀⠀⡈⠀⢠⠉⠑⠥⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⠆⣠⠊⠑⠠⣀⢀⡀⠄⠒⠈⠀⠀⠀⠂⠌⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠈⠐⢄⡈⠑⣤⠀⠀⠆⠀⠀⣠⠼⠆⠘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠏⠲⣄⣀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢌⣂⢐⠦⡀⠁⠀⢀⡰⠶⢃⡀⠁⠒⠠⣀⠀⠀⠀⠀⢀⠠⠀⠊⢀⡤⢺⠐⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠈⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡈⠑⠤⣀⠻⣀⡉⡼⠀⠀⠀⢀⣀⠦⠐⠈⣀⡠⠔⠊⠁⠀⠸⣠⡇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠉⠂⠤⠤⠄⠒⠈⢁⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢠⠌⠙⠤⠄⠀⠐⢒⣉⡠⠤⠛⠻⢅⡀
⣀⣤⠴⠖⠋⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠀⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡄⠀⠀⢨⣅⠀⠀⠀⠀⠀⠈⠙
⠧⣄⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡐⠳⡀⠀⠈⡄⠉⣒⣤⣤⠖⠊⠉
⠀⠀⢉⣲⠤⣔⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠗⠂⠔⠒⠈⣱⠀⣈⠢⠀⠙⠲⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠙⠛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⡄
⠀⠀⠀⠀⠀⠀⣠⠴⠛⣩⠽⠖⠒⠚⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⢠⡞⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠻⣆
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⣆
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡄
⠀⠀⣇⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⣀⡀⠈⠙⣦⡀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣧
⠀⣀⣻⡀⣠⡇⠀⠀⡼⢇⣠⠴⠋⠉⠉⠛⠶⠇⠙⢦⠀⠀⣸⠻⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⠉⠉⣿⠀⢻⡀⣸⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠋⠀⠹⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⢠⢀⡟
⢿⣄⣀⣿⠀⠀⠳⣿⠀⠀⣠⣶⠶⠶⠀⠀⠀⠀⠀⠀⠙⠁⠀⠙⠛⢷⣄⠹⣄⣀⣤⣀⠀⠀⠀⣸⠛⠓⢶
⠀⠈⠉⢹⡄⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠹⠃⠈⠁⠀⠈⢳⡄⢀⡿⠀⣀⣼
⠀⠀⠀⣨⣧⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠶⣄⠀⠀⠀⣴⣾⣿⣿⠛⠳⣆⠀⠀⠀⣀⡀⠀⠹⣾⠟⠛⠋⠁
⠀⠀⣰⠃⠸⣆⠀⠀⠀⠀⣼⠁⣿⣿⣻⣄⣼⣧⠀⢸⢿⣿⣿⣯⣶⣾⠿⡆⠠⠚⣏⠉⠀⣰⡟
⠀⢀⡇⠀⡆⠹⣆⠀⠀⠀⣿⠀⠈⠻⢿⣿⡿⣿⠀⢸⡄⠙⠛⠛⠛⠁⢰⠇⠀⠀⣿⠀⣰⠟⢦
⠀⢸⡇⠀⡇⠀⠘⢧⡀⠀⠘⢧⣀⣀⣀⣠⡼⠃⠀⠀⠻⣤⣀⣀⣀⣴⠟⠀⠀⠀⣏⡼⠃⠀⠘⡇
⠀⠀⢻⠚⣷⠀⠀⠀⠙⢦⣀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⡾⠋⢀⡟⢛⡟⠁
⠀⠀⠈⢦⠘⣧⠀⠀⠀⠀⠉⠛⠶⢤⣤⣤⡄⠀⠀⠀⠀⠀⣶⣤⣤⡤⠶⠛⠉⠀⢀⡾⢀⡞⠁
⠀⠀⠀⠈⠓⠾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⣴⡾⠟⠁⠀⠀⠀⠀⠀⢀⣾⡗⠋
⠀⠀⠀⠀⠀⠀⢸⡙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠳⣄⡈⠙⠓⠶⢤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⠶⠞⠛⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠲⢤⣤⣀⣀⣠⣽⡟⠉⠉⠉⡿⠦⣤⣤⣤⣤⣴⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠈⠉⠀⠘⡇⠀⠀⢸⡇⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣤⠞⠁⠀⠀⠸⢧⣤⣀⣀⣤⠟⠁
⣿⣿⠶⢿⠿⠷⣿⡷⣿⡿⢿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⢳⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣯⣿⣷⣈⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣽⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠟⣉⠉⣩⠍⣋⣷⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⢋⡉⢉⡭⣹⡇
⠀⠀⠀⠀⢀⠤⢤⣤⡤⣶⡾⢛⡀⢅⠓⢘⢰⣶⣿⣿⣩⡾⣿⠀⠀⠀⠀⠀⡠⠤⢤⡤⢤⣶⡟⣃⠨⡘⠆⡉⣾⣿⣿⡇
⠀⠀⠀⠀⠈⢦⡀⠙⣿⣿⢉⣈⣓⠠⣄⠘⢿⣿⣟⣵⡿⢀⣿⠀⠀⠀⠀⠀⠹⣄⠈⢻⣿⡟⣁⣙⣃⢤⡀⠹⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠉⠣⣿⠇⣹⣞⣿⣿⣿⣦⠘⡽⠉⠉⠀⢰⣿⠀⠀⠀⠀⠀⠀⠈⠙⢼⣿⢁⣷⣿⣿⣿⣷⡤⢋⠏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⣭⣭⣵⡿⠉⠉⡳⠔⠀⣠⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣩⣽⣭⣾⠏⠈⢉⠦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣤⣅⣀⠀⢈⣀⠀⢈⠿⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣤⣬⣀⡀⠀⣁⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⣶⣿⠋⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠷⣶⣶⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠲⠤⠔⢋⣰⣠⠶⠎⢀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠈⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣄⠤⠤⠒⠒⠲⠚⠀⠠⠀⣴⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠘⠃⠀⠀⡼⠤⠤⣌⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⠀⠉⢀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡰⠊⠀⠀⠰⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢐⣀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠎⠑⢄⡀⠀⠀⢇⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⡠⠊⠈⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⣄⠀⠜⠉⠑⢄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠈⠙⢆⣀⣀⠜⠑⢶⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠆⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢼⡄⠀⠀⠀⠸⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠀⡸⠓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⠆⢠⠀⠀⠀⠀⠀⢂⠀⠀⠀⠿⢤⣀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠃⠀⣠⢤⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠃⢠⠃⠀⠀⠀⠀⠀⠈⡆⠀⠀⢣⠀⠈⠙⠲⢤⣀⠀⢀⡜⠃⠐⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢀⠌⠀⢠⠎
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠎⠀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠙⠄⣀⠸⠆⢤⠀⠀⠀⠈⠉⠚⢇⣀⢀⡔⠀⠀⠁⠒⠠⢄⡀⠀⡀⠀⠀⠀⢠⠎⢀⢠⡆⢀⠠⠇⢄
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠠⠁⠠⠾⢤⣤⣤⣤⣤⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣀⠀⠈⠀⠀⠈⡆⠠⡴⠁⡐⠁⠀⠀⠈⣇⡠⠞
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠱⠦⠤⠤⠤⠄⠤⠄⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⠠⠾⡄⣰⠀⠀⢣⡀⠀⠐⠀⠉⠄⣠
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠒⠊⠉⣠⠤⠄⡀⠐⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠒⠂⣿⠈⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⢀⡀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⡇
⡜⠉⠀⠉⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠑⠦⠚⠁⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠶⣽⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠄⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠋⠀⠀⠀⡅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠒⠚⠠⠤⠤⠄⠀⠀⠀⢀⡀⠀⠀⠀⠤⠤⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠐⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢤⠀⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⠁⢠⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠸⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⢠⡀⣠⡀⢸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣿⡟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡋⠙⢫⣽⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢜⣿⡁⣀⡤⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣾⡟⠀⠉⡀⣿⣿⠻⢧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⢉⠉⢋⣵⠀⣤⣴⣿⠻⠿⠏⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡿⠛⠟⡅⠄⣼⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡉⣻⣿⠀⠀⢠⠤⠤⠤⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⢘⣀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣎⠣⣬⡙⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢻⡗⠚⡇⠀⠁⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠛⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠽⠟⠉⠵⠘⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡤⠖⠁⠀⠸⠃⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣹⡝⠷⣶⠶⠇⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠨⡟⠛⠁⠨⠟⠗⠶⠒⠗⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣜⣠⣤⣤⡄⠀⠀⡀⢸⠒⣦⠧⠀⠤⠀⢀⣄⣀⡀⠀⢨⡟⢦⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢳⣤⣶⣾⡵⣿⣍⢠⠲⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣭⣿⠘⢯⣤⣤⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⠁⠀⠀⠀⠀⠀⠀⢰⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠛⠛⠳⠶⣤⣀⣀⣤⠤⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⠋⡴⠄⠀⠀⠴⠂⠈⡏⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⣀⣀⣀⣿⠀⠀⢻⡝⢣⡟⠀⠀⣿⡀⠀⠀⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡰⢋⡴⠖⠒⠻⢦⣐⣒⠈⠭⢐⣒⡿⠟⠛⠲⠶⠾⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢄⠀⢸⡆⠀⡇⢸⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⠒⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⡃⣞⠛⣋⣉⣉⣹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡹⢋⡞⢡⠄⠀⣄⢲⡹⣝⡆⠀⠀⠀⠀⠀⠀⠀⢠⡿⢛⡻⣿⠟⣛⠻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡈⠀⠈⠀⠸⢘⡣⠘⠄⠙⠀⠀⠀⢰⣲⠀⠀⠀⠀⣿⠿⣯⡀⣇⠸⣿⢇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⠞⠁⡠⠀⢠⠀⠀⠀⠀⣄⠀⢠⡀⠀⠀⠀⠀⡠⣄⠀⣹⣧⢭⣥⠿⢷⡶⠻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⣀⡀⠀⠰⠁⢀⡾⠛⠛⠳⣄⣸⣤⣄⢱⡞⢙⡇⠈⠂⠊⠀⣿⢅⡲⣿⢘⣻⡘⠟⣻⡽⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣉⣹⢆⡴⠚⠛⣷⠶⣤⡟⠻⣄⣠⢼⡇⠙⠋⠁⠀⠀⠀⠀⠻⣮⣥⣾⣶⣾⠿⠛⠉⣾⣿⡞⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⡁⢾⠀⣰⠶⣿⡀⠘⣷⠀⠙⣷⠀⠻⡟⠓⣦⠀⠀⠀⠀⠀⠈⠉⠋⣉⠀⠀⠀⠀⠉⠉⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⠴⢿⡾⢧⣽⡄⠈⢧⠀⠈⢧⠀⠘⣇⠀⠹⣆⠘⡇⠀⢰⣷⠀⢀⡀⠀⣿⠀⠀⠀⠀⢀⣴⣶⠸⣷⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⠀⢀⡿⣞⠁⢻⡄⠈⢧⠀⠈⢷⠀⠈⠀⠀⠘⣶⠇⠀⢸⣿⠀⣼⣧⠀⣿⣀⣶⣶⣦⠘⠿⣦⡄⢿⣿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠙⣩⠖⠛⣰⠶⣿⡄⠈⠧⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⣿⣀⣿⠿⣷⣿⢿⣇⣾⣿⣄⣴⡾⠏⠘⠃⠰⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⠀⠀⢻⡄⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⠿⠟⣠⡉⢁⣈⠛⣋⣈⠛⣡⠀⣶⣾⣆⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣧⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢿⣇⣼⣿⣼⡟⢻⣿⣿⣀⣿⠻⣿⠏⠙⠁⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠈⠳⢤⣤⣤⣶⣋⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠙⢛⣿⠙⠿⠿⢃⣙⠛⣿⣠⣿⡶⣶⡜⢿⣦⣍
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢸⡁⢈⡇⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠙⣆⠀⢸⣆⠀⢿⣿⠋⣴⣶⡀⢸⣿⣧⣿⡇⣿⡆⢸⣿⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠉⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠘⣧⠸⣿⠀⢘⣿⡀⣿⣛⣿⡌⣿⡝⣿⣿⢸⣷⣾⢿⣦⣤⣼⡟
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣠⠏⠀⢿⡿⠛⢿⣇⣿⠟⠙⠿⢿⡇⠈⠛⠀⠛⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠤⠴⠚⠁⠀⠀⢸⣷⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠶⠶⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⠻⣆⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣙⠛⠿⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡼⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⡇⠀⢰⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡼⠁⠀⠀⣸⠁⠀⠀⣾⠀⠀⠀⡇⠀⢸⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡼⠀⠀⠀⠀⡼⠁⠀⠀⢠⡇⠀⠀⢠⡇⠀⠀⢰⠇⠀⢸⠙⡆⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⢀⣼⡇⠀⠀⠀⣼⠃⠀⠀⢀⡞⠀⠀⠀⡼⠀⠀⠀⣼⠀⠀⡼⠀⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢸⡇⢷⣠⡴⠋⢸⡇⠀⣠⠞⣿⠀⠀⣠⡞⠀⠀⠀⣼⠁⠀⠀⣰⠛⠶⠞⠁⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢀⣀⣀⡀⠈⣇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠛⠋⠁⠳⣤⠴⠚⠹⣄⣠⠴⠃⠀⠀⠀⡀⠀⠸⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸⡁⢨⡑⠦⠼⠀⠀⠀⠐⠳⠶⣤⣤⣤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⡤⠴⠾⠁⠀⠀⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠶⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⣀⣐⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠚⠀⠀⠀⠀⣀⣀⠀⠈⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⢠⠀⠀⠀⠈⠳⢦⡞⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢷⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣴⣏⠀⢀⣼⠀⠀⢠⡇⠀⢘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠸⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠛⠉⠛⠛⠻⡧⠶⠋⣯⠉⠛⢲⡦⣄⠀⠀⢀⣀⣀⣀⣀⣤⡤⠶⡟⠙⠒⣆⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⠷⠟⢦⡈⠓⢦⣀⡤⠞⠁⢙⣦⣤⠞⠁⠀⣰⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠳⠴⠟⠉⠳⣤⠖⠋⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡋⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡍⠉⢹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢘⠶⠲⡄⠀⢀⡡⠤⣄⡀⠀⠀⠀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⠀⣰⠋⠀⠀⢷⠀⡾⠀⠀⠀⠙⣦⢀⡖⠋⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠞⢀⡀⠀⠈⠑⢦⠀⠘⡇⣇⠀⠀⠀⠀⠈⣿⠀⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠷⠋⢣⡤⢦⣤⣘⢧⠀⠹⣿⡀⣀⠀⠀⠀⠸⡷⠿⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢧⠀⠙⢷⡉⠀⠀⠀⢀⡇⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠦⣄⣸⣽⠀⠀⠈⠳⣤⣀⣀⡜⠉⠉⠛⠛⠦⠤⠤⠤⠴⠶⠒⠒⠒⠋⠉⠉⠉⠉⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠛⠷⠦⠙⠋⠀⠀⠀⠀⠐⠛⠉⠀⠀⠐⢶⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⢛⣷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢤⣄⣀⡀⠀⠀⠀⠀⢀⡀⢄⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣺⠖⠋⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⢲⡶⠴⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡡⠚⠁⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⠞⢡⠞⠀⠀⠀⠀⠀⢀⡴⠋⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢉⢇⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⣰⠛⠶⠤⢤⣀⣀⡴⢯⡀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⢴⣃⠴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⡏⡔⠉⣉⣽⠂⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡰⢬⣥⠖⡋⡧
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠹⠃⡐⢦⡼⣂⡴⡋⢸⣀⣀⣠⣤⡤⠴⡖⠊⢉⣠⠟⠀⣈⡭⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠙⢦⡀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠚⠉⠒⠋⠁⠀⡼⣺⡅⠀⢠⠜⠳⣄⣏⡏⠉⠀⣠⠞⠁
⡔⠀⢀⡀⢀⡞⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡴⠃⠀⠀⠈⢻⡇⡠⠞⠁
⠇⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠉⠳⣤⣀⣀⣀⣤⠤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⡼⠳⠤⠤⠤⠶⠚⠉
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣀⣀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠙⠲⡦⠤⠤⠤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⢀⡧⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠇⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⢀⣀⡤⠖⠚⠉⠀⣳⠦⣤⠎⠠
⠀⠀⠀⠀⠀⠀⠀⠙⢗⣒⡶⣯⡉⠀⠀⠀⠀⠀⠀⡟⠹⡇⠠⠃⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠀⠀⠀⠀⠀⠀⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣣⣴⣾⠿⢛⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣾⣿⡟⣡⣾⣿⡿⠟⢛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⡿⣴⣿⡟⣡⣾⣿⣿⠿⠿⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠷⠿⠿⣿⣶⣦⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⢟⢻⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣏⡽⣿⣷⣶⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣒⣒⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⠸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⣯⡗⢾⠀⠹⣿⣽⣿⣿⢛⣿⣛⡟⣯⢞⡷
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⣛⣿⣷⠿⠛⠉⠀⢸⡆⠀⠈⠻⠿⠟⡟⠿⢿⠟⠋⣸⠇
⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⠻⡆⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠅⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡆⠀⠀⠀⣸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣷⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢿⠀⠀⢀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⡧⠐⢿⠂⠀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⡉⠑⢾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢡⠃⠆⠠⣷⡄⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡹⠁⠀⣸⠷⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢼⠘⠿⣲⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣁⣐⣂⣁⣤⢤⠛⢷⡀⠀⠀⠀⠀⣀⣤⣶⣦⡴⢁⣎⠉⠁⠀⠀⠀⢀⣴⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⢸⠃⠈⠁⠀⠘⠊⠉⠪⡑⠦⢐⡨⠿⣿⡯⣿⣷⣼⣯⣷⡄⠀⠀⣠⠏⠈⢯⣦⣿⠋⢹⡄
⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠀⣿⣿⣞⣯⣿⡼⢏⠳⢔⣫⡟⣴⠀⣸⢹⡀⢸⡾⡇
⠀⠀⠀⣛⡭⠭⠝⠒⠒⠉⠉⠉⠛⠓⠒⠲⠤⠤⠤⠀⣀⣀⣀⠀⠯⠽⣿⣛⣿⣿⡥⠄⠀⠉⢩⢎⡾⢹⠀⢸⡄⢳⢈⣧⡙⢦⡀
⠀⠀⢰⠋⡿⠿⢿⣶⣶⣶⣦⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢹⢀⡾⠯⠝⢻⣷⠁⠀⠀⠀⣇⣾⢁⡾⠀⢈⡇⢸⠸⢸⡎⣇⣷
⠀⠀⢸⠀⡇⠀⠀⠀⣀⣈⠉⠉⠉⠙⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡟⣿⣶⣿⣇⠀⠀⠀⠈⠉⠻⢧⣀⣼⣤⡷⠤⠼⠃⠈⠁
⠀⠀⢸⠐⡇⢀⡔⣫⠭⠭⡙⢷⡄⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣽⣻⠿⢿⣿⡆
⠀⠀⢸⠐⡿⣿⢸⠃⠀⠀⠘⡌⣿⣤⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣥⣹⣷⣶⣯⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣀⣀
⠀⠀⢸⢸⡏⠹⡜⢧⣀⣀⡴⢠⡿⢭⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⣿⠉⠀⠀⠀⠀⠀⠀⢀⣜⣹⡭⢷⣿⠋⠁
⠀⠀⢸⢸⡇⠀⠈⠒⠭⠭⠔⠋⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡤⣤⠤⠿⠀⠀⣀⣀⣀⣀⣀⢎⣾⡿⢣⡿⠁
⠀⠀⢸⢸⡇⠤⠤⢀⣀⣀⣀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇⣠⠔⠒⢉⣩⠭⠤⠒⠲⠏⢾⡷⢳⡿⠳⢤⡀
⠀⠀⢸⢸⡧⢤⣀⣀⣀⠀⠀⠉⠉⢹⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣧⡀⠔⠋⠁⢀⡀⠈⢀⡀⠤⠀⡉⠙⢂⣠⡏
⠀⠀⢸⢸⡗⠂⠤⠤⠄⣉⣉⠉⠓⢺⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢛⣏⠙⢛⣟⡲⠲⠶⠾⠗⠶⠚⠻⠿⣿⣏⠗
⠀⠀⢸⣸⡋⠙⠒⠂⠄⠤⠤⢈⣉⣹⡇⢸⠀⠀⠀⠀⢀⣤⣶⠋⠀⠀⠀⠟⠘⡆⠈⠻⢀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠛⠻⠿⢷⣶⣶⣤⣤⣄⣀⣸⡇⢸⠀⣀⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⠢⣀⣀⠀⠀⠀⠀⠀⣀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⣠⣴⡾⣿⣿⡿⠿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⢀⣼⣿⣵⣺⣽⢶⣟⣛⣛⣛⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⣾⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣏⣇⡤⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⢣⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣿⡿⠿⠿⠟⠋⢻⣿⣿⣿⣿⠀⠀⠀⠀⣿⢋⡴⠋
⠀⠀⠀⠀⠈⠻⢾⣿⡀⠀⠙⠁⠀⠀⠀⢰⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠛⠁⠀⠤⠴⠂
⠀⠀⠀⢀⣀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢀⢼⣤⠀⠀⠀⢠⣴⣧⠄⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠦
⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⢿⠿⠃⠀⠀⠐⠾⠯⠗⠀⢸⣿⡇⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣣⣄⠀⠀⣸⣿⣿⠀⠐⠚⠛⠀⠰⡤⠀⠒⠖⠂⠀⢸⣿⡇⠀⠀⠀⣼⢶⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢊⠝⠇⠀⠀⣿⣿⣿⣧⠀⠀⢠⡀⠀⠀⢠⠀⠀⠀⢀⣾⣿⡇⠀⠀⠰⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⡀⠙⠒⠒⠁⠀⣀⣤⣾⣿⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣉⣉⣛⣛⣛⣛⣛⣛⣓⣚⣛⣛⣛⣛⣋⣉⡩⠭⠽⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣠⠤⠬⠿⢦⡀⠉⠉⡩⠭⢍⡀⠀⠀⣴⣛⣛⠒⠲⢄⡈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⡔⢋⡠⠖⢊⣉⠭⢷⡀⣞⠰⣿⡆⢹⣀⠼⠓⠲⠤⡉⠲⣄⠙⣄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠋⠀⠊⣠⠞⠉⣀⠤⠦⢬⠟⢦⣙⡡⠞⠳⠒⠛⠲⣄⠈⠳⠈⠁⠈⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⠾⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⢧⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⠚⢳⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠘⡆⠀⠀⡀⠀⠀⡏⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⠃⢀⣞⠹⡄⠘⠃⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⠀⠀⠀⠀⠀⠸⡖⠈⠑⢄⠀⣼⢄⡀⡇⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠄⢀⠀⠀⢿⠁⣀⡈⢳⡇⣾⡇⡇⠀⠀⠀⠀⠁⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠳⡘⢿⢦⢳⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⣽⠇⣈⣀⣉⣓⠒⠲⠦⠤⠤⢤⣀⣀⣀⣀⣀⡀⠀⡰⢁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠯⠁⢀⡚⠥⠷⠾⠞⠃⠀⠀⠀⠀⠀⣀⣠⠤⠤⠞⠛⠋⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⠋⢋⠀⠃⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⡟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠃⡠⡞⠀⢀⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠻⠊⠀⣇⡠⠛⡇⢠⠞⣧⠞⣦⠴⠛⠉⠙⢣⢀⣹⠭⠭⠭⣝⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⡀⠈⠁⣀⡔⢲⣴⠾⠛⠁⠀⠀⠀⠀⠈⠳⣜⢆
⠀⠀⠀⠀⠀⠀⠀⢠⢋⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢚⡟⠑⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡸
⠀⠀⠀⠀⠀⠀⡎⠘⠀⠀⠀⠀⡰⠄⠀⠀⠀⠀⠀⠀⢀⣞⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⡴⠋⣀⡴⠁
⠀⠀⠀⢠⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⢴⠛⠀⣸⠋
⠀⠀⠀⡇⠀⠛⠉⠓⢲⡶⠤⠤⠖⠒⠏⠉⠀⣀⡴⠋⠿⣄⡀
⠀⠀⠀⠈⠢⢏⣂⡀⠀⠈⢁⣀⣀⣦⠶⠖⠋⠁⠓⠒⢲⡤⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣿⠟⠋⠀⣀⣴⣿⣤⡀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠻⠟⠛⠋⠉⠠⢶⣿⣿⡛⠛⣻⣦⠀⠀⣼⣿⡟⢿⣯⠻⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣶⣈⣉⣤⡿⠋⠀⠀⠙⣧⣜⣀⣉⣿⠆⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⢀⣠⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⢿⣿⣿⣿⣿⣤
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠈⠉⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⣣⠀⢿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠛⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣁⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⠏⠙⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⠿⠟⠋⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠷⣶⣄⠀⠀⠀⠀⠀⠀⣾⡏⠀⣀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⠻⣷⡀⠀⠀⠀⣴⡟⣀⠾⠁⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠶⠆⣼⣿⣤⣄⣤⣿⡇⠛⠀⠀⠀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢹⡏⠉⠉⠙⣿⣦⣴⣿⣿⣏⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣀⣠⣾⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠿⣿⣝⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠈⠻⢿⣤⠻⣯⡙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠁⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠙⣿⣆⠙⣦⡙⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠷⣶⣤⡈⠀⠀⠀⠈⢿⣷⡘⢿⣼⢿⣿⣿⣿⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⡈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣀⡀⠀⠙⣿⡆⠀⠀⠀⠀⠿⣧⡼⢻⣼⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣁⡄⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢼⣿⣏⣠⠀⢸⡧⠀⠀⠀⠀⠀⠻⣧⡼⢻⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡟⠿⣧⡀⠀⠈⠙⠋⢁⣼⠋⠀⢰⠆⠀⠀⠀⠀⣿⡄⠀⠈⠻⠟⠃⢀⣾⡇⠀⠀⠀⠀⠀⠀⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡇⠀⠙⠿⣦⣤⣤⣴⠿⠉⠀⠀⠸⢧⠀⠀⠀⠀⠈⠿⣧⣤⣤⣤⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠟⠛⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠛⠛⠿⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⠶⠿⠛⠛⠁⠀⠀⠀⢿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⡏⢙⡛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠿⢯⠛⣦⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣤⣤⡤⠚⠻⡿⠛⢠⡜⠀⢸⣤⠘⠛⣥⠀⠀⠀⠀⠙⠛⠛⠓⠢⣤⣤⢀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣿⠟⠛⠛⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣦⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣏⠀⠃⠀⠞⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠃⠀⡄⠀⠀⠹⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⢀⡆⠀⠀⠀⢠⡄⠚⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡠⠏⠀⠀⠀⠀⠀⠀⢀⠈⢀⠤⠀⠠⠤⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠀⠀⠀⠀⠀⠀⠈⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⠷⠶⠶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢉⠉⠉⠉⢙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⢿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⢸⠃⠀⠀⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⢠⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠘⠁⠀⠀⢠⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⢀⡜⠃⠀⠀⠀⢸⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡄⠀⠀⠀⠀⠤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢀⡁⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠑⢦⣄⠀⢸⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠈⠉⢶⣿⡁⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠇⣀⠀⠀⠈⣿⡇⠀⠀⠛⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⡔⠂⠀⢻⣧⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⢅⠉⢤⡤⠙⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣤⣤⣤⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠀⠈⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠦⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡩⠷⠒⠒⢼⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠱⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⠳⣄⠀⠀⡎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢧
⠀⠀⠀⠀⠈⠓⠬⣓⠢⢄⡈⢣⡈⠓⢶⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀
⠀⠀⠀⢀⡤⠤⠤⠀⣉⣦⠈⠉⠁⠀⠀⠙⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠌⡃
⠀⠀⠀⠀⢀⣉⣉⠉⠠⠦⠀⠀⠀⠀⠀⠀⠘⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⡟⠛⠛⠶⡒⢤⡀
⠀⠀⠀⠀⠙⢒⣒⣐⡒⠂⡄⡴⠒⠒⠒⠒⠒⠛⠒⣲⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⣧⣠⣤⣴⠡⠾⠇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⠛⠛⠛⠛⠉⠉⣩⣿⡇⣯⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠷⣿⡟⢏⠉⣲⠛⠦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠈⣿⠁⠀⡰⠋⠲⢄⡀⠀⠀⠈⣧⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠞⠀⠀⠀⠀⢘⣦⠞⠉⠀⠀⠀⠀⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡜⡇⠀⢀⡤⠤⠄⠀⠤⠤⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⡄⢰⡇⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⡖⠒⠤⠤⠄⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠎⠀⢠⠘⠇⡇⠀⢻⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠀⢸⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣸⠁⠀⠀⠀⢸⠀⡄⡄⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇⡇⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡆⠀⠀⠀⠀⡌⠀⢰⢁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡁⠀⠀⠀⢠⠃⠀⢸⢸⠀⠀⢣⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢇⣀⣀⡠⠋⠀⠀⢸⠘⡀⠀⠀⠙⠒⠒⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⢫⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢇⣀⠀⠀⠀⠀⠀⠀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢰⡉⠀⠀⠀⠀⠤⠤⣀⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠔⢉⠀⠂⠙⠛⠁⠀⠀⠈⣤⠦⣠⠤⣙⠲⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⡨⢄⣀⢤⠂⠀⣤⣀⠀⠀⠀⠀⠙⠲⢌⢳⡘⢆⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⠚⠃⣰⢋⡖⠁⠈⢉⠔⣩⠞⠻⠂⠀⠀⠀⠀⠀⠀⡀⠹⠀⡇
⠀⠀⠀⠀⠀⣠⠎⠁⠇⠘⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡀⠒⠚⠦⡀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠅⠀⢠⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⢿⡀⢳⠀⡆⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠃⠀⠀⠀⣜⠀⢀⡌⢀⡞⠀⣀⡀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠁⢠⠛⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⡾⠎⢠⠄⢀⣾⠃⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⡘⢆⠧⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠈⣵⡖⠈⡀⢰⠇⢠⠎⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡵⠁⠀⠚⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡸⠋⢀⢈⣧⠀⡄⠀⠀⠠⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣫⠴⢁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⢉⢀⠈⣴⠙⠄⠹⠀⡄⠇⠈⠃⠀⠀⠶⢶⠂⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠿⣿⠇⠀⠀⠤⠀⠀⠀⠀⢸⡴⠖⠛⠋⠛⠶⣤⣀⡀⠔⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠈⡐
⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠂⠀⡄⠁⠀⠀⠀⠒⠶⢄⠀⠀⠀⠀⡌⠀⠀⡤⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⢡⡀⠀⠀⠀⠁⠀⢠⠶⡘⠙⣄⠀⠀⠀⠀⢀⡤⠓⠢⢀⡔⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠆⠈⠁⠀⣆⠀⠄⢠⣈⠊⠀⠀⣟⢰⣂⠀⢰⠋⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢀⢠⡆⠀⡇⢀⠴⣿⠇⠀⠀⢰⡿⠀⠿⡀⣭⠒⠂⠤⠄⢀⣀⣀⠎⠀⢀⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠸⣯⣷⡘⠋⣿⠀⢻⡄⠀⠀⢰⡅⠀⠀⢁⠀⠀⢃⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠙⢧⠙⣇⠈⠳⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠑⠀⠀⠀⠀⢠⠂⢸⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠠⡏⠀⢸⠀⠀⡀⠀⣆⠀⠀⠃⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠀⢼⠀⡖⠁⣸⠙⠀⣦⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⣿⠀⣌⢁⡦⠀⣉⡳⣾⡆⠠⠤⠤⣄⡀⠀⠀⠀⣀⣀⣤⢼⡇⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢸⡇⢀⣿⢰⡽⠃⠀⠀⠀⠈⡍⠁⠀⠈⢹⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠘⠁⢿⡁⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⡀⠀⢈⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⠀⠀⠀⠈⠶⠒⠒⢺⡏⠉⠙⡃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠘⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐⣸⠤⡆⠿⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠾⠋⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣾⣿⣿⣶⣶⣶⣶⣿⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣲⡶⠶⠶⣶⣒⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⣤⣤⡇⠀⠀⠀⠀⠀⢠⡒⠒⠒⢲⠀⠀⠀⠀⠀⢀⡔⠒⠒⣎⠀⠙⠢⢄⣀⣠⠤⠚⠁⠘⡖⠒⠲⡄
⠀⠀⠀⠀⠀⡴⠒⠒⡟⠒⠒⠒⢳⡀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡆⠀⠀⠀⢀⠎⠀⠀⠀⠈⠙⠲⢤⢴⠋⡦⠤⠤⠒⠋⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⡇⡎⡜⠀⠀⠀⠀⠀⢳⠀⠀⠀⣠⠞⠒⠒⠀⠚⢧⣀⣀⣠⣋⣀⡀⠀⠀⠀⠀⠀⠀⠈⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⣧⠟⠀⠀⠀⠀⠀⠀⠀⢣⢠⠎⠁⠀⣀⣀⣀⠀⢀⡈⢳⠖⠤⠤⠬⠭⠭⣉⣉⠭⠶⣆⠇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⠋⠁⠀⠈⡆⣿⢱⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣤⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⡇
⠀⠀⠀⠀⠀⠳⠦⠤⠤⠤⠤⠤⠤⠴⢲⡏⠀⡇⠀⠀⠀⠀⡇⣿⢸⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠯⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⡇
⠀⠀⠀⠀⠀⢰⠤⠔⠊⠉⠉⠉⠒⠒⣾⡇⠀⡇⠀⠀⠀⠀⡇⢿⣸⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣶⡟⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⠙⠒⢄⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠷⡿⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠛⡆
⠀⠀⠀⠀⠀⣘⡒⠤⠤⠤⠤⠤⠤⠐⣻⡇⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠤⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⡅
⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠉⠉⠉⠁⠀⣇⠀⠈⠉⠐⠒⠒⠂⠉⠁⠀⡟⠒⠒⠲⠦⠤⠤⣧⠤⠲⣟⡿⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⠇
⠀⠀⠀⠀⠀⠳⠤⠤⢀⣀⣀⣀⠤⠤⠔⠻⠤⣀⣀⣀⣀⣀⣀⣀⣀⠴⠋⠙⠛⠓⠒⠲⠶⠾⠖⠚⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⡠⠠⣦⠀⢠⠀⠐⠆⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⢁⣀⡀⠂⠃⠔⠘⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠁⠀⠀⠈⠠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣘⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠆⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡋⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⢀⣀⣤⣤⣶⣶⣶⣯⣤⣤⣀⡀
⠀⠀⠀⠀⠸⢷⣦⡀⠀⠀⠀⠀⠀⠁⠀⣰⢋⣴⣿⣿⣥⣤⣈⣉⡙⠛⠛⢻⣿⣿⣵⡢⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⠀⠀⠀⣠⣤⣽⣿⣿⡿⠟⠟⠻⢻⣿⣯⣶⡶⠇⠈⠉⠽⣿⣷⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣿⣥⣴⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣒⠀⠙⠻⣿⣷⣕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠋⣉⣠⣤⣤⠤⠤⣴⣤⣀⣟⣻⠻⣿⣿⣿⣿⣿⡦⠍⠺⢿⣿⣿⣧
⠀⠀⠀⠾⠿⠛⠛⠋⠉⠁⢠⢾⠃⠈⣁⣀⣠⣤⣤⣤⣤⣤⣏⣉⡗⢀⢸⡏⠉⣻⣿⣿⣷⡉⠳⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡟⠀⠉⢉⣁⣤⣤⣤⣭⣴⡿⢷⡄⠀⠛⠛⠁⢠⣾⣿⠟⠿⣶⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢡⡶⠶⠛⠉⢩⡿⢯⠻⢿⣿⣦⣉⠀⠀⠀⢠⣿⣿⠟⠃⡀⠠⣽⣿⣿⡽
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣼⣧⡰⣤⣸⣆⠀⠈⠃⠀⠁⠈⠋⠁⠀⠀⠘⣿⣷⡀⠀⣙⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⡼⠀⠀⢀⣴⣿⡿⠥⡈⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣸⢯⣤⣽⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⢧⡀⠳⣿⣿⣿⣋⣉⣓⠢⡄⠀⣿⠀⠀⠀⠀⣀⣴⡿⠇⠘⠿⠿⢫⣵⣶⡌⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠿⣿⣭⣍⣉⡉⠙⢿⠃⢋⠀⠀⢀⡼⠋⠁⠀⠀⢀⣠⣶⢾⣿⢟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠠⠴⠖⠋⠀⠀⠊⠀⠀⠀⢀⣼⡿⢧⣌⡥⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣦⣀⣀⣀⠀⠀⠀⡀⣀⣀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⣴⠞⢿⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠿⠃⠈⠀⠀⠈⠉⠀⠀⠀⠉⠉⠉⠛⠒⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡆⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣰⣧⠀⣼⠇⠀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠀⢸⡇⠀⢰⡆⣰⣿⡀⢀⣼⠀⠀⢀⡀⠀⣀⣠⡀
⠀⣿⡀⠀⠀⠀⣴⠋⠉⢷⠀⠀⠀⢸⠉⠉⠁⠀⠙⠁⢸⡀⠀⠀⣾⣃⠀⢀⡼⣇⠀⠀⣾⢧⣠⠟⠈⠁⠀⠙⠋⢹⠀⠀⢸⡃⢰⠋⡀⢹⠄
⠀⠘⢷⣀⠀⠀⢧⠀⢑⡀⠳⡀⠀⠸⠀⠀⠀⣶⡄⠀⠀⠻⣦⠜⠁⠙⠳⠋⠀⠙⠒⠚⠁⠀⠀⠀⠀⢠⡄⠀⠀⠸⣄⠀⡸⠓⠋⠰⢆⡼⢰
⠀⠀⠀⠙⢻⡆⠈⠳⢤⢥⡄⢿⠀⡆⠀⠀⠀⠘⢿⣦⣄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣤⣴⡿⠁⠀⠀⠀⠙⠛⠁⣠⠞⠛⠛⣵⠋
⠀⠀⠀⠀⠘⣧⣀⣠⣤⠀⠹⣎⠛⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⡞⣡⡶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣤⣀⠀⠀⠀⠀⠀⠛⠠⣤⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠀⠉⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠊⠉⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠓⠠⠶⠦⠴⠶⠶⠒⠛⠋⠁⠀⠀⠀⠀⣠⡀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠚⠀⠀⠲⠂⣀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⢠⣀⠀⠀⢀⠰⠆⠉⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⠎⠀⠚⠲⠀⠀⢀⠶⠄⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡀⠀⢘⡓⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠛⠀⠀⠀⢻⡁⠀⣸⣄⣀⣀⣀⣀⣠⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠶⠄⠀⠀⠠⠴⠆⠉⠙⠢⠐⠋⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠉⠀⠐⠒⠒⣠⣤⣤⠴⠶⠶⠶⠶⠤⢴⣶⣖⠈⠉⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣿⠂⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⡿⠛⠋⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠙⠓⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣈⣀⣀⣀⠀⠀⣿⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠙⠋⠀⠀⠈⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡖⠁⠀⢠⢤⠀⠀⠀⠀⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢚⠀⠀⡜⢀⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣭⠃⠀⠙⠋⢁⠀⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠸⣦⣀⡼⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⡀⠀⡉⠑⠒⡄⢀⠀⠀⠀⡰⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢒⠏⠻⠤⠕⠀⢾⣿⣿⣿⢖⠝⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣴⣶⣤⣴⣿⣿⣿⡷⡍⣴⡁⠀⠀⠀⣀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣯⢟⠞⠱⡄⠈⠓⣖⠉⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡻⢿⣿⣿⣿⡿⡣⢯⠶⠚⠙⠦⣤⡼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣹⠊⠀⢀⢞⠞⡔⠛⠿⣬⡉⠁⠀⡶⠒⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠰⣇⢀⡰⡻⠃⠀⢸⠀⢀⣨⣿⡆⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠙⢦⠌⡐⣟⠀⠀⢠⠃⠀⢻⣿⡿⠙⠋
⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⣢⠾⠁⠀⡰⠁⠠⣾⣿⡟
⠀⠀⠀⠀⢠⡏⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠜⠁⠀⢀⠜⠀⠀⣤⣾⣿
⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡡⠋⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠀⠀⢀⠤⠒⠤⣾⠝⠀⠀⠀⠐⠛⠛⢻⠉⠙⣯⠉⠙⣇
⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⢀⠔⠡⢄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⡀⠀⡇⢇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠄⠀⠑⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢣⢀⡇⠘⡄⢸
⠀⠀⠀⠀⠀⠠⠴⠚⣉⠄⢀⠅⠀⡠⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⢣⢸
⠀⠀⠀⠀⠀⠀⠠⠭⠒⠭⡀⢄⣪⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠈⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠛⠉⠉⠉⠉⠉⠙⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠙⠉⠉⠑⢦⡀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠂⠁⠄⠳⢤⣤⣐⢷⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠉⠀⠀⢰⡄⠀⠀⢀⠋⣸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠀⠀⠀⠀⠀⠀⢑⡎⢰⢧⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡄⠄⠀⡏⠈⠀⠀⢠⡀⣱⠀⠈⠲⢤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠉⠉⠙⠒⠦⣄⠀⡳⣄⡀⠀⠀⢄⣠⡜⠀⠐⠈⠊⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⢀⡴⠁⠀⠠⡀⢀⠄⠀⢢⡀⠥⠐⠒⠒⠢⣄⠀⠈⠁⠁⢀⠹⠛⠛⡄⠣⠇
⠀⠀⠁⠀⠀⠀⠀⠙⢧⡀⠀⡘⠀⠀⠀⠀⠀⠀⡜⠀⠀⠁⠀⠀⠀⢣⠀⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡏⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡀⠀⢀⠀⢣⡀⠈
⠀⡄⠀⠈⠂⠂⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠈⡾⣄⠘⠃⠀⡄
⠀⠈⢢⡀⠀⠑⡀⠀⠀⠀⠠⡶⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠃⠀⠘⠌⠢⢄⣀⠇
⠀⠀⠀⠙⢄⠀⠈⠀⠰⠊⠈⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢀⣀⣠⠄
⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠜⠋⠁⠀⠈⠁⠐⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⡀⢀⣀⣀⣀⡀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈⠉⡟⠀⠀⠀⠉⠙⣇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢰⠃⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠠⠤⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣤⠇

⠀⠀⠀⠀⠀⠀⢼⠢⣄⠀⠰⡖⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠉⠒⠼⠆⠀⠀⠉⠒⠢⠤⠤⠤⠤⠶⠒⠒⠚⠛⠋⠉⠉⢳⡀⣀⣀
⠀⠀⠀⢠⣄⣀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢅⠠⠼
⠀⠀⠀⠈⢧⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠈⣛⠶⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⡴⠂⠀⠀⢀⣀⠤⡴⠂⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⡳⢛⡫⠿⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠸⣇⢶⠀⠀⢀⡤⠞⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠖⠛⠓⢦⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠈⠈⠙⠚⢁⣤⣒⣲⠆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠏⠀⠀⠀⠀⣳⣀⠤⠤⣌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠛⠋⠉⠙⠿⡽⠉⠒⣤⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠣⡀⠀⠀⣀⢧⡀⠀⠉⢀⠇⠀⠀⠀⠀⠀⠀⠀⡠⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣨⠍⡉⠁⠀⠙⠣⠉⠁⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠣⠤⠆⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢠⠀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣂⠔⠋⠀⠀⠀⠀⢸⣉⡹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⣀⣀⣠⠤⠖⠋⡙⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣀⣀⡀⠀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⡏⡟⠉⠈⠉⠉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⡟⠗⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⡿⠗⢾⣦⣀⣀⡀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⡤⠤⢤⣶⣶⣶⣶⡶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⡼⠓⣛⣿⣯⡙⡍⣉⠉⡍⢍⠙⣓⠺⣝⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⣸⠿⢭⣤⣈⣏⣀⣉⣔⣲⣌⣭⡾⠤⢗⣫⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣯⣍⣙⡒⠒⠲⠶⠶⠶⠶⠖⠒⠒⣋⣉⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠏⣇⠙⢮⡉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⡟⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞⠀⢻⠀⠀⠉⠲⣄⣀⠀⠀⠀⠀⠀⢀⣠⠼⠗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢾⡁⠀⠘⡇⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠺⠿⡶⠚⠁
⠀⠀⠀⠀⠀⠀⡄⠀⢀⣤⣤⣀⠀⢠⣞⣉⣙⠺⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣀⠀⢙⡶⣫⠶⠦⣌⣳⢚⣹⢹⡟⣿⣷⠦⣤⠼⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠂⠀⠀⣯⣍⡭⠞⡥⠖⠶⢤⡼⢋⡡⢫⣷⣿⣅⡠⢬⡛⣩⠴⢭⣳⡀⠀⠀⠀⠀⣀⡼⢣⡄⢀
⠀⠀⠀⠀⠙⢦⡤⠞⣱⠚⣹⡏⣰⢋⡶⢡⠋⡏⣽⡖⠲⠭⠗⣲⡶⠏⢝⢟⢛⡛⣫⠝⡴⡢⠢⠀⠑⠂
⠀⠀⠀⠀⠀⢀⣝⡻⢥⣞⡴⢧⣹⣼⣿⣜⣦⡳⢝⠻⣏⠙⣻⡥⣀⠀⠀⠐⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠐⠂⠁⠀⠒⠆⠉⠙⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⡾⠿⢿⣿⣿⢿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠫⣝⣿⡭⠶⣶⡿⠟⠻⠿⠿⢶⣭⣍⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢟⣡⣔⡲⠛⠉⠓⠂⠀⠀⠐⠦⡀⠀⠶⢶⣬⣉⠁⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣴⣏⣈⣥⣦⣤⣤⡤⣴⡶⣮⣿⣛⢶⣦⡀⣤⡀⠈⠁⠠⡄⡙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⣷⣾⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡶⢀⠀⠈⢳⡜⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠞⢹⣿⡅⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠦⠈⠛⠛⠛⣿⣿⣿⣄⠀⣿⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣷⠹⠀⢼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢰⣿⣿⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠸⣿⣿⣷⠀⣎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣾⣿⡟⠠⠚⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⡄⠀⣧⢹⣿⣿⢻⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⠃⠀⠒⡉⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠁⠀⣼⣷⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠻⡿⠀⠀⠁⡀⣀⠐⣀⣽⠀⠀⠀⢰⡋⡤⠐⠠⠐⢄⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⡐⠀⠀⠉⠐⠺⠿⠟⠋⠁⠀⠀⠀⠀⠳⢶⣿⡆⣀⣠⡄⠀⠀⠀⢻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠰⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡍⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠞⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢴⠟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⣝⠳⠖⣿⠇⠀⠀⠀⠀⢀⢺⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⣷⣄⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣰⣭⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠐⢀⡙⢷⣤⣀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢋⠀⠀⠀⠀⠈⢻⣄⠈⠙⠳⢶⣶⣶⣾⢿⡿⠋⡐⣹⣿⣯⣻⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⣆⡀⠀⠀⠀⠀⠀⠙⠳⠄⠀⠀⠀⠀⠀⠉⠀⠀⡿⢁⣠⣿⣟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡿⠿⠀⠀⠉⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⠛⠛⡿⢛⠁⢀⣿⡿⠹⡈⢿⣿⢻⠻⢿⣿
⣴⠾⠛⠉⠉⠉⠉⠙⠛⠛⠻⠶⠆⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡶⠚⠛⠉⠀⠀⠀⠼⠋⠀⢃⣌⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣭⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠕⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠋⠉⢀⣬⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠤⠄⠀⠀⠀⠀⠉⠉⠀⠀⢀⣠⠴⠚⠋⠁⢀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⠚⠉⠀⠀⠀⣠⣶⣿⣿⣿
⣦⣴⣦⣦⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣯⣭⣭⡉⠉⠉⢉⣭⣭⣭⠛⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣤⣤⣽⣷⣶⣶⣾⣷⣦⣤⣄⡈⡇
⠀⠀⠀⠀⢀⠤⠤⡀⢀⠀⣄⠠⠄⠀⠀⢸⣿⣿⠞⣡⡀⢹⣿⣿⣿⠁⣠⡈⣯⣿⣿⠀⠀⢠⠄⣀⠀⠄⢀⠤⠤⢄
⠐⠒⠒⢲⠃⠀⠀⠀⠀⠀⠘⠒⠚⠒⠒⠚⠛⠛⠒⠛⠓⠚⠛⠛⠛⠒⠛⠓⠛⠛⠛⠒⠒⠒⠒⡇⠀⠀⠈⠀⢀⠈⠖⠒⠒
⠀⠀⠀⠘⣆⢗⠀⣼⢠⣄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⡄⣠⠀⡸⣤⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡄⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠎⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⢆⠀⠀⠀⠀⠏⠑⡄⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡂⠀⠇⠀⠀⡘⠀⢠⠁⠀⠀⠀⠀⢀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠒⠂⠐⠂⠠⢎⢀⡈⠀⠀⣰⠷⠀⠊⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡴⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠒⠀⠉⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⡄⠀⢌⠢⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠡⠌⠦⢕⠚⠂⠀⠀⠀⡌⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠈⠢⠐⠀⠀⠀⣀⡼⠀⠀⠀⣀⠠⠄⠚⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⣨⢃⡹⠊⠁⠀⠀⣀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢘⡠⢶⠟⢹⠀⠀⠄⠒⠈⠁⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡖⠀⠀⠀⠀⠀⣸⠂⠁⠄⢜⠁⠀⠀⠀⣀⣀⣀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢡⡶⢀⠀⠤⠞⢙⣈⣤⣠⢭⠉⠁⠒⠤⢠⢡⠀⠀⠤⠠⡔⠠⠀⡠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡼
⠈⢆⠀⠀⠀⠀⠀⣐⡏⡀⠘⠀⠀⠀⠀⣈⡀⠀⠀⠀⢀⠟⠄⢊⣀⠐⡈⠑⣀⠌⠀⢠⠂⢄⠀⢀⠂⠀⠀⠀⠀⠀⡇
⠀⠘⣆⠀⠀⠀⡰⢹⡠⠉⠙⠂⠨⣉⢡⠀⠈⠦⠒⡖⠁⠀⠀⠀⢠⠀⠀⠉⠓⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢄⠀⠀⠄⠃⢇⠀⠀⠀⠀⠀⡄⢰⡄⠳⠈⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠇
⠀⠀⠀⠀⠒⠰⠀⠐⣚⣧⣐⣀⣀⣀⣈⠁⠈⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⢤⣴⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠰⡇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣿⣿⣿⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⣠⣤⣧⣤⣾⡀⢀⣧⣤⣤⣤⣄⡀⠀⠈⠙⢷⣌⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣠⠖⠚⢯⣁⢀⣀⣀⡀⠉⠉⠀⢀⣀⣠⠬⣝⠳⣦⡀⠀⠛⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡖⠁⠀⢀⡸⠋⠉⠉⠉⠉⢉⣻⠟⠉⠉⠁⠀⠈⣇⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⣰⡶⠚⠛⠛⠛⢣⣀⣠⢿⣤⣀⣀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠉⢹⠄⠈⢷⣄⣀⣀⣀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣻⣿⡶⠶⠤⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠆⠘⢧⣀⣀⣤⠀⠀⠀⣠⣼⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⠴⢧⣀⡀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠲⠤⠖⠁⠈⠷⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠾⡇⠀⠀⠹⡛⢻⡁⢀⡿⠻⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⠆⢨⠿⡟⢿⠄⠀⢀⣧⣄⣉⣉⣀⣀⣀⣀⡤⠤⠔⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠃⢪⡀⠀⢸⠋⢻⠛⡏⠉⡿⠛⠃⠀⠀⠀⠀⠠⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡍⠛⢿⡶⠾⠿⣶⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⣀⣤⡶⠛⠛⠛⠓⠒⠶⠶⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣁⠀⠈⢳⡒⣦⣀⣀⣀⠳⠤⠴⢶⡀⠀⢀⣖⣋⣀⣤⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠉⠁⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣈⣉⠶⡄⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠥⠤⢑⡄⠀⢀⡐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢉⡴⠓⣈⡉⢁⡒⠁⠀⠠⠠⠑⣹⣇⡠⠒⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡔⣫⠖⠉⠀⣀⡀⠀⣀⠀⠀⠀⠀⡨⣿⣧⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⣼⠁⠀⠀⠉⠈⠁⠈⠁⠀⠀⠀⣞⢀⡿⡧⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡈⢸⠃⠀⠀⠖⠲⠂⠀⠐⠒⠖⠀⠀⠈⠉⠀⡇⠃⢋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⢸⠈⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠠⢀⡇⠸⠀⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠸⠀⠇⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⠋⠉⠱⡀⠀⠄⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠤⠤⣤⡔⠈⣠⠶⠁⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣴⠂⠀⠈⣇⠀⠉⠐⠀⠋⠁⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣠⠏⠓⢦⣠⣀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠧⢠⠞⠁⢀⠀⠂⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡠⠆⠁⣠⡔⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣃⠈⠀⡠⠚⠄⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⢘⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠁⠀⢀⡌⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠛⣅⡠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠠⠃⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠰⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠱⡀⣰⡁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠑⣄⠈⠋⠁⠈⠋⠁⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠶⠉⠉
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠿⡄⠀⠀⠀⠀⢀⠤⢄⡀⠤⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠒⠒⠐⠢⡀⠀⡔⠁⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠟⠈⠑⠖⠉⠃⠀⠀⠀⠀⠈⠉⠳⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠈⣈⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠉⢀⣀⣀⡤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⢒⢒⡟⠉⠉⠩⣩⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣆⡠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠥⠟⢶⣉⣓⠻⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠭⣑⠢⣄⠀⠔⠊⠀⠐⠊⢭⡈⠳⡄⠈⠳⣄
⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡑⣄⠀⠀⠀⠀⠀⠈⠑⢄⠀⠳⡈⢆
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢆⠀⠀⠀⠀⠀⠀⠀⠱⡄⡃⢸
⢰⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣄⠀⠀⠀⠀⠀⠀⠀⣠⢃⠏
⢘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠳⣄⠀⠀⠐⠒⠋⡜⢣⠋
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢤⣈⡳⡄⣆⡆⣼⡵⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢉⠟⠁
⠈⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⠒
⠀⢣⠀⠀⠀⠀⡓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⡆⠀⠀⢸⣇⠀⠈⠉⠒⠒⠶⠶⠶⠖⠒⠚⠓⣤⣀⣀⠳⡀⠀⠀⡄⠀⠀⠀⠀⠀⡸
⠀⠀⠹⡀⢠⢸⣽⠖⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⣹⣿⡉⢱⡇⠀⠀⡀⠀⢠⠇
⠀⠀⠀⠱⡈⡆⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠁⣾⠁⢀⡼⠁⢠⠎
⠀⠀⠀⠀⠙⣧⢹⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡞⡜⣠⣾⠁⡴⠃
⠀⠀⠀⠀⠀⠘⢆⢫⠙⠲⠤⣄⣀⣀⣈⣁⣀⣀⡠⠤⠴⠒⠋⡿⣱⠕⢱⡧⠊
⠀⠀⠀⠀⠀⠀⠈⢦⢣⠀⠀⠀⠀⠀⢀⡝⠈⠂⣠⠠⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣰⣡⠈⠉⠉⠀⢤⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡹⠀⠀⠀⠀⠈⢆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢳⣠⣤⠀⠀⠠⠋⡀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⢠⠎⠀⠘⡆⢀⡾⠁⠈⠛⢦⡀⠀⡠⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡈⠈⠀⠀⠁⢄⠀⠀⢻⢸⡇⠀⠀⠀⠀⠱⡎⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⠐⠀⠀⠀⠀⢳⡀⠈⣎⠇⠀⠀⠀⠀⠀⢡⢀⡦⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠘⠙⠢⣷⡀⠈⠿⡀⠂⠀⠀⠀⢨⡄⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢻⠆⠀⠀⠙⣄⠀⠀⠀⡼⠲⠤⢴⣞⣁⣀⣀⣀⣀⣀⣀⣤⠤⠀⠀⠀⠀⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠠⣤⣍⡑⢯⠃⠀⠀⠀⠀⠙⠶⠞⠁⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⢤⣤⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⣠⠾⠟⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⣤⣀⣀⡀⣀⣀⡠⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡵⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⢋⡴⠋⠀⠀⠀⠀⠀⣠⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠉⢠⣎⡀⠀⠀⠀⠀⢀⠜⠁⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣠⣾⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢀⠏⠀⠉⠉⠁⢒⠖⡁⠃⠀⢀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠉⠹⠚⡉⠧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⢤⠁⢀⠐⢂⠀⣀⠄⢹⠀⠀⠀⠀⠀⠀⡀⠠⠔⠒⡄⠈⠊⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⡹⣄⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⣀⣠⠤⣹⡧⠞⠁⣄⡿⠒⠚⠛⣲⣄⢉⠄⡤⠐
⠀⡀⠀⠀⠀⠀⡼⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠞⠁⣠⠞⠁⠘⢾⣼⠀⠀⡠⠊
⢘⠁⠀⡔⠀⢸⠁⠀⠀⠀⠈⠓⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⠀⠀⠀⣀⣸⠔⠊
⠀⠀⠀⠃⠀⢸⡀⠀⠀⠀⠀⠀⠀⠉⠷⠶⠖⠚⠛⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⢰⠃⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣲⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠙⠛⠀⠛⠛⠀⠒⠒⠒⠒⣤⡀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣀⣤⠐⠪⣍⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣧⣀⣀⣀⡤⠴⠒⠋⠁⠀⠀⢀⣯⣹⡏⢀⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⣍⢓⡄⠀⠀⠀⠀⠀⠸⣇⣸⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠐⠂⠀⠀⠀⠂⠀⠂⠂⠀⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠘⠀⠃⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣀⡀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣹⣿⣿⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣶⠈⠙⠛⠛⣿⡆⠀⠀⠀⢠⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⣠⣿⠿⢾⣦⠀⠘⠷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣶⣾⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠉⠀⠀⠀⡀⠈⠉⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠸⣷⣀⣾⣏⣀⣀⣀⣈⣻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⣤⣤⣴⣶⣿⡿⠿⣿⣟⠿⠟⣿⡟⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⢿⣍⣛⣁⣿⠃⠀⠈⠻⠿⠿⢟⡁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⢈⣝⣛⡛⠁⠀⠀⣀⣠⣴⣾⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡈⢿⣿⡛⢿⡟⠛⣟⠉⢻⣰⣿⣣⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡙⢿⣾⣧⣤⣿⣴⣿⣿⣵⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣾⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣋⡉⠉⠉⠉⣉⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣯⡟⢻⣶⠏⠀⠻⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠁⠀⣀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠖⢻⣧⠘⢦⠀⠀⢀⣴⣿⠙⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⣿⠐⢧⡙⣦⣤⣤⡴⠟⣹⠏⠀⣷⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⠀⣿⠀⠀⣧⣘⣿⠤⣠⡾⠋⢠⡴⠿⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⣼⠧⠾⠦⡄⠈⠉⢻⡷⠯⠶⠾⠟⠀⠀⢀⣤⠄⠀⠀⢹⡇
⠀⠀⠀⠀⢀⡾⠋⠀⢠⡞⠁⠀⠐⢚⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠘⣧
⠀⠀⠀⢠⡟⠀⢠⣴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣿⡤⢶⡟⠁⠀⠀⣠⣴⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⢀⡿⠁⠀⢸⣷⣀⣴⣿⣿⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠆
⠀⠘⡇⠀⠀⠸⣿⡿⢹⠏⢹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢣⡀⠀⠀⣀⠔⠁⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠸⣿⠀⢿⣄⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡟⠛⠂⣠⣤⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⠾⠗⠚⠉⠁⠀⠈⣿⡄⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⠀⠀⠀⠀⢀⣤⣴⣿⡇⣀⣤⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣀⣴⠟⠉⠉⠛⠿⠿⠿⠟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣴⣿⠟⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠉⣿⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⠀⠀⣼⠃⠹⣇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⠀⢹⡆⢰⡏⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⠀⢸⣇⡿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⠸⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣾⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣯⠞⣹⣿⡇⠀⠀⠀⠀⠀⠀⡴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢰⡏⣰⣿⣿⣷⣶⣶⣶⣶⣶⡞⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠇⠀⠀⠀⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⡿⠛⢛⣿⣆⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⡶⠟⢻⣿⣿⣿⣟⣛⣛⠛⠛⣿⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠇⠀⢺⣿⣿⡀⠀⠀⠀⠀⠀⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠶⣶⣾⣿⣿⡷⣄⠀⠀⠀⠀⢳⣶⣶⣶⣤⣤⣤⣤⣤⣉⣉⣉⣿⣧⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣟⣮⠳⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⣿⡇⢹⡟⣿⢹⣿⣿⠸⣧⢹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣼⣿⡇⢸⡇⣿⡘⣿⣿⡇⣿⣾⣿⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇⢸⣧⢸⡇⠿⢹⡇⣿⢻⣿⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⡿⢿⣿
⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠇⢸⡿⢸⣿⢰⠺⡇⣿⢸⡏⠀⢹⣿⣿⡟⠋⠉⠙⠛⠛⠛⠛⠿⣿⣿⣆⠾⠛⠁⠀⠈⠙
⠻⠿⠿⠿⠿⠿⠿⠛⠻⠿⢿⡟⣠⣼⠇⣸⣿⣾⡇⣇⣻⡘⣧⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⣥⣿⣾⡿⠿⠿⢿⣿⣿⣷⣿⡧
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢁⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠷⠀⠓⠴⠃⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⠔⠋⢁⠤⡄⢠⠈⠒⢄⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠏⠀⠀⡎⠀⡇⠈⢆⣀⣀⡑⣄⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠋⣄⠀⠀⠓⠢⡱⣄⠀⠁⠀⡹⠈⡆⠀⠀⠀⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⠀⠈⠓⠤⠤⠚⠁⠀⠉⠒⠊⠁⠀⠸⡀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈⢇⠀⠸⡐⠻⠿⠟⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⡇⠀⠀⣀⠀⣀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠎⠑⡼⠀⢀⡇⠸⣿⢀⠠⠀⠀⠀⠸⣿⠀⠀⠀⢀⠇⠀⡸⣡⠄⠀⢣⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠀⠀⡜⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠴⣊⡠⠔⠁⠻⠂⡠⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠠⡀⠀⠀⠀⡏⢿⡧⠀⠀⡀⠀⠀⠀⠀⠠⣿⠏⠀⠀⠀⢀⠏⠙⠉⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⠜⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⢠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣥⠀⠀⠀⠀⢸⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣀⡙⠤⠤⠤⠤⢛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠸⣅⣀⣠⣄⠀⡨⠁⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠉⣀⡀⠉⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀⡀⠀⠀⠀⠰⠆⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣾⣿⣿⣶⡦⡔⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣏⡉⢿⣿⣿⡜⡞⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⣿⣿⣿⣿⣿⣿⣿⣇⢠⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⣿⣿⣿⣿⣿⣿⡏⡈⣘⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⢾⢿⣿⣿⣿⣿⣿⣿⡷⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠻⠿⠿⡇⠓⠦⠀⠀⠀⢐⣀⣀⣶⣤⠤⠔⣠⣴⠟⠂
⠀⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠃⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⠈⠉⠉⠄
⠀⠈⠙⠛⠛⠛⠿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠷⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠖⢻⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠀⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⠀⡀⠀⠀⠀⢘⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⡶⠶⣺⣿⡿⡿⠻⣻⡿⢛⣻⣿⠿⢟⡿⣿⠿⢟⣿⢿⣿⢿⣿⡿⣿⠿⣻⢿⡿⣿⡿⡿⠟⡿⢛⡿⣿⣿⠁⠀⠠⠀⠀⠀⠀⢠⣿⢸
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢷⡽⠋⠨⠚⠁⠈⣠⢞⢕⡴⡫⠟⢁⠔⡩⠚⣡⠖⠋⠁⣠⠞⠉⠩⠚⠁⢀⠕⠋⠀⠁⠈⠀⣠⢞⠉⠜⣻⡇⡀⠀⠀⠀⠀⠀⠀⢾⣿⢸
⠀⠀⠀⠀⠀⢀⡾⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠊⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣨⠖⠁⢠⡾⣻⠁⠀⠀⠀⠀⠀⠀⠐⣸⡿⢸
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣼
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢡⣿⡀⠀⠀⠀⢀⣠⣾⠟⣹⡇⣿
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⣻⡷⢀⡴⠾⠿⢋⣩⣴⣿⡇⣿
⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⡿⠻⠟⠟⠟⠉⠉⢿⡇⣿
⣿⣷⢄⣤⣀⣤⣄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⡿⣻⡟⣹⣿⣿⣿⣿⣵⣿⣫⢞⣿⣿⣿⣿⣿⣿⠞⢡⣾⢏⢞⡾⣿⢯⠟⡠⡃⢁⡼⠃⠀⠂⠀⡰⣠⢦⠆⡴⢀⠄⡠⠂⡀⡠⠂⡴⣰⡶⢣⣾⡿⠁⣰⢿⡇⣿
⣿⣿⢋⣞⡽⢃⣿⣿⡿⡟⡵⣫⣻⡿⡿⣿⡿⠋⠁⡴⡻⡣⣣⢟⡼⡳⠃⡴⠁⣠⠞⠀⠀⠄⢀⣜⡵⠃⢀⠞⢡⠊⠐⢀⠎⠄⢁⡞⢝⠊⠀⣱⠋⠀⣼⣯⣿⡇⣿
⣿⣣⢯⡞⣠⢏⣼⣟⢌⣞⣝⡽⣓⢟⣾⠝⠁⢠⣞⣟⣕⡽⣣⢟⠜⢡⠎⠀⡴⠋⠠⠀⡀⣠⢮⠞⠀⣠⠋⡰⠁⠀⡴⢣⠂⣠⠋⡠⢁⠌⡼⢱⠀⣼⣶⡿⠍⡇⣿
⣿⣧⠋⢜⣵⡿⣿⣿⣫⣪⠎⣠⠎⡪⠋⠀⣰⡿⣫⠞⢪⡞⡱⡃⡴⠃⢀⠞⢁⠔⢁⢜⡴⣳⠋⢀⣔⠁⠈⠀⢀⠞⡱⢁⡜⢁⡼⣡⢣⣞⠾⢁⢮⡿⢃⢎⠜⡇⣿
⠿⠷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⣜⣙⣃⣚⣛⣑⣃⣐⣋⣚⣚⣚⣀⣀⣁⣀⣋⣀⣉⣚⣙⣁⣀⣈⣡⣤⣾⣥⣭⣈⣰⣋⣀⣋⣀⣁⣁⣀⣁⣃⣘⣡⣡⣬⣬⡇⣿
⠀⠀⣿⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⣶⣻⠀⢴⣿⣿⣿⣿⡿⡿⠿⢟⡿⠿⠿⠿⠿⠿⠿⠿⣿⠿⡿⠿⡿⠿⢿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠿⡿⣿⣿⡿⠿⠿⠿⣧⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⢤⡤⣔⢲⣆⣲⠰⠯⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠸⢹
⠀⠀⠀⢀⠆⡃⡞⠙⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠡⠤⠠⠤⠐⠒⠒⠒⠢⠤⡀
⠀⠀⠀⠀⠀⢀⠃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⡸⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⢀⠠⠤⠐⠒⠂⠀⠀⠒⠒⠤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠐⠒⢲
⠀⠀⠀⠀⢰⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠄⢇⡀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⢸⠀⠠⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡖⠉⠉⠀⠘⠀⠁⣢⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⠠⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡁⢘⠀⠀⠀⠀⢈⠀⠈⠉⠁⠈⠉⠉⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⡀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠤⠀⡇⢰⠀⠀⠀⠀⠸⡀⢰⠁⠐⠒⠈⠂⢸⠀⠀⢀⠀⠀⠀⠈⠄⠈⡁⢀⠀⠠⠀⣊⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣧⠜⠀⠀⠀⢀⡔⠃⠈⠢⠨⠬⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⢌⠀⠠⠀⠀⠀⠊⠀⡀⠈
⠀⠀⠀⠀⠈⡗⠢⣤⠦⠒⠙⠀⠀⠀⢠⢊⡔⠊⠈⠉⠉⠁⠐⠒⢄⠀⠀⠠⠀⠀⠀⠀⠀⠀⡐⠉⠁⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠈⡄⠀⠀⠀⣀⣀⡀⠀⢘⢓⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⢬⢠⡈⡰⠂⠤⠀⡤⠠⠔⡨⡤⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⡜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠌⠡⠁⠀⢀⢂⣤⠀⠀⠀⠀⠀⠈⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠐⠒⠀⠉⠀⠀⣀⠠⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⠤⠔⠒⠋⢉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠋⠉⠀⠀⠀⣩⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠰⠟⠃⣠⠀⠀⠀⠁⣶⠄⠈⢳⣄⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠃⠀⠀⠀⢾⡿⢻⡆⠀⠀⣿⠀⠀⠀⠀⠹⣯⢸⡿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⢿⡀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡿⠀⠀⠀⠀⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⠀⠀⠀⠀⣀⣀⣤⡶⠿⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠛⠛⠛⢛⣋⣉⡁⠀⠀⣿⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⡄⠀⠺⣿⣿⣿⣃⡀⢀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣉⣿⣷⣾⢿⣥⣼⣿⠿⣿⠿⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⢰⡏⠀⠀⠀⢻⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⠿⠓⠀⠐⠿⠟⠉⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣆⡤⠔⠒⢈⣽⣋⡤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠡⠞⠁⠀⠀⠈⠉⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⠀⠀⠀⠀⠀⠀⠀⢛⣉⡠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠧⣀⢀⣴⠒⠒⠲⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠀⠀⢰⠃⠀⠀⠠⠂⡜⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣰⡁⠀⣇⠀⠀⡠⠔⠓⠒⢤⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⠀⡎⠀⠀⠀⢀⣾⣄⣀⡠⠓⠒⠢⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⣉⠛⠟⠀⠑⠦⠘⠻⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠈⣀⡄⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⢀⠈⡧⡾⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣥⠀⠀⠀⡏⠉⠁⠛⠓⠒⠒⠒⠀⠀⠒⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠸⡆⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⡠⠤⡀⠰⠒⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢠⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⢇⠀⠑⠸⡀⢰⠀⡰⢁⣍⡓⠦⣀⣤⣀⠀⡵⠠⣙⠇⠀⢀⠜
⠀⠀⠀⢰⠋⠛⢶⣄⡐⠧⢸⡰⢁⠂⠀⠉⠓⡞⠀⠈⠉⠳⡀⠙⢦⣀⣈⣣⣄⣀⠀⠀⠀⢀
⠀⠀⠀⠈⠑⠢⠤⠬⠥⠤⢄⡀⠚⡤⠀⠀⠘⠳⣤⡀⠀⢀⠳⠔⢣⡙⢹⢻⠁⠀⣷⡨⠭⠄⡁⠀⠀⠀⡴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⣅⠀⠀⣠⠛⠉⢧⣠⠾⠠⠄⠀⣙⠣⠤⡆⡰⠁⠀⠀⠀⠀⠀⠀⡰⠁⡰⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⢄⡨⢶⡊⠫⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡠⠄⠲⠊⣠⠴⠓⠉⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣽⠖⠦⣬⣉⣐⣀⠀⠀⠤⠄⠀⠐⢂⣹⠀⣀⡼⢃⠴⢤⡐⠶⠶⠦⠖⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⠀⠶⠀⢈⡁⢽⡏⢉⡉⠻⠟⢉⣉⡟⠸⣀⡠⠖⠀⠀⠈⠓⠤⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⠀⠛⠁⣤⡄⠘⠃⣠⡄⠘⠋⢠⣤⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⢆⣉⡁⣼⡟⠛⣟⣻⣿⣉⠩⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣒⠒⢒⣒⡀⣴⠛⡿⠭⠭⠭⢥⣏⣿⠠⠤⠔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠉⠉⠉⠉⠉⢹⣂⠊⠉⠉⠩⢤⠖⠃
⠀⠀⣀⣀⣀⣀⣠⠄⠴⡒⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⡒⠶⠒⣲⣄⣀⣀⡀
⠴⡭⠥⠀⠀⠈⠭⠭⠙⠴⠤⣤⠤⠤⣤⣤⣀⣀⠀⡀⠀⠀⢀⣠⣤⠶⡛⣩⣩⣁⣀⣀⣀⣀⡽⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⠄
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠆⠀⠀⠀⠀⠠⠐⠋⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣀⠀⠀⢀⡄⡀⠀⠠⠀⢠
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡤⠚⠁⠚⠀⠀⠀⠈⠁⠀⠀⣬⠃
⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠈⠃⠀⠗⠁⠀⠋⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⡀⠀⠛⠢⠄⠀⠀⠀⠀⠴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣁⣡⠄⡀⠈⠢⠄⡀⠀⠀⠀⣀⠐⠁⠀⠀⠀⠒⠂⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠉⠁⠠⡄⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠖⠢⠤⠠⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⡐⠀⠀⠐⠀⠀⠀⠈⠀⠐⢂⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠤⠄⠒⠂⠈⠉⠀⠀⠈⠉⠉⠉⠉⠉⠑⠒⠀⠀⠄⠠⠀⠐⠂⠀⠀⠉⠉⠁⠀⠈⠉⠉⠀⠀⠀⠉⠒⠒⠒⠢⡀
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡏⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢼
⠀⠀⠀⠀⣷⡄⠐⠢⠤⠤⠤⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⠁⠀⠀⠀⢨
⠀⠀⠀⠀⢣⠑⠤⢀⣀⣀⣀⠀⣀⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠀⠀⣀⠤⠖⠂⠉⠉⢉⠏
⠀⠀⠀⠀⠘⢮⢢⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠐⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠒⠒⠂⠀⠒⠊⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠈⠙⢮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⡼⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡟⣸⣿⣿⣿⢿⣿⠃⠙⣧⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣏⣠⣾⢿⣾⣿⣿⣿⣷⣟⡀⠻⣿⣇⣈⣤⣴⣞⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿
⡿⠉⢢⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠘⢷⠀⠙⠛⠟⠛⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠀⠀⠀⠀⠀⠈⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣶⣌⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⠋⠉⠒⠀⠀⠀⠀⠀⢀⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠎⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⠛⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣇⡼⢻⣿⣿⣧⡀⢰⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠁⣤⡤⠄⠀⠠⠄⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠤⣦⣄⠀⠀⠈⠙⠛⠻⣿⡎⠿⣿⣿⣿⣿⣾⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠘⣷⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⢤⣤⣙⣿⢿⠷⠀⠀⠀⠀⠀⠟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠙⠋⠀⢠⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣼⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡴⢦⣤⣀⠀⠀⠀⠀⠀⠀⣩⡿⠀⠀⠀⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢣⣤⣬⣿⠟⠋⠘⠺⢿⣿⣿⠃⠀⠀⠀⠁⠀⠚⠙⠻⠟⢉⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⢾⣧⣀⣴⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠠⣄⡂⠈⠀⠀⠀⠀⣀⣠⢴⣶⣶⣷⣶⣿⣿⣿⣿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠐⠚⠓⠿⠿⠛⡿⠟⠛⠛⠋⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠙⠛⠿⡟⠋⠉⠉⠉⠉⠻⣿⣿⣿⣿⠋⠀⠈⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠞⠋⠛⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆⠀⠀⠀⠈⠏⢻⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣿⣿⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⣤⣤⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠢⠀⠘⣿⡄⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠚⠃⠀⠀⢀⣈⣽⠇⠀⠀⣠⠀⠀⢠⡟⠀⠀⠀⡇⢹⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠦⣴⣿⡇⠀⠀⠀⢁⣳⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣶⣾⣶⣿⣿⣿⡇⠀⠀⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠟⠉⠀⠀⠀⠀⣠⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡃⠂⠀⣀⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⢿⡄⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣥⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢹⣗⡀⢰⠆⠀⠀⠀⠀⢰⣿⣿⣿⣿⠂⠀⠀⠀⢀⢾⣠⣿⣿⣿⣿⣿⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣷⣬⣶⣴⣾⣿⣾⣷⠞⠀⠀⠀⢠⢀⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠿⣿⣿⣿⣯⣝⣻⡿⣧⠀⠀⢀⣀⣀⣾⣿⣿⣿⣿⠀⠀⣠⠾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⡿⠿⠿⠿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠆⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⡴⠊⠉⠲⡄⠀⠀⠸⠁⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣇⠀⠀⠀⠀⠀⠰⡇⠀⠀⢶⡿⠀⠀⠹⣄⣀⣹⠟⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⣠⠍⠁⠀⠀⠀⠀⠀⠙⢢⣴⡾⠗⠀⠀⠀⠉⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠶⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠷⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠀⠀⣀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣴⠄⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠙⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠠⠴⠾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠰⢶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⢶⣤⣄⣀⣀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⣀⣠⣾⣿⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠙⠛⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠉⠁⠀⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⣀⣠⣴⣿⠟⠁⠀⢀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠠⡏⢳⣤⠶⠾⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⣧⠀⠹⣷⣄⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⢀⣀⣤⡦⡤⠤⠿⠋⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢷⣄⠈⠛⢷⣤⣀⣀⣼⡷⠿⠶⠶⠶⠞⠋⢉⣿⣦⣀⣠⣤⡖⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠙⢷⣤⣀⠈⠉⠛⠁⠀⠀⢀⣀⣀⣠⡴⠟⠁⠈⠉⠁⡜⢀⡎⢀⡀⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠙⠛⠿⣆⠀⠀⠀⣀⠉⠻⣅⠀⠀⠀⠀⠀⠀⠷⢼⣇⢸⣇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡀⠈⢻⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢯⣁⢀⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡿⠿⣿⡿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠟⠛⣿⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡆⠀⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠀⠼⠿⠶⣶⣦⣤⣤⣄⣀⣴⡶⠿⠛⠛⠋⠉⠁⠀⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⢀⣀⣀⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠟⠛⠛⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣯⡿⠙⣿⠛⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⠀⠀⠀⠀⡇⢰⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠱⠗⡠⠊
⠀⠀⠀⠀⠀⠀⣰⠳⢦⣨⠇⠀⠀⠀⢳⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢀⡏⠀⠀⢧⣀⠀⠀⠀⠘⡆⠀⠀⣇⢀⡞⠉⠉⠙⠓⠒⠶⢦⣀⡴⠊⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠉⢦⠀⠀⢹⡀⠀⢹⡈⢳⠀⣠⢤⣤⣄⣀⡸⠊⠀⣀⡴⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⡀⠈⡆⠀⣀⠷⠚⠺⣇⡏⢠⡟⠾⣄⡴⠋⢀⣴⡚⢧⡀
⠀⠀⠀⠀⠀⢸⡀⠀⡎⠀⠀⠈⢂⠹⡋⠁⢀⠴⠒⠋⠀⠸⣧⡾⢋⣀⣜⡧⠤⠽⠔⠛⠲⡄
⠀⠀⠀⠀⠀⠀⢧⠀⢣⠀⠀⠀⠈⡄⢳⠀⡎⡤⠀⠦⠤⣀⠈⢱⠉⠀⢀⣀⣀⣀⣀⣀⠀⢻⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⡆⠈⢆⠀⠀⠀⡀⠀⢿⢰⠁⠀⠀⠀⢀⠀⢸⠀⡎⠁⠀⠀⠀⠀⠈⡆⢸⠋⠁⠀⠈⠙⠲⣄
⠀⠀⠀⠀⢀⡔⠒⠻⡀⠈⠳⠤⠴⠃⠀⠈⠇⠀⠀⠀⠀⢸⠀⡽⠀⢰⠀⠀⠀⠀⠀⠀⢣⠈⡆⠀⢀⣄⠀⢠⠘⡄
⠀⠀⠀⠀⠀⢳⡄⠸⠥⣀⣀⣀⣀⣀⣀⠀⠈⣆⠀⠀⠀⡇⢰⢦⡆⢸⠀⠀⠀⠀⠀⠀⠘⡄⣧⠜⡅⠈⡇⢸⠀⢹⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠠⠤⢀⣀⣀⣀⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⠔⠁⠀⡆⠀⢧
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⢢⠀⠀⣇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡟⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠳⡤⠇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠉

⠀⠀⠀⠀⠀⢀⠄⣤⣤⣤⣤⣤⣤⣤⡄⠀⣠⣄⠀⠀⣠⣄⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠠⡀
⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡧⣾⠉⠙⣧⣼⠉⠉⣷⢸⡯⣤⢤⡥⣤⢤⡤⣤⣼⡈⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡌⣦⣤⣤⣤⣤⣤⣤⠇⣙⠷⠾⡁⣘⠷⠾⡋⣸⡷⡾⡾⠷⡷⢾⡷⣾⠾⠇⡀⣠⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠇
⠀⠀⠀⠀⠀⢠⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠉⠀⠀⠉⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣠⠤⠐⠒⠒⠂⠤⣄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡾⠁⠀⢠⠔⠉⣀⡤⠶⠶⠶⠶⢤⣀⠉⠲⡄⠀⠈⢷⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡾⠀⠀⡰⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⢆⠀⠀⢷⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⢰⠃⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⡆⠀⠘⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⣧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠸⡀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⠇⠀⢀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣧⠀⠀⠣⡀⠙⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢀⠜⠀⠀⣼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⢧⡀⠀⠑⢄⡈⠙⠲⠤⣤⣤⠤⠖⠋⢁⡠⠊⠀⢀⡼⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠒⠢⠤⠄⠠⠤⠔⠒⠉⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⡀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⢤⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠚⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠈⠹⠍⠉⠉⠩⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠍⠉⠉⠩⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢌⡉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠀⠤⠄⠤⠤⠤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠉⠒⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢪⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠁⠀⡴⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⢀⠎⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠎⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⡆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⡯⠯⠛⣛⣒⣒⣒⣋⣉⣟⡛⠒⠒⣒⣓⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠁⠐⠒⠒⠀⠀⠤⠤⣷⠤⠭⠤⠤⠤⢼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠬⠍⢁⣀⡐⠒⠒⠒⠘⠛⢲⡶⢾⣋⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡑⠒⠤⠤⠤⠤⠄⠀⣀⣀⣀⣈⡇⢠⣄⠤⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠒⢲⢲⠒⠒⠒⠒⠒⠒⡇⠀⠙⣏⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⠸⡀⠀⠀⠀⠀⢰⠃⠀⠀⢥⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⡇⡇⠀⠀⠀⠀⠈⠓⠒⠋⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣉⠓⠠⠍⠘⠛⣟⠒⠟⠁⠀⠀⢀⣠⠴⠚⠉⣆⡀
⠀⠀⠀⠀⠀⠀⢀⡔⡫⠛⠉⣩⠭⠿⣖⡦⠤⢀⣀⠈⢹⠒⠒⠒⠉⠉⠁⠀⣀⣠⢤⡶⠯⠭⢬⡑⢄
⠀⠀⠀⠀⠀⠀⢸⡚⠦⠾⠥⠧⠤⠤⠬⠅⠂⠨⠶⠾⢭⡯⠽⠯⠽⠿⠭⠭⠤⣈⣉⣸⣓⣀⣀⣙⣼
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠑⠂⠉⠉⠉⠁⠐⠒⠒⠤⠤⠤⠭⠥⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠬⠷⠘⠙⠚⠉⠛⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡎⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠈⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛
⠀⠀⠀⠀⠀⢰⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢀⣐⣂⣀⡀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⣇⣤⠀⠀⢀⡀⠀⢸⢇⢖⡴⠋⠀⠋⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⣀⣈⣸
⠀⠀⠀⠀⠀⠁⠀⠀⣠⠌⢯⡑⠒⠈⠋⠀⠀⢀⣠⠤⠤⠄⠀⠀⠀⣧⠀⠀⡾⠠⠢⠙⡄
⠀⠀⠀⠀⠀⠀⢰⢸⡇⠉⢉⣀⡀⠀⠀⠀⠞⢉⡠⠔⢺⡍⠃⠀⠀⠸⡆⢀⢧⠐⣲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢳⡅⢼⣻⡉⢭⡀⠀⠀⠀⠉⠻⣿⣿⠇⠆⠀⠀⠀⠙⠦⠞⠸⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠛⠛⢁⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡠⠀⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠠⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠙⠛⠃⠀⠀⠀⢀⡴⠁⠀⠀⠀⣠⠎⡇⠀⠐⠒⠒⠋⠉⠉⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠂⠑⣄⠀⠠⢄⣀⡠⠔⠉⠀⠀⠀⣠⠞⠡⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⠑⢦⣀⡀⠒⠂⠀⣀⠴⠋⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡀
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡉⠲⠙⠀⠄⠠⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⢹⢦⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠢⠑⣄
⠀⠀⠀⠀⢠⠇⠀⡀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⡴⡡⠒⠀⠁⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⡜⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢩⡝⡂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⡗⠾⠶⠖⠋⠀⠀⠀⠀⠀⠀⠈⢆
⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡣⡀
⠀⡄⠀⠀⠀⠀⠀⠀⣀⣠⡭⠴⢂⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡽
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡈⡎
⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡈⢃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠠⠀⡰⠁
⠀⠀⠑⢄⠀⠀⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢱⡀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢤⡀⠀⠀⠀⠒⠒⣒⡒⠒⠪⠍⠀⠀⠀⠀⢧⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠀⠀⠀⢌⣂⢪⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡠⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠒⠒⠊⠉⠀⠀⠁⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⢂⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡞⡩⢙⡄⠀⠀⢀⣀⠤⠚⠡⠀⢀⡡⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠈⠉⠉⠒⠂⠤⠤⠄⢀⠳⢬⡞⠡⠔⠚⠉⠄⠀⢈⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⣀⠀⠂⠀⠀⠀⠀⠀⠀⡆⠀⢀⣀⠥⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠒⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠈⠉⠉⠉⠐⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠠⣞⡙⡿⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡄⠘⣇⢱⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢪⣿⣇⠀⠀⠀⠀⠀⢀⠞⣳⠀⠀⣸⠙⣆⢸⠯⡁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⢀⡞⢿⠀⠀⠀⢀⡠⠔⠁⢠⡟⣠⠶⠃⢠⣷⡞⢦⡁⠀⢰⣜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣾⡀⢸⠀⠈⣾⣡⠀⠀⠠⣴⡆⠀⠀⠀⣴⠖⠋⢰⢻⡄⢷⠀⡼⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠓⠬⠧⠀⠘⠊⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠚⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠆⠀⠀⠀⠀⠠⠀⠀⢀⡼⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠀⠀⠄⠀⠀⠐⠉⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠁⡀⠀⠀⠀⠀⠠⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢦⣀⠈⠲⠄⣀⠜⠁⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⠉⠂⣄⠀⢱⠊⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠙⠛⠊⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠚⢻⠒⢻⠒⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡾⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡇⠀⢸⠒⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠉⣳⣄⣼⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢾⠗⠈⡿⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠚⠉⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠖⣖⣩⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⣀⡀⢶⣤⣬⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣿⡏⠀⠀⠐⣸⣯⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠗⠀⣈⣴⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣫⣠⠾⢿⣿⣿⠗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠘⡿⢃⠐⣻⣿⣿⠁⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⣼⣯⣥⣳⠀⢀⣼⣿⣿⣿⡁⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⡀⠈⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣟⣤⣿⣿⣿⣿⣿⡷⠃⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢻⠇⠀⠀⠰⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣽⣿⣿⣿⣿⣿⢿⣿⡷⠟⠈⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠁⠀⠀⠐⠀⠻⠓⠀⠀⠀⠀⠀⠀⠀⢸⣿⣎⠛⣿⣿⣿⣿⡿⡿⠛⠉⣆⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⡏⢉⠧⠴⠚⢿⣿⣿⠟⠁⠸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⣴⡄⠉⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⣄⠀⠀⠈⠀⠀⡱⠾⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⢲⠀⡘⣦⠘⠀⠀⠀⠘⣆⠁
⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣧⠈⣿⡇⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢰⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠒⠀⠳⠾⡾⢿⠓⠰⠂⣀⠀⣾⣷⢴⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢤⣾⣶⣵⣦⣤⣤⢸⡟⢶⠙⣿⣏⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠄⢻⣿⡿⣿⣿⡝⠋⣷⠈⠁⠘⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣽⣿⣿⡇⠀⠸⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣺⣿⠛⠟⠉⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣿⣶⠀⠀⠀⠀⠈⠋⠸⣯⠌⠠⠐⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠗⠀⠀⠈⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠀⠀⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠛⠛⢡⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠖⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠻⣷⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠹⡇⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⣾⣯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣶⠇⠀⢸⠀⠀⠀⠀⠀⠀⠙⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢉⠟⡀⠀⠸⡄⠀⠀⠀⠀⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠲⢦⣄⣀⣤⠞⠋⠉⠉⠁⠘⠦⣤⡤⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢀⡀⠀⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢆⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠞⠁⠋⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣄⣀⣀⣀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢠⣤⡤⠴⠶⠲⡆
⠀⠀⢸⣇⣀⣷⠀⠀⠲⣦⡴⠲⣦⡴⢦⡀⠀⠘⢻⡇⠀⠀⠀⣠⢤⣀⣀⣄⣀⣀⣀⣠⣀⣀⣀⠀⢠⡤⣄⠀⢀⣠⣤⣠⠄⣀⣤⣄⣤⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⢻⡏⠉⠿⢀⡄⠀⣿⠀⠀⣿⠀⠀⢻⡄⠀⢸⡷⠶⢦⡀⣁⣤⣿⠈⣿⠁⠈⠋⢸⡟⠁⠙⠁⢈⣠⣼⡇⣿⢤⣤⣿⠘⡧⣤⣼⡇⣴⠋⠛⣦⣴⠋⠙⣿
⠀⠠⠼⠷⠤⢤⣼⠇⣠⣿⣄⢀⣿⣄⠀⣈⣧⣀⢸⣁⠀⣸⢿⣁⢀⣿⣀⣿⣀⠀⠀⣈⣇⡀⠀⠀⣏⡀⣸⡃⡿⠦⠤⠟⢸⡷⠤⠴⠋⢿⡋⠙⡛⣧⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠈⠉⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣨⠋⠉⡄⠀⠀⠒⠒⠀⠀⠀⠀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠞⠀⠸⡁⠀⠀⢃⠀⠀⠀⠀⠀⠸⣦⠞⠁⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠸⣆⣤⡀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠐⠊⠙⠒⢦⡶⡀⠙⣷⡠⠆⠀⠀⠀⣿⠉⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢹⢷⠀⠀⢳⣄⠀⠀⢀⠿⢤⣤⣴⣋⠀⠀⢀⣀⣀⣀⣀⣤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠩⣭⣍⡑⢾⡟⠀⠀⠀⠉⠓⠴⠋⠀⠀⣤⡄⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡦⣤⣤⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣅⠴⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠂⠀⠀⠀⣀⣑⣈⣤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡡⠊⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡁⠀⠀⠀⠀⠀⠀⣠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⡴⠇⠀⢀⣀⡀⠀⢈⠁⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡀⠘⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⢠⠁⠀⠀⠀⠀⡄⡰⠊⣉⣠⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣨⣃⠔⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠙⠃⡀⢦⡽⣂⠴⠋⣸⣀⣀⣀⣠⠤⢴⠒⢋⣡⠗⠀⢙⣪⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠙⢄⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠉⠒⠛⠁⠠⠵⣻⠄⠀⡠⠛⢆⡞⡟⠉⠀⣠⠞⠁
⠀⢰⠂⠀⡄⠀⡾⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠞⠀⠀⠈⠉⣇⣠⠞⠁
⠀⠸⠀⠸⠀⠸⡇⠀⠀⠀⠀⠀⠈⠳⣤⣀⡤⠤⠤⠖⠒⠢⠤⣀⠀⠀⠀⠀⢠⠟⠒⠶⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⢤⣤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠹⡒⠂⠐⠶⠆⠀⠶⠆⠀⠀⢠⡀⠀⠀⢀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⠋⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣠⠄⠀⠚⠁⠀⠀⠸⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠥⣏⣓⡄⠀⠀⠀⠀⠀⢶⣀⡇⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠽⠤⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠬⡀⡨⠀⠀⠀⠊⠀⡖⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢱⠀⠊⠁⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⢀⠙⠐⢁⠤⢊⠂⠀⠀⣈⠂⠤⠊⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⣉⡉⠙⠀⡇⠀⠀⠀⠀⠀⠀⠈⡠⠊⠀⠒⠓⠊⠀⠀⠒⣄⠀⡂⡂
⠀⠀⠀⠀⠀⠀⠠⠐⠊⠁⠉⣉⢀⡐⡲⠛⣈⠝⢀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣞⠠⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⢄⣀⡠⠴⠮⠤⠤⠜⠀⡇⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠠⠤⢤⠐⠊⠥⣂⡈⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠃


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠙⠀⠉⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡁⠀⠀⠙⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠄⠀⠈⠀⠈⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⣉⣀⡀⠀⠀⠀⠀⠀⠂⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠒⠠⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠒⠚⠉⡉⡉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢲⢙⠠⢙⠃⠄⢡⡐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠰⠀⠐⣤⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠆⣀⠀⢳⡨⡀⠌⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠢⣁⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠉⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠉⠣⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢿⣿⣿⣿⣿⠟⠉⠁⠀⢄⠀⠀⠀⠠⢻⡿⠛⢉⠉⣿⡅
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢒⣒⣢⣤⣀⠀⠀⢸⣿⡟⢀⡸⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣴⠋⣽⣿⡿
⠀⠀⠀⠀⢀⠔⢉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣧⠈⠁⣿⣿⣿⣯⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⣰⣿⣿⠇
⠀⠀⢀⡔⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⠀⠒⠀⠀⠀⠀⠀⣴⣀⣴⣿⣿⠏
⠀⢀⠎⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁
⠀⡼⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠉⢹⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣀⡤⢐⢠⣿⠿⠋⠙⣷⣶⣴⣶⣄⡀
⢠⠇⢀⣿⣿⣿⣿⣿⣿⢫⡉⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢞⡾⠃⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀
⡞⠀⢸⣿⣿⣿⣿⣿⡇⢸⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡣⠊⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠁⠀⣿⣿⣿⣿⣿⣿⢀⠘⠱⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢰⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⢸⣿⣿⣿⣿⣿⡟⢸⣧⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣽⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⢐⠋⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⣼⡶⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠛⠿⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠶⠶⢶⣾⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣤⢀⡄⠘⠳⠄⠀⠀⠀⠀⠀⢸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⣿⣿⣿⠯⣿⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⣿⠟⠁⠀⣿⠀⠀⠀⣠⡜⠀⠀⠀⠀⠀⠀⢀⣴⣿⡀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣦⣴⣤⣶⣾⣿⣿⣿⣿⣿⠇
⠁⠀⠀⠀⠘⠧⠀⡰⠉⠀⠠⠤⠀⠀⠀⠒⠘⢸⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⣿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡋⠋⠛⠛⠛⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣿⣿⣿⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⣿⣿⣿⡿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⡉
⠀⠀⠀⠀⠀⢠⣾⣷⣆⠸⡆⠀⡿⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⣿⣿⡏⠀⣷⠀⡇⣀⣤⣤⣶⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣶⣧⡘⣧⡀
⠀⠀⠀⠀⠀⠸⣻⣿⡟⢀⡇⠀⣿⢻⣿⣟⣻⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡶
⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠀⠀⣿⣸⣗⣷⣹⣽⣿⣿⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡘⣿⡄⠀⠀⠀⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠙⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡘⣷⠀⠀⠀⠀⠀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡘⡇⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠓⠒⠶⠶⠶⢤⣤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠤⠤⠶⠶⠶⠟⠛⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⣾⠂⠀⠈⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⡏⠀⠀⠀⢃⠀⠀⠁
⠀⠀⢀⣀⣤⣤⣤⣤⣴⣾⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠸⠟⠿⠟⣿⣿⠿⠿⠿⠟⠻⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠈⠛⠿⢿⠿⠿⣿⠟⠛⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠠⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢁⠀⠛⠋⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⡀⠀⠀⠀⠀⣿⣟⠀⠐
⠀⠀⠀⠀⠠⣤⣤⣼⣶⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡄⠀⠀⠀⠀⠀⣒⣶⠶⠶⢶⣖⠀⢠⠀⠀⢸⡏⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠴⠶⠾⣿⣿⣿⣿⣶⣤⣤⣶⣴⣧⣤⠾⠯⠍⠀⠀⠀⠀⠀⠀⠀⡪⠭⠭⠭⠭⠵⠀⠈⢧⡀⠀⠻⠶⠂⠀⠀⠹⣿⣄⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣽⣯⡭⠭⠳⠶⠞⠓⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉