Koopa Troopa Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠤⣄⠀⠀⠀⠀⢀⠤⢤⡀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣾⡇⠀⠀⠀⣿⣦⣾⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⢛⡋⠀⠀⠀⢙⣛⠿⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠀⠘⠛⠡⡐⠒⣷⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⣄⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⡄⠐⣠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣧⠀⢲⣷⣤⣾⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⢻⣿⣿⣿⣿⣦⣄⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣾⣧
⠀⠀⠀⠤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⡀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⣤⣄⣀⣀⣤⣾⣿⣿⣿⣿⣿⣶⠊⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⡄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣇⣤⢾⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠁⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⡄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣆⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⡶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠉⠀⢠⡾⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣺⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠀⠈⢣⡄⠈⠐⠲⢤⣄⣀⣴⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣿⡷⠞⠀⠀⠀⠀⣴⣿⣦⣤⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⣀⣶⣤⣴⣿⡿⠛⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠒⠛⠋⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣤⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⠙⠛⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠙⠛⠛⠻⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⣀⡠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡠⠒⠊⠒⢟⡵⠒⠒⢪⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡸⠁⡔⠲⡀⢸⣄⡖⢢⠀⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⠀⢟⢫⣃⡼⢷⠿⠷⢤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠉⠲⡤⠋⠁⠀⠈⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡚⠉⠉⠉⢦⡀⠀⠀⠙⡦⢤⣀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⠋⠀⠉⠙⢶⣺⠋⠈⢱⡶⢄⠱⣄⠀⠉⢲⢤⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⡅⠀⠀⠀⠀⡸⠁⠀⣠⢧⡙⠦⣙⣢⡭⡭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣄⡀⢀⡜⠁⢀⡔⠉⠳⡙⠲⣤⠟⠋⠙⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣩⠏⠀⢀⢾⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡲⠼⡆⢀⡼⠁⠀⡰⠛⠦⣳⣄⣀⣀⣀⠀⠀⠀⠀⠈⡺⠁⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⢒⡟⠁⠀⣠⣾⣁⠀⠀⠀⠉⠑⠒⠚⠝⣿⠚⠿⣭⣀⡀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⠋⢈⡖⠚⠁⠈⠻⡙⠒⠤⢄⣀⣀⣀⠔⢧⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠸⡀⠸⡄⠀⠀⠀⢠⠗⠤⠤⠔⢶⣋⡥⠖⠚⢉⣷⠖⠋⠉⠲⡄
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠑⢄⣙⣒⣶⠖⠁⠀⠀⠀⠀⢯⣀⡤⠔⠚⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⣀⡠⢔⣺⠇
⠀⠀⠀⠀⠀⠀⠘⠳⢤⣀⣀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡤⠴⣒⡩⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⡸⠁⠀⠹⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣼⣷⢠⣶⡄⠀⢹⡇⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠙⠛⠿⣿⠸⣿⠇⠀⣸⠷⠶⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀⣤⡄⠀⠀⠈⠋⠓⠦⠶⠋⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⢿⡀⠀⠀⠀⠀⢀⣼⠟⠛⠿⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠉⠀⠀⣤⣶⡟⠳⣤⡀⠀⠀⣿⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⠛⠻⠶⠶⢶⡶⠾⢦⣴⠾⠛⢉⣿⣤⡈⠻⣦⣴⠿⣦⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣤⡶⠾⠋⢻⡀⠹⣆⠘⢷⡀⠘⣧⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⣇⣀⠀⠀⣀⣀⣤⢼⡇⠀⢹⠀⠈⣷⡀⢸⣧⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⠉⠉⢉⡟⢛⡛⠻⠀⠀⣾⡄⠀⠸⣇⣼⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣷⢿⣦⣤⣼⠛⠛⠁⠀⠀⠘⣯⣇⠀⠀⢻⣅⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⠿⠛⠻⢷⣝⣶⡆⡀⠀⠀⣰⡧⢿⣄⠀⠀⢹⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⡼⠛⠛⠛⠻⣿⡷⠚⠿⣿⣓⠒⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠛⠀⠠⠿⣿⠛⠷⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠶⠶⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢺⣟⠳⠦⠤⠤⠴⠶⢛⣚⣋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⠶⠖⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⢁⡄⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣠⡄⠈⠃⢀⣤⠀⠐⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣶⣎⡀⠀⢰⣶⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠀⠀⠉⠂⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢂⠁⠂⢤⠠⠒⠒⠀⠀⢃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠰⠀⠀⠀⠀⠢⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠂⠰
⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠤⣄⢠⡀⠀⠀⠀⣠⢀⡅⠐⠁⡘⠀⢁⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⢤⣀⠓⢖⠛⡁⠀⠀⠀⠀⠁⠀⠈⡀⠁⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⠦⣀⠀⠀⠀⠐⡃⢀⠈⠉⠒⠧⠀⠀⢀⠠⠊⠀⠀⠠⠀⠀⠀⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⢀⡀⢀⠄⠀⠀⠐⠀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠱⠃⠀⢠⠁⠈⠀⠀⠈⠐⠢⠄⠀⡄⠀⠀⣀⠠⠔⠠⠤⠤⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⡀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠁⠐⠠⠄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡀⠃⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⢀⠄⡤⠂⠰⠎⠀⠀⠐⡀⠀⠁⠐⠠⠄⣀⣀⡜
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠃⠄⡀⠀⠀⡀⠔⠋
⠀⠀⠀⠀⡸⠀⠀⠀⢃⠈⢄⠀⡀⠴⣁⡀⠒⠀⡏⠁
⠀⠀⠀⣸⠁⠀⠀⠀⠀⠑⢤⠉⠄⠀⠀⠀⠀⠰⠄⠀⠀⠠⠦
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠆⠄⢀⣀⠀⠴⠠⢄
⠀⠀⠀⠸⡄⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⡂⠀⠀⠈⠁⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠈⠓⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠶⠄⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠀⣤⠄⠂⠈⠉⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡀⠀⠀⠀⠀⠀⢀⡀⢀⡠⠄⠠⢄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣝⠒⠄⡀⠀⠀⢎⠐⠁⠈⡄⠀⠀⠑⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⢸⠋⠀⠙⣇⠀⠀⠁⠀⠀⠀⠈⢂⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞⠃⠠⠌⠇⠐⠂⠾⢦⣀⠆⠀⠀⠀⠀⡠⠖⠒⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⢠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⣀⢤⠊⠁⠀⠀⠀⢀⠊⠀⠁⠀⠀⠀⠀⠀⠀⢀⠔⠫⢐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⡠⠴⡈⠣⡀⠢⣀⠠⠔⢂⠋⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠀⠉⠪⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠐⠠⠀⠒⡁⠀⠀⠀⢠⡐⠂⠁⠘⠂⠁⡦⠂⠀⣀⠀⠀⠈⢌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠖⢲⣥⠖⠛⠉⠂⠈⠀⢀⡠⠔⠊⡇⠀⠀⡍⠁⠀⠂⠠⠦⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢠⠹⠠⠤⠄⠐⠂⠉⠀⠀⠀⢠⡦⠀⢠⡄⠀⠀⠀⠀⠀⢸⠥⢀⡀⠀⠀⠀⢀⡀⠀⠀⠠⣀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⣀⡸⠷⢀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⢆⠀⠀⠁⢀⡄⠀⠀⠀⠉⠒⠉⠀⠀⠐⠁⠀⠀⠉⢉⠶⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣉⠀⢠⠌⠁⠙⢦⠈⢦⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠳⣄⡀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠲⠞⠉
⠀⠀⠀⠀⠀⡀⠠⠤⠄⣀⡀⣀⣀⠜⠂⠉⢂⢸⡆⠀⠀⠀⠑⢄⣱⢀⣠⡐⠚⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⣄⠀⡄⠀⠀⠀⠀⠀⡀⠔⠈
⠀⠀⠀⠀⡀⠧⡀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠹⣇⠀⠀⠀⠀⠀⠇
⠀⠀⠒⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣼⠃⠀⡟⠢⡀⠀⠀⠠
⢀⠶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊⠀⡃⠀⠁⠀⠈⠦⢀
⠣⠤⠤⠀⠀⠀⠠⠤⠄⠀⠒⠚⠛⠒⠂⠠⠤⠟⢧⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠞⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⠤⠤⠤⠀⢈⠠⠔⠉⠤⠲⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠀⠰⢦⠈⢉⡀⠀⠀⡀⠀⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⠈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⡀⠀⠰⠶⣄⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣀⠄⢒⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡉⠓⡦⠀⠤⠤⢶⡁⠀⠀⠀⢀⠔⠉⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠢⠤⠤⠤⠤⡀⢸⠇⢹⠀⠀⠀⠀⠙⣶⡀⣼⡇⠀⠀⠀⠀⢀⡨⠗⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠓⠒⠠⢀⠀⠀⠀⠈⢻⡤⡶⠀⠀⠀⠀⠀⠈⠻⣯⡞⠉⠉⠉⠁⠀⠀⠄⠀⠖⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠒⠲⠀⠰⠀⢡⠀⠀⠀⠀⢠⠁⢙⠢⣤⠠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠈⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠆⠀⢰⠂⠀⢠⠀⠀⠀⠀⢠⠀⠈⡜⢀⠊⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠒⠬⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⡛⠀⠀⠀⠀⠘⣆⠀⠇⠈⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡠⠆⢰⠋⢢⠀⠀⢂⠀⠀⠀⠀⠆⢀⣆⡰⠟⠃⠀⠀⠀⠀⠀⠹⣧⣰⠀⠀⠀⠀⠀⢀⠌⠀⢀⠄⢄⠀⢠⠊⠱⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠒⣊⠀⠀⠀⠃⠀⢠⡴⠷⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⣀⠴⠃⠀⠀⠅⢀⠔⠀⠘⣂⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡀⡴⠊⠉⣉⣑⡢⣄
⠀⠀⡴⠒⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⡠⠔⠒⠒⢌⠁⠰⠋⠀⠀⠈⠹⣦
⠀⢸⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡸⠁⠀⠀⠀⠀⢱⠇⠀⠀⠀⠀⠀⠘⣇
⠀⠀⣃⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⡤⢇⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠇⠀⠀⠀⠀⠀⠀⢇⠀⣠⣤⡄⠀⠀⠘⡄
⡜⠉⠈⢣⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡈⠀⠘⡀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⣄⣠⡄⠀⠀⢸⣀⣸⣿⡧⠤⠤⣤⣇
⢣⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠈⠓⢤⡀⢱⠀⠀⢣⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸⠀⠀⢿⣿⡇⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠈⠱⢦⡀
⠀⢧⠀⠀⠀⡶⠈⠉⠲⢄⡀⠀⠀⠀⠉⠲⢆⠀⠀⢆⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢸⠄⠈⣄⣀⠀⠉⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⢀⣧⡀⠀⠰⡄⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠙⠢⢌⣆⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠉⠢⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢠⠀⠑⠄⠀⢈⣤⡀⠀⠀⠀⠤⠽⠦⣀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠀⠀⠘⢦⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⢧⡀⠀⠠⠀⠀⠀⠐⠠⡈⢦⡀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⢤⡀⠀⢄⠀⠀⠀⠀⢠⠽⠶⢄⡀⠀⠀⠙⢆⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⢸⠉⠑⠒⢤⣤⣀⡀⠙⢦⡀⠀⠈⠁⠀⠀⠀⢣⠀⠀⠀⠀⠈⣦⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⠈⠉⠁⢾⠉⠛⠒⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠈⠙⠓⢦⡀⣀⠀⠀⠑⣤⣄⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡛⠛⠂⠒⠾⠷⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⠙⠃⠾⢅⣀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⣸⠋⠓⠦⣄⣀⡀⠈⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠈⢣⡈⢙⠞⠉⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢒⠖⣒⠶⠶⢖⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡵⠋⠀⠀⠀⠀⡼⢍⠐⠒⠠⡤⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠚⠁⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢀⣎⠀⠈⢢⣀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡰⠧⣀⠑⣄⠀⠙⠮⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡧⠈⢁⠶⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠑⡌⠳⢄⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⣧⠔⠁⠀⠀⠀⠀⢀⠔⢹⠀⠀⠀⠀⠀⠘⡄⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠛⣄⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⣀⠴⢇⠀⠘⡄⠀⠀⠀⠀⠀⢱⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣬⠶⠚⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠱⣄⣣⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢄⡶⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠒⠰⢧⡀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠶⠉⠉⠉⠑⢤⣀⠀⠉⠒⢒⠮⠥⣀⡇⠀⠀⠀⠀⠀⢸⡇⠙⢦⡀⠀⠀⢀⢵⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⡁⡌⠀⠀⣠⡎⠁⠀⠉⠉⠉⢣⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡹⠖⠊⢡⡟⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⠀⠈⡆⠀⠈⢢⠀⠀⠀⠀⢀⡼⠃⠉⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠃⠀⢨⡆⢡⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠱⣄⠀⡠⠊⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡜⢸⠃⠀⠀⠇⠸⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠉⣳⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⢠⠃⡾⠀⠀⠀⠙⣄⢱⠀⠀⠀⠀⠀⠀⠀⠀⢉⠦⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣆⡀⢀⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠸⡆⣆⠀⠀⠀⠀⠎⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⠤⠤⠴⢇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠇⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠦⣀⣀⣀⡠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠓⠶⣄⣀⣤⠴⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠖⠛⠛⠛⢮⣋⡴⠒⠛⠓⢦⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡼⠁⢀⠤⣄⠀⠀⣏⠀⣠⢤⡀⠀⢳⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠃⠀⣯⣆⡼⡆⠀⣿⢰⢣⣧⠣⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢸⠀⠀⢧⣛⣱⣣⡤⠾⢚⡛⠛⠳⢦⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠈⠙⠦⣄⡼⠋⠁⠀⠀⠉⠁⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠿⣆⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣅⣀⠀⠀⠀⠈⣳⣤⡀⠀⠀⠀⢹⡛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡽⠁⠀⠀⠉⠛⢾⢛⡿⠁⠀⠉⣿⡶⢦⡀⢳⣀⠀⠀⠉⠛⣦⣄⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠏⠀⠀⠀⠀⠀⠀⢈⡟⠁⠀⠀⣼⠃⠳⣄⠙⢦⣌⡓⠒⣒⣫⠷⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⢷⡀⠀⠀⠀⠀⣠⠏⠀⠀⢀⣼⣍⠳⣄⡈⠛⠶⢾⣟⣙⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢳⣀⣀⣀⡴⠃⠀⠀⣰⠋⠀⠈⢳⡀⠙⢳⡶⠋⠁⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣯⢉⡟⠁⠀⠀⣰⡇⠀⠀⠐⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢾⣄
⠀⠀⠀⠀⠀⣤⣤⣀⣀⣷⠀⠀⠀⣹⠏⠀⠀⢀⡾⣁⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡼⠉⠳⣄
⠀⠀⠀⠀⠀⠈⠳⣌⠉⠹⣆⣠⠞⠁⠀⠀⣴⠋⠀⠈⠓⠮⣷⣶⠦⠤⠤⣤⣀⣀⣀⣀⣀⣀⡼⠃⠀⠀⠘⡇
⠀⠀⠀⢠⣴⡿⠛⠛⠛⣳⡏⠀⠀⠀⣠⣾⣧⡀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⣻⠏⠙⠯⢤⣀⣀⡀⣀⡶⠃
⠀⠀⢀⡏⡞⠀⠀⢀⡞⠁⠙⡦⠤⠞⠁⠈⠻⣎⠓⠦⢤⣀⡀⠀⠀⠀⠀⣠⡼⣇⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⣸⢻⠃⠀⠀⢸⡁⠀⢸⠃⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢉⣙⣳⡶⠞⠉⣀⣹⠦⠦⡄⢀⣤⠤⢤⣄⡀
⠀⢠⣯⡏⠀⠀⠀⠈⣇⠀⠘⣇⠀⠀⠀⠀⢀⡼⠋⠉⠙⠋⠉⠉⣷⠴⠞⠋⠉⠀⣀⡤⠿⠋⠀⠀⠀⠈⠹⡄
⠀⡾⣿⠀⠀⠀⠀⠀⠈⠳⣄⣈⣛⣲⣶⠾⠋⠀⠀⠀⠀⠀⠀⠸⣅⣀⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢾⡄
⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣀⣤⠴⣚⣭⠵⠖⠋⠁
⠀⠀⠈⠙⠳⢤⣄⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⡤⠶⠚⣋⣡⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠖⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠁⢸⠏⠀⠈⠻⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡄⠀⢠⡄⠀⠀⢱⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣀⣿⡏⠀⣿⡇⠀⠀⢸⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠋⠁⠀⠀⠀⠉⠳⣄⠙⠁⠀⢀⡞⠀⠉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⡴⠶⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡇⠀⠀⠀⠀⠀⠀⣠⠿⠷⠶⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠙⠂⠀⠀⢀⣠⠾⢧⣄⠀⠀⠀⠀⢻⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⣴⣿⣤⣤⣀⣀⣀⣤⣄⠀⠀⣘⡿⠛⢷⠀⠈⠛⢦⡀⠀⣸⡀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠉⣿⡉⠉⠉⠉⢁⣀⣤⢿⠛⠳⣄⠀⠻⣎⠁⠙⣆⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠛⠛⠛⠋⠉⠀⠸⡇⠀⠹⣆⠀⠙⣆⠀⠘⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⢿⣄⣀⣀⣀⣀⣤⡴⢞⡇⠀⠀⣿⠀⠀⢹⡄⠀⡼⠒⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣇⠀⢾⡄⠉⠉⠉⡟⠉⡉⠛⠃⠀⢀⣿⡄⠀⠀⣇⣰⠃⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣶⠞⢳⣄⣀⡼⠋⠛⠁⠀⠀⠀⢸⡿⡇⠀⠀⠹⡇⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⠶⠿⠿⣄⠻⣤⡀⠀⠀⠀⠀⣸⣃⢿⡄⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠈⣹⠿⠗⠶⠤⣴⡚⠭⠤⢴⣟⠦⣤⡴⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠱⠄⠀⠀⢈⣷⣦⣤⣈⣙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢤⣀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⡇⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣌⣉⣁⣹⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⣤⣉⠛⠒⠒⠒⠒⢋⣉⣉⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡔⠒⢆⠀⠀⠀⠀⠀⣮⠙⢦⡀⠀⠀⠀⠀⠀⢠⠖⢡⠼⢷⡇⠀⠙⢷⡀
⠀⠀⠀⠀⠹⣦⡀⠣⡀⠀⠀⢀⣿⠀⠈⢷⡀⠀⠀⠀⠀⡏⠀⣠⣄⠈⠇⢸⣷⠈⣧
⠀⠀⠀⠸⡇⢹⠛⢦⣌⠠⡀⢸⡟⠇⠀⠀⢻⡄⠀⠀⠀⣷⡀⣿⣿⣆⠟⠚⠛⠛⠿⢷⣤⡀
⠀⠀⠀⠀⠻⣦⣷⣤⠈⠻⣦⡙⠧⣀⠀⠀⠀⠹⣆⠀⣰⠉⠉⠈⢻⠁⠀⠀⠈⠁⠀⠈⠸⣷⡀
⠀⠀⠀⠀⢷⣄⣫⣙⠻⢿⣬⣝⠳⢤⡁⠢⡀⠀⠙⣦⣧⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⡸⣧
⠀⠀⠀⠀⠀⢹⡷⢿⣷⣿⣤⣉⡓⠀⠙⢦⠘⢦⡀⠸⣿⣦⠀⠀⠸⣇⠀⡀⠀⠀⠀⠀⢠⣽⠇
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣉⡿⠿⢁⣀⠜⠀⢻⣿⠦⠽⣿⣿⣤⡐⢝⠳⢷⣦⣠⣀⣀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣭⣉⣛⣛⣉⣠⣶⡾⠿⢶⡶⡾⢻⣿⣿⣽⣷⣤⣿⣿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠿⠿⣿⣏⠀⠀⣸⠋⠀⡼⣿⠻⣭⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⢹⣿⣤⠞⠁⠀⢀⡼⣿⢦⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⣀⡼⠛⠀⠀⣀⣴⡏⠀⠘⢧⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠁⠀⣠⣴⣾⠟⠁⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢄⡸⢧⣤⣶⣿⠋⠉⢻⣷⣦⣈⣦⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣄⡟⠠⣾⢿⠷⣿⠻⡙⠂⠀⣼⡿⠿⠛⣭⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣹⡇⢂⣿⡀⠀⠙⣧⡙⢶⡞⠁⠀⠀⠀⣿⡀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠘⣿⡇⠀⠀⠸⣿⠶⠃⠀⠀⠀⠀⠉⣡⡜⠉
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⡗⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢽⣿⣤⣄⢴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⢨⠏⠀⠀⠙⢿⡉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⡆⠀⢸⣿⡆⠀⢘⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠾⠦⣿⣅⡀⠸⡿⠃⠀⢸⡇⠀⠐⠿⣦⣄
⠀⠀⠀⠀⠀⠀⢠⣶⠟⠉⠀⠀⠀⠀⠀⠈⠉⠳⢦⣀⣠⠟⠁⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⡴⠶⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠳⠄⠀⠀⠀⠀⠀⢠⣾⡧⠶⠶⠿⠿⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⣶⣏⠙⠲⣤⡀⠀⠀⠀⠀⣿⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀⠀⣼⣯⣤⣀⣀⣀⣤⠀⠀⠀⠈⣿⡆⠈⣧⠀⠀⠙⠳⣄⠀⢀⣿⣀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠏⠀⠈⠉⣽⠉⢿⣷⣴⡶⠟⠋⠀⣠⠟⣶⣤⡀⠀⠙⢾⠋⠀⠙⢦⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⢀⣀⣠⡴⠞⠁⢸⠁⠈⠹⣆⠀⠈⣧⠀⠀⠈⢧⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠁⠀⢀⣠⢾⡇⠀⠀⢹⡄⠀⠸⡇⠀⠀⢈⡶⠞⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠤⠤⠤⠶⠖⣛⣉⡀⢰⡇⠀⠀⢸⣇⠀⠀⣿⠀⠀⡼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⣇⠀⠀⠀⢰⣏⠉⠉⠙⠛⠀⠀⠀⢸⡿⠀⠀⢹⠀⡼⠃⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢻⡄⠀⣰⠋⠉⠹⠋⠀⠀⠀⠀⢀⡾⢱⡄⠀⠸⣟⣁⡀⣰⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠾⣿⡆⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢧⠀⣼⠋⠉⢽⠋⠉⠉⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠈⢿⣦⡙⣆⣀⠀⠀⠀⠀⣠⣟⣠⠴⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⣛⡛⠶⣄⡀⠀⠀⠀⠈⣿⠙⢶⣽⣷⡶⠖⠛⢋⣉⣀⣤⠶⣿⣤⠴⠀⠀⠀⠀⠀⢀⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠊⠉⠉⠓⢦⡙⢦⡀⠀⠀⠛⠀⠀⠀⢹⡟⠛⠛⠛⠉⠉⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠙⢦⡙⣄⠀⠀⠀⠀⢲⣠⡿⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠱⣌⢦⡀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⡸⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⢀⠔⠋⠳⡝⢆⣠⠿⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⢀⡼⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄⠀⠀⣸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⣯⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⠀⢰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡼⠛⠋⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⠀⢀⠀⠹⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⣠⣶⣄⠀⣴⣎⢷⡀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠣⠼⡄⢿⠀⢠⠇⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⡁⣹⠀⢧⣼⢸⣇⡀⠀⠀⠀⠀⡼⠃⠀⠤⠀⠀⠈⠁⠐⠋⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⠛⠛⠋⠀⠀⠀⡀⡈⠙⣦⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⢀⣠⣤⠶⠾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢧⠀⠀⠀⠀⣀⠤⠄⢦⠀⠀⠀⣠⡞⠛⠲⢦⣄
⠀⠀⠀⠀⢠⠟⡹⢇⡠⠊⣁⠹⢿⡶⢲⡆⠀⠀⢀⠀⢀⣀⣸⠃⠀⠀⠈⠓⢦⣤⣮⣄⣀⣀⣄⣀⣰⡞⣏⠙⠦⡀⢨⠈⢳⡄
⠀⠀⠀⠀⣿⡰⠁⡜⠀⣴⣧⠀⠀⠐⣞⣹⣶⣖⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢬⡭⠥⠖⢫⠑⠄⠙⣆⠱⡀⢻
⠀⠀⠀⠀⣿⠳⣸⠀⢸⢛⠉⢡⠊⠉⠛⠁⢀⠉⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣧⢤⣤⣤⡴⣺⠀⢸⠀⢸⡄⡕⢺⠆
⠀⠀⠀⠀⠘⡶⠃⢀⣿⡆⠀⢹⣀⠀⠀⠀⠀⢠⠀⡄⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣢⢿⣄⣠⠃⠀⠀⠀⠺⡆⠀⢿⢀⡞
⠀⠀⠀⠀⠀⠛⠖⠿⢭⣤⣴⣾⣿⣷⣄⣄⣸⡼⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠋⠛⢮⣶⣀⣀⣠⣷⣻⣄⢀⣿
⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠃⠒⠢⡁⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠌⠀⠀⠈⠻⠉⠉⡿⠤⣌⣳
⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⠀⠀⠀⣀⣀⠀⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⡴⣦⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠙⣶⣶⣥⣤⣤⣽⡟⠒⠛⣉⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣈⣉⣳⣬⣁⣀⣨⣥⠤⠚
⠀⠀⠀⠀⠀⠀⠀⢙⣸⢃⡇⡇⣽⡇⠀⣠⣿⣗⡰⡇⣿⡇⢧⣍⡇⣿⣸⡃⣿⡇⢏⡇⡸⢯⣇⢇⡇⠿⣇⣕⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠑⠲⣴⠯⠭⣍⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡄⣴⠁⠀⠀⠈⢣⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⠲⠷⠦⣼⣿⠇⠀⠀⣤⢂⡼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠉⠁⠀⠀⠙⢦⠀⢸⣿⣿⠇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡾⠛⠛⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⢿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣴⠃⡸⠖⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢀⣠⣤⣤⣤⠔⠃⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⣼⠃⠀⠀⡠⠞⠚⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠾⠛⢧⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣏⣀⣀⢀⣰⠇⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⢲⣖⣀⠠⠜⢦⡀⣨⡿⠟⠋⠁⠀⠀⠀⠀⠀⠐⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠒⡀⠀⠀⠀⠀⢰⣏⡀⣀⣀⠤⣰⠟⠉⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣌⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠨⠀⠑⠁⠘⡄⠀⠀⠀⢰⠋⠉⠁⠀⠀⣿⡀⢀⣀⣤⡶⡟⠉⠀⠀⢳⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠈⠀⠈⠉⣽⠀⠉⠀⠀⠀⠈⠉⠉⣏⠀⠀⢸⡀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⠀⠀⠀⠀⣀⣠⡤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣇⠀⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣀⡠⠤⣲⣯⣥⣤⣀⣿⠒⠒⠒⠂⠁⣀⠀⠀⠀⢷⠀⠀⢻⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣼⡋⠀⠀⠀⠈⠻⣆⠀⣠⡾⠛⠉⠉⠓⢦⣸⡄⠀⢸⡄⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠘⣧⠀⠀⠀⠀⠀⢻⠻⣯⡀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡹⣇⠀⠀⠀⠀⠘⠢⠈⢻⣄⣀⡀⠀⣀⡼⠁⠀⣸⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠹⣆⠀⠀⠀⠀⠀⢠⣄⢹⡟⢿⠛⠿⠶⠶⠾⡟⢶⣶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⣧⡀⠀⠀⠀⢀⡿⠉⠀⠘⢆⠀⠀⢀⡴⠃⢸⠇⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣷⣄⣀⣠⡾⠃⠀⠀⠀⠀⢿⣛⢋⣀⠄⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⡴⠛⠉⠉⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⣀⣀⡴⣫⡏⣀⠀⢀⢀⢀⠀⢠⢠⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠛⠉⠐⠛⠈⠈⠐⠘⠁⠱⠌⠀⠀⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣦⣤⣶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠋⠉⠉⣿⡟⠉⠈⠙⢿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⢰⡟⠀⠀⠀⠀⠈⢻⣿⣟⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⣠⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣼⣿⣿⣧⠀⢰⣿⣿⣆⠀⠀⠀⠈⣿⣿⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣧⣀⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠉⠉⠉⠉⠉⠛⠿⣿⣿⡇⠀⢹⣿⣿⡏⠀⠀⠀⢰⣿⡏⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡜⣏⠉⠀⠀⠀⢀⣿⡿⠀⠀⠀⠈⠉⠛⠿⣿⣶⡀
⠀⠀⢠⣿⠟⠀⠀⠀⠀⢀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠻⢶⣶⣾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⢠⣿⠏⠀⠀⠀⠀⠀⣿⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆
⠀⣾⡟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣤⣤⣄⣀⣀
⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣤⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⠛⠛⠿⠿⢿⣿⣶⣤⣀
⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠉⠉⠉⠉⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣷⣤⡀
⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣤⣴⣾⣿⡟⠛⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⣿⣷⠈⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⢀⣾⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⣸⣿⣿⠟⢿⣿⡄⠀⠈⠙⢿⣷⣄⠀⠀⠀⠀⣾⣿⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣿⠏⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⠟⠁⠀⢨⣿⣷⣤⣄⡀⠀⠙⢿⣷⣄⣤⣾⡿⣿⣷⡄⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠃⠀⠈⠉⠉⠉⠉⠁⠀⣀⣠⣶⣿⣿⡏⠉⠛⣿⣦⠀⠈⠻⣿⣿⠉⠀⠀⠻⣿⣆⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣦⣴⣶⣶⣾⣿⣿⠿⠛⠁⢹⣿⡀⠀⠈⢻⣧⠀⠀⠙⣿⣧⠀⠀⠀⠹⣿⣇⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢿⣧⠀⠀⠈⣿⣷⡀⠀⠀⢹⣿⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠈⣿⣷⠀⠀⠀⢿⣿⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢸⣿⡿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣉⣡⣿⠇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⣿⣇⠀⠀⣿⡿⠉⢩⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⢠⣿⠟⠉⠉⠉⠛⠿⣿⠆⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⢻⣿⡀⣼⣿⠃⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣷⣄⠀⢹⣿⡄⠀⠀⠀⠀⣸⣿⣶⣶⣾⡿⠀⠀⠀⠀⠀⠀⢰⣿⢻⣿⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠙⣿⣿⡿⢿⣿⣦⣤⣀⣰⣿⠋⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠸⣿⡼⣿⠀⠀⠀⠀⠀⢹⣿⣄⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣯⣤⣾⣿⣿⣟⠛⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠄⢿⣇⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠛⠛⠿⢿⣷⣄⡹⢿⣶⡷⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⡿⣿⣆⠀⠀⠀⠀⠀⠀⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣾⣿⣿⣶⣶⣄⣀⣠⣤⣶⣿⠟⣋⣡⣴⣿⣿⣷⣄⣀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠉⠉⠉⠙⠻⢿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣯⡛⠻⠿⠟⠛⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠈⢻⣿⣿⣷⣶⣤⣀⣀⡈⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⣶⣿⠟⣿⣏⠙⠻⠿⠿⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣦⣄⣀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠛⠻⠿⠿⠿⠿⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣀⣀⣀⣼⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣿⡇⠉⠻⠿⣷⣶⣶⣶⣶⣶⣶⣶⣾⠿⠛⠛⠛⠛⠋⣡⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⣄⣀⣀⡉⠉⠉⢀⣀⣀⣤⣴⣶⣶⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⢶⣾⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠛⢻⣿⠁⠀⠀⠀⠀⠸⠿⣯⠏⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⣠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⣀⣿⣿⣟⣿⣿⣯⣤⣤⡙⡀⠀⠀⢠⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡶⠶⠿⣿⢿⣿⢟⣿⣿⣿⡿⣿⣿⣟⠀⠄⢀⣼⢷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠇⠉⣹⢀⡀⣸⢿⣿⣿⣿⣿⣿⣧⠒⢛⣿⣝⣶⣿⠸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣿⠀⠀⢹⡆⣠⣿⣿⢻⡾⣿⣿⣟⠃⠐⢿⣿⣿⣿⡟⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠈⢻⡅⢹⣧⣿⣿⣿⣿⣿⠙⠻⣿⣤⣴⣿⣷⣿⣿⣿⣷⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢺⣿⣦⣿⣿⣿⣿⣿⣿⣾⡷⣉⠛⠿⣿⣿⣿⣿⣿⣿⠀⠆⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣧⣭⣍⠛⢿⣿⣶⣦⠘⣏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠏⢼⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣉⣻⣆⡿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠉⠀⠛⠛⠛⠋⢹⡿⠛⣿⣿⠉⣿⣿⣿⡇⣸⡏⢸⡇⣿⣽⣿⣷⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⠀⠀⠀⠀⠀⠁⣸⣿⣿⣆⢹⣿⣿⢣⣿⠃⢸⣧⡧⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠈⢧⠀⠀⠀⠀⢾⣿⣿⣿⢹⣮⣿⣿⣿⣿⣇⠈⣿⣣⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣷⡤⣄⢨⡛⠂⠀⣼⣿⣿⡇⢸⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⣀⣷⣤⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⡏⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠷⣦⡙⣏⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⠇⠘⠛⠻⠙⠋⠕⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⡼⠁⠹⢷⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⣿⣿⡆
⠀⠀⠀⠀⢀⡾⠋⠛⠳⠀⠀⠈⠛⠛⠶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢸⡇⠹⣿⡇
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣾⣇⣼⠟
⠀⠀⠀⠀⠀⠘⠷⣶⢀⣠⠂⠀⠀⠀⠈⣩⣿⣿⣿⣿⡿⢷⠀⠀⠀⠉⠻⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⢨⣰⡏⣄⠸⠿⠃⡴⠋⠉⣿⣿⣿⡁⠈⠀⡀⠀⠀⠀⠙⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠻⠀⠀⡜⠁⠀⢠⣿⣿⣿⣧⣴⠀⠁⠀⢠⡄⠀⠘⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣠⣿⡿⠻⣿⣿⣿⠀⢠⠀⣸⣿⣇⣀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⢉⠧⢶⣿⢀⡇⠈⠛⠻⢿⣦⣤⣨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣟⣛⣻⣯⣙⣛⣟⣛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣯⣭⣿⣿⣯⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢠⣀⠀⠀⢠⡀⠀⠀⠀⣀⣶⠤⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⣤⠘⢦⡴⠊⠁⠠⠟⠉⢺⡁⠀⣀⡶⠟⠉⢀⡠⠒⠒⠤⡀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠘⠐⢓⠚⣓⠐⣟⠀⠀⠀⠀⢀⡤⠞⢿⣾⠋⠀⠀⢠⠃⠐⢠⠀⠀⠈⢳⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠁⠄⠻⠉⣆⠀⠀⠂⠈⢀⡤⠾⡿⠀⠀⠀⠁⠀⠀⣰⣀⠀⢀⣰⢷⡀⠀⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣧⠀⠀⢀⣀⠀⠁⠚⣲⠿⢷⣄⠀⠀⠀⠀⣀⣴⣀⡴⠾⠤⠤⢾⣿⣿⣿⢿⣿⡄⠑⡀⡷⠀⡬⡀
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠂⢹⣷⣀⣀⣽⢶⠒⣨⣶⣈⣤⢿⣷⣄⠀⠁⠀⣿⣏⠉⢈⣀⣴⣄⣿⡉⢁⠀⠈⠙⣦⣼⡧⠤⠴⠃⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠟⣀⢽⣿⡄⣼⣿⣿⣿⣿⣿⣀⣀⠽⢧⣙⣰⠛⢻⣷⣼⣿⣻⣅⣠⣈⠟⣠⡄⠀⠹⢧⡴⠀⡀⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡑⢰⣧⣤⣈⠻⠛⠙⠛⠉⠉⠉⡀⠠⠛⢿⡟⠛⢿⠛⠛⠻⣏⣭⡉⠃⠈⠁⠀⠀⠀⢠⡌⠁⠰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠡⣀⣠⠀⠐⠛⢙⡆⠀⠀⠀⠀⠘⡀⠀⠀⠉⠓⠲⢾⣷⢄⢲⡄⠀⠀⠘⢻⡄⠀⠀⠖⠀⠀⠀⠀⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⡁⢀⣆⠀⠁⠀⠀⠀⠀⢈⣴⣶⣶⣶⣦⣄⠈⢿⡄⠀⡃⠰⠀⠂⡼⡇⠀⠀⠀⠀⠀⠀⠈⠁⡈⠀⠈
⠀⠀⠀⠀⠀⡐⠂⠀⠰⠤⠀⠀⠁⢠⣾⣿⣿⡆⢆⠈⢿⣿⣿⣿⣿⣿⠂⠀⣿⣤⣙⠘⠃⣷⣾⠇⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠉⠨⢋⠴⠂⠀⠀⢰⡌⠛⣻⠿⢳⣶⡄⠈⠳⢿⠿⠃⡼⠀⠂⢠⢈⡅⠒⢠⣾⠃⠀⡆⠀⠀⠀⢀⡃⢀⠁⠂⠁⣰
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣤⣼⣇⠰⡁⢀⡘⢻⡿⣦⡴⠏⡠⠙⠑⠦⠀⣼⠉⠀⣷⠉⡏⠀⣼⣶⡦⠀⠀⣌⣀⠀⠁⠀⡰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠉⠀⠘⣿⣇⡊⠀⠋⠙⠛⠁⠂⠄⡄⠇⣸⣿⡹⠀⠘⣿⡎⠀⢠⣿⠟⠀⢀⣼⡿⠋⠀⢠⣾⡀⠁
⠀⠀⠀⠀⢸⠀⠀⣀⢰⣿⣆⠀⠀⠀⠀⠀⢸⣤⣘⣄⠁⢦⠐⠖⠞⢋⠐⡿⠁⠀⠀⠎⠀⢀⣿⠃⠀⢠⡿⠋⠀⠀⣴⣿⠟⠃⠀⣠⠎
⠀⠀⠀⠀⠸⡄⠀⡉⠙⣾⠉⢄⢠⣈⣰⢲⣾⡀⠈⠈⠀⢀⡦⠾⠊⣡⠌⠀⠠⠄⠈⠀⠀⠘⠁⠀⠀⠊⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠱⡔⠻⢃⠙⢴⠃⠀⠀⢮⢀⣼⠿⣞⠋⣀⣉⡃⠤⠂⠁⠀⣰⠃⢀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠰⠋⠀⢀⡤⠎
⠀⠀⠀⠀⠀⠀⣴⠖⠚⠓⠛⠀⠀⠀⢀⣁⠀⠑⡄⠀⠀⠀⠀⠀⠄⠀⣿⠏⣀⠴⠲⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠀⠀⡀⣇⣀⠀⠄⠀⣠⡀⠀⠈⣡⣴⠀⠣⣓⢲⣶⡆⢠⠀⣷⠋⠔⠁⣤⣴⡶⢶⣶⣾⡷⠃⠀⠀⢀⣼⣅
⠀⠀⠀⠀⠀⠀⠁⢀⢃⣀⠤⠤⡁⠀⠘⠛⠉⠀⠀⢃⣀⣉⡑⠈⠷⠖⢋⣠⡿⠋⠀⢀⣠⡉⠉⠰⠔⠂⣠⠊⠈⢹⣧⠀⠰⠁
⠀⠀⠀⠀⠀⢀⠘⠱⠈⠁⣴⣾⠇⠀⠀⠀⠀⠀⠈⠉⠱⣾⡝⢷⣶⠚⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠰⠿⠧⢤⡴⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢠⣀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⠀⠀⠀⢀⠐⠘⠀⠀⣿⣯⡌⣤⣤⠾⢿⣆⠀⣠⣤⣤⣀⣀⣀⣠⣤⣴⣶⡟⠉⢻⡷⣦⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣾⣦⡀⠀⠁⠀⠀⢀⣴⣿⣿⣡⣿⣉⠞⢿⡿⡻⢿⣿⠉⢿⡏⠀⠀⢿⣿⠟⠁⢀⡾⠤⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠛⢿⣿⣷⡾⠿⢿⡟⡉⣿⣷⣾⣟⣿⢻⣷⠀⢻⡅⢺⡇⠀⠀⢸⠁⢀⣤⠞⢀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠔⠉⠀⠀⣀⣀⣘⣿⣧⣤⢿⡟⣿⣟⣟⣛⣿⣤⣦⣍⠁⠂⢀⡾⠖⠋⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠃⢀⡀⠀⡀⠸⣿⣋⡼⢷⣟⡏⣹⣿⠿⡛⠛⠁⠀⢀⡞⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠈⠛⣿⣿⣬⠿⡏⠁⠀⠁⠀⢓⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢷⡀⠈⠳⡖⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢻⡷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠋⠉⠉⠁⠀⢐⣤⢴⣤⣀⠀⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢀⣯⣀⡴⣤⡈⠃⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣿⣤⡍⢡⢛⣋⡇⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⡇⢿⡼⢤⣌⣿⡇⡀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢼⡇⢀⣸⣇⣸⣿⣦⣿⣿⢁⠈⡤⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡏⡿⢯⡻⣿⣿⠇⢇⠎⠞⣵⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⢀⡙⠾⣿⣼⣯⣾⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢻⡎⣿⣿⣦⡀⠙⢿⡿⣯⣲⣶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢸⣷⢸⣿⣞⢿⣆⠀⡀⢈⠉⠻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⣿⣿⠀⣿⡄⣿⣿⣆⠹⣷⣼⣶⣷⣄⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡅⣿⣿⢹⠀⠘⣧⢹⣿⣿⣦⡘⢧⠉⡛⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣇⡟⣿⠈⢠⠀⠹⡆⠻⣿⣿⡳⣄⠱⣌⠃⢸⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢇⣾⣿⡇⢿⠀⣿⢰⠀⢻⣆⠹⣿⣿⣌⢓⢬⡛⢸⡇⠘⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣾⣿⢹⡇⠀⠀⣿⠸⣧⠀⢻⣷⡙⢏⢻⣿⣝⢦⢻⣧⠀⢹⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢰⡟⣿⢸⡇⠀⢸⣿⠀⣿⣆⠀⢻⣟⠦⠀⢿⣿⣮⢻⣿⠀⣼⡇⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣿⠘⠁⠀⢸⣿⡄⠘⣿⡄⢠⠙⣦⠀⠈⣿⠈⠋⠃⣴⡟⢇⢻⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿⠀⠀⠀⣿⣿⣇⠀⢹⣿⡀⢳⡄⠀⠀⣿⠀⢠⣾⢿⠁⠀⠈⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢱⠀⠀⠀⣿⡏⠹⡆⠀⢹⣧⠀⢻⣆⠀⠁⢀⣿⣷⡈⠀⠀⠀⠸⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⣆⠀⢻⡄⠀⠹⣷⣄⠘⢻⡿⣷⡄⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠘⠂⠀⢧⠙⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣏⢽⣟⠻⣿⢟⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⡀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠞⠏⠁⠀⡀⠸⠋⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠙⠻⠟⠋⠉⠛⠣⠎⠀⠀⠀⠀⢠⣧⠈⠀⢻⣄⣀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⡀⠀⢀⠀⠀⠀⣼⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⣦⠀⣸⣿⣂⡠⣦⠀⠀⢀⡴⠀⠀⠀⠀⢹⣿⣿⣦⣸⣦⡀⢀⠇⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣝⣿⣧⢻⡿⠉⠶⢹⡀⠀⢈⠟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡾⣿⡏⠀⢀
⠀⠀⠀⠠⣤⣤⣴⠤⣾⣏⠈⢄⠀⠘⠁⠸⢿⣿⡿⠿⠁⠀⣠⣷⡾⣄⣠⣀⠋⢀⣀⣷⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢼⣿⡃⠀⢠⣾⣿⣿⠟⠛⢿⣿⠴⠆⣼⣿⣦⣿⣿⣿⣿⣿⣿⡿⢯⣈⣽⣿⢏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⠀⠀⢀⢿⣿⣿⣦⣤⣿⣷⡆⠈⢸⣿⣿⣿⣿⣿⠟⣿⣿⠀⠀⠙⠿⠛⣿⣿⣷⠀⢠⡔⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⠶⣿⣿⠉⠸⠿⣿⠓⠀⠈⠻⣿⢿⡇⠹⠢⢵⡤⠀⠀⠀⠀⠻⢻⣉⣽⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⣠⣶⣶⣿⣶⣦⣤⡆⠀⠄⢈⡀⢠⣶⣾⣿⣿⣷⣦⣄⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣄⣶⣏⣤⡄⠻⣽⠇⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣴⣿⣿⣿⣿⣿⣿⣿⣧⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠻⠿⠋⠀⢀⣀⠀⠀⣿⣟⣭⣹⠛⠻⠭⣽⣿⣿⣿⣿⣿⣿⣏⠽⠛⠛⣭⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⣥⣬⣈⣿⣿⣿⡟⠀⠀⠀⢈⣻⣿⣿⣿⣿⣿⣉⠀⠀⠀⠙⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣷⡶⠶⠤⣿⣿⣿⢹⣍⣀⣀⣨⣽⣿⣿⣿⣿⣥⣬⣀⣀⣤⣿⣿⣿⡏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⣿⣉⠉⣿⡿⣇⣿⣷⣿⣿⡟⠉⠀⠀⠉⢻⣿⣯⣴⣿⣻⣿⡏⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠟⠂⠉⠛⣿⣽⣿⠿⠉⠀⢀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠰⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⡿⠿⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣷⣴⣿⣵⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⢉⠑⠋⣛⠛⣦⣤⣀⠈⠒⡄⠀⠀⢠⠐⠀⣀⣠⡴⠞⠓⣉⠛⡉⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠐⣧⣞⣛⣙⣩⣋⣲⢢⡀⠓⡄⢸⠀⠀⡀⢀⡜⢂⡼⢒⣋⣭⣉⣛⣳⣦⠂⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠉⠉⠀⠀⠀⠀⠈⠑⢿⢠⡸⡌⠀⠀⠰⢆⡤⡿⠊⠉⠀⠀⠀⠀⠈⡁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣂⣰⢀⠀⠀⠀⢀⣀⣠⣽⠃⡅⢀⡀⢀⠊⣹⣤⡀⣀⠀⠀⢀⣠⣂⣀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠞⠉⠽⠟⠁⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠈⠛⠷⠍⠻⠄⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⠀⠀⠀⠀⢀⠤⠤⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠈⠀⠈⠉⠃⠀⠤⠄⠀⠀⠀⠀⠠⠤⠂⠈⠋⠀⠀⠀⠈⠉⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⡅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠿⣦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠉⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠐⡦⣄⠲⠦⠱⠶⢦⣀⡴⠀⢀⡄⠀⠀⢠⡀⠀⠤⢁⡴⠶⠬⠤⠶⢢⢴⠂⠀⠀⣲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠷⠈⠀⠁⠀⠀⠀⠀⠀⠀⠈⠳⣴⡀⠁⠀⠀⢈⢡⣵⠞⠁⠀⠀⠀⠀⠀⠀⠈⠑⠄⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠨⠄⠤⢀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠀⠀⠀⠀⠲⠃⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠥⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠀⠄⠈⠉⠐⠋⠁⠉⠌⠁⠠⠄⢘⡆⢄⡀⢠⡄⠀⠤⠌⠥⠉⠈⠙⠂⠉⠁⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⠨⠇⡸⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠌⠉⣀⡀⡀⠐⠢⠬⠅⠁⠀⠀⠀⠀⠀⠈⠀⠩⠥⠤⠒⢀⣸⣀⢉⠱⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡸⠉⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢄⣀⠀⠀⣐⠼⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⢤⡺⠄⠀⠀⠿⢠⡤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠴⠂⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠲⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠀⡀⠀⠌⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣃⣀⡀⠂⠠⠄⠀⠀⠀⠀⠀⠀⠠⠁⠄⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢘⠃⠀⡤⠶⠾⠛⠋⠛⠛⠳⢦⣀⠐⠀⢀⠀⡀⠆⣀⠀⠀⢀⣠⠶⠞⠛⠛⠛⠻⠶⠦⡄⠀⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠇⠱⡄⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⠀⠠⡇⢡⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⡰⠋⠷⠄⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡀⢈⠳⠄⡀⠀⠀⠀⠀⠀⠀⢳⠀⢠⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⠴⢊⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⢀⠁⠖⠒⠿⠯⠽⠻⠐⠟⠋⠀⠀⠀⠀⠀⠀⠙⠛⠐⠿⠬⠿⠷⠒⠦⢉⠈⠁⢈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠⠤⠄⡤⠤⣶⣶⣶⣶⣄⣀⣤⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣶⣲⡆⠈⠙⣿⣿⣿⣿⠛⣍⣩⣽⣉⣝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⡽⠃⠀⠀⠁⠙⣿⠿⢷⣿⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⡜⠃⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣍⠐⠁⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸⣿⠀⠉⠛⠿⠀⡀⠀⠀⠀⠀⠰⠿⠿⠷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⢀⣴⢣⢼⡇⠀⠠⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⣀⣤⠀⣽⣿⠟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⠃⠈⢻⡌⣡⠺⣿⡿⠗⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠐⠉⠀⡰⠧⡄⣦
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣴⣿⠀⣢⣾⣏⡛⠁⠠⠁⠰⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢘⠀⠑⢾⠆⠐⣑⣿
⠀⠀⠀⠀⢠⠃⠀⠀⢠⡞⠁⡟⣰⣿⣏⠛⠁⠶⠤⠄⠠⣴⡂⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢤⣼⣶⡶⠚⡏⠸⡟⠉
⠀⠀⠀⠀⠸⠀⠀⡰⠋⠀⠀⡃⣿⣦⡍⠀⠚⠒⠒⠂⠀⠄⠈⠈⠉⠀⠀⠀⠈⠉⠉⠉⠉⠛⠃⠂⠼⠀⠀⠀⠀⡼⠏⠀⢀⡆
⠀⠀⠀⠀⣠⠤⠚⡁⠀⠀⢸⢠⡿⠶⠤⡀⣛⣗⠀⡇⠀⡑⠦⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⡆⠀⢸⠁
⠀⠀⠀⠸⢆⠄⠠⠀⠀⠀⡸⢸⣿⡷⣡⠭⠈⡉⠀⠀⢆⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣀⡾
⠀⠀⠀⠰⠼⠀⡅⠀⠀⠀⡇⠰⣶⢤⣙⣤⡠⣌⠲⣀⠀⡅⢀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢹⡇
⠀⠀⠀⠀⣧⠄⣀⠈⠀⢰⠇⠰⣿⡙⠷⠈⠋⠈⠑⠚⠁⠋⢁⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⡀⠀⠄
⠀⠀⠀⠀⢹⠀⡇⠀⠀⢸⠀⠐⣿⡁⢜⠠⠀⠀⠀⠤⣐⠺⢿⠛⠉⠀⠚⠴⠦⠀⠀⡌⠀⠀⢀⣠⠄⠋⠀⠰⣿⡈⠁
⠀⠀⠉⠁⢸⡀⡇⠀⠀⡄⢸⣦⣌⠳⠠⠀⠁⣿⣿⡦⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⣁⣴⡖⢀⡰⣄⠀⠗
⠀⠀⠀⠀⠘⡇⢀⠀⠀⠇⠘⣿⣿⣷⡀⠀⠀⣼⣿⣟⠀⠐⠀⠀⠀⠀⠀⠀⣁⣤⡀⠶⠚⠛⠿⠁⠈⠉⣤⠀⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠹⠀⡀⢸⠀⠀⢸⡿⠓⢄⡈⠀⠘⡷⠛⠀⣠⣦⡤⠴⠒⠋⠉⠁⠀⠀⢀⣠⣤⣶⣶⣦⣼⣿⠁⣜⣷⣾⠃⠀⠃
⠀⠀⢀⣤⠶⣷⠂⠙⣿⢠⣾⣿⡷⠀⢸⣧⣠⣆⡐⠆⠘⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⢸⡇⠀⠀⠈⠉
⠀⣄⠀⠀⠀⠈⠆⠐⠋⠈⠛⢿⣧⡀⣠⣈⠉⠉⠉⠲⠲⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⣿⣷⡀⢤⣀⠀⠀⡄⠀⠀⠀⠀⠈⠉⠙⠻⣿⡶⢤⣄⡀⠀⠈⠹⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀
⠀⣿⣻⣧⠻⠿⠘⣿⢆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠈⠉⠙⠒⠦⢤⣄⡀⠀⠀⠀⢀⠀⠀⠀⠀⣾⢠⠃⠁
⠀⣿⣟⢿⠉⠁⢸⡇⢸⣄⣦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⢇⠀⣀⠑⠎⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⢀⡆
⠀⢽⣾⣧⡀⠀⢸⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⡀⠃⡈⠉⠒⠃⠀⠀⠀⢸⠁⡄⠀⠀⠀⠀⠀⢸⣷⣄
⠀⠈⠉⢿⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⡠⢖⣩⠔⠉⣤⠶⠶⠀⠀⠀⢠⠏⠀⠁⠀⠀⠀⠀⠀⠒⢹⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣸⠀⢀⠴⣊⡬⠞⡋⠄⠒⠉⠹⣆⠠⠀⡀⠀⡷⡄⠴⡀⠀⢀⣦⡀
⠀⠻⣿⣿⣇⢠⠇⢀⣷⠀⠀⠀⠸⠧⠀⢀⡀⢀⣠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⡠⠺⢿⣿⣆⠈⡄⠉⠛⠻⠷⠶⢦⠤
⠀⠀⠀⠈⠉⢸⠀⣿⣄⣀⡀⠠⠀⢀⡚⢁⣄⠉⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠦⢭⣝⣧⣈⠠⠄
⠀⠐⢤⡀⠀⡇⠀⠸⣿⣿⡏⠘⠛⠿⣇⠀⠈⡏⠀⢈⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠘⠔⠂⠠
⠀⠀⠀⠙⠲⠀⠀⠀⠀⠙⠻⢶⣤⣤⣠⣦⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠷⠶⣧⠤⠦

⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⡐

⠀⠀⠀⠀⠀⠐⠂⠀⣉⣈⣀⣁⣈⣀⣐⠰⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠶⢐⣀⣉⣀⣁⣈⣈⡀⠀⠖
⠀⠀⠀⠀⠀⠀⢸⣏⡭⠀⢀⠀⠀⠈⡉⠉⡙⠒⠤⣅⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣀⡥⠔⠚⡉⠉⠈⠀⠀⣀⠀⠈⣍⣿
⠀⠀⠀⠀⠀⠀⣼⢸⢷⣥⡀⠃⡀⢈⠂⠘⢏⢳⡌⢸⠻⡲⣔⠂⡄⠀⠀⠀⠀⠀⠀⡄⢂⡵⡺⢻⠀⣔⢋⠟⠀⢊⠀⡆⠃⣠⣥⢿⢸⡄
⠀⠀⠀⠀⠀⢠⡏⡾⣾⠈⣉⠛⠛⢟⠛⠟⢻⠷⢿⣮⣆⣾⠈⣵⣄⠃⠰⠄⠴⠘⢃⣴⡅⣸⣇⣮⣾⠷⢿⠚⠟⢛⠿⠛⢋⡉⢹⡾⡆⣧
⠀⠀⠀⠀⠀⢸⣾⢁⢠⡀⢩⡀⢻⣀⣠⣤⣴⣶⣶⣦⣬⡙⠷⣧⣨⡹⠶⢖⠶⠾⣩⣁⡷⠟⣩⣤⣶⣶⣶⣤⣤⣁⣨⠃⣨⠁⢠⢀⢹⣼
⠀⠀⠀⠀⠀⠈⢿⣤⣌⡙⠶⠿⠿⢿⣿⠉⠀⠀⠀⠀⠈⠙⠳⡈⠻⣟⢐⠉⢑⢉⡿⠋⡰⠛⠉⠀⠀⠀⠀⠈⢹⣿⠿⠿⠷⠞⣉⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣽⣶⢍⡤⠀⠀⠀⠻⢧⣄⡀⠀⠀⠀⠀⢀⢹⣼⠈⢯⣧⣯⠏⢸⣼⢁⠀⡀⠀⠀⠀⣀⣤⠿⠃⠄⠀⠠⣬⢵⣶⡍
⠀⠀⠀⠀⠀⠀⠀⠀⣟⢳⡄⡠⠐⠀⠠⠀⠭⡉⠛⢋⠙⣡⣤⣞⡡⡾⠃⣇⡇⠻⡧⣛⣮⣤⡙⢉⠛⠋⡩⠅⠠⠀⠐⡀⠄⣴⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠹⣄⠰⠉⡝⡖⣶⣿⡤⣿⣿⠭⠑⠒⠙⠒⠠⠖⠦⠐⠚⠓⠒⠩⢽⣿⡧⣼⣳⡖⡞⡍⠱⢀⡼⢡⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⢤⣬⣭⠤⠶⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠳⠦⢬⣭⣤⠴⠶⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣔⣶⣒⣒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⢉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣽⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠋⣿⣏⡉⡺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠛⣿⠀⠈⣹⡏⠀⠀⢀⣀⣤⣴⡶⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣧⣴⣿⣿⡄⠘⠻⠶⠶⠾⣋⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣀⣀⣠⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢛⢻⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⢿⠈⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣿⣛⠻⠿⢿⣭⣽⣭⣿⠣⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢻⡀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠻⡉⠑⡀⣤⣤⣤⣤⣷⠀⠀⠀⠠⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠈⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣛⡁⠀⢀⣿⣶⣿⣿⣿⣿⣿⣿⢰⣷⣦⡀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣆⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣿⣾⣷⣸⡟⠻⣦⡀⠀⠀⠀⠀⠀⡜⢠⡀⣽⣦⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠻⣆⠀⠀⠀⣼⢁⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡄⠈⣻⠓⠾⠁⣠⣷⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣷⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⡀⠀⢀⢀⠄⣀⣄⢀⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⣙⠶⣿⠃⢾⡿⠫⢈⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⡇⠶⣄⠈⢁⠀⠚⢣⣿⣿⣿⣿⠉⠛⠿⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠻⢢⣼⣿⣯⡐⠒⣸⠏⢝⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣴⣂⣱⣾⣾⣿⠀⠀⠤⠀⠙⠃⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣤⣶⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⡇⠹⢿⡇⠹⠈⡏⠏⠿⠿⠙⠿⠿⠿⠙⠏⠁⠈⠁⢠⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣜⠱⠟⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣷⢿⡄⠀⢸⣿⣯⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⣿⣿⣿⣿⢹⡆⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⣿⣿⣿⣿⡏⣾⠁⠀⣰⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣸⣿⣻⡟⣿⠇⣯⢀⣼⣿⣿⠟⠻⣿⣷⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⡏⣿⣿⣿⡿⠀⢻⡟⠻⣿⠁⠀⠀⠈⢳⣿⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠘⣿⢀⢿⣿⣿⡇⡷⢸⡇⠇⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⡇⡂⣿⢸⢸⣿⡿⢸⡇⣾⢿⠀⡆⠀⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠃⢸⡟⢸⣿⣧⣿⠃⣿⢸⢸⡇⠀⡄⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣧⠀⠈⡗⠘⣿⢹⡿⢰⡿⢸⢸⡇⠀⡄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢰⠀⡇⡆⣿⣸⠇⣾⡇⡆⢸⡇⠀⡿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⣿⢸⠀⡇⡇⣿⠛⣸⣿⠇⡇⢸⡇⠀⣷⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⢸⠀⡇⡇⣿⡄⣿⣿⢠⡇⢾⣧⠘⡟⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⢸⠆⠃⡇⣿⠀⣿⡟⢸⡇⣼⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠙⠃⠀⠀⠀⠀⠀⢠⣿⠀⣿⢸⡃⠂⡇⣿⢰⣿⡇⢸⡇⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⠆
⠀⠀⠀⠀⠀⠀⠒⠲⠔⠐⠒⠒⠲⠛⢢⡆⠀⠀⢠⣿⠛⠀⣿⢸⡇⠁⡇⣿⢸⣿⠁⣾⣿⣿⣇⡄⣿⣄⠀⠀⠀⠀⢲⠛⠔⠂⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠐⠶⠀⠀⠂⠀⣨⢏⣾⢰⡇⢸⡇⡄⡇⣿⢸⣿⠀⣿⣿⣿⣏⣽⣻⠏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒
⠀⠀⠀⠀⠀⠓⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠈⠁⣸⣧⢸⡇⡇⡇⢻⢸⣿⠀⣿⡿⣻⣿⢸⠷⠀⠀⠀⠀⠀⠀⠀⠛⡤⠀⠀⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠛⠂⠀⣼⠏⠀⠀⠀⠛⣿⢸⣧⡇⡇⣸⠸⣿⠀⣿⡇⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⢲⡆⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠛⠛⠈⢙⣿⢇⠀⠀⠀⠀⠀⡹⣸⣏⡇⡇⢹⠀⣿⠀⣿⡇⣿⣿⠀⠴⠆⠀⢀⠀⢠⡀⠀⠀⠀⠉⠀⢘⡁
⠀⠀⠀⢀⠀⠛⠒⠒⠒⠋⠁⠀⠀⣴⢻⠏⠀⠀⣀⠀⠀⢠⡀⣿⡀⡇⡇⢸⠀⣿⠀⢹⣷⢸⡿⡇⠀⣠⠈⠛⢀⣤⠉⢁⠄⠀⠒⠞⢋⣡⡖⢀
⠀⠀⠀⡄⠀⠊⠠⠴⢶⠶⣤⡤⢾⣷⣾⠀⠐⠚⠛⢁⡀⢸⡇⣿⢰⠇⡇⣼⠀⣿⡇⢸⣿⡘⣧⣿⠀⠀⠒⠒⣿⣀⣀⣀⠐⠲⠶⢶⡞⠉⢛⣉
⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⣠⠀⠸⢿⡿⠀⠀⠀⠀⠀⢀⣾⢹⣿⢸⠀⡇⣿⠀⣿⡇⠈⣿⡇⢻⣾⣷⡀⠴⠊⠉⠙⠳⣿⡿⠷⠶⠚⠛⢱⠿⠛
⠀⠀⠀⠀⠤⠄⠁⠤⣤⠤⠤⠆⠀⣸⣇⠀⠀⠀⠀⠀⣼⡟⣸⡇⢸⢠⡇⣿⠀⣿⣷⠀⢿⣿⠈⣷⣹⣷⡀⠀⠀⠀⠀⠛⠓⠀⠀⢀⣠⣧⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣴⣶⣜⣣⡏⠀⢀⣀⣠⣾⡿⢁⣿⠁⡏⢸⠁⣿⠀⣿⣿⡀⢸⣿⣇⠸⣷⡹⣷⣔⠀⡀⠀⠀⠀⠈⠻⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠈⠉⠉⠀⢺⣿⡏⠈⠁⠀⠸⢿⣿⣿⣟⣡⣾⡟⢸⠃⣾⢀⣿⠀⣿⣿⣇⠀⢿⣿⣆⢹⣷⣽⣿⣷⡤⠆⠀⠉⠐⠦⣾⣶⡤⠐⠊
⠀⠀⠀⠀⠀⠄⠀⠀⠄⢸⣿⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⣠⠇⢰⡇⢸⣿⡄⢿⣿⣿⡄⠘⣿⣿⣾⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢠⣿⠏⠠⣿⣶⣏⣴⡿⢀⣾⣿⡇⣸⣿⣿⣿⣤⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠀⠀⢠⣿⠃⢠⣿⠏⠉⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡿⠁⣰⣿⠃⠀⠀⠰⠟⣿⣿⠛⠛⢻⣿⣿⣿⡁⠈⢿⣦⠀⠙⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣴⢤
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⡠⣯⠟⢀⣼⡿⠃⠀⢀⣶⣤⣤⣿⣿⣦⣾⣿⣿⣿⣿⣷⣄⠀⠹⣿⡄⠀⠈⢻⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡿⠃⢐⣏⡹⠀⠀⠀⢸⣿⣿⣿⢻⡏⠀⣤⣈⡿⣿⣿⣿⣿⠀⠀⠘⢏⣵⠀⠀⠈⢹⣛⣦
⠀⠀⢀⢀⡤⠀⠀⠀⣔⠛⠿⠉⠀⣰⣛⡏⠁⣀⠀⠂⠈⢁⣽⡿⠼⠷⠠⣿⢻⡀⠀⠨⠀⣀⣤⡄⠀⠀⠹⡛⣆⠀⠠⠈⠩⡟⢢⡀⠀⠉⠛⠀⠐
⠀⠀⠈⡜⠀⢀⠴⣾⠟⠋⠀⢀⠶⣿⠉⠀⣾⣿⣷⣦⣰⣾⣿⣄⣠⣤⣴⣾⣿⠟⢲⣾⣿⡿⠋⢀⣶⣦⡀⠈⠿⠛⣆⠀⠀⠀⠉⢻⠟⠀⡀
⠀⠀⢀⣤⣤⡿⠖⠋⠀⠀⣠⣾⠟⠂⠀⣠⠟⠻⠟⠋⣿⣿⣿⣿⣿⡦⣀⣸⣿⣀⣀⣼⣿⣿⣿⣿⡿⠟⠓⠀⣄⠚⢿⣤⡀⠈⠁⠈⠀⠶⢿⣤⡄
⠀⠀⣾⡀⠋⠀⢐⠂⠀⢾⣤⠋⠀⠀⣴⣯⣤⣄⣠⣴⡿⠟⣿⣿⣍⣀⠹⣿⣏⢉⡛⢻⣿⣿⣿⣧⣤⣤⢀⣀⣤⣷⣞⢀⣼⡀⠀⠐⠀⠀⠈⠳⣀⡀
⠀⠀⠈⠀⠀⠉⠉⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠀⠉⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⢒⣊⣉⣩⣭⣭⣥⣤⣤⣤⣤⣤⣭⣭⣍⣉⡁⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⣉⣭⣤⡶⠶⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣊⣩⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠤⠴⠶⠒⠒⠶⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣈⠉⠓⠢⣤⣀
⠀⠀⠀⠀⣤⣤⣴⡞⣩⣴⠾⠛⠉⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠤⠭⢷⣦⡄
⠀⠀⠀⢺⣏⢿⣿⣷⣉⠀⠀⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡀
⠀⠀⠀⠀⠻⡾⢿⣿⠉⢷⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⡼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣟⣾⣿⣿⠟⢋⣤⣶⣶⣶⣦⣌⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡸⣿⣇⠀⠀⠀⠀⠀⢰⣿⣿⡿⢛⣡⣾⠟⠉⠀⠀⠀⠈⠙⢿⣦⣌⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣿⡆⠀⠀⠀⢀⣿⣿⢏⣴⣿⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣿⡀⠀⠀⢸⣿⢡⣾⣿⡟⢰⣿⣇⣰⣤⡄⠀⣤⣆⣸⣿⡧⢻⣷⡌⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣷⠀⠀⢸⠉⣾⣿⣿⣧⠈⠻⠿⠿⠟⠀⢀⠛⠿⠿⠿⠁⣼⣿⣿⡈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣿⣧⠀⢸⠀⣿⡟⣿⣿⣷⣄⡀⠀⢠⣿⣿⡆⠀⢀⣀⣼⣿⣿⣿⡇⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⡆⠸⡇⢸⣿⢹⣿⣿⣿⢸⡀⠀⠁⠈⠁⣰⣯⣸⣿⣿⣿⣿⡿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢿⣿⡄⢱⠈⣿⡌⣿⣿⣿⠛⢿⣧⣤⣄⣥⡿⠋⣿⣿⣿⣿⣿⠇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⢿⣷⣈⣧⠘⣧⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣋⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡜⣿⣿⣿⣆⠹⣧⠹⣿⣽⣿⣶⣾⣷⣶⣾⣿⡿⢛⣡⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠹⣿⣿⣿⣦⠙⣇⠹⣿⣮⡻⣿⣿⣿⡟⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣷⡈⢧⡘⢿⣿⣿⣿⡿⣴⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣦⣳⣌⠻⣿⣿⣷⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣩⣭⣭⣽⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⡀⠸⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠟⢛⣛⣛⣧⡼⣿⣿⠿⣿⣿⣿⣿⣷⣄⡹⣿⣿⣿⣧⠀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢁⣴⣿⣿⣿⣿⣿⣧⢻⣿⣷⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢠⣿⣿⣿⡿⠿⠛⠟⣩⣌⠿⠛⢠⡆⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⢠⣶⡿⠀⢉⣠⡴⣾⠿⢁⢸⣿⣿⣿⣿⣿⣿⣏⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣏⠀⠈⢋⠉⢠⠴⣶⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢿⣿⣿⣤⣄⣀⠀⢒⢸⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠸⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⠜⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠸⣿⣿⣿⣿⣿⡿⣱⣷⠹⣿⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣿⣿⣿⣿⡇⣿⣿⣧⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⣿⣿⡿⠂⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⣸⣿⠁⠘⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢇⣿⠃⠀⠀⠁⡼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⡼⠃⠀⠀⠀⠀⠠⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠿⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⢿⡿⢿⠿⢿⠿⠋⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⢻⠟⠛⠻⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿
⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣴⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣄⡀⠀⠀⠸⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⢲⣿⣿⣿⡵⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⡀⢀⠴⣶⣀⣿⣾⣇⣶⠂⢈⠏⢙⣀⡃⠀⢹⣆⠀⢄⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡽⠀⠀⣘⣿⣿⣿⣿⣿⣟⡛⠣⠀⠈⡇⣠⣾⡟⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣡⡀⠀⠰⠁⠀⠸⠉⠙⣻⣿⠿⣿⡟⠛⣆⣈⣠⣿⣿⣿⣷⣿⣿⣿⡿⠃
⠀⠀⠀⢠⠀⢀⠀⠺⠀⠀⢀⡴⠊⠁⠀⠀⠀⠎⠙⢕⠢⠀⢠⠂⢢⡀⡏⡀⢸⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠉⢱
⠀⠀⠀⠀⢳⣼⣼⣄⣀⠠⠊⠀⠀⢀⡠⠐⠀⠀⠈⠈⠑⠄⠸⣧⣼⣄⢀⣀⠈⠻⣿⡿⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠻⢿⣿⠶⠊⣠⣶⡴
⠀⠀⠀⠀⣻⣿⣿⠯⡀⠀⠀⠀⡠⠁⠀⠀⠀⠣⠤⠘⢶⣴⣾⣿⣿⣿⣿⣿⣷⣶⡋⣀⣼⣿⠿⣿⣟⢿⣿⣷⣿⣿⣆⣀⠉⡗⢺⣿⣿⣿⡅
⠀⠀⠀⠀⠙⢫⣿⡶⡤⣤⣀⣈⣤⡤⢶⣶⣶⣶⣦⣤⣀⣾⣿⣿⣿⣿⣿⣿⣷⣤⢟⣛⣿⣥⣶⣿⣿⣄⣹⣿⡧⣹⣿⣿⠋⣠⠟⠿⣯⠃⠐
⠀⢀⡀⠀⠀⢸⣿⣿⣿⣞⢿⣿⣿⣿⡿⠿⠿⣿⣽⣿⣿⣯⣸⣿⣿⡟⢹⣧⣤⣶⣭⣽⡾⠿⠿⣿⣿⣿⣿⡟⣵⣧⣿⣿⡇⠀⠀⢀⡐⢀⡆⠃
⢰⡋⠈⠀⠀⠸⣿⣿⣿⣿⣦⡛⣻⣷⡿⠿⠷⢿⣿⣿⣝⣻⣿⣧⣼⣿⣿⣟⣻⣿⣿⡿⠼⠋⠙⠛⢛⢻⣿⣿⣿⣿⣿⣿⡇⠑⢦⣤⣴⠟⣽⢣
⠈⠳⣄⣀⠀⠀⣿⡟⣿⡛⢿⣄⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⢉⣩⣿⣩⣽⣿⣿⠁⢀⡐⣉⢷⡾⠹⡜
⠀⠀⠀⠀⠀⠀⠘⢷⣭⡼⣷⣿⣷⣄⡀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣯⣿⣶⣄⡀⠀⠀⠀⠀⣠⣴⣿⣿⣿⢦⣭⣿⠃⠀⠀⠑⡲⢺⠁
⠀⠀⠀⠀⠀⣇⠀⡘⢿⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⣶⣿⡿⠿⢶⣿⣶⣽⡿⣿⢿⣿⣿⣿⣿⣿⢻⣿⣿⢈⡿⠁⠀⠀⠀⢰⢁⡇⢀⣂⠔
⣴⠀⠀⠀⢰⠈⠉⠁⠀⠓⢭⣯⢼⣯⡿⣿⣝⣛⣷⡿⠟⠋⠁⠀⠀⠀⠈⠙⠻⢿⣮⣛⣛⣿⡿⣯⣿⢮⣽⡽⠊⠀⠀⠀⠀⠀⢸⣼⠀⠏⠉⠑
⠻⣤⣀⣤⠏⠀⠀⠀⠀⠀⠀⠉⠃⠀⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠂⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⡿⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡐⢚⡆

⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠠⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⢈⠁⡀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢯⣀⣨⠇⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠔⠀⠀⡀⢉⠀⠀⠀⠀⠄
⠀⠘⢶⣤⣀⡀⠐⠤⠀⠀⠀⠀⠀⠄⠀⠈⠢⠀⠀⠀⠀⠠⠀⠀⣎⢉⣎⣿⣇⣎⢉⡆⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⣀⣠⣴⠞
⠀⠀⠀⠙⠻⢿⣿⣟⣿⡶⠤⣤⣴⣶⣶⢶⣶⣦⡴⠦⣄⣀⡀⣠⠜⠿⠏⠀⠈⠟⠟⢤⡀⣀⣀⡤⠶⣤⣶⣶⣶⣶⣤⣤⣀⢴⡾⣿⣻⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣾⣦⣽⣿⣯⣿⣿⣿⣥⣴⣂⣀⣚⡏⡛⣷⣄⣰⠛⢳⣀⣴⡟⡫⡟⣃⣀⣢⣤⣽⣾⣿⣿⣿⣿⣥⣶⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠝⡙⢿⣷⡏⠀⠀⠀⠀⠀⠀⠉⠙⠯⣿⠀⡹⠉⠻⡀⢹⡿⠓⠉⠁⠀⠀⠀⠀⠀⠈⣻⣿⠟⡉⠽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡗⡟⢛⣝⡳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡤⠛⠣⣬⡋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣛⣝⠓⡗⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢚⡛⡮⡷⣭⣟⣻⢻⢶⢶⣶⣶⣾⣿⣿⣧⣀⣠⣷⣿⣿⣶⣶⢶⢶⢾⢻⣙⣯⡷⡯⣞⣛⢶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⣝⠶⢭⣲⣩⠻⢏⣟⣯⣽⡿⠟⠋⠁⠀⠉⠁⠀⠉⠛⠿⣿⣭⣏⣏⠿⢫⣱⣪⠵⢎⣭⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣝⣛⣛⣯⣽⠷⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠯⣭⣟⣛⣛⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣟⣁⠅⣀⡀⢠⡖⢀⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⣇⢸⠗⠁⣞⠀⡟⠈⢼⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⡄⠁⠀⠈⠀⠁⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⡄⠀⠀⠐⡲⠀⠀⢀⡿⠀⠈⢠⢖⡄⣴⠴⢠⣆⣴⢀⡄⣴⢀⢆⡴⣦⢠⠏⠀⠀⠰⣖⠀⠀⠀⢠⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠉⠁⠀⠀⠑⠁⠀⠈⠷⠤⠂⠓⠘⠀⠃⠀⠚⠁⠃⠘⠁⠿⠀⠘⠐⠃⠸⠀⠀⠀⠘⠊⠀⠀⠈⠡⠄⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢰⣟⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠃⣀⣴⡯⣀⣼⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⣿⣿⣿⣿⡀⠘⣛⣯
⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠙⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣤⣤⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣷⣶⣤⣤⣤⣴⣾⣿⣿⡗⢻⣿⠛⠀⣬⣹⠿⢿⣟⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⢹⠀⠘⠇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⢀⣰⣤⡄⣄⡤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⢠⣤⣆⡀⡀
⠀⠀⢿⡏⢀⡶⣹⠟⠛⢿⡔⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⡿⠛⡻⣏⢶⡀⢹⡿
⠀⢂⢘⣴⢉⣉⣥⡎⣼⢳⣧⡖⠙⠳⡂⠤⣠⣀⣀⠀⠀⠀⠀⢺⠁⠈⡌⠀⠀⠀⠀⣀⣠⣀⠤⢐⠖⠋⢲⣼⡞⣧⢱⣬⣋⡙⣦⠃⡐
⠀⢎⠀⢙⣾⣖⠻⣧⠠⣸⠸⡦⣬⡴⢃⠀⡠⠼⢛⠿⠷⣄⡤⢰⢨⡅⡇⢤⣠⠾⠿⡛⠧⢄⠀⡘⢦⣥⣴⠇⣃⠄⣸⠟⣳⣷⡛⠀⡱
⠀⠀⡟⠁⢈⡍⡿⠮⡷⣽⣦⣙⣲⣶⣶⣚⣅⣈⡸⠤⢵⠈⢳⣽⡡⢌⣯⡞⠁⡮⠠⠇⣁⣨⣓⣶⣶⣖⣋⣴⣯⢾⠵⢿⢭⡡⠈⢻
⠀⠀⠷⣤⣤⡷⠚⡏⢛⣛⢿⡋⠉⠉⠉⠉⠉⠉⠙⠷⣇⣁⠀⠈⡁⢈⠁⡀⣀⣸⠞⠋⠉⠉⠉⠉⠉⠉⢙⣿⣛⡛⢹⠳⠾⣥⣤⠾
⠀⠀⠀⠈⠉⢒⠖⢻⣉⣉⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⣗⠸⡆⡇⢸⢰⠏⣺⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⣩⣉⡛⠲⡒⠉⠁
⠀⠀⠀⠀⠀⠘⣌⣫⠉⢲⢾⣻⣾⡓⢦⡤⣠⣤⠤⢤⡤⣽⣆⣧⡰⢃⣾⣰⣯⢤⡤⠤⣤⣄⢤⡴⢚⣿⣿⡷⠶⠉⣝⣡⠃
⠀⠀⠀⠀⠀⠀⠊⣙⡏⢡⡎⢸⡧⠙⡯⣘⢴⡆⣷⣷⡒⠺⣿⠳⢜⡡⠞⣿⠕⡒⣿⣾⠱⡦⣻⢹⠋⣾⡇⢱⡌⢹⣋⠑
⠀⠀⠀⠀⠀⠀⠲⠩⢶⡅⠀⣸⢹⣆⡹⣌⢳⣼⠁⠺⠇⢘⡟⠀⠈⠀⠀⢻⡋⠸⠗⠈⣧⡞⣡⢗⢲⡇⣇⠀⢨⡶⠍⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢲⣏⣼⡇⡡⢽⣇⣿⠳⠦⠴⠋⠁⠀⠀⠀⠀⠈⠙⠶⠴⠞⣿⣸⡯⢌⢸⣏⣹⡖⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠚⢹⠃⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠘⡎⠓⠋⢀
⠀⠀⠀⠀⠀⠀⣤⠀⠀⣌⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣡⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢏⠠⠇⠀⢠⣠⣤⣮⠶⣶⣶⣶⣾⣤⣶⣖⡿⠀⢀⠁⠘⣗⢴⣦⣼⣶⣶⣶⡶⢮⣦⣤⣠⠀⠀⠧⢈⡇
⠀⠀⠀⢠⡜⣿⣾⠟⡯⠧⠄⠈⠐⠂⣄⢋⣉⢿⣿⣿⡧⣻⢈⢸⡣⣿⣿⣿⢏⣉⢃⡄⠒⠈⠠⠤⠯⡿⢟⣾⡟⣤
⠀⠀⠀⠀⢸⠉⡽⣧⣯⡙⠓⠒⠒⠒⠿⣬⣄⡀⠉⢿⣿⡝⠛⠛⣽⣿⠟⠁⣀⣬⡼⠗⠒⠒⠒⡚⣩⣣⡟⠍⢹
⠀⠀⠀⠀⢸⡦⣀⠙⣧⣱⡜⢿⡛⠋⠉⠉⠉⠛⢷⡘⣿⣧⣀⣠⣿⡟⣠⠟⠋⠉⠉⠉⠛⣻⠟⣴⣣⡟⢇⡠⢾
⠀⠀⠀⠀⠀⠑⢄⣈⠹⣏⢸⣗⠨⢂⢀⢀⡀⠀⠀⢻⣬⡃⠀⠀⣫⣼⠃⠀⠀⣀⣀⢀⠪⢑⣿⢈⣿⢉⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢿⡎⠨⢑⠒⡩⣟⣻⣾⣍⣈⢣⡻⡄⡼⣣⠋⣈⣽⣾⣛⡯⡑⣒⠩⠀⣾⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠦⣀⡑⣠⣿⣿⣿⣀⣿⠌⠁⠃⠃⠉⢼⣇⣸⣿⣿⣧⡒⣁⡠⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠿⠖⠛⠉⠀⠈⠓⠋⠀⠈⠙⠛⠾⠿⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣀⣠⣤⣄⣀⣤⣤⣴⣤⡑⢂⣀⡘⠛⠃⠀⠀⠀⠀⠀⣀⠀⠀⠐⠓⢰⡶⠶
⠀⠀⠀⠀⠀⠀⠀⢄⡀⢤⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠿⢋⡴⠫⠁⠀⠀⣱⢀⣄⣒⣈⣨⣷⣾⣿⣁⡴⠊⠀⠀⠠
⠀⠀⠀⣠⡤⠲⠋⠀⠀⣀⣨⣿⣿⣋⣤⣾⣿⡿⠿⠛⢉⣱⡖⣉⣀⠔⠲⠔⠚⣛⡋⠉⠉⠉⠉⣉⣩⡍⠙⠋⠐⠲⠤⢤⣤⡀
⠀⠀⠀⠉⠀⢀⡠⠶⢛⣿⣿⣿⡿⠿⠛⠉⣁⣴⡶⠞⠛⠉⢈⡉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⠋⠉⠀⠐⠲⠄⠠⣬⡈⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠠⠴⠾⣿⠟⣉⣤⡴⠾⠛⠋⢀⣀⣠⡴⠟⣠⣤⡖⣚⣷⣶⣶⣶⣾⣦⣶⣤⣤⣴⡾⣇⢀⡀⠀⣀⣀⠀⠀⠀⠀⠱⠀⠤⣤⣼⡷
⠀⠀⠀⠀⠀⠀⣀⠐⣠⣴⣿⡿⠋⣑⣡⡴⡾⠃⠀⢀⣀⣈⣩⣥⣴⣿⣿⡿⢿⡟⠛⠟⢀⠟⠁⠀⡀⠀⠈⠙⠛⠋⠙⠦⣤⣀⡂⣠⣿⣧⢰⣿
⠀⠀⠀⠀⠀⢀⢠⣾⣿⣿⠃⣠⣾⣿⣯⠔⠀⢀⣀⣬⡿⣛⣻⣯⣥⣤⣶⣶⣶⣾⣿⡇⣾⠂⠕⠀⠉⠀⠓⠲⠄⠀⠀⠀⠀⠉⠉⡁⣿⡿⠬⣿⡇
⠀⠀⠀⠀⠠⢮⣾⣿⣿⣷⣿⣿⣿⠏⠄⠀⠀⣈⢉⡽⣿⣿⣿⣿⣿⣿⠟⠿⠛⠻⠿⡛⠛⠒⢲⣤⣀⠀⠀⠀⠀⣀⠀⡀⠠⡀⠀⠀⢿⡅⢀⡟⠁
⠀⠀⠀⠀⣴⠿⣿⣿⣿⡿⢻⣿⡟⡰⠀⢠⡤⠟⣋⣴⣿⣵⣾⣿⡿⠟⣷⠞⠃⢉⣀⣨⣤⣄⡉⠙⠢⣝⣲⣦⡀⠉⠀⠀⠀⠀⠀⠀⣼⠁⢨⡇
⠀⠀⠀⣠⠃⢘⣿⣿⣿⣵⣿⣿⢀⠆⠀⢀⠴⠊⣽⣿⣿⡿⢟⣭⠄⢀⡤⠶⠟⢉⣉⠭⠥⠍⣛⠛⠢⣌⠳⣌⠙⢦⡀⠀⠀⣀⡀⣀⡽⣶⣿⠇⣀
⠀⡰⠛⠁⣠⡿⣫⣿⣿⣿⣿⣟⢸⠄⠀⢀⣴⣾⣿⣿⡟⣱⠏⢠⠞⠋⣠⠶⣞⠛⠛⠓⠶⢿⡏⠀⠦⣌⡑⢌⠓⠀⢧⠘⢠⣍⣹⢿⣣⣼⡇⢠⣿
⠀⠀⣠⠞⠃⣸⠟⣹⣿⣿⣿⣧⡌⠀⠔⠋⠉⠻⠛⢻⠀⢉⡤⠀⣠⣾⠉⢶⡌⠛⡘⠂⢤⠼⣧⠸⣄⠸⣧⠘⣆⠀⡞⠀⢠⣤⠀⢸⡇⡤⢂⣾⣿⠇
⠀⠈⠁⠰⠚⣁⣼⣿⣿⣿⣿⡏⡑⠀⠀⠀⠀⢀⣴⡇⠃⣸⠃⣰⠏⣙⡀⠛⡶⢲⣿⡿⢀⡠⢿⡄⠙⣇⠈⣇⠻⣸⣧⠀⠈⠃⠀⢸⡃⠀⢾⡧⠘
⠀⠀⠀⠀⢀⣤⣿⣿⣿⠟⣙⡈⠳⠂⠀⠠⠾⣿⢡⡗⢠⡟⣰⣿⣦⣬⣅⡘⠀⠛⠉⠐⠈⠁⠂⠳⣀⠉⠠⣿⠀⠁⡾⠀⠀⢠⠀⣿⠃⣿⡯⠓
⠀⠀⠀⠐⠋⢀⣻⣭⣶⣾⣿⣿⡟⣧⠀⠀⢠⠏⣼⡀⢾⣠⣿⣿⣿⣿⣿⡇⢀⡄⠐⠂⠀⠀⠀⠀⢿⣆⣐⢸⠦⡸⠂⠀⢀⡸⠀⡿⢄⣸⠁
⠀⠀⠀⠀⣺⣡⣿⣿⣿⠟⢋⣿⣇⡹⢇⢈⠃⣤⢹⡇⣾⣿⠙⠻⠿⠿⠋⣉⡀⠀⣾⣿⣿⣷⣤⡛⣸⡟⡌⣸⣤⡏⢰⣶⠈⠁⣸⠁⢀⡏⠀⣦
⠀⠀⣠⡞⠁⢹⣿⣿⣿⣧⣾⡿⣿⣶⣼⠏⣸⢃⣾⡇⣿⠁⢤⣦⡴⣠⣾⣿⣏⠀⢿⣿⣿⣿⣿⣟⡟⡸⢀⣿⣼⠁⡀⠀⠀⣀⣯⡀⣼⠁⠀⠘⠁
⠀⠐⠋⢀⡴⠛⣿⣿⣿⣿⠛⣿⢿⣿⠏⡼⠃⣼⠿⣧⢿⣷⠈⠁⣾⣿⣿⣿⡅⠛⡀⠛⠻⠿⠋⣹⡑⠀⣼⣿⡿⠀⣠⡄⠚⣿⡟⢀⣿
⠀⠀⠀⡿⠶⠾⢻⣿⣿⠇⣴⣿⣿⢃⡼⠀⣼⠁⠀⣿⣿⣿⣄⣃⡬⠁⠛⠋⠁⠚⠋⠀⠀⣋⣸⡃⢁⣐⠉⣾⢧⠚⠛⠛⣻⡏⠀⣾⡃
⠀⠀⠀⣠⠴⠾⢛⣿⣿⣿⣿⡟⣡⡞⢡⣼⡇⠀⠛⢿⣿⣏⡘⠛⠷⣶⣼⣄⣴⡿⠋⢓⣤⡞⡱⠁⣰⢋⣴⡟⠈⠻⠁⣰⡏⠀⢸⡿⢃⡀⠿⢷⡀
⠀⠀⠘⢡⣶⣾⣿⣿⣿⡿⢋⣼⠏⣠⣿⡿⠀⡄⢀⣀⣷⢌⢙⠓⡾⣴⣬⡷⠇⢘⣿⣿⠟⢈⣡⣾⡡⢛⠹⢦⣤⣄⣴⡟⠀⣰⠏⠀⠀⠀⠀⢠⡇
⠀⠀⠰⠟⠩⣿⣿⡿⠛⣡⡞⠁⣠⣿⣿⠁⠈⢀⣤⣤⢇⠀⠘⠓⡷⡶⠶⠟⣱⡿⢋⣥⡶⣿⠿⢉⡔⢯⣤⠚⠁⣠⠟⠀⣠⡟⢠⡄⠀⡴⢀⡹
⠀⠀⠀⠀⠀⣹⠟⠁⣴⠏⠐⢳⡟⢿⣿⣴⡶⢿⡏⠹⣷⣶⣤⣀⣀⣄⡠⡴⠏⠀⢈⣽⠟⡡⠖⠁⠄⠀⠉⣶⡼⠃⠀⣰⢋⣤⡟⠀⠰⠓⠉
⠀⠀⠀⢰⣿⢁⣠⡞⠁⢠⣶⡿⢻⣿⠛⢡⣶⣾⣧⡘⢻⠿⠛⣿⣏⣥⡤⠴⠞⢋⣩⣴⣾⣤⣶⡀⠀⣠⣾⡟⠂⣤⠞⠁⠸⣧⣤⢀⣀
⠀⠀⠀⠈⢡⡞⠋⠀⢀⣾⡏⠀⢸⠏⢸⣿⡁⢸⣏⣃⡞⠛⣛⣷⡤⠶⠖⢿⣟⢋⣷⣦⡼⠟⣿⣁⣹⠟⠁⢀⣼⡋⠀⢀⣀⡘⠻⠂⠸⢤⣆
⠀⠀⠀⢰⠏⠀⠀⢐⣾⠃⠀⣴⠋⢀⡼⢯⢿⣿⠏⢀⡿⣿⡛⢁⣞⢋⣐⣿⣿⣦⢉⣭⡶⣻⣿⣿⠏⠀⣠⣿⡟⠻⢶⣄⠉⠉⠑⢦⣄⢀⣈⡄
⠀⠀⠀⠀⢇⣠⣦⣾⠇⠀⢾⡿⠀⢼⡇⣰⣼⡟⡷⠘⣷⣽⣷⣮⠿⠿⠟⢋⣠⣤⣿⣽⣷⠿⢻⣇⡀⣰⣿⠛⠝⣤⣄⠙⣆⠀⠂⠸⣿⠉⠈⡟
⠀⠀⠀⠀⠀⣼⠿⠁⠀⢐⣿⠃⠀⣿⠀⣿⠜⢿⣧⣀⣿⡟⣷⣤⣶⠖⣾⠏⠛⣩⢮⣟⠫⣭⣓⢾⣍⣽⢻⣿⠀⣿⣿⠆⢸⡆⠀⠀⠈⠓⠊⠁
⠀⠀⠀⠀⣼⠁⢠⠶⠈⣤⡿⠀⢸⣏⠀⣿⣧⣼⢻⣿⣿⢿⢾⣏⣸⣆⣁⣤⠞⣩⣾⡟⢒⡾⠷⢆⢜⡥⣿⣧⠀⣿⡟⢸⡿⠀⣶⣂⠶⠶⠶⡄⠁
⠀⠰⠀⢘⡇⡀⠀⠶⣿⡟⠀⠀⣾⡟⠀⡿⣿⣿⠀⢿⣧⣼⣤⣿⣿⣿⠟⢁⣴⣿⣧⣼⠛⠻⣧⣠⢻⡅⣿⣿⠀⠈⢿⣿⣀⠀⠈⠁⠀⠠⣴⠇⣠⡤⡀
⠀⢠⠀⠨⣝⣳⣃⣴⡿⠁⠀⠀⣿⠍⠀⣿⡌⣿⠄⠘⣯⣿⣿⠃⠀⠉⠉⠛⠛⠻⣿⣯⠤⠀⣽⣿⢸⡇⠸⣿⣿⣄⡀⠈⠛⢉⠑⠒⠢⢤⣀⡀
⠀⢀⠀⢀⡾⡿⢉⣿⠃⠀⠈⢰⣿⠀⠀⣿⠀⣿⡆⠘⣿⡿⠻⠶⣦⣄⣀⡀⠀⢠⣿⠟⠀⢰⣿⣿⡏⣧⢰⣿⣏⢮⠙⣿⣷⣦⡻⢦⣤⠄⠈⢹⣷⣦
⠀⠀⠀⡸⢀⠀⢸⣗⣂⣤⣀⣸⡟⠃⠀⡯⠀⣸⡇⠀⠘⣷⡆⢀⡠⣻⣿⣿⣿⣿⠛⠀⠀⣸⣿⣿⣧⣸⡼⣿⣿⢸⡆⣿⠙⣻⣷⡀⠉⢻⣿⣿⣏⠉
⠀⠀⠀⡇⠀⠀⢿⡿⣿⣿⠇⣸⠇⠸⠀⣿⣉⣿⡏⠀⠀⠙⣷⣌⢷⠘⠟⠻⢿⣿⠀⠀⢀⣿⣿⣿⣿⣷⡶⠞⠃⣰⢇⡿⢀⡇⢸⣿⣄⡀⠛⢻⣿⡷
⠀⠐⠈⣷⠆⠄⡘⢠⣿⣿⣶⠋⢀⣀⠀⢼⡟⣻⣋⡀⠀⢀⣻⣟⠀⠌⠘⡄⠀⣏⠀⠀⣼⢿⣿⣟⠉⣀⣠⣴⣾⠏⠼⠁⣸⢗⠸⣿⡿⣿⣦⠀⠈
⠀⠠⠀⠁⠺⢴⣇⣼⡏⣿⡟⢀⡾⠋⣹⣿⣇⠘⣿⡇⠀⠈⣿⡿⢶⠆⡀⠀⠀⣿⠀⢀⡇⢸⣿⣿⡿⠋⠉⠙⠀⡠⠞⠃⠉⠑⢦⣀⣀⡠⢿⣻
⠀⠰⠀⠀⠀⣼⣿⣿⣵⡏⠀⠁⠀⠐⠿⣿⣥⣽⡿⠛⠀⠀⠈⠺⠧⣤⠑⠊⠀⠀⠉⠉⠀⠛⠙⠺⠶⠦⠔⠿⠋⠀⠀⠀⠀⠀⠀⠙⠛⠓⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⢠⠣⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⢣⠀⠀⠀⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠷⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠀⢀⠀⠀⠀⠴⠏⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠘⢦⣅⠀⠀⠂⠀⡆⠀⠢⡆⠀⠀⠀⠀⠀⠀⠀⠀⡬⠂⠀⡄⠀⠂⠀⠀⣥⠞⠀⢀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠤⠀⠙⠦⠀⠂⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠠⠞⠁⡠⠄⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⣝⣀⣉⣠⡤⠴⠚⠛⠛⠛⠒⠲⢤⠴⠒⠚⠛⠛⠚⠲⠤⣄⣈⣁⣘⡥⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠁⠀⢀⣀⣤⠤⣄⣀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⣀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢾⡉⠀⠀⠀⠀⠈⠑⣧⠀⢠⡟⠉⠀⠀⠀⠀⢈⡹⠆⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠓⠶⠤⠤⠖⠊⠁⠀⠀⠙⠒⠦⠤⠴⠒⠋⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣀⣀⣀⣀⡤⠤⠶⠒⠲⠦⠤⣄⣀⣀⣀⣀⡤⠖⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⠖⠉⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠁⢦⡀
⠀⠀⢸⠻⡄⠇⠠⠃⠀⠠⠐⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠠⡀⠀⠣⠀⠇⡼⢻
⠀⠀⢺⣆⡈⢓⡶⢂⠐⠆⣶⡀⢌⡓⣄⠀⡀⠀⠀⠀⠀⠀⣠⠨⢠⡀⠀⠀⠀⠀⠀⠀⣀⡔⣋⠄⢰⡆⠖⢀⠲⣖⠋⣀⣿⠂
⢠⠛⠉⡉⠙⢺⣧⡈⠂⠰⢈⣓⠨⢉⡙⠏⣢⣤⣀⡀⢀⠤⠸⣆⡾⠠⢄⢀⣀⣠⣤⠚⠟⣉⠁⢰⣋⠸⠂⠈⣠⣿⠚⠉⡉⠙⢣
⠀⠄⠈⣹⡧⠭⢽⣎⡏⢁⢀⠙⣶⢸⠀⠞⠠⡘⡉⠩⢺⡄⢠⠜⢬⠂⣼⠮⠉⡙⡀⠉⠆⢸⢲⡟⢁⣀⠉⣏⣾⠯⠥⣺⣉
⠀⠀⡈⢇⠉⠳⣧⠙⣇⢷⡶⠶⠶⠖⠚⠚⠲⠶⣄⣠⠀⡟⢿⣀⣸⠏⡅⢠⣁⡴⠶⠚⠚⠓⠲⠶⠶⣶⢇⡟⢩⡵⠋⢁⢏⡀
⠀⠀⠤⠺⠈⢀⣏⡁⠙⢟⢧⠀⠀⠀⠀⠀⠀⠀⠈⢷⣚⠿⠚⠨⠜⠺⢟⣰⠏⠀⠀⠀⠀⠀⠀⠀⣠⢟⠟⠁⣉⣇⡈⠸⠢⠄
⠀⠀⠰⠀⠀⠁⠀⠉⣘⢠⡿⡑⢦⣀⣀⣀⣀⣀⣀⠈⣷⢸⡟⠀⠘⣿⢰⡏⣀⣀⣀⣀⣀⣀⣠⠔⣹⣧⢘⡈⠁⠀⠁⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠺⠖⠚⡽⡿⠢⢄⣟⡉⢡⠈⡫⡄⠙⡇⢟⠾⢞⠇⡟⠁⡨⡋⢠⠉⣙⣏⠥⠾⡿⡝⠒⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡵⠃⠘⠲⠿⠒⠋⠸⣄⣉⡔⠀⠀⠙⠁⠀⠐⣌⣁⡼⠉⠓⠺⠷⠒⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢹⠂
⠀⠀⠀⠀⠀⢰⡄⠀⠀⠠⠑⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠡⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⠄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠊⠀⠀⣠⠀⢀⠴⢀⣴⣤⣀⡉⠓⠀⠀⠀⠘⢷⢶⠖⠐⣊⡉⠴⠒⠉
⠀⠀⠀⠀⠀⠀⣴⠁⠸⣿⣿⣦⡑⠀⢀⡀⠈⢹⡿⠯⣆⠉⠛⠛⠫⠁⠀⠀⠀⠀⠀⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⣿⣿⠇⢰⢿⣽⣶⣾⣿⣷⡇⠠⣁⠤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠾⣿⣿⠅⡀⠈⣸⣿⣿⣿⣿⡿⠁⠀⠀⠓⠀⠐⢢⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣘⠂⠀⠀⡜⠁⠀⠁⠀⣿⡆⠋⠄⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⣤⣄⠀⢠⢖⣹⠀⡀⠀⠄⠀⠀⣄⠀⠁⠀⢀⣐⠓⠄⠘⠂⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠈⢹⡟⡀⠀⠸⠋⣠⠽⡖⢀⢊⡄⣴⣹⣺⠃⣿⢿⣷⠀⠀⡀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⣾⣷⡁⠀⣀⣾⣧⠀⠀⠀⠘⠈⢻⣿⣿⣠⣿⣦⠹⣧⡇⠘⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣷⠀⢿⠃⠀⠁⠀⠘⠀⠀⢸⣿⣿⣿⠻⣟⠀⣿⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⢄⡀⢰⣄⠀⠀⠀⣿⣿⡇⠙⠟⠙⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⠀⠀⣸⠏⡋⠀⠀⢠⠸⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⠀⠀⣿⠇⠈⠀⠀⢈⠢⡟⠄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢺⣀⣼⠏⠀⢀⡀⠀⣸⠃⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣏⠀⠀⠈⣠⣞⡰⣇⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⠋⠀⠀⠙⠿⠆⢹⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠣⠽⠯⢿⣛⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠉⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⣼⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⡀⠑⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⣁⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⣤⣄⡀⠉⠙⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣡⠞⠋⠁⠀⠀⠀⢀⣴⣾⣿⣿⣷⣤⡀⠀⠀⠀⠈⠉⠛⠦⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⡵⠋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣼⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⡟⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠋⠉⠉⠙⠻⢿⣿⣿⣿⣿⣦⠀⢰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡇⢸⠇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢠⣤⣤⣤⣤⠀⠀⣴⣶⣤⣤⡄⠈⣿⣿⡇⣼⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⠇⢸⣿⣿⠁⣿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣀⠉⠛⠋⠀⡴⢦⠈⠙⠋⠁⣠⣿⣿⡏⢀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⣠⡄⢠⡄⣰⣄⣠⣿⣿⣿⡟⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏⠙⠋⠙⠋⣹⣿⣿⣿⠏⠀⠀⢸⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣇⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣯⣡⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⠧⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠘⢶⣶⣶⣖⢦⣴⣶⣶⣶⣴⣶⣦⡀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⡶⣶⢲⢶⢖⣶⣶⣶⠊
⠀⠀⠀⠀⠀⠀⠹⣿⣭⣻⣿⣝⣻⢮⣽⣿⢿⣿⣝⣧⣴⢠⣖⣤⣾⡷⣿⣯⡟⢿⣿⣿⣿⣿⣭⣞⣽⠃
⠀⠀⠀⠀⠀⠀⠀⢺⣽⣿⢯⣿⣿⠟⠺⠿⢶⣿⣿⣯⣿⣽⣿⣿⣛⣷⣯⠾⠿⠛⠷⣾⣹⣻⣹⡿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢲⢿⣻⣇⣿⣦⣀⡀⢀⣙⣾⢝⣟⣿⣹⢷⣟⣿⣁⠀⣀⣠⣾⢿⣿⡟⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣏⠭⣭⣿⣯⠿⣿⣿⣽⠟⠻⣿⢻⣏⣿⣝⣛⣛⣿⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣼⢟⣿⣧⣽⣿⣩⡾⠁⠀⠀⠹⣿⣿⣗⣯⣻⣿⣿⣿⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠋⠙⠛⠉




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣴⠆⣤⡌⡄⠀⠀⠠⠆⠀⢛⠀⠂⠀⡀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⣤⢠⣦⡀⢁
⠀⠀⠀⠀⠀⠀⠀⣀⠁⠄⠀⠀⠀⠈⠀⠀⠀⠰⠈⠀⡇⠀⠀⠀⠈⢠⡄⣰⠀⡔⢈⠰⠄⡇⢰⡈⡄⢀⠀⠀⠀⠠⢠⣿⡿⠈⢿⣷⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠠⠐⠁⠄⠀⠀⢣⠀⣀⡡⠄⢇⠸⢀⠃⣂⢉⠥⢢⠀⠰⠀⠀⠆⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢀⠔⠀⡠⠂⠀⠀⠀⠀⠆⡙⠡⠂⠤⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣄⠀⢀⠔⠁⠀⠘⠁⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠘⠀⢀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠈⠀⣠⠀⠀⠠⠆⠀⢀⠮⢀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠠⡀⠈⠉⠈⠉⠛⠶⢖⣤⣤⡄⢰⠆⢀⠧⣀⠀⠃⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⠀⠸⠿⣠⡾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠠⠠⠀⠤⠀⠄⠠⠠⠄⠐⠁⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⢸⢸⣻⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣀⢤⡤⢉⠀⡁⠀⠐⠁⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠿⡿⠿⠋⠀⠠⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠘⠀⣐⠶⠋⠀⠀⠑⠈⢤⣸⣷⠀⣾⡇⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠞⠛⠀⠛⠳
⠀⠀⠀⠀⠀⠀⢀⠁⣽⠀⠀⠀⠀⠀⠀⠄⡈⠉⠀⠉⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠲⠂⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠮⣁⠋⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⠂⠔⠀⠀⠀⠀⠈⠢⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢐⠄⡄⠀⠀⠀⣦⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠁⡠⠠⡀⠀⢀⣿⠀⠀⢨⡀⠆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⢦⠇⠀⣸⣇⡁⠀⡦⠖⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡼⢖⣢⠞⣑⠀⣣⡀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢰⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠔⠁⠂⡐⠚⠀⠈⠀⠈⠁⠀⠁⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠋⠁⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣀⡀⠀⢀⣀⡀⢀⣀⣀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠤
⠀⠀⠀⠁⠀⠀⡀⠐⠁⠀⠀⠠⠀⠤⠈⠀⠀⠀⡎⠰⠌⡟⢡⣌⢹⡏⣡⡌⢿⢁⡇⠀⠀⠀⠴⠆⠂⠀⠀⠀⠈⠄⠀⠀⠀⢀⠠⣇⡸⠌⣖⣀⡄
⠀⢀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠶⠆⠀⡄⢀⠀⣿⠰⠆⣡⠘⠿⢠⡀⠿⢃⣬⣸⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠁⠊⣤⣄⠀⣤⡄⠁⠀⠂
⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡰⠀⠊⠀⠘⠛⠛⠛⠻⠾⠛⠙⠓⠛⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠀⠀⠀⠀⢉⣁⣀⢉⡁⠀⠀⠐
⠀⠰⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠘⢤⡒⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠹⢿⡿⠥⠀⢀⠀⠀⠀⠄⢂⠘⠿⠿⠞⠁⡀⡠⠁
⠀⠀⠠⡂⠁⠀⣀⠀⠀⠀⠀⠁⠀⡀⠄⠀⠤⢀⠀⠈⠂⠀⠀⠐⠀⠆⢂⠉⠉⡀⠒⣀⡉⠆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠉⠓⠓⠐⠋⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠎⠁⠀⠀⢒⠀⢱⠀⠀⠀⠀⠀⠉⡀⠻⠃⠘⠟⢀⡀⢀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⠀⢀
⠀⠀⠀⠠⠊⠀⠀⠁⢀⠀⠐⠸⠂⣶⡅⠀⢴⣦⠀⡃⠂⠀⠀⣁⡑⢉⠴⢘⢃⡐⠣⡈⢀⡔⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡠⠀⠐⠄
⠀⠀⠀⡀⢰⡆⠠⡦⠀⠀⠀⠙⢆⡉⠀⠀⠐⢁⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢼⢠⠈⠀⠁⡄
⠀⠀⠀⠈⠐⣀⢂⡐⠁⠀⠀⠀⠀⠀⡩⠁⠙⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢁⠔⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡰⢨⠀⠀⠀⣀⠱⡀⠀⠀⣠⣶⣶⣶⣶⣶⣦⣄⠀⠀⠀⣊⠀⠀⠀⠀⠛⠃⠀⠀⢘⢀⠀⠀⠀⡌⠀⠀⠀⢌
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠁⠀⠃⠇⠀⠀⠀⠐⠂⠂⠀⢸⡇⡊⠇⡎⠋⡦⢙⣼⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡀⠄⠸⣒⠐⠒⣦⣘⠦⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠘⢷⣶⣷⣶⣷⣶⣿⠟⠁⠀⠀⠂⡀⠀⠀⠀⠀⠀⡠⠁⡐⠀⠁⠀⠐⠂⠈⠓⠁⠐⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⡀⠈⠢⠀⣽⠟⠉⠉⠉⠁⠀⠀⠀⠈⠈⠀⠂⢈⡀⠄⠊⡀⠆⡀⢂⠀⡀⠀⠀⠀⠠⠐⠀⠄⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠈⠁⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠀⢠⢂⠰⠁⠀⠀⠴⢀⣀⠀⣀
⠀⠀⡀⠐⠄⠀⠀⠀⠀⣀⠀⠄⠠⠀⠂⠀⠀⠈⠀⠀⠀⠀⠂⢠⠀⠀⠀⡀⢨⡘⡏⠎⠀⠀⠀⠀⠀⢔⠄⠀⠀⢀⠀⠁⠆⠀⠀⠜⠂⠒⠀⠀⠘⠦
⠀⠀⠱⠜⠀⠂⢄⣐⠂⠅⠀⠐⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠂⠁⢀⠂⠁⢑⡊⠁⢑⡒⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠾⠇⠀⠰⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡌⠀⠀⠻⠛⠤⠻⠛⠀⠀⠀⠀⠀⡈⠢⡐⣀⠆⡠⠀⠀⠀⠀⠂⡀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡄⠀⡀⠀⢀⠀⠀⠀⠇⠀⠀⠀⢶⣤⡺⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⣗⡈⠡⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠂⠂⠘⠐⠁⠀⠀⠀⠀⠀⡄⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⢀⠄⠀⢀⠀⠀⠀⢄⠀⠀⡀⠀⠀⠀⠀⠀⢄
⠀⠀⢱⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠁⢀⣄⣀⠀⣠⣠⠂⠜⠀⠀⠐⠁⠀⠀⠀⠁⢄⠀⠈⠁⠁⠐⡀⠔⠁⠀⡀⠀⠁
⠀⠀⠈⡆⠀⢠⠞⡁⠀⠈⢿⣿⡿⠁⠀⠀⠀⣠⠾⠻⣿⠿⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠈⠂⠀⠢⠰⠀⠀⠀⠀⠀⡥⠀⠂⠀⠋
⠀⠀⠀⢣⠀⣾⠘⠡⠘⠃⣸⣿⡇⠀⠀⠠⢠⠇⣴⣄⠀⢠⣶⠈⠇⢀⠀⠀⠀⢀⠄⢠⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠈
⠀⠀⠀⣬⣶⣿⣿⣤⣴⣾⣿⣿⣿⠿⠆⠈⠚⢀⠈⢥⣶⣤⠁⢀⠜⠊⠀⠀⠀⠈⠄⠰⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠈⠒⠄
⠀⠀⠀⠘⡍⢹⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠁⠰⠉⠑⠂⠀⠀⠀⠀⠀⣰⢊⣁⡎⢱⡆⠒⣈⠐⠂⣀⣰⣀⢀⣰⡄⠀⠀⠀⡀⠀⡀⢀⠀⢀
⠀⠀⠀⠀⢃⠀⢻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⠔⠀⠀⠀⠀⠓⡀⠀⠀⠀⣇⠸⠤⠁⢠⡄⠀⠻⠀⠶⠦⢈⣿⠸⡇⠀⠀⠀⡄⠌⠠⠑⠈⠴⠅⠠
⠀⠀⠀⠀⠘⡀⠀⠙⢿⣿⣿⢿⣯⠙⠷⠄⠀⠀⣠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠶⠶⠿⠾⠷⠿⠶⠞⠶⠶⠟⠿⠾⠁⠀⠀⠀⠏⠔⠀⠉⠉⠁⠢⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠃⠀⠐⠀⠀⠛⣦⣀
⠀⢠⡶⡶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠁⠀⠐⡀⡐⠀⠀⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⡶⣦
⠀⢾⡠⠀⠀⠘⢷⣤⡴⠶⠟⠛⠛⠛⠿⠷⠶⢦⣼⠷⠀⠀⠂⠀⢨⠀⠀⠂⠀⠰⢿⣤⠶⠶⠿⠟⠛⢛⠛⠷⠶⣤⣴⠟⠀⠀⠠⣸⠆
⠀⠘⢷⡄⠤⠖⠂⠁⡀⠃⠈⠁⢀⠀⠀⠀⠘⠀⠠⢉⠉⠐⠒⠂⠀⠀⠒⠐⠈⢉⠡⠂⠘⠀⠀⠀⠀⠀⠉⠀⠃⡀⠁⠒⠢⠄⣴⠟
⠀⠀⢸⡇⠉⢌⠁⠈⠀⠠⠤⣄⣤⡵⠶⢦⣵⣄⠀⠀⠀⠢⠀⡀⠀⠀⡀⠠⠂⠀⠀⢀⣴⣥⠶⠶⣥⣄⡤⠤⠀⠀⠀⢉⠄⠁⣿
⠀⠀⠀⣿⠀⠄⠀⠰⢥⣶⡾⠛⠁⠀⠀⠀⠀⠉⠻⣶⣤⡀⠠⠰⠀⡰⠀⠀⣠⣴⡾⠋⠁⠀⠀⠀⠀⠙⠻⣶⣦⠵⠀⠀⠀⢸⡏
⠀⠀⠀⠘⣧⠈⡀⠀⠀⢉⠛⠶⢦⣤⣤⣤⣤⣴⠞⠋⠉⠁⢀⠰⠀⠐⢀⠀⠉⠉⠛⢶⣤⣤⣤⣤⣤⠶⠞⢋⠁⠀⠀⢈⢠⡿
⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⢀⡔⠈⠉⠈⠐⣄⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠘⢷⣄⠁⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣠⡶⠶⠶⢦⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣀⣀⣀⠀⠀⢀⡀⠴⣈⣴⠟⠁⠀⠀⠀⠀⠙⢷⣌⡰⠄⣀⠀⠀⢀⣀⣀⣠⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣭⣬⣥⣤⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠦⣤⣭⣬⣵⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢴⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⣿⣿⣿⣿⣏⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⡀⠀⠀⢀⢀⡷⢠⣼⣿⣿⣿⣿⡟⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣾⣿⣿⣿⣿⣿⣶⣶⣾⣶⣦⣄⣒⣒⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⣿⢿⣟⣿⣿⣿⠿⢿⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠊⠙⣿⠛⠛⠛⢫⣿⣿⡇⠀⣼⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠣⠄⢀⣈⣓⠈⠳⣠⣴⣾⣿⣿⣯⣿⣷⡀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠚⠒⢀⣈⣭⣛⠷⡹⣿⣯⡙⠿⠭⠿⢿⣿⣦⣄⡙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠊⠀⢀⣈⠄⠂⢀⣡⡶⢟⣫⡥⠀⢀⢀⠻⣯⠈⠄⠀⠀⠀⠀⠈⠙⠻⣦⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢛⠃⠀⣠⠞⠋⣀⣴⢞⣫⡵⠞⣫⡿⣧⣿⣾⣿⡆⢹⡆⠙⢆⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⢶⡶⠔⠁⡠⠞⠋⠊⠈⠁⣠⣾⣫⣾⣿⣿⣿⡟⢿⡄⢳⡀⢆⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠺⢷⡔⠈⠀⠀⢀⣤⣴⣾⣿⡿⢫⣾⠇⣿⣿⣿⠘⣿⡀⣷⠀⠙⢮⡓⢤⡀⠀⠀⠀⠀⠀⠀⠠⠭⠃⡄⠀⠀⠀⢠⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠴⠿⣿⣇⡙⢿⣿⠿⠟⠋⣭⣴⣿⡟⢸⣿⣿⢹⣇⢹⣧⠘⡄⢢⡀⣹⠞⠃⠵⠤⡀⠀⠀⢢⣄⣴⣖⣷⣤⣀⡁⢀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠐⡲⠞⢿⣟⣿⠦⠤⣤⣿⣿⣿⣿⡇⣼⣿⣿⡿⣿⠈⣿⠀⠘⡄⠻⠃⠀⢀⡀⠀⠤⠚⢛⣀⠀⠀⠀⠀⠉⠤⠶⢛⣋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡆⠙⢿⣾⣟⣧⣀⣠⣾⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣠⣾⣷⣾⣿⠉⠁⠘⠂⢡⣠⣤⣴⠾⠛⠉⠓⠒⠠⠟⠃⠬⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⡴⠸⣿⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⡿⣧⣤⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢯⢼⡇⣿⣿⣿⣿⣧⣟⢻⣿⣿⡿⠿⣿⣭⣽⡿⢴⠟⠔⢷⣹⡇⠻⢿⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣽⡿⣶⣿⣿⣿⣿⡟⡌⢸⡟⣽⣿⣿⡟⣿⡿⣿⣴⡦⠄⠶⠋⠑⠈⠁
⠀⠀⠀⢀⠀⠀⡐⠀⠀⡼⠀⣿⣧⣿⣿⣿⣿⣿⡅⠀⢸⢃⣿⣿⣿⢱⡜⢷⠘⢿⣿⣆⠀⠀⠢
⠀⠀⠀⠀⢀⠀⢈⡦⠊⢀⣠⣿⠃⢻⣿⣿⣿⣿⠁⡇⣿⢸⣿⣿⣿⠈⣿⣦⡀⠀⢻⣿⣷⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣄⠀⠠⡀
⠀⠀⠀⠈⠂⣄⠉⣠⣾⣿⡟⠁⠀⣼⣿⣿⣿⡇⢀⣇⠀⢸⣿⣿⣿⣇⢻⣿⢷⡼⣆⠻⣿⣦⡙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⣀
⠀⠀⠀⣐⡤⠜⣹⣩⡟⣫⠀⠀⠀⠹⣿⣿⣿⣧⣼⣯⠀⠰⣿⣿⡞⣿⣦⡙⠧⢻⣜⠆⠹⡻⣿⣶⣄⡙⠃⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠋⠀⠠
⠀⠀⠀⠈⠉⠛⠋⠉⠁⡉⠀⠀⣠⣮⣙⣿⡏⢻⣿⣿⣧⠀⢻⣿⣧⠹⣿⣿⣦⢤⣛⣦⡀⠱⡈⢻⠛⣿⡷⠦⣌⡁⠀⠀⠀⠀⠀⠀⠚⠁⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢉⢻⣿⣟⢷⣸⣿⣿⣿⣷⡌⢿⣿⣶⣿⣿⣷⣶⣮⡙⠻⣦⠰⡸⠂⡿⢻⣆⠀⠠⣀⡀⠀⠀⠀⠀⠠⠖⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣮⡻⣿⣄⢘⣿⣿⣿⣿⣿⡜⣿⣏⣿⣿⠟⣥⣝⢿⣷⡈⢧⠹⣶⣄⠀⣹⡶⡀⠀⠉⠓⠦⠀⠀⠀⠀⢒⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠙⡙⠃⡽⠋⣴⢿⡿⢛⣭⣿⣿⣿⣿⢻⣧⣼⣷⡜⢎⢿⣷⡈⣇⠹⠻⣮⢿⣧⡿⠄⠈⠑⢦⣀⠀⢤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠛⠀⠸⣦⣉⣤⣤⣾⣿⣿⣿⣿⣿⣿⡙⢻⣿⣿⡌⢮⣿⣷⣹⣦⢳⣬⣿⣿⣇⠸⣆⢢⡀⠉⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠡⡄⠘⠛⠿⠿⣿⣻⣿⣿⣿⣯⣾⠃⠀⠈⠙⠛⠦⣜⣈⣛⣛⣳⡛⠛⡧⣻⣆⣿⣦⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⡿⠄⠀⠓⠒⠊⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠖⠒⠒⠒⠒⠒⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣡⣤⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣩⠔⠒⠛⠛⠓⠒⠤⠌⠉⠛⠛⢗⠒⠒⠲⠶⠦⣤⣀⡀⣼⠁⢀⢀⣴⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠠⠊⠀⡠⠁⣼⠿⠋⠑⢦⣀⠀⠀⠠⡀⡄⠀⠑⣄⠀⠀⠀⠀⠈⠙⠣⠶⣾⣿⠇⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠐⢀⠔⠋⠁⣸⡣⠀⠀⠀⠀⠉⠟⠦⢀⠈⠻⡀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡿⠑⢦⡀⠀⠀⣠⡿⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣟⡇⠀⠀⢀⡠⢔⣦⣀⠼⢷⡄⠈⠀⠀⠀⢳⡄⠀⠀⠀⠀⣼⠁⠀⠀⠙⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠟⢯⡅⠀⠐⢿⣿⡯⠃⢰⣿⣆⠀⠀⠀⠈⣿⡄⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢻⠟⢱⣦⠈⣉⠁⣠⡯⢻⣿⡄⢠⡆⠀⢹⠏⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⣶⣆⠀⠀⠀⢈⣽⣿⣷⠹⣿⡇⢨⡇⣰⠘⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡸⣿⣿⣧⣶⡞⠀⢠⠟⠉⠀⣿⢣⠟⢁⡿⠀⢸⣿⣾⣷⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⣞⢿⣿⣿⠡⠄⠂⠀⣰⡟⠁⠀⠀⠘⠃⢠⡿⣿⡟⠻⡄⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣕⣈⠻⣇⣠⣀⣼⠏⡔⠀⡄⠀⠀⢠⣿⡟⢻⡇⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣿⣿⣽⣮⣝⠿⡟⣸⡇⠀⣿⠀⠀⣿⠋⠀⢸⠃⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⠀⠀⠇⠁⣿⡇⢹⣶⡇⣿⡇⠀⡔⠀⣼⠇⠀⠀⠘⠀⡆⠀⠁⠀⢀⡰
⠀⠀⠀⠀⠀⠀⠀⢰⠟⢠⠸⡆⠀⠀⠀⡫⢷⡞⡓⠀⢛⣃⡰⢁⣼⢿⠃⢠⠀⠀⢠⡇⠀⡀⡄⠈⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⢗⣠⠎⠀⠀⠊⡞⠁⢀⢠⣾⡑⢿⣿⣿⣤⣿⣄⣺⣾⡇⠘⣸⣧⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡭⠖⠉⠇⠀⢀⠀⢀⡱⡀⣰⡟⢦⣙⡶⡹⣿⡿⢿⣿⠿⣿⣿⣾⡿⢿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⡀⢀⠀⠀⠀⢸⠀⢸⠹⠿⠧⢙⢳⣞⠓⠁⡌⠃⠀⢀⡄⠀⠉⠁⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠚⣧⣾⣗⠄⣠⡿⣦⡟⣦⠀⠐⠀⠙⠀⠀⠰⡇⠀⠀⠈⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠁⠉⠛⠉⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⠀⠈⠉⠉⠛⠒⠲⠤⠤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⠉⠲⠤⣄⣀⠀⠀⠀⣠⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢌⡃⠀⡀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⠀⢧⡉⠒⢍⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠒⢬⡲⢤⣭⠢⡍⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡐⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡛⡆⡈⠑⠯⣛⠾⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡈⠂⠄⡀⠀⠀⠀⠀⢉⢁⢢⢃⢠⣆⢨⠙⠢⣀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣁⠃⡀⠀⠀⠀⠸⡄⢸⡈⡀⡈⡄⡔⡈⡓⢤⡀⢤⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠯⠭⠳⢤⣙⠢⠀⠆⡇⠀⠇⠁⠁⢃⢠⢡⢱⢂⢍⠲⣌⠧⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠉⠤⠬⠭⠭⠓⢦⣀⠃⠀⠸⠀⣸⠸⡘⠈⡈⡘⠘⡼⢨⠑⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⣀⡀⠀⠨⠭⠄⣰⠟⠢⣅⠃⠅⡆⠇⠃⢡⠃⢃⠣⠇⣫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠐⠒⠒⣂⡀⣽⠋⠀⠀⠀⠙⠾⣵⢰⠸⠸⠐⠸⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠠⠤⠤⠤⢠⡖⠁⠀⠀⠀⠀⠀⠀⠈⠛⠦⣇⣆⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠤⠉⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠄⠉⠉⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢒⡒⢪⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⠀⠀⠀⣀⣒⡒⠒⢒⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠐⠒⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠁⠀⠀⠀⠤⠀⠀⠩⣭⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠂⠀⠀⠀⣀⡁⢐⡀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢀⠈⠀⠀⠀⢀⠀⠲⠂⠐⢂⡾⠁
⠀⠀⠀⠀⠀⠀⡰⠁⠔⠀⠀⠀⠀⠠⠤⠤⠤⠤⢤⡟⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢈⣀⣒⣒⡂⣀⡞
⠀⠀⠀⠀⠀⠈⢦⡀⠀⢀⡐⠒⠒⠀⠀⠐⣐⠞
⠀⠀⠀⠀⠀⠀⠀⠑⢶⡀⠠⠤⠍⠥⠀⣒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢭⣭⣭⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⡠⢐⡡⢉⡤⣊⠕⡣⠔⡠⢒⡠⠊⡡⠒⡡⠐⡠⠔⡡⢖⡠⢖⡠⢂⡠⢂⡠⠄⡠⢀⡠⢀⡄⠀⠀⢀
⠀⠀⠀⠀⠈⠐⠀⠈⠔⢩⠔⣡⢔⡡⢊⠕⢋⠔⡩⠔⣁⠔⢉⠔⢋⠔⢊⠔⣋⠔⣩⠔⣉⠔⢉⠔⢉⠔⢊⠔⣉⢔⡡⣪⢞⡵⢋⠔⡡⢊⠔⡡⢀⢔⡠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠈⠀⠘⠁⠈⠁⠈⠁⠀⠁⠈⠁⠈⠁⠘⠁⠈⠁⠘⠁⠘⠁⠚⠁⠉⠘⠁⠉⠀⠁⠈⠀⠁⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠛⠉⠉⠩⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣀⡀⠀⠀⣾⣿⣿⣦⠘⡏⣿⡀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⣿⣿⣷⠀⠀⠻⣿⣿⡿⠀⠃⣹⡇⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠻⠿⠃⢰⣤⡄⢨⣭⢀⡎⢐⣿⣷⠀⠀⠈⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⡇⠘⠛⠃⠀⢡⣿⣿⡿⢻⣿⣷⡀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⢰⠘⣿⣿⠇⣼⣿⣿⠟⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⠉⠛⠀⠐⠉⢉⣾⠛⢃⠀⠀⠀⠀⠃⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣇⠀⢀⣀⣀⣴⣿⠃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢀⣄⠉⠁⠙⠏⠻⢿⣿⣿⣷⣿⠃⡰⠈⠀⠀⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠈⠁⠀⠀⠐⣄⠀⠉⠛⣿⣿⠀⠀⠀⠀⢠⠾⠄⠄⠠⠤⠄⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⠀⠐⠚⢿⡄⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⠤⠤⠦⠤⢿⣄⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇⠈⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠄⡠⣠⣤⣤⣘⣦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡓⣷⣿⠋⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠘⠀⠀⠈⠀⢀⣝⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠁⣤⣷⣿⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢹⡎⠉⠉⠉⣠⣾⡛⠉⠂⠀⠀⠀⢄⠠⠀⠃⠁⠀⢠⠆⢰⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢳⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⢈⡇⠀⡐⠀⢠⡏⢀⡇⠀⣿⠈⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠉⠀⠀⠀⠀⡟⠿⠇⣶⠀⠀⠈⠀⠸⠀⠀⢿⣄⣀⣈⠹⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⠀⠀⠈⠛⠛⠧⠀⢀⠀⠀⠀⠀⠁⠀⠀⠸⡇⢸⣷⠀⣶⠂⣠⣾⣿⣱⣿⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⡀⠀⠀⠀⢀⠠⠈⠀⠀⠀⠀⠀⠁⠈⠽⠀⡟⢰⠏⠀⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣧⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣶⡶⢾⣷⣶⣶⣤⣦⠶⣢⣄⠂
⣿⣿⣿⣄⠀⣀⠀⠘⣿⣷⡀⠀⢀⢆⠀⠀⠀⠀⠀⢤⣤⣤⣀⣸⣰⠀⠰⠦⠀⠀⣰⡟⡇⠈⢀⣴⣿⣿⣷⢻⡟⣾⣿⣿⡇
⣿⣿⣿⣿⣾⣿⠀⠀⠏⣿⣿⣦⠀⢲⣤⠀⡄⠀⡄⠉⡋⢉⣻⣿⣏⣤⠀⣠⢤⣾⣿⣤⣐⣼⣿⣽⡿⣿⣿⣧⣵⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣤⣷⣄⠈⠃⣿⣳⡄⠈⢣⡀⠀⡇⠀⡗⢿⣈⢛⣀⣿⠶⣷⡿⠛⠋⠉⠙⠻⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢾⣿⣿⢻⣿⣟⢷⠀⣧⠀⡇⠀⠀⡏⠉⢀⣴⡏⠠⡆⠈⢷⠄⠀⠘⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠉⢶⢲⡌⣿⡌⢸⣿⣿⣷⣾⣶⣿⣾⣶⣷⣿⣾⣿⣿⣦⣅⠐⣶⡅⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠶⠀⣴⣿⡇⢙⣿⣿⣿⡿⣿⡟⠻⠿⠿⣿⣿⣿⣿⣿⠿⠳⡌⠁⠀⠀⠀⡀⠈⠙⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⡇⠀⠀⠈⠀⢿⣿⣿⣿⣿⣿⡇⢈⣿⡇⠀⠀⠈⡏⣿⣿⡿⠁⠀⠀⠈⠒⠀⠀⠀⠈⠓⠤⠀⠈⠻⣿⣿⣿
⣿⣿⣿⡟⢡⡄⠀⠀⠀⠀⠀⢠⣀⢠⣤⢀⣙⣛⣠⣽⡇⠀⠀⢀⡅⣿⡟⠀⠀⣠⠀⠀⠀⠀⠶⣄⠀⠀⠀⠀⢸⡆⢸⣿⣿
⣿⣿⡟⠀⠈⠛⠦⠀⠀⠀⠀⠨⣿⣆⢿⡇⣿⣿⡿⠏⡏⣠⠠⠾⠓⠁⠀⣠⣾⣿⠁⠀⠀⠀⠈⠘⠀⠀⠀⡠⠋⠀⢸⣿⣿
⣿⣿⡿⡆⣤⢀⣀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣹⠛⢻⣧⡵⠀⢀⡀⣰⣤⡄⣧⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣼⣿⣿
⣿⣿⣷⠇⣿⢈⣿⠰⡄⡀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠿⠣⠎⣸⣼⣯⣿⣇⣿⣸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢹⣧⣿⣿
⣿⣿⣿⠠⣿⣸⡿⠀⠈⠉⠀⠀⢰⣌⢻⣿⣿⣿⣉⣉⣁⢩⡝⢿⣿⣿⣿⡟⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⢸⣿⣿⣿
⣿⣿⡏⢰⣿⣿⠃⠀⡄⡀⠀⠀⡀⠻⣿⣿⣿⣷⣦⣶⣶⣴⣶⣿⣿⣿⣿⡓⣾⡏⢠⡇⠀⠀⠈⠀⣀⡀⠹⣿⣿⢸⣿⣿⣿
⣿⣿⣧⠘⣿⣿⡀⠀⡇⣇⠀⠘⣎⠀⣬⡿⣿⠶⠿⠧⠤⠤⠤⠤⠤⠹⣿⡿⢟⠴⠋⣀⣀⡀⠀⠎⠀⠈⣶⣿⣿⣾⣿⣿⣿
⣿⣿⣿⣆⢻⣿⡇⠀⡇⠸⣆⠀⢸⡀⠇⠸⣾⡶⠔⠶⠶⠶⠶⠶⠿⣟⣿⠀⠋⠐⠋⡀⠔⡡⢣⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣿⠁⠀⡇⠀⠈⠀⠀⠳⡀⠀⠻⣷⣶⣶⣶⣶⣲⣤⠈⠉⠀⠀⠀⠠⠁⠀⠀⢡⠈⠆⢀⣨⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠈⡇⠀⠀⠀⡀⠀⠙⠳⣤⡀⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠟⣿⣿⡸⠀⠀⠳⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠀⠀⠀⠀⠀⠀⢀⡠⠎⢀⣶⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿
⠋⠠⠾⢿⣿⣿⣇⡀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢸⡈⢯⠻⣿⣯⡄⠀⠹⣿⣿⣿⣿
⠐⠁⠀⠀⣈⡭⠛⠈⠳⣤⡀⠨⡑⢦⣄⠀⠀⠀⠀⠀⠙⠳⠦⠤⠀⠀⣰⠏⠁⠀⠀⠀⡈⢧⠈⠳⣌⡙⠷⣤⣄⠘⠿⠋⠙
⠀⠠⠒⠋⠁⠀⠀⠞⠀⠈⠻⠄⠈⠲⢌⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⡀⠀⠀⠱⠄⠑⠄⠈⠛⠦⣄⡉⠓⢦
⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⠲⠃⠄⠰⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣬⣉⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠒⠛⠛⠓⠲⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⢛⣩⣤⣶⣶⣾⠛⠋⠉⠀⠀⠉⠙⢛⣿⣶⣶⣤⣍⣙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⣴⣾⣿⣿⡿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⣿⣿⣿⣦⣍⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⣾⣿⣿⣿⣿⣿⣟⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⣿⣿⣿⣿⣿⣷⣦⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⣾⣿⣿⣿⣿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⣿⣿⣿⣿⣿⣌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣿⣿⣿⣿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⠄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⢰⡸⣿⣿⣿⣿⡯⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢽⣿⣿⣿⣿⡟⡄⣠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⡆⣧⢿⣿⣗⠪⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⣺⣿⣿⢱⠃⣿⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣇⢸⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡟⡟⢸⣿⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⣿⠀⣿⣿⣿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣿⢿⣿⢿⡇⢸⡏⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⠀⣿⣞⠻⠿⠶⠯⣥⣤⣀⠀⢰⡏⣆⠄⠀⠀⡀⠀⠀⡄⣾⣆⠀⢀⣠⣤⡭⠷⠾⠿⣫⣿⠀⣿⡇⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣿⠀⣿⡿⠟⠒⠀⠀⠀⠀⠠⣄⣼⡁⠽⢰⢸⣠⡇⣿⣦⠐⠇⣿⣠⡔⠀⠀⠀⠐⠒⠚⠿⣿⡀⣿⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⣿⠠⢋⣶⠀⠀⣀⣀⠀⠉⠉⠙⠻⠷⣦⣾⣿⣿⣿⣿⣿⣶⠾⠟⠛⠉⠉⠁⣀⣀⠀⠀⣰⣌⠇⢻⣇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⢰⣿⣧⣾⣿⣿⣿⣿⣿⣶⣄⡈⡑⢄⡹⣿⣿⣿⣿⡟⣡⢄⡁⣠⣴⣾⣿⣿⣿⣿⣿⣮⣻⣦⠘⣿⠃
⠀⢀⠀⡀⠀⠀⢀⠀⣀⠀⢠⣏⣀⢨⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣝⣷⣽⣿⣽⣿⣽⣯⣵⣫⣽⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣏⣀⡙⣦⣀⢀⠀⠀⠀⡀⢀⡀
⠀⠈⠈⠁⠈⠁⠈⠉⠉⠀⢹⣙⣿⡏⡛⢙⣿⣿⣿⣀⣘⣻⣿⣿⣿⣿⣿⣿⣋⠈⠉⢛⣿⣿⣿⣿⣿⣿⡛⠄⣘⣿⣿⣿⡟⠙⠋⣿⢇⣿⠁⠈⠈⠀⠀⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⠇⣤⣰⡌⣿⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢰⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡄⠛⠈⠲⠶⠿⢿⣛⣛⣿⣿⣟⣛⣣⣿⠏⣸⣿⣿⣷⢘⣿⣮⣛⣛⡿⣿⣟⣛⡻⠿⠷⠖⡃⠘⢁⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣿⣿⠀⠀⢽⣿⣭⡁⠀⠉⠉⢻⣿⣿⣿⣿⢃⣿⣿⣿⣿⡎⢿⣿⣿⣿⣿⠋⠉⠁⠀⣭⣿⣿⠇⠀⢸⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠉⠉⣿⢇⣾⣿⣿⣿⣿⣿⡌⣿⡍⠉⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⣹⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⡷⣦⣦⣤⣶⣲⠦⢆⣀⡀⠀⠀⢸⡜⣿⣿⣿⢿⣿⣿⢇⡟⠀⠀⠀⣀⡐⡤⣶⡶⣤⣤⣶⢾⣻⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣻⡿⣿⣿⣛⣥⣾⣿⣿⣿⣷⣆⠈⠙⠘⠿⢻⡟⠿⠏⠚⠁⢠⣾⣿⣿⣿⣿⣮⣛⣻⣿⡿⣿⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⢿⠉⠉⠁⣿⢿⣿⢽⡇⠀⠀⢀⠘⠇⡀⠀⠀⢠⣿⣿⡿⣿⡇⠛⢻⣿⢿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀⠀⠀⠁⢹⡿⣷⣼⣷⣆⣼⣬⡀⠀⣧⣹⣤⣼⣿⣴⣿⣿⠇⠀⠈⠈⠀⠀⠀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⣿⡏⢿⢻⠙⢿⢳⡶⢻⠋⡟⢹⠙⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣾⣼⣆⡘⠀⡇⢸⣀⣧⣼⡾⠻⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣾⣿⣿⠟⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡆⠀⠉⠋⠘⠉⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⡖⠛⠛⠉⠓⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⠁⠀⠀⠀⡞⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡀⡤⠌⠷⠓⠒⠒⠒⢦⣄⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠋⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⣀⣀⣀⣀⠀⠀⢀⣠⣤⣤⣤⣼⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣉⣿⣟⣹⣿⣿⣏⣿⣧⣾⣿⠿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⢿⣿⣿⣿⠿⠛⠛⠛⠿⠿⠿⠋⠐⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠤⠐⠒⠒⠒⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠉⠁⠀⠀⡠⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠈⠲⢄⡀⠀⢀⡠⠜⠁⢸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡇⠐⢄⠀⠀⠈⠉⠁⠀⡠⠒⠑⠈⠗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠃⠀⣷⣶⣤⣅⡢⢄⡀⢐⣈⣤⣤⡄⡴⠃⠈⢷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡌⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣀⡀⠀⠻⣿⠿⠟⠋⠙⠉⠛⠿⣿⣿⡿⠁⢀⣀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠁⠀⠀⢃⠀⠀⠀⠀⠃⠀⠀⠀⢠⠃⠀⠈⢻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠘⡄⠀⠀⠀⠄⠀⠀⢀⠆⠀⢀⡠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⢵⣄⣀⣤⣀⡤⠤⠾⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣷⣶⣄⠀⠀⠀⠀⢀⡾⠋⠁⣤⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⡻⣿⣿⣿⣿⣿⡆⠀⣠⠴⠋⠀⠀⠀⠓⢤⡀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠑⢻⣿⣿⣿⣿⢿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⣿⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⣹⣻⣿⣷⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠃⠉⠙⢻⣿⣿⣿⣟⢿⣿⡿⣿⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⢦⣄⡀⠀⣠⡤⠂⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢺⣾⡇⡀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⠇⠁⠑⠺⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⢿⣿⣿⣿⣻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠹⣿⢿⣿⣿⣹⣿⣿⣿⣿⣿⣿⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⠟⠀⠉⠨⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⢉⣑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢈⠉⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠸⠿⡯⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⡀⢿⡇⠀⠀⡇⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢠⠔⠐⠠⢄⠀⠀⠀⢠⠀⠈⠛⠀⠈⠃⠀⠀⡇⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠑⢄⠀⠈⡆⠀⠀⣴⣿⣦⠀⠀⠱⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠃⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢄⣀
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠈⠑⢤⡀⠀⠲⣄⣀⣼⠾⠋
⠀⡜⠀⠀⠀⠀⠀⢀⠔⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⢎⣀⠀⢀⡄⣀⡴⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⢆⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣥⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠈⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠡⠄⠀⠀⠀⣀⣤⣶⣾⡿⠿⠟⠛⠛⠛⠛⠿⡿⢿⣷⣶⣤⣀⠀⠀⠀⠂⣤⢀⡀⢦⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠘⡏⠀⠀⠀⣀⣴⣾⡿⠛⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠉⠻⢿⣿⣦⡀⠀⢅⢈⡗⠈⠇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠊⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⢠⠾⠿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠉⠻⣿⣦⡈⠉⠈⠋⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣤⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡟⢣⣽⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣯⣷⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⢹⣟⣻⢉⣹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⣿⣿⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣏⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⢸⣷⡃⠘⢾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡆⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡄⠘⠀⢻⡄⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⣷⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⡄⢰⡁⠀⢲⡆⠁⠀⠀⠈⠁⣟⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠘⣿⣿⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠂⠀⠀⠀⠀⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠐⢄⠀⠛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣷⣶⣆⢶⠀⠈⠃⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡟⠋⠸⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠰⠞⠁⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⣿⡀⠄⠀⢄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⣾⣿⣷⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣷⣤⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠈⢿⣿⣿⣿⢫⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⡏⢣⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠘⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣻⡿⠁⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⣶⣤⣶⣤⠀⠀⢸⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡿⠀⠀⠀⢠⣶⣶⣶⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣇⣹⡿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣾⣶⣷⣾⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⢸⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠂⠀⠀⠀⠁⠂⠀⠀⠀⠀⠐⠈⠀⠀⠀⠐⠁⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠔
⠀⠀⠀⡀⠀⠲⣄⠀⠘⢅⠀⠀⠀⠀⠐⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠁⠀⠀⠀⠀⡨⠃⠀⣰⠖⠀⢀
⠀⠀⠀⠙⠄⠀⠉⠐⠄⠀⡀⠀⡀⠉⠂⢀⣂⣀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣐⡀⠒⠁⢀⠀⢀⠀⠠⠂⠁⠀⠠⠋
⠀⠀⠀⠀⣿⠦⣄⣁⣀⣠⣬⠴⠚⠛⠉⠉⠉⠉⠉⠉⠓⠦⠴⠚⠉⠉⠉⠉⠉⠉⠛⠓⠦⣥⣄⣀⣈⣠⠴⣿
⠀⠀⠀⠀⢻⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠘⡇⠀⠀⠀⢠⡴⠚⠉⠉⠀⠀⠉⠉⠳⢤⡀⠛⠛⢀⡤⠞⠉⠉⠀⠀⠉⠉⠓⢦⡄⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⢀⣤⡷⠀⠀⢾⣄⡀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠉⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⣀⣀⣀⣠⡤⠴⠶⠒⠒⠶⠦⢤⣄⣀⣀⣀⣀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠉⠒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⠏⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⣤⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡞⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⢸⣿⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠘⣿⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠹⣿⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡼⠠⢀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣸⢣⠃⡜⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢰⣇⡏⣰⠁⠀⠀⣼⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢡⣟⡞⢠⡏⠀⠀⠀⡇⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣾⡟⠀⡾⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⡄⠀⣿⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⣾⡿⠁⣼⠁⠀⢀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢻⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢃⣾⠯⠁⣰⠃⠀⠀⢸⡇⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⢁⣾⠋⢠⢠⠏⠀⠀⠀⣾⢡⠃⣠⢀⡇⠀⠀⠀⠀⠀⢠⡇⠀⢸⡄⠀⠀⠀⢀⣉⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡿⠿⠋⢀⡾⠃⢠⢇⠏⠀⠀⠀⢠⡟⠘⢠⡏⢸⡇⠀⠀⠀⠄⠀⢸⡇⠀⢸⡇⢀⣴⠞⡫⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠠⠋⠀⢠⡟⠌⠀⠀⠀⠀⣾⠃⢡⡟⠀⣸⠀⠀⠀⣸⠀⠀⢸⠁⠀⢸⡇⣾⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⡞⠀⠀⠀⠀⠀⣸⡏⢠⠟⠀⢀⡟⠀⢀⢠⡇⠀⠀⠞⠀⠀⠈⣷⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢡⠏⠀⠀⣼⠁⢠⠃⣾⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢀⣤⡀⠀⠀⢠⣿⡵⠃⠀⠀⠀⠏⡠⠃⢰⣇⣴⣶⣶⣤⡀⠀⠀⠀⢻⣦⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⠟⠙⢻⡄⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⣻⡷⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠀⠀⠈⣿⣿⠋⣠⣴⠶⠶⠶⣦⣄⠀⢸⠿⠛⠃⠀⠀⠻⠦⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡼⠁⠀⠀⢰⣿⡷⠟⠉⠀⠀⠀⠀⠈⠻⣷⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠿⠿⠿⣿⣿⣿⣿⡿⠿⠶⠶⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠔⠊⠉⠁⠀⠉⠙⠒⠤⢄⣀⣀⣀⠤⠒⠉⠉⠉⠉⠑⠒⠤⠤⠔⡄
⠀⠀⠀⠀⠀⠀⡠⠾⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠐⢺⠋⠙⢢
⠀⠀⠀⠀⢠⠎⠀⠀⢇⠀⠀⠀⠀⠐⠂⠀⠒⢀⠀⠀⠀⠀⠀⠄⡁⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⡇
⠀⠀⠀⠀⡎⠀⠀⠀⠘⡀⠀⠘⠤⣀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠔⠁⠀⡜⠀⠀⠀⣰⠁
⠀⠀⠀⠀⢳⡀⠀⠀⠀⢣⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠀⠀⠀⠒⠒⠒⠀⠉⠀⠀⠀⢰⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠣⡀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠢⢤⡀⠀⠀⠀⠀⠀⡠⠃⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠑⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋⢀⡞⠁
⠀⠀⠀⠀⠀⠀⣠⠖⠒⢲⡧⣄⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣠⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠻⠤⠤⠊⠀⠀⠉⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊⠁⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠉⠉⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⣾⡇⠀⠀⢸⣿⡄⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠘⢿⠇⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⢠⣖⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠀⠀⠀⢀⡴⠞⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠒⠠⠄⠀⠀⣀⣀⣀⣀
⠀⠀⠀⣀⠀⠀⠈⠁⠀⠶⠄⠠⠠⠶⠚⠃⠀⢠⡅⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⠓⠚⠓⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠈⢀⣀⣠⣄⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠑⠶
⡘⠁⢠⠟⠉⠛⠳⠓⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢤
⠁⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠸
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠡
⠀⠀⠀⠀⠘⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢐⠿⡏⢓⡠
⠀⠀⠁⠀⠀⣿⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠠⣼⠃⡏⠁
⠀⠀⣀⣠⣼⡟⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⣾⣧⣀⠶⠛⠉
⠀⠀⠉⠁⠀⠀⠀⢠⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠿⠿⠓⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠅⡀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢳⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡆⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⣷⣦⣣⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣄⠀⢀⠀⠀⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡆⠀⠀⠀⠀⠀⣻⡆⠀⠀⠀⠀⠀⠀⣼⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷⡍⠠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢴⣅⣀⣀⠰⣭⣧⠀⠀⠀⠀⠈⠀⠉⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⣿⡄⠀⠀⠀⠀⠠⠆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠦⣀⣀⠀⠀⡀⠀⣾⣁⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣀⡇⢸⣿⣿⣄⠠⡿⣆⣠⣤⡴⢤⠄⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣘⣿⡿⠛⠂⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠄⠈⣤⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⢦⣿⡴⠈⠁
⠀⠀⢠⣦⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣠⣶
⠀⠀⣼⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣼⡆⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣦⣴⣿⣿⣤⣾⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣛⡇
⠀⠀⢹⣿⣻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⣾⣻⣟⣿⠃
⠀⠀⠀⠻⣟⣿⣿⣻⣿⣶⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣾⣿⢿⣿⢿⡿⠋
⠀⠀⠀⠀⠈⠛⠳⠿⣿⠿⠿⠿⠟⠛⠉⠉⠙⢿⣿⣿⠟⠉⠉⠙⠛⠻⠿⠿⢿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠉⠉⠉⠉⠉⠑⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⣠⠤⠶⠦⢄⠀⠀⣠⠴⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠱⡞⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⢀⠇⠀⠀⠀⠀⢰⡃⠀⠀⣴⣶⣶⠀⠀⣴⣶⣶⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢇⠀⠀⠻⠿⠟⢠⢆⠹⠿⠟⠁⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⠖⠁⠈⠓⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⡏⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠑⠒⠒⠊⠉⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⠤⠒⠊⠉⠀⠈⠉⠒⢤⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠉⠉⠉⠉⠉⠙⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⡴⠛⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠟⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢯⡀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣠
⠀⠀⣀⣤⢤⣄⡀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣀⡀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⠙⠦⢤⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⡼⠉⠀⠀⠀⠙⠶⣄⠀⠀⠀⡸⠁⠀⢸⣿⣿⣆⠀⠀⠀⣠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡼⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠁⠀⠀⠀⠻⣿⣿⣷⠖⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠋⠁
⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠒⠒⠒⠊⠁
⠀⠀⣷⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⠉
⠀⠀⠈⠛⠛⠁⠘⣆⠀⠀⢀⡾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢷⡀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢩⣳⣀⣀⣤⣀⣀⣞⡍⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⣤⣧⢀⠴⡏⣻⠛⣟⢹⠦⡀⣾⣤⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠐⣷⠆⢋⣹⡋⢸⠟⣶⠻⡇⢙⣏⡙⠰⣾⠂⣸⡀
⠀⠀⠀⣷⣤⣤⣤⣤⡶⢾⣟⣽⣿⣿⣷⣾⣯⡁⣾⠓⠉⠚⣷⢈⣽⣷⣾⣿⣿⣯⣻⡷⢶⣤⣤⣤⣤⣾
⡤⠀⣤⣽⣿⣿⣟⣿⢟⣿⡿⠷⠿⠿⣽⣿⣿⣿⣿⣢⣀⣔⣿⣿⣿⣿⣯⠿⠿⠿⢿⣿⡻⣿⣻⣿⣿⣯⣤⠀⢤
⠣⡉⠁⠻⣿⣶⣶⣶⣿⣅⠀⠀⠀⠀⠀⠙⢯⣽⣿⣿⣿⣿⣿⣯⡽⠋⠀⠀⠀⠀⠀⣨⣿⣶⣶⣶⣿⠟⠈⢉⠜
⠀⢀⡥⢾⣿⣟⣻⠿⣿⣿⣿⣶⣦⣤⣴⣶⣏⣿⣷⣿⣿⣿⣾⣿⣹⣶⣦⣤⣴⣶⣾⣿⣿⠿⣟⣻⣿⡷⢤⡀
⠀⠘⠮⠃⠙⠢⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣿⡿⠋⠁⠀⠈⠙⢿⣿⣿⣟⣿⡿⣿⣿⣿⣿⣿⣿⠔⠋⠘⠵⠃
⠀⠀⠀⠀⠰⡏⠿⠉⠓⠻⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠟⠚⠉⠿⢹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠄⠀⠀⢀⣰⡿⠁⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠓⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⢠⡶⠦⠆⠤⠤⠄⠐⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡄⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠂⠻⣷⣄⠀⠀⠀⠀⢀⣼⣿⣿⠂⠀⠀⠈⠹⣷⠀⠀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⣿⣷⣦⠀⠉⠻⣿⡿⠏⠀⠀⠀⠀⢀⣿⠟⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠈⠛⠛⠋⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣀⡴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠠⣄⣀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⣿⠋⠉⠀⢻⠁
⠀⣠⠞⠃⠈⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠷⠶⠏
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠛⠛⣳⡄
⠸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⣤⣤⣔⠚⠀⠀⣰⠋⠀⠁
⠀⠀⠀⢧⠀⠸⣦⠀⠀⠀⢻⡿⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠈⠳⠾⠟⢱⣄⡀⣸⡷⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣛⣻⣄⣀⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣼⡟⠉⠳⢦⣤⣤⡴⠞⠋