Trickster Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣀⠰⣾⣿⣿⣿⣿⣿⣷⠀⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⢠⡿⠿⣿⣿⣿⣿⣿⡌⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⣠⣶⣤⣍⣙⠛⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣾⣿⣿⣿⣿⣿⣿⣶⣆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⠙⣿⣿⣿⠋⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣈⣉⣡⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠟⠋⠀⠀⠀⠀⢠⣤⣀⣀⣀⢀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣦⣬⣍⠉⣭⣤⣴⣶⣿⣶⢸⣿⢁⣤⠈⣿⠋⣤⡌⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⠈⣿⣄⠻⣿⣿⣿⠟⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⢿⣷⣤⣤⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠈⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢀⢀⡀⡀⠀⢠⡀⡄⢠⡀⣀⢀⡀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠈⠀⠁⠈⠁⠀⠈⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣻⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢿⣛⣯⡽⠿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⢻⢞⣭⢞⣫⡽⢖⣛⣭⠭⠭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⢫⠞⣫⠵⣛⡭⠶⣒⣛⣭⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡵⣋⡵⢖⣫⠭⠶⣒⣒⣂⠀⢲⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⡥⢚⣩⣴⣾⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣴⣿⣿⣿⣿⣿⣿⡿⠁⠀⢸⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠟⠿⣿⣿⡇⠀⠀⢀⣠⡖⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠘⠍⠂⠀⣴⣿⣿⣷⣤⡈⠇⠀⢠⣤⣤⡒⣦⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠥⣤⣶⣶⣧⠀⠐⠀⠈⠛⠉⠀⠙⠀⠄⡀⠀⠈⠛⠛⠀⠀⣾⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⠟⠛⠀⠀⣠⡤⢀⡏⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⢸⡏⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠋⠋⠀⠀⠒⢤⣬⣷⣾⠃⢀⠀⡀⢸⠅⠀⠠⢌⣙⣿⠛⣷⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠉⠓⠉⢀⠀⠀⠗⢠⡾⣀⠀⢠⣽⠋⣹⡿⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⠀⠀⢀⣀⣼⡆⠀⠀⢸⠀⠈⠓⠒⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠘⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠈⠉
⣀⢀⡀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⠀⠀⠀⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠶⠏⠀⢿⠛⡄⠀⠀⠀⠀⡜⢻⠃⠈⠷⠶⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠛⠛⠀⡀⠀⠀⠤⣼⢶⡄⠀⠀⠀⠀⣴⢺⡤⠀⠀⠀⡀⠘⠛⠃⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠉⠀⠰⠀⠀⠱⠄⠀⠈⢿⣹⡇⠀⠀⠀⠀⣿⣹⠏⠀⡀⠰⠀⠀⠰⠀⠈⠉⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠁⠀⠐⠀⠀⠈⠀⠠⠰⡗⠐⠊⠉⠀⠀⠀⠀⠂⠈⠉⠒⠒⡟⠀⠀⠈⡀⠀⠐⠀⠀⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠐⠒⠀⠀⢄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠠⠄⠀⠀⠂⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠙⠀⠀⠚⠂⠀⠀⠁⠀⠀⠀⠀⠐⠶⠦⣴⣤⣶⡤⠶⠖⠀⠀⠀⠀⠀⠁⠄⠐⠒⠂⠀⠐⠀⠐⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠐⠀⠀⡈⠀⢀⠂⠀⠀⠀⠀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠂⠀⠉⡀⠀⠒⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⢿⡟⣿⣿⣿⣿⣿⣿⣦⣀⣀
⠀⠸⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢀⣴⣿⣿⣿⣿⣿⣿⣯⣿⣿⠿⠟⠿⠿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠘⢿⣿⣿⣦⣀⣀⣤⣶⣾⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣀⣤⣄⡀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⣿⣿⣿⣥⣶⣾⣿⣿⣏⣹⣿⠂⢀⡸⠟⠀⠈⣿⣿⣿⣟⣿⡆⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣛⣋⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢶⣿⣤⢀⣴⣾⡿⢋⣿⠏⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠻⣙⣯⣬⣥⣿⣄⠋⠿⢿⠇⠉⠙⠀⣤⠉⣀⡀⠾⠋⠀⢿⠁
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣴⣿⣿⣿⣿⣿⣯⣿⣟⣉⣽⣆⠀⢉⣶⣀⡀⣰⣿⣿⣿⣤⡀⣴⠀⣷⣿⠄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣾⣿⣿⣷⣤⣴⣿⣿⣷⣽⣿⣿⣿⠰⣿⣿⣏⠐⠛⢹⡿⠉⣩⣿⣿⣤⣯⡝
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢿⣿⡿⠶⠶⢟⣿⢿⣿⣿⣿⣿⠃⠀⢘⣁⣤⣤⣀⢀⣤⣾⣿⣿⣿⣟⠁⣄⢀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⡘⣿⣿⢿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣾⣿⡿⢛⠛⠋⠈⢹⣿⢸⣿⡍⠙⢢⣿⣿⡄
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡜⢿⣿⣿⣷⣬⣭⣤⣤⠀⣄⠀⣿⣿⣧⢾⡿⠀⢀⣸⣿⢠⣿⡇⢀⠉⠙⠃
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡘⢿⣿⣿⣿⣥⣀⡈⠙⠻⠿⠷⠾⢋⣦⣿⠀⢹⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⡿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⠈⠻⣿⣿⣿⣿⣷⣿⣾⣴⣦⣾⣿⣿⠟⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣤⡈⠻⠿⣿⣿⣿⣿⡿⠿⠟⢋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣷⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⢻⣿⣷⣄⡀⣠⣾⠟⠉⣹⣿⣿⣿⣿⣿⣿⡿⣿⣦⡀⣀⣼⣿⣿⠃⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠻⣿⣿⣿⣿⣧⡀⠀⠸⣿⣿⣿⣿⡿⠉⢠⣿⣿⣿⣿⣿⡿⠁⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠙⠛⣻⣿⡿⣶⣄⣻⣿⣿⣿⣴⣷⡿⣿⣿⠛⠋⠁⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣾⢿⣿⣷⡄⢉⠛⠻⠟⠛⡋⠁⣼⣿⡿⢿⣶⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢠⣬⣿⣿⣆⠳⣆⢀⠶⢃⣾⣿⣯⡄⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⢸⡷⣀⣛⠿⠷⢤⣤⠶⡻⢛⣛⣻⢷⠀⢸⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⢸⡇⠈⠻⠿⢿⡇⠰⣿⠿⠿⠉⣿⣼⠀⠀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⠀⣿⣦⡀⠀⢀⡄⠀⠈⠿⣄⣠⣼⣿⣿⠀⠀⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣆⠀⠈⢿⡙⠀⣉⠓⠄⢠⣶⣿⣿⣿⣿⠏⠀⢀⣿⣿⣿⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⣿⢻⣿⣿⣆⠀⠈⢻⠄⠻⣌⣉⢉⣉⣽⣿⢳⠋⠀⢀⣾⣿⣿⡿⣿⣷⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⠃⠸⣿⢻⣿⣆⠀⣿⡀⠀⠠⠤⢥⣴⣾⣿⣿⡇⢀⣾⣿⡿⢘⣷⣌⡻⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡆⢰⣿⠈⠹⣿⣦⣿⣷⣄⠀⠀⢀⣿⣿⣿⣿⣧⣾⢏⣿⢁⣰⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⣿⣿⣦⠀⢿⣿⣿⣿⡉⠛⠛⢻⣿⣿⣿⣿⡿⢃⡾⠏⣼⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣧⡘⣿⣿⣿⣷⣤⣀⣼⣿⣿⣿⣿⣵⠏⣀⣼⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡟⢻⡿⢻⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠶⣤⣄⡀⠀⠀⣀⣤⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠐⢶⣿⣿⣿⣷⣤⠻⠁⠀⣾⣿⣿⣭⣌⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠈⠙⠛⠋⠑⠃⠀⠀⣿⣿⣿⣿⢩⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢠⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⣾⢤⠀⠀⠀⠀⠀⣀⣀⣤⣦⣾⣿⣄⣾⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⣀⣉⣻⣟⠁⢹⣿⣿⡀⢀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣰⣿⣿⠤⢠⠀⠀⠉⠉⢙⣻⣿⠋⠻⣿⣿⠃⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠆⠀⣿⣿⠃⢠⣿⡄⠀⠀⠐⠻⣿⡃⣠⣾⡿⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠰⡀⠘⠗⠀⢿⡟⠛⠾⠷⣶⣤⣾⣿⣿⣿⠁⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⡀⢃⠀⠀⠀⠸⡇⠀⠀⠀⣼⣿⡿⠁⠀⠁⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠆⠀⢤⠀⠈⠀⠀⠀⢻⣿⣧⠠⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⡀⣠⣤⠀⠀⠀⠀⠉⠙⠛⠿⠿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠀⠀⠻⠿⠂⡀
⠀⠀⠀⠸⣦⡀⠈⠉⠛⠿⣿⣿⣧⠀⠀⢠⣴⡏
⠀⠀⠀⠀⠙⣿⣷⡄⠀⠀⠈⠫⠗⠀⠰⣿⣿⠿⠆⠀⢀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠻⣁⣤⣤⣼⣄⠀⠃⠀⠈⠁⡀⠀⠀⠙⠀⠈⠋⠀⠘⠻⠿⠀⢀⣀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⢀⠈⢹⣿⠿⠛⠀⠀⣴⡆⣸⡇⠀⢀⡄⢀⡄⠀⡄⠀⠀⠀⠀⠀⠙⠿⠷⠛⢀⣤⣤⣄⡌
⠀⠀⠀⠀⠀⠀⠸⣆⠜⠛⠀⠀⠛⠶⣶⣿⣿⠃⠀⡘⠁⠈⠀⠈⡵⠀⠀⢞⠀⠀⠀⠀⠀⠀⣼⠿⡿⡞
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠄⠠⣸⠄⢀⠀⠿⠀⠀⣀⠒⣶⣶⣯⡠⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣤⣀⣀⣀⣀⣤⣾⡀⣄⢠⠀⠀⡇⡄⠸⢀⣴⡇⠀⢀⣽⡿⢦⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠓⠜⠆⠇⠂⠐⠄⠀⠸⢇⠈⠳⠤⢿⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠈⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⠖⠀⠀⠘⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠒⠂⠀⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠠⠐⠀⠐⠢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠱⢿⣿⣷⣾⣶⣌⠹⣷⡆⠀⠀⣀⣤⣤⣤⣀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⣤⣤⠀⠙⠻⠿⠿⠟⡣⠃⠀⠀⣾⣿⣿⣻⣍⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⢶⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⠑⠈⢀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠏⢠⡿⠟⢫⣼⠃⠀⠀⠀⠀⠀⣤⣄⣀⣠⣄⣾⣿⡀⢀⡾⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣿⠟⣁⠖⢻⣟⠀⠀⠀⠀⢀⠀⠀⠉⢫⣿⠿⠛⢾⣷⣾⡇⣿⠃⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡄⣼⡏⠀⣤⣿⣲⡇⠀⠀⠈⠉⠛⢿⣿⣾⣶⣤⣽⣷⣿⡇⣿⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣹⡄⢹⣧⣾⠍⣾⣿⣇⠀⠀⠀⠠⣴⣶⡿⠓⠀⢘⣷⣿⡿⣱⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⣇⠀⣿⡯⢸⣿⡇⠙⢦⣦⣄⡀⠘⣿⣇⣤⣴⣿⣿⠟⣰⠟⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⣽⡅⢸⣿⠀⠀⠀⠉⠛⠛⠻⢛⣭⣿⣿⡿⠉⢸⡟⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣿⣧⡼⣿⡆⠀⠀⠀⠀⠀⣰⣿⣿⡿⢿⣧⣠⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⠃⠀⢻⡄⠀⠀⠀⢸⣿⣿⡏⠠⡸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡅⠀⠈⣷⠀⠀⠀⣸⠏⡻⡂⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠂⠀⢻⢻⡇⠀⠀⣽⡆⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠊⢠⡟⠀⠀⠀⣾⠁⠀⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡀⠀⠀⣾⠀⠀⠀⢀⣿⣤⡀⣷⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣴⢇⣀⣰⡇⠀⠀⠀⠘⣿⣿⣿⣿⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂⠁⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡆⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⢰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠁⠀⠀⠀⠀⢸⠝⠿⠭⡅⠀⠀⠀⠀⠈⠻⠏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣦⣤⣄⡀⡄⠀⠚⠈⠒⠀⡠⢀⣤⣤⣴⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣹⣿⣿⣿⣿⡇⠀⠠⢤⠀⠀⣼⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⠶⠯⢿⠿⠃⠀⠀⠀⠀⠀⠸⢿⡿⠽⠾⣿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠃⠀⠀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡠⠀⠀⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣀⣠⣥⡄⠀⠀⠀⠀⠀⠀⠈⢳⣄⠃⠻⣿⠏⠈⣠⠟⠁⠀⠀⠀⠀⠀⠀⣤⣤⣄⡐⢦⣄
⠀⠀⠀⠀⠡⡀⢀⣀⣀⣀⠀⠸⠃⣾⣿⡿⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⠻⠇⠀⣀⣀⣀⠀⣠⠁
⠀⠀⠀⠀⠀⢻⢟⣿⣿⣿⣆⠀⠈⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠙⠾⠹⠾⠃⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⣿⣿⣟⠿⠃
⠀⠀⠀⠀⠀⠈⠹⡛⠟⠟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⠃⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠛⣻⠋
⠀⠀⠀⠀⠀⠀⠀⢹⠣⣄⣸⣷⣶⣶⠞⠁⢀⠀⠀⡀⠽⢿⣶⡄⠀⠄⣀⣀⠀⣀⡀⣀⣀⢰⠀⣠⣾⠿⠇⡀⠀⢀⡀⠙⢳⣶⣶⣿⡇⣠⢺⠇
⠀⠀⠀⠀⠀⠀⢀⠈⠀⢻⣿⣇⣨⣿⣾⠋⠀⠀⠀⢳⢦⡀⢉⡃⠀⡄⣿⣿⣿⣿⣿⣿⡇⠀⠀⢛⠁⢠⢴⡇⠀⠀⠈⠻⣶⣯⣀⣽⣿⡇⠘⢀
⠀⠀⠀⠀⠀⠀⠀⢡⠠⠘⠻⣿⣁⠈⢻⣿⠃⢀⣴⠊⠈⣷⠀⠀⠀⠁⡜⠻⢿⣿⡿⠛⢇⡀⠀⠀⠀⡟⠀⠳⣄⡀⠹⣿⠋⢁⣸⣿⠛⠠⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠛⠚⠋⠀⠀⠀⠘⠀⢀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠑⠀⡀⠐⠃⠀⠀⠈⠙⠒⠛⣹⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠃⠀⠀⠀⢶⣶⣶⣶⣶⣄⠀⣴⣶⣶⣷⣶⠆⠀⠀⠀⠸⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠙⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣻⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣳⣿⣿⣟⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢱⡇⠀⣿⡟⠙⠯⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣖⣢⣿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣍⣁⣸⣿⢇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⣇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⡀⠀⠈⢻⣿⣿⡀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⣠⣿⣊⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡄⢀⣿⣿⣿⠁⠤⡷⣤⣀⣀⡀⠀⠀⠀⠠⢤⣠⣾⠟⠋⠉⠉⠉⠉⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣿⣿⠿⠁⢸⡿⠛⢿⣤⣤⡟⠹⢟⣥⠶⠄⠀⠀⠀⠀⠹⣷⡔⠂⡄⠀⠀⠲⣄⣠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠀⠀⠀⠀⠀⣸⣿⣾⣿⢩⣏⡆⠰⢋⣇⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⡇⠀⣦⠤⢧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⣶⣶⠄⠀⠛⠉⠙⠃⠈⠀⢠⣀⠀⢿⠀⠀⠀⠀⢄⡀⠀⣸⣤⣬⡇⠰⡧⠄⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⣞⡿⠃⢠⣏⣄⣶⣀⡀⠀⠀⣾⠇⠶⠀⢰⡀⠀⠀⠀⠀⠀⢿⡀⣨⡇⢸⣦⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠛⠁⠀⣬⡻⠿⢿⣿⡀⠀⠂⢹⣴⢾⣿⠊⠁⠀⣀⣀⡀⠀⢸⡟⠉⢀⣸⠏⠀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣼⠏⠀⠀⠘⣿⡇⠀⢸⣎⢳⣼⣇⣀⡀⠀⠉⠁⠀⠀⠘⣿⡳⣌⡉⠹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠄⣠⣤⡀⢿⠇⠀⢀⡟⠋⠉⠀⠀⠀⠀⠀⠀⣀⣄⠀⣹⣿⣏⣩⣼⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠟⠉⠈⢻⡿⠀⠀⠈⠁⣀⡶⠚⠉⠛⠶⣄⠀⠘⣿⣿⣿⣯⡀⠉⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣨⣇⠸⣿⡀⣹⡟⠁⡀⠄⠀⣹⣿⣄⠀⠀⠀⢀⣹⡷⠄⢸⣿⣳⣟⣹⠆⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠰⠿⠋⠀⣉⠹⣦⡉⢷⠈⠛⣷⠁⠐⢠⣿⠟⠛⠛⠛⠛⠛⠉⠀⠀⠘⣇⣻⠏⠁⡰⣶⣄
⠀⠀⠀⠀⠀⠀⢠⡷⠜⠉⠷⠘⠉⠀⠈⣶⡆⠘⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡇⣰⡛⠷⠈⢽⣾
⠀⠀⠀⠀⠀⠀⠀⢰⢠⣤⠀⠀⠀⠀⠀⣿⣧⢦⣤⣤⡀⠄⢠⡶⠚⠃⢀⣴⡀⠀⠉⠙⡏⠀⠀⠀⠀⡆⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠾⣄⣀⣠⡀⠀⠀⠀⢿⡟⢸⢿⡿⣤⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠁⠀⠀⣀⡀⠓⠘⣄⣴⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠻⢿⠀⡏⠺⠋⠃⠀⣸⡿⡟⠀⠀⠀⠀⠀⢯⡙⢉⠶⣶⠻⠛⢿⣠⣽⣿⣶⡖⢻⣴⣆⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠓⠀⢠⡴⠀⠀⢀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⢸⣹⣧⠀⠛⠻⢿⣿⣱⡆⣼⠁⠛⠉
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣞⣣⠞⠀⢰⡆⠉⠁⠈⣃⡄⠀⠀⠀⠀⠀⠀⡸⠟⠛⢻⡙⠿⡆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠉⠻⠟⠀⠀⠈⠉⠁⠀⠀⠀⠈⠁⡶⠀⠙⠋⣤⣾⡄⠀⢀⣾⡁⠀⠀⣸⣻⣦⣄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣄⡀⠀⡀⠀⠘⠀⠀⣀⠀⠀⠀⠸⡇⠀⣀⠴⢿⣿⣿⣿⣿⣿⣷⢄⡠⢻⣿⡏⣿⣷⣾⡟⠛⠿
⠀⠀⠀⠀⢿⣤⡖⠀⠀⠀⡤⠀⢠⠉⡐⠙⠶⣀⢾⠃⢁⢀⠧⢿⣿⣿⣿⠀⠀⠀⡇⣾⣉⡙⢿⠋⠉⠓
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣄⣼⠧⣾⣷⣄⠀⠘⡆⣠⠊⠏⢀⣾⣿⣧⣤⣄⠀⣶⠀⠹⠟⠉⠒⠉⠉⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢰⡧⠁⠀⠀⠁⠈⣿⣁⣨⣿⣇⠀⢹⡁⡆⠐⢾⣛⣿⣿⣿⣿⡽⠷⠶⠶⠦⠄⠀⠀⠀⠀⣀⡀⠀⠁⠐
⠀⠀⠀⠀⠰⠶⠶⠆⠿⠅⢰⡉⢾⡿⠿⠉⣡⠄⢺⣟⣋⡡⠀⠈⢿⣿⣿⡿⠿⠿⠿⠛⣻⠿⠿⠗⠂⠼⠛⠁⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢘⡀⢀⣾⡟⠁⢀⠀⢁⣥⣰⡏⠉⠐⢂⣀⡐⣚⢃⣀⣥⡶⠖⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠆⠀⠠⡿⠷⠿⠐⠂⠛⠋⠉⠙⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠄⠁⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⢿⣿⡟⠀⠀⠀⠀⠀⠉⠙⢩⡀⠢
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠈⠛⢻⡿⣟⣿⢫⡀⠀⠀⠀⢀⣤⣾⣿⠀⠠⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠲⢏⣩⠤⡁⠀⣇⣴⣿⣿⣿⣷⣤⣀⠀⠈⢿⠂⠀⠀⣀⣒⣀⣀⠐⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣿⠋⠀⠀⠀⠀⢢⡀⠈⠂⠀⠀⠀⠸⣿⣿⣿⣫⡾⢿⣿⠓⠒⢺⠀⠀⠀⠙⠿⣿⣿⣿⣦⡙⡿⠀⠀⣠⣤⣤⣤⣀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣶⣶⣾⣷⣶⣧⡀⠀⠉⠀⠀⠀⠈⠫⠋⠁⠀⠈⠉⠀⠀⢠⠀⠀⠀⠀⠀⠈⠙⠛⠩⠂⠀⠀⢸⣿⣿⣿⣭⡙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢛⣿⢿⣿⣿⣿⡇⠀⠀⠀⠄⠀⠀⠀⢠⡼⠀⠀⠀⠀⠈⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣟⣿⢛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣾⣿⡷⠚⠛⠁⠇⠀⠀⢠⣾⣶⡆⠋⣸⠃⠀⠀⠀⠀⠀⠈⢁⡅⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣿⠀⠀⡠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠴⠏⠿⠃⠀⠀⠐⠶⣦⣈⡁⣉⣻⣴⣾⣿⠀⠘⠀⠀⠀⠀⠐⣿⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⣿⣆⣼⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⡿⢧⡟⠀⠀⠆⠀⣶⠀⡠⠛⢁⡗⠀⠀⠉⠙⢷⣶⣿⣧⣀⣿⣿⣿⠀⡁⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⢠⡆⢀⠀⢸⡞⠀⣰⠟⣇⠀⠀⠀⢤⣶⣾⠟⠉⢹⣷⡿⠋⠠⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⢸⠀⠀⠀⠀⣿⠀⠈⠳⣷⣄⣀⣻⣯⣤⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣆⣀⣀⡀⠀⡀⢀⣠⣤⣴⣷⣿⢸⡄⠀⠀⠀⠏⠀⠀⠀⠀⠉⠉⢉⣡⣾⣿⠋⠀⡀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠛⢡⣢⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠿⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡛⠋⠁⠀⠈⢑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⡜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣀⡀⠀⠀⠀⠀⢿⣿⣤⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠙⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣒⢦⠀⢠⣶⣟⣿⠀⠀⠀⠀⠀⢘⣻⣷⣦⠀⢠⠖⣒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡆⠀⠙⠛⠿⠛⠁⠀⢸⡿⠿⢹⠀⠀⠀⠀⠀⢸⡙⠿⣿⠀⠀⠙⠻⠟⠛⠁⠀⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣷⣀⡌⠃⠀⠀⠀⠀⠈⢠⢠⣇⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⠀⠀⠀⢈⣙⣿⣉⣿⡧⠀⠄⠀⠀⠀⠀⣿⣇⣹⣟⣉⡁⠀⠀⢠⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠆⡆⠀⠀⣭⡿⢁⣽⠇⠀⠀⠀⠀⠀⠀⢻⣇⠘⣯⡅⠀⠀⡔⠘⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⢸⡝⠒⠤⠤⠟⠹⠏⠀⡀⠀⠀⠀⠀⡀⠈⠿⠝⠧⠤⠴⠲⣽⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠎⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣶⡄⠀⠀⣴⣶⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡆


⣿⠿⠿⠿⢿⠟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣛⣛⣿⣟⣿⣿⣿⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⣤⡒⡀⢨⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣻⣿⣿⣿⣷⣦⡀⠁⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣦⣈⠙⠻⣿⣿⣿⠇⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣯⢛⣦⣈⠻⠿⣟⡓⠓⠸⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⣧⡝⢽⣿⣝⠳⣦⣤⣭⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠋⠁⠘⠇⠀⠀⠉⠛⠧⠀⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⠀⠀⠀⣄⣀⣀⠲⠴⢀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠒⠹⣯⣔⡌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⡀⠨⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠘⢆⠀⢀⠸⣿⣦⣄⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣖⣿⣿⠋⣿⣿⡆⠀⠑⠤⣌⡉⠉⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⠖⠒⠄⠀⣿⣿⣆⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠋⠀⠀⢸
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣹⣿⣦⡁⠈⠃⠀⢠⣹⣿⣿⣿⣾⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⡤⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠤⠬⠟⣟⡻⢦⡀⣠⣷⣟⣽⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡾⣷⠚⣰⣤⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣾⣿⣟⣻⣿⣿⣽⢹⣷⣶⣾⣽⣳⣽⣾⢿⣯⣷⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠘⣧⠁⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⡟⣻⣇⣠⣽⣿⢭⢛⡿⣻⣾⣿⣿⣽⣿⡿⣯⣿⣿⣻⡄⠀⠀⠀⠀⠀⠀⢿⠀⠘⠆⠀⠘
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⣿⣿⣿⠙⣿⣿⣯⢿⣿⣷⣿⢟⢻⣟⣿⢹⡟⣿⣹⣿⣞⣿⣿⡛⠁⣹⣿⣿⣷⠀⠀⠀⠀⢠⢼⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣻⢿⣿⡟⣷⣽⣿⣿⣯⣽⡿⠿⢏⣙⣼⣄⡍⣿⡍⠩⣿⣿⣿⠿⣿⠷⣿⣿⣿⡿⢆⠀⠀⡄⠹⣄⠑⠶⠠⠀⠞⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡿⣿⣟⣓⣦⣿⢀⣿⣿⣿⣋⣷⣿⣿⣿⣛⠿⡿⢿⢦⡉⢿⠀⣙⣽⡿⢕⣷⠀⣿⣿⣿⣿⠊⢀⠜⠁⠀⠈⠉⠒⠄⠈⠐⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣷⣿⣿⣓⣿⡟⠋⣾⣿⣿⡷⣿⣿⣿⢿⡿⣷⣚⠀⠀⣰⣆⠐⣦⠈⠋⠊⠹⠁⢠⣿⣿⢟⢏⣀⣥⠐⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠈⠀⠀⠀⣼⡯⣽⣞⠟⠡⠟⠳⠦⣄⠨⣽⠈⣤⣾⡿⢿⣿⣿⣟⠀⠁⠀⠈⠁⢸⠃⠀⠀⠀⣠⠢⠀⣿⣧⡾⣿⠁⠁⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⣯⠟⠋⠡⠚⢓⠀⠀⠀⠙⠒⠾⣦⡀⠉⡙⢘⡋⠉⠁⠀⠀⠀⠀⢀⠎⠀⠀⡄⠀⠁⠄⢰⢿⢧⡾⠃⠀⠀⠀⠂⢀⠆⠀⢠⠃
⠀⠀⠀⠀⠀⠀⡁⠂⠀⠁⠀⢀⢎⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⡤⠃⠀⠀⠀⠱⡀⠀⠄⡘⢰⣸⣦⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⠈⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⣀⠉⠀⠠⠄⠐⢈⠂⠀⠱⠤⢣⣉⠉⠀⠀⠀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⡀⢀⠀⠐⠒⠒⠀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⡶⠖⡶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠥⠄⠒⠀⠀⠘⠀⠀⠝⣆⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢤⡀⠀⠀⡀⠉⠁⠈⠁⠓⢄⢀⡾⣟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⢛⠀⠀⢸⣷⡳⣤⠃⠀⠀⡤⣤⡤⠤⠈⠀⠁⠸
⠀⠀⢸⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡖⢀⠘⠋⠀⣿⠷⠟⠿⡿⣄⡤⠀⢇⠌⡭⡑⠀⠐⠪⡺⠢⡀
⠀⠀⠀⢻⣏⠉⢛⡲⠶⠶⣶⠒⠒⠀⠉⠀⠀⣠⠂⡜⣦⣤⡿⡒⠂⠐⠳⠾⠕⠚⠛⠳⣅⠘⢦⡉⠀⠹⢄⠣⡀
⠀⠀⠀⠀⠙⢷⡀⠍⠀⠀⠈⠓⠀⠀⢀⣠⠾⠁⠀⠿⣿⢄⣠⠿⡵⢰⠠⠤⡀⠉⠀⠀⠈⠢⣀⡏⠠⣀⠂⠀⣬⢢⡀
⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣶⣶⠶⠾⠟⠡⠀⠀⠀⠂⠘⠂⠀⠀⠀⠈⡳⢤⣉⠑⠦⡄⠀⠀⢹⡙⢶⡉⢳⡇⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠶⠀⠀⣾⠠⠆⠀⠀⠀⠀⠀⠀⡇⡨⠀⡄⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣑⣠⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠂⠉⢉⡇⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣯⡀⠙⣿⣷⣄⠁⠘⠓⣌⠻⠶⢶⣬⣽⡶⠀⠀⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠠⠀⡁⠀⠀⣠⣤⣦⣤⠀⠬⠓⣿⣿⣴⣴⣿⢿⣾⣣⣄⠀⠌⠑⢄⠀⠉⢹⡧⢀⡀⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⡤⠀⠙⠀⢢⣀⡐⢀
⠀⠀⠀⠀⠀⠀⢹⠇⠴⠇⠐⠶⠶⢼⣿⠁⣼⣷⠟⢻⣿⢛⣿⣿⠶⣶⡶⠀⠈⠀⠀⠐⠠⢀⣠⡉⠀⠘⠃⠈⠀⠀⠀⠰⠆⠀⠈⡟⢻⣦⣀
⠀⠀⠀⠀⠀⠀⠏⣼⠃⡆⠀⠀⠀⠸⣷⣿⣿⠃⠀⠹⣿⣿⡿⢷⣦⣬⣁⣄⠀⠀⠀⠀⠀⣠⣉⡒⣰⠂⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠶⢿
⠀⠀⠀⠀⠀⠀⣸⣿⣤⣧⠀⠀⠀⠘⣿⣿⠁⠀⠀⠰⢿⡟⠀⠸⣿⣷⣿⡿⢠⡼⠃⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀
⣀⡀⠀⠀⠀⠀⠉⣽⣿⣿⠀⠀⠀⢀⣿⡟⠀⠀⠀⢀⣿⠂⠀⢰⣿⣿⣿⣿⡏⢻⠓⡶⠀⣀⣀⠈⠀⠀⠀⢀⠀⠀⠐⡀⠀⡒⠀⠀⡇⢠
⡏⠀⠀⠀⠀⢠⠀⠀⠻⣿⣦⠀⠀⣼⡟⠓⠀⠤⠤⢼⣟⠃⠀⠘⣿⣿⣿⣿⡄⠀⠀⢀⣠⣿⠟⠛⢻⡶⣇⠈⠀⠀⠀⠀⠀⠁⠀⢀⣁⠀⣤
⡇⠀⠀⠀⠀⢡⠀⠀⠀⠈⠡⣉⢰⣿⣦⣀⡀⢠⣤⣄⣹⣷⣆⠀⣵⣿⢻⣻⠇⠀⠀⣿⣯⣿⠆⣤⣀⣰⣿⣿⣧⠀⢀⠀⠀⢀⡴⠛⠁⣠⣶
⡅⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣦⣼⣿⠋⣽⢿⣿⠟⠛⣑⣿⠿⠋⠙⠓⠆⠀⠙⢧⡀⣀⠉⠻⣿⡿⢿⣟⡀⠸⠍⠀⠘⣷⣁⠀⣿⣿
⡇⠀⠀⠀⠀⠐⠀⢀⣤⣤⣤⣼⣿⣿⣿⣿⠉⢀⣠⣾⣿⣦⣄⣿⣟⠀⠠⠀⠀⢠⣶⡀⢸⣿⠛⣄⠀⢨⣇⠊⢙⡣⢤⡤⠀⠀⠹⣯⡔⠃⣿
⡇⠀⠀⠀⠀⠀⣡⣾⣿⡟⠙⠛⠟⢿⣿⡿⣷⣌⡈⠻⣷⣦⣽⡇⢿⠘⢦⡰⣾⠛⠉⠁⢈⣿⠉⣽⣁⣼⣫⣤⣿⠋⣀⠀⠀⢤⠀⠈⢻⣾⣿
⠃⠀⠀⠈⠀⣼⣿⣋⣉⣉⣰⠂⠄⠀⣼⢱⢿⣿⡏⢰⣿⡜⣯⣿⣿⣲⣴⡿⢿⢩⣖⠒⠚⢻⣿⣿⣿⣺⣶⣿⡿⢋⣥⣀⣴⠂⢠⡔⣿⣿⣿
⡗⠀⠀⠃⣸⣿⣿⠻⠏⣻⣏⠶⣾⣀⠀⢻⠈⣿⡷⣿⣯⣿⣮⣽⠋⠍⠀⢀⠤⡾⠛⣛⣻⣿⣿⣽⣿⣿⣿⣏⢰⡟⢿⣏⠀⠀⠈⢿⣿⣿⣛
⡇⠀⠀⣱⣿⣿⠃⢠⣴⢏⣩⣇⡘⣀⠉⠋⠀⢸⣿⣧⣿⣩⠟⠛⠤⣤⣶⣿⣦⣾⠛⠉⣹⣿⣟⣿⢻⡿⣿⣿⡟⠻⣦⡉⠀⠀⠀⠘⣿⣿⣿
⠃⠈⠀⣿⣿⡇⢠⣟⣵⣾⣿⠿⢷⣿⠶⣤⣤⠜⣿⡿⠿⢿⣿⢠⡟⢶⠫⠉⠉⢠⡠⢶⡿⣿⡿⠿⢿⠛⠟⡉⢳⡔⢻⣷⠄⢠⠀⠘⢩⡿⣽
⡀⠀⠀⣿⣿⠃⢿⡿⣿⢷⠸⠀⣾⣀⣀⢈⣵⠀⢿⣿⡴⢛⡿⢻⣝⣿⣉⢀⡴⢋⡁⠀⣤⣾⣄⣀⣀⣀⣠⣤⣾⣏⡿⣿⡇⠈⠀⢰⢳⡗⣘
⡄⠀⠠⣟⣿⣷⠀⣤⠛⡜⠀⠀⣽⣿⣷⡞⠁⢀⣾⣏⢷⣾⣷⣾⠛⠋⣬⣤⣽⢿⣷⣼⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣸⢰⣿⣿
⡄⠀⠀⢸⣷⣿⢸⣿⣦⡇⠀⠀⢸⣿⡉⣇⣀⣾⢿⣏⢿⣿⣿⣿⣦⣼⣿⣿⠟⣿⣿⣿⣿⣏⠙⢿⣿⣷⣼⣿⣿⣿⣿⣿⠀⢿⠀⢸⠈⣿⣿
⠀⠀⠀⣼⣿⣿⣦⡘⣿⡇⠠⡄⢸⣿⠁⣿⣿⣿⠸⣿⢺⣿⣿⡿⠋⣿⠀⣿⠀⣿⣿⣿⣿⣿⡀⢷⣿⠿⣿⠿⡟⢹⣿⣿⡄⠐⠄⠈⠀⢿⣿
⠂⠀⠀⣿⣿⣿⣽⢻⣿⣷⣀⣇⢸⣿⠀⣿⣿⣿⣷⣿⡟⣿⣿⣇⣨⣿⡀⠀⠀⣿⣿⣿⣿⣿⣻⣶⠟⢸⣿⣾⡏⠘⣻⣧⡄⠐⠀⠀⢴⠟⣿
⠀⠀⢠⣿⣷⣿⢹⢸⣿⡟⠻⣿⣿⣿⡇⢼⣿⣹⡿⣿⡄⢻⣿⣿⠿⣿⡇⠈⢸⣿⣿⣿⣿⣿⣿⣿⣤⣸⣿⣿⡇⠀⢻⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⡏⠀⠈⣿⠀⠀⢸⡿⠸⡇⢸⣧⣽⣇⠸⣇⠈⠛⠁⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⡇⠀⣿⣿⡇⠀⡄⠀⠰⠸⣿
⠀⢠⣿⣿⢻⡇⠀⠀⣿⠀⠀⢸⡇⠀⡇⠸⠉⠛⠉⠀⠹⡆⠀⠀⠀⢸⡇⠀⢠⣿⣿⣿⣿⣿⣿⢿⡿⢹⣿⣿⠃⠀⢹⢿⡏⠰⠇⠀⠂⠀⣿
⠀⢸⣿⣿⢺⡇⠀⢀⣿⠀⠀⢸⣇⠀⣟⣠⠀⠀⠰⡆⠀⠙⡄⠀⠀⢘⡧⢀⣸⢿⣿⣿⣿⣿⣿⣺⣿⢸⣿⣿⠀⠀⣼⣿⣷⠈⠀⠄⠀⠀⢹
⠁⣸⣿⣿⢻⣇⡀⢨⣿⠀⠀⢰⣿⣀⣿⠉⣦⣄⣴⣿⡴⠟⣁⣦⣠⣼⣗⠀⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠹⢾⣿⠔⠀⠀⠀⠀⣸
⠀⠙⣿⣇⣾⣣⣄⣠⣟⠋⠁⢈⣿⡁⢘⣷⣿⣿⣿⡿⠋⠉⠉⡀⠀⠀⢿⠀⠿⢿⡿⠉⠻⠿⢿⠿⣿⣿⣿⣿⣄⡀⢀⣼⣿⣷⣤⣄⣰⢠⢻
⡄⢀⡀⠊⠛⣿⣿⣏⠻⠷⡟⢩⠌⠉⢹⣿⣿⡏⠙⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⠀⠀⠀⠈⠒⠈⢿⣿⣿⠟⠛⠛⠇⠤⠉⠀⠀⠉⢉⣶
⠁⠀⠀⠀⠀⣿⣿⠁⠀⠀⡧⠠⠄⠀⣸⣿⠻⣿⡀⠀⠀⢠⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⡃⠀⠀⣿⡟⠀⠀⠀⣦⡀⠀⠈⠠⣴⢋⣁
⠄⠀⠀⠐⠀⠈⣿⡄⠀⠀⡆⠀⠐⢀⣿⣿⠀⠘⣿⡆⠄⠀⠆⠀⠀⠀⢸⡇⠁⠀⠀⠀⣀⠀⠀⠀⠀⠰⣿⡇⠀⠀⢸⣿⣷⣄⡄⠀⠀⡍⠉
⢀⡠⣶⠀⢶⡄⠸⣧⠀⠀⣿⣶⣤⣼⣿⡇⠀⡀⠈⢿⣶⡀⠀⢀⠀⠀⢸⣧⣯⣁⡀⠀⠈⠀⠀⠀⠀⠘⣿⠇⠀⢠⣟⣿⣿⡏⠘⣁⠀⠀⠬
⠲⠶⣛⠳⢈⣁⢠⣿⣧⠀⣿⣿⣿⣿⠿⣷⡟⣡⡶⣾⡿⣿⣶⣼⣏⣷⣈⠙⣦⣼⣿⡇⠀⠀⠀⢈⣀⢸⣿⠀⠀⣄⠋⢹⣿⡄⣾⣁⠁
⡀⠀⠉⠸⠧⣴⣷⣿⠛⠉⢛⠛⠉⠀⠁⠙⠛⢋⡙⠛⠛⠛⠛⠻⠉⠉⠛⢋⠉⢛⣉⡀⢐⣷⣶⣶⣦⣾⡏⢀⡀⢀⣉⣸⣟⠀⠗
⠁⠀⠀⠀⠠⣴⣿⣏⣠⣀⣨⡅⠀⠀⠀⠀⠀⡀⠉⠀⠠⠀⠠⠀⠀⠀⠠⠅⠐⠿⣿⠙⣿⣿⣧⣤⡼⠿⣦⣾⣿⢿⣽⢿⣿⠟
⠀⠀⠀⣌⠀⠀⠀⠉⠛⠉⠛⠉⠁⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⢿⡿⢧⣤⡟⠉⠁⠀⠉⠉⢥⠀⠐
⠤⠀⠒⠀⠦⡀⠒⠀⠀⠀⠠⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠘⢻⠟⠺⠷⠿⠏⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣠⡄⠀⠀⠀⠂⠀⠀⠒⠂⠀⠀⠀⠒⠂⣰⡄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣄⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡏⠀⣤⢶⡄⠀⠀⢠⣶⠠⣦⠰⣦⠄⣼⡇⢁⣡⣦⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠁⢀⣐⣀⣈⣉⣠⠤⠵⠶⠀⠠⠄⠀⢸⡇⠀⣿⠈⠀⠰⠂⢀⡿⠀⣻⡀⣿⠀⢸⡇⠈⣷⠈⠁⠀⠀⠚⠃⠐⢂⠠⠆⠈⣀⠐⡀⠀⣋⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣽⣿⣬⣿⣮⣤⣤⣴⣾⣿⣤⣽⣯⣿⣤⣾⣥⣬⣿⣦⣤⣤⣤⣤⣤⣤⣬⣤⣴⣦⣤⣤⣭⣤⣤⣤
⣿⣿⣿⠛⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⢀⡄⣾⡟⣴⣶⢖⣤⣄⣀⠀⠀⠀⠀⣴⣿⣶⡍⠀⠉⢳⣶
⠀⠀⠀⠀⠀⣤⡹⠇⠙⠃⢻⡇⣾⣿⣿⠿⠳⡄⠀⠀⣥⣶⣶⣦⣄⠀⠈⠁
⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⢿⢏⣴⣿⣿⣿⣧⠀⣿⠟⣫⣭⣭⣄
⠀⠀⠀⠀⢿⡷⠀⠀⠀⠀⠀⠀⠀⣿⡿⢛⣩⣭⣭⣅⢠⣾⣿⡿⠿⠿⢧
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⢋⣴⣿⡿⠿⢛⡛⠊⢋⣥⣶⣶⢷⣶⣦
⠀⠀⠀⠀⣠⢰⣿⡟⣴⣿⡟⣫⣤⣼⡿⢋⣴⣾⣿⣿⣿⣿⣿⢿⠃⢻⣿⣿⡇
⠀⠀⠀⢰⣿⣜⢿⡇⣿⠏⣼⣿⡿⠛⣡⣿⣿⠟⢋⣉⣉⣉⣤⠀⠀⢸⣿⣤⣙
⠀⠀⠀⢠⣙⠛⠀⠁⠻⢸⣿⢿⡀⠻⠿⠟⢡⣶⣿⣿⣿⣿⡇⠀⠀⢺⣿⢿⣿⡇
⠀⠀⠀⠘⡿⠀⠀⠀⠀⠈⠟⣸⣿⣦⡀⠀⠀⠙⠿⣿⣿⣿⡀⠀⠀⢀⣠⡶⠈⠥⡀
⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⠻⡟⣿⣿⣦⠀⠀⠀⠘⢿⡷⠅⠰⣴⣿⣿⣧⣄⡀⢻⠂⠀⣤⣤⣀⡲⢤⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢿⡟⣃⣠⣤⣤⣥⡀⠠⡆⠀⠻⠿⠟⠉⠻⠏⢹⠀⠀⠈⠙⠿⠿⠎⠁⢠⣾⣿⡳⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢹⣿⣿⣿⠇⠀⠀⠀⠀⡀⣴⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⡾⡿⠁⠀⣀⡀⠔⠿⣃⣴⡇⠀⠀⠀⠀⣴⠀⠀⠀⠀⢦⣤⣤⣿⣀⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠁⠀⠀⠀⠉⠻⣿⣿⡿⠁⠐⠀⢄⡠⠛⣰⠀⠀⠐⢶⣿⣥⣽⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠀⡾⠙⢤⡀⠘⣿⣋⣨⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣄⣀⣀⣀⣠⣤⣶⣧⠆⠀⠀⠇⠀⠀⠉⠛⠉⢽⠟⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠿⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠸⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠿⠋⠀⠀⠀⠀⠉⠛⠿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠛⠛⠉⠉⠉⣉⡵⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠐⠛⠒⠒⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠛⠂⠀⠀⠀⣀⠠⠤⠤⠤⠦⠤⠤⠤⢤⣀⣀⠀⠀⠀⢀⣀⣀⣤⠬⠭⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠔⢩⠃⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⢄⠀⠀⣷⠀⠀⠀⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⡆⢀⡇⠀⠀⣀⣀⣀⣈⣙⡛⠀⠀⠒⠋⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠤⠬⣍⠓⠾⠀⠀⠀⢸⠁⠀⠛⠀⢸⠍⣹⠈⠉⠉⠉⢣⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⠚⠲⢤⡀⣠⠞⠦⣅⣀⠀⢀⣠⠎⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠙⡇⠈⣹⢧⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠉⣀⠤⠤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠁⢱⢤⡀⠀⠀⠀⠀⠀⠁⠀⢤⣀⣀⣀⡼⠀⣸⠁
⠀⠀⠀⠀⠀⢀⡠⠖⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡴⠁⢀⠏⠓⡦⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⢏⢀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⣠⣋⣀⣘⣁⣰⡭⠭⠟⠒⠂⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⡤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⢦⣀⡤⡔⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣹⢃⣀⢿⡄⠀⠀⠀⠉⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡴⡊⠁⢀⡔⠐⠒⠒⠙⢷⠋⠀⠀⠀⢸⣤⠟
⠀⠀⠀⠀⠀⠀⠙⠲⠥⠞⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠸⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⣤⢤⡤⠴⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣸⣸⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡲⣾⣤⣤⣽⡤⢤⣷⡖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠭⠟⠚⠉⠑⠺⠿⠾⠟⠃

⠀⠐⠂⠀⡄⢳⢄⠀⠀⠀⠀⠀⢼⠂⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠀⡴⠣⠁⣴⡇⠀⠀⠐⠀⠀⠠⣈⠞⠈⣆⠀⢸⢂⣼⣿⣿⣬⣀⣀
⠀⢀⠀⢰⡅⠘⡇⠈⠀⠀⠀⠀⣸⠀⢠⣾⡿⣿⡇⠀⠀⠀⠀⠀⡼⠃⠀⡰⢁⠃⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠀⣸⠜⣽⠟⣿⠻⣿⣿
⠀⢸⣧⡘⠀⢀⠙⡀⠀⠀⠀⠀⡏⣰⣿⠟⠁⣿⣇⣤⣶⣾⣿⣿⡷⣤⡐⠀⣸⢒⠠⠀⠀⠀⡔⡇⠀⠀⢀⡄⠀⣿⣼⡿⢀⡏⠀⠑⠘
⠀⠸⣿⣧⠀⡀⠀⠸⡀⠀⠀⠀⣳⣿⠏⠀⣠⣟⣻⣿⣿⢫⣬⣄⢩⣷⡟⣆⡧⠈⠀⠀⢀⠎⠀⠀⠀⠀⢸⣷⣴⣿⣿⠃⢸⡏
⠀⠀⠸⣿⡄⠀⠀⠉⢿⡄⠠⠀⣿⡟⠀⠘⣿⠐⣿⣿⣿⡈⣙⡋⠠⣭⠠⣿⡱⠀⠀⢠⠏⠀⠀⠀⠀⠰⠇⢀⡻⣿⠁⠀⣾⠁⠀⢡
⠀⠀⠁⣿⣧⠀⠀⠀⣸⣷⠀⠀⣿⡇⠀⠀⠙⣦⣽⣿⣿⣿⣿⣦⣠⠄⠂⢺⠄⠀⢠⠏⠀⠀⠀⠀⠀⣽⣾⣥⠔⠋⠀⢰⣿⠙⢶⢾⡇
⠀⠀⠀⢹⡏⣷⡀⠀⢸⣿⡅⢠⣿⡇⠀⣐⣶⣿⠿⠿⢿⣿⡟⠻⣿⣿⣿⡿⠀⣠⠋⠀⠀⠀⠀⣠⣾⣿⣿⠃⠀⠀⢀⣿⣿⡆⠀⢠
⠀⠀⠀⢸⡇⣿⡇⠀⢰⣿⠃⠸⣿⠀⣸⣿⣿⣿⣶⣶⣿⣿⣿⣦⣤⣤⣿⣀⣰⠡⠴⣶⣾⣾⣿⣿⣿⣿⠏⠰⠀⢠⣾⡿⣷⠈⠀⠀⣻
⠀⠀⠀⣸⣷⣿⠂⠀⢸⣿⠀⢀⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣉⣩⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⢏⠅⠀⠀⢀⣾⡿⠁⠈⠀⢀⣼⣿
⠀⠀⢀⡟⣿⣿⠀⢀⣾⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣼⣿⠿⢡⠟⠁⠀⠀⣾⣿⠃⠀⠀⢠⣿⡿⢹
⠀⠀⢸⠁⣿⡏⣠⣾⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠉⠒⠶⣶⣾⡿⠛⢀⡏⠀⠀⠀⣼⣿⠏⠀⠀⢠⣿⡟⡰⠿
⠀⢀⡏⠀⣿⣷⣿⢧⠤⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⢀⠀⡇⠀⠀⣴⣿⠏⠀⠀⣴⣿⠏⠀⠎
⠀⠸⠧⠄⣿⣿⡇⠘⣦⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⣿⣿⣿⡆⢀⠂⣴⣶⣦⠀⠀⠀⠈⣃⠀⢠⡇⠀⣰⣿⠏⠀⣠⣾⡿⠃⠀⠁
⠀⠈⠀⠀⣿⣿⣿⠀⠹⡄⠀⠀⠀⣿⣿⣿⣿⡏⠀⢸⣿⣿⣧⢤⣾⣿⣿⣿⣿⣄⠀⠀⠑⢄⢸⠀⣴⣿⠏⢀⣾⣿⠟⠁
⠀⠰⠤⢀⣿⣿⣿⡄⠀⢳⠀⠀⠀⢿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠙⢾⣿⢋⣴⣿⠟⠁⠀⠀⠀⠀⠀⡠⠚
⠀⢀⠀⢸⣿⣿⡿⠸⢄⠀⢇⠀⠀⠘⣿⣿⣿⣧⣠⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠻⢷⣆⡀⠀⢱⣿⡿⠃⠀⠀⠀⠀⠀⢀⢞⠱⡀
⠀⠀⢀⣿⣿⣿⡇⠀⠀⢵⢼⡀⡈⠀⠙⣿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣶⣶⣦⣤⣦⣤⣤⠻⠿⢶⣶⣿⠛⠻⢶⣖⡂⠀⢀⡿⡘⡄⢳
⠀⠀⣿⣿⣿⣿⢇⠀⠀⠀⢧⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠻⢿⣿⣷⡀⠀⠀⠙⢧⡀⠀⠈⠉⠋⣿⠀⣡⠘⢉
⠀⢸⣿⣿⣿⡇⠘⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣟⠿⠿⣿⣿⢿⣷⣦⠀⠈⢻⣿⣿⣶⣄⣀⡀⠙⣷⣄⠀⣸⡿⣍⠀⠀⠈
⠀⣿⣿⣿⣿⠁⠀⠈⢧⡀⠀⠀⠀⢀⠀⠀⠀⡠⠞⠤⣈⡁⠀⠀⣰⣿⡇⠀⠀⣿⣷⡆⠈⣧⠙⢿⣿⣿⣿⣶⣌⣿⣿⣿⠃⠈⠳⢤
⠀⣿⣿⣿⡿⠀⠀⠀⠈⢐⢄⠀⠀⠀⠘⢦⡞⠁⠀⠂⠈⠃⢠⣾⠟⣼⠈⠃⠀⣿⣿⣷⡀⣷⠀⠀⢙⡿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⡝
⠀⢻⣿⣿⠀⠢⠀⠀⠀⠈⢢⠣⡀⢀⡴⠋⠀⠢⠀⠀⠄⣴⣿⡋⢸⠏⠒⠄⠀⠿⣿⣿⣧⢹⠀⠘⠁⠀⡈⣻⣿⣿⣿⣿⣿⣿⣷⣄⣱
⠀⣿⣿⣯⠀⢀⠀⡀⠀⠀⠀⢣⡸⠎⠀⠀⠀⠀⠀⢈⣼⠟⠁⣇⣾⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄⠂⣀⠞⠠⠃⠎⠳⣌⢉⠉⢻⠿⠿⢟
⠀⣾⣿⣿⠀⠈⠀⠛⠐⢂⠄⡟⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠠⢸⠇⠀⠀⠄⢀⡠⣾⣻⣿⣿⡁⠌⠡⠀⠀⠘⠀⠀⠟⣦⠹⣤⣀⢃⡘
⠀⢻⣿⣿⠀⠀⠀⢠⠜⠇⡄⠀⠀⠀⠀⠐⠀⢰⣿⠃⠀⠀⢀⡏⠀⢀⠤⠚⣡⠞⠁⣸⣿⣿⡗⠈⠀⠀⠀⠀⠀⠀⠀⠁⡷⡀⠈⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠙⠲⠤⠤⠖⠒⠒⠛⠛⠛⠛⠋⠒⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣌⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⡴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⢸⣧⠴⠖⠛⠲⣄⠀⣄⣻⠉⠓⠲⠿⡍⠓⠲⠦⠿⠏⠛⠲⢦⣄⣀⠀⢦⣤⣀⠀⠑⠦⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠢⢾⡀⡮⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢀⡞⠠⠌⠉⠓⠛⠀⠉⠓⠦⣌⣷
⠀⠀⠀⠈⠛⣟⣛⣒⣒⠒⠒⠠⠤⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⠶⢦⠀⠉⠁⠀⠀⠀⣀⣀⠀⢀⠴⠒⠓⣄⢠⠞⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⡠⠔⣊⣁⣀⣀⡤⠤⣤⠤⣤⣤⡬⠍⠛⠛⢛⣿⡿⠶⠶⠶⠦⠈⠧⣄⣀⣀⣠⡄⠀⢠⡴⢋⡽⢩⠅⢰⠋⠀⠀⠀⡜⠈⣦⠄⠀⠀⣸
⠚⠋⠉⠉⠉⢻⡅⢠⠖⢾⠉⠹⡇⠀⣵⣔⣚⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢻⡉⣵⢻⠀⢾⠀⠀⠀⡰⠃⠀⠈⢟⠛⣛⡡⠤⣄
⠀⠀⠀⠀⠀⠀⠳⠜⠢⡀⢣⠀⡇⠐⢋⣁⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣⣸⣿⠀⠀⠑⠒⠊⠁⢀⠀⠀⠨⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠉⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠳⣼⡄⠀⠀⠀⠐⢦⠉⠄⠀⢀⣀⣀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣇⢠⣤⣤⣀⣀⡀⣀⡰⢛⣶⡤⢾⣿⡆⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣙⡆⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠒⠛⠋⠉⠉⠹⣄⣈⣉⢙⡄⠀⠀⠉⠀⠁⠸⡅⢀⣼⠉⠹⣆⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⡟⠉⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣏⡃⠀⠀⠀⠀⠀⠀⠘⠲⢋⡆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢇⠀⠀⠀⠀⠀⠀⣎⠁⠀⠘⠛⠊⠁⠀⠀⠀⠀⠀⡀⠈⢧⣀⠀⠀⠙⠳⢤⣤⣤⣤⣤⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⠀⡾⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⢀⣨⣽⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⢶⢲⠊⡿⠀⠀⠀⠈⠓⠦⣄⣀⣀⡀⠀⣰⠁⣧⠔⣹⢿⣿⣤⡀⠀⣴⢮⣉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣦⠁⠀⠀⠀⠀⠀⢰⠃⡏⠉⢩⠏⠁⠀⠉⠀⠀⠀⠙⠦⣙⢺⢁⣤⡨⢏⡁⣹⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⣯⣄⡀⠀⠀⠀⠀⢿⠾⠗⠂⢺⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢰⡶⣄⣹⠁⠀⠀⠀⠀⠉⠉⠒⠒⠒⠀⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⡜⠻⠏⡟⡿⡲⠶⡾⢶⣊⠻⠟⠻⠇⠉⢳⠀⣀⡠⠤⠒⠊⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⢀⣃⡤⢖⡽⠿⣧⡀⠈⢑⣲⣭⣛⣓⠶⢲⡏⣟⡓⠒⠶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⣴⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢭⠥⣶⣛⡃⢂⣴⡧⠯⢬⢤⠖⠒⢺⠻⡉⠁⠀⠈⠉⡿⠿⣷⣶⣶⣤⣤⣍⣉⣉⠒⠒⠲⠦⠤⢴⠋⣠⣾⠟⣋⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡸⠀⢸⠉⡏⠻⣳⠀⢸⠾⡆⠀⠀⡇⢧⠀⠀⠀⠀⢹⣤⣤⣤⣌⣉⣉⡛⠛⠻⠿⠿⢿⣶⣶⣶⠾⣋⡥⠊⢹⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⠇⠀⢸⠰⡇⠀⠉⠀⠈⠉⠁⠀⠀⢳⠼⡄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢤⣄⣼⡧⠚⠁⠀⠀⢸⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⣎⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⡈⠃⠈⠀⠀⠘⠀⠋⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⠀⠀⠀⡈⠀⠁⠔
⠀⠀⠄⠀⡄⠁⢐⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⣄⠑⠀⠁⠀⠀⠁⠘⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⠀⠀⠀⡀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠈⠡⠈⡀⢀⣉⡀⠀⠀⠀⠐⠢⠀⠀⠤⠈⢀⠈⠠⠂⠀⠐⠊⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠒⠀⠀⠀⠒
⠀⡐⠠⢀⠀⠀⠠⠠⠀⠀⠤⠀⠀⠀⠀⠙⠒⠀⠑⢀⠁⠀⠀⠁⡈⣁⠀⣈⣁⠀⠀⠀⠀⣄⠒⠈⠁⠁⠀⠠⠁⡔⠂⡀⠐⠈⠠⠀⠀⠁⠀⠀⣈
⠀⠀⠀⢨⠐⠁⠀⠂⣀⠒⠦⠀⠀⠀⠀⢉⡀⠀⢁⠠⠀⠀⠀⠠⠀⠄⠀⠠⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⢏⡠⠀⠀⠀⠀⠀⠀⡀⠢
⠃⣠⠀⡀⢀⠀⡄⢳⡈⠓⠀⠀⠀⠀⢂⠀⡤⠂⠀⠰⠀⡅⠈⠀⠒⠀⠀⠢⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠄⠈⠀⠃⠸⠀⠃⠈
⠀⠀⠀⠁⠘⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠄⢠⠀⠀⢀⠐⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠃⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⡰⢀⠀⠀⠀⠀⠀⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠄⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⢠⣤⢀⠀⡀⣴⣤⡆⠀⣀⣶⣶⣶⠀⣠⣀⣄⠀⠀⠀⣰⣧⣶⣤⡄⠀⠀⢀⠀⢀⣾⡆⠀⠀⠀⢰⡆⠀⣀⡀⣴⣦⣴⣤⣤⣤⠀⢠⡇⠀⠀⣤⣤
⣿⣿⣿⣿⣿⣿⣿⡇⣠⣿⣿⣿⣿⣷⣿⣿⣿⣤⣤⣾⣿⣿⣿⣿⣧⣶⣶⣾⣰⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢸⣷⣶⡆⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⡦⠐⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠈⢻⣿⢿⡿⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⠀⠀⠀⡀⠐⡈⣋⠂⠀⠀⠀⠀⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢻⣿⣿⣶⣄⠀⠀⠀⠀⢨⣴⣿⣿⡿⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠭⠿⡿⠟⠀⠀⠀⠀⠸⢿⡿⠯⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠀⠀⣀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⣴⣶⠆⠀⠀⠀⠀⠀⠘⢧⣁⣙⣛⣉⣵⠋⠀⠀⠀⠀⠀⠐⣶⣦⣌⢲⣀
⠀⠀⠀⠀⢦⡴⣶⣶⡄⠈⠸⠛⠛⠁⠀⠀⡀⠀⠀⠀⠀⠈⢿⣻⢟⡿⠁⠀⠀⠀⠀⢢⠀⠀⠈⠛⠛⠃⠉⢠⣴⣶⢦⣴
⠀⠀⠀⠀⠈⠾⠿⡿⢷⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠐⢋⠀⠀⠀⠀⠀⠀⠀⡾⢿⠿⠷⠃
⠀⠀⠀⠀⠀⠈⡖⢄⢸⣧⣴⡶⠂⢀⠀⠀⠸⢷⣦⡀⠀⢀⣀⢀⡀⢀⡀⠀⢀⣴⡾⠏⡀⠀⡀⠐⢶⣦⣼⡇⣀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⡬⣷⣟⡀⠀⠘⡦⡄⠉⠁⠀⢸⣿⣿⣿⣿⡇⠀⠘⡏⢀⢴⡃⠀⠀⣙⣶⠥⣼⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠙⢿⣤⣌⣿⣁⡴⠊⠀⢻⠀⠀⢈⠌⠉⠻⡿⠋⠁⡀⠀⠀⡟⠀⠙⢦⣈⣿⣁⣤⡿⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠈⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠏⠀⠀⠀⠻⢿⣿⣿⡆⢰⣿⣿⣿⠟⠀⠀⠀⠘⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓


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

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⡀⠠⠖⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠱⠀⠐⠁⠠⠴⠒⠶⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢠⠀⡠⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠤⠉⠀⡀⢀⠴⠛⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠈⠀⡇⡌⡀⠠⠂
⠀⠀⣠⠆⠀⣀⠘⠉⠁⣤⣀⠈⠁⡤⠀⢀⣉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⠤⡁⢠⢀⡜⠐⠕⡊⠀⠠
⠀⠀⠀⣠⠞⠁⣠⠴⠂⢀⠀⠠⠄⠀⡀⠈⠉⠀⠘⢦⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠠⠄⣉⣁⢈⣛⣿⣿⣷⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⠀⠃⠆⠴
⠀⠀⠀⠋⡠⠀⡀⠀⠐⠋⠀⡀⠀⣀⠈⠃⠀⠀⠐⡄⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠄⢀⢄⠼⡟⡿⠁⡝⢅⠁⡀⠈⠀⠀⠀⠀⣐⡂⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠘⠁⣰⠁⡀⠀⡀⠘⠁⠀⠙⠀⢠⠀⠘⡄⠸⠄⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠈⡁⢈⢊⠄⢋⠈⠈⠂⠁⠀⠁⠀⠀⠀⠀⢒⡀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠋⣸⠁⠸⠁⢀⠀⡄⠀⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀⠦⠑
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⡞⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣤⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠛⢉⣀⡀⢆⠈⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠣⠀⢿⠀⠀⠈⠀⣴⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡤⠬⣻⡾⠷⠆⠈⠉⠀⠀⠀⠀⠀⣠⣀⠀⠈⠳⢄⠀⢸⠀⠘⠀⡼⠁⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡔⢡⠎⡿⡙⠂⠲⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⡀⢀⡠⠀⠀⠂⣀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠈⠀⠘⢀⡇⠃⢠⠀⠁⠀⠀⠀⠀⠀⠈⠛⠛⢋⡁⠀⠠⢍⡾⢄⡒⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠈⠘⠆⠁⠀⠂⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠈⠉⢀⣠⠖⠁⠀⠁⡀⠀⢠⡀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠐⢃⡀⠐⠂⠀⠈⠁⠀⣴⠃⠀⣿⠀⠀⠓⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠐⠀⠈⠀⠀⠤⠈⠓⠀⠀⠀⠀⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋⠀⠊⡀⠀⠀⠀⡄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠡⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢈⠀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡟⠀⠀⢸⣿⣧
⠀⠀⠀⠀⣄⠀⠀⠠⢾⣿⣿⣿⣿⣿⡇⠀⠀⠈⠉⣡⡄⠂
⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠙⠛⣿⣿⢷⠀⠀⣠⣾⣿⣄
⠀⠀⠀⠀⠀⠙⣿⣿⡦⠀⠀⠀⡈⠚⠉⠀⠈⢿⣿⡿⠛⠀⠀⣠⠀⠀⣀⡂⠀⣀⣀⣀⠆
⠀⠀⠀⠀⠀⠀⠈⠛⣁⣤⣴⣶⣾⡄⠈⠁⠀⠀⠉⢠⠀⠀⠀⠉⠀⠀⠉⠁⠀⠈⠻⠿⠀⠠⣤⣄⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢉⣿⣿⠿⠇⠀⠀⣴⣦⠂⣿⠀⠀⢀⡀⠀⡄⠀⢀⡄⠀⠀⠀⠀⠀⠙⠿⢿⢦⠛⠀⣠⣤⣄⣀⡌
⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠻⠅⠀⠐⠶⢭⣬⣷⣿⣿⠀⠀⠸⠃⡐⠃⠀⠈⡥⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⢀⠀⡀⣼⠀⠀⠀⣴⠁⠀⠀⠠⣤⣀⣀⣿⠁⣩⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠀⡁⡀⢻⠋⠀⣿⠀⢈⡴⠀⠀⠲⣤⣿⡛⣿⣾⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣶⣶⣶⣾⣿⣿⡄⢧⡘⢇⠁⠸⠐⠀⠈⠰⠿⠷⣤⠀⢾⣾⣫⣽⣿⢇⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠊⠃⠋⠈⠀⠛⠀⡀⠒⠲⠒⠠⠒⠛⠻⡿⠛⠁⢪⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⣈⠉⠉⠀⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣴⣤⡄⠀⠀⠀⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠀⠀⠀⠀⠈⠙⠿⠇

⠀⠀⢀⠄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔⠀⢀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠁⠣⢀⠀⠀⢀⡀⠀⠀⠀⣸⣤⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⣶⣿⣶⣼⣿⣷⣧⣾⣼⣿⣷⢂⣤⡀⠀⠀⠀⠀⡠⠔⠋⠀⢠⠖⠋
⠀⠀⠀⠀⡄⢄⠀⠀⠀⠀⠀⠀⢤⡈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣋⡤⠀⠀⠀⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣤⣞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠂⡈⠁
⠀⠀⠀⠠⠤⠤⠴⠒⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠐⠚⠙⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣱⣿⣿⣿⠻⠿⠋⠉⠿⢻⣿⠃⣼⣿⣿⣻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢈⣩⣽⡒⠀⠘⠋⠀⢘⣩⣤⣼⢿⣿⣿⠛⠁⠀⠀⠀⣀⠀⢈⡀⠄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠈⠁⠛⠋⠀⠀⠀⠀⠉⠀⠛⠉⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠁⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣤⡤⠄⠀⠀⠠
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⡀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣷⣦⣄⡑⠀⠠⠀⡀⠀⠀⠀⠀⠀⠈⠠⠀⣈⣤⣾⣷⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣴⡶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⡛⢦⣀⣀⣀⣀⣠⣞⡉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠀⠉⠉⠉⠉⠉⠘⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠐⠿⠽⠆⢠⡄⠘⠉⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠠⣤⠼⠧⡤⠄⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⣴⠀⠀⠀⠀⣧⠖⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢠⠀⡴⠚⠛⠛⠋⠉⣿⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠁⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⡿⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⡼⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⢀⣼⠈⢳⡀⠀⠀⠀⠀⠀⠀⣷⢰⠇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠐⣿⣀⡀⠈⠳⡄
⠀⠀⠀⠀⠐⠟⢃⣠⣴⣿⣴⣶⣿⠟⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠓⢿⣤⣶⡴⠿⠖⠲⠶⠶⠶⠖⠋⠁⠈⡿⠀⠀⡷
⠀⠀⠀⠀⠀⢰⣿⡿⣟⠛⠛⠉⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣾⢹⣦⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣌⡓⠿⢶⣤⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉


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

⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⡆⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⡆
⡄⠀⠀⠀⢌⠓⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⣀⠀⠀⠀⠀⠀⡷⢄⢸⣇⣀⣀⡀⠀⠀⠀⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⣀⠀⠀⠀⠀⠀⡷⢄⢸⣧⣀
⡤⠐⠀⢠⠼⠁⢀⣴⣿⣶⡿⠂⠀⠀⠈⠛⠿⣿⠀⠀⠀⠄⠀⢸⣿⣋⣻⣭⡤⠐⠀⢀⠼⠁⢀⣴⣾⣶⡿⠂⠀⠀⠈⠛⠻⣿⠀⠀⠀⠀⠀⢸⣿⣋⣿
⡶⠀⢀⠼⢦⠀⡌⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢿⣏⠙⢷⡦⠀⢀⡼⢦⠀⡌⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢿⣏⡙
⠶⠚⠉⠀⢸⠀⠀⢀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠻⣿⠛⠲⠚⠉⠀⢸⠀⠀⢈⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠻⣿
⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⢠⠉⠲⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⢠⠁⠲
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣦⡹⠆⢀⣤⣶⢤⡞⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⡽⠆⢀⣤⣶⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠁⠀⣸⣿⣿⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⢿⠃
⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢛⡄⠀⠀⠀⣀⣀⣀⣿⠀⡰⡇⠀⠀⠀⠀⣀⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠀⣀⣀⣀⣿⠀⡰⡇⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠘⢿⣿⣷⣦⠀⠸⠅⡀⠀⠤⣬⣽⣟⣻⣿⠁⠁⠀⠀⠀⠸⠿⠛⠁⠀⠀⠀⠘⢿⣿⣷⣦⠀⠸⠅⡀⠀⠠⣬⣹⣟⣻⣿⠃⠁⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⢈⠉⠉⡀⢀⡼⣇⠀⠠⣾⡿⢉⣿⡟⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡀⢠⡼⣇⠀⠠⣮⡟⢉⣿⡟⠁⠰
⠀⠀⠀⠀⠀⠀⠩⠐⠀⠀⢸⠁⠈⠑⠶⠾⢛⣿⠏⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⠀⠀⢸⠁⠈⠛⠶⠾⢓⣿⠏⠀⢀⠄
⠤⠀⠀⠀⠀⠀⠀⠈⠁⠠⠈⠀⠀⠀⠀⡀⠀⠁⢠⠆⠡⠀⠀⠀⠀⠀⠀⣀⠤⠀⢀⠀⠀⠀⠀⠈⠁⠠⠈⠀⠀⠀⠀⡀⠀⠀⠠⠆⠡
⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣷⢶⣶⣤⠀⠺⣡⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣷⢶⣶⣦⠀⠛
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡻⣿⢿⡇
⡄⠀⠀⠀⢊⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⣀⠀⠀⠀⠀⠀⡷⢀⢸⣇⣀⣀⡀⠀⠀⠀⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⣀⡀⠀⠀⠀⠀⡷⣀⢸⣧⣀
⡤⠀⠀⢀⠼⠁⢀⣴⣾⣶⡿⠂⠀⠀⠈⠙⠿⣿⠀⠀⠀⠀⠀⢸⣿⣋⣿⣭⡤⠀⠀⢀⠼⠁⢀⣴⣶⣶⡿⠂⠀⠀⠈⠛⠿⢿⠀⠀⠀⠀⠀⢸⣿⣋⣻
⡶⠀⢀⡼⢦⠀⡌⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠈⢿⣏⠙⢿⡶⠀⢀⡼⢦⠀⡌⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢿⣯⠙
⠒⠚⠉⠀⢸⠀⠀⢀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠻⣯⠛⠒⠚⠉⠀⢸⠀⠀⢈⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠻⣯
⡀⠀⠀⠀⠈⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠉⠒⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠠⠀⠲
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣦⡹⠆⢀⣤⣶⣤⡞⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣦⡽⠆⢀⣤⣶⢤⡾⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠁⠀⣸⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠁⠀⣸⣿⣿⠿⠁
⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢛⡂⠀⠀⠀⣀⣀⣀⣿⠀⡰⡇⠀⠀⠀⠀⣀⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠀⣀⣀⣀⣿⠀⡰⡇⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠘⢿⣿⣷⣦⠀⠸⠅⡀⠀⠤⣬⣽⣟⢻⣿⡃⠁⠀⠀⠀⠸⠿⠛⠁⠀⠀⠀⠘⢿⣿⣷⣦⠀⠸⠅⡀⠀⠤⣬⣹⣟⢻⣿⠃⠁⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⡀⢠⡾⡇⠀⠠⣾⡟⢉⣿⡟⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡀⢠⡾⡇⠀⠠⣾⡟⢉⣿⡟⠀⠰
⠀⠀⠀⠀⠀⠀⠸⢈⠀⠀⢸⠃⠈⠛⠶⠾⢋⣿⠏⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠀⠀⢸⠁⠈⠛⠶⠾⢛⣿⠏⠀⢀⠄
⠤⠀⠀⠀⠀⠀⠀⠘⠁⠠⠈⠀⠀⠀⠀⠀⠈⠁⠠⠔⠱⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⠀⡀⠀⠁⢠⠒⠡
⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⣷⢶⣶⣦⠀⠺⢳⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⣶⣶⣶⣦⠀⠺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⣍⣙⣒⠲⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣤⣄⣀⣈⠉⠉⠛⠲⡄⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘⢃⡆⠰⢀
⠀⠀⠀⠀⠀⠀⣠⠞⢁⠴⣫⠖⠙⠋⠉⠉⡙⢶⠀⢳⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢻⠦⠼⣧⣴⢋⡄
⠀⠀⠀⠀⠀⢺⠁⡴⡋⠚⠁⠀⠀⠀⠀⠀⣷⢸⡇⠘⡇⡛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣤⣾⣙⡴⡆⣹⠶⡶⠆
⠀⠀⠀⠀⠀⠘⡆⢷⢻⡀⠀⠀⠀⠀⠀⠀⠸⠀⣧⠀⣷⠛⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⣿⠶⣤⠾⢷⣅
⠀⠀⠀⠀⠀⠀⢻⡘⡇⢃⠀⠀⠀⠀⠀⡀⢘⡴⠋⣰⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡼⠇⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠈⣧⠹⠮⣤⣀⣦⣠⢞⡵⠋⣠⢞⣿⣃⡶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠲⠦⢤⣄⣉⣉⠉⣠⣾⣵⣋⣼⣿⠂⡼⠸⠇⠈⠉⠙⠓⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢰⠀⠀⠀⣾⠁⠘⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡿⠓⠀⢬⣿⠾⢥⣴⡋⣿⢲⠶⠦⣭⣽⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣷⢤⣀⣰⢿⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣿⣛⠛⢯⣡⠏⢘⣤⣄⣀⡀⠉⡟⣆⠀⠀⠀⠀⠀⣀⠀⠀⣸⠀⡉⢁⡈⠋⢴⡁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡏⠀⠀⠀⣼⠁⠀⠀⢻⣟⣲⡿⠀⡏⢹⠉⠉⠓⠛⢦⣤⡀⠀⠀⣁⣠⣾⣁⡀⢡⡌⡵⠀⢠⠿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣠⡀⢁⡟⠰⣠⣼⣯⣉⡐⢶⠇⠀⠀⠀⣿⠯⠉⣝⣾⣡⡆⣠⠒⡶⣦⣤⣇⣷⡀⠈⠉⠀⠀⠀⣿⢀⣄⠀⢀⣈⣳⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠦⢹⡷⠞⠷⣾⠫⠷⢄⣀⣹⡿⠴⠄⣀⣸⠃⠀⠀⢈⣧⠤⢿⣟⢀⡇⠸⣀⣄⣉⠉⠛⡶⣄⠈⠁⣿⠋⠈⠳⣼⡅⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠀⣨⡴⠿⣩⠜⠆⢽⠛⡒⠒⠀⢸⡋⠙⠓⢮⡏⠀⠀⠀⡼⠛⠃⠰⣿⠿⣶⣿⣇⢸⡏⡏⠙⠓⠿⠤⣤⡏⠀⠀⢀⡘⣇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⠒⣦⡞⠳⣥⠀⠀⠀⠉⠛⠛⣿⣛⡓⠠⣴⠃⠀⠀⢀⣿⠯⠉⣹⣦⣞⡠⣠⠶⡶⣤⣄⣘⣹⡄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠼⠧⠀⠘⠂⠀⠀⠀⠀⠐⢯⣁⠉⣳⣟⣀⠀⠀⣼⠁⠀⠀⢸⣇⣈⣿⡟⢀⡇⠿⢀⡈⠉⠙⢳⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⠈⠉⣽⠯⠬⣄⣳⡇⠀⠀⠀⡾⠓⠂⢨⣿⠾⢴⣧⡏⢹⠛⡗⠺⠷⢧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠐⣧⡀⠁⢰⣇⠀⡴⠙⢦⣤⢤⣿⣒⠂⢠⣼⠃⠀⠀⢀⣿⣛⠛⢿⣅⡜⢀⢁⠤⣄⣀⡀⠹⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣷⣄⣸⢻⣀⣠⣴⠟⠁⠰⣏⡉⠉⣳⡏⠀⠀⠀⣼⠁⠀⠀⢹⣯⠽⢷⣮⠀⡟⢹⠉⠙⠛⠳⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣦⣀⣠⡽⠈⣁⠀⠉⢽⣁⣀⣉⣀⣀⡉⠉⣽⣯⣍⡐⢶⠇⠀⠀⢀⡿⠛⠀⠰⣿⣼⣡⡞⢠⡆⠴⡀⠀⠋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⢦⠠⠄⢉⠘⠃⠀⣠⠞⠛⠉⠉⠁⠶⠷⣤⣀⣹⡿⠶⠄⣀⣸⠃⠀⠀⢠⠧⠜⣿⢭⠿⣷⡾⣧⡾⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⣃⡤⣌⠀⡤⠦⠬⣷⣄⡀⠀⠀⠀⠀⠀⠀⢸⡋⠙⠓⢮⣏⣀⠀⠀⡼⣃⣴⣏⣒⢲⠄⣷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⡾⠋⠀⡈⢳⣇⠀⠀⠀⠮⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠭⢭⣜⣳⠃⠁⠠⢼⡷⢦⣴⠿⢮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣤⠼⠋⠀⠀⠀⠘⠃⠀⡿⠄⠀⠹



⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠎⠉⠁⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠉⠱
⠀⠀⠀⠸⠀⠀⠄⠠⠄⣀⠉⠢⡀⠀⢀⡠⠔⠒⠈⠉⠉⠀⠀⠀⠈⠉⠉⠐⠂⠤⣀⠀⡠⠒⠁⣀⠤⠄⢤⠀⠀⡇
⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠙⢢⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣖⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠇⠀⡇⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀
⠀⠀⠀⢰⠀⢃⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⡄⢀⠇
⠀⠀⠀⠀⠣⣈⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣡⠊
⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠈⠒⡄⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠉⠉⠁⠒⠂⠤⣀⠀⠀⠀⠈⢢⡀
⠀⢀⠜⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣀⣀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡐⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠑⣄
⢠⠊⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⡠⢊⣤⣤⡙⢢⠀⠀⠘⡀⠀⠀⢀⠁⠀⢠⠊⣤⣤⡉⢢⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠈⢢⡀
⡟⠒⠤⠄⠠⠤⠒⠁⠀⠀⠀⠀⠑⢌⠿⠿⡡⠊⠀⠀⠀⠃⠀⠀⠘⠀⠀⠐⢌⠿⠿⣃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⡃
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡦⢄⡀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠀⠀⠀⠀⢀⡠⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠁⠒⢠⠄⣀⣀⣀⣀⡀⣤⠔⠒⠉⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠀⠀⠀⡀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀⠀⠀⠀⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠈⠀⠀⠀⡇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡘⠀⠀⢠⠁⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠇⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⠀⠀⠀⠀⠙⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠸⣱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢘⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣤⠤⠐⠊⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡜⠃⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡉⡄⠀⠀⠀⠀⠀⡇⠇⠉⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⢰⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⢀⠀⠀⠀⠴⠤⠀⢒⠖⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⢁⠈⡆⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡄⢸⠀⠀⠀⠀⡌⠸⠀⠀⠀⠀⠀⢠⡅⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠘⡄⣬⡄⠀⠀⢆⡁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⣇⠌⠀⠀⠀⡼⢠⣆⣀⠠⠆⠂⠈
⠀⠀⠀⠀⠀⢀⡠⠇⠀⠈⠢⣀⢨⠁⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠸⠀⠀⣠⠎⠀⠘⠥⣀
⠀⠀⠀⠀⠀⠃⡴⠂⣀⠀⡠⢋⠋⠀⠀⠀⠀⠀⠈⠑⣤⠤⠤⢀⣀⡠⣤⠞⠀⠀⠀⠀⠀⠀⠱⡚⢇⠀⢠⡀⢲⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⢀⠃⠀⢀⠀⠀⠀⢠⣄⠀⠈⠃⠀⠉⠉⢸⠃⠀⢠⡄⠀⠀⠀⡀⠀⠑⡀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠋⠢⣀⡠⠊⠀⠁⠈⠁⠀⠀⠀⠈⠒⠒⠁⠲⢄⣀⠜⠑⠦⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⡆⣤⡀
⠀⠀⠀⠀⠀⠀⣤⢼⣿⣵⣿⢛⣴⣦⣄⡀⠀⠀⢠⡿⠿⠧⠀⠀⢹⣷
⠀⠀⠀⠀⢠⡦⠉⠀⠉⠘⠇⣿⣿⠿⣛⣽⣄⠀⢠⣾⣿⣿⣦⡀
⠀⠀⠀⢀⣭⡀⠀⠀⠀⠀⠀⠹⣣⣿⣿⡿⠿⠧⠘⢟⣥⣶⣶⣶⡄
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⣿⢋⣵⣶⣿⣿⣆⣾⠿⢛⣋⣉⣙⡀
⠀⠀⠀⠀⠀⣠⣴⠖⣤⣶⡤⠀⢰⣿⡿⣋⣭⣤⣭⣤⣾⣿⡿⢹⣿⣿⡄
⠀⠀⠀⢠⣏⢿⣿⣸⣿⢋⣾⣿⡿⢏⣼⣿⣿⠿⠿⠿⠟⡉⠁⢸⣿⡛⠷
⠀⠀⠀⣘⢿⠎⠛⢹⡇⣿⣿⡋⢴⣾⡿⢋⣤⣶⣿⣿⣿⠁⠀⢸⣿⣿⣷⡄
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⢡⣷⣄⠀⠀⠙⠻⣿⣿⣿⡇⠀⠀⠀⢉⣽⠙⣃
⠀⠀⠀⣠⣥⡀⠀⠀⠀⠀⢸⡿⢿⣷⣄⠀⠀⠀⠹⣛⠟⠀⢀⣾⣿⣧⣀⠈⣷⠀⢀⣀⣀⡀⢤⡀
⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠁⢿⡉⣃⣀⣠⣀⡄⠀⠠⠀⠘⢿⠟⠛⠻⠍⠀⠀⠀⠙⠻⠿⠦⠛⢀⣴⣶⡶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢻⣿⣿⣿⠀⠀⠀⠀⠀⣰⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⢸⠿⠿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠿⡏⠀⠀⣄⡀⠻⣏⣴⡇⠀⠀⠀⢀⡞⠀⠀⠀⠰⢤⣤⣿⣀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠿⣿⠿⠃⠀⠀⢆⠘⢁⡆⠀⠀⣓⣾⠧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠂⠘⠀⡇⠑⢤⣀⢻⣅⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⣤⣤⣤⣴⣶⣿⣷⠂⢀⠀⠁⠀⠀⠀⠀⠘⠋⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠟⠁⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⠖⠀⠀⠀⠙⠿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡇⠀⠀⠀⠀⠀⢰⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠘⠛⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡹⡄⠀⠀⠀⣀⣄⣄⠀⠀⢀⡟⣼⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣦⠀⠀⠀⠀⠀⠈⢷⣳⡀⠀⠀⢻⣿⠛⠀⠀⢸⢃⡇⠀⠀⢰⣿⠃⠀⠀⢀⡴⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⢀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣸⠁⠀⠀⠟⠁⠀⠀⣠⢋⡴⠋⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⢦⡀⠀⠀⠀⠈⢻⣇⠀⠠⡿⡄⠀⠀⣿⡏⠀⠀⠀⠀⠀⣠⣾⠗⠁⠀⠀⠀⠀⢀⣴⣮⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⠀⠀⠀⠀⢻⣆⠀⢧⢱⠀⢠⣿⠀⣠⢿⠀⢀⣴⠟⠁⠀⠀⣴⣤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠹⡄⠘⣿⡄⢸⠇⢀⣿⠋⠰⠋⠀⠀⠀⠀⠈⠿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⢭⣍⣓⡶⢤⣄⣀⠀⠑⢦⣑⣆⠉⠀⠘⠃⠀⠀⠘⠁⠀⢀⣠⣶⡿⠇⠀⢀⣀⣠⡤⠴⢖⣒⣛⣩⡭⠽⠇
⠀⠀⠀⠀⠀⢠⣴⣶⣲⣤⡄⠀⠀⠀⢠⡤⢬⣉⡙⠛⠛⠿⠷⣶⣬⣙⡃⠀⠀⠀⠀⠀⠀⠀⠐⠛⢩⣥⠶⠾⠿⠛⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⢖⣒⡿⠟⠛⠃⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢠⣈⠉⠓⠒⠾⠃⠀⢰⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣉⣥⠴⠒⠋⠉⠀⠀⠀⠀⢀⢔⡿⠃⣠⠆⠀⢀⠀⠀⣷⡀⠈⠻⣷⣄⡀⠀⠀⠀⠈⠛⠃⠁⠙⠻⣟⣒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠀⣀⣄⣄⠀⠘⠋⢀⣴⠏⠀⠀⢸⡆⠀⠹⣿⠀⠀⠈⢻⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣧⠀⠀⠀⠀⢻⣿⠓⠀⠀⣠⡿⠃⠀⣴⠀⠈⣷⠀⠀⢉⣄⣄⠀⠀⠙⢦⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠀⢀⣠⡤⠀⠀⠀⢠⢞⡟⠁⠀⣸⣿⠀⠀⣿⡆⠀⢸⡿⡟⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟⠀⠀⠀⣰⣫⠏⠀⠀⢀⣇⡇⠀⠀⢹⣷⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠹⣦⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⢰⡿⠁⠀⠀⠀⠸⠾⠁⠀⠀⠸⣿⡆⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠘⡎⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠄⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣤⣠⣤⠀⠐⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠁⠈⣿⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣽⠟⣫⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠖⢋⠁⠀⡀⠀⡀⠀⣀⡀⠀⠂⣀⡀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⡀⢀⡐⢟⠿⠟⠋⢀⣚⠿⠿⢿⣻⠿⠗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣾⠃⣾⠃⣾⠃⣸⡟⢻⡇⣼⠏⣻⠇⣼⠋⢿⠆⣼⠟⣿⠇⣼⠇⠙⣿⠛⢱⡟⢰⡟⠻⡷⢰⡟⢻⡿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣴⡟⠸⣯⣴⡟⣠⣿⣠⡿⢡⡟⢠⡿⢰⡟⠀⠀⣴⣿⣴⡟⢠⡿⠀⢸⡏⠀⣾⠃⣾⠃⠀⠀⣾⠃⣼⠇⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠁⣾⠁⠀⣿⠃⠀⣹⠏⠉⠀⣾⠃⣼⠇⣾⠃⣀⠀⣼⠃⣼⠇⣼⠇⢀⣿⠀⢰⡏⢸⡏⢀⡀⢾⡿⢶⡿⢰⡟
⠀⠀⠀⠀⠀⠀⠸⠏⠰⠟⠀⠰⠏⠀⠠⠿⠀⠀⠀⠿⠶⠟⠀⠿⠶⠟⠰⠟⠠⠿⠰⠟⠀⠸⠇⠀⠿⠁⠸⠷⠾⠃⠾⠃⠾⠃⠸⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣄⡀⢲⣶⣶⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣽⣛⡶⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⠄⢀⡿⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡙⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⣤⣿⣄⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠉⠋⠀⠱⡟




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠛⠉⠙⠿⣿⣶⣤⣄⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣰⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠙⠻⢿⣷⣶⣶⣶⣶⣾⡿⠿⣿⣇
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⣼⣿⠟⠉⠁⠀⠀⠀⢻⣿
⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⣠⡆⠸⣿⡇
⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣷⡙⠻⣶⣤⣀⠀⠀⣀⣀⣠⣤⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⣾⣿⡇⠀⢿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣷⡀⠈⡙⠿⣿⣿⣿⡿⢿⣿⡄⠀⠀⠊⠁⠀⠀⠀⣀⣴⡿⠟⠁⠈⣿⡀⠘⣿⣇
⠀⠀⣿⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠻⣷⡀⠈⠳⣄⠉⠛⠿⣦⠹⣧⠀⠀⠀⠀⣀⣴⡾⢿⣉⡀⠀⢠⠀⠸⣧⠀⢹⣿⡄
⠀⠀⠸⣿⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⢷⡀⠀⣿⡇⠈⠀⢠⣴⣾⠟⠋⢤⣬⢿⣿⣷⡏⠀⣀⣹⣧⠀⢻⣿
⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⡇⠀⠀⠀⣸⣿⡆⣿⣇⠀⠀⣾⣿⠀⠀⠀⠀⠉⠂⠋⠉⠓⠀⠈⢿⣧⣀⣾⡟
⠀⠀⠀⢸⣿⡆⠀⠀⠈⣿⣿⠀⠀⠀⠀⢸⣷⡀⠀⠀⣙⣛⠁⢸⣿⠀⢀⣿⡏⠐⢦⡀⠀⢀⣰⣶⠦⠄⠀⠀⠀⠻⣿⡟
⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠹⣿⣄⠐⣿⠙⣧⢠⣿⠀⣸⣿⠇⠀⠈⢻⣶⡿⠻⣇⢠⣶⣄⠀⠀⠀⢻⣿⡀
⠀⣤⣶⠿⠟⠃⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⠻⣷⣿⠀⣿⣿⠏⢰⣿⡟⠀⠀⠀⠸⠟⠁⠀⠈⣃⣈⡉⣷⣄⠀⢸⣿⡇
⢸⡿⢀⣤⣤⣤⣤⣤⣴⣿⣿⣧⣄⡀⠀⠀⠀⣿⡄⠘⠛⠀⣿⠇⣠⣿⡟⠀⠀⠀⠀⢀⣠⣴⣞⣛⣉⣍⣉⣻⣿⣷⣼⡿
⢸⢇⣾⠟⠛⠛⠉⠉⠉⢙⣻⣿⣿⣿⣷⣦⣄⠈⠿⣶⣶⡾⠋⡴⠟⣿⣷⠀⠀⠀⠀⠉⠉⠙⠓⠿⢬⣍⣋⣙⣹⢿⣧
⣼⡿⠃⠀⢀⣠⣴⣾⡿⠿⠿⠛⠛⠻⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠐⠶⢤⣤⣤⣈⣉⣁⣼⣿
⠁⢀⣴⣾⣿⣿⣭⣄⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄⠀⠀⠀⠀⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣧⡀
⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠉⠛⢿⣧⡀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣤⣀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⣄⡀⠀⠘⣦⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠙⢿⣿⣷⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣈⣻⣷⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣆
⣿⣷⠿⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣿⣿⣷

⠀⠀⠀⠀⠀⣶⡰⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⣿⣷⠈⢣⡀⠀⠀⠀⢀⡄⢸⣿⡿
⠀⠀⠀⠀⠀⢼⣿⣄⠸⠄⣶⣶⠆⣼⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠘⠟⠋⡁⡰⢻⡗⡶⣈⣻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⢀⢤⡄⣼⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠫⠃⠠⣿⣿⡿⠿⠿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣬⡋⠿⣀⣿⣾⠐⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢳⡄⢸⣿⣡⣾⣿⣿⣿⡄⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢹⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⣎⠙⢿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣇⣀⠀⢠⣿⣿⣿⠻⡧
⠀⠀⠀⠀⠀⢰⡇⠀⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⢿⢿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣟⠁⠀⠸⣼⣿⣿⣎⣿⣿⣿⣿⣿⡟⣿⣿
⠀⠀⠀⣈⡤⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⢿⣿⣠⣘⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⢀⡀
⠀⠀⠀⠾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣟⣻⣿⣋⢱⠶⠶⠦⣤⡄⣀
⠀⠀⠀⠀⣼⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⡶⠊⠉
⠀⠀⠀⠀⠈⠙⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠞⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠉⠉⠉⠉⠉⠉⠋⠁⠈⣿⣿⣿⣮⡉⠛⠛⠿⠿⠿⠿⠤⠄⠤⠀⢀
⠀⠀⠀⠀⠀⠐⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠙⢲⡀⠀⣰⠞⠛⠛⠋⠙⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣔⣩⠤⣤⠀⠀⠀⠹⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠛⢉⡴⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⢀⢈⡗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⠛⠁⠀⠀⠉⠓⠦⣄⣀⠐⢶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣄⡉⠳⣄
⠀⠀⠀⠀⢀⡀⠀⠀⢹⡄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣓⠮⣷⣄
⠀⣠⠶⠛⢉⢉⠙⠲⣼⠃⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠠⡀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠙⠂
⢰⡇⠀⠀⠉⠘⢳⠀⠹⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠻⣦⢹⡀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⢰⣤⣀⠀⠙⢷⡀
⠈⠛⠲⠤⠤⠤⢤⡄⠀⠀⠀⣀⣤⣴⠲⡖⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧⠁⠀⠀⠀⠀⠀⣀⣀⠘⣆⠙⠳⢦⣽⣏⠓⢦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⣾⣿⣿⣇⠀⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣇⠀⠀⢀⣀⣉⣉⣉⠀⠸⡆⠀⠀⠀⠉⠀⠀⠈⠛⢾
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⠀⠻⣇⣹⣾⠀⢹⣆⠀⠀⠀⢷⠀⠀⠀⠀⠀⣿⡴⠚⠉⠉⠀⠈⠉⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢧⣈⡷⡁⢸⢦⡀⠀⠘⣆⡀⠀⢠⢀⣿⠘⠀⠀⠀⠀⢀⣠⠞⠀⠀⢧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡄⠀⠀⠙⢦⣼⠧⣄⣹⢦⡀⠀⠉⠓⢶⡛⠙⠲⠴⠶⠖⠛⠉⠀⠀⢀⣠⠼⠛⠉⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⠻⣦⡀⠀⠀⠻⣦⣼⠳⣦⣽⡷⣤⣜⡂⢃⠀⠤⠆⣀⠀⢠⠢⠀⢀⢤⣀⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⠀⠐⣿⠻⢦⡐⠈⠀⠙⠓⢾⣟⠛⢮⣿⠛⠶⣦⣌⣀⣁⠀⠈⠈⠀⢈⠉⠉⠙⡏⠁
⠀⠀⠀⠀⣠⠞⠀⠀⢰⠃⠀⠀⠸⣆⠀⠙⠦⣀⠀⠀⠀⠈⠙⠺⢤⣉⣳⠟⠤⢤⣼⣉⣉⡟⠉⠈⠀⠀⠀⣿
⠀⠀⠀⡴⠃⠀⠠⠽⡟⠀⠀⠀⠀⠙⢆⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠉⠙⠒⠾⠧⢤⣯⡀⠀⠀⠀⠀⣰⠇
⠀⠀⡼⠁⠀⠀⠀⣸⠃⢀⣠⣤⣶⣚⡿⠃⠀⠀⠀⠀⡼⠻⡝⠛⠲⠦⢤⣤⣀⣀⣀⣀⣈⣀⣀⣠⡤⠞⠋
⠀⡼⠁⠀⠀⠀⠀⣯⠶⢋⣿⡿⠟⠛⢦⣀⣠⠖⣲⠴⠗⠒⠛⠒⠶⡞⣽⣶⠀⠀⠈⠈⠉⠉⠁
⡼⠁⠀⠀⠀⠀⠀⠛⠋⠉⠀⠙⠦⣄⡀⠉⠉⠈⡉⠓⠶⣄⠀⠀⠀⡗⢙⠿⠶⣦
⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣧⠀⠠⡈⠓⢦⡼⠀⠀⠀⠈⠋⠙⢺⠃
⠀⠈⢳⡈⠙⠓⠶⠦⣤⣄⣀⣀⣀⠀⣼⢈⠳⣄⣸⠦⡼⠃⠀⠀⠀⠀⡴⠒⠻⡆
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢉⣉⠉⣿⢸⡀⠀⠁⠀⠀⠀⠀⠀⢰⡋⠙⢶⡞⠁
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠐⠀⠀⠀⠿⢿⡈⢧⠀⠀⠀⠀⠀⠀⠐⣺⡛⠛⠻⣄
⠀⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠳⢄⠀⠀⠀⠀⠀⠹⡟⠲⠶⠋
⠀⠀⠀⠀⣼⢷⣶⢤⣤⣀⣀⣀⢀⣀⣠⣏⣀⣙⣦⣤⣀⣀⢀⣀⡴⢃
⠀⠀⠀⠀⠙⠶⢍⣛⠳⠤⣬⣭⣏⣁⣀⣀⣭⢿⣩⣃⣈⣉⡿⠁⢀⡾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣄⠀⣠⣶⣶⢿⠀⠀⠀⠀⠀⢸⣗⣶⣦⡀⢀⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠐⠻⠿⠿⠟⠌⠀⢸⣿⣿⢫⠀⠀⠀⠀⠀⢨⠟⣿⣿⡇⠀⠕⠿⠿⠿⠓⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⢀⠛⠀⠀⠀⠀⠀⠸⢣⢀⣿⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠁⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠿⡀⠀⠀⠀⠒⠾⡿⠛⣿⡇⢀⡀⠀⠀⠀⠀⣾⡿⠛⣿⠶⠂⠀⠀⠀⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠧⢠⠀⠀⠘⣛⡿⠋⣿⠇⠀⠀⠀⠀⠀⠀⢻⣟⠹⣿⡛⠁⠀⢀⢀⠗⠀⠀⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⣼⢧⣀⣈⣙⡷⣾⡟⠀⢀⠀⠀⠀⠁⠀⠘⣿⡷⣟⣉⣀⣀⢞⡆⢀⠰⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠈⠀⢹⡁⠀⠉⠁⠀⠉⠁⠐⠃⠀⠀⠀⠀⠑⠀⠈⠁⠀⠉⠁⠀⢸⠃⠈⠀⠆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⠀⠆⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⣄⣠⣀⠀⠀⠀⢀⣀⣄⣤⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠙⠛⠛⠇⠀⠀⠿⠛⠛⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣉⡛⣋⣝⣛⣟⣋⣿⣻⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠾⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣀⣁⣈⡛⠷⣦⣄⠀⠀⢀⣤⣶⣶⠶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠸⢿⣿⣿⣿⣿⣶⢘⠁⠀⠀⣿⣿⣿⣷⣶⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠀⠀⠘⣿⣟⣟⢿⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⠼⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⣸⡇⡀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣿⣷⡀⣼⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠀⠈⢀⡇⠀⠀⠀⠀⠀⠈⠙⠻⣿⡿⠋⢻⣿⣻⡀⠈⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣴⣿⡇⠀⡀⠀⠀⠒⠒⠶⣾⣷⣷⠶⣾⣿⣿⡇⢠⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⣼⣿⡿⠋⢁⣿⠀⠀⠀⢀⣠⣶⡿⠋⠀⣩⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠃⠀⠹⣿⠁⢀⣿⣿⣄⠠⣀⠀⠈⠻⣧⣤⣾⣿⡟⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢡⠘⡄⠀⠙⠀⠘⣿⠈⠉⠙⠛⠛⣿⣿⣿⠟⢿⠟⠀⠀⠄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⢱⠀⠀⠀⠀⠻⡀⠀⠀⠀⣸⣿⣿⠋⠀⠀⠀⠀⠚⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠇⠀⠠⡄⠀⠀⠀⠀⠀⢸⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⣰⣤⣶⡿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠎⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⠀⠀⠸⣀⡀⡄⠀⠣⢄⡀⠰⣅⠀⠀⠀⠀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⢠⠀⢀⠜⠀⡀⠜⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠁⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠈⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣀⣤⣶⡆⠀⠀⢸⣿⣿⣷⣦⣄⣙⣿⣿⣿⣿⣏⣠⣴⣾⣿⣿⡇⠀⠀⢰⣶⣤⣀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠙⠉⠀⠀⠀⠀⢸⣿⣿⠟⠉⢻⣿⣿⣿⣿⣿⣿⡟⠉⠻⣿⣿⡇⠀⠀⠀⠀⠉⠋⠀⠀⠀⢸⣿⣿⠅
⠀⠀⠀⠀⠀⠸⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⡿⠿⠿⢿⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢋⣥⣴⣶⣶⣶⣶⣶⣬⡙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⢀⠞⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠈⠒⡦⠀⠙⠉⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡔⠁⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠶⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣄⣸⣿⣧⠀⠀⠀⠀⣀⣈⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡁
⠀⠀⠀⠀⢀⣾⠟⢉⡉⠙⢻⣿⠀⣤⣞⣋⠠⠤⠤⣤⣹⣿⣧⡀⠀⠀⠀⠀⣴⣿⠟⣉⣁⣉⣛⢶⣄⠀⠀⠀⢠⣾⠛⢉⣉⡛⣦
⠀⠀⠀⠀⣼⠃⠀⢀⣹⡆⠸⠟⢸⡏⠀⠀⠀⠀⠀⠀⠈⠙⢻⡇⠀⠀⠀⣾⡿⠟⠛⠉⠉⠀⠀⠀⢛⡇⠀⠀⠀⠁⠀⣾⡟⠀⣼
⠀⠀⠀⠀⣿⡀⠀⢾⣯⣀⣠⡀⠈⢛⠲⠴⠤⠤⠤⠤⣼⣿⣿⠇⠀⠀⠀⣿⣦⣤⠀⢀⣀⣀⡠⠴⡟⠁⠀⢀⣀⣀⣀⣿⣷⢀⡿
⠀⠀⠀⠀⠸⠇⢀⣼⡿⠟⠋⢁⠀⠀⠘⠒⠂⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⣹⡿⠟⠛⠀⠩⠥⠄⠚⠀⠀⠀⠀⠉⠛⠿⣿⣧⡼⠃
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⣀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⣾⠋⠀⠐⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠺⠶⠿⣦⣤⣠⣾⡿⠉⠛⠋⠀⠀⠀⠀⠀⣠⡶⠛⠉⠈⠉⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠘⠛⠛⠲⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣤⣴⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠉⠉⠉⢹⡇⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⣀⣀⣼⣷⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠛⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠼⠛⠉⠙⠛⠻⠷⢶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠋⠉⠀⢠⡿⠁⠀⠈⠙⠻⣦⠀⠀⢻
⠀⠀⠀⣠⣴⠞⠋⠉⠉⠉⣀⣤⡶⠾⠶⣦⡀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣶⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⢸⡅⠀⢀⣀⣤⣄⣼⡇⠀⠘⣷⣤⣄⡀
⠀⢀⡾⠋⠀⠀⠀⠀⠀⢾⣏⣁⡀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠞⠋⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⣦
⢰⣿⠗⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣷⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡄⢹
⣿⠃⢀⣴⠇⠀⣠⣤⣄⠀⠀⢀⣠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⣤⠖⢻⡇⠀⠀⠰⣆⠀⠈⣿⡿
⣿⠀⢸⡏⠀⠀⡏⠀⠙⠷⠾⠛⠋⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠀⠀⠸⠀⠀⠀⠀⣿⠀⣰⡟
⠸⣧⣼⡇⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣤⣤⣤⣤⣴⣾⣿⡆⠀⢺⡆⠀⠀⣠⡿⠞⠋
⠀⠀⠈⠙⠶⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣤⣿⡧⠀⠈⠻⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⠄⠐⠒⠒⠶⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠉⠉⢹⡿⠉⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⠉⠙⠙⠛⠛⠛⠋⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣀⣀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠉⠙⠛⠿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣤⣤⣤⣶⣶⡶⠟⠋
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠙⢶⣴⣦⠤⠞⢹
⠀⠀⠀⠀⠀⠐⡆⠀⡀⠛⠀⠀⠘⣲⣤⠤⠤⠤⠤⠤⠠⢄
⠀⠀⠀⠀⠀⠀⡈⣁⢹⡀⢸⠀⢠⣜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒
⠀⠀⠀⠀⠀⠀⠐⢌⠛⠰⢆⡠⠗⠝⣉⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⣿⣟⣏⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠘⢦⡀⢀⠀⢀⠀⣀⣠⣤⡾⠉⠀⠀⢀⣼⡻⣿⣦⡠⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠮⣷⣾⡿⠛⠛⠉⠉⢸⣇⣀⣤⠶⣿⢿⣧⡘⢿⣧⣄⢀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠘⣿⣧⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠘⠛⢿⣮⡻⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⢹⣿⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠉⠻⠿⣿⣿⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠛⠚⠿⠃⠀⠄⠰⠿⠇⠀⠀⠀⠰⠿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠠⠄⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⣀⣠⢴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡐⣶⣞⠯⠋⠛⠛⠋⠉⡄
⠀⠀⠀⠀⠀⠀⠀⡮⣼⢽⣽⠛⠄⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠠⠿⡯⡗⣺
⠀⠀⠀⠀⠀⠀⠀⠁⠯⠣⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⠄⠂⠁⠀⢠⠏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⢀⠀⠀⠱⠄⠔⠀⠀⠀⠀⢀⡟⣐⠢⡕⠀⠀⠀⠋⠀⠴⠆
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈⠉⠀⠐⢀⠀⠀⣀⣀⠀⠀⡾⢑⠄⠍⠀⠀⠀⢀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⣼⢁⡾⡕⠀⠀⢰⢁⠔⡈⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠀⣸⠃⠚⠿⠂⠀⠀⠉⠉⠀⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀⠈⠀⠀⡤
⠘⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠁⢨⠁⡀⠈⠀⠰
⡇⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⢸⠎⠠⠠⠤⠃
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠤⠄⠆⠈⠀⠠
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠘⡉⠉⠉⠉⠉⠉⠹⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡤⠴⢦⡤⠶⣅
⠀⠀⠀⠀⠀⠀⠀⠰⣀⠂⡙⠓⠄⡍⢃⢠
⣶⣶⠶⠶⠶⣶⣶⣶⣿⣶⣿⣿⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠿⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠛⠿⣷⣄⠀⣀⣴⣾⠟⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣏⣽⣷⣤⣹⡍⣼⠋⢉⣉⠷⣆⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠒⣀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠟⠀⣸⠃⣿⣺⣿⡆⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⢀⣾⠟⠛⣿⡟⠁⠀⠀⣀⠀⠀⠀⠀⠈⠷⠶⠶⠾⢷⣦⣿⣏⣉⣀⡤⠟⠀⠀⢸⣿⣤⡀
⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡇⠀⠀⠀⠘⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀⢘⣿⠀⢻⡆
⠀⠀⠀⠀⠀⣾⠇⠀⠀⣿⠁⠀⠀⠀⣠⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⢰⡿⢸⡇⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⢀⣿⡇⡠⢿⡄⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⢸⡇⣼⠁⠈⢻⣦⡀⠀⠈⠛⠿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣽⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣼⡇⠀⠀⢸⡏⠈⢦⡔⠁⠈⠻⣦⣀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⢿⣿⡿⠿⢿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⢠⡄⠈⢿⣤⠊⠑⢦⣄⡀⠈⢙⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⣿⣧⡀⢷⡀⠈⠛⠷⠶⠶⠶⠿⠛⠋⠁⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡏⠻⣤⡙⠶⣤⣀⡀⠀⠀⠀⠀⢀⡴⠛⠋⠛⢦⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠠⣿⣧⣤⣌⣛⠶⣤⣀⠀⠀⠀⢀⣴⠋⠀⠀⡀⠀⡈⣧⣤⣤⣤⣶⠶⠿⠛⠃⢠⠇
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠈⠉⠙⠛⠿⣯⣝⠓⣶⠞⠁⠀⢀⠀⢻⡀⡂⢹⡀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠘⠿⠶⣶⣶⠶⠶⠶⢶⣤⣀⠙⣿⠋⠀⠀⢀⣼⡇⠸⡇⢷⣼⠛⠛⠛⠛⠛⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠳⣤⣀⣤⡶⠋⡘⢣⠀⣷⡞⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣘⢉⡽⠀⢠⠃⠈⡷⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣠⡏⠀⣼⣁⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⢿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⠈⠉⠉⠉⣹⠋⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣧⠀⠀⢰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣼⣷⡤⠹⣧⢀⡿⠀⠠⣾⣧⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⡤⠶
⣤⣤⣀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠉⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢙⣿⠶⠟⠋⠁⢀⣼
⢳⡀⠉⠛⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣠⡿⠃
⠀⠻⢦⣀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⠾⠛⠻⠿⠶⣶⡶⠶⠶⠶⠶⠶⢶⣦⣤⣤⣤⣤⣴⡾⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣹⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠹⣧⡙⣧⣼⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠛⠛⠋
⠀⠘⢿⣜⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣦⣿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠒
⠀⠀⠐⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠘⠋⠀⠘⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⠰⠿⣿⡒⠀⠀⠀⠀⠀⠀⠋⠛⣼⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⢹⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀⠀⢀⣀⣿⠛⠀⣀⣀⣀⡀⠀⠀⠈⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⣻⣿⣿⡿⠻⢿⣿⣭⣹⣿⢦⡄⢠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣾⣿⣿⣿⣷⣦⣤⣿⣿⠛⠃⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠹⣿⣿⣿⣿⠋⠉⢻⣿⢿⣿⠟⠉⠀⠀⠀⠈⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣌⠛⣿⣿⣿⣦⣤⣿⡿⣦⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⢿⣇⠀⠀⠀⢐⣿⣿⣿⣿⢟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣽⣏⣹⣿⣿⣶⣄⡀⣾⣿⣿⣷⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣯⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⢿⠻⣿⢻⣿⣿⣿⣿⣿⣿⡷⣈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⢀⠀⠀⣀⠤⠶⠿⠟⠁⠀⠼⠠⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠚⠉
⠀⠀⣢⠶⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⣿⣿⣴⣾⣿⣧⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠻⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⣀⡀⣀⠐⠲⢶⣶⣄⡙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⡄⠀⠀⠀⡤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠘⠐⡄⠀⠀⡇⠀⡄⠀⠇⠀⠜⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠁⠀⠀⢦⠀⠁⢠⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠀⠀⠈⠳⠀⠀⠀⠘⣧⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠦⡀⠀⠀⢢⡘⣇⠀⢿⠀⢸⠀⠀⡾⠀⢀⡴⠁⠀⠁⣠⠖⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⠙⢦⣄⠀⠙⢦⡱⠈⠆⢸⡄⠈⠀⣸⠃⢠⠞⠀⢀⡴⠛⢁⠴⠊⠁⠈⢀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠈⠑⠦⡈⠳⣄⠈⠀⠇⠀⢠⠇⡰⠋⢀⠔⠋⠀⣀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⠀⠒⠶⢤⣄⡈⠁⠈⠣⡀⠀⠀⠎⠘⠀⢀⣠⠴⠞⠋⠀⠀⣀⣀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣤⣄⣀⣀⡀⠀⠉⠙⠒⠤⠈⠀⠀⠀⠔⠚⠉⠀⠀⠐⠂⠉⠉⠉⠁⠀⣀⣀⡈⠉⠉
⠀⠀⠀⠀⠀⠈⠉⣉⣁⡀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠀⠀⡀⠀⠀⠀⠀⠄⣀⡐⠒⠒⠲⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⠥⣒⠂⠀⢀⡴⠊⢠⠇⠀⢣⠀⠀⠀⠉⠙⠲⠦⣭⣑⠀⠀⠀⢀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⢀⣠⡤⠖⠋⠁⣠⠞⠋⡠⠃⡞⢰⠀⡀⢧⠈⢦⡀⠑⢤⣀⠀⢀⠀⠀⠀⣀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⠁⠀⠀⠀⠚⠁⢀⡞⠁⣼⠁⢸⠀⢃⠈⢧⠠⡙⣄⠀⠈⠳⢦⡉⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⣠⠏⠀⢰⠃⠀⣿⠀⢸⡀⠈⢇⠙⠌⠳⡄⠘⠦⡉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢀⠔⠀⠰⠃⣰⠁⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⠀⠀⠁⠀⠀⠀⠀⠁⢀⠀⢿⠀⠀⠀⢡⠀⠠⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠎⠀⢀⠃⠀⠀⠘⠀⠈⠀⠘⡄⠈⠃⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⡆⠀⠀⢰⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣈⣷⣤⣤⡾⠻⠿⡶⣦⡀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣶⠛⠉⠁⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⠃⠀⠀⠀⢀⣶⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠛⠶⠶⣤⣤⣤⣭⣿⣙⣁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠦⣤⡀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⢀⣀⣀⣉⣛⣛⡁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢻⣄⠀⠀⠀⠀⢀⡶⠈⣷⠀⣠⠾⡏⠉⠁⠀⠈⠉⠉⠛⠶⣤⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠙⠓⠶⠶⠞⠋⢀⣰⣿⢾⡇⠀⢷⡀⠀⠀⠀⠀⢠⡆⠀⠀⠉⠛⢻⣿⢃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠿⣯⡵⠀⠈⠙⠶⠦⠴⠶⠋⠀⠀⠀⠀⣠⣼⠟⠋⠉⠉⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⠟⢶⣿⠉⠻⣦⣤⣤⠶⠲⡆⠀⠀⠀⢀⣀⣤⡴⠞⠋⠁⣾⠁⠀⠀⠀⠀⣠⠶⠶⢶⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠚⠛⠛⠉⣽⠃⢠⡿⢧⣤⠴⣿⣸⣧⣀⠀⣿⣶⡶⠛⠙⢿⡄⠀⠀⠀⢸⣧⡀⢠⣤⣀⣼⠋⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣾⣁⣀⣠⣤⣤⠤⠶⠟⠛⠉⠁⠀⠀⣠⠟⢻⡄⠀⠀⢸⡏⠳⠏⠈⠉⣿
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣄⣤⣤⣤⡶⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠈⣷⠀⠀⠈⣷⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋⢹⠀⠀⠀⠀⢹⡄⠀⠀⢹⡄⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣤⠴⠶⠶⠶⠚⠛⠋⠉⠀⠀⠀⠀⣸⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠈⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣿⠀⢀⡿⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣿⣠⠞⠁⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠶⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠞⠋



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣨⠄⡀⠀⠘⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡖⠀⠁⠀⠂⠂⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⣠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠘⠀⠀⠀⠀⠴⠉⠈⠹⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢨⠀⣠⡀⠀⠄⠘⠀⠀⠀⢀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡍⡉⠂⠀⠀⢀⣉⢀⠁⠀⠀⠀⠀⠀⠐⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡦⣿⣿⣀⡀⠀⠈⠀⣀⣑⠈⠀⠀⠀⠀⢦⣿⣷⡖⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣶⡇⣫⣿⣿⠀⣠⡶⣬⡛⢴⡦⠐⠁⠀⠀⢰⣦⣬⣭⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡿⢏⣽⣝⣱⣶⣿⣿⣿⠿⠿⣥⡿⡄⠸⡶⣷⣿⠏⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⡗⠀⢿⣿⢻⣥⢄⠀⠀⠂⢀⡂⠐⣷⡆⢈⣿⠿⢀⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣧⣧⣀⠀⡀⠉⠙⡆⠀⠀⡇⡐⡏⣧⣹⠗⣬⡟⠾⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣁⡛⠿⢉⡇⢰⣟⣠⣶⣶⣧⣥⣷⣧⣧⣿⣀⣀⢨⣣⡼⣋⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⣿⠛⣩⣷⠸⣿⣯⢝⡉⠉⡉⠉⢛⡛⢧⣍⣻⣾⡿⠿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⠁⠚⣿⣷⡀⢻⣧⣬⣥⣬⠀⢠⣵⡥⠂⠛⣻⣭⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡶⣿⠋⢻⣿⡉⠟⠰⡟⠛⠿⠇⠀⠸⣟⠀⠀⢸⣛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⣨⣛⣿⠧⠘⣼⣇⠀⠀⠀⠀⠀⠹⠦⡞⠯⠓⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢹⡀⠠⠋⣰⠀⠀⠀⠀⠀⠋⣼⠗⠴⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠏⠀⠘⠋⠑⠠⢀⡀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⠀⢡⡄⠀⠀⠀⠀⠀⠈⢁⣠⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣿⣧⣤⣤⣤⠦⢦⣾⣿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠛⠐⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠠⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠀⠀⠀⢣⠀⠀⡇⠘⠀⠇⠀⠀⡠⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⢇⠀⡇⠀⡜⠀⢀⠜⠁⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠆⠇⢠⠑⠠⠋⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠉⠉⠉⠉⠀⠒⠂⠠⠀⡀⠱⢄⣀⣶⣔⣴⢀⢈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠁⠀⢀⣀⡀⠠⠤⠤⠤⠄⠈⣀⡀⣰⣿⣿⠛⠘⢟⣫⡤⠀⠀⠒⠒⠀⠀⠉⠉⠉⠀⠐⢀⠀⠂⠀⠰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⠣⣼⣿⣧⣀⣀⢙⠿⠄⢀⠂⠒⠐⠦⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠈⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡠⠒⠊⠁⠀⣀⠤⠰⠆⡉⠘⡽⡟⣻⡟⠿⢧⡀⢠⡌⠁⠢⢄⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⢀⡀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⣠⠔⠋⠀⢀⡴⠊⡁⢊⡄⠠⠂⢉⠁⣄⠀⠠⡀⠑⢤⡀⠉⠢⣀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠀⡔⠁⡜⠀⡙⠀⢠⠀⠈⢆⠀⠈⢢⡀⠙⢦⡀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠔⢁⠞⠀⡜⠀⢀⠃⠀⢸⠀⠀⠈⢆⠀⠀⠱⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠊⠀⢰⠁⠀⢸⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠱⡄⠀⢀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⡎⠀⠀⠸⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠾⠀⠀⠀⡀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠎⢀⠘⠀⢦⠁⠃⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⢀⠎⠀⡀⠀⠰⠀⠀⠁⡀⠀⠒⢀⡈⠀⠐⠀⣁⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠂⠂⠈⠁⢀⠁⠈⠀⠄⠁⠀⢠⠀⠌⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⠐⣀⡁⢀⠀⠘⠀⡀⠀⡀⠀⠰⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠂⠀⠆⢀⡁⠀⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠁⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠂⠀⠠⡀⠱⠈⠀⠀⠠⠀⠰⠀⠄⠀⠠⠀⠀⠈⠀⠀⠀⠂⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢸⣳⣄⠀⠀⢀⣴⣓⡄
⠀⠀⠀⠀⠀⢐⠷⡿⣳⠖⣾⣉⣿⡀
⠀⠀⠀⠀⠀⣸⡇⢀⣤⣀⡉⠉⠉⡇
⠀⠀⠀⠀⠀⠹⠧⣌⢻⡿⠿⣆⠀⣱⡀
⠀⠀⠀⠀⠀⢿⢚⣿⣁⠀⢀⡠⢀⣿⣱⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠘⢛⢙⣿⣷⣤⣤⣤⣤⠤⠆⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠠⠃⢀⣼⣿⣿⣍⡥⠈⠰⠀⠀⠀⠀⠤⢄⠄⡀⠐⣽⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡄⣨⣿⣟⣿⡿⠓⠖⠂⠈⠐⠀⠀⠀⠘⢢⠙⠀⠨⣿⣿⣿⣿⣩⣶⣿⣽⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠨⠛⢿⣿⣿⡄⠀⠀⠀⠐⠀⠀⠄⢀⠀⠂⠀⠎⠡⣹⣿⣯⢿⣿⣿⣷⡾⢧⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⢘⡰⡶⢾⣾⣿⣤⣤⣀⣤⣤⣤⡶⠗⠀⠀⠀⠀⠀⢻⠉⠛⠿⣿⣏⣭⡙⢋⠿⣝⢻⣿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⡿⢿⣿⣿⣿⠿⠿⠛⠛⠻⡿⣷⢀⡀⠈⣀⠀⠀⡜⠀⠀⠀⠈⠙⢿⣅⣨⠿⢮⣱⣊⣿⣟⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠡⠙⣷⣿⣷⣤⢤⣌⠁⠀⠀⠀⠀⠀⠀⠉⠻⣶⣞⣻⠿⢿⣿⣿⡿⣿⣿⣿⡇⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠛⢿⣞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣌⡙⣿⡝⠁⣿⣯⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⠇⠀⠠⠾⠿⠇⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠶⠿⠟






⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠀⢲⡀⠀⠠⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⠶⢺⠋⠀⠀⣤⡤⠄⠀⣀⠄⠆⣐⣠⢂⠠⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡘⠀⠀⠀⢳⣦⠤⠄⢀⡠⠔⢫⠸⠒⠁⠀⠀⠈⣗⢈⡆
⠀⠀⠀⠀⠀⠀⠈⠉⢁⣤⠾⠤⢤⢀⠞⠉⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠈⡟⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⠿⠶⠷⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠠⠔⠁⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡤⠴⠖⠒⠒⠒⠒⠒⠶⠦⢼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠔⠚⠛⡿⠻⠿⠿⢿⣿⣿⣶⣦⣤⣤⣀⣰⠀⢀⣠⣤⣶⣿⠿⠟⠛⠉⠁⣧⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⠡⠿⠟⠉⠉⠉⠉⣿⠀⠈⠟⠿⠿⠃⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⠢⢀⡀⠀⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠠⡤⠀⠀⠀⠉⠒⠤⠤⠀⠠⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠹⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢱⡀⢨⡗⠒⠶⠤⣤⣤⣤⠤⡤⣤⠤⠐⠒⠚⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠙⢾⠁⠀⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠱⢦⡀⠀⣏⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣧⠔⣆⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⣄⣀⡀⠀⠀⣷⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠃⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⣿⣷⣄⠀⠀⠀⢀⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⡟⣻⡿⠋⠀⠀⠀⠀⠀⠉⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡤⠀⢀⣠⠖⢾⣷⡀⢶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡿⠀⠀⠈⠙⠂⠀⣿⣇⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣶⡀⠀⠀⠀⠀⠀⠀⠉⠁⢻⡶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣥⡤⠐⢶⣦⣤⣤⠤⠼⠋⠹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡄⠀⠀⠉⠛⠋⠀⡇⠀⠀⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣸⣿⡿⠛⠂⣿⣤⣀⠀⣄⣠⣾⡆⠀⠀⢸⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⢿⣷⣶⡶⠻⠉⢿⣿⡿⠋⢈⢿⣶⣶⣾⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣠⠙⢿⣄⡐⠖⢀⠟⠀⠀⡀⠀⢿⣿⣿⣿⣿⠇⠊⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠙⠿⣿⣧⣄⣉⠛⠾⣯⣤⣀⣤⣤⣶⣶⢶⣿⣿⡿⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⢸⡆⠈⡛⠿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡟⢿⠁⣗⡀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠁⠀⠀⣏⡉⠉⢉⠉⣿⣿⣿⡟⣣⣟⡃⣸⢇⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⣿⠂⠀⠀⢀⠃⠛⠀⠀⠀⣿⣿⣯⡾⣿⣿⡁⣻⣶⣴⠖⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⡇⠀⡄⠀⠰⠀⠀⠀⠀⠀⡿⢿⠟⠟⠁⢈⠇⣿⢹⣿⣟⣡⡁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠂⠀⢨⣿⠀⠀⠀⢀⠀⣀⠀⠀⠀⣿⡘⣀⠀⣠⣾⣧⡟⠀⣽⣋⠉⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠙⠀⠀⠀⣿⣷⣿⡇⣿⠿⢿⡇⠀⡛⢁⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⢻⡔⠀⠀⠁⠀⠀⠋⠭⣿⠟⡉⠀⣆⣠⣾⣧⣾⡏⠙⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣰⡿⠏⢹⣏⠀⠐⠀⢸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠀⡠⠀⠀⠀⠀⠀⠸⠿⠆⠸⠀⠨⠿⢿⡟⠁⢀⣸⣿⢧⠀⠠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣅⣴⣿⣿⣿⣾⣧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⣿⣿⠹⠛⢻⣿⣿⣿⡿⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠸⣏⣠⠀⠈⣿⣿⣿⠇⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠐⠃⠐⢹⣻⣿⡆⣠⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣄⣣⣄⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⠉⠀⠀⠀⠂⠀⠀⣨⡾⢿⣿⣿⡟⠃⠀⢠⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠒⣆⣀⢀⣾⣿⣶⣶⣿⣶⣼⣷⣶⣿⣿⠀⠈⣿⣿⣇⠸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠉⠋⠹⢿⣿⣿⣿⣿⣿⡿⣿⢻⣿⣿⠀⠀⠘⣟⣿⣏⠀⠀⠐⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣧⠇⣽⠀⡿⣿⠀⠀⠀⡹⣿⡿⠃⠀⠀⠙⢣

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠃⠀⠙⣿⣿⣦⣀⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⣶⣶⣿⣿⡿⣿⣿⡿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠀⠀⠀⠀⠐⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡀⠀⠀⠀⠀⠈⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣁⣬⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠋⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⢠⣘⡀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡘⠀⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⢀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠿⠻⠇⣼⣿⣦⠀⢴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠟⠛⣀⣶⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠈⠁⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠁⠀⠐⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠚⡃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⡀⠠⡀⠈⠀⠄⢀⠀⠁
⠀⠀⠀⢀⠀⣐⠈⠓⠔⢁⠣⠆⠀⠨⢀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⣀
⠀⠀⠀⠤⠀⠀⠲⠀⡠⠘⠀⠀⠁⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣂⣄
⠀⠀⢀⠄⢉⠖⠀⠨⢵⠀⢠⠆⠀⠒⡀⢀⡤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⡀⠂⢠⠄⡈⢀⢁⠈⠀⠀⠀⡄⢁⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠂⠄⠀⠉⣀⠠⢠⠘⠀⢁⠀⠄⠑⠀⢂⠂⠀⠀⠀⡀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠐⠀⠁⠔⠀⠀⠆⠀⡆⡀⢘⠁⠀⡐⠀⠀⠀⠀⢀⡠⠌⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠁⠄⡐⠀⢀⠉⠀⠐⠁⠀⠀⠀⣀⣐⠂⠄⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡀⡀⠈⠁⡀⢀⠄⠦⡂⡇⠀⡀⠀⠀⣈⠉
⠀⠀⠀⣀⠐⡀⠑⠫⠠⠁⠀⢠⡀⠁⡪⠠⢊⡀⠀⠠⡂⠀⡠
⠀⢀⡀⠀⠣⣀⠨⡀⢄⠄⠃⢀⠇⡠⢀⠀⠙⡒⠂⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠤⠀⠂⠀⠀⣢⠄⠀⠀⠀⠀⢂⣀⠀⠚⠀⠬⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡆⠀⡄
⠀⠀⠀⠀⢲⠌⠠⠂⢀⠀⠀⠀⠀⠒⢀⠂⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠄⡘⢙⢠⣇⡴⠃⠞⣀⢠⠄
⠀⠀⠐⡀⠰⠉⡊⠀⢉⠌⠀⠌⠀⠨⡂⠙⢈⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡲⠓⢕⠈⢳⡎⢸⣟⢃⠎⢐⣵⣡⠄⣀
⠀⠀⠀⠀⠒⢀⠆⠔⢀⣰⠀⡄⠐⡤⠀⠐⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠦⢌⣳⣄⣐⢀⡳⠘⢁⠠⠴⠋⣩⡴⠋⣁⡀
⠀⠀⠀⠀⠀⠐⢁⠄⠀⠉⢠⠀⡄⢆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣁⣶⠆⣁⣉⣁⠁⠀⠀⢐⡐⠋⠉⠁⠐⠒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠛⣉⡩⢉⠖⢀⠀⣈⠒⣄⡉⢉⡻⠏⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠞⡿⢂⡠⢚⢉⠆⣮⠀⡨⠉⢿⡌⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠖⡹⠁⡖⣼⠀⢨⡧⠹⠾⣳⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⣴⡜⢉⡇⣌⠁⢰⣳⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁

⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠄⣀⣀⣀⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⣿⢸⠹⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⢹⣿⣿⣿⣿⣿⣁⠉⠉⠀⣀⣴⣾⡇⠘⠀⡁
⢸⣧⣤⣶⣶⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⠉⠀⠉⣿⠀⠀⢹⣷⣯⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⡄⠀⠒⡗
⢸⣿⣿⡟⠛⠻⣿⣇⠀⠸⣷⡤⢄⠄⠀⠻⠀⠈⡇⠀⠠⠀⠀⣯⣀⠀⢸⣿⢼⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⠁⠀⠀⠁
⢸⣿⣿⣄⠀⣠⣿⣿⠀⠀⠄⠄⠎⠀⠀⡀⠀⠀⠃⠰⠀⠀⠀⢾⠉⡿⣿⣧⣬⠼⠿⣿⣿⡇⠀⠀⠀⣿⣿⡀
⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠄⠀⠀⠀⠀⠋⠀⣠⣤⣄⣤⣤⣄⣶⠀⠀⢼⠀⠃⠀⢄⣿⣿⡇⠀⠀⠐⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⢚⣦⣾⣦⠀⠀⠐⣿⣿⠿⠹⣠⣾⡯⠴⣶⠎⠀⠀⠈⠈⠋⣽⡇⠀⠀⠀⣿⣿⠇
⢸⣿⣿⣿⣿⣿⣿⣿⠠⣾⣇⠠⡟⠁⠀⠀⠀⢹⢏⣤⣈⢿⠇⠀⠼⠋⠀⡤⠀⠁⢻⣘⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⡇
⢸⣿⣿⣿⣿⣿⣿⣿⠲⣾⣦⣌⢷⣶⠀⠀⣠⣾⣿⣿⡟⢻⡿⣭⣿⡿⠒⣶⣄⣤⣾⣿⣿⣿⣷⡄⠀⢿⣿⣧⠀⠀⡇
⢸⣿⣿⣏⣹⣿⣿⣷⡄⣽⣿⡆⢈⣿⡇⢠⣿⣿⣿⣿⠀⠀⣧⠉⠛⠃⣴⣿⣿⣻⡿⣿⣿⣿⣿⣿⣄⣿⣿⡀⠀⠀⡀
⢸⣿⣿⣿⣿⣿⣟⠁⢠⣾⡟⢳⣸⡿⠋⣨⣿⣿⣿⣿⡀⠠⣿⣬⣴⣠⣿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣶⣀⣿
⢸⣿⣿⣿⣟⠙⣷⠘⢛⠻⣿⡆⠙⠀⣸⣿⣿⣿⣿⣿⣷⣾⠿⠿⢿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⡿
⢸⣿⣿⠟⣿⣄⠙⡂⣸⡇⣨⣿⣉⣭⣿⣿⣿⠛⠛⠉⠉⢻⣿⣦⣧⣀⣴⣶⢼⣿⣿⢿⠁⣴⣿⣇⠁⣿⣿⢿⡯⡅
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣦⡌⢹⡜⢉⠙⣿⣿⡐⣿⣿⣼⣿⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠘⣿⠛⠉⣽⣿⣿⣿⣿⣿⣿⣿⡋⠙⠙⢶⣾⣯⣮⠁⣿⣿⣟⠛⢿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣀⣴⣿⣿⣿⣿⣿⡿⠿⢿⠇⠀⢠⣿⣿⣷⣾⣿⣿⣿⠀⣵⣾⡿
⢸⣿⣿⣯⣿⡏⢽⣷⣻⣿⡻⠿⠿⣿⣿⡟⢸⣿⢻⠏⢛⣿⣿⣻⣿⣿⡇⠀⢈⠀⠀⡿⠙⣿⣿⠋⠓⠉⠛⠀⣿⣿⣧
⢸⣿⣽⣿⣿⣿⣾⣿⣿⣧⡿⣿⢒⣿⣿⠀⣸⡇⠀⢀⣾⣿⣼⣜⣿⡿⢀⣿⣿⡆⠀⣹⠀⠈⣥⠀⠀⠀⠀⠀⣿⣿⣿
⠘⣿⣿⢿⣿⣿⣿⣇⣿⣿⠿⣽⣿⣿⣿⣀⡿⣠⣤⠈⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⡿⠀⠀⢸⡆⠀⠀⣶⣾⣿⠆
⠀⣿⣿⠋⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟⠹⣿⣿⡿⢹⣿⣋⠅⠀⣩⣿⣿⡿⠟⠁⢀⡿⠶⢖⣻⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠋⠃⠰⡽⣿⢿⢿⣿⡿⢿⣿⣿⣿⠳⠀⠁⠀⠀⠘⠛⠛⠛⠋⠁⠀⠀⠀⠲⠶⠛⠛⠉⠉⠉⠁⠀⠈⠉
⠀⠀⠀⢠⠀⠀⣉⢸⣟⡿⠩⣯⡃⣿⢆⡤⣾⣦⡄
⠀⠀⠀⠙⠛⠋⠉⠙⡁⠄⣒⣨⠿⠷⠛⠒⠛⠉⠁


⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢻⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡼⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣈⢉⡃⠡⢂⢨⢛⣴⡆⠐⠶⣧⢀⠀⣤⣧⣤⠷⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⣢⡀⠹⢄⣭⢼⣤⡡⣾⣧⣀⣱⣦⣵⣤⣾⣿⣟⡩⠪⠹⠇⠀⠺⣶⣾⣿⣿⣿⡷⠚⣻⠋
⠀⠀⠀⠀⠀⠀⢐⡿⣾⢟⢹⣿⣤⣆⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣾⣯⣽⡿⠉⠋⢡⠾⠋
⠀⠀⠀⠀⠰⢿⢯⢟⡀⣴⢖⡿⣱⣿⣿⣿⣿⣟⡛⣿⣿⠌⠉⡉⠉⠉⠑⢋⡩⢿⣿⣹⣿⣿⣯⣿⠗⠀⠠⠌
⠀⠀⠀⠀⠰⠰⣶⣴⣶⣶⣮⣼⢿⣿⣿⣿⣿⡟⠀⠹⣷⣦⠀⠀⠀⠀⠀⠀⠈⣿⠛⢿⣥⣿⣿⣟⡴⠜⠁⠀⠀⢒⠀⠀⠀⢀⠠
⠀⠀⠀⠐⠛⠛⠛⠙⣻⣵⣿⢿⢷⣿⣿⣿⣿⣧⠀⠀⠹⣿⣷⣦⣤⣶⣤⣠⣼⣿⣾⡿⠿⠿⠿⢿⠄⠒⠚⠂⠀⠈⠹⠿⠿⠞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣷⡆⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠶⠶⠧⡾⠶⠦⠀⢀⣀⢠⣇⡐⠂
⠀⠀⠀⠀⠀⣠⣴⣿⣯⣛⢻⣿⡿⣿⣽⠛⠿⢿⣿⠿⠟⠳⡯⠟⠂⠁⠈⠛⡙⠙⠛⠓⠿⠛⠛⠓⠿⠿⢿⡿⠛⠻⠿⠯
⠀⠀⠀⢀⠈⠛⣿⣿⣿⣿⠟⠳⢶⣤⣹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠
⠀⠀⠀⠘⠦⠤⠼⠟⠋⠁⠀⠀⠀⠀⠉⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠖⠒⠂⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⣦⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣈⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⠄⠀⠤⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣰⢠⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣡⡶⠂⠀⡀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠠⠉⠘⠋⠁⠁⠀⢸⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣧⡀⠀⣀⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠴⠎⠀⠀⢀⣤⠦⣴⣶⣿⣿⣟⣾⣿⣿⣿⣿⣿⣏⢻⣿⣻⡆⠉⠛⡻⢿⠟⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⣀⠀⣼⠃⠀⠀⣽⣿⡟⣶⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠶⡿⠀⠀⠀⣿⣿⡇⠻⣦⣤⣤⣤⣤⠀⠀⠀⠈⣿⣷⡀⠀⠰⣥⣄⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠃⠀⠀⢸⣿⣿⡅⢠⣦⡤⠀⢹⠣⣴⣾⠆⢀⣿⣿⣷⣄⠀⠀⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢀⣤⣿⣿⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣤⣀⣀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠤⠤⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⣾⠇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠤⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠂⡀⣀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⣐⣂⣀⡀⠙⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠐⣦⣤⣤⣶⣦⣤⡀⠀⠸⠿⢁⣴⣿⣿⣿⣿⠟⠁⠀⠀⠘⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡙⠿⠂⠀⢀⣤⣶⣶⣴⣤⣾
⠀⠀⠀⠀⠀⠀⢙⢃⣤⣿⣿⣿⣷⠀⠀⠀⠎⠙⠛⠛⠋⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠈⠙⠛⠛⠋⠕⠀⠀⠀⣿⣿⣿⣿⣤⡘⡃
⠀⠀⠀⠀⠀⠀⠈⢻⠛⡻⢿⣫⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠘⢁⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣽⠿⢟⢋⡋
⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣼⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⣀⣼⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡆⢠⣷⣷⣴⣦⣤⡴⠆⠀⠀⠀⡄⠀⢸⣆⠀⢀⣾⡇⠀⢀⠀⠀⠀⠲⣶⣤⣶⣶⣿⢿⡀⣸⠃⢿
⠀⠀⠀⠀⠀⠀⣄⠀⡃⠀⡟⣿⡏⠉⠛⡿⣊⣀⣀⡠⠄⠀⠁⣐⢀⡿⠀⠈⢿⡀⣆⠘⠀⠠⢤⣀⣀⣱⢿⠋⠉⣻⡿⣿⠀⠘⠀⡠
⠀⠀⠀⠀⠀⠀⠈⡄⡇⠀⢿⣿⣷⡶⠾⣿⡿⠛⠁⠀⠀⢀⢸⣧⣸⡇⠀⠀⢹⣠⣿⠀⠀⠀⠀⠈⠻⣿⣿⠷⢶⡟⣿⡿⠀⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⢻⣿⣅⠀⠉⢿⣿⡳⠀⠀⣶⢰⡟⢹⣧⡆⢠⣼⠋⢿⠀⣆⠀⠐⢞⡿⡏⠁⠀⣨⣿⠏⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠄⠀⠻⣿⣿⠶⢾⣯⣀⣤⣾⠿⠉⡇⠈⣿⠀⠘⣿⠀⠸⠙⡿⣦⣤⣀⣽⡧⢶⣿⣿⠋⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠘⢿⠓⠀⡀⠉⠀⠀⠈⠆⢡⠀⣟⠀⠀⣿⢀⠄⠘⠁⠀⠀⠉⠀⠐⢻⠿⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣦⡴⠈⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⠃⠀⠀⠀⢀⣠⣴⣤⣴⣦⡀⡀⠀⠀⠀⠘⢿⣟⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠉⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠉⠛⠻⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠉⠉⠛⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡦⢦⡀
⠀⠀⠀⠀⠀⢠⡆⣿⣯⣿⡟⣵⣶⣦⣄⠀⠀⢠⠿⠛⠷⠀⠀⠸⠷
⠀⠀⠀⠀⣸⠎⠉⠀⠀⠙⢸⣿⠟⣫⣭⣥⡄⠠⣿⣿⠿⠷⠄
⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⢫⣾⣿⠿⢛⣛⠀⢋⣴⣿⣿⣿⣆
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠸⢫⣴⣿⣿⠿⢷⢸⢟⣩⣥⣤⣭⣄
⠀⠀⠀⠀⡀⣶⡿⣡⣾⡿⣂⣤⣿⠟⣩⣶⣾⣶⣶⣿⣿⡏⢸⣿⣿⡄
⠀⠀⠀⣼⣧⢿⡇⣿⢏⣼⣿⠿⢋⣾⣿⠿⢛⣉⣙⣩⡄⠀⢸⣿⣌⡑
⠀⠀⠀⣬⡛⠁⠁⠛⢸⣿⢣⡘⠛⠟⢡⣾⣿⣿⣿⡿⠀⠀⠸⢿⠿⣿⡀
⠀⠀⠀⠸⠁⠀⠀⠀⠈⠃⣾⣿⣦⡀⠀⠈⠛⢿⣿⢇⠀⠀⣀⣴⡞⠀⢠⡀
⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠻⢩⣿⣿⠀⠀⠀⡈⠒⠑⠀⢰⣿⣿⣷⣦⠄⠃⠀⠰⣶⣦⣜⢲⠀⢀⣀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⢶⣾⣷⣶⡀⠈⠀⠈⠙⢁⠀⠉⠀⠀⡀⠀⠀⠉⠉⠁⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣿⠛⠛⠀⠀⣴⡦⢀⡏⠀⠀⠀⠘⡅⠀⠀⢀⣀⣀⣰⡏⠉⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠁⠉⠀⠀⠙⠲⣴⣿⣿⠇⠀⠀⡄⠸⠇⡀⠀⠠⣭⣟⣛⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⠀⠀⠚⢠⡞⢆⡀⠰⣾⢋⣹⡿⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⣀⣀⣀⣀⣤⣴⣆⠄⠀⠸⠀⠀⠙⠓⠋⠽⠏⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⡿⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⠆⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠰⠚⠛⠛⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⠛⠓⠆⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⣤⣤⣤⣭⡂⠀⠀⠀⠀⠀⠀⢐⣩⣤⣤⣤⣄⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋⠀⠀⡇⢸⠀⠀⠈⠛⠿⠿⠿⠛⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠙⠒⠶⠶⠖⠒⢫⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠲⠶⠶⠖⠋⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⠀⠀⠀⠀⠀⠈⠃⠙⢦⣄⣠⡴⠋⠘⠁⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢆⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⡰⣱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡣⡀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠋⠉⠁⠀⠀⢀⢜⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠂⠀⠀⠀⠈⠑⠒⠒⠒⠒⠊⠁⠀⠀⠀⡐⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡂⠀⠀⠀⠀⢑⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⡅⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡀⠠⠄⠀⡇⢠⢄⠀⠠⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢀⢀⠈⢷⠀⢠⠀⡇⠀⡄⢀⡏⠠⠂⠀⠠⠀⠃⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠢⠠⠈⣆⠈⠀⡇⢰⢀⡞⣀⠑⢀⡴⠃⡀⠀⠀⠀⠀⠀⠰⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠐⠈⠀⢈⠳⣄⢣⡸⣾⣇⣧⡇⡼⡰⢂⡴⠋⢀⠁⠊⢀⢀⡀⠤⠀⣰⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣅⣓⢮⣳⣷⣼⣿⣿⣿⣿⣵⡯⠞⣉⣤⠴⠚⠋⡁⠀⠀⣴⣿⣶⡖⠈⡓⣤⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠤⢬⣙⣛⣿⣿⣿⣿⣿⣿⣿⣾⡯⠥⠄⠂⠀⠀⠁⢀⡞⠉⡿⠛⠓⠼⣉⡽⠈⠑⠢⢤⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠋⠉⢉⣉⣭⣿⣿⣿⣿⣼⣿⣿⣿⡿⠯⣛⣉⠉⠉⢉⣱⣏⡄⠀⠀⣤⠴⡆⠉⠀⣶⣾⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣀⣡⠴⠚⣩⢷⣿⣿⣿⣿⣿⣿⣟⠯⣓⠦⣄⣈⠀⣲⡟⠀⣙⠦⠀⠧⢄⡜⠃⠀⠛⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠒⠉⢀⠀⡢⢋⡴⢫⢾⡏⣿⡟⡿⡌⢯⡳⣌⠓⠀⢈⣽⠉⠉⠙⠃⠀⠀⠀⠈⠀⠰⡖⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢈⡴⠋⠐⠁⡞⠰⠞⡇⠺⠹⡄⢂⢈⠳⢤⣽⣽⣿⣯⠀⠀⠀⣶⣿⣤⠀⠘⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠈⢀⠀⠂⡼⠁⠂⠀⡇⢀⠁⠱⡀⠡⣰⣯⡄⠈⠋⢉⡀⠀⠀⠉⠻⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠸⠁⠐⡰⠀⡇⠘⡀⠀⠡⡰⠳⡴⠃⠀⠠⣞⠉⢏⠀⣠⠞⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⡇⠀⠀⠀⣜⠁⠀⠀⢀⣀⡀⠘⠋⠉⠀⠈⡷⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⣏⠙⠢⣄⠾⣿⠧⠀⢐⠖⢶⡜⠁⢸⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡲⢄⡙⠳⢄⡀⠈⢱⣿⠈⡁⢸⠀⠩⠀⡼⠀⠄⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⠟⠒⢤⣉⡶⠧⠘⣆⢱⢸⡄⠇⣼⢁⠕⢁⠾⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣰⢇⢄⡘⠀⠈⡙⢮⣳⡘⣿⣾⣿⣸⣧⢣⡔⣫⠠⠈⢀⡠⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡴⠁⠀⡀⣉⠓⠲⢬⣶⣽⣿⣿⣿⣿⣿⣷⡿⣊⡥⠖⣚⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⠞⢀⣀⣀⣁⣀⣈⣉⣶⣾⣿⣿⣿⡿⣿⣿⣿⣿⣟⣋⣁⣀⣉⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣠⠏⠀⠀⠀⠀⡄⠀⠐⢔⣺⡿⣿⣿⣿⣿⣿⣿⣿⣯⣝⡛⠢⠄⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡰⠃⠀⠀⠀⢀⢀⡤⠔⠚⣉⡤⢺⠿⣿⡿⣿⣿⡟⢿⣯⡻⢍⢓⠲⢤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡜⠁⠀⠀⠀⠀⠈⠁⠀⠐⠨⣈⠜⠣⡊⡿⠹⢿⠙⡽⡌⠣⠙⢦⡁⠠⠂⠈⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⢀⠈⣸⠁⠸⢸⠀⠡⠹⡄⠊⡂⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢨⠃⠀⠄⢸⠀⠢⠀⢱⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⣿⣿⣿⠟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡤⢤⣀⢀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠁⠀⡿⠛⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⢀⡌⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠸⠏⢀⡄⠀⠀⠀⠙⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣤⣤⣄⠀⠛⠂⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠛⠟⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣿⣿⣿⢠⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣮⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡌⠢⠀⠀⠀⠀⢠⡆⠀⠀⢠⡀⠀⠀⠀⠀⠔⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⢻⣿⣿⣿⣆⠀⠀⠀⠀⡼⣇⡀⢀⣼⣇⠀⠀⠀⠀⣴⣿⣿⣿⢏⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢿⣿⣿⣿⡆⠀⠀⣰⣧⣬⣍⣩⣤⣼⡄⠀⠀⣰⣿⣿⣿⡏⣼⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⠀⠁⠀⠀⠹⠋⠀⠀⠉⠀⠀⣿⣿⣿⣿⠇⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⣿⣇⢻⣿⠸⣿⣿⣿⡿⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⢿⣿⣿⣿⢰⣿⠏⣿⣿⠃⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢿⣿⣷⣬⠑⠈⠉⠁⠀⠀⠀⢠⣧⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⠉⠀⢫⣥⣾⣿⡏⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡞⠹⣄⠀⠀⣰⠏⢳⡀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡤⠊⠀⠀⠀⠀⠀⡀⠀⢠⡞⢀⡄⠙⣷⡾⠃⢠⡀⣳⡀⠀⢀⠀⠀⠀⠀⠀⠁⣼⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⠤⠤⠤⣄⡀⢷⡀⠈⠙⠾⣇⠀⠸⠃⠀⣼⠗⠋⠁⢀⡎⢀⣤⠤⠤⢤⠄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⢤⣄⡉⠢⠑⠀⠀⠀⠈⠳⡀⢀⠞⠁⠀⠀⠐⠋⠘⢁⣠⡄⠀⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡀⠘⡇⠀⠀⠀⡆⠘⣿⣿⣷⣦⡀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠁⡀⠀⠀⠀⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠘⣷⣄⣧⠀⠀⠀⢻⡄⠘⢿⣿⣛⣉⣂⠀⠀⠀⣄⣠⠀⠀⠀⣔⣉⣻⣿⠟⠁⣰⠇⠀⠀⠀⡟⣠⣿⠁
⠀⠀⠀⠀⠀⠀⡄⠘⣯⠻⣦⠀⠀⠀⠻⣄⠀⠉⣹⣿⣿⣇⠀⠀⠘⠁⠀⠀⣼⣿⣿⡋⠁⢀⣴⠏⠀⠀⢀⡼⢛⡿⠃⣠
⠀⠀⠀⠀⠀⠀⠙⣦⡘⢧⡈⠓⠄⠀⠀⠈⠛⠷⠶⣦⡙⢿⠀⠀⠀⠀⠀⢠⡿⢉⣴⠶⠟⠋⠁⠀⠀⡤⠊⢀⠞⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡈⠂⠀⠀⠀⠀⠀⢤⣴⠀⠃⠘⡆⠀⠀⠀⠀⢸⠃⠈⠐⣦⠄⠀⠀⠀⠁⠀⠂⣁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣅⠂⠀⠀⠀⠀⢤⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠀⠀⠀⠀⠐⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⢦⡻⡄⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⢠⢏⡔⢀⡀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⠙⢷⣿⣦⣝⣷⡀⠀⠘⢿⣿⣿⡿⠀⠀⣠⣾⣫⣴⣿⠾⠋⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⣿⣿⣿⣦⣄⣀⣬⣭⣀⣠⣴⣿⣿⣿⣿⣷⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⣿⣭⣍⡹⣿⣿⣿⣿⣿⣿⣿⡿⢋⣭⣭⣷⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣮⣿⣿⣿⣿⣿⣟⣵⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠙⠿⣦⣄⣀⣀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠛⠛⣿⠀⠀⠀⠀⣠⣤⣶⡶⡾⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⢀⣿⠀⢀⣴⡿⠟⠋⠉⠀⠈⠁⠀⠈⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣰⣿⠟⠀⠀⠀⠀⢠⣴⣷⠶⠶⢤⣦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡟⠀⠐⠠⢤⣾⠟⠁⠀⢀⣀⠀⠈⠻⣦⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣵⢟⣿⣿⣧⠀⠀⣠⡴⣶⣶⣦⡀⠀⠀⢀⣼⠗⣿⣿⡧⠬⠁⢐⣿⠇⠀⣠⡿⢛⠏⠑⡄⠀⢹⡗⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠾⠛⣹⡟⠀⠈⢿⣦⣿⣩⣿⣥⡶⢀⣼⡟⠰⣿⣿⣯⡁⠀⢻⣿⠁⠀⣿⡁⠘⠶⠚⠁⠀⣼⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣰⠟⠀⠀⠀⠺⠭⠛⠿⠋⠉⠀⢸⣿⠀⠀⢿⣿⣿⡴⠤⠼⣿⡇⠀⠈⢻⣦⣤⣄⣤⡾⠛⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⣼⡟⢷⡠⣈⠻⣿⣿⣶⡖⢻⣿⣦⠀⠀⠀⠈⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠁⠀⠴⠚⠛⠳⡀⠀⠀⠀⣠⠟⣼⡟⢿⣿⣿⣦⡙⠿⠿⢿⣿⣿⣿⣿⣿⣴⣶⣄⣄⣠⣤⣾⡿⠛⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢦⣤⢤⣶⣶⡄⠁⠀⢀⡴⠋⠘⣿⠃⢸⡿⠁⠉⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⣿⣿⠿⠛⠋⠀⣀⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⠏⠀⠀⠀⠀⠈⡇⠀⠀⣿⡴⠛⢙⡷⠟⠛⠷⣶⣤⠦⣤⣤⡤⠤⢤⢠⣀⣠⣤⣶⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⣸⠠⠀⠀⠀⠀⠀⢹⠀⠀⣿⠀⣴⠏⠀⠀⠀⠀⠈⠛⠷⢶⣶⣶⡶⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣤⣀⡀⠀⠈⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣠⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣄⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣾⣿⣾⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⠁⠀⠀⣤⠋⢰⠆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠴⣀⡀⢘⡄⠀⠀⠀⠀⢾⡷
⠀⠀⠀⠀⠀⠀⠀⠆⠈⢁⠄⠀⡄⠢⡀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠰⠒⠚⠃⠀⣠⠒⡄⠀⠀⠀⠀⡠⠄⠀⠳⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠃⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⡆⢑⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⢀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠂⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠁⠀⠀⠐⠉⠀⠀⣀⠠⠤⠐⠒⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⣀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠁⠀⡐⠁⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠰⣛⡆
⠀⠀⠀⠀⠀⠀⠀⢎⡹⠆⠀⢀⣀⠀⠀⠀⠀⠀⠉⠂⢄⡀⠀⠈⠂⠐⠀⠃⠀⡼⠀⠂⠀⡀⠄⠊⠁⠀⠀⡒⡦⠀⠀⠘⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠉⠀⠰⠹⠀⠀⠀⠈⠀⠄⠀⠀⡀⠀⠰⠀⢀⠄⠊⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡠⠜⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠢⡀⡈⠀⠀⠁⠐⢀⣀⠠⠔⠒⠀⠈⣠⡀⠀⠀⠉⠉⠀⠀⠀⠀⢈⣂⡀⢈⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⠉⠀⠀⡀⠀⠉⠉⢉⠉⠀⠀⠀⠄⠀⠀⠀⠠⠀⠊⠁⠀⠄⠀⠀⠀⠀⠛⠋⠀⠀⡀⠀⠀⡞⢳⠀⠈⠀⠑⠚
⠀⠀⠀⠀⠀⡤⠴⢻⣀⠀⠀⠇⠸⠀⠀⠀⠁⣀⡀⠤⠀⠒⠐⠂⠈⠀⠈⠀⠀⠀⠀⠈⠉⠍⠉⠁⠐⠒⠒⠠⠤⣀⡀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡱⢀⡠⠊⠁⠀⠀⠤⠄⠒⠉⠁⠀⠠⠄⠀⠀⡠⠔⠀⠁⠌⢰⢀⠂⠠⡍⠂⠀⠐⠂⠀⠀⠀⠀⣀⠀⠈⠉⠒⠤⡀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⡀⠀⠀⣀⠤⠒⠀⠀⠐⠁⠀⠀⠂⠌⠀⠀⠀⠀⢂⠀⠀⠀⠑⠄⡀⠀⠀⠸⣅⠗⠀⠀⠀⠀⣈⠑⢄⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⡠⠄⠀⠰⡉⡳⠀⢀⠊⠠⠀⠘⠀⠀⠀⠆⢠⠒⢢⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⢈⣤⠀⠀⠀⠀⠀⡠⠃⠀⡄⠀⢠⠀⠀⠀⠘⡈⠓⠁⠀⢀⡀⠈⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡤⠊⠑⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣉⠆⠀⠀⠈⠛⠁⠀⠀⢀⠜⠁⠀⡘⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣤⠗⠀⠀⠀⠀⢨⣇⠤⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠊⠠⣀⠀⠰⠀⣀⠔⠁⢀⠴⠁⠀⠀⠀⠁⠀⠀⠀⢴⣷⠀⠀⢰⠀⠀⢰⠋⢳⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢐⠁⠀⠠⠊⠁⢀⠔⠁⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⣀⡠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠳⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣀⠀⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⣀⠜⠢⢤⠀⠀⠀⠀⠀⠀⠸⡿⠆⠀⠀⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⡤⠤⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠈⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣷⣄⠀⠀⠚⠻⢿⣿⣿⣿⣿⣿⠁⠀⠀⠘⠛⠟⡀
⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠙⠻⣿⣿⢿⡀⠀⠀⢀⣤⣾⠃⠀⠰⣄
⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠙⢭⠒⠀⠘⣴⣿⣿⣿⣶⣤⣀⢻⠀⠀⢠⣤⣤⣌⡑⢦⡀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠈⢛⣁⣀⣤⣤⣬⣆⠀⠠⠄⠀⠙⢿⠟⠋⠉⠛⠈⠸⠀⠀⠀⠉⠻⠿⠟⠌⠁⢠⣾⣿⣟⠶⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠉⢛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡿⠟⡓
⠀⠀⠀⠀⠀⠀⠀⣆⠀⣼⣿⠋⠉⡁⠀⢀⠰⣿⡗⢁⡿⠀⡀⠀⠀⠈⣡⠆⠀⠀⠀⢠⣀⣀⣀⣿⠀⠠⢺
⠀⠀⠀⠀⠀⠀⠀⣯⡴⠁⠉⠀⠀⠈⠙⠲⣤⣶⣿⣿⡇⠀⢀⠀⡀⠀⡿⢀⠀⠀⠤⣤⣈⣿⠛⢻⣷⡇⠘⢀
⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⡀⠀⠀⢧⠂⣠⢺⡀⠀⠀⣤⣽⠟⠛⣿⡿⠃⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⠘⠀⠀⠀⡿⠀⠑⠷⣤⣸⡿⢴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣴⣦⣤⣤⣴⣶⣿⣿⡟⠂⢀⠀⠃⠀⠀⠀⠀⠀⠀⠛⠃⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠿⢿⡟⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⡟⠁⠀⠀⠀⠘⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⢦⣀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⠛⢦⣄⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀
⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣹⠆
⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⠋
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁⠀⠸⣇
⠻⠶⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢻⠙⠦⠤⠐⡒⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢶⡒⠐⠁⠀⠀⠀⢰⠁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠚⠁
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢙⡆⠀⠀⠀⢀⣈⣱⡄⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢸⣦⣠⡟⣧⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢀⡾⠤⠒⢦⠀⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠉⠀⠘⠋⠽⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿⢠⣄⣀⠀⠀⠀⠀⣤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢸⡇⠀⠀⠀⠀⠀⣠⡞⠋⠈⢷⠀⠀⠀⠀⠙⣧⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠋⢠⣶⠋⠐⠂⠀⣼⠗⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣀⣀⣀⣀⣤⣄⡀⢶⡈⢇⠀⠀⠀⠀⣼⠃⣰⠏⣿⣦⢀⣴⣄⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⡇⢸⡧⠸⣆⠀⠀⢠⡯⢰⡏⠀⠀⢿⠟⠁⠘⠋⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡞⢿⣁⢹⡄⠀⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡾⠁⠘⣧⢂⢻⡀⢸⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡿⠀⠀⠀⢸⡎⠌⢳⡾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⢻⡔⠀⢳⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⡆⠀⠀⠀⠀⠈⣷⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢻⣄⠀⠀⠀⣠⠟⣇⠂⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠙⠛⠋⣁⡼⢻⡄⡀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠛⠉⠀⠀⢷⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠁⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠄⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣘⡀⢹⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠛⠋⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⡸⠛⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⢨⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⡬⠯⠉⠙⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠲⣁⣴⠞⠉⠐⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⣀⣰⠀⠠⣤⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠛⢉⠀⢽⡾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⢍⢢⡀⡀⠀⠀⣻⣿⣧⣠⠐⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣧⣁⣉⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⠉⠛⠿⢿⣿⣿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⢪⣺⣭⠙⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣸⠱⣿⣿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣷⢶⠈⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⣿⣯⣀⢄⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣘⣋⣴⣿⣿⣿⣿⣤⢤⣶⣾⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⠤⡌⣤⣴⣾⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡿⢽⣗⡚⠒⠲⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠄⢸⠟⣛⡟⠛⠿⠿⠿⠟⠲⠦⠖⠣⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣀⣀⡤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠛⠉⠁⠀⠀⠴⠾⠗⠒⠒⠒⢒⣲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣱⠶⠀⢀⣠⡤⢖⠲⡄⠀⠀⠀⠚⠉⠉⠉⢉⣭⣶⠶⠄
⠀⠀⠀⠀⠀⢀⣠⠶⠺⣍⠀⢀⣠⠟⠛⠉⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡭⠵⠦
⠀⠀⢀⡤⣞⠭⠖⠒⠒⠦⣤⣋⠤⠤⠶⠲⣶⢺⠃⠀⠀⠀⠀⠀⣠⠖⠒⠲⡶⠿⠖
⢠⠖⠉⠀⢯⡉⠉⠉⣋⡽⣇⠀⠀⠀⠀⣀⡴⠋⠀⣴⢀⡤⠀⠘⠡⠾⢋⡾⠁⠀⠀⠀⠀⡀
⣏⠀⠀⢀⠤⠽⠯⣍⠁⠀⠈⠱⠒⠉⠉⠉⠉⡤⢾⡵⢋⣡⣾⣭⣽⠿⣥⣀⣀⠀⠀⣖⢼⢹⣞⡆
⠘⠦⣀⠘⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠉⠀⢀⣀⡀⠀⡿⡶⠒⠦⠬⠭⣈⠉⢸⠁
⠀⠀⣆⣿⠿⣶⡶⣶⠒⠲⣖⠒⠛⠋⠉⢡⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠓⠲⠤⣄⡀⠈⠙⠃
⠀⠀⠈⠑⠚⠄⡆⢘⠒⣚⠭⣦⠀⣠⠔⠁⠱⡄⠀⠀⠀⣠⠴⠦⢄⡀⢀⣀⣠⡔⠊⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠋⠥⠟⠀⠀⠀⠙⠛⠁⠀⢀⣀⡹⣄⡀⡼⠁⠀⠀⠀⠈⠙⠷⠀⣷⡾⠛⠛⠛⠭⣭⣕⣦⡀⠀⠀⠀⠀⠀⢀⡔⡄⠀⢠⢌⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠯⣗⡢⢽⣷⣄⡀⢀⡾⣓⠦⠤⠖⠉⠉⠛⢿⢿⣼⠁⠀⢹⡧⠐⠆⠐⢀⢠⠾⠀⠖⠶⢬⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢥⣌⣓⣬⡣⣙⣼⡇⠀⠉⠒⠦⣄⣀⣠⠞⠮⠟⠤⠴⠚⠀⠀⠁⠀⠈⠈⠲⡆⢸⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣭⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠀⣀⡀⠀⠀⡀⢠⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠉⠉⠁⠀⠈⠉⠋⠉⠙⠛⠋⠉⠉⠉⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠐⠂⠸⠇⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠂⡐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⢈⠠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣄⢀⣠⠖⠀⠀⠈⣑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢘⠊⠉⢿⣧⡀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠘⠙⠒⠛⠷⠄⢀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠁
⠀⠀⠀⠀⠀⡀⠀⠅⠚⠁⠀⠀⠀⠀⣠⡴⠟⠛⠀⡖⠀⠀⠀⠐⠀⠀⠐⠀⠃⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠈⠒⠠⠤⢤⠻⠺⠓⣚⣁⣴⣄⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠄⠈⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠙⢉⢄⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡄⠀⠀⠀⢐⡛⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠻⠤⣤⣾⡿⡆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠏⠙⠦
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠉
⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣶⣅⠐⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣛⠛⢿⣿⣿⠇⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⣿⢙⣤⣙⡟⠯⡛⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠎⢹⠟⠏⠷⣶⠒⣦⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢄⡈⠁⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠏⢯⣤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⣤⡀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡾⣿⣷⠈⠠⡀⠹⠿⢣⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣆⠀⢨⠗⠀⢸⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⣩⣠⡀⠈⠀⣠⣽⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⢠⡆⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣽⢧⣔⣚⣽⣝⣓⣴⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣌⠛⠁⠻⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⡎⣽⣿⡟⢿⢿⣽⣿⣿⣻⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⠀⠠⠀⠈
⠀⠀⠀⠐⠀⠀⠀⠀⣠⣿⣿⣾⣿⣟⢻⣿⣿⣿⣿⡾⣟⡿⡿⢸⣹⣿⢿⣿⣿⣟⢈⣿⣿⣷⠀⠀⢀⡴⢯
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⢿⠏⣿⣿⡿⣿⣿⣿⣵⣼⣷⠊⢷⡀⢿⣿⣿⡟⠹⣿⣿⡏⠄⢀⠆⠱⠤⠑⠀⠈⣄
⠀⠀⠀⠀⠀⠀⣸⢿⣽⣿⢻⣿⠿⣼⣿⣿⣿⣿⣿⣿⡯⠌⠁⢩⡈⣥⠘⠯⠱⡟⢸⣷⡷⠇⢠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⣯⣶⠟⠉⠛⠃⢤⠽⡏⢡⣾⡿⣿⣿⡏⠀⠀⠈⠁⠏⠀⠀⢠⠂⢸⣿⠼⡿⠉⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢸⠋⠀⠊⣩⠀⠀⠀⠉⠓⠄⠀⠍⠉⠀⡀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠸⠻⡞
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠑⠀⡠⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢅⠻⠻⠆⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⠀⠀⠀⠤
⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⢀⣀⣐⣀⠀⠀⠀⡀⠀⠀⣀⣀⣀⣀⠒⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣷⡴⢶⣶⣦⣄⠀⠘⠟⣴⣾⣿⣿⡿⠋⠀⠀⠀⡃⠀⠀⠉⠻⣿⣿⣿⣦⠹⠗⠀⢠⣴⣶⣶⠶⣾⠃
⠀⠀⠀⠀⠀⠨⣵⣾⣿⣿⣿⡀⠀⠈⠂⠉⠉⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠉⠉⠁⠃⠀⠀⣿⣿⣿⣿⣦⡭
⠀⠀⠀⠀⠀⠀⢰⠘⠫⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠛⠫⠋⡼
⠀⠀⠀⠀⠀⠀⢸⡏⢆⠀⣼⣧⣠⣀⣀⣠⠀⠀⠀⠀⠀⣧⠙⠛⣤⠁⠀⠀⠀⠀⣠⣀⣀⣄⣸⣿⡀⢠⠚⡇
⠀⠀⠀⠀⠀⡀⠀⠇⢸⣷⣿⠛⠻⣿⣋⣀⣀⡀⠠⢂⡄⢹⠇⠈⣿⠰⡀⠇⢀⣀⣀⣉⣻⡿⠛⢻⡷⣿⠀⠇⢠
⠀⠀⠀⠀⠀⠈⡄⡆⠸⣿⣿⡶⠾⣿⡟⡋⠀⠀⠀⢸⣧⡿⠀⠀⢸⣠⡿⠀⠀⠀⢉⡻⣿⡿⢶⣾⣿⠗⢀⠀⡌
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⣧⣀⠉⣿⡿⠂⠀⣶⣸⠏⣿⡴⢰⣾⠉⣿⣸⡄⠀⠻⣿⠏⢀⣤⣿⠏⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠡⢄⠀⠀⠈⢿⣿⡙⠻⠶⠶⠟⢻⡀⡄⢹⡇⠀⣿⠀⠇⡸⠻⠷⠶⠟⢋⣽⡿⠋⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣀⠀⠙⠁⠀⠀⠀⠀⠀⠁⠓⡜⢅⣀⠻⠜⠄⠁⠀⠀⠠⠤⢤⠏⢁⣀⣤⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠘⠋⠀⠰⠀⠀⠀⠀⡂⠄⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡿⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⢺⣿⣦⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠋⠀⠀⠀⣤⣶⣾⣿⣿⣿⣶⣦⡀⠀⠀⠙⠻⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⠿⠿⠟⠛⠛⠻⠿⠿⠇⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⢴⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⢡⠀⠀⠀⠀⠀⠀⠀⠌⠀⣀⢀⣀⠈⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⣀⣀⠀⠈⠀⠄⠙⢄⠀⠈⠀⠀⢀⠑⠀⠠⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣎⢣⡀⠀⠀⠁⠈⠄⠀⠹⠢⢄⠀⠀⠀⡀⡀⠤⠐⣠⠎⠌⠀⠆⠀⠀⠘⡣⠀⠀⡴⠃⠿⢫⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠘⢛⠓⡖⠒⢀⠤⡀⢀⠀⠃⠀⢁⡤⢞⡊⠢⠰⣯⠏⠰⠀⠀⠀⠀⡀⢀⠡⢊⣩⢤⡀⠄⠃
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠳⣌⡀⠀⠀⠀⠀⠀⢄⠁⠈⡱⠔⠁⠀⠡⠀⠆⠈⡄⣈⡔⠹⡏⠢⡈⡏⢀⠃⠌⠀⠀⠔⢁⣦⡾⠋⠀⠘⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠐⠉⠑⠢⢄⡙⢷⠀⠀⠀⢀⠀⠀⠀⢻⢦⠂⠀⠀⣆⣘⣀⠿⠯⠴⠾⠆⠀⠘⠓⠒⠒⠒⠂⠚⠐⠻⠏⠂⠀⠀⢀⠠⠐⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⣠⠔⠉⠳⣄⠀⣹⠼⠓⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠚⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠤⢀⣔⠁⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡈⠉⠉⠉⠓⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣄⠀⣄⢀⣀⠤⠒⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⢀⠐⡀⠀⠀⠀⠀⣀⣠⣤⢶⡖⠒⠛⠋⠋⠋⠛⠟⠛⢉⠟⠛⠛⡿⢿⡁⠀⠈⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢐⣪⠴⠒⡹⡝⢤⢡⢹⡌⣇⢨⣤⣤⣧⣀⡾⡼⣠⢆⡄⠈⢦⠿⠬⢿⣦⣀⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⡛⠻⠿⠀⢢⢂⣧⢧⣈⣍⣉⣠⣸⣁⣠⣁⠈⣀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢌⠂⠱⡏⠸⠋⠉⠘⢹⠉⡳⠑⠁⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠠⠀⠄⡂⠄⠂⠠⠐⠀⢀⠀⠀⠀⢀⠀⠀⠡⢂⠠⠐⠀⠂⠠⢀⠠⠂⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠈⠀⠀⠀⠂⠀⠀⠀⠈⠔⠀⠄⠈⠐⠈⠔⠈⠀⠈⠀⠨⠐⠁⠊⠐⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠁⢹⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠱⠁⠀⢸⠏⠉⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠯⠴⠇⠂⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢴⠶⠶⢖⠲⢤⣠⠖⠉⠀⠀⠀⠀⠀⢀⠜⠉⢧⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⠐⠋⢀⡞⠁⢀⡤⠴⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠆⠀⠀⡤⠍⠐⠤⡀⠈⡄⠀⡾⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⣤⣤⣄⣀⡀⠈⢦⣶⣴⠇⠀⠈⣇⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠉⠉⢻⠲⠶⣏⣀⣀⣀⣘⣆⣀⡤⠴⠚⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⠀⣀⣀⣀⣧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢠⣈⣦⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠒⠛⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶⢀⢀⣤⣤⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢣⢠⡉⡈⡄⡀⡖⣶⠦⣴⢿⡶⢠⢷⣿⠁⠀⠀⠀⠀⠀⠀⡀⣾⢿⢸⡏⡟⢻⢸⣿⡿⣿⢷⣦⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⡀⠈⠋⠁⠁⡀⠐⠀⡀⢠⠠⣠⡘⡀⠀⡟⠀⠀⠀⠀⠀⠀⡄⠳⠼⠸⡃⠁⠣⢸⣟⣻⣿⣿⣆⣏⣿⡄⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⠃⣜⢠⢌⠠⠀⠀⡀⠁⢁⢨⣥⣁⡁⣼⠁⠀⠀⠀⠀⠀⢰⢳⡶⡆⣯⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇⠁⢡⢿⢸⠇⠇⠳⠹⢸⢿⠾⣾⠛⣟⣿⠀⠀⠀⠀⠀⠀⣼⣲⡄⢇⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⠁⠆⢈⠀⠈⠀⠀⠀⡀⠠⠠⣀⢠⠽⢽⡿⠀⠀⠠⠄⡀⠀⠍⠭⠭⠭⣙⠺⣛⡛⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣧⢠⢱⡇⡞⡀⣦⡼⢠⡁⡌⡀⠳⡈⠃⠀⢀⠨⠀⠀⠀⠀⠀⠀⠀⢛⣰⣏⠻⠏⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠑⢇⢣⠋⠁⠀⠤⠀⠰⠄⠀⠀⠀⠀⠀⠐⠀⠘⠛⠂⠀⠈⠉⢉⠉⠐⢠⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠘⡄⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠙⠶⣶⣦⡀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⢀⣤⣀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⢀⡀⠀⢀⣀⡀⢀⣀⠀⢼⣄⢳⣷⣄⠀⠀⠈⠻⣶⠶⠽⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣦⠀⣼⠁⢻⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠐⠖⠈⣧⡀⠈⠙⣋⣭⠹⢄⣀⣥⣾⠿⡟⡆⠀⠀⠀⠀⠀⠀⠀⢠⣴⡉⠙⠻⣦⣄⠀⣴⣫⣿⡿⠛⠀⠷⠶⠎⠰⠆
⠀⠀⠀⣦⣤⣦⡀⠀⡘⣃⡀⠀⠀⠰⠂⣞⣀⣴⣯⡶⠺⠷⠄⢳⡍⠷⠀⠒⠂⠞⠃⠀⠀⠀⠀⠀⠈⠻⠿⠦⣄⣀⣈⠖⠋⠙⠻⣧⠒⡰⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠸⣿⣿⣿⣷⠟⠉⠛⠢⠤⠀⢸⣿⢿⣟⣦⡀⠀⠀⠀⠀⠉⠻⠶⠄⠰⠶⠶⠶⠤⠤⠀⠀⠀⣀⠢⠀⠀⠀⣀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢿⣿⣿⠏⣦⠀⠀⠀⠀⠀⣾⣿⡿⣧⠿⣿⡦⣤⣤⡀⠀⣤⣤⡤⠤⠤⠀⠀⣠⠀⠀⠀⢠⡈⢄⠀⠙⠛⠀⠀⠀⠀⠀⢠⠀⣤⠀⠀⠐⠂⠰⡖⢲⡆
⠀⠀⠀⢠⠜⠈⠁⠀⠈⠓⣄⠀⠘⢼⣿⣿⡇⢻⡠⣿⠱⣿⣿⠓⢷⡟⠀⠀⡆⠐⠀⠀⠀⠀⠀⠓⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠁⠈⠐⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⢾⣄⠒⠘⠿⠿⠃⠀⠀⢸⡦⣿⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠘⠻⠋⠁⠀⠈⠓⠢⠄⠀⡀⠀⠀⠈⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠆⠀⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⣄⠀⠈⠁⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠨⠃⢀⠀⢸⠟⠉⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⠀⠀⡆⠀⠂⠰⠉⠁⠀⠀⠀⢠⣅⣂⡠⠥⠷⡄⢧⡀⡆⠀⠀⠀⣀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣀⠀⠀⢎⢧⠀⠀⠘⠀⠀⠁⢀⣠⣤⠾⠂⠀⠀⠈⠙⠛⠳⢤⣈⠀⢣⠁⢡⢀⡤⠀⣉⣠⠤⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣀⡀⠀⠀⠣⠀⠀⠀⠐⠈⠡⣞⣁⡀⠀⠀⠀⠀⠑⠀⡄⠀⠀⣁⡀⠁⠲⢋⣔⣪⠥⠤⠶⠶⠶⠶
⠀⠀⠀⠀⣤⠀⠀⠀⠀⢀⣀⡀⠈⠉⠉⠐⠀⠀⠀⠀⠀⠒⠒⠒⠊⠉⠐⠀⠰⠘⡆⠀⠁⠤⠘⠁⠀⠀⠀⢁⠀⢐⠲⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠃⢠⠖⠁⡀⢠⠀⢢⠑⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠴⠋⡀⢀⠆⠀⠀⢀⠈⢆⠀⠉⠳⣄⠀⢀⠀⠀⠀⠀⡤⠀⢀⠌⠀⠃⠘⠀⡀⢣⠀⠈⣦⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡌⠀⠀⠀⢸⠀⠀⢣⡈⠲⣄⠀⢸⠀⠀⢠⡞⠁⠀⠈⠘⠀⠀⠀⠀⠇⠀⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠑⠀⠈⠳⣾⠀⣰⠏⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠈⠂
⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠐⠀⠀⠀⣟⣷⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠁⢳⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠈⠀⠀⠀⠀⡼⠃⢀⠎⠙⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⡜⠑⠢⣄⠀⠀⠘⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⢀⣤⠀⠀⠙⠢⡀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠂

⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠙⢦⡀⢠⠖⠛⠛⠉⠉⠉⠛⠓⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⣴⡭⠴⡦⠀⠀⠀⠳⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠈⢁⠔⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⣤⣼⠉⠲⢤⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠓⠦⢬⣿⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣦⣌⡳⣄
⠀⠀⣠⠤⠶⠦⣄⠈⡇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠻⢦
⢠⠏⠀⠀⠂⢲⡄⢻⣄⠴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳⢦⡙⡆⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⢀⡀⠀⠈⠳⣄
⠘⠦⣤⣀⣀⣀⡁⠀⠀⠀⠀⣀⣠⢤⠄⠀⠀⠀⢸⠀⠀⠀⠈⢳⡙⠀⠀⠀⠀⠀⠀⠀⢻⠓⠦⣄⣻⡗⠦⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣧⠀⢀⣴⣿⣿⡃⢘⣆⠀⠀⠀⢸⡆⠀⠀⠀⠀⢷⠀⠀⠀⢐⣚⣉⠁⠈⢧⠀⠀⠉⠙⠀⠈⠙⠮⣷
⠀⠀⠀⠀⠀⠀⠘⣆⠘⠀⢻⡉⣳⡏⠘⣦⠀⠀⠀⢧⠀⠀⠀⠀⢸⡧⠖⠋⠉⠉⠉⠙⣶⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠙⣅⣹⢎⣈⡷⣄⠀⠘⣆⡀⠀⡄⣸⠃⠁⠀⠀⠀⣀⡴⠋⠀⢹⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⢷⡄⠀⠈⠳⣤⠛⢦⣸⠶⣄⠀⠉⠓⢾⠉⠑⠒⠒⠚⠋⠁⠀⠀⣠⠴⠛⠉⠀⣹
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠙⢦⡀⠀⠀⠑⢴⡉⠲⣼⡓⠦⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⡤⠞⠁
⠀⠀⠀⠀⢀⡴⠃⠀⡤⠀⠀⢇⠙⢦⡀⠀⠀⠈⠚⠧⣉⣳⠧⣉⣳⠖⠤⢤⡄⢠⣤⠤⠄⠀⠀⢷
⠀⠀⠀⣠⠏⠀⢠⣴⠃⠀⠀⠘⣆⠀⠈⠳⢤⣀⠀⠀⠈⠉⠒⠶⢯⣉⣙⡞⠲⡴⠃⠀⠀⠀⠀⣼
⠀⠀⡰⠃⠀⠀⢉⡏⠀⠀⣀⣀⣬⣧⠀⠀⠀⠈⢻⢶⠦⣤⣀⣀⠀⠀⠈⠉⠙⠷⠀⠀⠀⣀⡴⠃
⠀⡼⠁⠀⠀⠀⢸⣡⠴⣿⣿⡿⢯⡀⠀⣠⢤⣀⣯⡬⠧⣄⣠⠍⢭⡙⠒⠒⠒⠒⠒⠚⠋⠉
⣰⠁⠀⠀⠀⠀⠿⠷⠚⠉⠳⣄⡀⠙⠛⠃⠚⠧⢤⡀⠀⠀⢸⠞⢿⣦⣄
⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄⠠⡈⠲⢤⣽⠀⠀⠈⠓⠓⢶⡞
⠀⠘⢮⠉⠓⠲⠤⣤⣀⣀⣀⠀⠀⡏⡹⢤⣘⡦⡼⠇⠀⠀⠀⢠⠖⠚⣆
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⢉⣉⢹⡇⣇⠀⠉⠀⠀⠀⠀⠀⣾⠉⠓⡶⠃
⠀⠀⠀⠀⠈⢳⡀⠀⠐⠀⠒⠒⠛⢧⠸⣄⠀⠀⠀⠀⠀⢈⣿⡉⠙⢧
⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠀⢀⡷⣌⠢⠀⠀⠀⠀⠀⢹⠙⠒⠋
⠀⠀⠀⠰⣟⠿⣗⡶⠶⠤⢤⣤⠤⠾⠦⣤⡽⡶⠤⢤⣤⠔⠋⣄
⠀⠀⠀⠀⠈⠓⠲⢭⣙⡒⠚⠶⢶⣒⣒⣺⣷⣧⣤⣴⡏⠀⣼⣹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⣠⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠉⠀⠀⠀⠀⠀⠀⠀⠈⠠⠠⠀⠀⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠡⠀⠄⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢈⠃⢀⠀⠰⠄⠀⠉⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠐⠂⠐⡀⠀⠀⠸⠀⣀⡀⠐⠀⢉⠐⠀⠀⠀⠀⠀⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠂⡀⠀⠀⠀⠁⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠈⢣⣄⣠⡖⢺⡄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠄⠀⣾⢀⠠⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠀⠀⡀⠐⠀⡀⠁⣨⠋⢀⠈⢠⠄⠁⣆⣴⠶
⠀⠀⠀⠀⠀⢠⡄⢀⠈⢠⠈⢁⡅⠈⢉⠒⠂⠀⠤⢠⠤⠤⠲⣄⢦⣤⢀⣀⣀⡤⠬⠹⢁⡄⡉⠒⢨⠀⢁⠏⡄⠀⠶⠠⠄
⠀⠀⠀⠀⠀⠀⡘⢫⡐⠉⠘⠈⠁⠂⠛⠡⠄⠗⠀⠆⢀⠀⠠⠀⠀⠀⠈⠀⠀⢰⠀⠀⠀⡀⠉⠲⠀⢘⠋⠠⠀⠙⠀⡄
⠀⠀⠀⠀⠐⠂⠳⡀⢰⡚⢠⠸⠆⣀⠶⢀⠰⡇⢰⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠓⢤⠙⠚⠀⢀⠐⠃⠆⢀⠈⠁
⠀⠀⠀⠀⢀⢠⡆⠁⢸⢷⠀⢠⣆⠁⣤⠈⢀⡇⢈⠀⠀⠀⠨⠀⠀⡀⢀⠀⠀⢸⠨⢠⠀⠁⠤⠀⢼⠖⠂⡄⠀⠦⢀⠘⠀⠆
⠀⠀⠀⠀⠈⠀⡃⠆⢹⠈⡌⠀⡅⠛⢀⠐⠀⡅⢠⠀⠀⠀⢀⠀⠀⠀⠐⠀⠃⠘⢠⠀⠀⠆⢀⠀⠸⡦⡄⡀⠃⢀⠀⠀⠆⢀

⠀⠀⠀⠀⠠⠘⡇⠀⢸⠀⠳⠄⠋⠶⠙⠀⠄⢃⠀⡀⠁⠀⢈⠀⠀⠀⠀⠂⢀⠀⠰⠀⠁⠠⠀⡀⠀⣧⡄⠃⠤⠈⠐⠂⣄⠈
⠀⠀⠀⠀⢀⠠⡀⠁⢸⠀⣾⢀⠖⢤⠸⠆⡀⠛⠀⠀⠐⠀⠐⠀⠀⠁⠲⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⢨⠁⠶⢀⠘⠀⡄⠁⠐
⠀⠀⠀⠀⠈⠀⡀⠆⢰⠄⢠⡀⢠⡄⠠⣆⠀⢰⠄⠀⠀⡀⠊⡀⠀⠐⠀⠄⠀⠠⠀⠘⠀⠀⡀⠚⢤⢸⠁⢰⠈⠀⠄⠀⠶
⠀⠀⠀⠈⠲⠎⠁⣤⠈⠀⠋⡙⠀⠀⠁⢀⠘⢀⡄⠀⠀⠁⢠⠀⠀⠀⡀⠂⠀⠄⢁⠀⢲⡀⠁⠀⠠⢘⠰⢀⠀⠃⡀⠁⣠
⠀⠀⠀⠀⢀⡀⠧⣀⠀⠃⠦⡇⠀⠛⠤⠋⠠⠘⠇⠀⠀⠄⢠⠂⣁⠀⡅⠘⠀⢀⠀⠄⠀⠙⠘⠠⠆⣠⢤⠈⠀⠆⠁⠒⢀
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠃⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠁⠈⠀⠀⠈⢂⠘⠀⠠⠀⠀⠰⠆⠀⢳⠀⠀⠂⡀⠃⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠞⠛⠲⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣫⣥⣶⣿⣿⣿⣿⣿⣿⣷⣆⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣵⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠻⣿⣿⣿⣿⣿⡿⠀⠀⠤⣤⡀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⣯⣿⣽⣯⡙⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢿⣿⠛⠛⠿⠅⠀⠘⢿⣾⡏⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠁⠀⠉⠉⢣⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠞⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⣦⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣶⣦⢀⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⠀⡀⠀⠀⢰⣤⡆⠀⠀⠀⠀⣴⣶⣶⣿⠀⠀⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠁⣿⣀⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠵⡿⠀⠀⡾⠀⢿⡀⠀⠀⠀⠿⠋⣙⡿⠙⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⡇⠀⣤⣥⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⣰⠇⠀⠀⠻⢦⣄⣀⡀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⡿⣿⣿⣄⡐⠂⣰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⣿⠀⢀⣤⠶⢚⣛⣛⣻⣿⣇
⠀⠀⠀⠀⣠⡾⠛⠉⠀⢰⣯⡴⢿⡇⠀⢈⡿⣟⢻⣧⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⠿⠟⠛⠋⠉⠉⠉⠉⠛⠳⣦⣀
⠀⠀⠀⣰⠟⠀⠀⡀⠀⢠⡋⠀⠈⣿⣴⠛⣆⡿⣿⡇⠙⢧⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⢰⡏⠀⠀⢠⣿⣷⠀⢷⡀⡸⣸⡇⣸⠋⣇⣿⢠⠀⡼⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⢀⣼⠇⠀⢀⣽⣇⡿⢀⢌⢷⣱⣳⣿⡿⠀⢹⣿⣷⣱⠃⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣐⢽⣿⣆⢦⡀⠀⠀⠀⠀⠀⠀⡈⡀⡀⠀⠀⠀⠀⠀⣿⡇
⠀⣾⠃⠀⠀⣺⣿⣾⢁⣎⣾⣿⣷⣿⣿⡷⢚⣾⣿⣿⠃⢰⣟⢸⣧⡞⢻⡄⠀⠀⠀⠀⠀⢸⡟⠀⣐⠦⣽⣿⣿⢸⣿⠀⠀⠀⢀⢆⢣⣱⣿⣿⣦⣤⡀⡔⣦⣿⠇
⢸⡇⣤⣾⣿⣿⡿⠷⢾⣿⡿⠟⠋⠉⠀⠚⠿⣿⣿⡇⠀⣾⣷⣼⣿⢧⡀⠉⠙⢳⡀⠀⣀⣿⠁⠀⠦⢭⣿⣿⣿⢸⣿⣧⣼⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⡇⠙⠋⠁⠀⠀⠀⣸⣷⡇⠀⠀⠀⠀⠀⣓⠢⢭⣿⣤⣿⣋⡥⡦⠘⠇⠀⠀⠈⢻⠟⣿⡟⠀⠨⣭⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠏⣽
⢹⣿⣿⣴⣴⣼⣿⣿⡼⣧⣿⣶⣤⣤⣄⣒⡠⢉⣲⡾⣿⣿⡖⠒⡗⠀⠀⠀⠀⠀⡞⣼⣿⠇⠀⣒⣲⣿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣸⣿⣿⡿⠟⠛⣉⣩
⠀⠙⠷⠿⣿⣿⣿⣷⣶⣿⣿⡟⢏⠋⠙⣿⡟⠛⠋⠁⢸⣿⡿⣭⣭⣤⣤⣤⣤⣾⣗⣿⣿⠀⠠⠬⣽⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢻⠉⢀⣶⡞⣋⡟
⠀⠀⠀⢠⣿⣿⣿⣿⡿⣿⠓⠌⠀⠀⠀⣿⠁⠀⠀⣶⠺⣿⡧⠘⣿⠁⠀⠀⠀⠘⠻⢿⣿⠀⠀⠭⣿⣿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣞⣉⣉⠙⢿⡄
⠀⠀⠀⣾⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣤⣿⣟⡀⣿⠀⠀⠀⠀⠀⠀⢸⡏⡤⠀⠛⠉⠵⠛⡉⡍⢠⢠⢰⢰⡄⡽⡎⢷⠀⠀⠸⣍⣤⣈⠻⡏
⠀⠀⢰⣿⣿⣿⣟⠦⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡷⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⣿⣤⣤⣄⣤⣦⣷⣿⣼⣾⣾⣿⣷⡇⣿⢸⣄⣀⠀⠀⣈⣥⠞⠁
⠀⠀⣼⣿⣿⡿⠮⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣿⠯⢠⡿⠀⠀⠀⠀⠀⠀⠀⢻⣜⡻⠿⠿⠿⠿⣿⣿⣿⣿⠿⠿⢿⣣⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⡿⠭⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⡟⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡿⠷⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣛⡂⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣇⡀⠀⠀⢸⡇⣠⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠿⣿⣧⣿⣮⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⣿⣿⡏⠁⠀⠀⠀⢠⡿⢿⣿⣿⡟⠀⠀⠀⠀⢻⣯⣭⣿⣋⣛⡒⣶⣦⡄⠀⠀⢀⣿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⣿⠃⠈⢿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠉⢸⡇⠀⠀⢸⡿⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡀⠀⠀⠀⠀⠀⠲
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠀⠀⠀⠀⡌⠀⡇
⠘⡏⠋⠉⠉⠉⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⡱⠤⠶⠶⠃⢀⠁
⠀⠹⡄⠀⠀⠀⠀⠀⠒⠊⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠅⠀⠀⠀⠀⣎⠀⠈⠀⠓
⠀⠀⠹⣄⠀⠀⠲⡄⠀⠰⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⡀⠀⠀⠹⡀⠛⠂⠀⠀⠐⢤⡀
⠀⠀⠀⠈⠳⣄⡀⠈⣑⠤⠀⠀⢀⣀⣀⣈⠉⠒⠢⠤⠤⣄⣀⣀⣀⣠⠤⠤⠔⠒⠒⠉⠁⠀⢁⡥⠀⠀⠀⠈⠐⠢⠄⠀⠀⠀⠉⠲⠤⡴⠒
⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠚⠋⠉⠀⠀⢈⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠡⠄⡄⠈⢁⣲⠢⠭⠤⠤⠤⠚⠳⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢤⠁⢠⣤⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣘⡇⢀⠤⢖⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⢀⣠⡴⠓⡖⠒⠒⠦⠬⠤⠤⠤⠖⠒⠋⢡⠃⡎⠀⠀⢱⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠐⡅⠀⠳⢄⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠱⠀⠠⠈⢇⠑⠀⢄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠼⠞⠁⠈⠙⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣳⠃
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿