Woods Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⠀⠀⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣴⣿⡆⠀⠀⠀⠈⠳⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣼⢿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣴⣿⣔⣿⣷⠞⢀⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⠟⣿⡿⠆⠠⠉⡀⡅⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⢋⠀⠀⠁⢦⣴⠀⠀⠀⠀⠁⢈⣀⣼⣷⠀⣠⣄⡀⠀⢤⡀
⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠱⢦⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠊⡅⠺⠗⢘⣤⢘⣪⢻⣿⣿⣲⣼⣿⣿⣿⣤⣿⣦⣄⡀⢀
⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣾⣇⣿⡿⣿⠆⠀⠀⠀⠀⠀⠀⢀⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶
⠀⠀⢘⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⢋⢀⠘⠋⠛⠨⠒⠀⠀⠀⠀⢰⣿⣺⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⢬⣿⣿⡆⠀⠀⠀⢸⡄⠀⠀⢀⣀⠄⢀⢸⣧⢂⠾⣦⡆⢰⣾⢷⣤⡄⣨⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷
⠀⠘⢺⣿⣿⣅⡀⠀⠤⣿⡯⢤⣴⡥⢋⣔⣿⣿⣿⡄⢀⠍⠀⠀⣤⣗⡽⣃⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠹⢿⣿
⣀⣩⣿⣿⣿⣋⡄⢀⣲⣾⣷⣿⣋⠠⣿⣿⣿⣿⣿⣷⠁⠀⢀⡘⠽⠳⣋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠋⣴⣮⣿⣦⣿⣿
⣈⣉⣉⣿⣯⣥⣶⣖⠾⣿⣿⡿⠏⣐⣿⣿⣵⣼⣿⣟⣃⡀⢸⣇⢀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡽⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣾⢿⣿
⣬⣽⣿⣿⣿⣿⣥⣬⢻⣿⣿⣿⣟⢬⣿⣿⣿⢞⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡽⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠿⠿⠛
⠹⡿⡿⣿⡿⡿⠿⠻⣿⣿⣿⡛⠛⠛⢿⣿⣿⣿⡋⠙⣿⣿⣿⣿⣏⠻⠿⠿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠟⠉⠀⠛⠋
⠀⠀⠁⠂⠐⠀⠀⠀⠀⣾⣿⠀⠀⠀⠜⣿⣿⠏⠀⠀⠀⠉⢹⣿⠛⠀⠀⠀⠹⢿⡇⠀⠀⠈⠋⠛⠻⠿⠿⠟⠛⠋⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠘⠋
⣶⣶⡶⢶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢾⣙⣧⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣹⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⣏⣟⡙⠛⠂
⠀⠀⠀⠀⠀⠈⣉⣶⣿⣿⣧⠘⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣜⠿⠋⠉⡉⣶⣤
⠀⠀⠀⠀⣰⣿⡿⣿⠇⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠐⠛⠛⣁⣀⠀⠓⣤⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⣡⣾⢛⢃⣯⣬⣟⠛⠛⠁⡄⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣿⣷⣤⣤⣬⡿⢿⠒⢻⣿⣭⣼⡄⠀⢸⡄
⠀⠀⠀⠀⣄⣸⣿⣿⢾⣿⣿⣯⣤⣄⣰⣷⡀⣨⣷⡀⠈⢻⡧⠀⠀⠀⣦⠀⠀⠀⠀⢰⣧⡀⢠⣙⣿⣿⣿⠁⣘⡷⠶⠐⠯⠉⠻⣿⣿⣏⣼⡇⠀⠀⣀
⠄⠀⠀⣼⣿⡿⢿⣿⣧⣿⣿⣿⠹⢿⣿⣿⣿⣯⣿⣀⣤⢽⣿⠁⠀⠐⣿⠂⠀⠀⠐⢿⣿⣿⡈⣻⣿⣿⣿⡁⠀⠀⠀⠀⢀⡀⠹⣿⣿⣿⣿⣇⡀⠀⣼
⣄⣤⠲⢸⣟⠂⠉⠉⠀⣿⠛⠛⠛⢲⣾⣿⣿⣿⣿⡿⢗⣾⣿⡏⠙⢺⣿⠦⣀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣯⣭⡥⠆⢐⠀⢻⣿⣿⣿⣿⣿⣿⣷⠀⢐⣾
⣷⣿⣿⣾⣷⣔⠂⠀⠀⣿⢀⣶⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⣶⣆⣾⣐⣾⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿
⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⡄⣠⣼⣤⣰⣆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣼⣿⣿⡿⠿⠋⠍⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⠶⠾⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠄
⣿⢟⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠂
⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⢻⣿⡯
⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢄⣙⣿⣿⣿⡇⣾⣀⣴⣶⠀⣤⣤⠀⣠⡄
⣷⣬⠡⠄⠀⢀⣴⣥⣀⠀⠘⠀⠆⠀⠀⠈⠵⢿⣴⡀⣀⣈⣝⣿⣿⣿⣿⣿⣿⣗⣘⣿⣿⣿⣿⠿⠂
⣿⣻⡂⠀⢄⣈⣿⠛⣏⣬⣰⣮⣶⣶⣦⡄⠠⢦⣿⡟⢻⠿⣿⣿⣿⣽⣿⣷⠿⣿⣿⣿⣿⣿⣿⣂⠀⠀⠀⠀⠀⡀⠄⡀⡄
⣿⣿⡿⠀⢦⡹⣿⢿⡟⠙⣻⣿⣿⣿⢧⣠⢻⣿⣽⣿⣟⣻⣶⣾⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⠟⢀⣀⠀⠰⡇⣽⠖⠷⡿⠇
⡟⠛⢻⠗⠘⣦⢇⡀⢿⣿⡏⣯⡉⣿⣋⡛⢻⣷⣄⣿⣟⣿⢷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠁⠈⣻⣿⡿⠃⠈⠁⢁⠀⠁⠀⢀⢀
⢀⠀⣄⡷⢶⡿⣮⣈⣆⣹⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣸⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣤⣄⠀⠀⠀⠀⣤⣤⣠
⠋⠻⣧⣁⣪⣟⣛⡻⠷⠇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⠃⢛⡗⠸⠟⠙⠻
⠿⠟⠟⠉⠉⠉⠛⠋⠀⠀⠢⠀⠀⠓⠀⠈⠋⠢⠈⣛⣩⡼⠤⠉⠉⠿⠋⢠⠆⢀⡤⠀⠀⠀⠀⠽⠀⣸⠥⠿⠏⠉⠉
⣤⣤⣆⣠⣀⢀⠀⢀⣴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣠⠀⠀⠀⣀⠀⠀⣀⠀⣀⠀⠀⠀⢀⣠⣦⣤⣴
⣿⣿⣷⣾⣿⣡⣷⣭⣿⣭⣷⣴⣶⣾⣣⣤⠄⢀⣠⣤⣤⣦⣤⣤⣤⢀⡤⣄⢀⠀⠀⢩⣀⣺⣦⣾⢿⣽⣯⢷⣵⣺⣰⣿⣲⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⡖⣻⡭⣽⣿⣿⣿⣦⣿⣿⣿⣿⣯⣷⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠔⡀⠀⠀⡠⠀⠉⠁⣀⢀⡀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈⠀⠀⠀⠄⠀⠀⠤⠀⠈⠘⠉⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠝⠂⠀⣁⠢
⠀⠀⠀⠀⠀⠀⢊⡁⠀⢀⡄⠊⣀⠔⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣁⠆⠀⠀⢀⣠⠈⠀⠀⠀⠁
⠀⠀⠀⡠⣀⡀⠧⢈⡡⠙⢤⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠀⠀⠁⣤⣒⠋⠉⠀⢈⡠⠀⠀⣈
⠀⠀⠰⠀⠞⠡⠞⠂⠱⡤⠆⢡⠀⠈⠀⠀⠀⠀⠀⠀⠂⡠⠄⠀⠀⠀⠀⠀⠐⠀⠀⡀⡀⠀⠐⠛⠉⠀⢈⣥⠶⠚⠁⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡯⢃⡤⣀⠔⠊⠁⠀⠀⠀⣠⠴⠦⠀⠁⠁⢀⣀⠔⣞⡭⠦⡉⠂⠀⠀⡢⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉⠁⠴⠉⠊⠀⢠⡀⠠⠂⠀⣈⡤⠀⢀⣀⠴⠚⡽⠂⠉⠄⠀⠀⠀⠀⠀⠈⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢭⢻⣰⠀⠐⢶⢤⠀⠀⠄⠒⠉⠀⣠⠴⢊⣡⡴⠾⠋⠁⡀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⠲⠶⠾⠇⢠⣄⠀⡷⠀⠀⠀⠀⣨⣔⠾⠋⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣚⣉⡡⣶⡀⢸⣏⠈⠉⠛⠗⢀⣠⡶⣻⠟⠉⠀⠠⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢠⠏⠘⠁⠬⠀⠚⣗⣀⣀⣼⣭⢔⣚⠁⠱⢊⡁⠀⠀⠀⠀⡘⠉⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⠀⠞⢋⡀⢈⡫⠀⢸⣈⣯⣗⠉⢰⡜⠖⢿⠀⠖⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⢀⠚⢰⣧⣈⠳⡀⢻⡿⠉⡡⠠⢸⠍⡆⠀⠁⠰⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠑⠲⠾⠭⠤⠖⠛⠁⠀⣳⢄⣆⠚⣥⣰⠟⡾⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣈⡥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠮⠄⠱⠾⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠖⠋⠁⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣪⣐⡀⣠⠴⠚⠁⠀⠀⣀⠴⠂⠀⠊⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⠀⠀⠀⣠⠴⠋⢲⢄⣏⠁⠘⠉⠀⢀⣀⡴⣶⠯⠞⠉⠀⠐⠀⠀⠀⢀⠠⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⠈⠘⢂⠜⢉⡱⠍⣦⣀⢀⣀⡭⣦⡉⢤⠀⠀⠀⡀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠐⠲⡆⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡃⣡⢾⢄⡷⣿⢿⡿⠛⠀⠀⢻⡄⢀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠄⣻⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣤⣹⣖⣯⡴⠂⡈⠁⠀⠀⠀⢀⣇
⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣠⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠠⠞⣿⣿⠟⢃⣿⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠇
⠀⠀⠀⠈⢰⣶⣿⡿⢧⠴⣃⠠⣿⣿⣇⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠈⠀⠙⠂⠉⠻⠿⢿⠀⡦⠀⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠃⣀⠀⠀⢴⠀⠀⠀⡀⠀⠙⠈⠛⠀⠀⢠⣿⣿⣦⣤⣠⣠⣀⣼
⠀⠀⠀⠀⠀⠐⣿⡄⠀⣿⡇⠀⠀⣷⠀⠀⠀⠀⠀⡂⠸⢟⠿⠹⠋⠿⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⣼⣿⣿⢀⣿⣟⡀⢸⣿⡀⣇⠀⠀⣀⠃⠀⢀⠀⠀⡀⠀⠀⠁⠙⠙⠟⠿⡷⣄
⠀⠀⠀⠀⢨⣿⣿⣷⣾⣿⣿⣥⣽⣿⣣⣿⡀⢠⣿⣸⣧⠀⠀⢰⣿⡀⣧⠀⠀⠀⠀⢰⡛⢫⡛⣿⣦⣄
⠀⠀⠀⢐⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢠⣾⣿⣟⣿⣵⠀⠀⠀⣼⡇⣾⡇⠈⠉⢻⡧⢄⡀
⠀⠀⢀⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠻⣛⣿⡏⣉⣲⡟⢰⣻⣿⣿⣿⡧⠀⣀⣿⣷⣿⣿⠀⠀⠀⠀⠀⠉⠂
⠀⠀⢈⡻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠄⠀⠀⠀⣝⣿⣿⣿⠏⢠⡀⢫⢿⣿⣿⣿⡤⣼⣿⣿⣿⣿⣖⡀
⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⡃⣠⣀⣰⣴⠾⣿⠿⠿⠋⠀⠀⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢀
⠀⠀⠮⣿⣿⣿⣿⣿⣿⣛⣉⣁⣀⣀⣠⣤⣤⣤⡴⠃⠀⠀⢀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠤⣿⡿⠿⣿⣿⣿⣿⠉⢻⣿⠈⣿⡇⠀⢸⣥⡅⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣆⣸⣿⣤⣽⣯⣤⣿⣇⣤⣴⡾⣧⣤⣤⣤⣥⣤⡜⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⠟⠁⣾⣿⠿⡿⠧⠿⠾⠓⠖⠛⠙⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣨⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣿⠀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⡞⢿⣿⡏⠂⠈⠹⣿⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⢸⣿⣷⣜⣿⣧⡀⠀⣀⠈⠀⠈⠀⠀⠀⣀⡤⠀⠀⠀⠀⢀⣠⠦⢀⡀
⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⣴⠀⣠⣀⣴⣿⣿⠇⣠⣿⣿⣿⣿⣻⣿⣷⡀⠀⠀⠀⠀⢀⠀⢘⣿⡟⠀⠠⠂⠘⣯⠟⠀⠠
⠀⠀⠀⢀⣶⣿⡇⠈⢡⣴⣿⡿⠀⠀⠙⢻⣿⣿⡄⢿⣿⡷⠘⠛⠹⠿⠿⣿⣷⣶⡀⠁⢀⡀⢩⡿⠁⣄⡴⠀⠘⠛⠀⣰⠀⠁
⠀⠀⠀⠘⠻⠟⠓⢰⢾⡏⡟⡷⢘⣀⢰⠈⣻⠹⠗⣮⠁⢠⣇⠀⠀⠀⠀⠉⠙⠻⣿⢶⣤⡍⡘⢰⡎⢸⣧⠀⣾⡄⣦⢼⣧⣀
⠀⠀⠀⠀⣴⣴⣧⣿⣿⣿⣷⣷⣿⣿⣾⣶⣿⣷⣿⣿⣇⢸⣿⢀⣆⠀⠀⡀⣀⣀⣤⣶⣽⣧⣦⣾⣧⣼⣿⣶⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣍⣛⠛⠛⠻⢿⣿⣿⣟⣹⡿⣎⣿⣿⡟⣿⣿⣾⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠃⠀⠀⠀⡀⣿⣿⣿⠋⣥⢣⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⢻⣿⣿⡿⡿⣿⣿⣿⣿⣿⠿⠟⠛⠛⢉⠉⠁⠀⠀⠀⡄⠀⣿⣄⣸⣴⣿⣽⣁⠙⠳⢋⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠈⠻⡏⣯⢿⠆⣻⣿⣿⣿⣋⠀⣴⡄⠀⠀⣾⡀⠀⠀⣠⣤⣿⣾⣿⣿⣿⣿⣷⣿⣷⣦⣆⣈⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠱⢟⡈⢰⣬⣿⣿⣿⣅⡄⣻⢗⡀⢰⣿⣷⡂⠀⠈⠉⠉⠙⠙⠛⠛⠛⠛⠻⠿⠿⠟⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢶⣮⣿⣿⣿⣽⣊⣿⡟⣁⡠⣿⣯⡖⡀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣿⣾⡥⠂⠀⡀⣺⣿⡾⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡓⣀⣙⣿⣿⣿⣏⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡅⣬⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠉⠉⠁

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

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⢶⣶⡶⣶⡶⡶⢶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⣿⣿⣷⣦⣾⣤⣦⣴⣶⣾⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠭⠈⠉⠿⠯⠿⠏⠀⠀⠹⠩⠾⣿⣿⣽⣽⣽⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⣉⠉⠁⠁⠀⠀⠀⠀⠀⠀⡀⠀⠠⠘⡉⠉⠛⠺⢶⠲⡀⠀⠈⣻⣻⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡿⠿⠭⠍⠁⠀⠈⠁⠀⠀⠴⠦⠤⠤⠶⠶⢶⣶⣿⡾⠷⠗⠓⠒⢈⠀⠐⠿⠟⠫⠍⡈⠌⠑⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⠿⢦⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠌⠐⡂⠀⠀⣡⡀⠀⠀⣚⣨⣛⣿⣿⣿
⢸⣿⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠬⠥⠤⠴⠶⠤⠅⠀⠀⠈⠀⣴⣂⣀⡇⠈⠋⣉⣯⣿⣿
⢸⣿⣯⣭⡍⠉⣉⡉⠁⠀⠀⠀⠠⠤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣤⠄⠀⠀⠀⢙⠛⠉⠙⠛⠉⢛⡿⣿⣿⣿
⢸⣿⣿⣿⣶⠾⠻⢿⢓⣚⣒⣂⣩⡩⠅⣔⡀⣀⡄⡠⣤⣾⣧⢛⣟⣵⣛⣽⣧⢐⡞⢬⣵⢞⣲⣷⣶⣶⣿⣿⣿⣿⣿
⠈⠋⠙⠛⠛⠿⣿⣿⣭⣿⣿⣿⣧⣿⣿⣿⣟⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⢸⣦⣀⣦⣼⣦⣽⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡛⣿⣾⠻⣿⢻⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢧⡿⢫⡖⣿⠿⢻⣻⡟⣿⣿⣿⣿⣿
⢸⣿⠙⣿⠛⡟⢏⢤⡻⠸⢛⠑⡞⡋⣯⡔⡈⢣⣿⣷⡿⣫⣝⣻⣿⡜⠧⠈⣭⡜⠇⣸⣆⠛⣰⢏⠹⡇⣩⣾⠻⣟⣿
⢸⡟⡇⢡⡟⠀⣻⠎⠁⢤⠛⠇⢱⡇⢟⢸⣻⢨⠑⠿⠇⣏⠣⣙⡯⣿⣦⢰⡜⢣⡀⠟⡁⠾⣫⢶⡇⣰⠵⡚⣰⠼⢨
⢸⣇⡇⢸⠀⡀⢠⡖⡀⠰⡇⣰⠘⠃⡌⣄⢁⡸⡽⣹⡄⡖⢾⣻⣿⣿⡿⢰⡿⢸⠃⣾⣿⢰⣏⡾⠀⡀⣴⡆⢠⣾⢃
⢸⡿⡿⢿⣥⣿⣀⣱⡗⢀⢁⣻⠆⠀⣇⣏⡀⣸⣇⡿⣀⣻⣞⡉⠿⠟⡥⠬⠥⡼⠀⣻⣷⣼⠟⠁⣼⣧⡿⢧⣼⠿⠞
⠈⠃⠀⠀⠀⠉⠉⠉⠋⠙⠛⠻⠖⠒⠞⣛⡉⠍⠂⠄⠀⠐⠚⠉⢀⡠⠝⠛⠋⠒⠠⠢⠟⠛⠒⠛⠋⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⢠⡄⠐
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠐⠂⠀⠀⠀⢀⣈⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠐⠂
⠀⠚⠂⠀⠀⠀⠠⠒⠀⠀⡀⠀⠚⠀⠀⠀⠀⢀⡀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⢀⠀⠀⠉
⠀⠁⡤⠀⠀⠉⠀⢀⡀⠈⠁⠀⠀⠀⣊⡵⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⢀⣀⠀⠠⠤⠀⠀⣀⠀⠀⢒⣶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⣠⡀⠀⠀⠐⠒⠀⠀⠀⠀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠂⢀⡂⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⡀⠐⢄⠀⠀⠀⠀⠀⠄⠤⠉⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⠉⠹⠀⠂⠀⠀⠀⣄⠀⠀⠠⠄⢆⡀⣈⠤⢨⠈⢉⡀⠀⠀⠀⣀⠒⠒⠀⠛⠃⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢤⡤⠀⠀⢀⠁⠂⠐⠒⠶⣤⡤⣟⠅⠐⠢⣞⣭⠁⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢢⢜⡈⡁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⢿⣾⠍⠀⠀⠘⠁⠰⠃⠉⠉⠩⣷⡿⠒⠂⠀⠥⡴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣉⣸⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠽⠚⠤⠅⠀⠀⠀⠀⠘⠀⠈⠀⠨⠴⠛⠯⠁⠈⠛⠋⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠈⠟⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⢶⡇⠀⠀⠀⠀⠠⣿⡄⠀⠀⢰⣿⣦⠀⢀⡆⠀⠀⠀⠀⠀⢰⡄⠀⠨⣿⠇
⠀⠀⠀⠀⠀⠒⠂⠘⠀⠚⠒⠒⠀⠈⠃⠐⠒⠂⠃⠘⠂⠐⠛⠀⠐⠀⠐⠒⠒⠋⠀⠒⠀⠀⠛⠛⠀⠈⠁⠀⠚⠀⠀⠀⠈⠓⠓⠐⠂⠀⠀⠓

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢸⣿⣿⣿⣿⣹⣿⢽⣿⠉⢹⣿⢹⣿⣿⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣎⣻⣯⡍⠭⣽⣿⡏⣿⣿⡿⣠⠄⣿⣿⣿⡇⣿⡿⠁⢼⣿⠇
⠀⢸⣿⣿⡿⣿⠃⣿⠌⣿⠀⠘⣿⠠⣿⡏⠀⢤⢸⣿⡗⣼⣿⣿⣿⡇⠘⣿⣿⡀⠀⠘⣿⡇⣿⣿⣷⠃⠀⣿⣿⣿⠇⣿⡇⠀⠀⢿
⠀⢨⣿⣿⡗⣿⠀⣿⡇⣿⠀⠀⣿⠃⣿⣧⠀⠈⣼⣿⣇⣸⣿⣿⡟⡇⠀⢹⣿⠁⠀⠀⣿⠀⣿⣿⡏⡀⠐⣿⣿⣿⠀⣿⠇⠀⠳⣼
⠀⠘⡏⣿⣿⢿⡆⢻⡇⣤⠀⠀⣿⠀⣿⣿⠀⠀⢽⣿⡇⠙⣿⣿⡇⡏⠀⢸⣿⠀⠀⠀⣿⠀⣿⣿⡇⠻⣼⣿⣿⣿⢠⣿⠀⠀⢰⡇
⠀⠀⠃⣿⣿⢸⡇⢸⠆⢄⡄⠀⣿⠀⢻⣿⣤⡀⣼⣿⡇⠀⣿⣿⡇⣧⠆⣸⣿⠀⠀⠀⣿⠀⢻⣿⡇⠀⠈⣿⣿⡏⢸⣿⠀⠀⢸⡇
⠀⠀⠀⣿⣿⢸⡇⢸⡇⢸⡇⢸⣿⠀⢸⣿⠀⠙⣿⣿⡇⠀⣿⣿⡟⠁⠀⠙⣿⠀⠀⠀⣯⠀⢹⣿⡟⠀⠀⣿⣿⡇⢸⡏⠀⠀⢸⡇
⠀⠀⠀⢻⣿⢸⣧⢸⡇⢸⡇⠀⣿⠀⢸⣿⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠀⣿⠀⠀⠀⠧⠀⢸⣿⣧⠀⢀⣿⣿⡇⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⢸⣿⢸⡇⠘⠇⠀⣿⠀⠘⣿⡆⠀⣿⣿⡆⠀⣿⣿⡇⠀⠀⠀⣿⠃⠀⠀⣄⡀⢸⣿⣿⠀⢸⣿⣿⡇⣾⡇⠀⠀⢹⡇
⠀⠀⠀⢸⣿⢹⡇⢸⡇⠀⠀⠀⣿⡆⠀⣿⣇⠀⣿⣿⠀⠀⣿⣿⡇⢠⠀⠀⣿⠀⠀⠀⣍⠙⢻⣿⡿⠀⢸⣿⣿⠂⣿⡇⠀⠀⢾⡇
⠀⠀⠀⢸⣿⠀⡇⢸⡇⠀⠀⠀⣿⡆⠀⣿⡏⠀⣿⣿⠀⠀⣿⣿⡇⢸⠀⠸⣿⡆⠀⠀⠀⠀⢸⣿⣷⠀⢸⣿⣿⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠄⢺⣿⠄⣧⢸⡇⠀⠀⠀⣿⠃⠀⣿⡇⠀⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⣿⡇⠀⠀⢰⠀⢸⣿⡿⠀⢸⣿⣿⠀⣿⡇⠀⠀⢸⡇
⠀⠀⡄⢸⣿⡆⣿⢸⣿⠀⡄⠀⣿⠂⠀⣿⡇⢠⣿⣯⠀⠀⣿⣿⣇⢸⠀⠀⣿⡇⠀⠀⢸⠀⢸⣿⣿⠀⢸⣿⣿⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠇⣸⣿⠇⣿⠸⣿⠀⡧⠀⣿⡄⠀⣿⡇⢸⣿⡏⠀⠀⣿⣿⣿⠈⠀⠀⣿⡇⠀⠀⢸⠀⢸⣿⣿⠀⣾⣿⣿⠀⣿⣿⠀⠀⢸⡇
⠀⠀⠂⣿⣿⠇⣿⠀⣿⠀⡇⠀⣿⡇⠀⣿⡇⢸⣿⡇⠀⠀⣿⣿⣿⢸⡆⠀⣿⡇⠀⠀⢸⠀⣻⣿⣿⠀⣿⣿⣿⠀⣿⣿⠀⠀⢸⡇
⠀⠀⡀⣿⣿⡇⣿⢰⣿⣀⣿⠀⣿⡇⠀⣿⣇⢸⣿⡇⠀⠀⣿⣿⣿⢸⡇⠀⢿⡇⠀⠀⢸⠀⣿⣿⣿⠀⣿⣿⣿⠀⣿⣿⠀⠀⢀⡇
⠀⢸⠀⣿⣿⡇⣿⢸⣿⢀⣏⠀⣿⡇⠀⣿⣿⢸⣿⡇⠀⠀⣿⣿⣿⣸⡇⠀⢸⣧⠀⠀⣸⠀⣿⣿⣿⠀⣿⣿⡇⠀⣿⣿⠀⠀⢰⡇
⠀⢸⠀⣿⣿⡇⣿⣼⣿⠀⣿⠀⣿⡇⠀⣿⣿⣾⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⣿⡆⣿⣿⣿⠀⣿⣿⡇⠀⣿⣿⠄⠀⣸⣿
⠀⣾⣦⣿⣿⣇⣿⣿⣿⡇⣿⢠⣿⡇⠀⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⠀⡀⣿⣿⣿⣿⣿⣀⣿⣿⣧⠀⣿⣿⣷⣿⣿⣿
⠀⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣼⣿⡇⢰⣿⣿⣿⣿⣿⡷⠔⣿⣿⣿⣿⣿⣶⣾⣿⡖⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇⠀⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣤⣼⣿⣇⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡏⣀⣤⠴⠖⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠛⠛⠓⠶⢤⣤⡤⠤⠤⠤⢤⣤⣤⣤⣤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⣿⠉⠁⠀⠠⠤⠐⠂⠈⠻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠁⣨⠶⠛⠛⠛⠛⠛⠛⠶⣌⠛⢦⣀⠠⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠤⠀⠐⠤⣤⣤⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⣴⠋⠀⢣⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⣇
⠀⠀⠀⢸⡇⢀⡼⠋⠀⠀⠀⠀⢀⠖⠋⠓⣄⠀⠀⠈⢿⡄⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⠴⠶⠶⠶⠚⠛⠛⠉⠉⢀⣽⠂
⠀⠀⠀⢸⠁⣸⠇⠀⠀⠀⠀⠀⠌⠀⠀⠀⢸⡄⠀⠀⠈⣿⠀⢿⡄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⠂⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⠤⠴⠒⠺⣥⡀
⠀⠀⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠘⢷⣤⠤⠤⠤⠴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡬⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⣰⠏⢠⡏⠀⠀⠀⠀⢀⡴⠋⠉⠙⡄⠀⢹⠀⠀⠀⠈⣧⠀⢸⡇⠀⠠⠤⠤⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⣿⠀⢸⡇⠀⠀⠀⢰⠋⠀⠀⠀⠀⢹⠀⠸⡇⠀⠀⠀⢿⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡤⢼⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⢠⡇⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⢸⡇⢸⡇⠀⠀⠘⡇⠀⣴⠀⠀⣰⠇⠀⢸⠁⠀⠀⠀⢸⡇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠶⠶⠛⠋⠉⠀⣿
⠀⠀⢸⡇⠀⡇⠀⠀⠀⡇⠀⠈⠓⠋⠁⠀⠀⡞⠀⠀⠀⠀⣿⠀⣿⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⣿⠐⢿⡄⠀⠀⢻⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢰⡏⢰⡿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠀⠀⣠⣤⣤⣤⣤⣤⡤⠤⠴⠶⠶⠖⠒⠚⠛⠛⣻⠇
⠀⠀⠀⢸⡇⠈⣷⠀⠀⠸⣆⠀⠀⢀⡠⠞⠀⠀⠀⠀⢀⡾⠁⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠤⠤⠶⠶⠒⠒⠛⠋⢉⡟
⠀⠀⠀⠀⣿⠐⠙⣧⠀⠀⠈⠓⠊⠉⠀⠀⠀⠀⠀⢀⡞⠁⣶⡟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⣀⣠⡾⠁
⠀⠀⠀⠀⢸⡆⠀⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣁⣠⣤⡤⠶⠖⠒⠋⠉⠁⠀⠀⢀⣀⣠⣤⠶⠖⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠻⣄⡀⠇⠉⠛⠶⠶⢤⣤⠴⠖⣋⣴⠟⠛⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠉⠉⢁⣀⣀⣠⣤⣤⠶⠶⠶⠚⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⣤⣤⣧⣤⣴⣶⣟⣉⣀⣀⣀⣤⣤⣤⠤⠤⠶⠶⠶⠒⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢰⡀⣤⡾⠤⣀⣦⣀⡀⣆⣀⣴⠀⠀⠀⢠⣀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⢨⠶⣇⣀⡀⠀⠀⠀⢀⠆⠀⠀⡄⢺⡄⢀
⠀⣋⡿⠐⣷⢄⣹⣿⣿⣣⠟⡿⢤⠼⣿⢟⡌⡬⣦⣶⣢⣋⠕⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⣽⡄⢐⢻⠙⣫⣍⢠⢹⣧⣴⡶⡏⣱⠺⣶⠧⢿⠆⣼⡄
⠀⣩⠷⡌⠱⡀⠽⠻⣏⠨⣆⠀⣲⠃⣿⠅⣲⠁⣻⠏⠡⠼⠶⠞⢠⣄⣠⣀⡠⣶⡀⢀⣠⣤⣤⡜⣋⡶⢓⢹⠠⠾⡁⢾⣦⢷⠀⢿⣏⠓⣀⡙⢌⢙⡖⢊⣿⠻⢷
⠀⡿⡈⢳⣦⡟⣱⠇⢹⣤⠿⡊⣹⠀⣵⡆⢨⣻⠖⠚⣶⠛⣼⣍⠘⣿⠏⢻⡇⡛⢻⣿⣏⢿⡿⢃⣯⣉⣄⢯⢀⣽⡄⠺⡿⢄⠻⠾⣿⣆⠰⠐⡄⠙⣏⢁⡗⢷⡞
⠀⠌⢀⣾⣤⠃⢲⣆⡨⡿⢠⠃⣩⣬⢄⠐⢻⡷⣊⡬⢉⣾⠁⣡⠙⠫⣭⠞⠃⠡⠌⢻⡕⣨⢏⢉⠵⡀⢽⡧⣹⡀⢸⡀⠳⡀⠳⢤⣬⡁⡸⡆⠙⠀⢣⣜⠧⠜⠳
⠀⠀⠀⠈⠃⠒⠯⢄⠑⣀⠀⠈⠉⠐⠏⠻⠋⢅⣄⢳⡟⠀⠳⣌⡟⠳⠍⠃⠐⠒⠀⠈⠀⠀⠀⠈⠀⠑⢚⠓⠉⠑⠾⢌⠑⢾⡃⣈⠃⠭⠠⠧⠀⠇⢤⡡⠆⢨⡙
⠀⠀⠀⠎⠀⠼⢀⠈⠑⠤⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡇⠘⣧⠀⠈⢟⠢⠉⣍⠛⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢲⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡆⠀⢷⣶⠂⢠⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣅⡀⠿⣿⣾⠛⢿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣿⣿⣿⣿⡿⠿⠋⠀⠈⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠿⠛⠉⠉⠀⠀⠀⡒⠀⠚⢶⣾⡻⣿⣯⣿⠁
⠀⠀⠀⠀⠺⠟⣿⣿⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠯⠀⠁⠀⢆⠀⠀⠀⡇⠀⠀⠀⣀⠟⠻⠾⣯⠻⠄
⠀⠀⠀⠀⢀⣾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⣤⠀⠃⠀⢠⣤⣶⣀⣄⠻⠂
⠀⠀⠀⠾⣛⣿⣿⣿⣏⠙⠛⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣷⣆⡀⠀⠀⠘⣦⣧⡶⣾⡿⢃⣹⣿⣷⣄
⠀⠀⠀⣰⣿⢻⣿⣿⣿⠿⣶⣆⠀⠀⠀⠀⢀⣴⣿⣿⣄⠀⠀⠀⠀⠀⣠⣼⣿⣿⣷⡦⠙⢟⣹⣿⣦⣴⣿⣿⣿⣷⡄⡉⠛⠿⢿⣿⣸⣷⡀
⠀⠀⠊⠁⢰⣿⣿⣿⣿⣷⣦⡈⠀⠀⠀⠀⣠⣴⣿⣿⣄⡀⠀⠀⠀⠀⣨⢟⠏⠉⠀⠐⠠⠛⠛⡿⣻⣿⣿⡿⠏⣄⠀⠀⠀⣠⣀⣀⣹⠏
⠀⢀⠴⢿⣿⣿⣿⣿⣿⣿⣿⡛⠆⠀⠀⣈⣩⣼⣿⣿⣿⡝⠀⠀⠀⠀⠀⠀⠀⠐⢄⠰⠠⢨⣤⣭⣭⡁⢸⡇⠀⠀⠡⣠⣾⣿⣿⣍⠿⣶⣤⣄
⠀⢀⣴⣾⣿⣿⣿⣿⣧⠛⠿⣿⡆⠀⠀⣩⣵⣿⣿⣿⣮⠉⠂⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣝⡋⠾⠿⢿⣾⡇⠀⠀⠀⣿⠟⢉⠁⠀⢘⣿⣿⢿⡃
⢠⡿⣿⣿⢿⣿⣿⣿⣿⣷⣄⠈⠻⠆⢈⣿⣿⢿⣿⣿⣿⣏⡑⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠩⠀⠀⡄⠀⢹⡇⠀⠀⠀⣿⠀⠀⢣⣰⣿⣿⣿⣛⠁
⠀⠀⠛⠁⣸⡿⢿⢻⣿⡿⣿⣷⡀⠠⠟⢉⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢲⠂⠀⢸⡇⠀⠀⠀⣿⠀⠀⠘⣏⣱⣿⣿⣮⡅
⠀⠀⠀⠀⡟⠀⣾⠀⠘⡇⠉⠈⠁⠀⠴⠿⢿⡿⢿⡟⢿⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⠀⢸⣇⠀⠀⠀⡟⠀⠀⢈⣷⣿⣿⣿⡻⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⢸⡟⣹⢿⢿⠿⣆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⢸⣿⠀⠀⠀⡇⠀⠀⢸⡇⠁⣸
⠀⠀⠀⠀⠀⠀⣿⡄⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⢸⣧⠀⠀⣀⣠⣴⣶⣶⣤⣤⣄⣀⣸⡇⢀⣀⣸⣇⣀⣸⣿⡀⠀⢸⡇⠀⠀⢸⡇⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⣿⠿⡫⠟⠉⠀⣈⡅⣩⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠶⠋⠩⠚⡫⠥⠒⠉⢀⣠⣴⣷⣶⣿⠿⠛⣉⣽⡿⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠋⠝⠋⣁⠀⠀⠀⠀⢀⣠⡤⣶⣾⣿⣿⡿⠟⠋⠀⠀⠠⠚⢁⣠⠖⠋⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣋⠭⠀⠀⠄⠒⣊⣉⡤⠔⠊⠁⣀⣬⣶⣿⣿⡿⠛⠁⣠⣊⠤⠂⠀⠀⠐⠉⢵⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⡋⠁⠀⠀⠀⠐⢉⠁⠀⠀⠀⢀⣤⣾⣿⠿⢛⡅⠀⠴⠞⢉⡠⠔⠀⣀⢀⣠⣾⣿⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠓⢬⡲⢝⢷⣤⣤⠶⠊⠁⢀⣤⣶⣾⣿⠿⠋⢡⠖⠉⠀⠀⠠⠚⠁⣀⣴⣿⣵⡿⠟⠉⠖⠉⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠉⠈⠳⣽⣧⡀⠀⢀⣵⡿⢟⡫⠄⣀⣤⠴⠒⠁⢀⣤⣦⣴⣾⣿⠿⣛⣽⣯⡤⢚⣡⣴⠿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠉⠀⡠⠙⠄⠀⡸⣿⣧⣶⣯⠵⠊⠡⠖⠋⢉⡠⢀⣤⣴⣿⣿⠿⣛⣩⡴⠞⠿⠋⣡⠾⣿⠟⠁⣪⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣻⣧⠶⢶⠖⠒⣻⣿⣏⣉⠉⠻⣿⣦⡀⣀⡤⠖⣋⣠⣾⠿⣋⡭⣶⠟⠋⠁⡀⠀⠀⢚⡅⢀⣡⣴⡿⢟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣇⠀⡔⡠⢋⣼⣿⠁⠈⠉⠂⠀⣈⢿⣿⢋⣴⠞⠋⠁⠀⠊⠁⠜⠁⣠⠖⡉⠀⢴⣾⣯⣴⠿⡛⠁⠚⢩⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣰⣿⢋⡜⢿⣟⣆⠘⣷⣿⣏⣤⢤⣤⣿⣭⣤⣶⣶⣟⡋⣠⣶⠵⠖⠋⠀⣀⣤⠾⢛⡿⣀⣶⣿⠟⣋⡵⠊⣠⠖⠙⣩⣴⠿
⠀⠀⠀⠀⠀⠀⣼⣱⠃⠈⠉⠞⠻⣿⣾⡿⢿⡫⠁⣼⣿⣿⣿⠿⢇⡈⢯⣻⡏⠀⠀⠠⠒⠋⠁⣴⣴⣾⠶⣟⠉⠀⢜⣩⣴⢟⣡⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⣿⠃⢀⡤⣠⣴⣿⠿⡋⠄⠈⠁⣰⣿⣿⠟⠁⠔⠉⠁⢨⣿⣷⣠⡤⠖⣚⣥⡾⠟⣋⣵⠾⢃⢀⣠⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⣿⣀⣴⣿⣿⣿⣿⡇⠣⠀⣰⣡⣿⠟⠉⠃⠀⣀⣰⣀⡀⢸⣿⣁⣴⠾⢛⠥⠖⠋⢡⠴⣞⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⣧⠀⠀⠀⣿⣷⣶⣶⣶⣶⣿⣾⣶⣶⣾⣿⠋⢀⣠⡤⠖⣁⣤⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣼⣿⠿⢿⣭⣉⠉⢁⠐⠳⠲⣸⣻⡟⠉⠀⠀⢈⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣭⡓⠄⠀⣿⣧⣴⣦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⣦⢀⣷⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶⣄⡀
⠀⠀⢀⣴⣦⣾⣿⣾⣿⡏⣾⣫⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣾⣿⡿⠟⢸⣷⣄⡀
⠀⠀⠁⠓⡾⣛⣿⢻⣿⣻⡿⣟⡿⣿⠽⠇⠀⠀⠀⠀⢀⣰⣦⣾⣿⣿⣿⣿⣋⣴⣾⣿⣹⣿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠁⠈⠉⠀⣈⡉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣠⠢⣄⣾⠗⡙⢆⣠⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢠⣾⠏⣰⣿⣏⡼⠁⠀⠛⢁⠈⠢⠀⠀⠀⠀⣠⣴⠆⣤⣀⣀⢀⣠⡴⣾⣁⠴⠃⠶⢤⡀
⠀⠀⠾⠛⡫⢺⠿⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠛⠁⠀⠑⠍⠛⠀⠈⠀⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀
⠀⠀⠀⢀⣠⣔⣹⣿⣮⣴⣿⡿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠟⠓⡢⣒⠆⢦⡀
⠀⠐⠚⠻⠿⢟⣿⠿⢛⡛⠉⠉⠭⠝⣛⡻⠷⠤⠄⠀⠀⡤⡖⠚⢦⡃⠁⠺⠥⠟⠠⠚⠿⡩⠒⠄
⠀⠀⠀⠀⠉⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣽⠏⣿⣿⣦⡀
⠀⠀⠀⠀⡠⠁⣠⣦⣞⣴⣤⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⣷⡯⣿⣏⣸⣿⣿⡟⣻⣿⣦⣄⡀
⠀⢀⣴⠬⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠂⠀⢠⣶⣿⣷⣾⣿⣷⣿⣿⣿⣿⣯⣾⠿⠿⠷⠿⠿⠶⠤
⠀⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⣰⣣⣱⣄⡴⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣅⣀⣀⣴
⠀⠀⢠⣴⣷⣼⣿⡿⠿⢿⡿⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣴⡿⣿⡿⣬⣽⣿⣶⣶⣤⡄
⠀⠰⢋⣵⣦⣀⣠⣾⣷⣬⣤⣶⣬⣽⣮⡓⠀⠀⠀⢠⢾⡿⠿⣿⢛⣿⣷⡚⠁⠙⢾⣿⣇⣙⠿⢥⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢀⣴⣿⣟⠃⠀⠀⠳⡄⠀⠀⠀⠀⢀⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠈⡳⣄⠀⠀⣠⣾⣿⣿⣿⠂⡤⠀⠀⠈⢦⡀⢀⣴⣿⡿⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠁⠀⠙⣿⣷⣾⣿⣿⣿⣄⣱⠆⠀⠀⠀⠀⠈⠻⣿⣿⡻⣇⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⡌⠦⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⡥⠂⠀⠀⣠⣾⣿⡟⣿⣿⡇⠈⠁⣼⠀⠀⠀⠀⠀⠀⢀⠀⢤⣹⣦⣄⡀⠀⠀⠀⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠚⠛⠛⠛⣛⠛⠛⣤⠐⠛⠛⠛⠐⠿⠿⠷⠾⠤⠶⠄⠀⠀⠀⠀⠀⣿⡇⠈⠙⠛⠛⠛⠃⣰⡆⠸⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⡀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠉⠙⢦⣠⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠀⠀⠀⢀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠙⢍⣉⣛⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣠⣤⣄⠀⣠⣾⣿⠛⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠑⢴⡛⠻⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣣⡄⣠⡾⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠙⢿⡟⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⣿⣿⡿⠁⠛⠁⠀⠀⠀⠀⠀⠀⣿⣷⣄⢀⡀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⣱⣿⠏⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣦⣷⣄⠀⢀⣈⠧⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃⣾⣿⠃⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣦⣙⣻⣮⣶⣄⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⡉⢀⣤⣤⣥⣾⣿⣷⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣥⡙⠿⠦


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇⠀⠈⢦⡀⠀⣀⣴⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣠⣾⣿⡿⠛⠃⠀⠀⠀⠹⣾⣿⣿⣆⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣾⣿⠙⠷⣾⣿⣿⣿⠶⡆⠀⠀⠀⠀⠀⠈⢿⣿⢿⡀⠀⠳⣤⡞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣿⣿⡿⠛⠀⠀⠙⣿⡿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠈⠻⣷⡁⠀⠀⠙⣿⡆⠈⢷⣄
⠀⠀⠀⠀⠀⣠⡴⠛⣡⣾⠟⢛⠁⠀⠀⠀⠀⠈⠻⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠈⢻⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠈⠑⢺⣿⣿⣿⣦⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠶⢦⣤⣄⣀⠖⡛⠛⠛⠛⠛⠛⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠷⠦⣶⣄⣀⠉⠢⣄⠀⠀⠀⠀⠀⠈⠙⠻⢷⣷⣤⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⣀⠀⠀⠀⠉⠙⠿⣦⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠠⠄⡀⠀⠀⠀⠠⠽⣟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⡠⠔⠒⠂⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡾⠿⠛⠛⠛⠛⠛⠶⢦⣤⣳⡀⠀⠀⠀⠐⠊⠉⠀⠤⠄⠀⠞⠐⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢥⣤⣀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣷⠀⠀⣠⡶⠿⠶⠾⠶⣶⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠄⣙⣧⠀⠀⠀⠀⣼⢫⣀⣨⣥⣤⣤⣭⣗⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣀⠀⠀⢈⡃⠀⠀⢸⣷⠟⠋⠉⠀⠀⠀⠀⣉⠽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠉⠻⢿⣦⣀⣿⡄⠀⠘⢿⠿⢶⡄⠀⠀⠀⡠⠤⠒⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⣽⠿⢟⣛⣉⣉⣩⣭⣙⣶⣄⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⣀⣉⣉⣉⠲⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣛⠭⠖⠊⠉⠁⠀⠀⠀⠀⡬⣻⣿⣿⠷⠾⠟⠻⠶⢷⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡿⠗⠢⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠺⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠀⠀⢸⡆⠀⠀⠀⠀⣰⣿⣄⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢠⠏⠀⠈⣙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⢹⡄⠀⠀⠀⠀⠈⢀⡠⠘⠯⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠠⠖⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠰⣄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠸⠃⠠⡀⠀⠀⠀⢀⣴⠟⠀⠀⣀⠀⠀⠀⠀⠹⠻⡄⠀⠀⣰⡧⠀⠀⢋⡄⠀⠀⠀⠀⣠⠀⢼⠁
⠀⠀⠀⠀⣀⡄⠀⠀⠰⣿⠀⠀⠀⠀⠈⠀⠀⠠⣿⢧⡄⠀⠀⠙⠲⢀⣄⠀⠀⠀⠀⠀⠈⠙⠋⠄⠀⠈⢉⡀⠀⠀⠈⠛⠀⠀⠉
⠀⠀⠀⠛⠿⠄⠀⠀⠀⠀⠀⠀⠰⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠀⠀⠑⠀⠦


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣤⡀⠀⠀⠂⠀⠀⣰⣆⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣥⠙⠷⡄⠀⠀⠀⢼⡿⠃⠀⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡀⠀⠀⠀⠺⢿⣧⢰⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣴⣃⡀⠀⠘⠛⢿⣷⡀⠀⠀⠀⠀⠻⡦⠈⠙⠒⢀⣤⠀⠀⠀⠀⠀⣠⡆⠀⠈⠛⠻⠧⠀⠀⠀⢀⣼⡗⠆⠀⠀⠀⠀⣠⡆
⠀⠀⢞⣿⡿⠁⠀⠀⠀⠀⠙⢷⣤⣙⠓⠆⠀⣈⠁⠀⢠⣾⣯⡀⠀⠀⢠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡉⠀⢦⠀⠀⠀⠀⠙⠷⡄
⠀⠀⠈⣿⣧⣄⠀⢠⡀⠀⠀⠀⠉⠀⠀⠀⠺⠷⠀⠀⠀⡉⠉⠈⠀⠀⠀⠀⠉⠉⣁⣴⣿⠃⠀⠀⠀⠻⠿⣷⣄⠈⠑⠄
⠀⠀⠀⠁⠙⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠘⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣼⡀⢀⣼⡿⠃⠀⣼⠃⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣿⣿⣦⣾⢿⣎⡄⠚⠁⡸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⡀⠆⠰⡋⠙⣿⣿⣷⣀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⠟⠁⠀⠀⢁⣼⣿⣿⡧⣸⣿⣿⣿⣿⣇⢸⠃⠒⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⢀⠈⠀⠀⢸⣧⠎⠀⠀⠐⠀⣿⣿⣿⣇⣿⣿⣿⡿⠛⣿⠆⠀⠀⠨⣿⣿⣟⣻⣿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⢃⣶⡟⠉⠀⠀⠀⠀⠀⠀⢺⣿⣿⡏⠀⠛⢿⣷⣾⠶⣁⠀⠀⠀⢘⣿⣿⣿⢿⣿⣿⣯⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢉⡸⠿⠃⠋⠉⠀⠀⠀⠀⠀⢰⣆⠀⢸⣿⡻⣦⣵⣤⣸⣤⡛⢿⡾⣀⢀⡀⢀⠙⢿⢹⡇⣩⠙⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠈⢰⠀⠀⡄⢀⣷⢀⣦⢸⡆⢸⣿⠀⣆⠙⠟⠀⠙⠠⠀⡈⠛⠀⠈⠁⢸⡇⠿⠷⠈⢼⠇⠿⠼⠱⠏⣰⡧⣀⣧⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⡆⣿⣦⡄⣿⣼⣿⠿⠟⠛⠋⠉⠙⠳⢿⣠⡄⠀⣄⡄⢰⣧⠸⠇⠸⢰⣸⠷⠂⠀⣤⢰⢀⡄⣠⠀⠀⠟⠐⠀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⢀⡄⣾⣧⣿⣷⠿⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠿⢿⣾⣿⣼⡄⢉⠙⢩⡀⠀⢀⡿⠚⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢾⠇⠀⠀⣸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣿⡄⠄⠀⠀⠀⠀⠀⢀⡼⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠑⢸⣿⣿⣄⢦⠀⠀⠀⡠⠋⡄⣻⣿⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⢭⣸⣿⣦⣄⣀⠀⠀⠀⠀⡴⢫⠖⡂⠸⣿⣿⣿⣿⣷⣮⠭⠀⠊⠐⣱⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠬⠷⠛⢉⣜⣧⡝⡻⢦⣀⢀⡠⠃⠃⠀⠄⠘⢿⣿⣿⣼⣿⠁⠀⠀⠀⣴⡛⠋⢛⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⠞⣉⣩⣍⢻⣦⣿⡟⠀⢠⡄⠀⠀⠀⠀⣿⣿⣿⣿⣧⡞⠋⠛⠁⠁⠘⠉⣿⣿⣿⣿⣵⣳⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠒⢲⡃⠨⢻⣷⠟⣻⠁⠀⠠⠿⣦⣀⠤⢤⣤⣿⣷⣿⣿⣿⣆⣀⠁⠐⠊⠄⡛⢿⣿⣿⣿⣿⣷⣿⣿⡄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⢐⠝⠉⣰⡏⣠⡄⢀⡀⠀⠨⢑⠤⡀⠉⠉⠙⢹⣿⣿⣷⣤⡄⠀⠀⠀⠀⠉⠒⠻⣿⣿⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠐⠁⠀⠀⠁⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⣺⣿⣿⡿⠿⣿⡆⠀⠀⠀⠀⠀⣐⣿⣷⣾⣿⣯⠅
⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⢀⣠⠄⡀⠀⠀⣀⠀⠰⠿⠛⡛⠓⠸⠯⠿⠽⠓⠒⠖⠻⡍⠛⠉⠉⠁⠐⠚⠒⠱⠶⠦⠆⠲⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⢀⡉⢀⢀⡀⠀⠁⠀⠉⠁⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠲⠂


⣿⣾⡾⠿⠾⠾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣾
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⢊⣿⣦⣀⠀⣠⣤⣴⠏⠁⠠⣟⣿⣿⠟⣦⠤⢤⣀⡠⠚⠛⢩⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢐⣩⣿⣯⣭⠿⠗⢃⣀⣠⣤⣄⠈⠻⠿⠅⠀⣠⣿⣿⣟⡢⠀⠻⡿⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠛⠋⠉⠀⠀⠐⠋⠉⢠⣾⣿⠧⠀⠀⠀⠘⠻⠿⣏⣻⡯⠀⢀⠀⠀⣾⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢶⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⠜⠂⠁⠀⠘⠿⣿⣿⣟⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣆⣀⣠⣠⣠⣀⣠⣸⠀⠀⢀⣠⣤⣄⣀⣠⣤⣀⣀⣀⣀⣀⢀⣰⣷⣀⠀⠀⡀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉⠉⠉⠹⠯⠭⠭⠽⣛⣛⣿⠿⣟⡛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠻⠯⠭⠭⢝⣓⡚⠻⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒⠒⠪⠭⠭⣛⡳⠦⢄⡀⠀⠀⢈⣉⡉⠉⢉⣍⣑⣒⡂⠤⠄⣀⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠉⠹⠗⠶⠎⠉⠒⠀⠀⠀⠈⠉⠉⠉⠛⠛⠭⠍⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⡴⡶⡛⣻⠉⠉⣿⢍⢟⢻⢶⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢿⡌⡜⡰⢸⡄⠀⣿⡌⣆⣣⠷⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡀⠀⠀⠀⠙⠲⠭⣝⣛⣻⠁⠀⣸⠭⠽⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡟⠀⠀⠀⠀⣀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢠⡇⠀⢠⠇⢀⡤⠖⠊⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⢶⣟⣹⡟⠉⣟⢒⣶⡄⢸⠄⠀⠀⠀⡞⠀⠀⡼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢭⡇⠀⣯⣭⠭⠴⠋⠀⠀⠀⡼⠁⠀⢰⠃⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⡏⠀⠀⠀⠀⠀⡝⠙⡶⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢳⠀⠀⠀⠀⠀⢀⡏⠀⠀⡾⠀⠀⠀⠀⢀⡞⠀⣸⠁⠀⣀⡠⠔⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⢧⠀⠀⠀⠀⣸⠀⠀⢠⡇⠀⠀⠀⢠⠎⠀⣰⠃⢰⢋⣁⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠦⣄⠀⠳⡀⠈⢳⡀⠀⠀⡏⠀⠀⢸⠀⠀⠀⢠⠏⠀⣰⠃⠀⠈⠮⣓⣮⠋⣱⠦⣄⠸⡄
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠙⢦⠹⡄⠀⠹⣄⢸⡇⠀⠀⢸⠀⠀⠀⡟⠀⢠⠇⠀⠀⠀⠀⡸⠃⣰⠯⠭⠭⠞⠁
⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⢀⣀⡠⠞⠀⠹⡄⠀⠘⢾⡇⠀⠀⢸⡀⠀⣸⠁⠀⢸⠀⠀⢀⡤⠊⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠉⢇⠀⠀⠀⠀⢳⡀⠀⠘⡇⠀⠀⠈⣇⠀⡟⠀⠀⣿⣠⠖⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠳⢤⣀⣀⡀⢧⠀⠀⢳⠀⠀⠀⢹⡄⡇⠀⠀⢸⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣹⠺⡆⠀⠈⢧⡀⠀⣰⠃⢧⠀⠀⢸⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⢧⣀⣀⡞⠉⠉⠁⠀⠘⠦⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣴⡄⢍⡻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢁⡼⠋⠙⠄⠻⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⣤⠞⣫⠶⠌⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣻⠏⠀⠞⠀⠀⠀⠀⠀⠘⢿⢿⣿⢿⣷⣤⣤⡾⠋⠁⠈⠀⠀⠀⠒⠾⢿⣷⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⢦⠙⢿⡙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⣠⠖⠋⢀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠂⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠂⠀⠈⠉⠓⠦⢄
⠀⠀⠀⠀⠊⠁⠀⢀⣼⣦⡾⠷⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠷⠶⣶⣤⣤⣄⣀⣰⣷⠀⣤⠀⠀⢀⣿⡀⡀⠀⠀⠀⣀⠀⣤
⠀⠀⠀⣀⡤⠖⣚⣭⠥⠶⠖⠚⢛⣛⣻⣿⣿⣟⡛⠛⠓⠲⠶⠤⣤⣀⡀⠀⠈⠉⠙⠛⠻⠿⢿⣦⣄⣾⣿⣶⣿⠀⠀⢀⣿⣠⣿⣆⣀⣠⠔⠂
⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠠⢤⣀⣀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠚⣻⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⡾⣿⠃⠀⠹⢦⡀⢀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼⡁⠀⠀⢷⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠔⢸⡇⣁⣹⡿⡏⠈⠀⠐⠀⠙⠃⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣧⠐⠀⢷⡄⠐⠺⠀⠀⠀⠀⠙⣄⠀⠀⠀⣀⡤⣴⠋⠀⠋⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢼⣿⡖⡠⠞⢣⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⠛⣶⠛⠀⠀⠀⠀⠀⠂⠤⠀⡀⠀⠀⠀⠀⠀⠑⠀⢰⡀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⢽⣿⣅⠀⠀⠀⠓⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣺⣇
⠀⠀⠀⠀⠀⠀⠩⣽⣿⣭⣄⣄⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢠⠀⠀⠀⣼⣇⠀⣘⢾⣿⡁⢀⢀
⠀⠀⠀⢠⣧⢄⣫⣶⣿⣟⡿⢿⣿⠿⠾⠷⠆⠄⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⣀⢀⣂⣼⣇⣼⣵⠌⠀⠧⢐⣿⣿⡀⠬⢭⣿⡖⠚⣀⣰⣀
⠀⠀⠀⠀⠀⣲⣽⣿⣛⣛⣃⢤⠖⢶⡶⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣾⣾⠿⠧⣽⣻⣿⣗⠀⠈⠉
⠀⠀⠀⠀⠀⣬⣭⣿⣿⣿⣴⠂⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠾⢧⣾⠿⣿⡂⠀⠘⠃
⠀⠀⠀⠀⠰⠶⠤⣄⠀⠈⠉⠐⠻⠆⠀⠀⣀⠀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠚⠲⠆⠀⠀⠒⠂⠁⠀⠈⠠⠤⠷
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠓⠀⠛⠛⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠣⠶⠶⠠⠤⢠⣀⣀⣠⣄⡀
⠀⠀⠀⠀⠈⠉⠋⠽⠛⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢈⣡⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⡿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢤⡀⠒⠶⠾⢦⣔⢾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣇⣤⣤⣤⣈⠻⢶⣶⣦⣶⣶⣤⣄⣤⣤⣴⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣿⣿⣿⣿⣿⣧⡄⠀⠉⠛⢻⣿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠶⢂⣤⣴⣶⢶⣾⣶⣶⣦⣾⠀⢠⡿⠛⣻⣟⣠⣿⠷⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡌⢿⡿⠟⠀⠃⠀⠈⣠⣾⣩⣹⡿⠑⡆⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⣶⠄⠀⠀⠀⠀⠀⠿⠿⠻⠛⠏⠁⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢦⠀⠸⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⡰⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣠⣶⣧⣄⠃⠀⠀⣸⣶⡀⢸⣄⠀⢀⣠⠎⠀⠊⠁
⠀⠀⠀⠀⠂⠀⢣⡀⣰⡈⠛⢹⣿⣯⠤⢦⣿⣿⣷⣼⣧⣖⣼⡃⠄
⠀⠀⠀⠀⠀⠀⠻⠋⠉⣿⣿⣻⣿⣿⣷⣄⣿⣿⢉⠙⡀⠟⠋
⠀⠀⠀⠀⠀⠀⠺⣟⠁⠈⢻⢛⣾⣿⠋⡙⣛⠋
⠀⠀⠀⠀⠀⠀⠈⠟⠓⠀⢠⠨⠅⠈⠁⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⣸⣄⠀⢆⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⢳⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⣡⣿⠈⠢⠀⠀⢱⣤⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⠏⠀⠀⠀⠀⢀⡴⠟⢁⡀⣴⠿⠋⠀⠀⡀⠀⠀⠈⢿⡀⠀⢀⣠⣤⣶⢿⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⢤⣴⣾⣿⠿⠟⠋⡳⠀⠀⣁⣴⣭⣶⣿⣯⡀⠈⠓⠦⠀⠀⠻⣆⠀⠀⠀⠙⠤⠈⠉⠻⠟⠀⠷⢤⣀⠙⠢⢀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⣠⠶⠾⢿⣿⣿⡿⠟⠛⠉⠙⢶⡄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠳⠤⠀⠀⠀⣠⣤⣤⡿⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢻⣿⣿⣿⣿⠿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣷⣿⣿⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⣴⣼⣿⣿⣿⣿⣿⣿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣷⣤⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣟⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⣻⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⢰⣿⡿⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡏⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣿⡇⣿⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡾⠛⣷⡙⠀⢠⢸⡟⣰⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣽⡧⠆⢼⣇⣿⣿⣾⣧⣿⠋⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⣿⣿⣣⣀⣾⣿⣿⣿⣿⣿⣿⡿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡇⣸⣿⢟⣼⣿⠻⣿⣿⣿⣿⡇⠀⢻⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣶⣿⠏⣼⡿⠃⢘⣿⣿⣿⣿⣿⡀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣽⡟⢁⣼⡟⠑⣀⣿⣿⣿⣿⣿⣿⣧⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣡⣵⡿⠿⠷⠾⠿⠛⠛⠁⠀⠈⠻⠿⣄⣀⡂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠼⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⣠⡞⠁⣼⣿⣷⣄⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢊⡠⣾⡷⢦⣤⣲⠞⠋⠀⠀⠻⣿⣯⠙⠛⣲⣦⣾⡁⠁⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠤⠾⡿⠟⠋⠉⡠⠶⠖⣻⣷⡄⠈⠉⠀⣴⣿⡿⣷⡮⠀⠘⠃⢙⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠙⠓⠍⠀⠀⠀⠀⠈⠉⠺⣇⠢⠔⠀⠘⣿⣿⣿⡷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⠀⢀⢀⣉⢀⣀⡄⠀⠀⣀⣀⡀⠀⣀⡀⠀⠀⠀⠈⢀⣴⣄⠀⠀⠉⠙⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠉⠭⠿⠿⠿⣟⣛⣿⠿⣷⡒⠛⠛⠉⠉⠛⠛⠛⠛⠛⠫⠯⠽⠿⢟⣓⡲⠿⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⠒⠫⠭⢝⣓⠢⠄⠀⠀⠀⣉⣉⣉⣉⣭⣐⣒⡂⠤⠄⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠒⠒⠒⠒⠒⠛⠛⠒⠉⠀⠀⠀⠀⠀⠉⠉⠉⠙⠂⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣦⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢊⠭⠙⠉⠉⠙⠓⢄
⠀⠀⠀⠀⣀⣤⣤⡤⣤⣴⣷⣿⣽⣿⣿⣿⣿⣻⡿⣿⣇⠀⠀⠀⠀⣀⠤⠒⣒⢀⣴⡂⠁⡡⠋⠁⠀⠀⠐⠲⠢⢄⠀⠂
⠀⠀⣰⣿⣿⣷⣾⣷⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣼⡏⠀⠀⠀⡼⠁⠀⠀⠀⠈⠁⠠⢊⠀⠘⠀⠀⠀⠀⠀⠄⠈⡀
⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⡀⠶⠄⠀⠀⠐⠁
⠀⠀⠘⢿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⠀⠀⠀⠈⠙⡄⠀⠉⠀⠂
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⢻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⢐⣠⡀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣿⣶⣢⣴⣦⣄⣀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡁⣀⢀⠈⠉⠳⠛⠀⠂⢤⢴⣶⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣟⣿⣿⣿⣿⣿⣿⣭⣼⣿⣿⡆⠀⠀⠀⠀⠀⠈⠃⠀⠀⠨⠉⠖⢄⡀⠀⠀⠀⠛⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣿⣟⣿⣿⢿⣿⣿⠿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠆⠤⢤⠤⢤⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠛⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠑⠀⠀⠀⠂⠠⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠄⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣿⣿⣿⣿⣿⣿⣷⣦⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡇⢠⡄⢸⠀⠀⠂⡛⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠛⠸⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠨⣛⣦⣷⣾⣴⡀⢰⠟⠋⠁⡈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠐⠀⠀⠀⠘⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠁⠰⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡾⠛⢻⡆⠸⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠀⣴⣿⠷⠟⣀⡀⠀⠀⠀⣤⣤
⠀⠀⠀⢀⣴⣷⣄⠄⠀⣿⣶⡴⠛⠀⣠⡈⠰⣾⣥⢠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠆⣼⣿⣿⣶⠗⠀⠀⠀⠀⠀⢿⣟⡀
⠀⠘⠛⠻⠉⣀⣩⣶⣿⣀⣵⣦⡀⠔⠛⠑⠶⣦⢉⡿⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣴⣿⢏⣤⣾⣿⣿⣿⡅⠘⢋⠀⠄⠀⠀⠀⠉⠠⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠂⠙⠋⠡⠟⠉⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠘⠛⣵⠞⢉⣤⣾⣿⠟⡿⢻⠟⣁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢻⡟⠁⢈⠀⠁⠀⠉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⡀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⣰⡟⢁⡄⠙⠛⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣥⠄⣴⣿⠟⠋⠀⠄⠀⠀⠈⢶⣾⡅⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⠟⣼⣿⣿⣿⠿⣦⠀⠀⠀⠀⠀⠛⢇⠀⠀⠻⡅⠀⠀⣸⣇⣰⣦⣀
⠀⠀⣀⠀⢀⣴⢂⡔⡿⢏⣴⣾⣿⣿⣿⠗⠀⡈⠁⠀⠀⢀⡀⠀⡈⢢⣄⡀⠀⡄⢰⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠚⠋⠀⡉⠉⠁⠠⣶⣿⡿⠋⢉⡿⠃⠀⢰⣧⠀⢰⡂⢸⣇⠀⣧⣰⡍⠉⣻⣿⣾⣋⡀⣾⣴⡷⠀⢀⡀⠀⠀⠀⢀⠔⠃⠀⠰
⠀⢠⡄⢀⣧⠀⠀⠺⠻⠋⣀⠀⠈⠀⣀⠀⣼⣿⣄⣼⣷⣾⣿⣿⣿⣿⠿⠾⠛⠋⡙⢛⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢤
⠀⣸⣿⣼⣿⣆⠀⢀⣆⣠⣿⣶⣧⣶⣿⣿⣿⣿⠟⠻⠟⡿⠛⠋⠁⠀⠀⠀⣼⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢷
⠚⠛⠛⠛⠛⠿⠦⣞⣉⡁⠀⠈⠠⠄⢛⠙⠋⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠎⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣤⣦⣀⣿⣄⠀⢰⣷⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠛⠛⠻⠿⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠢⠀⠀⠈⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠢⠄⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠐⠠⠄⣀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⣴⡆⣼⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡥
⠀⠀⠀⠉⠛⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⠀⠀⠀⡾⠻⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⡀⠀⢀⡀⠀⠛⠁⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡥⠄⠀⠀⠀⠀⠀⠀⠀⠠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣴⡀⣄
⣧⠀⠘⢁⡄⢁⣸⡇⠀⠀⠀⢠⣧⠀⣠⠀⠀⠉⡉⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠴⠤⣄⠀⠀⠀⠀⠀⠀⢀⡰⠂⠀⠘⠀⠲⡄⠀⡀⠀⢠⣿⣿⣿
⣿⣇⠀⠞⠀⣼⢿⣷⠀⠀⠀⢸⠿⢦⣿⡁⢰⣿⠓⢰⡞⠂⠃⠀⢀⡀⠀⠸⠇⠄⠀⠀⠀⠀⡀⠂⣀⡀⣠⢇⢈⡕⠀⠄⠄⢀⡄⡁⠰⠃⠀⠀⠉⠿⠿
⠉⠁⡀⠀⢸⣿⡿⠀⠀⠈⠁⠀⠀⠸⠿⠳⠀⠙⠀⠘⠁⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒⠃⡄⢀⣥⣵⣤⢠⣀⡆⠰⠂⣽⡧⠉⡄⢠⡄⢀⠀⣠
⢠⡎⠀⡀⣼⣿⣏⠄⢀⠀⠀⠀⠠⣄⢠⣶⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠀⠀⣿⣷⣼⡄⠸⣇
⠎⠀⣰⣗⠀⢤⠀⠀⠈⠻⠲⠒⠢⠀⠀⠙⠓⠂⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠰⠄⠈⠉⠹⠿⠧⠀⠐
⠀⠀⡿⠒⠆⠀⠉⠀⠠⠄⠀⠀⢦⡄⠀⠀⠀⠐⠤⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠲⣦⡤⣘⡒⣀⣀⠀⠴⡆⠀⠀⠈⣧
⠀⢼⣿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠈⢵⡄⠠⠀⠐⣶⣤⣀⣽⣶⠤⠄⠰⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠷⠤⠴⡾⠿⠟⢿⣄⠀⠐⠢⣶⡀
⠀⠐⠂⠀⠀⠀⠤⠀⠒⢲⣤⣶⡾⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠶⢦⣤⣀⣉
⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠴⠶⠛⠛⠛⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠻⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⠥⠒⠀⠈⢉⡭⠙⠒⠢⠄⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢀⠤⠰⠃⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡚⠉⠐⠉⡉⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠿⠃⠀⠀⣀⠾⠁⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⡂⠔⠀⠠⠴⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠴⠶⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⣠⣤⣴⣶⠶⣿⣿⣿⣷⣦⣄⡀
⠀⠀⢠⡤⢴⣿⡻⠿⡿⠟⣿⠿⠙⣻⡿⠷⣤⣀⠀⠀⡀⣀⣀⣀⣀⣠⣤⣶⣶⣷⣿⣿⡏⣿⣏⣺⣗⣿⣿⣷⡿⢽⣿⡛⡛⣿⣽⡿⣦
⠀⠀⠘⠁⠈⠸⠾⠽⠿⠷⠞⠷⠶⠿⠦⠾⠙⠩⠿⠿⠛⠛⠻⠍⠙⠙⠛⠿⠋⠃⠑⠾⠿⠒⠛⠓⠊⠃⠀⠉⠑⠛⠀⠐⠋⠩⠭⠉⠋
⠀⠀⠀⣀⠀⣄⢀⣀⠸⠶⠀⠔⢠⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⣀⠠⡤⠤⠠⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣀⢀⢀⠀⠀⡀⢀⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠐⠀⠂⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠉⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠢⣄
⠀⠀⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠴⣿⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠙⡗⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤
⠀⠀⠀⠀⠰⠝⣒⣶⢠⡿⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠐⠄⠀⠀⠈
⠀⠀⠀⠀⠈⢁⢻⠿⣿⣟⣿⣗⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠚⠀⠀⠈⡷⠀⠀⠀⠈⠰⡐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⢓⡀⠂⠀⠀⠐⠛⠛⠛⠒⢚⠂⠒⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠉⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠈⠙⠒⠲⠶⠶⢶⣤⡤⠤⠴⠂⠀⠀⠀⢉⠒⠤⠄
⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣖⣉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣁⡀⠀⠀⠠⠤⢤⣄⣀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠂
⠀⠀⠘⠛⠟⠛⠏⠋⠉⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⣀⡀⠀⠈⠉⠙⠛⠛⠛⠛⠓⠲⠿⠟⠉
⠀⠀⠀⠀⢤⠀⡤⠄⠀⠘⠂⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠙⠟⠃⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠚⠉⠉⢉⡭⠭⠭⢭⡛⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⡡⠤⢤⡀⠀⠀⠀⠀⠈⠆⡀⣀⡠⠟⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣫⠤⠤⣀⠀⠑⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⡔⠉⠀⠀⠀⠈⡧⠀⠀⢠⠖⠉⠉⠉⠉⠉⠙⡆⠀⠀⢠⠞⠉⠉⠉⠉⠲⡀⠈⢦
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠣⠤⠤⠤⠖⠊⠁⠀⠀⡇⠀⠀⠀⠀⢀⣀⠴⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇⠀⠀⠱⣄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠑⠒⠒⠉⠉⠉⠀⠀⢀⣠⣀⡀⠓⠤⠤⠖⠚⠁⠀⠀⢀⠤⣌⢧
⠀⠀⠀⢰⠃⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢘⡞⡆
⠀⠀⠀⢸⠀⠀⠀⢇⠀⠀⠀⠀⣀⡤⠔⠋⣀⣀⣠⡤⢤⡤⢤⢤⠤⢤⠤⣤⠤⢤⣤⣀⡀⠀⢰⠂⠒⢦⠀⠈⠉⠀⡇
⠀⠀⠀⢸⡄⠀⠀⠀⠁⠀⠈⠉⣠⡴⠚⠛⢅⡀⠈⢦⠀⡇⢸⢸⢀⡆⡴⠁⡠⢊⠴⢊⡩⠓⢦⡑⠦⠞⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠳⡀⠞⠉⢱⠀⣠⠚⠡⢤⣉⡒⠦⢌⡓⢄⠣⣡⢈⣘⣸⡜⣡⡮⢖⡥⣚⡥⠔⠊⣁⡙⣦⠀⢀⠖⣳⡼⠁
⠀⠀⠀⠀⠀⠘⠢⡀⠁⠀⣇⣉⣉⣐⣒⣺⣷⣶⣦⡭⠟⠋⠉⠉⠉⠉⠙⠛⢯⣭⡶⠶⠟⠛⡒⠒⢸⡇⠐⣲⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⣦⡐⠒⢒⣒⡦⠽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣭⣉⣉⣩⣭⡴⠾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠃⠀⣀⣴⣿⢿⣀⠈⡤⢋⣤⣥⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠔⣿⣷⣿⣿⣿⣿⣾⣿⠿⢿⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⠿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣀⠀⠀⠀⢠⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠛⠁⣹⣿⣟⣷⣄⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠀⠀⢀⣼⣿⣿⣿⣿⣼⣿⣦⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡏⣘⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⠃⣸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣠⣿⣿⣿⣿⣿⠟⠉⠻⣿⣿⣿⠁⢀⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⣰⢀⣘⣿⣿⣿⣿⠏⢡⣤⡖⣾⣿⡅⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⣿⣿⣛⣿⣶⣿⣿⣏⣸⣿⣴⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠻⣿⣿⣷⣿⣦⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣭⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡷⠶⠤⣴⣿⣿⡿⢿⣿⣿⣿⠿⢿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣀⣿⡁⣿⠚⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣿⣿⡇⠀⣾⡿⢫⣤⣿⣿⣿⣿⣿⡟⣿⣿⣟⣣⣾⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣝⣻⠉⣀⢀⣏⣀⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⣟⣿⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣾⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠰⠛⠙⣿⠃⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠉⠁⠀⠀⠀⠀⠀⠀⢹⠛⢿⣿⠁⠀⣸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⠀⣻⣿⣲⣿⠻⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⣯⢹⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣛⣲⣶⣦⣴⣶⣿⣷⣽⣿⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠁⠈⣀⣀⣤⣼⣿⣿⣿⣿⣻⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠴⠿⠛⠛⠛⠛⠛⠂








⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡛⠏⠙⢻⡋⠉⡟⠛⢛⡛⠏⣻⡿⢛⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢠⣾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠂⢠⣿⣿⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⣠⣿⣿⣿⡃⠀⠀⠻⡿⢿⣿⣿⣿⣿⣇⠀⠀⠀⣄⡀⢀⢰⣆⢐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⣰⣿⣿⣿⣿⡷⠀⠀⠀⠙⢮⡻⣏⣻⣿⣿⣷⣶⣴⣿⣿⣿⣾⣿⣿⣿⣇⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⢂⣴⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠙⢯⡻⣾⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣦⣿⠀⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⡘⣧⣽⣟⣿⣿⣿⣿⢿⣿⡷⣦⣬⣿⣿⣿⣿⣿⣦⣾⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⠛⢷⣌⡛⠂⡀⣠⣤⣍⣲⣾⣿⣿⣿⣿⣿⣯⣥⣶⣧⡶⣿⣿⣿⣿⣿⣽⣿⣿⠀⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡆⣸⣿⣿⣿⣿⣿⡿⠛⠷⣤⡟⢿⣷⣮⡙⠋⣻⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡄
⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡌⡿⠒⢿⣿⡗⢦⣿⣿⣿⡿⠋⣵⣾⣿⣿⡟⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢁⣿⢷⣤⣽⣿⣿⣮⣿⣿⣿⣾⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣶⣿⣧⣤⣿⣿⡟⠛⠁⠈⣹⣿⣿⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠙⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣾⣿⣿⣿⡿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡿⠛⢻⣿⣿⣿⡟⡟⢿⣿⣿⣿⣿⣿⣯⣴⣶⣿⠟⣿⢉⣹⣿⣿⣿⣿⣿⣿⣷
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣅⣹⣿⣿⡛⢿⣷⡍⠛⠿⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣛⠻⢿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡿⢟⠛⣹⣯⣽⣿⣿⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠿⠟⠛⢃⣤⣶⣿⣿⣿⣵⣄⣤⣴⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⠀⠰⣠⣶⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⡉⠧⠀⣶⣿⣿⣿⡋⠉⠉⠉⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠓⠃⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣃⠢⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⢿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⡿⢿⣿⣿⣿⠟⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠋⠸⡟⠛⠀⠀⠀⠀⠈
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏
⣿⣿⣿⣷⣦⡄
⣿⣿⣿⣿⡟⠁
⣿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠹⠟⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠘⣿⣧⡀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⠁⠀⣤⣟⣉⣳⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣠⡀⠀⣠⠋⠀⠀⠀⠳⣖⣒⣒⡛⢦⣠⣦⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠘⣷⡞⠁⠀⠀⠀⠀⠀⠈⠳⣖⢀⡔⠁⢿⣛⣆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡞⠁⠀⠀⡿⠷⣄⠀⠀⠀⠀⠀⣠⡤⡼⠋⠀⠀⢠⠘⣞⠓⢦⡀⠀⠀⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠨⣿⡅⣠⠞⠀⢠⣦⠀⠙⠿⢭⣍⣓⣦⣀⡞⠁⠈⠀⢀⡀⠀⣾⡆⠀⣶⠀⠹⣄⠀⣿⡇⢸⡇
⠀⠀⠀⠈⠉⠙⠛⠾⣇⠀⠀⢺⣿⠀⠀⠀⠀⠘⣖⣒⣛⠢⠄⠀⠀⢸⣇⠀⣾⣷⠀⣿⠇⠀⣹⣦⡽⠥⠾⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⣹⣿⡁⠀⣷⠀⠀⠈⣷⠐⠀⢰⡇⠀⣽⡯⠀⣹⣭⠤⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⠆⢸⣿⠀⠀⠀⠈⠀⠀⢘⣗⡤⠴⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡏⠁⠈⠉⠀⠙⠲⠤⣶⡾⢭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠙⠊⠉⠉⠉⠒⠒⠒⠲⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⢠⣄⠀⠲⢰⠙⠃⠀⠠⣦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⡀⢠⡄⡂⠀⠀⢻⡇⠘⣷⠄⠰⣄
⠀⠀⠀⠀⠀⠀⢙⠀⢘⠃⣤⠀⠀⠀⠀⠢⠌⠃⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⢸⠏⢀⠘⠀⠟⠀⠀⠈⠀⠀⠁⠀⠀⠘⠇⠀⡀
⠀⠀⢀⠀⠇⠀⡸⠀⠈⠁⠘⢧⣼⠀⠆⠀⡄⠘⠆⠃⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠄⢰⠆⠠⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠛⠀⠀⠀⠐⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠁⠀⠞⠀⠁⠀⠁⠀⠀⠀⠀⠀⡔⠀⠘⠗⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢸⢿⣧⡀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⠀⠈⠀⠙⠲⣄⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠻⣷⠀⠘⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠈⠏⠀⠀⠀⠈⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⠧⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦⣄⠄⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⠀⠀⠒⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠼⠃⠂⠀⡴⠄⠀⠘⠻⣆
⠀⠀⠀⠀⠴⡁⠸⡇⠐⠀⢰⠀⠀⡄⢀⠀⠀⣈⠋⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⢠⠀⠀⠑⠦⡀⠙⠳⣄⡀
⠀⠀⠀⣀⡀⠉⢠⡀⠀⠀⠆⠀⠐⠁⠀⣄⠀⠇⠀⠈⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠈⠙⣿⣿⣷⣔⠁⠀⠀⠀⠀⠙⢶⣤⡀⢻⣷⡀
⠀⠀⠆⠀⠰⠄⠀⢳⠀⠂⡠⠀⠀⠀⠀⠈⠦⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⠿⠷⢄⠀⠀⠀⠀⠀⢿⣷⡎⠉⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⣿⣿⢀⠑⣤⣶⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠄⣠⣾⠏⢻⣿⡇⠈⢀⠀⠻⡇⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⣿⣿⣿⣭⣿⣷⡄⠀⠀⠄⠀⠀⠀⠀⠀⣀⣤⣖⠀⠀⠀⠀⣀⣤⡾⠂⢤
⠀⠀⠀⠀⠀⠀⢠⡄⡀⠀⠀⢀⡴⠀⢠⣤⣾⣿⣿⡟⢀⣼⣿⡿⣿⣿⣻⣿⣿⣄⡀⠀⠀⢈⠀⠀⠀⣸⣿⡍⠀⠠⠋⠈⣭⡾⠀⠀⠂
⠀⠀⠀⠀⡶⣾⣿⡁⠈⣠⣾⣿⡿⠀⠀⠈⢻⡟⣿⣧⠸⢟⣿⢗⠚⠛⡵⢟⠿⢿⣿⣶⣆⠀⠀⢀⡈⢹⠟⢀⣄⡼⠀⠀⠛⠁⢀⣆
⠀⠀⠀⠀⡿⠿⠛⠆⡦⣿⣉⡋⡷⢸⣁⢰⡌⢸⡈⠿⢠⡍⠠⣸⣄⢀⠀⠀⠀⠈⠉⠋⢿⡷⣦⣤⠁⡚⣰⡎⣸⣧⡀⢸⡆⢴⡤⣿⣇⡀
⠀⠀⠀⠀⣰⣴⣶⣾⣿⣷⣿⣷⣷⣿⣿⣾⣷⣾⣿⣼⣿⣿⡄⣽⣿⢀⣆⠀⠀⠀⠀⢀⣀⣰⣎⣿⣨⣤⣿⣇⣸⣿⣥⣾⣿⣾⣿⣿⣿⡯
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣤⣅⣙⠛⠛⠿⠿⣿⣿⣿⣿⣿⢻⣿⣍⣿⣿⣿⢟⣿⣿⣿⠇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⢈⠙⣿⡿⣿⠷⣻⢹⣸⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠹⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⣰⡄⠀⢄⣸⣼⣳⠙⢻⣦⠿⢼⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢻⣧⢿⡟⣿⡹⣯⣿⣿⣿⡿⠛⠉⣅⠀⠀⠀⣶⠀⠀⠀⠀⠀⣶⣤⣿⣿⣾⣿⣾⣿⣿⣗⣅⡀⠸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⢸⡇⡿⢁⠺⣿⣿⣿⠷⠀⢸⣿⡀⠀⣔⣿⣂⠀⠀⠛⠿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⣀⡘⢿⣿⣿⣷⡷⠚⣸⣯⣶⠀⠺⣿⡿⡂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣷⣿⣿⣿⣿⣧⣾⣿⣥⣶⢹⣿⡿⣣⡄⠀⠀⠀⢠⣿⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⠞⡁⠀⢠⣨⣿⣟⣣⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣠⢬⣽⣿⣿⣿⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣯⡄⣶⣽⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣀⡀⢄⣀
⠀⠀⠀⠀⠈⣿⣿⡁⠀⠈⢁⡟⢦⡀
⠀⠀⠀⠀⠀⠉⢿⣿⣄⠤⠀⠁⠠⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣄⠀⠈⠁⢜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⡀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠸⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⣄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣗⠐⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡌⣙⠘⣿⣿⣳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡧⡀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣷⡆⢀⣾⣿⣽⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣱⠅⠀⠀⠈⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⠆⠀⣈⡍⢿⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣠⠀⠐⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡊⠘⠻⣿⣷⣿⣿⣿⡽⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⡀⣺⣿⣿⣿⣿⣹⠷⡋⢿⣿⣿⣿⣿⣿⢿⢿⡇⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠀⠀⠸⢻⣿⣿⡿⡿⢣⡌⠠⠇⠀⠉⢻⣿⣿⣿⡖⠫⠣⠀⠼⡁⢠⣤⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠄⠀⢀⣿⣿⡿⢣⠑⠈⠁⠁⠄⠀⠀⣈⣿⣿⣿⣇⣀⠀⠈⢓⣯⣽⣿⡆⢬⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠂⠀⣸⣿⣿⡗⠋⠀⢀⣀⣀⡊⠀⠀⢀⣾⣿⡿⠛⠛⠛⠛⠛⠛⠻⢿⡏⠼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⡀⠀⣼⣿⣿⣶⣴⣾⣿⣿⣿⣿⡆⠀⢠⣿⣿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠐⣿⣿⣿⣿⣿⣿⡽⠿⠁⣿⡇⢀⣾⡛⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⣿⣿⣿⣿⣿⡿⠉⠈⡰⡿⢁⣾⢿⠃⠡⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣿⣿⣿⣿⣟⢃⠀⣠⣿⢃⣯⣿⣆⡩⣶⡿⠁⠀⠒⠒⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀⣀⠘⠻⣷⣿⣿⣿⣿⣿⣧⣴⠟⣃⣰⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⡄
⠀⠀⠀⠀⣀⣀⡀⠀⠈⢉⡙⠛⠿⢿⣿⣿⠿⠿⠿⣿⣼⣽⣾⣿⣏⣙⡿⠰⠶⠶⠤⢦⣤⡀⠀⠀⠀⠀⠀⢀⡀⢈
⠀⠀⠀⠀⢀⣠⠽⠃⠀⡼⣿⣶⣶⢄⣀⠁⠢⠄⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠼⢐⠃⠈⠲⣆⡀⠀⣸⡿⠉
⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠹⠌⠋⠁⠘⠀⠀⠛⠳⢿⣄⠰⠿⢷⠶⠦⡄⠠⠠⢀⣤⣄⣀⠀⣬⣿⣶⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠⢀⠀⠀⠀⢀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢉⣠⢤⣧⣌⠉⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠘⣧⡁⠈⢢⡀⠀⠀⢠⠞⠛⠂⢈⠿⠗⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠤⠘⢧⣼⣤⡄⢠⡿⣿⡿⠖⠛⠒⠶⠬⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠷⠟⣷⣿⡇⠀⠀⠀⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⠈⠓⠀⠀⠚⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣋⣽⣿⣧⠴⣶⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣰⣿⣆⣀⣠⣤⣁⣤⣠⣶⣄⣀⣄⣉⣈⠁⢀⣀⠐⠂⠈⠛⣦⣄
⠀⠀⠀⠀⣰⣏⡴⡛⠛⠉⠀⠈⠪⣻⠁⠙⡭⡹⢹⣹⢹⡙⡏⣉⢋⢋⠟⡯⢴⣦⣀⡉⡤⠈⢣⡀
⠀⠀⠀⠀⡟⠁⠀⠀⠄⢀⠂⠄⡐⠈⢶⣄⠌⣱⡾⣏⣧⣕⣼⣏⣡⣪⢊⡴⠓⠉⡹⠶⣗⣄⠀⢱⡀
⠀⠀⠀⠀⢷⡀⡀⠀⠀⠠⠬⣒⣬⣑⣦⣜⡷⠿⠷⢿⣿⣿⣿⣿⣥⣿⣥⢤⣴⣮⣴⣵⣮⠿⣌⣋⠡
⠀⠀⠀⠀⠀⠻⣤⠒⣒⠺⠷⢶⣬⣽⣻⣿⡇⠀⠀⢰⡟⢿⣿⣶⣦⣤⣄⣀⡤⠍⠉⢉⡬⠟⢻⣥⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⢬⣥⠶⢒⣛⣻⣿⠀⠀⠀⢸⠀⢸⣧⣙⠡⡉⠙⡻⠽⢍⠛⡻⠶⠄⣼⡛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣿⠀⠀⠀⠀⡆⢻⣿⣿⡀⠀⠀⠈⠂⡀⣑⣌⣢⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⡇⣼⠻⢽⣧⠴⠖⠚⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠠⣶⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⣠⠂⣤⡀⡻⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣰⣵⣿⣾⣿⣿⣷⣮⣬⣢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣀⡀⣤⣀⣼⠟⠋⠁⠸⠸⡟⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢈⣗⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⡏⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⣀⣻⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠀⠉⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡤⣿⢸⢸⡇⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⢸⡇⡇⢿⢤⡄
⠀⠀⠀⠀⠀⠀⣰⢾⠀⡏⣸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣇⢸⠀⡷⣆
⠀⠀⠀⠀⣶⣾⣿⢸⠀⡇⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⢸⠀⡇⣿⣷⣶
⠀⠀⠀⢰⣿⣿⡙⡿⢠⡇⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⢸⡄⢷⢃⣿⣿⡆
⠀⠀⠀⢸⣿⠹⡇⡇⢸⡇⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⡇⢸⢸⠏⣿⡇
⠀⠀⠀⣸⡿⠀⡇⡇⢸⢡⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡈⡇⢸⢸⠀⣿⣇
⠀⠀⠀⣿⡇⠆⠀⠇⢸⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠸⠀⢸⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⢸⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⡇⠀⠀⠀⢸⣿
⠀⠀⢠⣿⠃⠀⠠⠀⡟⠸⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠇⢷⠀⠀⠀⠘⣿⡀
⠀⠀⢸⣿⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⡇⠂⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢸⠀⠀⠀⠀⣿⡇
⠀⠀⢸⡟⠀⠀⠀⠀⠁⠀⠒⠬⣕⣒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⣪⠥⠒⠀⠈⠀⠀⠀⠀⢻⡇
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠤⠤⣄⣈⠙⠲⢽⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⡯⠖⠋⣁⣠⠤⠤⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⣿⡇⠀⠠⠂⠀⠀⠀⠐⢒⠰⠬⠭⢛⣲⢬⣛⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣛⣥⣖⡛⠭⠥⢆⠐⠒⠀⠀⠀⠐⠄⠀⢸⣿
⠀⢠⣿⡇⡀⠀⠀⠀⠤⢀⣒⠂⠤⠬⣁⡒⠲⢭⣛⡻⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠿⢟⣋⠭⠖⢒⣉⠥⠤⠐⣒⡀⠤⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣂⠒⠦⠬⣉⣒⠲⠦⢍⣙⠒⠦⢭⣙⠲⠦⣉⡓⠦⣙⠻⣿⣄⠀⠀⠀⠀⣠⣿⠟⣋⠴⠚⣩⠴⠒⣋⡩⠴⠒⣋⡩⠴⠖⣒⣉⠭⠴⠒⣒⣿⡇
⠀⠈⠻⠿⠿⣿⣶⣦⣬⣉⡓⠒⠦⠍⣑⠒⠤⠉⡒⠠⠉⠒⠀⠑⠠⠙⢷⡀⢀⡾⠋⠐⠊⠁⠒⠉⠄⢒⠩⠥⠒⣊⠩⠴⠖⢚⣉⣥⣴⣶⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣷⣶⣤⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣡⣤⣶⣾⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣶⣦⣤⣀⡀⣠⣾⣿⣿⣿⣿⣷⣄⢀⣀⣤⣴⣶⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠟⠛⠋⠉⠉⠙⠛⠻⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣿⣿⣟⣦⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⡀⠀⢰⣷
⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⣾⣿⣿⣟⠉⠻⢿⣗⣦⡀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⠀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣾⣿⣼⡇⢀
⠀⠀⠀⠀⣧⣿⣧⣦⣾⣿⣻⣛⣽⡿⢷⣄⣀⣰⣼⣿⣷⣾⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠾⣿⣭⣉⡙⠛⠿⠟⠙⢺⣿⣿⣷⣿
⠀⠀⠠⣤⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠈⠉⠛⠛⠿⣿⣽⣿⣿⡷⠦⠀⠀⠀⢀⣚⣏⣽⣿⠿⠿⠿⠿⠷⠶⠤⠤⣤⡤⢤⠴⠶⠿⠿⠿⠿⠿⠿⢦



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣄⣀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢀⣠⡀⢀⣴⣷⣦⣤⣤⣴⣿⣵⣾⣿⣿⣿⣦⣀⡀⣀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣄
⠀⠐⠛⢿⣿⣿⠉⠙⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠉⠋⣹⣿⣿⣿⣷⠴⠀⠀⠀⢀⣾⣷⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⣷⣄
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⡠⠔⣁⣻⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠏⠱⣴⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣛⣋⣳⣄⠀⠀⠀⠔⠋⠹⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠹⡶⡀⠀⠀⠀⠀⠀⠘⠁⠀⠂⠀⠀⡠
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠶⠀⠠⠞⠃⠀⠰⠧⠄⠀⠀⠀⠀⠀⢀⠀⢀⣀⣴⣂⢀⣤⣄⣀⣀⣤⣄

⢄⠀⠀⠀⠀⠀⠀⣀⢤⠶⡄⠈⠀⠀⠀⠀⠀⢹⡖⢄⠉⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⣠⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠃⠀⠀⠰⢦⡄⠄⠀⠀⠙⠐⠁⠀⠀⠀⠀⠜⢁⠀⠉⢊⡁⠁⠀⠈⠀⠀⢀⠤⢀⡽⠁⢹⠀⠀⡰⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⡀⣄
⠀⠄⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡠⢋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⣸⠴⢄⣀⣤⡤
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠄⠀⠀⠀⠠⠞⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠄⠷⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠄
⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⠁⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠠⠀⢀⡀
⣖⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⡾⠉⣿⠒⢾⣿⡉⢿
⠘⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠐⣀⡿⣀⠀⠸⠙⢿⡈
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠠⠈⠀⠀⠰⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⠀⠀⠀⠸⠋⠜⠓⠊⠋⠀⠊⠭
⠀⠀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠐⠂⠁⣀⢀⣤⢀⠀⠀⠀⠀⢀⠄⢀⠀⠀⡠⠀⡀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠐⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠈⠁⠀⠀⠀⠰⠀⡀⢰⡆⠆⠀⢄⠀⠀⠀⠀⠀⠀⠀⠐⠃⠆⢈⠀⠀⠀⠀⠀⠈⠀⠉⠁⠠⠄⠐⠂
⠀⠀⠠⠀⠀⠀⠃⢠⠀⢠⠄⠀⠀⠀⡀⠈⠁⠀⢀⠀⠠⠀⢄⠠⠈⡄⠠⠐⠀⠠⡀⠄⠠⠀⠀⠦⡀⠀⠀⠀⠈⣱⠀⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠰⠆⠓⢠⢧⠀⠀⡀⠰⠀⠃⠀⠂⠀⠠⢤⠮⠀⠈⠻⣂⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⢀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠈⢴⡇⠀⠀⠀⠈⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠠⠀⡠⠭⢠⡀
⠀⠀⠉⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂⠀⠀⠀⠀⠀⠀⠀⢤⡤⠩⠂⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⣤⣀⣈⣂⣀⣤⡀⠀⠀⢀⠆⠀⢀⠀⠀⠀⠀⠀⢀⣆⣲⠀⣶⣤⣤⢴⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡌⢡⡞⠁⠹⣿⡌⣧⠀⠀⠈⠁⠀⢀⠀⢀⣀⡀⠀⠈⠀⠁⡀⠁⠀⠀⠈⠈⠁
⠶⠆⠐⠰⢏⠼⠤⠶⠠⢳⡜⠂⠀⠀⠀⠠⠈⠛⠃⡠⠠⠁⠀⠀⠀⠰⠒⠄⠀⠀⠀⠀⢄⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣠⠀⢠⣴
⣶⣤⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠏⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠘⠋⠀⢀
⣷⠦⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢠⠀⠀⠀⠀⠀⠀⠀⣤⠀⠂⠄⠀⠀⠀⠀⠂
⠻⠂⠀⠀⠀⠀⠀⢈⠀⠀⠀⠄⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡄⠉⠀⠙⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠿⣦⣄⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠑⠀⠀⠀⠀⠀⠀⠰⠄⡀⠀⠀⠏⢀⠀⢀⠈⠀⢀⠀⠄⠀⢀⡆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠠
⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⠓⠚⠟⠿⠛⠛⠛⠋⠁⠀⢀⠎⡀⠺⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡀⠀⠀⠠⠀⠀⣤⡏⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⡂⠀⠑⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⢿⢿⠻⠏⠿⠿⢿⢿⣷⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠟⠛⣉⣡⣌⣤⣤⣤⣴⣶⣶⣄⣄⠁⠛⠙⠿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢋⣈⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣬⣍⡻⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡦⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⣟⠋⣟⠈⣋⠡⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⢟⡀⣽⡧⠈⠘⠁⠴⢧⣬⣿⡿⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣛⣽⣧⣶⣙⣿⣿⡆⢰⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠏⣇⢈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⣂⣉⣙⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠀⠁⢸⡟⠻⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣴⣷⣾⣿⣿⠿⣿⣿⣿⠿⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣡⣤⣶⣶⣤⣄⣀⠢⡀
⠀⠀⣠⢔⣾⣿⡿⠋⢀⣉⣤⣶⣤⣴⣦⣴⣶⣤⣮⣙⡻⠆⠀⠀⠀⠀⠀⣧⠀⢠⣿⠀⠀⠀⠀⢀⣠⠀⠀⡰⠶⠿⠿⣿⠿⠿⠿⠿⠿⠿⢿⣷⣝⢢⡀
⠀⢰⣵⢿⣫⣥⣤⣾⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡇⠐⢸⣿⡀⠀⠀⠀⣿⣷⣶⣷⣶⣶⣶⣷⣶⣷⣶⣛⣿⣻⣗⣾⠶⠤⣬⡙⣆
⠀⡟⢋⣼⣿⣿⣿⣻⣻⣤⣬⣥⣿⣿⣾⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⣿⡁⠀⠀⠐⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⣖⠠
⠀⠰⣽⣿⣿⣿⠿⠿⠛⠉⠛⠻⡯⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⢹⡇⠀⠀⠀⢋⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⢿⠿⠟⢻⣿⣿⣿⣿⣷⣞⠃
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢡⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⠀⠀⡷⠀⠀⠀⠀⠉⠛⠛⠋⠉⠉⠀⠉⢻⡾⡇⢰⣾⣧⣶⣿⣿⣿⣿⣿⣿⣆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠀⢸⣷⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⠇⠸⠟⠉⠙⠻⠿⢿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣴⣾⣹⣟⡻⢿⣛⣿⣻⣦⣄⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣣⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣾⡷⢿⡿⠿⣿⡿⢛⣿⣿⣛⣋⣋⠃⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣀⡟
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⣿⠠⠈⠳⡀⠀⠀⠀⠀⠠⢽⣠⠀⣷⣾⣿⣿⣿⠿⣿⠎⡿⣿⣿⡿⠿⠇⠀⠀⣿⡇⣀⠀⠀⠀⠀⢠⡀⣟⠉
⠀⠀⢸⠇⠃⣠⠀⠀⠀⠀⢒⠀⢁⠀⠘⣿⣧⠸⣗⠀⠀⠀⠀⢈⢻⣠⡄⣿⡙⣿⣿⣿⠀⣹⢷⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢹⠄⠀⠀⠀⠘⡿⠟
⠀⠀⠠⡴⠆⠉⢀⣄⠀⠀⠘⠁⠀⠁⠀⠈⢿⣦⡙⡄⠀⠀⠀⢉⢹⣀⣀⣿⠃⠁⠋⢻⠀⣿⢸⠀⢠⣿⡀⠀⠀⢀⣾⡯⠀⡰⠀⠀⠐⣿⣾⣷
⠀⠀⠀⠀⣤⣤⣼⡍⠀⠀⠰⠡⠤⠁⢁⠀⠀⢻⡗⠈⢮⣂⠀⢉⣹⡅⠉⣿⠂⠀⠀⢸⠀⡿⢸⠀⠲⣿⡅⠀⢀⣾⡿⠁⢰⠃⢀⣤⣄⣸⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠛⠉⠀⣤⡖⠀⠀⣲⠀⠘⣀⠀⠀⢻⡀⠘⠿⢣⠈⠙⡧⠄⣿⠀⢸⠁⣼⢀⣷⡟⠀⣛⡇⠀⢀⣾⡏⢠⠰⡞⠀⠀⠀⣿⠛⢋⠀⠒⠡⡀
⠀⠀⠀⠀⠀⠀⠒⢃⡱⠁⠤⡄⠰⣂⢼⠿⠀⠀⣀⢷⠾⢤⡈⣷⣴⣧⣤⣿⢠⢺⣧⣿⣾⠃⡇⠀⠉⡟⢳⣿⠟⠀⠀⡘⠀⢸⣇⣠⡶⢀⡈⠛⡀⠒⠑
⠀⠀⠀⠐⠶⠀⠰⢣⠕⠀⣺⣏⠈⢛⠂⢀⣤⢀⣽⣷⠶⢼⡗⣼⢿⣿⣿⣯⡼⢸⣿⡹⡟⢠⡍⠙⢀⣧⣾⢏⡜⠀⡾⢠⡄⠾⣾⡇⠀⢺⠔⡠⢁⢀⡀
⠀⠀⠐⠷⠈⢠⡄⠉⢙⡚⣥⢾⡧⠰⢂⣌⢙⠉⣰⣟⠀⣺⣿⣸⣷⣿⣿⣿⡇⣿⣿⣷⢂⡼⢳⣾⣿⡟⢇⡾⠑⢸⠋⠀⡗⠀⠘⣇⣟⡀⢛⢀⡘⠚⠃
⠀⠀⠀⠀⠀⠛⢼⠧⠘⡻⠏⡄⢱⠀⠀⣻⢦⣤⣽⣯⠀⣿⠏⠁⠉⣿⣿⣿⣿⣿⠋⣡⣾⣧⣿⣿⡿⢀⢾⠃⢠⣾⣷⣴⣳⡾⢄⠈⠀⡟⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣁⢐⠀⢛⠀⡽⠃⣩⣤⣿⣿⣧⣠⣿⣦⣠⡀⣿⠋⠈⣿⡧⢾⣿⣿⣿⣿⣿⠃⠀⠠⢀⣼⣿⣿⣿⣿⣿⣾⣧⣼⣇⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣶⣧⣶⣿⣿⣿⣻⣟⣹⣿⡉⠈⢷⣿⣆⣼⠏⠀⠀⢠⣾⣿⣿⣟⠀⠄⠀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡷⢦⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠿⣮⢹⣿⣠⣴⣶⣿⣿⣿⣿⣿⣤⣶⣿⡿⣿⣿⢿⣿⣿⡿⠿⠿⠋⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠛⠿⣿⣉⣻⡿⢿⣿⣿⡿⠧⣀⣀⣾⣎⠹⣯⣀⡀⢹⠈⠛⠛⠀⠈⠉⠋⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠁⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠈
⠀⠀⢀⡀⡀⡀⠀⢀⣀⠀⢀⣀⠀⠀⣀⠀⠀⢀⠀⠀⢀⣀⠀⢀⡀⡀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⣀⡀⠀⣀⠀⠀⢀⠀⠀⢀⠀⠀⠀⣀⡀
⠀⠀⢸⣅⡇⡇⠀⡏⢸⠀⢸⠘⡇⠀⡇⡧⠀⢸⠀⠀⣎⣸⠀⢸⣷⡇⠀⡇⣧⠀⠀⠀⠀⡇⣳⠀⢸⢸⠆⢰⡇⡇⠀⣼⢰⠀⢸⠀⠀⢸⡀⠀⢸⡙⠃
⠀⠀⠘⠧⠧⠇⠀⠳⠞⠀⠸⠼⠁⠀⠷⠃⠀⠾⠀⠀⠇⠸⠀⠸⠸⠇⠀⡷⠃⠀⠀⠀⠀⡷⠃⠀⠸⠼⠀⠈⠧⠃⠀⢿⠏⠀⠸⠀⠀⠸⡀⠀⠀⠹⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣮⣍⣤⣍⣽⣏⣼⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠸⢿⣄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠱⡁⠀⢻⡇⠀⢡⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠘⡄⠀⠃⠀⣻⠃⠀⢈⡟⠀⠀⠀⠀⣠⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢀⣴⡿⠗⠠⠀⠀⡀⠴⠁⠀⠀⠈⢷⡀⠢⣠⡞⢹⡇⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠂⠀⠠⣠⠟⠁⠀⢠⡖⠀⢨⠀⠈⢄⠀⠀⠀⠀⡇⠀⠻⢇⠀⠹⡆⠀⠡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠀⠀⠀⠚⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⢨⠗⠀⢠⠎⢃⠀⠀⠀⢃⠀⠈⠢⠀⢿⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠀⠀⠐⠁⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⣠⡟⠀⢈⡀⠀⠀⠀⠀⠀⠀⠁⠀⠡⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠏⠁⠀⡸⠃⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠀⠀⢰⡄⠀⠀⠒⠒⠶⠤⠤⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠒⠒⠂⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣺⣿⣷⣂⠐⣿⣗⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣿⣿⣃⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠠⣿⣿⣿⡅⣸⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠀⠂⠤⡀⠀⢀⣽⣿⣿⣧⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⣻⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⢴⣿⣿⣿⡷⠦⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⡛⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠒⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠁
⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⢿⠿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⣾⣿⣿⣿⠟⠋⠁
⠀⠀⠀⢸⣿⣿⣿⠃⠀⣀⣤⣴⣶⣦⣤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢸⣿⡿⣡⣴⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣶⠀⣠⣿⣶⣦⣄
⠀⠀⠀⠀⢩⣾⣿⣿⣿⣆⡀⠀⠉⠙⢿⣿⣿⣿⠛⠿⠟⠛⠛⠃⠠⢏⢻⣷⡆⠉⠿⠦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠅⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠈⢠⠀⠀⠈⠙⠐⢹⠀⠀⠐⠞⠀⠀⣀⣴⡴⠿⠶⠶⢤⠠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⣻⠁⠀⠀⠀⠀⢀⣀⡐⠀⠀⠀⠀⠠⠤⠁⠈⠁⠀⠀⢸⡿⠩⣐⠒⠂⠁
⠀⠀⠀⠀⠘⢿⣿⣿⡿⠿⠊⠀⢀⣜⡁⠠⣔⢶⣶⣿⣟⣅⣄⢸⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠡⠖⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⣶⣛⣫⣤⣿⣿⣷⣿⡟⢿⣷⢊⠰⠿⠇
⠀⠀⠀⠀⠀⠀⠸⠀⢀⣈⣴⠾⠋⠛⠉⠉⠉⠉⠉⠉⠛⠻⠿⣦⣄⠀⠀⢀⣀⣀⣀⣴⠿⢶⣦⣤⣤⠤
⠀⣤⣤⣰⣦⣠⣴⡶⠟⠋⠁⢀⣴⣦⣉⣿⣅⣀⠀⠀⠀⠀⠀⠀⠙⠻⣏⣲⣿⠿⠟⠙
⠐⠋⠉⠛⠓⠋⠉⠓⢦⣁⠀⠠⠁⢼⣿⣿⣿⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⡀⠠⠀⠙⠳⠉⠟⡿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣾⣛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠹⢷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠙⣆⠀⠀⠀⠀⠀⠀⢠⠞⠛⠓⢤
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠑⢶⣄⠀⠀⠀⣀⣴⡗
⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠐⢰⣿⣄⣀⣤⠞⣃⡠⠖⢊⣡⠞⠁
⠀⠀⠀⠀⠀⡞⠀⠀⣴⠞⠛⢢⠀⠀⠈⡇⠀⠀⢀⡇⠀⣠⠋⠉⢳⡄⠀⠈⢷⠀⠀⢨⠀⠀⠉⠉⢉⣠⠞⠉⣠⣶⠟⠁
⠀⠀⠀⠀⢸⠃⠀⢰⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⢸⠁⠀⢏⠀⠀⢈⣿⠀⠀⢸⡇⠀⠘⣷⣶⣶⣚⣉⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠸⡆⠀⠀⠳⣤⡤⠞⠀⠀⢰⠃⠀⠀⠸⣇⠀⠈⠀⠀⠞⠃⠀⢀⡾⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠘⢦⣀⠀⠀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⢦⡀
⠀⠀⠀⠀⠀⡀⠀⠈⠉⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⣰⠿⠁⠀⠀⠈⣧
⠀⠀⠀⠀⡼⠋⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠚⠉⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⢀⣿
⠀⠀⠀⢰⠃⠀⣿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠊⠉⠉⠙⡆⠀⢀⡾⠁⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠘⡇⠀⠘⠂⠠⠄⠚⠉⠀⠒⣤⣀⠀⠀⢀⡏⠀⠐⣿⣤⣤⣤⡾⠁⠀⣾⠁⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⢦⣀⡀⠀⢀⣀⣀⡤⢖⣛⡽⠇⠀⠸⣧⡀⠀⠈⠉⠉⠉⠀⢀⣼⠇⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢳⡝⠒⠤⢤⣤⠴⠖⣫⠋⠀⠀⠀⠀⠘⢿⣦⡀⢀⣀⣠⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁

⣟⠉⣹⣛⣟⣛⣛⣻⣟⣛⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣛⣛⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡿⢿⡟⠇⠀⣤⡀⠤⡴⣴⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⠉⠁⠀⠀⠁⠀⡄⠀⢁⠀⠀⠀⠒⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠈⠀⠀⠸⠄⣶⣇⣄
⠀⠀⠀⠀⠀⠀⢤⠀⢻⠘⡀⠘⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠙⠏⠁
⠀⠀⠀⠀⠀⠀⢀⡆⠘⢁⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠰⡄⡄
⠀⠀⠀⠀⢸⡴⠈⢠⡆⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢤⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⢀⣾⡟⢀⠈⡁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠆⢀⣀⣴⣶⣶⠶⠒⠀⡤⢸⠀⢸⣿⡇
⠀⠀⠀⢸⣟⠃⠆⠀⠁⠀⠀⠀⡸⣶⠂⢀⣤⣤⡤⠶⣶⣺⣿⡿⢹⠵⠛⠁⠀⠀⠔⠀⠀⠃⢸⣿⣧
⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⣀⣼⣧⡴⢞⣩⣿⣿⣾⣾⣿⡿⠟⢭⣶⣾⣶⣤⡤⠂⠀⠀⠀⠱⢼⣿⡇
⠀⠀⠀⠘⣿⡇⣀⠀⠀⢸⣿⣿⣟⡉⠉⠉⠓⣹⣛⣥⣤⣤⣴⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⢿⣿⡀⠀⠀⠈⠛⠻⠛⠿⠦⠠⠞⠛⠛⠛⢿⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠈⣿⣷⡄⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⣀⠕⠀⠂⠀⠀⠀⢀⣸⣿⡿
⠀⠀⠀⠀⠀⠈⢿⣷⣨⠃⢄⡑⠀⠢⡄⠀⠀⠀⠀⠆⢰⡅⠀⣾⣿⡀⠀⠀⠀⠀⠞⢹⡏⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⣜⠁⢦⢰⣿⣇⣤⣤⡀⠁⠀⢻⠆⠘⠿⢷⣶⣤⣄⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠖⣻⣿⣯⡿⣿⡭⠛⢺⣇⠀⢚⡡⠀⢀⡟⢩⣧⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢀⠁⣷⣿⣿⣿⣿⣄⣴⠾⠷⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣴⣤⠂⠀⠀⢀⣿⣿⣷⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⠀⠊⠁⠀⠀⠀⠚⠁⣿⣿⣿⣿⣆⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣦⡀⠙⠳⠌⠺⢦⣄
⠀⠀⠀⡀⣀⣀⣀⣄⣀⣀⢀⣀⣀⠀⣀⢀⡀⢀⣀⡀⠀⡀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣷⣦⣀⡀⠀⠉⠁
⠀⠀⣩⣍⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠰⠿⠿⠿⣦⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣦⠀⠤⠄⠀⡀⠀⠤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⡨⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠍⠶⠀⢀⠀⠀⣸⡴⠪⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡈⠀⠀⠀⠘⢃⠀⠀⠀⢈⠤⠄⢀
⠀⠀⠀⢶⢀⠀⠠⠀⠆⢠⠀⡅⠳⣀⠂⡈⡀⡈⠙⠁⠀⢠⡆⠀⠀⠀⠐⠀⠀⠐⢄⠀⠀⠐⢀⡀⠁⠀⠀⡀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠈⡱⠂⠀⣈⠀⠁⠈⡄⠙⠀⢁⠈⠤⡃⠁⠄⠀⠀⠀⠀⠀⠈⢀⠐⠀⠀⠀⠀⠀⠀⠀⠅⠄⠨⠀⢀⠀⡘⠁⠀⠀⠉
⠀⠀⠀⠤⠀⠶⠀⠐⠈⠈⠐⠡⢀⡀⠀⠀⠀⠂⡀⠀⠦⠀⠄⠀⠀⠀⠀⠀⠈⡁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠴⢤
⠀⠀⠀⠬⠄⠀⠀⢀⠀⠀⠀⠠⠀⢠⠀⠀⠀⢀⡀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠈⢡⠀⣀⡆⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠠⠆
⠀⠀⢀⠈⠂⠀⠀⠹⢆⡀⢀⡜⠀⠈⢄⠀⢀⠼⠇⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠈⠢⠉⠀⡂⠄⠀⠄⠈⠳⠐⠀⠀⠰⡌
⠀⠀⠀⠈⢔⢀⣀⠀⠀⠈⠁⡨⡄⠨⡀⠉⠁⠀⢀⣀⣐⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠣⣀⠂⢠⠂⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠙⠀⡀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤⢄⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢂⣹⡤⠂⠀⠴⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⠀⠀⠀⠀⠀⣀⠀⡀⢸⢠⠔⠈⠼⠂⠐⠑
⠀⠀⠀⠀⠀⠀⠄⠴⠀⠀⡀⡀⠐⡠⠴⢡⠈⠂⠠⡄⢀⠂⠀⠀⠀⠀⠀⢄⠰⠌⠂⡀⠀⠀⠀⢡⠀⣀⠂⠀⠓⡐⢎⡘⠈⠘⠀⠃
⠀⠀⠀⠐⠈⠁⠠⠋⠀⠄⠑⢁⡐⢀⡀⣘⠐⠀⡀⢅⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢪⠴⣆⢹⡈⠂⡄⢐⠁⠁⠆⢈⠂⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠜⠐⠒⠂⠀⠠⠆⠎⠃⠰⠁⠀⠀⠀⠙⢇⠀⠀⠀⠀⢐⣒⠒⢢⡤⢩⠎⠈⣆⡜⠙⢄⡞⢈⢲⠎⠹⢶⣊⣁⠄⠂
⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⠀⠀⢂⠀⠀⡁⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⠈⠆⠈⠈⠁⠊⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢀⠆⠀⠀⢀⠃⢸⡀⠀⡠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣓⠂⠈⡄⠀⡔⠀⢪⡍⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠉⠉⢠⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠙⠀⣠⠚⠐⢃⡀⠁⠀⠁⠀⠸
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠄⠀⠀⡀⣇⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡰⠈⠆⡀⠀⠀⠠⣭⠄⠀⠑⡬⠨⢀
⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⢦⠀⠀⠈⠉⠀⠀⠤⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⣄⡀⠀⠀⠀⠀⠀⣀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢯⢺⠀⢧
⠀⠀⠀⠀⠀⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡕⢸⡆⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠖
⠀⠀⠀⠀⠀⢱⡘⢝⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠎⢸⡧⠀⣇⠀⠀⠀⠀⠀⠀⠀⣠⢔⢋⡴⡝
⠀⠀⠀⠀⠀⠀⢣⠠⠱⢦⠉⠳⣄⠀⠀⠀⠀⠀⢠⣿⡳⡑⢸⡯⠀⢸⠀⠀⠀⠀⠀⣠⢪⠸⢈⡞⢰⠇
⠀⠀⠀⠀⠀⠀⠈⢯⠭⢌⠻⣆⡄⠓⢄⠀⠀⠀⢸⣸⣝⠆⢸⣿⡂⠸⡄⠀⠀⣠⣺⡞⡏⣴⠏⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢯⣯⣗⡘⢿⣄⡈⠳⣄⠀⢸⣿⡣⡁⢸⣿⡢⠀⡇⢀⢜⣟⢧⢃⣾⠏⢀⡞
⠀⣤⣀⠀⠀⠀⠀⠀⠀⢻⣿⣒⡀⠹⣷⡄⠈⢦⣸⣿⣝⠀⢸⣿⡯⠀⣧⢮⣿⠘⢠⣾⠃⢀⡞
⠀⠘⣌⠫⣓⠲⠤⣀⠀⠀⠙⣿⣖⡤⠘⢿⣴⡄⢻⢿⡢⡁⢸⣿⡯⠀⣿⠟⠇⣠⣿⠃⢠⠎⠀⠀⠀⣀⣀⠤⠠⢒⢂⠴
⠀⠀⠘⢎⢮⣙⢦⣌⠙⠲⢤⡈⢺⣿⡥⠈⢻⣷⣿⡼⣝⠄⢸⣿⡏⢰⡿⠘⣰⣿⠋⢠⣋⢤⣶⢿⠳⠚⣩⠝⢊⠕⠁
⠀⠀⠀⠀⠳⣳⣦⣌⡛⠧⣤⡈⠓⠿⣿⣧⡀⠹⣿⡏⠪⡁⢸⣿⠁⣸⡇⣰⣿⠃⣴⣿⠾⠛⣈⣤⠶⠋⣠⠔⠁
⠀⠀⠀⠀⠀⠈⠛⢟⣶⣤⠈⠙⠳⣦⣌⠙⢷⣂⠙⣿⣑⠄⢸⣟⠀⣿⣿⡿⢁⡾⠉⣀⣴⠞⠋⣁⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⢿⣶⣄⡀⠉⠻⢶⣌⠳⡜⣯⡃⢸⠇⣼⣿⢋⠴⣋⣴⡾⠋⣡⠔⠊⠁
⠀⠀⠀⠀⢀⡀⣤⠤⠤⠤⠤⠬⠿⠿⠷⢦⣤⣈⠓⠾⣮⣧⢸⣿⣿⣿⡵⢞⣋⡥⠶⠯⠴⠖⠒⠒⡒⠒⠂⡐
⠀⠀⠀⠀⠀⠉⠒⠤⢭⣉⣩⣝⡛⣒⣒⢒⡒⡒⡨⢹⣾⣿⣿⣿⣿⡿⢿⣟⣓⣒⣚⣛⣋⣭⣭⡥⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠿⣛⡿⠟⣋⣭⠴⠊⣿⠑⢶⣬⣀⠲⢯⡛⢯⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠵⠊⣡⣶⠿⠛⠁⠀⠀⣿⡀⠀⠈⠚⠿⢶⡌⠑⠌⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠈⠉⠁⠂⠓

⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⢠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠰⢈⡘⠀⠀⢀⡿⠃⠀⠀⠀⠀⠰⣮⠇⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⡀⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⠀⣐⠀⠑⣤⠀⠉⠀⠎⠀⠀⠀⠀⠀⠙⠄⠀⠀⠂⠀⣁⣤⣤⣶⣟⣛⣓⠒⢒⠋⠀⠤⢶
⠀⠁⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢩⠐⠘⠿⣧⣦⣄⣈⠀⠀⢀⠀⠀⠈⠁⠀⠀⢀⣼⣿⠿⠁⠀⠀⠀⠀⠈⠈⠃⠘⠒⠒
⠀⠀⠀⢸⡇⠀⠀⣓⠀⠀⠀⠀⠀⠬⠀⠀⠀⠈⠁⠉⢿⣻⣳⣶⣶⡄⠦⠴⡄⡐⠋⠉⠁⠀⠀⡆⠄⠀⠀⠆⣀⡀
⠀⢀⠀⣸⡇⠀⣸⢸⣃⢀⢠⢷⡒⠾⠀⠂⠀⠒⠀⠰⠈⠓⠒⠈⢀⠆⠀⠀⠀⠈⠑⠂⢤⠀⠀⠁⠤⠈⠀⠦⠸⣽⠳
⠀⢸⠀⢸⣏⠀⢼⠍⠋⢺⠛⠉⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠐⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠪⠘⠂
⠀⠘⠋⠩⠁⠐⠉⠤⠀⢸⡁⠁⠀⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠
⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⠀⢀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⣿⠀⠀⣴⡿⣻⢿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⣀⣿⣀⣾⠋⡼⠹⣼⡆⠀⠀⠀⠀⡸⡇⡴⢿⣾⡥⣶⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠿⠛⠛⠛⢹⠁⠐⢷⣸⣿⠀⠀⠀⢠⣿⠙⠏⠀⠿⠁⠀⠈⠹⡼⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⡠⠀⠀⠀⠀⡼⢋⠏⠀⠀⠀⢿⠯⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠁⢸⡀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠜⡽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠐⠉⠉⠁⠀⠀⠀⠀⢧⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡂⢻⠶⠚⠀⠰⠁⠀⢀⠀⠀⢀⠀⠀⠀⠈⢷⣄⠹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠿⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠳⠦⣤⣔⣀⣐⣄⣨⠶⢏⠀⠀⠀⠀⠀⠙⠷⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢹⡄⢀⡴⠃⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡴⠒⠛⠉⠀⠀⠀⠀⠀⣀⣴⣋⣤⣤⣄⡀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣾⣏⣀⣠⡴⠒⠒⠒⢋⡽⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡟⠀⠈⠆⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡀⠀⠀⠀⠀⠀⢠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢠⣤⠽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣙⣦⣐⣀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⣠⣄⡲⡶⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⣍⣩⣤⣬⣭⠿⢾⣦⣀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⣰⠖⠛⠛⣉⣀⣀⣀⡤⠞⠛⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣩⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡄⡤⣘⣁⡀⠀⢄⡤⡞⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⡈⢀⠋⠐⠋⢣⡭⣤⡾⠁⢀⣀⠨⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠲⠀⢶⡰⣤⠅⠃⣦⡗⢀⢂⡤⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⢉⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠙⠄⠃⠸⣧⠘⠢⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⢻⡶⠓⠆⠀⠋⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢀⠺⠃⠁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⡄⠀⡀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠈⠂⠀⠀⠀⠀⣼⣅⠂⠀⠼⣄⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠁⠁⠰⠀⠀⠀⠀⣀⠄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠴⠚⡤⠀⠄⠀⠀⡀⣠⠄⢸⡧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡇⢀⠳⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠠⠞⠠⢶⣦⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⣈⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠒⠀⢠⠀⠀⠀⡀⠊⠀⠀⠐⠂⠈⠀⠀⠀⠀⠑⠀⠆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢐⣿⣿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⢰⡇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⢠⣿⡟⠋⠀⠀⠀⠄⠀⢀⠒⠀⠀⠁⢀⣾⠀⠀⠀⠀⠀⠀⠺⠟⠛
⡆⣼⢃⠀⡀⠀⢠⣶⠟⠀⠀⠀⢀⣀⠀⢰⣿⡟⠁⢀⣼⡿⠇⠀⠀⢹⠀⠀⠀⠀⠒⢂⣄⠀⣸⠟⠁⠀⣤⣾⣷⡄⠲
⠷⣿⠄⢰⡇⢀⠆⠻⣷⡀⠈⠀⠀⠁⠀⠘⣿⣥⣶⣿⣿⣧⣄⠀⠀⠈⠃⠀⠀⠀⠀⠾⠟⠀⠉⣀⠴⠊⠉⠈⢠⠀⠀⣈
⣳⣿⠏⠹⣧⢹⠅⣤⠸⠿⠧⢀⠐⠀⠀⠀⠉⠙⠛⠙⠟⠉⠻⠟⠀⣆⠀⠀⡴⠇⢀⠀⡀⣠⠀⡀⠀⡗⢠⠀⣾⢠⣆⣿⣾
⣿⣿⣧⣠⣿⣿⡄⣿⠀⡀⢰⠹⣦⢦⡀⠀⣀⠀⠀⣀⣆⣀⠀⢠⢀⠉⠀⢠⣤⣦⣸⣴⣇⣿⣾⣿⣾⡟⢺⣿⣿⣾⣿⣿⣺
⠈⠉⠉⠉⠙⠿⢷⣿⡾⠟⠛⠙⠛⠋⠁⠀⠛⠁⠀⠉⠙⠛⠙⠋⠛⢃⠀⠈⠻⠟⠿⣿⣿⠿⠿⠟⣿⣦⠾⠛⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣲⣖⠂⠁⠀⠀⠀⠀⠀⣀⣀⠐⠂⠠⠀⣄⣀⣠⣤⣤⣀⣀⣀⠀⠀⠀⢀⣁⣀⣤⣶⡶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⢤⣤⣄⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⠀⠉⠉⠉⠉⠉⠙⠻⠿⠟⠉⠁⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣥⠶⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠺⣿⣤⣀⣁⣨⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠤
⠀⠀⠈⠀⢀⣦⡄⢙⣿⣷⣶⣿⣿⣿⠿⢿⣿⣿⣿⡿⠗⠚⠃⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⣤⠤
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣥⠀⠀⠈⠙⠻⣿⣥⠈⠉⠀⠉⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⠶⠦⢤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⠀⠀⠀⠉⠒⠄⠀⠀⠉⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠰⢶⣄⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⡀⠄⠂⠊⠀⠀⠀⠀⠀⠐⠛⠻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⢀⣠⡤⠄⠀⠢⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⣤⠴⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣃⠀⠀⠀⣠⣺⣵⠾⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣀⣀⣤⣤⣤⣴⣶⠎⠉⢷⣤⣾⣿⢟⡁⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⠀⢀⣰⡾⣿⡿⡆⠀⢸⣿⣟⣩⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠊⠄⠇⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠈⠈⢿⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⠇⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣼⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⠀⠛⠿⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣠⣄⣀⣈⡑⠠⠀⠀⠀⠀⠀⠐⢀⣴⠏⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢽⠀⠹⣦⡿⠓⠼⠖⠀⠀⣮⣿⠔⠈⠀⠀⠀⠂⠺⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠄⡲⣶⡞⠉⢱⡶⠚⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢹⡀⠀⠀⢈⣧⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠳⠆⠴⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⢀⡴⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣀⣤⠶⢋⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣠⡾⠋⢀⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⢰⣿⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠙⢷⡄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⢀⣼⠇⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⣠⡿⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡵⠚⣁⣠⣤⣴⣶⣶⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⠲⠶⣦⣤⣀⠐⠲⣄⠙⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⡾⠃⣿⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠀⢀⡿⠒⢶⡄⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⠃⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⢿⡟⠀⠀⣴⠋⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⢀⡞⣀⣤⠞⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠉⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠙⣷⡴⢶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⣤⣤⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢀⡾⠃⣠⠏⣸⣻⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣸⠃⣰⢿⣞⣋⣥⡤⠶⠶⣻⣷⣀⣀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠸⣤⣏⣸⣿⠷⠶⠶⠞⢛⣽⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢮⣭⣋⣉⠳⠶⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢴⣦⠈⡄⠀⠀⠀⣶⣶⠏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣀⡠⣶⡀⠀⠀⠀⠀⠀⠀⠛⣷⡈⢀⢀⠠⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣉⠝⠻⢸⣷⠀⠀⠀⠀⢡⣠⣿⣨⡿⣦⣹⣾⣔⣾⠇⣿⠆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢻⠟⣻⡛⢤⠀⠉⠀⠈⢳⡄⠉⠛⢿⣿⣷⣬⣿⣿⣿⣷⣾⣯⢠⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣸⠼⠿⠃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⠛⠛⢿⣿⣿⣿⣿⣾⣿⡿⢂⡔⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠐⠈⠉⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠿⣿⣿⣿⣿⡿⠃⡷⠷⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⣀⡀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⣧⢰⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠒⠀⠉⠁⠀⠀⠀⠀⠈⠀⠁⢉⣤⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠅⢼⣿⣿⣿⣿⣿⣿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢄⣉⠐⢶⣄⠀⣠⠀⠀⠀⣀⣾⣿⠀⠐⠀⠀⠀⠁⠀⠄⠉⠀⠢⠹⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠣⡀⢄⠈⠑⢬⣷⠆⢴⣻⡟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⢡⡄⡀⠀⠉⠩⠀⡄⡣⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠃⢀⠈⠜⣧⢀⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠖⢹⣟⠈⢀⠀⠐⠀⠁⢀⢰⣿⣿⣿⣿⣿⣿⡿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠣⣼⣿⣇⣤⣌⣦⣤⣠⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⣿⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢆⠀⠶⠾⠁⠊⠀⡻⡿⠿⣿⣿⣿⣟⣿⣿⡿⣖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠈⠙⠈⠓⠋⠀⠈⢸⠁⢙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



⣿⣿⡿⠿⣿⠿⢿⠿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠲⠶⠶⢤⡀
⠀⠀⠀⠀⠀⣠⠔⠁⠀⠈⠁⢓⡤⢔⠦⡄⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢀⡀⢠⡄⢀⠈⠳⣄
⠀⠀⠀⠔⠁⠠⠀⣀⢀⣠⠴⠋⢀⣤⡗⡏⢛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⡉⠻⣄⡙⢌⣧⢛⣠⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠋⢀⠶⡀⠀⠑⠀⡷⠁⠽⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⣘⡳⣌⠳⢸⣇⢊⣥⠀⣀⢙⠆
⠀⠀⠀⠀⢀⠖⠋⠁⡸⢡⡀⠳⠾⡓⢠⣅⠔⣂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢀⡈⠙⠊⢕⢾⡧⠊⢣⡼⣁⢮
⠀⠀⠀⢀⠏⢀⠤⠔⡡⠃⠈⠳⠴⠚⠳⣇⠴⠭⢿⠏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡳⠼⣶⣝⢬⡟⢁⣚⠴⢃⡎
⠀⠀⠀⣮⠔⣋⡴⢊⡔⠋⠑⠈⠁⣀⣤⣇⡴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣾⠗⠶⢭⣺⡗⢩⠟⠒⠋⠁
⠀⠀⠀⠐⠿⠛⠉⠉⡀⣀⡀⣠⠴⠃⢀⠟⡠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣄⠀⠀⢹⡟⠁
⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⠗⠃⢀⣠⣾⢫⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣟⠷⣄⣸⣇
⠀⠀⠀⠀⠀⠀⠉⠉⠫⠶⣷⣾⠿⠿⢛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠿⠿⠯⠍⠙⠓⠢⠉⠛⠻⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣋⣓⠖⢋⣿⢧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣱⠯⠱⡶⠶⣶⣤⡤⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⣀⡠⣳⠋⠀⠹⢱⣷⣋⣉⣯⣍⣙⡻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠵⠀⠀⠃⠄⠀⠀⠐
⠀⠀⠀⠀⠀⢠⡟⣰⢃⠂⣾⢣⡿⢈⢉⣉⣛⣋⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠂⠐
⠀⠀⠀⠀⢠⢟⡴⡡⠃⣠⢞⣾⡿⠖⠋⣅⠀⢌⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠢⢄⡃⠀⠈⠃⠐⠉⠉
⠀⠀⠀⡠⣻⢋⠔⢁⡴⠓⢁⣸⣦⠞⢓⢦⣑⢄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢡⠖⠢⡤⡗⣄
⠀⠀⠀⠈⠁⠀⠀⠉⢀⠀⠉⣨⣯⣎⢀⡀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢔⠕⠒⠋⢀⡃⠀⠉⠁⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠴⠛⠂⠀⠀⠈⠉⠁⠢⠀⠄⠈⢑⢄
⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⠯
⠀⠀⠀⠀⠀⠀⠨⣿⣯⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣾⣿⣿⣭⣄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⠯⣂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣌⣛⣾⣿⣿⣿⣷⡂
⠀⠀⠀⠀⠀⢠⣽⣿⣿⣋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⣈⡹⣿⣿⣿⣿⣯⣤⠄
⠀⠀⠀⠀⠀⢒⣾⣿⣿⡿⠎⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣷⡰⣮⣙⣿⣿⣿⣿⣿⣍⡴⠄
⠀⠀⠀⠀⣘⣻⣿⣿⣿⣟⣉⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣶⣬⣿⣿⣿⣿⣿⣿⣥⣶⡆
⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⠷⠖⠒⠒⠀⠀⠀⠒⠊⠁⢀⣀⣶⣿⣷⣚⣿⣿⣿⣿⣿⣿⣿⣭⣇
⠀⠀⠀⢀⡘⣿⣿⣿⣿⣷⡧⠖⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢉⣻⣿⣿⣿⣿⣟⠛⠁⡀⠀⡀⠀⠀⠀⠀⠀⢀⠠⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡁
⠀⠀⠀⠈⠉⠉⠻⣿⣿⣿⣿⡛⠋⠀⠀⡷⢰⡀⢰⠀⡇⣸⣖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣾⣍⡇⢰⣶⣷⣾⣷⣾⣤⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡲⣿⣿⠀⠁⠉⠁⠀⠀⠀⠀⠘
⠀⠀⣰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢻⠏⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⡠⠊⠁⠈⠠⠀⠀⠀⠰⠆⠀⠀⠀⠀⡀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣴⡿
⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠁⡀⢀⡀⢀⡀⡄⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣦⣾⣿⣿⠇
⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠈⠉⣹⣿⡟⢿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⣘⣦⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⣀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠈⠂⠀⠀⠐⠘⠲⠄⠀⠀⠀⠀⠤⣸⣿⣿⠀⣿⣀⣠⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠾⠿⢿⣶⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡌⠀⣀
⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⣿⡄⢀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠈
⠀⠀⠚⠦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⡴⠚⠡⠀⠈⠳⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢀⣀⣤⣀⣀
⠀⠂⠀⠀⠀⠀⠁⠂⡀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠙⠐⠂⠄
⠀⠀⠀⡄⢀⢀⡀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢰⣧⣤⣴⠿
⠤⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣆⠀⠀⠃⠀⠀⣠⣔⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠉
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢄⡀⠀⠙⣿⣧⣀⡄⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠋⠉⠙⢦⡀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣤⣈⡻⠏⣁⠺⣿⠖⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠤⠒⠒⠻⠟⠛⠛⡿⡿⢉⡤⢺⣿⣏⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠲⠍⠀⠀⠀⠀⢀⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠈⢹⣿⣷⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠚⢯⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠓⢠⣴⣟⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⣰⡿⠀⣴⠦⣶⣎⠃⢋⣽⣿⣿⣿⣿⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢼⣷⣳⡛⠁⣆⡜⠀⠏⠋⣠⢸⣿⣿⣿⣿⢯⡿⠂⢹⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠆⣎⣿⣿⣠⠾⠿⢁⣴⠆⠀⠏⠆⣿⣿⡿⣦⣾⠁⢠⣿⣿⠿⣿⣆
⠀⠀⠀⠀⠀⠀⢴⣿⣿⣰⣿⣿⡿⣿⣶⡀⠘⢃⡴⠀⣤⡲⢿⣿⣿⡿⠋⠀⣼⣿⣿⣿⣿⡛⣦⡀
⠀⠀⠀⠀⠀⣀⣼⣯⣹⣾⣿⡿⣿⣼⣿⡻⣦⣿⠧⠀⣹⣿⣵⠿⠋⠀⠀⢺⣿⣿⣿⣿⣿⣷⡼⣷⣄
⠀⡀⠀⣤⣾⣟⣿⢿⠿⣿⣿⢻⣾⡝⢻⣷⣾⢷⣠⣦⡿⣋⡅⠀⠀⠀⢀⣾⣻⣿⣿⣿⣿⣿⢿⣬⣿⣷⡀
⠀⣠⣾⡿⣷⣾⣿⣿⣶⣿⣿⣿⠙⣿⣾⣿⣮⣧⣹⣷⡏⢿⡶⢀⣄⣴⣿⣿⣿⣿⣿⢿⣿⣿⣜⣿⣾⣿⣿
⢸⣿⣯⣿⡿⣿⣿⣿⠸⣿⣿⡏⠀⢽⣿⣿⠃⣿⣿⡿⠀⢸⣷⡟⢹⣿⣿⣿⣿⠃⣿⣿⣿⡏⣿⣿⡿⢻⣿
⢸⡟⢹⣿⠇⢸⣿⠇⠀⢻⣿⠁⠀⠨⣿⡟⠀⢸⣿⠇⠀⠈⣿⠁⠐⣿⠇⢻⡿⠀⢸⣿⣿⠁⢻⣿⠇⠘⣿
⢸⠇⠘⡿⠀⠀⣿⠀⠀⠘⡏⠀⠀⠀⢸⠃⠀⠈⣿⠁⠀⠀⠁⠀⠀⠛⠀⠘⠃⠀⠀⣻⡟⠀⠘⡿⠀⠀⢿
⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠉⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢸⠀⠀⠀⢠⣤⠠⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠀⠀⠀⡏⠁⢠⡀⠀⠀⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢾⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡈⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠉⠂⠀⠀⠀⠀⠀⡡⠀⠀⢼⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠐⠲⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠⣼⠟⠁⠰⣒⡒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠦⡄⠀⠣⣇⢀⡀⣤⣀⡼⠧⢶⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣂⣈⣀⠰⠋⠰⡆⠐⡠⠠⠌⠟⠀⠹⠈⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠉⠀⠀⠔⠂⣀⢙⣆⠀⣴⢶⠾⠷⠶⠒⢀⣤⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠿⠟⠒⠉⠁⠀⠀⠐⠂⠑⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⠼⠿⣿⣿⣷⣬⡿⣿⣿⣯⣥⣿⣿⣧⢧⡾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣏⣽⣸⡷⢴⠀⠀⠀⣀⠀⢀⠈⣥⠄⠀⠀⠄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⢋⣿⣷⣯⣴⢀⣤⡄⠀⠀⠀⡙⠀⠀⢀⠀⠀⠀⠀⠀⢈⠻⠦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣹⡿⣼⣿⡿⡟⢁⡈⠡⠄⠦⡄⠂⠛⢓⣤⠈⠀⡤⠙⣦⣀⠀⠒⡾⠁⠀⣴⣾⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⣶⡄⠀⣀⣹⣿⣧⣿⣽⡷⣁⣩⡀⢼⡿⠀⠀⠀⠀⠈⡁⠀⢠⣴⠿⡧⢀⣀⠀⣀⠰⣤⣿⡛⣿⠿⠓⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀⢛⣣⡾⣧⣸⣿⣿⢿⣿⣿⣶⣾⢟⣋⠛⠃⠀⠀⢀⠀⠐⣞⣿⣚⢫⣽⣡⠀⠄⣿⣠⣼⣶⣿⣿⣿⣯⣰⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⡸⢿⣦⣫⣥⡴⣿⣿⣿⣿⢿⣟⣧⣟⣶⣌⢠⣍⣣⠀⠈⢇⢀⣿⣿⣿⣿⣿⣉⣷⣶⢶⣿⣿⣿⣿⣿⣿⡿⠋⢿⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠸⣿⠛⣏⣻⣤⣼⣿⣻⣏⣩⣿⣬⠉⣹⣿⣦⣽⣄⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⡈⣾⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠳⠿⡶⣯⡟⢲⣷⣿⣿⣭⣿⣏⠛⣷⠞⠃⣀⡺⠁⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣮⡉⠻⠇⠻⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⣘⠀⠀⣽⡟⣿⢿⣿⡟⢿⣾⡇⢀⣤⠯⡰⠼⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣲⠖⢒⠈⢰⣾⣿⣿⣗⢈⣷⣥⠿⠋⣠⡤⢸⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⡳⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⠛⣏⣴⣀⡸⠛⢹⣿⣿⠉⡁⠀⠀⣴⣦⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠿⠉⠁⠀⠀⠀⠘⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣾⠏⠑⢫⢹⣿⣇⡴⢯⡿⣥⢦⠉⡀⢠⣿⡿⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣊⣤⡀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⢿⣿⠇⠸⠃⣯⠀⠀⡀⢈⣿⣿⠶⠌⢿⡟⢿⣿⣏⣿⣿⣿⣿⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⣄⠠⠂⠀⠀⣿⠀⠈⣽⣿⣿⡟⠀⡀⠀⠻⣈⡿⠐⠛⠀⠤⠀⠀⠀⠉⠉⠓⠲⣄⠀⠀⠀⠘⢠⡄
⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠿⢈⠀⠀⠀⠀⣿⠀⣰⠋⠀⠁⢤⠀⣀⠀⢴⣿⡄⢀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⣾⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⣿⠉⠀⠀⠀⠀⠈⢧⠀⠐⢋⡄⠀⠀⠀⢹⣶⠃⡀⠂⠠⠲⢷⣾⠀⡸⣥⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠿⠆⠀⠀⠀⢀⠀⣦⣤⡏⠁⠀⠀⠀⢸⡟⠀⠠⠄⠀⠀⠄⣿⠈⢟⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿
⠀⠀⠀⠀⠀⠀⠰⠄⢁⣦⠀⠀⠓⠀⣰⣾⡿⠂⠀⠙⣿⣾⠀⠀⠀⢸⠇⠀⠀⠀⠈⠐⠄⠹⣆⢸⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠁⠀⢁⡴⣟⣿⣦⠀⠀⣀⠈⠁⠀⠀⠀⠀⣿⣿⡇⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢫⣄⣿⣹⣯⣴⣆⡉⠉⠀⠀⣤⣀⢘⣿⣿⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡄⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⡆⠙⣦⠦⠈⣻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠩⡄⠘⠃⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣽⣿⣿⣿⣿⠿⠶⠆⠀⠹⡿⣿⣿⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠁⠀⠀⠀⠀⠀⢰⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⡀⣦⠤⠙⠂⠠⣿⣯⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⣾⠀⠀⠀⢐⣃⡀⢀⣿⡟⠀⢈⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡹⠿⣿⡆⠀⠀⢀⡾⠁⠀⠀⠀⠈⠁⠀⠈⣿⣿⡀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢏⣘⣿⣿⣤⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠘⢦⣿⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣩⠌⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠹⡇⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⣿⣧⠀⢰⡇⠀⠀⠀⠀⠀⣀⠀⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣞⣿⡇⠀⠇⠀⠀⠉⠀⠀⠀⠀⠀⢰⣿⣧⠀⠸⡇⠀⠀⠀⠀⠀⣟⠘⣷⠀⠀⠀⢀⣿⣧
⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢬⣿⡇⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠐⣿⣷⠀⢈⡗⠀⠀⠀⠀⠀⡏⠀⣿⠄⠀⠀⠠⣿⣇
⣿⣦⣟⣿⣶⢤⣶⡄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⡇⠀⢰⡀⠈⠀⢸⠀⠀⠀⠀⠀⣿⣿⡇⠈⣿⡀⠀⠀⠀⠀⠋⠀⢿⡀⠀⠀⢸⣿⣟
⣿⣿⣿⣿⣿⣾⣿⣇⣼⠂⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⡄⠀⠈⡁⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⡇⠐⠛⠁⠀⠀⠀⡄⠀⠀⣿⡏⠀⠀⣿⣿⠋⢀⡀
⢹⢿⣿⣿⣿⣽⡏⠳⣾⣟⣢⢤⣄⣼⡍⠛⠠⠀⠀⠀⢛⣲⠀⣿⡇⠀⠀⠇⣠⡀⠀⠀⠀⠀⠀⠀⢿⡿⠃⠀⠀⢀⣰⣶⣆⠀⠀⣠⣟⣿⣿⡟⢹⡇⠉⠈⠓
⠀⢠⡸⣿⡷⠉⡁⢘⠻⡇⠀⠘⠋⢀⣀⣰⡶⠀⠀⠀⠸⠉⣿⣿⣇⠀⠀⠜⠋⠶⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠘⠃⢿⢻⣶⣿⣿⣿⢿⣿⣿⣏⣷⣿⠀⢈
⢀⡢⠤⠀⠀⠈⠁⠈⡃⣁⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⢉⠀⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⡌⠡⠀⠈⠀⠀⠀⡈⢅⠉⠙⡼⠛⠛⠉⠀⠁⠄⠸
⠀⠙⠀⠀⠀⣎⠀⠀⢡⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛⣶⡄⢀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠁⠀⠐⡀⢠⠖⠀⡞⠛
⡁⠀⢀⡠⠴⡄⠃⠀⣘⠀⠃⡈⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⢜⣿⠉⠁⢀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂
⠀⠀⠈⠀⠐⠀⠀⠁⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⡆⠀⠀⠀⠀⠀⠲⠁⡄⠃⠀⠀⠤⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠰⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡤⠚⠁⠀⠀⠀⠀⠊⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠛⠛⠉⠉⠉⠉⢹⣏⠉⠛⠛⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠆⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠁⠀⠙⣷⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣴⠿⠁⠉⣻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⢿⣆⠀⠀⠀⣠⡿⠀⠀⢀⣤⣶⣶⣦⣄⠉⠉⠀⠀⠀⣾⠏⠁⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠙⠻⠿⠟⠛⠁⠀⢀⣾⠋⠀⠀⠈⢹⣧⠀⠀⠀⠘⣿⡀⠀⠀⠀⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⢿⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣿⣄⣀⣀⣀⣸⣿⡀⠀⠀⠀⠈⠻⠶⣶⠾⠟⠁⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⣀⣿⣧⣶⣶⠿⠿⣿⠛⠛⢻⡟⠛⠛⣿⠟⠛⠛⢻⡟⠻⠿⢷⣶⣶⣤⣄⣀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡿⠿⣿⡉⠁⢻⡄⠀⢻⡀⠀⠘⠇⠀⠀⠹⠀⠀⠀⢸⠃⠀⠀⡾⠀⠀⢨⡟⠛⠻⣷⣦⣄⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠈⠳⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⡼⠋⠈⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣯⡉⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣾⣇⣀⡀⣠⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠰⣄⢀⣘⣧⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣷⣿⡟⠀⠀⠀⠀⢀⠈⣿⣷⣾⣿⡿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠸⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠈⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⢰⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⡄⡸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡗⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⣸⢱⠁⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⢰⠃⣼⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⢀⡏⠀⠃⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠈⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣄⣀⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠳⣦⡀⠀⠀⠀⣠⣴⣿⣿⡿⠂⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠈⠻⣶⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⡄⠈⢻⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⡠⠀⠀⠀⡰⠟⣫⣿⠟⠃⠈⣿⠀⠀⠀⠀⠘⣆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⣰⣾⣿⣷⠟⠁⠀⠀⠀⣠⣾⠟⢁⠔⠀⠀⠸⠀⠀⠀⠀⠀⠘⣷⡀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀
⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⢰⡟⠁⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣄⠀⠉⠛⢷⣦⣄⡀⠀⠀⣠⣶⣿⣿⠋⠹⣿⣄
⠀⣠⣾⣿⡿⠋⠟⠁⠀⠀⠀⠀⠀⠸⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠈⠀⢉⣷⡾⠛⢉⡿⠁⠀⢷⡙⣿⣷⡀
⡾⠛⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠈⠁⠀⠀⠀⠻⡜⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣧⠀⢀⡀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⠀⠀⣧⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠶⠶⠾⠿⢤⣼⣇⣀⡀
⠀⢸⣇⠀⡆⠀⠀⠀⠀⢰⠀⣾⣷⠀⠀⣀⣾⣦⣼⡿⠶⠶⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓
⠤⢿⣿⣼⣿⣤⣤⣤⠤⠾⠶⠿⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠋⢻⠿⠋⠀⠀⠉⠻⢿⣟⣿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠀⠛⢻⠉⠉⢹⡴⢼⡉⠉⢹⠖⠉⠉⣴⠋
⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⢤⡀⠈⢿⠉⢙⠀⠀⠉⠉⠉⠀⢠⠄⣯⡀
⠀⠀⠀⠀⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠳⣄⠠⠤⠄⠀⢀⡴⠋⠀⢿⡎⡆
⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠁⠀⠀⡠⠜⠛⠈⢭⡥⠟⢀⣤⠔⠊⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠢⡇⠀⠀⠀⡔⡛⠀⠀⠀⢀⣀⡿⠆⡈⢸⠀⢀⡠⠊⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢘⠣⢄⣰⣶⠚⢬⠀⠸⡇⠸⠖⠉⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⡀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠁⠉⠊⠳⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠿⠀⠀⠀⠠⣸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡇⡔⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⠀⡖⠤⢤⣃⣀⣀⠀⠀⢀⣃⣈⣀⣀⠀⠀⢸⡤⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⠬⠦⠀⠘⠆⠐⢯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⢤⢤⣃⣀⡸⠄⠀⢠⠤⣀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠈⠉⠉⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣄⣀⣀⣀⣤⣼⣀⣀⡀⠀⠀⠀⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡤⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢻⣷⠈⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⢺⡀⠀⠀⣿⠟⠁⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢸⡇⠀⣀⠀⠀⡀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⠀⠀⠠⣿⣿⣿⡿⠀⠸⢇⣀⣿⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⢿⣿⣿⣧⢰⡿⠞⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣶⠀⢸⣿⣿⣇⠘⠁⣠⠖⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⡤⠀⠻⣿⣿⣶⣶⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣴⣿⣿⣿⣧⣿⣸⣤⣾⠿⠋⠉⠁⣾⠿⠿⠃⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣼⡟⠃⣿⠇⠐⠟⠃⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠈⢻⣿⣷⣾⣄⢁⡀⠲⠀⠿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣇⠀⣱⡟⠃⠉⠉⢙⣤⠜⢺⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡿⠷⠾⣷⣆⣹⣧⣼⣿⣿⣿⡿⠿⠟⠛⠃
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠘⠟⠙⢣⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⢸⡇⢠⣾⡅
⠀⠳⣄⡀⢸⣿⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⢸⣿⣾⠟⠁⢀⡄⠀⣠⡆
⠀⠀⣿⡇⠘⣿⡁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣽⣧⡴⠞⠋⢀⣶⡟
⠀⠀⠉⢁⡴⠈⠉⠀⠀⠁⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠻⣿⣿⣿⣿⡟⠉⠁⣠⠶⣠⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠑⣄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣆⠈⠻⣿⣿⣷⣮⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⠃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⣿⡇⠀⣹⠿⠟⠋⠉⠀⠀⡀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡜⢷⢀⣿⠀⠀⢠⡤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⢁⣴⡗⠉⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠉⠀⠀⠀⠠⣤⡀⠀⢸⣿⣿⠉⢿⣿⣯⡘⠻⣧⣴⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⣨⢻⣶⣌⣿⠃⠀⠀⠹⠁⣠⡾⢛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣿⣿⣠⣆⠀⠀⠿⠄⢹⣿⡏⠀⠆⠀⠀⣴⣟⣵⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣆⡀⠀⣤⡀⢻⣿⠀⢺⡆⠀⣼⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠻⣿⠉⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡟⠉⠉⠉⠉⠀⠹⣿⡾⣿⣇⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⢰⣦⣤⣄⠀⠈⠁⢸⣿⣤⣿⠇⠀⠀⢀⣤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣰⠁⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⢻⣿⣿⣷⣤⡀⠈⠻⣿⠇⢀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡇⢰⡿⠻⣿⣿⡿⢡⠀⣿⢠⡜⢣⣾⠏⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠜⠁⠀⠘⢿⠇⢸⡄⠸⣿⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠈⠁⠀⣴⠁⠈⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⢠⠆⠀⢀⣤⡀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠙⢧⡀⢰⣈⣷⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⣾⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣶⣼⣿⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠀⠀⠀⠺⣿⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣏⠈⣿⣿⣇⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⠈⣤⠀⢀⣴⡀⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⡀⢃⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⢷⣄⡹⡀⢸⣿⠇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠘⡷⠗⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣤⠀⠀⠀⣉⠛⣿⠈⠻⠀⢈⡟⠀⠀⠀⠀⠿⠃⢠⠀⣀⠀⠀⠾⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣘⠿⣤⡄⠀⠀⠸⠟⠀⠀⠀⠀⠀⡂⠁⢠⣿⡆⠘⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⣿⣷⣽⣿⣴⡀⢰⡆⠀⠀⠀⢠⣾⣿⣿⠀⢻⣿⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠈⠻⣿⣿⣿⠀⢸⡇⢠⡆⢠⡿⠋⠈⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠻⣿⣤⣤⡆⢀⣶⠏⠀⣶⣾⣿⡀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠈⠙⣿⠁⠈⣡⡄⠘⣿⡇⢀⠁⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠸⡀⢀⣿⡇⢸⣿⣇⡈⣡⡭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⢰⣧⠘⣿⡇⠈⢻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣄⠀⢸⣿⠀⠸⠇⠀⠏⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣿⣿⣿⢈⣿⠀⠀⠀⠀⢸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢻⣿⣇⠈⠸⠿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⢹⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣼⣿⡀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⣰⡿⠟⠀⠀⠀⠀⠀⢀⠀⠠⠀⣠⡄⠀⠀⠀⠀⠠⣿⠿
⡄⣼⠁⠀⠀⢀⣤⠎⠀⠀⠀⢀⠀⣠⣾⡟⠁⢘⣿⠇⠀⠀⣇⠀⠀⠀⠠⢀⠀⠀⣿⠇⠀⣠⣴⣆⠠⠄
⢷⣿⠀⣼⠀⡄⠻⣆⠀⠁⠀⠉⠀⠘⣿⣤⣾⣿⣧⣀⠀⠀⠙⠀⠀⠀⠀⠾⠇⠘⢁⡤⠚⠉⠉⡉⠀⢀
⣻⣿⠈⢿⣸⡇⣤⠸⠿⠄⡀⠀⠀⠀⠈⠙⠋⠛⠁⠉⠆⠀⡆⠀⠰⠇⢀⠀⠀⡄⡀⠀⣞⢠⢠⡇⣤⣾⢸
⢿⣿⣦⣾⣿⣇⣿⢀⣄⣆⣳⣦⡄⠀⣤⠀⢠⣴⣤⣄⣠⣄⡀⠀⣦⣦⣼⣾⣼⣿⣿⣿⡏⣻⣿⣿⣿⣿⡸
⠀⠀⠀⠀⠉⠛⠻⠟⠃⣉⣉⡀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠉⠄⠀⠈⠋⠛⠛⠛⠋⠉⠻⠷⠉⠉⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠠⢤⣶⣿⡃⠀⠀⠀⠀⠀⠐⠲⠄⠀⠉⠐⠒⠒⠻⠷⠶⠶⠦⠤⣤⣤⠶⠖⠻⠿⣿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢀⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣇⡀⠀⢙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⠒⠤⠤⠄
⠀⠀⠀⠀⣠⡄⠙⣶⣦⣾⣿⣿⠿⢿⣿⣿⣿⡷⠴⠓⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣉⣛⣿⣧⣈⣁⣀⣈⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣄
⠀⠀⠀⠀⠠⠿⠛⢻⣿⣿⡗⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⣿⣿⣿⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣷⣶⣶⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠷⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠒
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⠃⠀⠰
⠀⠀⠀⠀⠀⢠⠀⣠⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⢺⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡴⠊⠁
⠀⠀⠀⠀⣰⠏⠀⠆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣤⡿⠏
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠨⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⢡⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⣿⣯⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⢷⣦⣄⡁
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣧
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠘⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡤⣶⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠐⠀⢉⠑⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠫⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠈⠄⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⢀⣄⣸⣃⠀⢀⡠⣀⡠⣀⣀⠌⣀⣀⡠⣀⣠⣀⣌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⢛⡉⠉⠉⠈⠉⢻⠀⠈⠉⠀⠀⢀⣋⣀⣠⠠⠭⠝⢛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⣏⠤⠤⠢⡤⢄⣈⡇⠀⠀⠀⠐⢈⠀⠀⢀⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⣠⠤⣶⠤⢤⣤⣄⣀⣈⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢣⠀⠱⠊⠉⠙⠢⢬⠀⠀⡏⡉⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢀⠏⢠⡢⠀⢀⣤⠴⠆⣧⠀⣷⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣠⢀⠎⠀⢆⡤⢾⡇⠀⢷⣳⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠐⢁⡎⠀⠀⠈⠒⠤⡇⠀⢸⡃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡀⠀⠀⠀⠀⡘⣇⠀⠀⠀⠀⢠⠇⠀⠘⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡃⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠸⣤⣄⣠⢇⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⣠⡞⠁⣼⣿⣷⣄⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢊⡠⣾⡷⢦⣤⣲⠞⠋⠀⠀⠻⣿⣯⠙⠛⣲⣦⣾⡁⠁⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠾⡿⠟⠋⠉⡠⠶⠖⣻⣷⡄⠈⠉⠀⣼⣿⡿⣷⡮⠀⠘⠋⢙⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠙⠓⠍⠀⠀⠀⠀⠈⠉⠺⣇⠢⠔⠀⠘⣿⣿⣿⡷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⠀⢀⢀⣉⢀⣀⡆⠀⠀⣀⣀⡀⠀⣀⡀⠀⠀⠀⠈⢀⣠⣄⠀⠀⠉⠙⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠉⠭⠿⠿⠿⢟⣛⡿⠿⣿⣒⠛⠛⠉⠉⠛⠛⠛⠛⠛⠩⠯⠽⠿⢟⣓⡒⠿⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⠒⠫⠭⢝⣓⠢⠄⠀⠀⠀⣉⣉⣉⣩⣭⣐⣒⡢⠤⠄⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠒⠒⠒⠒⠒⠛⠛⠂⠉⠀⠀⠀⠀⠀⠉⠉⠉⠙⠂⠤

⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⣴⡛⢆⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⣦⡀⢤⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⡇⢀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⣰⡋⠔⡉⣸⣧⡚⠃⠀⢿⡦⢀⠀⠀⠀⠀⠀⠀⢚⠀⠣⠙⣇⠈⠉⠷⢯⠀⢿⢀⠀⠀⠀⠀⠀⠀⣠⠎⢈⠷⠄⠈⠀⣰⠏⠇⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣆⠐⠾⣿⠈⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣏⣷⣔⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣿⠀⢶⣿⡁⠢⣠⣶
⠀⠀⠀⠀⠀⠰⠦⡽⠿⠿⠼⠿⠶⠾⠿⠶⠯⠤⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠦⠀⠀⠀⠀⠀⠐⠛⠚⠛⠛⠛⠛⠛⠒⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢠⣖⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡂⡀
⠀⠀⠀⠀⠀⠀⣀⢼⣇⡈⢿⣿⠑⢾⣇⡑⢄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡞⢿⣿⣶⣥⣾⣓⠤⡀⠀⠀⠀⠀⠀⠀⣠⢠⣾⠃⢿⣿⣧⠈⣤⡄
⠀⠀⠀⠀⠀⠘⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠾⠿⠿⠿⠿⠿⠷⠿⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠶⣀⠤⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⡀
⠀⠀⠀⠀⠀⠀⡄⢤⣾⡇⠹⣗⣀⣓⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠏⠛⠃⠐⠻⡦⠌⢀⠿⢤⡀⠀⠀⠀⠀⠀⠀⣀⢤⣴⡇⣻⣿⠀⣼⣻⡄⢀⡀
⠀⠀⠀⠀⠀⠘⠙⠛⠋⠛⠛⠛⠙⠋⠛⠙⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠋⠛⠛⠛⠉⠉⠛⠛⠙⠙⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⢦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣯⠑⣄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡎⣅⢤⡤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⡆⠹⠿⢭⠀⠐⢋⣈⠦⢤⡀⠀⠀⠀⠀⠀⢀⣤⣾⣦⠻⢿⣿⣦⠚⢿⠧⡳⠄⠀⠀⠀⠀⠀⠀⢤⠚⠀⠙⠡⠞⠈⠡⠿⠛⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⢾⣇⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⡀⠀⠀⠀⠀⠀⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠰⢿⣆⣀⡀⢴⠇⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠴⠛⢁⡈⣀⠀⠀⠀⠀⠀⣿⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠠⠀⠀⠘⡖⠀⠀⠀⠀⢢⠃⠀⠀⠀⠑⠀⠀⠀⠒⠀⠀⠀⠀⠀⣺⢉⡁⢁⠀⠀⠀⠀⡏⠀⠀⢰⠉⢳⠀⠀⠀⠀⢀⡤⠴⠈⠉
⠉⢳⠀⠀⠀⠀⠘⣇⠀⠀⣼⠋⠀⠀⢀⣀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢎⠀⠈⠀⠼⡇⠀⠀⠀⠤⠀⢰⠉⠀⠀⢱⠀⣰⠋⠩⠬⢀⣈⡀
⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢁⠀⠀⠀⠊⣿⠳⠄⠀⡎⠀⠚⠃⠀⡏⠀⠀⠀⠁⠀⡜⠀⠀⡆⢸⠙⢄⠀⠙⠋⠁⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠅⠀⠀⠀⠀⡀⣿⠀⠀⠀⠘⢏⣀⣀⣀⡸⠀⡠⠒⣆⠀⢣⠀⠀⠁⢀⡇⢀⡀⠀⢀⠴⢤⡀
⠀⣀⠀⠀⠁⠀⡴⠒⠲⣀⡀⠀⠀⢠⣿⡍⠉⠀⠀⠀⠀⣿⠅⠀⠀⢀⡀⠈⠁⠀⠀⡎⢁⠀⢙⢲⠀⠉⢹⡏⠉⠠⣗⡺⠀⣡⢞⡦⡉
⣾⡏⠀⠀⡤⠼⠀⠀⢄⢭⣙⡤⢤⡈⠙⡗⠂⠁⠀⠀⠈⢸⠀⠀⠰⠁⠛⠉⡇⠀⠀⣆⠀⠀⣀⢸⠀⠀⠸⠇⠀⠀⠀⠀⠀⢔⣽⣕⠄
⣾⡇⠈⠀⡇⠀⡠⠔⠚⠮⡡⠅⠸⠧⡄⠃⠀⠀⠀⠀⠀⡈⠁⢠⣟⠀⠉⠀⠉⢳⠀⡜⠀⠀⠀⢸⠀⠀⣄⠠⠂⠀⠰⠇⠀⡨⢺⡎⢅
⣿⡍⠉⠀⠘⠠⠀⠀⠀⠀⠀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢴⡄⠀⠹⡄⢹⡀⠠⠀⣸⠀⠀⡟⠀⠀⠀⠀⠀⠀⢘⠙⡮⡓
⠉⡗⠒⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⢨⣵⠀⠀⠀⢸⠣⠀⠀⠀⠢⡴⠃⠀⠙⠒⠚⠁⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠈⠃
⠀⠑⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⠖⠒⢲⠖⠚⠁⠀⠀⠀⠸⠴⠀⢰⠚⠒⢦⡶⠲⢤⠤⢄
⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡠⡄⠀⢠⠇⠀⠶⠀⠀⠄⠄⢈⢀⡤
⠁⡀⠀⠁⠀⠀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⢀⣀⠊⢀⠐⠀⠂⡀⠀⠀⠡⠞⠤⠂⡌⠈⠀⠙⣆⠈⠁⠀⠈⠉⠉⠉⠉⠀⣻⠧⠠
⢘⠧⢘⣇⠀⠀⢠⠐⠃⠀⠀⢋⠁⣆⠀⠀⠐⣤⡔⣀⠉⢁⣀⣀⡈⢁⠆⠐⠀⠀⢸⠀⠈⢀⡀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠧⠠
⣨⠔⠋⢠⡇⡄⠛⠀⠊⠁⠀⢀⠀⠀⣇⠐⠔⠓⠓⠒⠒⠒⠒⠒⠛⠁⠀⠀⠀⠀⢸⢁⣤⠀⠀⢾⠀⠀⠀⠀⠀⠂⠂⠀⠐⠿⣧⡀
⡀⠐⣀⣈⡶⠀⠀⠉⠉⠀⠀⠀⠀⠐⠂⠃⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡀⠀⠄⣸⠀⣄⠄⠀⠁⠀⠀⠰⠀⠀⠄
⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣡⢶⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣶⠚⠁⠀⠓⣈⣀⠀⠀⠀⠀⠀⠀⡴
⠀⠏⠀⠀⠀⢀⣴⠀⠀⠀⠅⠀⢠⠔⢠⣄⠀⠀⢈⢕⣽⣓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⢀⣰⠁⢼⣤⣐⡀⠀⠀⠀⠄
⠋⠀⠀⠀⠀⠋⣾⢷⣄⠀⠀⠀⠘⠁⠈⠓⢣⠄⠀⡫⣺⣎⠕⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⠀⠀⢣⠀⠙⠀⠤⡈⠉⠷⢲⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠌⣼⡀⠈⠀⠀⠀⢀⣴⠙⡆⠀⠀⠀⡪⢸⡦⠑⠂⢀⣘⡟⣀⣿⠂⠀⠀⠀⠀⢸⠉⠠⠄⠀⠽⠇⠤⠀⠙⠧⡄
⠀⠀⠀⠀⠀⠊⣿⡆⠀⠀⡴⠲⠋⠁⢰⠇⠀⠀⠀⠀⠀⠁⢀⣀⡏⠀⠙⠲⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠁
⢱⠀⠀⠀⠀⢠⢿⠃⠀⠀⢡⡀⠈⣀⣈⣳⠀⣄⠛⡶⢉⠁⢸⠀⢀⠤⢤⣀⡈⢸⣄⠀⠀⢀⣾⢦⡀⠀⠀⠨⠁⠀⠀⠀⡠⠤⠔⠒⠴
⢘⠀⣀⠀⠀⠁⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠀⠰⠛⣻⠀⠉⠀⠀⠀⠀⠀⠂⠀⠄⠐⢂⠀⢒
⠀⢼⣛⠀⠄⠀⠘⠃⠀⠠⡞⣕⠄⠀⢰⡄⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠑⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠟⠿⠿⢿⣿⣿⣿⣷⣷⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢀⠀⠀⠉⠀⠐⠀⠀⠉⠈⠉⠙⠿⠿⠿⠛⠟⠛⠻⠿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⣀⡼⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠆
⠀⠀⠀⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀
⠀⢀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡎⠀⠀⢿⣿⣿⣿⡇
⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣻⣿⣿⣿⡀⠠⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣇⠀⠀⠻⠿⠁⣤⣴⡆⠀⠀⠀⠀⠀⣀⣐⣿⣿⣿⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠙⠋⣀⣤⣰⣶⣶⡿⠟⠃⠀⠀⢀⣤⣜⠉⢽⡷⣿⣋⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠍⠁⢸⣿⣿⠀⠀⠀⠀⠴⣷⣽⡋⠉⠀⠀⠈⠉⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢚⣿⡆⠀⠀⠀⢶⡦⢀⡉⠀⠰⢦⡀⣠⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢹⢿⣷⠀⠀⠀⠀⠀⠉⠁⠠⠄⠈⣃⠙⠃⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠛⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣷⠀⠀⠀⠀⠀⠒⢲⣦⠀⠀⠈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⡄⠀⠀⡀⠀⠀⠾⠃⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢺⣿⣧⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡏⣿⣿⡄⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⣿⣇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⠍⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢷⢫⢺⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⢶⡟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢟⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣽⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⢀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠆⠠⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠛⠁⠉⠄⠂⠀⠄⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠚⣸⢹⣾⠀⠀⠀⠀⠀⢀⡄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢻⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⠄⠀⠀⠈⠣⠀⠀⠀⠙⠂⠀⠀⡀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡀⣀⠀⠇⣠⢀⡆⢼⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣿⣿⣟⣷⠈⢠⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠄⠀⠛⠰⠀⠼⠤⢄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⡻⢇⠼⠁⡾⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠾⣶⠤⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡅⢉⡰⠋⠾⠘⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡞⣷⠒⠃⠀⠠⢀⠴⠀⠀⠀⠀⠀⣿⡇⠸⠗⣁⣤⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣇⣶⣿⣿⣿⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣃⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠰⢫⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡈⠳⣦⣠⡄⣆⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⢵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣟⡷⡂⠉⣉⣿⡿⠷⠞⠋⠉⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠲⠤⠀⠰⣞⢀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⣿⡌⠙⣡⢖⡄⡬⠁⠘⢏⣁⣘⡅⠃⠈⢷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠰⠴⢄⠁⡠⠌⠀⠐⠚⠂⠆⡄⡀⠘⠣⢍⡴⠆⢶⡄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⢶⣧⣀⠶⢴⣠⠁⣋⠀⢦⣽⠀⣉⠀⣅⠹⠦⢨⡅⡔⢋⣀⡛⠁⠻⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⠗⢉⠀⠞⣁⡈⠋⣠⠆⠉⠶⣯⠾⢖⡏⠀⠉⠀⠻⢇⠰⣤⠂⢈⠇⣍⡁⢻⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠣⣌⠀⡄⣼⠓⢇⡄⢰⠆⠲⡟⣻⣾⡼⡣⡴⡃⣴⣘⠗⢋⡀⠡⠶⣈⠿⠄⣤⢻⡠⠄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠜⠃⡛⠒⢧⢁⡇⣤⠡⣀⡴⢚⣿⠋⣄⡈⠀⣧⡶⡴⢫⡆⢸⣇⣳⣠⠛⠂⣷⠈⣀⢡⠴⠂
⠀⠀⠀⠀⠀⠀⣠⣯⣁⠐⣦⡇⣵⠀⠓⢦⣅⡚⣉⢾⠏⢸⣇⢀⣡⠒⣷⣸⠇⣴⢟⡉⢡⠦⡀⠰⠈⣣⠬⠋⠠⠄⢆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠴⣂⠀⡡⠀⡿⠠⠿⠻⣏⣠⢹⡶⢿⡟⢇⡈⢢⠟⣾⠃⣄⡁⣶⣀⠃⣤⠈⠩⢇⢀⣼⡈
⠀⠀⠀⠀⠀⠀⠀⠘⢷⠆⠉⠤⠙⠋⠉⣁⣒⠒⠾⢷⣄⡶⠘⣿⡄⣴⢫⣾⢃⣶⢉⠍⠓⣈⠠⠯⡀⠷⠘⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢹⡅⢘⠋⢳⡎⠑⢯⡭⢈⣽⠾⣇⡙⢷⣦⣿⣿⣿⣟⣡⣤⣿⣿⣚⢹⣧⠐⠛⢿⣠⣤⠀⠉⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠘⢁⣭⠥⠄⣁⠠⠆⢀⠻⡙⠷⠈⠻⣦⡹⣿⣿⣿⣿⠟⢿⡆⠰⠟⢓⠈⠴⠆⢣⡍⣀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⣀⠋⠘⠗⠐⠀⣤⡄⠈⢿⣿⣿⣿⠇⠀⠀⠁⠀⣤⠛⠁⠀⠰⠆⠀⢫⣤⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠉⡀⢠⡜⠇⠀⢐⡗⣀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠵⠆⣠⡄⠁⠀⠀⠶⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠘⠁⠙⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠓

⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⣿⡦⣄
⠀⠀⢺⣿⣿⣯⠀⠀⢰⣷⡄⠀⠀⢀⣴⡿⢹⠀⠘⣿⡈⠑⢦⡀⠀⠀⣀⣴⣿⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠳⣄⠀⡀⠀⣴⡄
⠀⢰⣿⣿⣿⣿⣷⠀⣾⣿⣧⠀⢀⣾⡛⠁⠀⠳⣄⠸⣇⠀⠀⠉⢳⣾⣿⣿⣿⠇⠙⢦⡀⡀⣠⠞⣷⣤⢶⡿⠁⠀⠈⣾⣿⣠⣿⣿⡀
⠀⣼⣿⣿⣿⣿⣿⣶⣿⣿⣿⣴⠿⠿⠿⣷⣤⡀⠈⠀⠙⢦⠀⠀⣸⣿⠟⠋⢸⡆⠀⠀⠙⣍⠓⢄⡘⣿⠳⠅⠀⠀⢀⣿⣿⣿⣿⣿⣆
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⢧⠀⠀⠀⠈⠃⠀⠙⠚⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿
⠈⠀⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠀⠀⠠⠄⠴⠶⠶⠶⠶⠤⢤⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣾⠿⠛⠛⠻⠓⠒⠒⠀⠀⠈⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠲⠟⣛⡲⢖⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠄⠒⠒⠂⠤⠈⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣁⣤⡶⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡞⣾⣾⢛⡵⠊⣀⡀⠄⠀⠀⠀⠀⠀⠄⢀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⠟⡠⠊⢁⣀⣀⣀⠀⠀⠀⠀⢸⣼⡰⠀⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⢫⡿⢈⣴⣿⣿⣿⣿⣿⣷⣄⠀⠈⡟⡏⡇⡆⣿⡇
⠀⠀⠀⠀⠀⢸⢹⡇⣾⢁⣿⣿⠟⠋⠙⠪⣿⣿⣿⡆⠀⠁⠀⠇⠃⣿⣷⠀⠀⠀⠀⠀⠀⠀⢰⠳⣄⠀⣼⠆⡀
⠀⠀⠀⠀⠀⢹⢺⢀⡏⣼⣿⠃⠀⠀⠀⠱⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⠰⣦⡙⢇⣀⣹⣆⣀
⠀⠀⠀⠀⠀⠸⣼⠈⡇⣿⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣄⠘⠷⠈⠁⠨⠉⢙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠱⠇⠀⣿⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⡇⡇⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠐⣨⡙⣦⠹⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠠⣧⢿⠀⠘⡇⡇⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢼⣿⣿⠏⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠀⠀⢰⠀⡄⣼⡿⠁⠀⠀⠀⠀⠀⠀⣰⢏⠁⠀⠀⠐⡄⣠⡁⡀⢀⣀⣠⠀⠀⠀⠘⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠈⡇⠀⠀⠸⢠⣼⣿⡟⠀⠀⠀⠀⢀⣴⠞⢣⣄⣀⣀⠀⠁⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⢻⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢻⣷⠃⠀⠀⢀⣾⣿⡿⠀⠀⢀⡤⠺⣍⣭⣵⣿⣿⣿⣟⠉⠒⠀⠙⣛⣛⣋⣥⣴⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⣿⠀⠀⢀⢸⣿⣿⣇⣠⠾⡩⢂⠳⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡿⠀⠀⣼⣿⣿⣿⠟⡁⣈⠐⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠀⢣⠈⢇⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢠⣇⣿⣿⣿⠋⠠⠀⠐⣠⣾⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣶⠀⠈⣧⠈⠣⣀⠀⢀⢞⡽⠋⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⠃⠀⠀⢀⣾⣿⣿⣿⡿⠏⠁⠸⣿⣿⣿⣿⣿⣿⣧⡀⠈⠳⣤⣈⣹⣟⣿⠁⠄⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⡏⠀⠀⠀⢊⣿⣿⣿⡿⠟⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀⠈⢉⡉⠁⠀⠈⠭⠁⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⡿⠀⠀⠀⢾⣿⣿⣿⡿⠁⠀⣀⡴⠞⠉⠉⠀⠈⠉⠻⡿⠿⣿⣿⣿⣿⡭⢥⣴⣶⣶⣮⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣰⣿⣿⡿⠛⠡⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠾⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇⠀⠀⣨⣿⣿⣷⣶⣶⣦⣤⣶⣞⡀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⠄⣀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣷⠀⠀⠀⢙⣛⣿⣿⣿⣿⣿⣿⣿⣿⢿⡋⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣇⠀⠀⠀⢤⣺⣿⣿⣿⣿⣿⣿⣿⣿⣦⠔⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⣿⣿⣄⠀⠀⠈⣭⣿⣿⣿⣿⣿⣿⠿⣋⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣦⣀⣉⠉⠉⣙⣻⣭⣷⣿⣿⠯⠭⠭⣛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣯⣥⣬⣭⣭⣭⣥⣤⣤⣄⣐⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠻⣈⣹⠂⡄⠀⠄⠀⢈⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡉⣸⣆⠀⠀⠀⠀⣈⣀⠐⢆⡜⠁⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠁⠀⠈⢹⠌⠣⣤⠀⢠⠟⠀⣠⣎⠀⠀⠀⠀⣤⠔⠚⠁⠀⡌⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠄⠀⠀⠸⣆⠀⣿⠀⣾⠀⣾⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠠⠉⢠⠀⠄⠀⠈⣿⠀⣛⣾⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⢀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠰⡄⠘⣿⠀⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡤⠒⠈⠉⠉⠀⠀⠀⠀⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣤⠀⠾⠀⢾⣳⣟⢀⡀⠀⠀⠜⠀⠀⠀⠀⣀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠈⠀⢀⣌⡀⠻⣾⣦⠅⠀⠀⠀⠀⡴⠞⣩⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠻⠀⢣⣴⣛⡏⠀⠀⠀⢀⡼⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣄⠀⢿⣽⣿⡟⠀⠀⢀⡞⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡄⢸⠟⣿⠀⠀⠀⠾⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⠶⠖⠀⠀⠀⠀⡇
⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣼⣄⣿⠇⢀⡜⢠⠇⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠁⣂⣴⠯⣠⠶⠀⠀⠀⠀⠀⢀⠀⢠⠄⠀⢠⠌
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⣿⣽⡆⡸⣀⠁⡀⠀⢀⣠⡴⠛⠁⠀⠠⠀⠀⠀⡐⠚⠉⡁⠠⠈⠀⠴⠛⠀⠀⢠⠛⣀⠄⠀⣰⢿⠆
⠸⣿⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⣩⣤⣾⢡⡏⢠⢁⡾⢛⣉⠤⠴⠃⠀⢀⣠⠴⢧⣴⠦⣄⠤⠔⠂⠉⢠⠆⠀⣀⡴⠰⠁⡤⢺⠃⡟⠠
⠀⣿⠀⣰⠄⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⢸⡟⢿⡟⢠⣿⣾⡗⠋⠀⠀⣠⡤⠚⢁⡤⠦⣻⡶⠓⠊⠀⠀⠀⠀⣈⡠⠚⠏⢀⢁⣼⠁⣼⢟⡿⠁⢠
⠀⣿⣴⡟⡁⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠈⢛⣼⡇⢸⣿⣿⣤⢴⠿⠋⣠⠀⠺⣿⡴⢊⣡⣾⠧⠤⠤⠄⠒⠀⣹⡤⠀⠀⠻⣥⠀⠀⣠⠜⢀⣠⣈
⢰⣿⡿⠀⣧⠂⠀⠀⢀⠀⠀⠀⠀⠀⣰⠀⣬⠛⣷⣿⣿⠏⠀⠀⠀⠀⢠⠀⢰⣿⡶⠉⠁⠀⠀⠐⠀⠀⠀⠀⠙⠓⠀⠀⠄⠱⠂⢀⡏⡰⠳⡏⠄
⢸⣿⡇⠈⡇⠀⠐⠂⠈⢠⠾⡃⣤⢀⠀⣀⢿⣼⣼⣿⡟⠀⡀⠀⠀⣠⣌⡠⠾⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠰⡗⠸⠁⡤⢸⢃⡆
⢸⣏⠀⠀⠀⠀⠀⠀⢰⢘⠀⣻⣆⢠⠀⠀⢸⣗⣿⣿⡿⠶⢷⡄⠠⠟⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡦⠐⠀⠈⠁⠉⠰⠁⠰⠀⠀⢀⣤⣼⡟
⢸⣿⠇⠈⡗⠂⠀⠘⠋⠀⠀⢹⡏⠀⠀⠀⢠⡉⠨⢾⡀⠀⠀⠀⠘⠄⠀⠀⣤⣤⠀⢀⣠⡔⢢⡄⠛⠚⠁⠀⠁⢠⣬⡆⠀⠀⠃⠀⠀⢀⣸⣿⣧
⣞⠋⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡙⠂⣼⡁⠀⠀⠀⠀⠀⠀⠈⠑⢶⠆⣡⣄⣂⣙⢣⣄⣷⡜⠦⠌⣁⡛⢠⡀⡄⠠⠆⢠⠦⢩⡿⣿
⣿⣷⠀⠰⠐⠀⠀⠀⠀⠀⠀⢰⣆⣄⢚⣿⡀⠈⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠖⡿⠉⡉⠉⢃⣱⣾⠁⣼⣕⣿⣷⢎⣴⠋⠀⠈⢈⣿⣿
⠉⣼⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⢀⡈⠈⠉⠁⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠘⠡⣺⣥⡿⣿⣿⢯⣶⣿⡏⣽⣓⠛⠉⣘⠀⢳⡟⢹⣿
⢋⡈⠀⢡⡄⠀⠀⠀⢻⠀⠀⠰⠈⠉⢥⣤⡀⣀⡠⢿⡇⢀⣤⠴⢾⣡⣾⠁⠉⠶⠟⠁⠉⠁⠾⠉⠀⠹⣿⠚⢛⣿⡟⢲⣟⠲⣤⣾⢹⣶⣞⢺⣿
⢸⣷⡆⠺⠇⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⢻⡹⠋⠀⢈⠉⠀⠀⠈⠙⠛⢲⠞⢠⡄⠀⠀⠀⠀⠀⠀⡄⣀⢀⡈⡿⠋⠒⣿⢘⣿⣿⣿⣿⣿⣿⣿
⡀⣩⠁⠀⠐⠀⠆⠀⢸⢀⠀⠄⠀⢠⡤⠂⡠⠀⠘⠓⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⠉⡄⠀⣤⣄⠠⣌⢣⠉⠀⠁⠀⠀⢸⣿⢹⣿⣿⣿⣿⣿⣩⣿
⠀⠉⠀⠉⢃⠀⠀⠄⢀⡁⠀⣀⡰⢦⡀⠀⠀⣀⣠⢀⣀⣤⣠⠀⠁⠀⠉⡠⠄⠀⠀⠁⠀⠘⠛⠀⠘⣷⣷⣦⢶⣤⡾⣶⠹⣾⣿⣿⣿⣿⣿⣿⣿
⢤⣾⠿⣦⡿⡄⠠⣀⠈⣉⢘⡈⠉⠚⠃⣰⣀⠉⣱⢀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢠⡄⢀⠀⠀⢰⣾⠇⠛⠈⠁⠬⠽⠟⠉⢦⠟⢛⣻⠿⠿⠿⢿⠿
⠉⠀⢲⣾⠁⣀⠂⠀⠰⠀⠘⠃⠙⡿⠖⣿⢉⡋⢻⣻⣷⣴⣿⠁⠚⠀⠀⢀⣿⣿⣷⡾⢀⡀⡼⠟⠘⢤⡀⢰⠤⣴⣦⣴⠀⠀⠙⠁⠀⢀⡐⠀⠴
⣄⠐⠀⡇⠀⡉⠀⠀⠀⠀⠀⠈⠀⣇⣼⡉⠛⣷⠋⠉⢿⡉⠉⠶⣦⠷⠿⣿⣿⣿⣿⡋⠈⠓⠀⠀⠀⠀⢠⠀⠀⢉⠈⡀⠀⠛⠛⣦⣀⡀⠉⠀⠐
⢽⡟⠲⠧⠀⠀⠋⠀⢀⠀⠀⠀⠀⠘⠑⠦⠀⠀⣄⡒⣀⠄⠀⠀⠀⠀⠀⠈⠉⠀⠈⢋⡀⠀⠀⣦⠃⡀⠀⠀⠐⠂⠈⠀⠒⢦⣟⡁⠤⠄⠰
⠀⠁⠀⡀⠚⠖⢧⠀⢼⡀⢀⠀⠜⠀⡋⠘⠸⠇⢀⣄⠉⠀⠁⠈⠻⠃⢠⡤⠀⠰⠶⣮⡄⠀⠀⠀⠀⠁⠀⠁⠈⣶⣤⣶⣦⣤⡿⠃
⠿⠓⠆⠁⠀⠀⠐⠚⠣⠄⠺⠄⠐⠂⢠⠀⠀⠀⡀⠡⠴⠀⠰⠒⠂⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠐⠈⠉⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⣨⡄⢁⠀⠀⠀⠀⠀⡃⠀⠐⠀⠀⠀⠀⠀⠰⡔⠘⠇
⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⣟⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣥⣚⣀⣀⠆⠴⢶⡔⠒⠚
⠠⠀⡄⢨⠉⡤⢤⠄⠀⠃⠀⡅⢠⠐⠂⠀⣤⠉⠺⠉⢠⠕⠚⠃⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣄⠀⣰⠦⡴⣿⣏⡄⡀⠓⠘⠃⠰⠷⠀⢡
⠀⠀⠆⠈⠀⡰⠂⠀⠀⠀⠀⣴⢁⠤⣁⠀⠀⠀⠐⠀⠀⠐⠈⡅⣟⠠⣯⢰⠀⣿⠂⢴⡃⠀⠀⢿⣠⡼⠋⠙⢃⢦⠛⡻⠆⠹⠧⣤⡟⢷⣀⣦⠉
⠀⠀⠀⠂⢠⠁⠀⠀⠐⠁⣀⠄⢨⠐⢀⠂⠴⠀⠀⡀⠚⠃⡐⠛⠉⠠⢤⠀⠀⢀⡰⠀⢹⠀⠶⠈⠉⡛⠒⠒⠉⣀⣘⠛⣤⠒⢘⠋⠁⡀⣿⡀⠈
⠀⠀⠀⠐⠃⠖⠀⠰⠀⠀⠁⠀⠸⠀⡠⠀⠰⠀⠂⡁⠐⠂⠀⡐⠁⡀⠀⠀⠀⠈⠀⠀⠠⠀⡋⡠⢦⠉⠀⢀⠘⠿⠈⢰⡄⠀⡈⠁⢸⣧⠛⢭

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣤⣠⣤⣴⣶⠞⠻⡝⢻⡍⡓⡒⣦⣄⠤⣄⣀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠉⢻⠳⡈⢣⠀⠘⡀⢱⠀⠹⢸⠙⡆⠈⢿⣧⠈⠙⣷⣆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⢸⠀⢀⣇⢸⠀⢰⡿⢠⠃⠀⡜⢹⠀⠀⡸⢹⡀⣀⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣴⣷⣿⠟⢠⡾⢣⡿⢠⢿⢇⡟⣠⡞⣡⠏⠀⡠⠃⡼⠀⠀⠀⠉⠓⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⢰⡃⡘⠀⣼⠀⣿⠀⢨⡇⢸⢸⣥⡞⠁⣔⣙⣠⠞⠁⠀⠀⠀⠀⣠⣿⢀⡠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⡇⣷⠀⢹⠀⠻⠀⠈⡇⠈⣼⠻⡇⢠⢿⡿⠁⠀⠀⠀⡐⢒⣯⡤⠛⠁⠀⠀⠀⣱
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠁⠀⡇⠸⠀⠸⠀⠀⡆⢀⢳⠀⣿⠀⣇⠇⢈⠇⠀⠀⣠⣞⡠⠊⠀⠀⣀⣠⢄⡒⠚⣿⠁
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⡇⠀⡇⢠⡇⢼⢺⡄⣿⣦⡿⠀⡞⢀⡴⣊⣥⢯⡤⠒⠈⠉⢁⣀⠾⠛⠊⠁
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣶⢤⣗⣰⡃⣰⣃⡼⠤⠞⠛⠚⠉⠉⠓⠋⠉⠉⠉⠉⠛⠚⠉⠉⠉⠉⠁
⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠀⠁

⠈⠁⠈⠐⢁⠀⠀⠁⠈⠈⠁⠈⠈⠀⠒⠀⠑⠈⠈⠀⠈⠂⠀⠈⠀⠁⠈⠐⢂⡈⠂⠁⠉⠀⠁⠉⠀⠀⠈⠂⠁
⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠀⠛⠠⠀⠀⠀⡀⠠⢔⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢄⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠁⠀⠀⠒⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠆⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣢⠞⣦⣶⣲⢶⡖⢰⣒⡖⣞⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠸⣦⡞⣿⢻⣼⣯⡬⣭⠇⠿⠀⠠⡄⠀⠀⠀⠀⠀⠀⡀⠑⡠⠠⠠
⠀⠁⠎⡜⡁⠈⠂⠀⠀⠀⠀⡀⠀⠀⢰⣟⣶⣲⢶⣶⡒⣿⣼⣶⢰⡴⣖⡂⠀⠀⠀⠀⠀⢀⠀⢷⡟⢸⠇⣠⡄
⠀⠀⠈⠀⠀⠀⠀⡀⠀⠐⠠⠰⠀⠀⠸⠃⠿⠹⡾⠻⠥⠿⠻⠿⠼⠳⢼⠇⠀⠀⠀⠀⠠⣼⣧⠈⠃⠘⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡠⡄⠀⠠⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠉⠉⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⠀⠀⠀⢠⠠⡀⠈⡁⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠀⠀⠈⠓⣧⢀⢠⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⢉⠀⠀⡠⠀⠀⠠⠤⢀⠀⠀⢱⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠨⠀⠙⠆
⠀⠀⠀⠀⠀⠁⠁⠫⡀⠀⠀⠀⠄⠀⠋⢀⠉⡚⡀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠀⠈⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠁⡸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⡆⠀⠠⠀⠀⠀⠠
⠀⠀⠈⠀⠐⠠⠀⠀⠀⢧⠁⠄⠀⡀⠂⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠚
⠀⠐⠀⡄⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⠀⠀⠐⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠐⢸⠀⠀⠀⠀⡀⢸⠀⠀⣶⠀⠀⠀⠀⠞⠸⠀⡌⠀⠈⠀⠄⠀⠀⢀⠀⠇⠀⠰⠆⠙⠀⠀⠈⠁
⠀⠸⠛⠃⠀⠀⠀⠌⠀⠁⠀⠁⡄⠀⢠⠀⠤⡀⠀⠸⠀⠀⠀⠀⠀⠀⡈⠀⠀⡄
⢠⡆⠀⠈⢀⣩⠀⠐⣀⢀⠰⣇⡀⠀⠀⠠⠀⢠⡄⢾⣀⠀⠀⠀⠀⠀⠒⠀⠢⢐⠀⡀⠀⠀⠀⠰⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡴⣣⢾⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢻⠁⣰⡯⠖⢁⣴⠁⠀⠈⠢⣀⣴⣻⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⠁⠀⠀⠈⠀⢀⡴⠃⡜⠀⢠⡀⠀⠈⠂⠀⡟⠀⠈⠀⠀⡆
⠀⠀⠀⠀⠀⣀⠀⣿⢰⡆⣿⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢠⡧⢰⡀
⠀⠀⠀⠀⠀⣿⣬⣿⣿⣷⣿⣿⡆⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⢠⡇⢰⡿⠾⠛⠋⠉
⠀⠀⠀⠀⠀⡄⠀⠈⠉⠛⠿⠿⠧⠼⠧⠀⠀⠀⠀⠀⠀⡄⢠⣇⣰⣿⡧⠞⠋
⠀⠀⠀⠀⢠⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⣊⠭⠂⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠛⠻⠆⢀⣷⠀⡀⠄⠀⠒⠄⠛⠉⠀⠀⠀⢀⠀⠀⠀⡆⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⢖⢧⡄⢀⠀⠀⠀⢀⣾⡄⠀⢸⣧⠀⢰⠀⠀⢰⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⢹⠀⠑⢄⠀⠈⠉⠓⠠⠾⠿⠆⠿⠦⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⢀⡠⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣾⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⢏⣿⣿⣹⣿⣿⢿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⣿⣿⣿⣿⡿⡟⠀⠤⢥⣉⢘⣨⠿⠬⣝⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣾⣿⣿⣿⣿⠁⠀⠒⠉⠀⠅⠀⠀⠊⠓⠀⠘⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠀⠰⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢼⣿⣿⣿⡄⠀⠀⠀⣘⡆⢠⣅⡀⠀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢘⣿⡏⢹⣷⡀⠐⠉⡢⣜⣃⠟⠙⢸⣿⣿⡿⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣿⡇⢀⠀⠀⠀⠀⠀⠀⠙⢧⣦⡀⠑⠦⠔⠀⢀⡾⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⡸⠙⠳⠦⣤⡶⠶⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⢀⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⣾⣿⣷⣄⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠛⠛⠛⠙⠋⠉⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⠦⠀⠐⠀⠛⠻⠇⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠉⠁⠀⠀⠀⡀⠰⣾⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣆⠀⠀⠀⢰⣶⣷⣦⢳⣼⣾⣿⣷⣆⠀⡀⠰⣴⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣟⠻⠃⠀⠀⠈⠋⢿⣽⣿⣷⣿⣏⣿⣋⣿⣷⡇⠺⢻⣦⡄⠀⡷⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⣰⣶⣶⣶⣿⣿⣧⣴⣿⣿⣿⣿⣿⣷⣀⣿⣿⡅⠀⢠⠙⢀⣀⡀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣰⣧⣀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠰⠂⠀⠀⣠⡈⠹⢿⠛⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣼⡿⠋⣉⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠙⠛⠀⠀⠁⢊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠸⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣹⣷⣿⣖⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣿⣿⣿⢦⠀⠀⠀⠘⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠁⠈⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣳⣦⡤⠶⠞⢳⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣶⣶⣤⢺⡿⠷⣿⡿⠋⠐⠚⠿⣿⣿⣿⣿⠿⠉⣠⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠙⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠙⠛⣿⠃⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠇⢰⡟⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⠍⠀⣼⣿⣿⣿⣿⠙⠀⣾⡂⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⢸⣿⣿⣿⣿⣆⠀⠉⢠⡆⣀⠀⠀⡰⠃⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠻⣿⣿⠿⣿⠂⣴⡿⠧⠾⣆⣸⣿⣿⣿⣇⣀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠶⠶⠿⠆⢭⣴⣴⣶⡭⠿⠿⠿⠿⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛
⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠴⣦⢴⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠋⠄⡿⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠂⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣤⡀
⠀⠀⠀⠀⠲⠃⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⢠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠤⢷⣆⠀⡀⢠⡖⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠁⠈⠀⠉⠀⠀⣤⢄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⢤
⠉⠐⠂⠀⠤⠀⡑⠒⠆⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠀⠀⣀⣀⣀⡀
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠿⣆⠒⠂⠉⠉⠉⠁⣠⣄
⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡠⣠⠠⠄⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠐⠄⠀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣒⠀⣀⡈⠉⠀⠰⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠋⠉⠀⢠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠋⠻⠿⢦⣄⣂⠀⣀⠀⠀⠀⣀⡀⠉⠉⠁⠀⣀⢠⠄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠷⣶⣦⡤⠀⠉⠻⠿⠦⠤⠾⢥⣤⣴⣄⣀⣠⣏⣿⣄⠀⠀⠛⠉⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠙⠛⠛⠛⠿⠷⣶⣷⣴
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣸⣿⣯⡀⠀⠀⡀⠀⠀⠀⢀⣴⣿⠹⣿⢦⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣿⣿⣿⣄⠀⣾⣷⠀⠀⢠⣿⠏⢣⡀⢹⡇⠈⠳⢄⣀⣤⣾⣿⡑⣄⠀⠀⠀⣀⣀⢀⣠⡾⠉⠣⣠⡄⢰⣷
⠀⣽⣿⣿⣿⣿⣨⣿⣿⣇⣰⣿⣦⣄⠀⠑⠄⠻⡀⠀⠀⣻⣿⡿⢻⠀⠈⠳⡴⢼⡁⢻⡯⡻⠀⠀⠀⣿⣿⣿⣿⡇
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠙⠳⠄⠀⠀⠙⢀⣶⠟⠁⠀⠘⡇⠀⠀⠘⠆⠉⠢⠧⠀⠀⠀⢰⣿⣿⣿⣿⣿⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡇
⠀⠀⠉⠉⠉⠉⠉⠈⠉⠀⠀⠀⠐⠒⠒⠒⠒⠳⠶⢾⣷⣦⣤⣤⠤⠤⠤⠴⠾⠿⠋⠙⠛⠛⠓⠀⠀⠈⠉⠉⠙⠉⠁