SMOKING Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡎⠛⠛⠛⠉⠁⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠛⠉⠀⣀⣤⣀⣀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠻⠿⠃⠀⠀⠀⣠⡤⢭⠏⠁⣴⡟⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠓⠀⠀⠀⠀⠀⠀⠀⡠⢧⡤⠤⠇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠛⠸⣇⠀⠀⠀⠀⠰⠿⣄⠠⠤⢤⣶⡟⠀⠀⠀⠀⠀⠀⠀⢀⡆⠰⡆⣴⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡎⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⢿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⣿⣿⣿⣿⢦⣀⠀⠀⠀⠀⢠⣄⠀⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣩⣿⣾⣿⣿⣿⣷⣯⣝⠲⢤⣀⣀⣹⡷⠚⣯⡀⠀⠀⠀⠀⠀⠀⢀⠁⠊⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠧⣙⢻⣿⣿⣿⣦⣀⣩⣭⣤⣾⣿⣿⣶⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⣫⣴⣶⣶⣶⣤⣍⡛⠿⣿⣿⣿⣿⡟⠁⠈⢻⣿⣿⣶⡀⠀⠀⠀⢳⡄⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡷⠂⠀⠙⣿⣿⣿⣦⡀⠀⢻⣿⣿⣿⣦⠀⠀⠚⠀⡀⠀⠀⡠⠋
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠛⠓⠂⠀⠹⣿⣿⣿⣿⣦⡈⢿⣿⣿⣿⣷⠀⠀⣦⣌⠀⠰⠃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠦⠀⠘⣿⣿⣿⣿⣿⣾⣿⣿⡿⠃⠀⣮⣿⣿⣷⣄⠐
⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠆⠀⠀⠈⢿⣿⣿⣟⠻⠟⠙⠃⠀⢀⣽⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣿⣿⡿⣿⣎⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣁⠀⠀⠀⠀⠹⣿⣿⣿⣦⡄⠀⢀⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠓⡀⠀⠀⠀⠹⣿⣿⡿⠁⢠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⣿⣿⣿⣿⣿⣟⣻⣯⠥⠀⠀⠀⠀⢹⣿⠇⠀⣤⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢿⡇⣿⣿⣿⡉⢿⣿⣷⣤⣛⣿⣿⣿⣿⣭⣭⣥⣒⣢⠭⠂⠀⠀⣰⣿⡄⠠⣽⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠁⠙⣿⣿⡁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⡤⠄⠀⢠⠏⣿⣷⠀⡙⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠻⣿⣿⣷⣌⣙⢻⣿⣿⣿⣿⣷⣶⣂⡘⠀⣿⣿⡆⠹⣮⡻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣍⡛⠿⣿⣿⣿⣿⣿⣉⡀⢿⠹⣿⢷⣌⢻⣮⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡍⠻⢿⣿⣿⣿⣷⣦⣶⣤⡈⢿⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⡈⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠡⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠞⠀⢠⠀⠐⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⡡⠀⠐⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣀⠔
⠀⠀⠀⢰⡁⠀⣠⠶⠴⠁⠀⠀⢀⠀⠀⠀⡀⠃
⠀⠀⢀⡠⠁⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠠
⠀⠀⠫⠃⠀⠀⠀⠀⠉⠂⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⢀⡔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⠀⠂⠄⠀⠀⠐⠢⣙⢆
⠀⠀⠀⠀⠀⠠⣀⣀⠔⡀⠤⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⠒⠚⠿⠇⣦⠀⢀⡀⠻⠄⡉⠳⢤⠆⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠒⠀⠀⠀⠀⢂⠀⣀⣴⠈⣼⣍⢀⠀⠀⠀⠀⠐⠂⢁⡐⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠰⠤⠀⠤⣄⠀⢀⠀⠀⢀⠀⠀⠆⠀⠀⡙⠀⠉⣻⡄⠀⢀⣤⣤⠀⠄⠈⠀⠈⠦⣆⠀⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠀⢰⠀⠌⡆⠀⠀⠀⠘⠋⠀⠀⢸⣷⣤⣯⡶⠀⠀⠀⠲⡌⣤⡄⣧⠀⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠉⠀⠀⠀⠈⠂⢠⡄⣠⠞⠃⠘⠳⣄⢀⠀⠀⠀⢸⠁⢯⣌⠉⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠔⠁⠀⢐⠈⠁⠉⠀⠀⠀⠀⠀⣀⢀⡤⠈⠙⠓⠀⢼⡆⣀⡾⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⡀⠀⠀⠀⠑⠠⠀⣀⣠⣤⡶⠏⠈⠉⢧⠀⠀⠀⠀⠀⡀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡟⠀⠀⠀⠀⠀⠀⠁⠈⠹⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠈⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⠤⠀⠂⠈⠀⠰⠴⡶⠀⠀⡀⠀⠚⠿⡀⠀⠀⠀⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⡰⠁⢠⠐⢢⣿⡀⠀⠀⠀⣨⢿⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠀⠀⠀⠐⠛⣇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢠⡿⠛⠁⢀⣶⠿⠛⠉⢁⣴⠿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢀⠶⢀⣽⣷⣶⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⢻⠏⣿⣷⡦⢶⠅⠁⠀⠀⠀⠀⠀⢢⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢾⣄⣈⣗⡈⠄⠀⠰⡅⠀⠀⠀⠀⠀⡠⢋⠌⠠⡜⠋⠕⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠨⢙⡟⢣⠀⠀⠀⢰⠀⠀⠀⣀⠞⣁⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⣄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣸⠧⣀⣴⣄⠀⠈⡆⠀⠘⠁⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢹⠂⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡌⡇⠀⠸⡀⠀⢱⣤⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠏⠀⢀⡀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠟⠟⠃⠀⠀⠀⠀⠐⠈⠀⠀⠀⠐⠚⠤⢠⣭⣥⣤⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠫⣠⠔⠁⠻⠿⠟⠁
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡀⠀⢀⠀⢀⢀⠀⢀⠀⡀
⣿⠿⠿⠿⠿⢟⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣻⣟⣿⣿⣟⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣰⣶⣶⠒⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⡙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠹⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⡧⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⡍⢹⣿⣿⣷⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣤⢤⡀
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣛⣿⣿⣿⣿⣿⣟⣹⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⡏⣹⣿⣿⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠟⠻⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡧⢘⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣏⠀⠀⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠟⣿⣿⣿⣿⣿⣿⠇⢀⣤⣸⡇
⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣥⣤⣠⣤⣾⣿⣿⡟⠉⠁⣰⣿⣿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⢉⡛⢿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⢹⣿⠟⠹⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠈⠛⠻⠷⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠭⣟⣋⣡⣤⣀⣀⠀⠀⠙⠐⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⡿⠋⡁⢠⣿⣷⣶⣦⠀⠀⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣇⠀⠐⢞⠡⣿⡏⢻⣟⢉⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⢹⣿⣿⣿⡆⠀⠈⠀⠙⠂⠤⢈⣉⡋⠛⠛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢻⣎⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⢉⠀⠀⠀⠀⠀⢀⡘⢶⡀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⠙⣿⣿⡟⠳⣤⣀⣀⠀⠀⠀⠀⠙⠓⢶⡇⠀⠀⠀⠀⠀⠘⢃⣸⡇⣊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠻⢾⡽⣵⣾⠀⠀⠀⠀⠈⠉⠲⢤⣀⣀⡀⢀⡽⠀⠀⠀⠀⠒⠀⠻⣯⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣀⡐⠂⠢⣄⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣟⢠⣗⣤
⠀⠀⠀⠀⠀⠀⢠⣾⡿⢋⣿⣿⡑⠀⠀⠛⠋⠀⡀⠘⢷⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⢻⡇⢹⡇⠠⢐⣄⢠⡀
⠀⠀⠀⠀⠀⣴⡿⠋⠀⣿⣿⣿⣷⡄⠀⠀⣠⣶⣤⣀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡁⠀⢸⡇⡸⣧⠀⣿⠀⣴⠂
⠀⠀⠀⠀⠾⠋⢠⣴⣿⣿⣿⣿⣿⣷⡄⠀⢹⣿⣿⣿⡿⢿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣸⣿⣿⣿⣠⣿⢰⣿
⠀⠀⠀⠀⠀⠀⠸⣯⣻⣿⣿⣿⣿⣿⣿⣦⠈⢿⠟⠋⠀⠀⣯⣥⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⠀⣿⣿⣿⣿⣧⣿⣽⡟
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⠛⠿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣾⣷⣤⣌⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⣷⣈⣻⣟⠙⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⣿⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣤⠀⠈⠁⠰⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⢀⣾⣿⣿⣦⣄⣀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠘⠛⠋⠻⠿⠿⡏⠃⠉⢀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⢿⣿⠿⣿⣿⠿⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠉⠀⠀⠀⠈⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠿⣿⡿⢿⡿⢿⡿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣶⣼⣶⣾⣤⣾⣷⣴⣿⣥⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠄⠈⠀⠀⠈⠂⠀⠰⠀⠀⠄⡃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡀⠀⠀⠀⢀⣀⣀⣀⡠⠒⠀⠠⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⠀⠀⠀⠀⠀⠁⠒⠋⠀⣋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣠⠐⡄⠀⠀⠀⡴⠾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠠⡀⢈⠄⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡆⢀⠖⠘⡀⠄⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⠂⠀⠀⠰⣀⡀⠒⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠉⠀⠀⠀⠠⡀⠀⠀⠈⠑⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⡀⠅⠀⠐⠀⠀⠀⡞⠢⢄⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠂⠀⠀⡀⠁⠀⠘⢶⡀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠄⢴⠐⡀⠆⢃⠀⡀⢂⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠙⢾⡄⠱⠀⣄⢂⠐⠊⠰⡃⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⡀⠀⠉⠐⠀⠀⢠⢀⠂
⠀⠀⠀⠀⢀⠎⠀⠀⠀⡁⠀⠀⠈⢇⠀⢋⠘⢃⠀⣀⠒⠵⠲⢴⠃⠉⣱⠀⠀⠀⠀⠠⠤⠄⠀⣸⠀⠀⠀⠀⠈⠙⢥⠀⠠⠬⠧⠀⠀⠀⠠
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠉⠀⠚⣇⠈⠠⠀⢞⠡⠂⠎⠁⡼⠓⣞⡁⠉⠢⢤⠄⢠⣀⡀⡄⢠⡀⠀⠀⠀⢀⠄⠀⣤⣖⣂⣀⣀⣙⣓⡦
⠀⠀⠀⡞⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠠⠈⠠⡀⠀⡜⠀⢰⠁⠈⠉⠒⢲⡄⣸⠀⠅⢠⡼⠃⠀⠀⢰⠀⠉⠏⠀⠉⠉⣉⡉⠉⠁
⠀⠀⠘⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠑⢀⠀⢰⠁⠀⢸⠀⠀⠀⠀⠀⠉⠓⠚⠛⠁⠀⠀⠀⠀⢀⠃⠀⠀⠄⠣⠀⠉⠀⠀⠠
⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠁⠀⡐⠀⠀⠳⣄⡀⢰⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠐⠊⠁⠀⠀⠀⠤⠠⠄⠁⠈⠉⠠⠐⠂
⢰⡃⠀⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⠊⣉⣉⠻⣿⣦⣤⡔⢊⣁⣉⣉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠉⠒⠤⢄⣀⠀⠀⠀⠀⡄⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠘⠒⠐⠒⠀⠐⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣖⣚⡋⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠛⠛⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠨⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣼⣷⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠻⣉⣤⣵⣦⣴⣤⣤⣤⣤⣴⣦⣴⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣴⡾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠿⠿⠛⠉
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢀⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠻⢿⣿⣿⠀⠀⠀⠀⠀⠀⠄⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⣀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢐⡂⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠙⣿⣿⣿⣿⣿⣷⡌⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠿⢿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠀⠀⠀⠰⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣟⠉⢸⡷⠑⢿⠇⠻⡄⠀⠀⠀⠀⠀⠀⠀⠉⣙⠃⠀⠀⠀⠀⢀⠂⣋⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣸⡓⠄⠀⠀⠐⠶⣦⣀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠠⠀⣹⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠉⣹⠟⠶⠦⣤⣤⡤⠁⠀⠀⠀⠀⠀⠀⠢⣈⠙⠄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣷⣮⣝⣻⠷⣿⡑⣤⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⣀⠰⣿⣆⠈⠂
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠿⠛⠿⣿⣿⣦⣿⣾⣿⣶⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡟⣳⣦⡀⠑
⠀⠀⠀⠀⣴⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠿⣐⡉⢩
⠀⠀⠀⢸⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡳⣦⣄⠀⠀⠀⠀⠀⠀⠈⠈⠛⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣿⣮⣽⣷⡄⠀⠀⠀⠀⠀⠀⢶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⡳⠀⠀⠀⠀⠀⠐⢶⣉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠐⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿
⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⡀
⣿⣟⣉⡿⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡲⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣟⣫⣅⠤⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡽⢶⣒⣦⣶⣾⣿⣿⣿⣷⣶⣦⣀⣤⣤⣤⣤⣬⣥⣶⣶⣶⣶⣦⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⢻⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⡟⡿⠟⠛⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡟⡜⢙⡆⠙⢁⠐⠀⠀⠀⠀⠀⠀⠀⡠⣭⡟⠙⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠩⡻⢿⣿⣿⣿⣿⣿⣿⣿⡱⢈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠏⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠏⠀⠀⠤⠤⠀⠀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠀⠤⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡾⠀⣤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠄⠈⠀⠀⠈⢣
⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣦⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠈⠂⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠻⠄⠀⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠈⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠒⠀⠀⠀⠀⠀⠀⠉⠱⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠁⢀⡀⠤⢄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⡔⠀⠀⠁⣀⣀⠇⠀⠀⠐⠄⠀⠀⠀⢠⠆⠐⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠧⠀⠀⡏⠉⠁⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⢀⠄
⠀⠀⢠⣆⠁⠀⠐⠂⠀⢄⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠐⠃⠀⠀⠀⠀⡂
⠀⠀⠈⠂⠀⢠⠀⠀⠀⠀⠈⠁⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰⠣⠄⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠜⠀⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠤⠀⣀⣈⡁⠂⠀⠀⠈⣑⠮⣕⡀
⠀⠀⠀⠀⠀⠐⠤⠤⠤⢊⡅⠒⠀⣀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢈⠩⠏⠹⠟⠈⠤⠀⢤⣄⠙⠂⠠⠈⠲⡴⠠
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⡀⠀⣀⣶⠁⢰⣩⠀⡀⠀⠀⠀⠀⠒⠁⢀⡄⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠲⠤⠀⠤⣄⠀⠀⠀⠀⠀⡀⠀⡠⠃⠀⠀⠝⠀⠈⢻⣇⠀⠀⣠⣤⣠⠀⢄⠈⠀⠀⠳⣤⡀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⠊⠠⠀⠠⢱⠄⠀⠀⢀⠺⠟⠀⠀⠈⣿⣖⣸⣣⡆⠀⠀⠀⠦⣤⢠⡄⢹⡀⠘⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠐⠃⠈⠀⠈⠠⠴⠀⠐⠋⠀⠀⠀⠘⠄⢀⣄⢀⡰⠗⠉⠿⢌⡀⠀⠀⠀⠀⢸⡋⢿⣊⠑⢢⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠤⠐⠐⠒⠈⠁⠈⠀⠀⠀⠀⢨⣁⠘⠲⠦⠀⢸⡅⠀⢹⡇⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⡀⠀⠀⠀⠁⠀⠀⠀⠘⠄⠀⠂⠀⠀⢀⣠⡴⠖⠒⠾⡈⠄⠀⠀⠀⠈⠃⠘⠉⢁⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢳⠄⠀⠀⠀⠀⠀⠀⠇⠻⢿⣿⡍⠀⠀⠀⠑⠀⠀⠀⠀⠀⣲⣤⣴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⢠⠴⠶⠆⠀⡀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢤⡯⠤⠔⠒⠀⠁⠀⠀⠈⠓⢻⡁⠀⠠⡀⠐⠉⠛⣦⠀⠀⠀⠀⠃⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠢⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣⠓⠀⣦⠈⣡⣿⡆⠀⠀⠀⣀⣼⢻⡷⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡐⠀⠀⠛⢻⡀⠈⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢰⡟⠙⠉⢉⣼⠿⠛⠁⠁⣠⢶⠎⠽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢀⡾⢀⣱⣿⣷⡞⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡜⢻⠋⣿⢗⣶⢖⠇⠀⠀⠀⠀⠀⠀⠀⢢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣅⠀⣹⣂⠡⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡔⢁⠃⢠⡙⢛⠕⠁⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈⡟⢿⠛⣄⠀⠀⠀⢰⠀⠀⠀⠀⣰⠋⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⢠⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⡀⢀⣤⡀⠀⠘⡆⠀⠀⡏⠀⠘⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠈⠻⣟⢻⡄⠀⢹⠀⠀⠰⣄⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⠀⠁⠀⣰⣿⣿⠁⠀⠈⠀⠀⠠⠔⠀⠀⠀⠈⠀⢾⠀⠰⠳⠂⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⢠⠆⠀⡩⢻⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠎⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣉⣟⣛⣉⣏⣅⣍⣻⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣭⣽⣭⣿⣿⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣰⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⡿⣿⣿⣾⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⠈⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠁⠀⠠⠶⢦⣤⡄⠀⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⡉⠛⠉⠀⠀⠀⠰⠞⠒⠓⠀⠸⡟⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡌⣁⠀⠀⠀⠀⠀⣀⡄⠛⠛⠒⢃⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠊⠁⢻⡆⠀⠀⠀⠀⠟⠓⠀⠓⣰⢿⠏⠀⠀⠀⠀⠀⠀⣤⣴⡇⠸⣇⢿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠉⠩⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣜⣿⣿⣿⣯⣳⠦⣀⡀⠀⠈⢷⣤⣀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣾⣿⣿⣿⡻⣿⣿⣿⣮⣉⠓⠲⠶⠛⣁⣽⣦⣄⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⣋⣉⣙⡂⠻⢿⣿⣿⣷⣾⣿⣿⠿⠛⢿⣿⣧⣀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣶⣈⠙⠻⣿⣿⣷⣄⠀⠈⢿⣿⣿⣷⡀⠀⠀⣼⠃⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄⠀⠸⣿⣿⣿⣷⣄⠈⣿⣿⣿⣿⣆⠀⠠⣀⠂⠀⢠⠏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀⠙⣿⣿⣿⣿⣷⣼⣿⣿⣿⠏⠀⡰⣿⣷⣄⠉⢀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡀⠀⠙⢿⣿⣟⡿⣿⣿⠿⠟⠀⠀⣻⣿⣿⣿⣷⣆
⠀⠀⠀⠀⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠆⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣷⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠄⠀⠀⠀⢻⣿⣿⣿⠇⢀⣴⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣶⣶⠴⣶⣒⡀⠀⠀⠀⢹⣿⡏⠀⡈⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣿⢻⣿⣿⣿⠻⣿⣿⣦⡹⢿⣿⣿⣿⣟⣛⣛⡩⢥⣐⠆⠀⠀⢀⣿⡀⠀⣹⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠈⢻⣿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣃⣀⠀⠀⡾⣿⣷⠀⠹⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠙⢿⣿⣿⣭⡛⠿⣿⣿⣿⣿⣿⣤⠄⠘⠃⣿⣿⡆⢲⣜⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⡻⢿⣿⣿⣿⣿⣿⡛⠀⢻⠹⣷⣶⡙⢷⣜⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣜⠻⣿⣿⣿⣿⣷⣬⣤⣍⡘⢿⣦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⣹⠆⠀⠟⠉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⣷⡀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠉⠀⠉⣩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢁⡀⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⣰⡟⣡⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢏⣾⣿⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣣⣿⡿⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠛⣵⣫⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣾⣿⣿⡿⠛⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣿⣿⣿⡙⢿⣿⣧⣄⣀⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣿⣿⣿⣷⣷⣿⣷⣟⢉⣭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣻⣿⣿⣿⣿⢷⣬⡁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢹⣿⣿⣟⣯⣿⣿⣿⠇⢰⠁⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣷⡀⠈⠙⠿⢿⣿⣿⣿⢀⠇⢀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠙⣿⣿⣿⣦⣿⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣄⠀⠀⢻⡿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡁⠀⠀⢻⣿⣏⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣮⣝⠿⢿⣶⠗⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣤⣀⠀⠘⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿
⣦⣴⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⣂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡦⠀⠁⠁⠀⠈⠁⠩⠛⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡄⠀⠀⠀⠀⡴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠄⠠⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠁⠂⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠂⠀⠠⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⢈⠦⡀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠀⠀⠄⠀⢀⠀⠀⠰⣌⠀⠀⠁⠐⠟⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢰⠀⡀⠘⠀⢀⢀⠀⠘
⠀⠀⠀⠀⢀⠜⠀⢀⠀⠈⠺⡄⠁⡠⡌⠀⠂⠐⡁⠐⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠄⢀⣁⠐⠀⠰⡠⠂
⠀⠀⠀⢀⠊⠀⠀⠀⠀⢀⠀⣣⠀⠁⠁⢀⠴⡉⠫⠝⣏⣀⠼⢄⠀⠀⠀⠒⠒⠀⠆⠀⠀⠀⠀⠀⠒⢐⣒⣐⣀⣀⣀⠂
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⢄⠀⠁⡔⠁⡟⠢⠤⢀⡊⠈⡆⠆⠃⣽⠀⠀⠀⠐⠤⠜⠛⠲⠶⠶⠶⠚⠁
⠀⠀⠜⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠳⡀⠠⠀⠁⡴⠀⠀⡇⠀⠀⠀⠘⠾⠠⠦⠊⠁⠀⠀⠈⠆⠀⠀⠠⡀⠘⠃⠀⡀
⠀⡔⠀⠀⠀⠉⠀⠀⠀⣀⠁⠀⠀⠀⠑⣄⡈⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠔⠂⠀⠀⠠⠀⠀⠀⠀⠁⠀⠄
⢸⡁⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠈⠉⠛⢿⡦⠤⠒⠁⠉⠉⠉⠉⣉⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⠓⠢⠤⣀⣀⠀⠠⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢞⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠯⣲⣤⣴⣶⣾⣿⣷⣤⢤⣀⣀⣀⣠⣤⣽⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⢟⡛⢿⣿⡟⠹⠟⠋⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡏⡎⣹⠂⠉⠄⠀⠀⠀⠀⠀⠀⠀⠤⠾⠏⠁⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠪⢛⣿⣿⣿⣿⣿⣿⣿⡝⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠉⠨⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠡⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠈⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⡶⠀⠀⣀⡀⠀⠈⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡯⠀⢀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠰⣶⠆⢠⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠳
⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷⡈⢦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠈⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠙⠂⠀⠁


⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⡴⠖⠒⠒⠲⠶⠶⣶⣾⠶⠆⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⣧⢸⣷⠄
⠀⠀⠀⠀⠀⠀⠀⢿⠰⡿⠛⣀⠀⡀⠀⣀⣀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⣾⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣭⣭⣭⣭⣥⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⢀⣼⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠛⠋⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠐⠒⣳⣦⠆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠠⠉⠻⠇⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠉⠉⠛⠛⠋⢻⣿⣿⣿⣿⣌⠓⠀⠀⠀⠀⠀⠀⠀⠠⣂⣀⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⡛⢻⡿⡇⠀⠀⠀⠀⠀⠀⣀⣨⠈⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡄⠾⡃⠙⠁⢑⣀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⣶⡈
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡀⠀⠀⠀⠙⢳⠦⣤⣀⣀⡠⠀⠀⠀⠀⠀⠀⡙⢃
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣯⣝⣻⢿⣗⣄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡴⣀⠠⢿⡄⠁
⠀⠀⠀⠀⣾⣿⣿⣿⠿⠛⠛⠛⠿⠷⣾⣿⣿⣾⣿⡧⠀⠀⠀⠀⠀⠀⠀⠰⣿⡽⣶⡄
⠀⠀⢀⣾⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠌⠴⠂
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣻⣦⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠂⠀⠀⠀⠀⠶⣍⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣃⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⢽⣛⣛⣛⣙⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣾⣿⣧⣭⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⡀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⢻
⠀⠀⠀⠀⠀⢸⡷⣿⣿⣿⣿⣿⣿⣿⣯⣻⣏⢹⣿⣿⣾⣿⣷⣶⣶⣷⣶⣶⣶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣯⣽⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠏
⠀⠀⠀⠀⠀⠘⡇⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⡽⠉⠹⠿⠛⠋⠁
⠀⠀⠀⠀⠀⢀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⡀⣴
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣉⣀⣤⣿⣿⣿⠟⠋⢡⣾⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⠟⠊⢙⠻⣿⣿⣿⣿⣿⡿⠟⠀⠀⢹⣿⠓⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠉⠛⠛⠟⠛⢿⣿⣿⣿⣿⣿⣿⣶⡬⢽⣉⣥⡤⢀⡀⠀⠈⠐⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣧⠘⠛⢡⢀⣾⣿⣿⡷⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠿⣿⣿⣿⡄⠀⠹⠸⡿⠀⠻⠤⠤⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠾⣿⣿⡇⠀⠀⠀⠀⠈⠐⠒⠢⠤⠄⠀⠀⠀⠀⠀⢢⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠳⣿⣿⠣⣀⣀⠀⠀⠀⠀⠰⠦⣴⠁⠀⠀⠀⠀⠻⠂⡟⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢾⣿⣵⡋⠀⠈⠁⠈⠓⠦⣀⣀⡀⣈⠆⠀⠀⠀⠂⠐⢿⡛⡏
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣦⡀⠂⠢⣤⡌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢸⡃⣿⡤
⠀⠀⠀⠀⠀⣰⡿⠋⣼⣿⣆⠀⠀⠉⠀⠀⠈⠻⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠈⣷⠈⡇⠀⣵⠀⡆
⠀⠀⠀⢀⡾⠏⣀⣼⣿⣿⣿⣇⠀⠀⣿⣷⣦⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣯⣼⣷⢰⡟⣸⠇
⠀⠀⠀⠈⠀⢨⣟⣿⣿⣿⣿⣿⣧⡀⢻⣿⠿⠋⢙⣟⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡆⢰⣿⣿⣿⣞⡗⣿⠇
⠀⠀⠀⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣷⣄⠁⠀⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡿⢿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣉⢷⣶⣬⣉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢹⣦⣹⣏⠙⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠸⣿⠁⣽⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣼⣧⣄⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡟⠿⠿⢿⣿⣿⠿⠙⠂⣴⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠈⠀⠈⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠻⠟⠹⠛⣿⠛⣻⡿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣩⣿⣿⣶⣿⡄⠀⣤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠙⠿⣷⣦⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⢛⡉⠽⣿⣿⣿⣿⣷⣄⠀⢬⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⡠⢂⣤⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⢼⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢶⣏⠀⣠⣾⣿⣿⣿⣿⣿⣿⢿⣿⣦⡑⠂⠙⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠿⢸⣿⣿⡿⠛⠛⠟⢫⣥⣄⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣿⣟⡁⠀⠘⢣⡬⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣸⠁⢸⣿⡿⠇⣠⡶⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠁⠀⡿⢀⠋⠁⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⢫⣹⣿⣿⣧⣿⡏⢿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠠⢼⣿⣿⣶⣶⣶⣤⠀⢠⣶⣿⣿⣯⣿⣿⣿⣿⡿⠈⢳⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⢿⣿⣿⠀⢸⢿⣏⠙⠉⠉⠈⣿⣿⠇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡿⠀⣤⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠈⡿⠀⠾⠀⠹⣦⣀⠠⢀⣿⡿⠀⠀⣯⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⣯⣽⠃⢀⣿⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣦⡀⠀⢀⠀⠀⢸⣿⠀⠻⡿⠃⠀⢰⣿⣿⠟⠙⠷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⡯⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠸⠿⢆⠜⠁⠀⢸⣿⡇⠀⠀⠀⠀⣨⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣠⣾⡇⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⢲⣶⡤⢶⣖⣲⣿⡇⠀⢤⠀⠀⠀⣸⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣈⠉⠀⠀⠈⣹⣿⣿⣶⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡎⠉⣿⣿⣿⢈⣧⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣟⣿⣿⣿⣿⣿⣿⣾⡟⠀⢴⣿⣿⣿⣿⡃⠻⢶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⣿⣿⣏⠱⡄⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡐⠆⠈⠀⢻⣿⣧⠀⠀⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡟⠁⠀⠀⠀⢸⣿⣿⡞⣿⣯⣿⣿⣿⢻⡄⠀⠀⠀⢈⡟⣿⡆⠀⠀⠸⡿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣾⣿⠧⠀⠀⠀⡄⢸⣟⡻⣷⣿⡏⠫⠈⠉⠀⠙⠀⠀⠀⠈⡁⢻⣿⣄⠀⢸⠀⠀⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣹⣿⣿⡏⠀⠀⠀⠀⠃⣼⣿⡁⡿⢻⡇⠀⠀⢫⣿⡆⠀⠀⠀⠀⠓⣼⣿⣿⡄⢸⠀⠀⠈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣹⣿⣿⠇⠀⠀⠀⠠⠀⣿⡿⡇⠀⢠⡄⠂⠀⠀⠉⠻⠀⠀⠀⠀⠘⠻⡿⣿⣧⢸⠀⠀⠀⠈⣿⣿⠻⢆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⠇⡇⠀⠈⠁⠀⠀⠀⠁⠀⢧⠀⠀⠀⠀⠀⡇⠘⠟⢸⠀⠀⠀⠀⠈⠘⣦⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣸⣿⠟⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⣸⠀⠀⠀⠀⠀⠀⠙⠂⠸⠟
⠀⠀⠀⠀⠀⣰⡿⠃⠀⠙⡟⠁⠀⠀⠀⠀⠀⢸⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢟
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣾⡄
⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢿⣿⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡉⠀⠤⠄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢿⣎⡠⣊⠁⠀⠶⣬⡀⠊⠚⣯⣠⠾
⠀⠀⠀⡍⡶⣤⣶⣅⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠳⢶⠂⡙⠢⢌⠂⠱⣎⣏⢳⣇
⠀⠀⠀⠳⣤⠀⠛⠓⠾⢄⡀⠀⠈⠑⢲⣤⡄⠀⠀⠀⠈⠻⣆⠈⢻⡆⢸⡟⠀⣻⡀
⠀⠀⠈⣷⣤⠀⠀⠀⠀⠈⢻⣀⢀⠀⠀⠀⡠⠇⡀⠀⠀⢘⣿⠀⣿⡇⢸⣿⠀⣿⡇
⠀⠀⠀⠿⠿⠛⠓⠒⢦⣤⣼⠉⠀⠀⠑⠒⢷⣦⣽⡄⠀⠘⣿⠀⣿⡇⢸⣷⠀⣿⡇
⠀⠀⠀⠀⡀⠀⠀⠀⡾⠟⠿⣷⠀⠀⠀⢀⣿⠛⠛⣿⠀⢸⣿⠀⣿⡇⢸⢿⠀⣿⡇
⠀⠀⠀⣟⣁⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⣿⡆⠈⣿⠀⣿⡇⢸⣿⠀⢹⡇
⠀⠀⠀⠀⠈⠳⡀⢸⠀⠀⠀⢿⡀⠀⠀⣸⠃⠀⠀⢻⡇⠀⣿⠀⣿⡇⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⡸⠀⠀⠠⢸⡇⠀⢀⡏⠀⠀⠀⠘⣿⡶⠿⢴⣿⣷⣾⣿⣶⣿⣷⣶⡆
⠀⠀⠀⢀⡀⢸⣿⠵⠀⠀⠐⣸⣿⠀⡼⠀⠀⠀⠠⠞⣿⣷⣶⣶⣶⣤⡤⠈⠉⠑⠟⠛⠛⠀⠀⠂
⠀⠀⠀⢸⣇⣸⣿⣦⡀⣤⣤⣿⣿⣷⡕⢀⠀⠀⢀⠀⣹⣿⣇⣿⠿⣿⣻⣶⡶⠓⣶⣦⡴⣼⠒⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⠿⢷⣴⢶⣶⣿⣾⣿⣿⠻⢶⡆⣿⡿⢭⠅⣾⣯⣽⣾⣿⣤⣤
⠀⠀⠀⣿⣿⡛⠛⣻⡿⠫⠭⠿⠨⠭⠿⠿⠁⣿⡟⠋⠙⠙⠀⣉⣀⣯⣀⣀⣀⠀⢀⠀⣼⣿⠿⠿
⠀⠀⠀⣉⣉⣉⣉⣙⣤⣖⣦⣀⣀⣀⣴⣶⣦⣏⣀⣄⣀⣸⣭⣿⣽⣧⣤⣼⣿⣷⣿⣟⣁⣀⣀⣀
⠀⠀⠀⠿⠿⠭⠽⠤⠤⠤⠤⠤⠽⠿⠷⠿⠥⠼⠷⠿⠶⠤⠤⠶⠤⠤⠴⠴⠽⠷⠤⠤⠽⠷⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⡔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠼⠻⠿⠙⠻⠏⠷⠻⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠓⠶⠁⠀⠠⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡖⡆⠀⠀⢠⠂⠀⠘⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠂⢀⠤⠀⠀⠀⡀⠀⠀⣀⠼⣀⡆⠁⠀⢀⠰⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠂⠀⠀⠐⠂⠀⡄⠂⠀⠐⡠⠞⠑⠚⠋⠀⠀⠈⠱⠀⠀⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣓⠆⣀⠀⠀⣠⠾⠀⠀⢀⠀⠀⢣⠀⠀⠀⠈⠈⡎⠒⣊⢁⡠⠄⠀⢀⠀⠀⠀⠀⠀⠣⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⡄⠀⠀⠀⠀⣸⠁⠀⠀⠉⠘⠀⠀⠈⡴⢃⠀⡼⠁⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣜⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⡀⠀⠀⢀⠀⠀⢠⠤⢀⣀⣠⠂⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⢾⠀⠀⠀⠁⠀⠀⠀⢠⠀⣠⠟⠀⠙⠒⢶⣁⡴⠋⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠂⠀⠀⢈⣠⠏⠐⠠⠚⠙⠂⠀⠀⠤⠖⠉⣲⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀⣀⡤⠤⢾⡁⣠⠀⠀⠴⠂⠀⠀⡀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⢏⡀⡀⠀⠀⠀⡶⠃⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⠉⢹⠂
⠀⢠⠖⢦⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡛⠛⡇⠀⠀⡞⠉⠰⠿⡄
⣠⠞⠛⠉⠛⡏⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠾⣇⠀⠀⠛⠛⠛⠛⠁
⢷⣄⣠⡤⣴⠥⡴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠖⣦⠀⠀⠘⠒⠒⠒⠋
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⠴⢧
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛
⠀⣿⠀⠀⢸⠀⡇⠀⠀⣠⡄⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⣤⣤⣀⡀
⠀⣿⠀⠀⢸⠀⣇⣀⣰⠃⠸⠋⡰⠟⢛⣧⣠⡞⠉⠙⠋⡉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠉⠓⢦⡀
⠀⣿⠀⠀⢸⢠⠋⠘⠉⠁⠀⠀⠀⠐⠋⢱⡜⠀⠀⠀⡐⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠙⣄
⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣆
⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠈⢀⠇⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⠶⠤⢤⣀
⠀⣿⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⣿⠀⠀⢸⠈⢢⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢠⣞⡀
⠀⢻⠀⠀⢸⢰⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠈⡅⢹
⠀⣿⠀⠀⢸⠸⠤⠤⠤⠤⠤⠤⠤⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢸⠀⠄⠀⡧⠼⢶
⠀⣿⠀⠀⢸⢤⣤⣤⣤⣤⣤⣤⣤⣬⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⣣⠀⣠
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⠀⠀⠀⢀⣸⣀⣀⣀⣀⣀⣀⡀⠀⠀⣇⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⣿⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⢸
⠀⠙⢶⡶⠾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⠞
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⢒⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠒⠒⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⣀⡠⡀⠀⠾⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠋⠈⣭⢀⠂⠁⢀⣗⠒⠄⣀⠀⠀⠀⠀⢀⡠⠒⠉⢁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣽⣿⣷⣀⣀⡙⠘⠀⠀⠀⠘⠋⠲⡄⠹⠶⠤⢐⠏⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡀⠀⠀⠀⡠⠊⠀⢰⣿⣿⣿⣿⣿⣿⢿⠟⠀⠀⠘⠃⠀⠦⡀⣀⠔⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣪⢻⣝⠣⠤⢶⡎⠀⠀⣀⠼⣿⣿⣿⣿⣿⣿⣶⣶⣿⡄⠀⠀⢀⢀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⣷⣻⡿⠿⢿⣏⣶⣯⡀⠀⢙⡿⠛⣻⠋⠀⠀⠀⠉⠳⡤⠊⠁⠀⠀⠀⠀⠀⠈⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠀⣀⠀⣾⣗⠋⠉⠿⡍⠁⠲⠶⣅⡀⠀⠀⡀⠀⠀⠈⢦⠀⠀⠐⠂⠤⢀⡀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⠞⣋⣭⣄⣯⢄⠀⠀⠀⠘⠲⢤⣙⢦⡙⢄⠀⠇⢀⠔⠀⠀⢣⠀⠀⠀⠀⠀⠈⠉⠒⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠴⡻⠃⢠⠄⣈⠑⠛⠒⠶⠤⡀⠀⠀⠉⡿⢋⠑⢄⣡⡊⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡻⣄⣀⠛⠐⠉⠀⠀⡀⠀⣆⡈⣑⠤⣀⢁⠃⠀⠀⠀⠙⠀⠀⢦⠀⠃⠀⠀⠀⠀⢠⠔⠒⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣛⣆⡙⢯⣳⣤⣦⠮⠤⠤⠋⠙⣌⠠⡺⠦⠤⡀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⡠⠄⠈⠙⠓⠒⠚⠻⡒⠒⢦⡀⠀⠙⢄⢀⠜⠀⠑⠒⠤⣈⡀⢀⡠⠊⢀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⡔⠁⠀⠴⠋⠀⠀⠀⢰⠀⡴⣞⣂⡟⣽⣦⡙⣄⢀⡔⠁⠀⢀⡠⢤⠆⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠳⢭⣥⠤⣿⡏⣧⠜⠁⠠⠔⠊⢁⠔⠁⠀⠀⡠⠊
⠀⠀⢀⣌⡠⠔⠒⠃⡉⠉⠐⠒⠒⠤⢄⡀⠀⠀⠀⡿⣇⡏⠀⠀⠀⠀⠀⠊⠀⠀⠀⠊
⠀⠀⠛⠒⠨⠷⠤⣀⠉⠂⠀⠀⠀⠀⠀⠈⠑⠤⡀⠈⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⣀⠀⠀⠤⢀⣀⣀⠈⢦⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠐⠒⠠⠤⠼⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⣀⡀⠀⠀⠀⡀⠀⣀⠀⠀⡀⣄⠀⠀⢀⡀
⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠷⠶⣆⡉⣒⡄⠀⠈⠻⢦⣄⣈⠉⠹⣧⡄⠸⡇
⠀⣬⢱⣷⣦⢾⡁⠐⠐⠂⠀⢀⠀⠀⡀⠀⣈⠈⠉⠸⡶⢛⣔⠀⢀⠻⡌⢶⣹⡇⢠⣿
⠀⠻⣆⣄⠈⠛⠢⠶⣤⣀⠀⠀⠈⠒⢠⠀⣈⠃⠀⠀⠀⠀⢹⣧⠀⠰⣿⠀⢹⡟⠀⢻⡆
⠀⣷⣌⣁⠀⠂⠠⠀⠘⢻⣇⡀⡀⠀⠈⠀⠈⢰⡄⠀⠀⠀⠘⣿⡇⢸⢿⠀⡟⣷⠀⠿⣷
⠀⢛⢿⣯⠤⣆⣤⣄⣀⡐⣿⠋⠑⠘⠦⠤⣄⣀⠒⣷⡀⠀⠀⣿⡇⢸⣿⠀⠃⣿⠀⡀⡿
⠀⠀⠁⠀⠀⠀⠀⣽⣿⣷⣿⡀⠀⠀⠀⠀⣬⣿⣿⣿⣧⠀⢠⣽⡇⢠⣽⠁⣆⣿⠀⡒⣇
⠀⠀⠀⢀⠀⠀⠀⠉⠉⠉⣽⡇⠀⠀⠀⠀⢸⠀⠀⢹⣿⠀⠘⢿⡅⠠⣿⠀⣃⣿⠀⡯⡟
⠀⠠⠀⡀⠂⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠐⠙⠀⠀⢸⣿⡄⠀⣿⡄⢸⣿⠀⡇⣿⠀⠅⡇
⠀⠀⠀⠀⢻⡀⠐⠀⠀⠀⠸⣿⠀⠀⠀⡀⡇⠀⠀⠀⣿⣇⠀⣻⡃⢸⣿⠀⣇⣿⠀⢄⡷
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⡆⠀⢀⠀⠀⠀⠀⠀⢹⣿⡾⠿⢗⣿⣿⣷⣿⣿⣾⣿⣿⣿⣶⡆
⠀⠀⣀⠀⢸⣿⢰⠀⠀⠀⢆⣼⣧⠀⠀⡀⠀⠀⠀⡠⢺⣿⣷⣴⣶⣶⣤⣤⡄⠋⠋⠳⠿⠛⠛⠛⠂⠒⠐
⠀⠀⣿⡇⣼⣿⣆⠀⣀⣀⣿⣿⣿⣷⡤⠀⠀⠀⠀⠀⠊⣿⣿⡍⣿⣿⣿⡷⢶⣤⡤⠶⢦⣦⣀⣄⣶⠴⠂
⠀⢶⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣇⣠⣀⣀⣴⣧⣾⣿⣿⣛⣩⣥⢸⣟⡜⠻⠃⢴⡿⣛⣯⣶⣿⣀⣀
⠀⣼⣿⣟⣻⡿⣿⣿⡿⠻⢿⠍⠽⢶⣷⣶⡎⣿⣽⣿⠟⠿⡿⠐⠲⠋⢼⠏⠁⠉⠙⠛⠛⠛⠻⣿⣿⣿⣿
⠀⢿⣿⣿⣤⣴⣾⠛⢉⡉⠉⠀⠉⠉⢉⣉⠁⣿⠟⠀⠀⠀⡠⢤⣴⠶⡾⠓⠒⣦⣶⣂⣤⣄⡾⠛⠋⠉⠉
⠀⣤⣤⣤⠤⡤⠼⠶⠯⠽⠶⠶⠶⠶⠾⠿⠷⠿⣦⠶⡤⠦⠷⠶⠿⠶⠷⠶⠶⠿⣿⡿⠿⠿⠶⡴⠦⢤⣤
⠀⠚⠚⠓⠒⠛⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠒⠛⠛⠛⠒⠒⠒⠛⠒⠒⠒⠒⠚⠒⠓⠒⠒⠒⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⢆⣤⣦⣄⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⣴⢎⣵⣶⣮⣿⣶⣠⣤⡀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢾⣿⠿⠿⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⠙⢻⣿⣿⡉⢻⣿⣿⣿⣇⡀⠀⠀⠀⠀⠘⣿⣯⣶⣿⣟⠿⠟⠻⣿⣟⡛⣿⣿⡀⠀⠀⠀⠀⢾⣿⣷⣴⣾⣷⣎⡻⣿⣿⣿⣵⣿⣿⣦
⠀⠀⠀⣾⣿⣿⡟⠉⠻⠗⠈⢩⣿⡏⠹⠃⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⣴⣿⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢙⣿⣿⣋⣀⣶⡆⠀⢻⣿⣤⣶⠶⣆⠀⠀⠀⠀⠀⠀⣰⣞⠿⣻⣿⡿⠛⣵⠿⠿⠿⠋⠀⠀⠀⠀⢰⣿⡀⠈⠉⣻⣿⣿⣿⣿⢿⣿⣟⣉⡀
⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⣙⣛⡛⢉⣽⣶⣤⣙⣿⣦⠀⠀⠀⠀⠀⠀⠈⠛⣵⣶⣼⣿⣿⣧⣝⣿⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠉⠁⠀⠻⠿⠿⢿⣿⣿⣧⣾⡦⠀⠀⠀⠀⠀⠀⠘⢿⡿⠻⣾⣿⣟⠛⠿⠿⠟⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⡟⢿⣯⣿⡿⣿⣦⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠳⠄⠀⠀⠀⠀⠀⠀⠸⠏⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⠉⠛⠋⠀⠹⠾⠟⠛

⠀⠀⠀⠀⢀⡾⠿⣿⣦⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡄⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣆⣀⣠⣤⣤⡀
⠀⠀⠀⢀⣈⡓⠂⢈⣟⡛⠛⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣏⣿⣟⡛⣟⣛⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⢿⣿⣿⠿⣾⣿⢿⣾⠟⠻⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⢋⣼⣿⣽⣿⣿⣃⣄⡀⠀⠀⠀⣀⡀⠘⠿⠿⠿⢿⣽⣿⣿⣿⡇
⠀⠀⠀⢰⣿⣿⡷⠀⣩⠶⣶⣾⣿⣤⣄⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣸⣿⣿⣷⣦⣄⠺⢏⡙⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠉⢉⢷⣮⣥⣶⣼⣿⣿⠇⠛⠀⠀⠀⠀⠀⣴⣶⣼⣽⣿⣿⣧⢌⣛⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣄⣉⣛⣿⣷⣿⣿⣿⣯⣤⡀
⠀⠀⠀⠀⠰⣿⣿⣿⡿⠛⢛⣥⣴⣶⣾⣿⣷⠄⠀⠀⠻⠿⡿⠿⣋⣿⣱⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠨⣿⣿⣟⣻⠿⠿⠟⢋⡁⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠈⣟⢿⣧⣾⣿⡿⡿⣹⣿⢿⣸⡿⠀⠀⠀⠀⠀⠀⢾⣿⠛⠿⠿⠿⣿⣿⣷⢬⡀⠀⠀⠀⠀⠉⠉⢿⠿⣿⣾⢿⣿⣿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠖⠛⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⢟⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⣸⣿⣿⣿⢇⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣷⢄⣤⣄
⠀⠀⠀⢀⣼⡿⣫⣭⣭⣻⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠻⣾⣿⣿⣶⡄
⠀⠀⠀⢋⣴⣶⠿⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⣠⣿⣿⡿⠿⣿⣿⡝⢿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⣘⣛⣡⣶⣶⣜⣿⢿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣛⢛⣭⣝⣬⣿⣿⠀⠀⠀⠀⠘⠿⣿⡿⢃⣴⣶⣶⡙⣿⣷⣿⣿⣿⣿⡇
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣭⡵⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠸⠿⢿⣿⣦⣻⣿⣿⣿⣿⠟
⠀⠀⠻⣿⣿⠿⣿⡿⣿⣿⣿⣿⣿⣋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⣼⣿⣇⣠⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣷⣴⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣫⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢶⣬⡿⠟⠛⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⢒⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⢀⣀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠴⠒⠋⣹⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣼⣀⠤⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠉⣀⣠⠤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣁⡤⠔⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⢢⡤⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣴⣋⣀
⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⡞
⠀⠀⠀⠙⢶⣖⠒⠒⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠈⢻⡳⡀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠾⢶⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢳⡘⢦⡀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⠋⠀⠠⡏⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢧⠘⣷⡄⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢰⢄⡔⢆⡇⢱⡄⠀⠀⠱⣄⣀⣄⠀⢲⣠⣾⣟⣤⠖⠀⠀⠀⠀⣀⠔⢹
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠙⠋⢈⣡⠬⠽⠯⠥⠧⢞⠓⠂⣸⠋⢉⣿⣿⣿⣟⠁⠀⣀⣠⣴⣊⠅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡴⠊⠁⠀⠀⠀⣀⡠⠴⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢸⡍⣛⣛⣓⣲⡶⠤⣄⣀⡤⠖⢿⡇⠀⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⡯⣟⡻⣿⣽⡟⠛⠉⠁⠀⠀⠀⠁⢀⡤⠚⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡧⣔⣲⢏⢌⡝⣳⣢⣤⣀⣀⣠⣶⣯⣴⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡵⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⡿⠛⠓⢤⡀
⠀⢶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠿⣛⡛⠛⠒⠲⠶⣶⣤⣤⣤⣴⣶⠟⣛⣛⣛⣛⣛⣿⣿⣿⣛⣛⣛⣛⣛⣛⣭⣭⠤⠤⠤⠴⠶⠞⠳⣄
⠀⠠⠤⠤⠬⠭⠭⠿⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠉⠉⠉⠉⠹⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⡀⢳⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⡤⢴⣶⣶⣶⢦⣄⡤⠖⠛⣛⣛⠲⣄⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡫⢵⣾⠯⠿⢿⣡⠞⠛⢀⠈⠙⢦⠝⠁⠀⠚⠛⠛⠲⠬⠺⠿⣿⣯⣭⣭⣟⠙⠷⡆
⠀⠀⠀⠀⠀⠀⠀⡰⢫⡴⢿⣴⠥⡨⠋⠉⠀⠽⠚⠋⠚⠀⣄⠀⠀⠀⢠⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠈⠻⡅⠀⠿⠷⢶⣄⡀
⠀⠀⠀⠀⠀⢠⢞⢵⠏⠴⢖⡟⠀⠁⠀⡀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⡇⠀⢠⠁⠀⢠⠀⢀⠤⡀⠉⠀⠈⠙⠂⢳⠇
⠀⠀⠀⠀⠀⡇⣏⠄⣠⢠⠀⠠⡆⠀⡄⡇⠀⠀⡀⡇⠀⠀⢸⠆⠀⠀⠸⣄⣸⠇⠀⣼⠀⠀⡾⢰⠁⠀⡇⢀⣶⠀⠀⠀⠸⡝⣶⣄
⠀⠀⠀⠀⠀⠑⣼⣧⡌⢀⡀⠀⠘⣎⠀⠙⠄⢀⠇⠹⠐⢲⠤⠁⠁⠀⣠⣺⣁⠀⠀⠀⠀⠀⠃⠸⣀⠔⢁⠞⢸⠀⢀⡔⢠⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠣⣿⣛⠷⢉⠆⠀⠘⠆⠀⠀⠀⠀⠀⠀⢠⠀⢠⡀⢀⠯⠭⠿⠀⣰⣤⣄⠀⠘⠀⠀⠀⠉⠀⢸⡠⠋⢀⣴⣯⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⡠⣠⡄⣠⣤⣄⣈⣈⣽⣷⣾⠨⢸⣿⡧⠀⠀⠀⣿⣿⣿⡿⠤⠤⢄⣀⣀⣀⢤⣌⣁⣰⣿⣿⢛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢽⣿⣿⠇⡾⡿⣏⠀⠀⡿⡟⠀⠘⡿⣇⣀⣀⡀⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣾⣟⠛⠻⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⠀⠀⣳⡇⠀⢀⣿⡇⠀⠀⠀⠿⢿⣤⡅⠀⠀⠈⢹⣿⣿⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢻⣿⣿⢀⠀⣿⡆⠀⠘⣿⡧⠐⠀⠀⠀⠨⠄⠃⠀⠀⠀⣸⡿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⠀⣸⣿⠋⠀⢀⣿⠄⠤⠄⢫⣄⡀⡀⠀⠄⠀⠀⠀⠀⠀⠀⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠿⢿⡄⠀⣼⠃⠂⢠⣶⠁⠀⣁⢡⡜⠉⠤⠀⠀⢽⠀⠄⣿⠀⣀⠀⠀⣶⣦⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣽⣿⣿⣿⣦⣾⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣾⣿⣿⣵⣼⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉
⠀⠀⢀⢀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠘⠂⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠛⢷⡄⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢿⡄⠀⠀⠀⠀⢠⡿⠙⣧⡀
⠀⠀⠀⠀⣾⠛⠛⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠈⠛⠶⣦⣤⠶⠟⠁⠀⠈⠻⣦
⠀⠀⠀⠀⣿⠀⠀⠀⢹⡇⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣾⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠙⣷
⠀⠀⠀⠀⣿⠀⠀⠀⢹⣧⡶⠟⠛⠉⠉⠛⠻⢶⣄⠀⢿⣄⠀⠙⠻⣦⡀⠀⠀⠀⠀⢀⣴⠿⠋⠀⣠⡿
⠀⠀⠀⠀⢻⠀⠀⢀⡾⠋⠀⠠⠶⠶⠶⠶⠀⠀⠙⢷⡀⠉⠻⣦⡀⠈⢿⡶⣦⣴⢶⡿⠁⢀⣴⠟⠉
⠀⠀⠀⠀⣿⠀⠀⢸⣧⣤⣀⣤⣀⣀⣀⣠⣀⣀⣀⣼⡷⠶⠶⢿⡿⠶⠛⠁⣾⠃⠀⠻⠶⢿⡿⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⣿⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⣿⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⠀⠀⠀⣿⡄⠀⠀⠀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠏
⠀⠀⠀⠀⣿⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⢶⣤⣴⠾⠷⣦⣤⡴⠟⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⢹⡏
⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡀⣀⡀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⡀⣸⡇
⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢁⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⢠⡈⠤⠒⠒⠒⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣯⣄⣴⡀⠀⣹⢀⣀⣀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⠉⠈⠛⠿⡿⠞⠋⠈⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡚⢿⣷⡀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢃⣿⣿⠇⠀⣿⠀⠰⠞⠛⢉⡉⡁⠀⣀⣉⠉⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡏⠀⠀⠤⣿⠀⠰⡃⠁⢈⣿⠀⠀⠰⣇⠀⢀⣿⣾⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠐⢸⡆⠀⠙⠒⠛⠃⠀⠀⠀⠙⠛⠋⠁⣿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⣿⠐⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠋⠀⠀⠀⠀⠈⠀⢻⡇⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠⠏⠀⠁⠀⠀⠀⠀⠀⠈⢉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣤⣤⠄⠀⠀⢀⠎⡀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡖⠸⣧⡀⠀⢼⣷⠀⠀⣠⠋⠀⠀⠀⢀⣀⠀⠀⣿⣿⣿⡀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡀⠀⡀⠀⠀⢀⠔⠈⠀⠀⠃⠀⠀⢍⠉⠙⣿⠉⠉⢁⠀⠀⠐⠐⠈⠁⠀⠀⢿⣿⡿⢉⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⢌⣶⠞⡙⢣⡀⠀⠀⠀⠀⠀⠀⢷⡀⠛⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⣡⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠐⡾⢸⢀⣐⢀⠴⡆⠐⠂⢄⠀⢸⠀⣧⠐⠃⠀⠈⠁⠁⠀⠀⠀⠀⠀⠐⠀⠀⢰⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡟⠁⢍⠘⢎⣀⡁⠀⠐⡀⠀⠀⠀⡸⠀⠙⠀⠀⠀⠀⠀⠀⠄⠀⣴⣶⣦⠀⣀⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠜⣩⡅⡾⡒⢧⡀⠀⠀⠀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠚⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠛⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠠⢤⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⢀⣤⠖⢛⣉⠁⠀⠀⠀⠉⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⢠⡞⢁⡞⠋⠀⢀⡴⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣾⠀⢸⡇⠀⠠⡏⠀⠀⠀⠈⠉⠳⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡖⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⢻⡄⠈⠳⣄⠀⠙⢦⠀⠀⠀⠀⣠⠇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠙⡆⠀⠈⢳⠀⢸⡇⠀⢀⡞⠁⠀⠙⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⣼⠁⠀⢀⡞⠀⡼⠀⠀⢾⠀⠀⠀⠠⣄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁
⠀⠀⠀⠀⢰⡇⠀⢀⡞⠀⢸⠃⠀⠀⠈⠳⠄⠀⠀⠈⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⡏
⠀⠀⠀⠀⠈⣧⠀⠘⣇⠀⠈⠧⣄⡀⠀⠀⠀⣀⣠⡴⠃⢠⡏⠀⢀⣤⣤⣤⣤⡤⠴⠶⠶⠏⠀⠀⠀⠀⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠘⢦⡀⠈⠳⠦⠀⠀⠙⢦⠀⠀⠉⠀⣀⡴⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠦⢤⡄⠀⠈⡇⠀⠀⢸⡇⠀⠀⠀⠀⢸⡷⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠀⠀⠘⢧⡀⠀⠀⠀⢸⣇⣀⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠘⠶⣤⣀⣀⠙⠶⠶⠖⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠈⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣤⣀⣀⣀⣀⣠⡤⠶⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⣤⣤⡀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⣿⡀⢈⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⠛⠋⠁⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⡀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠁⠐⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡤⡀⠈⠀⢸⠀⠀⡇⠀⢀⠚⠠⡀⢃⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣑⠀⠀⠢⢄⠀⡀⢟⢦⡁⠡⣉⠄⠬⠓⠂⠉⠉⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢋⡀⠀⠠⣤⠴⠁⠈⠀⠙⠁⠀⢀⢀⡀⠠⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⠉⠐⠀⠬⠄⠒⠠⠒⡡⠚⢀⠔⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢠⡦⡄⠀⠀⢩⡉⠉⠉⠀⠀⠀⠐⠒⠠⠀⢀
⠀⠀⠀⠀⠀⢀⡐⠇⡣⠈⠁⢀⡆⠤⠜⠋⠁⠀⡀⠀⠀⠀⠀⠞⠀⠄⠀⠀⠠⠶⠦⢤⣀⡀
⠀⠀⠠⠄⠀⠊⢡⠈⠁⢀⠄⠉⡀⡀⠀⠀⣄⡴⠗⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢀⣀⡄⠤⠉⠀⠐⠀⠄
⠊⠀⠀⠀⠀⠀⠃⢀⠴⠃⢠⠀⠀⢐⠀⠈⠁⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⢉⡉⢳⡀⡀⠀⠀⠁⠢⢌⡁⠀⠠
⠀⠀⠀⠀⠀⠀⣠⠆⢀⢠⠤⠥⢠⠃⠀⢀⡼⠠⠯⢴⡞⠀⠀⠀⠀⢴⡋⠀⠀⠐⠀⢁⡌⠀⢱⡄⠀⠀⠀⠀⠙⠦⡀
⠀⡀⠀⠀⢀⠔⠀⠀⠀⠈⠑⠦⢄⠀⢠⠆⠀⠀⠂⡀⠀⠀⠀⣀⠈⠀⠁⠀⠠⠀⡀⢀⡆⢀⡴⠀⠀⠀⠀⢀⣠⣤⠕
⠸⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠤⣀⡠⠐⠒⠓⠻⠖⡲⠎⠀⠑⢄⣀⣤⠲⢖⣿⠁⠀⠀⠀⠀⡠⢂⠞⢁⠃
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠁⠀⠀⡠⠛⠈⠉⠙⠓⠀⠀⢀⡠⠈⠐⠁⡠⠞⠠⢄
⠀⠀⠈⠉⠉⢀⡠⠖⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠂⠁⢺⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡁⠈⠁⠀⠠⡀⠀⢠⣇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠟⠀⢼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠐⠄⠀⠀⠀⠀⠐⠀⣤⡄⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠔⠋⠑⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⠀⠀⠈⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⢸⠀⠀⠀⡇⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠉⠉⠉⠛⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⣄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠄⣴⣦⣤⣄⠀⠀⠈⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠛⠁⢠⣿⣿⣿⣿⣷⣤⣸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⡀⠀⠘⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣶⣿⣯⣽⣿⡀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠉⠻⣿⣿⣿⣿⣷⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠙⡩⠁
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠁
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠹⡿⠋
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⢺⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣿⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⣠⣾⣿⣤⣿⣿⣿⣃⣰⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣼⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⣿⢉⠟⠀⠀⠈⠉⠙⠿⢿⣿⣿⣿⣏⣹⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⡀⣄⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢷⣤⣤⣄⣠⣉⣈⢳⡿⠿⠿⠟⠀⢀⣴⢿⣿⣿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⡿⣿⣿⠁⢸⣿⣿⣷⣧⣌⡓⠒⣿⣿⣿⡟⣻⡿⠟⠋
⠀⠀⠀⣀⣴⢲⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣟⣻⣿⣿⣅⠋⢻⣿⣷⣀⡀⠈⠉⠉⠃
⠀⢦⡾⣿⣿⣿⣻⣿⣿⣿⣹⣿⣿⣶⣿⣿⣿⣟⣿⣿⢻⣶⣿⣿⣿⣿⣿⣆
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠈⠻⣿⣻⣿⡿⢿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⣿⣿⣿⣟⣹⢿⣿⣿⡿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⣿⣿⣷⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⢲⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢀⡠⠞⠉⡠⠊⢀⣮⠴⠚⠋⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠋⠀⢀⠞⠀⣠⠊⢁⣤⠴⠶⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⣠⠿⠖⠚⠉⠉⠉⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⠒⠐⠤⡀⠀⠀⣤⣼⠃⢰⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⡠⠒⠐⡆
⠀⠀⠀⠀⠀⠈⠑⠒⠒⠛⠀⢰⠏⠀⡷⠋⠀⠀⢀⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⠓⠒⠒⠁
⠀⠀⠀⠀⠀⡔⠋⣹⠖⠀⠀⡯⢀⣼⠁⠀⠀⠀⠈⣋⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡆⠈⣇⠀⠀⠰⢎⡋⢳
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⣏⡠⣿⠀⠀⠀⣰⠋⠁⠀⠀⠙⢦⠀⠀⠀⡀⠀⢀⡤⠲⢤⠀⢹⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣹⡀⠀⠀⣇⢀⣀⣠⣤⡤⢾⠀⠀⠀⡁⠀⢸⠧⠤⢤⠇⢸⡗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⢧⠀⠀⠻⡉⠀⠀⠈⣠⠏⠀⠀⠀⢧⠀⠀⠑⠒⠊⠀⠀⣇⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠦⣈⡄⠀⠀⠉⠓⠒⠛⠁⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⣠⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣠⣾⣿⣟⣲⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⣏⡹⢿⣿⡟⠀⠀⠀⢀⡴⠋⠀⠀⠀⡞⢙⡧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠸⡆⠀⠀⠀⠀⠉⠙⠻⠁⠀⢀⡴⠋⠀⠀⠀⢸⠉⠑⣺⠀⡸⠓⡄
⠀⠀⠀⠀⣀⣤⣄⡤⠖⣋⡭⠖⠛⠛⢻⠖⠒⢷⡀⠀⠀⠀⠀⠀⠀⢀⣤⠏⠀⠀⠀⠀⠀⠸⡈⠘⠟⠚⣧⠜⠁
⠀⢀⡴⠋⠁⠀⠀⠀⢾⠁⠀⠀⠀⠀⠈⠛⢻⡟⠙⠶⢤⣤⣤⡤⠶⡋⢳⡄⠀⠀⠀⠀⢀⡴⢁⡤⠂⠈
⠀⠈⢣⡔⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠿⣦⣤⣤⣶⡭⠖⠋
⠀⠀⠈⠓⣾⡁⡼⠓⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠈⠉⠁⢀⡀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡼⢋⢿⡄⠀⠀⠀⠀⣿⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣠⠟⡵⠃⣠⢷⡀⠀⠀⣀⣿⡄⠀⠀⠈⠉⠙⠛⠓⠶⠦⡾
⠀⠀⢀⡾⢫⠞⢠⣿⢷⡊⢧⠀⣺⣏⠈⠳⣄⣠⠀⠀⠀⠐⠢⢤⣾⠁
⠀⣰⠟⡴⢧⠀⡼⠋⠀⢷⡈⣋⡽⠛⠶⠚⠁⠉⠛⠶⢤⣤⡴⢾⣿⠶⢤⣀
⣾⣣⣎⣠⡼⠛⠁⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠆⣸⠇
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠓⠞⢿⢿⠛⠓⢲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣝⣒⣒⡒⠄⠤⠤⠏⣀⣈⣳⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣛⣛⣛⣷⣶⣾⣶⣿⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠿⠁⣾⡿⢫⣴⡿⠃⣰⡿⡛⠀⣀⣠⣴⠟⣰⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⢸⠟⢀⣴⠟⢁⣶⢏⣾⢁⣠⡽⠛⠁⢸⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⢀⡿⢀⣿⠋⢀⠜⢁⣼⣿⡋⠁⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣼⠃⡾⠃⢠⠊⣰⠟⠁⠙⠿⠷⠶⠂⠀⢸⣿⡇⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⡏⢸⠇⠀⠃⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠇⠘⢀⠂⡼⠁⠀⡆⠀⠀⠀⣠⣤⠶⣶⣾⣿⣷⣦⣤⣬⣧⠉⠻⣿⣿⣿⡏⠀⢠⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠎⡸⠁⠀⢰⡇⠀⠀⣾⡏⠀⠀⠀⣿⣿⣶⣭⣿⣿⢿⣷⡀⠈⢻⣿⣿⣿⣯⡶⠋⠀⠀⠀⠀⠀⠀⣰⣿⠟⣽⣿⣿⣿
⠇⡜⣰⠁⠀⢠⣿⠃⠀⢸⣿⡇⠀⠀⠰⢿⡏⠉⠉⠻⣿⣿⣾⢿⣦⡀⠙⣿⣿⡿⠀⠀⠀⣀⣀⣤⠴⣞⣫⡥⠾⢟⣿⣿⣿
⢀⢡⡇⠀⠀⠀⣿⠀⠀⢸⣿⣿⣮⠀⠐⢸⣿⣄⠀⠀⢰⣿⣿⣿⣿⣿⡀⠈⠻⣿⠻⠟⢛⣛⣭⣶⣿⣿⣿⣃⣴⣿⣿⣿⣿
⢏⡾⠀⠀⠀⠀⣿⡀⠀⠈⢻⣿⣿⣧⣀⢸⣿⣏⢳⣶⣼⣿⣿⣿⣿⣿⣷⣀⣴⣿⣿⣿⣿⣿⣿⠟⠛⣛⣛⣛⣻⣿⡿⢿⣿
⡾⠁⠀⠀⠀⠀⢻⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⡀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣙⣋⣛⣁⠀⣀⣰⣿
⠃⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⣀⣤⣴⣾⠿⣿⣿⢿⣿⣿⣿⡿⣿⣯⣽⣶⣿⣵⣾⣿⡿⢿⣟⡻⣿⣿⣿⣿⣿⣿⣯⠛⢛⣻
⠀⠀⠀⠀⠀⠀⡿⠀⢀⣶⠟⢉⣿⣿⠇⠀⣿⣿⠘⢿⣿⣿⣿⣦⣄⣤⣭⣉⣴⣶⣶⣭⣷⣾⣿⣿⣿⡿⠻⣍⣤⣴⣴⣿⣿
⠀⠀⠀⠀⠀⣼⣇⣴⡟⠁⠀⢺⡟⢿⣧⣤⣿⣿⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣿⣿⡿⠀⠀⠀⠈⠀⠀⠉⠉⣴⣿⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⣿⣿⣿⣍⠛⢻⣿⣦⣍⡙⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣺⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⠉⠳⠆⠻⣿⣿⡧⣄⠀⠉⠛⢿⡻⢦
⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣗⣒⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⠶⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣀⣀⡀⠀⠈⠉⠙⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀⡀⠀⠉⠙⠓⠶⠤⣤⣄⣀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣤⣀⣀⡀⠉⠉⠙⢻⠇⠉⠉⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⡟⠀⠀⠀⠀⢰⡟⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣾⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⢛⣉⣉⣉⣉⣉⣻⠃⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢠⡖⠛⠉⢉⣥⠤⠤⢿⣋⡛⠲⠦⣤⣤⣬⡟⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢦⣀⠈⠳⢦⣄⣈⡛⠲⠶⠀⠈⠉⠛⠲⠶⠦⠴⠟⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⢿⠶⠦⣤⣭⣉⣉⣛⣛⣛⣀⣀⣀⣠⣤⠴⠞⠋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⡿⠉⠉⠉⠉⠉⠀⠈⣧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡼⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣀⣠⡤⠶⠚⠉⢹⠀⠀⠀⠀⠀⠘⣇⠀⠀⣀⣤⣄⣀⣸⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡴⠞⠁⠀⠀⠉⠛⠲⢤⣄⣀⡀⠀⠀⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠈⠓⠲⠾⣭⣁⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣛⣿⣿⣿⣟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⣿⣿⣿⣿⣿⣷⣶⣄⣀⣐⣺⣿⣿⣿⣿⢿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢃⣤⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡿⠛⠋⠉⠉⠉⠉⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⡏⠀⠀⠚⠓⠶⢶⡄⠀⢀⣤⣤⣤⣤⣌⢹⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠈⠃⠀⠀⠋⠛⠙⠛⠃⠀⣾⣟⠿⠶⣴⡿⢻⡿⡛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⣾⠱⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⢰⣃⠀⠀⠓⡄⠀⠀⠀⠒⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣾⣿⠀⠀⠀⠀⣀⠘⢿⡛⠛⠛⠙⢦⠀⠀⢀⣶⣏
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⣽⣿⣿⣿⣿⣄⠦⣀⣁⡴⠁⣸⠃⠐⠀⢰⢦⡄⠀⣼⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠛⢼⡿⣿⣿⣿⣿⣿⣿⣦⠁⠀⠁⢠⡟⠒⠒⣦⠩⡀⢁⣼⣿⣿⣿⣿⣿⣩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⡻⢿⣿⣿⣿⠏⠀⠀⠀⣸⡀⠒⢉⠀⢐⣵⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣌⠙⢿⡟⠀⠀⠀⢀⠟⠁⠰⠋⢀⣾⣿⠇⠈⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢟⠥⠀⠀⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠒⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⣩⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⡿⠁⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⠌⣰⣿⣿⣧⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⠛⣽⣿⣿⠏⠁⡀⠀⠀⠀⢠⣾⡅⢀⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⣤⡀⠈⠳⢄⣠⣿⣿⡇⢸⣿⣿⣿⣿⠀⢠⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⠇⠘
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠿⣿⣿⣶⣄⠀⠉⠙⣿⣇⣿⣿⣿⣿⣿⢀⡿⣹⢇⣿⡿⣽⢿⣿⢏⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣮⣟⢿⣿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⣼⢣⠏⣼⠟⢱⠋⡟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⠃⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠁⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠦⠐⠀⢀⣴⠄⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠁⠀⠉⢉⡉⢬
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣄⣠⠠⢄⠠⣀⠀⢠⠄⣤⡠⡀⠄⠀⠀⠀⠦⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠀⠤⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⡐⠀⢘
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠂⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣤⠀⣠⠀⢀⠀⠀⠠⠠⠤⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⣠⣄⣤⣀⣀⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣴⣦⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣶⣠⠔⠋⣽
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠿⣿⣿⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠉⠀⠀⢸⣿⣦⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⠀⣀⣆⣛
⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣿⣤⣼⣿⡿
⠀⠰⣿⣏⡁⢳⣀⣠⣶⣶⣦⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋
⠀⢸⣿⣭⣴⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣴⣿⡗
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠁⠂
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡴⠒⠚⢫⣥⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣴⠶⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⡀
⠀⠀⣤⣴⠶⠚⠋⠁
⠀⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠚⠋⣉⣉⡂⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⣛⢙⡃⠀⠔⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠛⠛⡗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠰⠤⠤⠤⣾⡀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠙⡏⠙⠉⠉⠉⠉⠋⠉⠋⢹⠁⠀⠀⠀⠙⡋⠙⠉⠙⠉⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠀⠀⠀⠀⠀⠁⠀⠙⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠀⠀⠀⠁⠄⠁⠄⠁⡇⠀⠀⠀⠨⠈⠠⠀⠠⠀⠁⠀⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠁⠃⠁⠀⢱⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠅
⠀⠀⠀⠀⢰⠂⠀⠛⠁⠘⠀⠘⠁⠊⠃⠀⠑⠘⠋⠉⠀⠈⠉⠁⠘⠀⠘⠀⠈⠑⠈⠓⠈⠛⠀⢲
⠀⠀⠀⠀⢸⠀⠀⠀⣏⣀⣀⣈⠀⠀⠀⣇⣀⣀⣈⠀⠀⠀⠀⣏⣀⣀⣀⡇⠀⠀⣇⣀⣀⣈⡇⢸
⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⡀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⠀⢸⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⢇⣀⣀⣀⠀⠀⠀⣇⣀⣀⡀⠀⠀⠀⠀⢇⣀⣀⣀⠇⠀⠀⣇⣀⣀⣠⠇⢸⣀⣀⣀⣀⣸⡀⠈⠳⡀
⠀⠀⠀⠀⢸⠀⠀⠀⡠⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⡠⠄⠄⠤⡀⢸⠒⠒⠒⠦⠤⣘⡄⠀⢣
⠀⠀⠀⠀⢸⠀⠀⠀⠧⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠃⢸⠆⢀⣤⣶⡄⠈⡇⠀⢸
⠀⠀⠀⠀⠘⠒⠂⠒⠀⠒⠒⠐⠂⠒⠐⠒⠒⠀⠐⠒⠀⠀⠐⠃⠀⠒⠒⠒⠂⠂⠐⠒⠀⠒⠂⠚⢰⠋⠀⠀⠘⠀⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠢⠤⠔⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⢤⠖⠉⠀⠉⠳⡤⠒⠐⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣤⡤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⢾⠇⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⠛⠛⠛⠛⠒⠒⠒⠒⠚⠀⢀⣠⠜⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠓⠢⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠉⠑⠀⠓⠒⠓⠚⠉⠉⠁⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠉⠉⣺⣿⠟⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢿⡟⠁⠀⠌⠛⢿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⣦⣄⣐⣀⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡏⡇⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠓⠒⠒⠚⠀⠀⡇⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⣤⠀⢀⣧⢧⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠀⠙⡉⢸⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⡞⠁⠀⠀⢱⢠⡅⠀⠀⠀⡄⢢⡜⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡴⠋⠀⠀⠀⠀⢸⣼⣀⠀⠀⣠⣀⠘⠇⠏⠜⣧⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠗⠀⠀⠀⠀⠀⠓⠚⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢛⡛⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠦⠈⡁⢉⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠄⠲⣤⠀⡀⡤⢈⠕⠂⠠⣠⠞⠉⠉⠳⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⡠⠄⢀⠠⠔⠉⠀⠁⠀⠀⠀⠶⢎⣽⠿⠾⠇⠐⢿⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣬⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣤⠀⠄⠐⠀⠐⠀⠀⠀⣠⢀⡳⢶⠖⠉⠃⠀⣤⣔⡻⠃⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠂⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷⡏⠀⠀⣀⠤⠄⠀⠀⠉⢀⣤⣄⣀⣤⡀⠀⠀⢀⠦⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠷⠀⠀⠀⠀⠀⣀⠀⡢⢤⣾⣾⠁⠀⠐⠁⠁⠀⠀⠐⢶⣛⣿⡏⠛⠙⠻⠻⣶⢶⡿⡒⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢀⣀⣠⠖⠾⠞⠁⠀⠀⠈⢹⡀⣠⣆⣴⣶⠶⠤⠿⠺⢾⣛⣩⡗⠂⠀⠐⠄⠀⠀⠀⠀⣠⣀⠀⠁⠤
⠀⠀⠀⠀⠀⠀⢀⣽⣿⣷⣿⣿⣥⡄⠀⠀⠀⠀⠀⢀⣸⣧⣾⣿⣟⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠋⠙⡏⠁⠀⠀⠀⠀⠀⢸⣿⡛⠟⠛⠛⠃⠀⠀⠀⠀⢀⡀⢠⠔⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠁⠀⣣⢀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⢉⡁
⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⢉⠘⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠊⠑⠐⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣟⠃⠀⠀⠀⠸⡀⠆⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠆⠀⣸⠎
⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⣿⠂⠈⠀⠀⠀⢀⠈⣸⣿⡇
⠀⠀⠀⠀⢰⣿⣿⡇⢦⡀⠀⠀⠀⠀⠄⢰⡀⢀⣿⣿⡀⠒⠀⠀⠀⠈⣁⢸⣿⡇⠀⣀⡀
⠀⠀⠀⠀⣾⣿⡻⠨⠑⠁⠀⠀⠀⠀⠀⣄⢣⣾⣿⣿⣷⡇⠀⠀⠀⠀⢰⣼⣿⡇⠀⣿⡇⠀⣀⣀⡀
⠀⠀⠀⣼⣿⣿⡇⣄⠀⣄⠀⠀⠀⠄⠀⢨⣿⣿⣿⣿⣿⣷⣶⣶⡆⢰⣾⣿⣿⣿⣤⣿⣇⣀⣿⣽⣿⣀⡀
⠀⠀⣰⣿⣿⣿⣿⣿⣷⣿⡓⣄⠀⡐⠠⡎⢿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣟⣾⣿⣿⣿⣿⣿⣽⣿⣿⡇⠀⠀⠀⢀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣭⠻⢕⠌⠳⢌⠘⠿⠿⡿⣿⣿⣿⣿⣿⣻⣿⣿⣷⣝⣛⣛⣿⣿⣻⠿⢿⣿⣷⣿⡓⠛⠘⢛⣃⣘⡂
⢸⣿⣿⠛⠛⠛⠛⢻⣿⣿⣿⣕⠄⠀⢀⣀⣁⣴⣶⣾⣾⣦⣾⣿⣿⣷⣿⣄⣀⡀⣷⣾⣿⣿⣿⣵⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠃
⠈⠋⠉⠀⠐⠐⠘⠛⠛⠛⠛⠛⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠐⠒⠒⠒⠒⠒⠒⠲⣞⣛⣻⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⢛⣛⠛⠛⢛⣻⣿⣛⣛⣛⠛⢛⣛⠛⠉⠉⠉⠉⠉
⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢰⣂⣀⣀⣀⠀⢐⡆⠀⠀⠀⠀⠐⠀⠉⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠉⠉⠉⠉⠓⠃⠀⠀⠀⠀⠀⢀⠎⢠⠎⠀⠀⠀⠀⠀⠔⠒⠒⠒⠒⠠⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡜⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠔⠀⠀⠀⠀⢀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⣿⣿⠟⠀⠀⢮⣀⣈⣉⣉⣉⣩⡇⢀⣄⠀⠀⠐⠀⠢⠤⠤⠤⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠳⢄⠁⠀⠀⠀⠀⠈⠀⠀⠀⠛⣻⡟⠀⠀⢻
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠀⠀⠀⠀⠀⠀⠀⣴⣏⣀⠀⠀⣸
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣠⣿⣿⣿⣧⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣾⣧⣄⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠹⣇⠀⠀⣴⣏⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣀⣀⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗⠀⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⣴⣋⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣉⣉⠡⠊





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠌⣉⡀⠠⣤⠀⢠⡊⠠⢖⡀⢼⠄⢉⣦⠀⣴⠀⢀⠈⢁⣐⠂⠲⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⢰⠒⡆⠺⠇⠸⠀⠑⣃⣀⠑⠀⠈⠆⠀⠈⠀⠈⠀⠉⠁⠀⠙⠠⠜⠀⠠⢊⣀⠋⠇⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⢀⣩⠤⢶⡊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣡⠶⠊
⠀⠀⠀⠀⠀⠀⠀⣀⠀⡰⢄⡀⠈⠉⢿⣓⣳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡿⠂⠉⠀⠀⢢⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢢⡕⠐⠢⢀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠈⠁⠀⠄⠿⣍⠤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠐⢚⡱⠆⠠⣤⠶⣀⠜⠀⠀⠠⣀⠳⢦⠀⡰⠋⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⣀⠁⠀⠀⠀⠒⠃⡁⠀⠀⠘⣩⢿⢾⣛⣧⠀⠈⠙⠉⢣⠈⠀⠀⠀⠀⠄⠀⣇⠀⡀⠘⠁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠐⠒⠈⠀⠀⠘⠓⢋⠈⣁⠀⠀⠐⠈⠈⠀⠀⠉⠀⠀⠀⠀⠀⢰⡹⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠠⠡⠔⡒⢈⡉⠆⠀⠀⠀⠘⠀⢀⠼⠇⠄⠀⠀⠄⠀⠀⡀⢀⣀⠉⠀⠀⠈⠀⠀⠉⠀⠀⣀⣀⣀⡀⣀⠬⡁⠙⣻⣄⡀
⠀⠀⠀⠀⠀⠲⠂⠠⠈⠩⢁⡰⣿⣷⣄⣤⣄⢠⡼⢧⣀⣀⣈⣀⣀⣰⣏⣀⡀⠀⣱⡄⡀⠀⠀⠀⣠⠔⢉⡠⣴⣚⡛⡄⠈⠁⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡀⡀⠑⠀⠉⠱⠋⠀⠈⠉⠉⠘⡷⠀⠉⢿⣩⠙⠛⣽⠉⠍⠻⠀⠉⠣⠘⡆⢀⠜⢁⡄⠀⣞⣴⣿⠏⣷⣴⡤⣤⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠐⣈⣥⣠⣴⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠇⠀⠀⠈⠀⢠⣹⠀⠀⠠⢤⢾⠏⣠⡏⣴⠚⣿⣃⣀⣨⠷⠿⠿⣋⠿⠆⢹⡀
⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠃⠀⠀⠀⠀⠀⢀⣰⠇⠀⠀⠀⠀⠀⣦⠀⡄⠀⢹⡦⠠⠴⢀⣾⣀⣹⣿⣿⡐⠮⠙⠛⣉⢛⡋⡌⠁⠄⢐⠟⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⢀⣠⣠⣴⠿⢇⠐⠿⣶⣴⢤⡙⢻⡿⣿⡛⠋⠛⠲⠶⣶⣿⡗⠶⣶⣤⣤⣤⣶⣆⠲⠅⠊⣹⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠷⠂⠛⠛⠳⠆⠀⠤⠞⢺⠿⢦⡈⠁⠘⣓⡾⠛⠛⠁⠀⠀⠲⠀⠓⠲⠦⠤⣤⣤⣤⡤⠴⠾⠿⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⣶⣿⠻⣿⣿⡁⠀⠀⠈⢹⣿⣷⣦⣄⣀⠀⠄⣀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠛⠋⠀⢛⣍⣿⣷⣽⣿⣻⣦⡀⠀⣹⣿⣿⣿⠿⠉⣭⡘⠉⠉⠉⠉⠩⣬⣥⣼⣿⣿⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣠⢏⣨⣧⠀⠀⠸⣿⣿⠟⢁⣸⣿⡻⣷⣤⠾⠿⠃⠀⠀⠀⣀⣤⣤⣤⠄⣴⢚⣲⡿⠿⠆
⠀⠀⠀⣀⠴⠇⠀⢀⣤⠦⡈⢻⡷⠿⣿⡋⣿⠟⠁⠀⠀⠈⢘⣟⣀⣠⢜⣻⣿⣤⡙⢲⣶⠷⣶⣶⣿⣿⡿⣿⢳⡊⠛⠛⠛⠁
⠀⠀⠐⠁⠀⠲⠶⠿⠋⠐⠓⠚⠃⢰⢃⣿⠃⠀⠀⠀⠀⣨⢿⣷⣾⣿⣿⣿⣶⣿⠿⣭⣿⣿⣿⣿⣿⣿⣥⡟⢽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⡏⠊⠀⠀⠀⠀⠀⠙⣂⣶⣿⣟⡯⢶⣽⣋⣹⢿⣻⣿⣿⣿⣿⠋⣵⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣽⣇⠀⠨⣴⣾⣷⣄⢺⣿⡿⢯⣶⣯⣭⣕⣺⣿⣿⣿⣿⣿⣿⣿⣤⣬⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡌⡛⣿⣿⣿⣿⣼⣿⣿⣿⣽⣟⣿⣽⣿⣻⣿⣿⣿⣿⣿⣿⡿⢛⣱⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⢿⣿⣷⣈⠙⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢤⣴⣿⣿⡟⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠭⣟⠳⣼⣿⣿⣿⣿⣷⡉⢿⣿⠿⢿⣿⣿⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⠈⢽⣿⣿⠃⠀⣠⠔⣣⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠷⣈⠛⠝⠛⠿⠿⠷⠌⠿⠆⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣻⣯⣛⠛⠚⠛⠋⠚⢁⣀⣤⡼⠿⠶⠶⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⡶⢶⣶⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣴⣾⣯⣭⣭⣍⣥⣴⡷⠀⠒⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣎⠙⠒⢺⣿⣿⣿⣿⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠟⠛⠛⠉⠁

⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⣀⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠊⠁⣤⠴⠶⠶⠚⠁⢀⡤⠤⠄⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠀⠀⢂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⣈⠁⠀⠀⠀⠀⢠⣌⠀⠀⠀⣀⠀⠄⠐⠒⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⢐⠛⠀⠀⠀⠀⠀⢸⣽⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢸⣿⠀⠰⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⢘⣛⡅⠀⠀⠀⠀⠸⠻⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⢾⠐⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⡁⠀⠀⠀⠀⢘⠸⠀⢘⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠀⢸⠠⡿⠄⠀⢀⣤⢤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠤⠤⠀⠤⠤⠇⠐⠻⠷⠀⠒⠚⠀⠘⠉⠀⠀⠀⠀⠀⠀⢀⣶⣶⡖⡜⠘⠐⠒⠒⠒⠂⠀⡼⢉⡿⠿⡶⣻⡀
⠀⠀⠀⠀⠀⠠⠲⠷⢦⠀⠀⠀⠀⢀⠀⠀⠠⠄⠀⠐⠂⠀⠀⠀⣠⢤⣤⢄⠀⢀⠀⠀⠸⡿⢁⠀⠻⠀⠀⠀⠀⠀⠀⠊⣡⢮⠃⠀⢅⠸⠗
⠀⠀⠂⠀⠄⠃⠀⠀⠈⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡴⡻⢻⡛⢿⡇⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⢀⣀⠀⠈⠁
⠀⠠⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠞⠻⠿⠿⡇⠀⠀⠀⠀⠀⠚⠉⠀⠃⠀⠛⠂⠀⠀⣀⠀⠀⠀⠚⠛⢻⡀⠀⠀⢀⠀⠃⠀⠀⡾⠿⠿⠛⠀⠀⠀⠈⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠌⠀⠠⠁⢡⡀⠀⠀⠀⠀⠀⡤⣰⣟⣿⡶⡄⠀⠀⠀⠀⠀⢠⠌⠂⠠⢆⠀⠀⠀⠀⠀⠀⣠⠁⠡⠀⠈⢦⡤⠀⠀⠀⠂
⠀⠴⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⢠⠞⠑⠂⢩⣍⠙⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢰⣿⡿⠿⢶⠀⠀⠁⠀⢀⡤⢶⣶⠤⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠈⠁⠀⠀⠀⠈⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣄⠀⠀⢠⡀
⠀⠀⠛⠗⠠⠀⠘⠃⠀⠀⠀⣼⠋⠉⠈⣁⣼⡄⠀⠀⠀⠃⠀⠠⠄⠀⢶⠝⠿⠿⠿⣶⠀⠀⣀⠀⠀⠀⣸⢓⠉⡁⣴⠀⠀⠈⠁⢀⣿⣳⡷⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂⠙⠊⠀⠠⠄⠀⠀⡀⠀⢀⠺⠠⠈⠀⠀⠠⠈⠷⠀⠀⠀⠀⠀⠁⣄⠀⡃⠿⠄⠀⠀⠀⡨⢿⠈⡧⢨⣧⠀⠀⠀⠂
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⡛⣉⣙⢛⣟⣋⡿⢻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⠤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠉⠀⠀⠀⠠⢤⣤⣄⡀⠀⠀⠙⢿⣿⣦⣴⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⡏⠀⡀
⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠄⠀⠀⠀⣴⣀⣀⣀⠀⣀⠀⠀⠴⠤⠀⠀⠀⠀⢀⡒⠛⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⠟⢳⡀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⠏⠀⠘⠋⠉⠉⠁⠀⠉⠉⠉⠉⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⠀⠀⠀⢀⣇⠀⠀⠀⣸⡀⠀⢠⡇⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢳⢤⣸⠦⠤⠚⡀⠽⠶⠶⠾⠤⠛⠒⠒⠓⠹⠷⣯⠉⢹⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡧⠤⣄⠀⠀⠈⡆⠀⠰⠖⠁⠀⠰⠆⠀⠀⠀⠀⠀⠐⠆⠀⠀⠈⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠳⠬⠹⢦⣀⣀⠇⠀⠀⡠⡀⠀⠀⠀⠀⢀⠐⢄⠀⠀⠀⠀⠀⠊⡀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠉⠊⠑⠂⠐⠂⠁⠀⠀⠈⠁⠀⠀⠴⠪⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⣏⢳⡄⠀⠀⠀⠀⠀⢀⡼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⢀⣠⠴⢋⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣠⣾⠀⠀⠀⠀⡄⣠⠛⠦⢤⣀⡐⠶⠤⠤⣤⠤⠤⠶⠶⠒⠛⠉⠀⢀⡼⠳⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠋⠁⠘⠦⢤⠤⠞⣹⠃⠀⠀⠀⠈⠉⠉⠙⠛⣿⠦⣤⣤⠴⠚⠒⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠠⢀⣠⠴⠒⠒⡆⠴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢇⠀⠀⠀⢣⣰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡆⠀⠀⠸⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⡆⠀⠀⢿⠀⢻⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣘⣆⠀⠸⠦⣤⠭⣽⣦⣤⡞⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣠⠔⠊⠉⠁⠈⠓⠒⠒⠚⠒⠲⠦⠞⠀⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠷⠶⠶⠶⠶⠤⠤⢤⣤⣤⣤⣤⣤⣄⣸⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠈⠀⠀⠀⡀⠐⠀⠀⠀⠶⠛⠉⣁⠶⠁⠐⠀⠀⠈⠀⣴⣟⣣⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⢠⠐⢈⠠⠐⠀⠀⠀⠀⠂⠀⠐⠋⡁⠄⠂⢁⣤⣦⣴⡌⠁⠀⠀⣀⣀⣀⣴⣫⣽⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤⡰⠌⠀⠀⠀⠀⡀⠄⠊⠁⢀⠠⠐⠉⠀⠀⠀⠄⡀⠀⣴⣲⣧⡶⣿⣿⣿⣿⣿⢞⣴⣿⣿⡿⣿⣷⣿⡟⣩⣴⣾⠖
⠀⠀⢀⠠⠔⢀⠀⡠⠄⠀⠐⢾⡥⠖⠖⠓⡔⢄⠢⡁⢀⠂⠀⠈⡠⠀⠀⠀⠑⢊⣩⣽⣺⣿⣿⣿⣾⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⢋⣿⣾⣿⣿⣿⣿⣯
⡓⠌⠀⠠⢌⠠⡁⣤⠶⠃⣌⠠⡑⠄⠂⡔⣈⣦⣳⣴⣂⣠⣄⣄⣴⡶⣆⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⣾⣿⣿⣿⡟⢹⣿⣿⣿⡿⢿⣥⣿
⠉⠀⠠⡄⠦⠳⢉⣠⣰⣍⢤⠁⣔⡆⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠋⠉⠈⠉⢙⣿⣿⣯⡟⢧⣾⣿⣿⣿⣿⢃⣿⣿⣿⣿⣷⣿⣷⡝
⠄⠂⠀⠀⠀⣔⣮⣳⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⢀⣴⠟⠉⠈⠻⣇⢸⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣻
⠀⠀⣀⣰⡅⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣀⡀⠀⠀⠘⣼⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⠿⣛⢧
⠒⠝⣮⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠘⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⢸⣿⣿⣿⣿⣿⣿⣭⣶⡿
⣑⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⡀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠀⠈⠀⣼⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡽⢞⡋⢅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣶⣶⠖⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣥⣶⣾⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣛⡭⠥⠤⣄⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠿⠛⠛⠋⠉⠉⠁⡀⠀⠀⠀⠀⠉⠻⠶⣿⣯⣂⣤⣀⣀⣀⣀⣀⡀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟
⠀⠀⠀⠠⣄⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⡉⠻⣿⣿⣷⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣈⡀⠐
⠃⢀⣄⠀⢀⠈⠐⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⣠⣯⣽⠿⠿⠟⠉⠁⣸⣿⢿⣿⡟⠓⢤⣄⡀⣀⡄⠀⢀⣾⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠛⠀⠀⠄⠀⠀⠁⢂⡤⠤⣖⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⢀⣿⡏⢸⣿⡀⠀⠀⠀⠉⠓⠢⢾⣿⣿⣿⣿⢿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⢩⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡾⣿⣄⠀⠀⠀⠀⠀⠀⣀⡼⠿⣿⣿⣿⣛⣛⣛⣻⣤⣬⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠢⡈⠉⠁⠠⠀⠀⠀⠀⠀⠀⠚⠲⣶⣤⣖⣩⣷⠶⠛⠂⣀⠤⠖⠒⣨⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣿⣿⡿⠿⣿⣯⢴⠂⢉⡠⠴⠚⠉⠁⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠰⠐⠂⠛⠛⠺⣉⠙⠻⠟⡻⠋⢛⣻⠿⢿⣛⠉⠀⡀⠴⠞⡉⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡄⠀⠀⠀⠀⠈⣡⡬⠕⠚⠉⢀⣠⠴⠚⠛⣁⣀⣠⣴⣾⣿⠿⠿⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁

⠀⠀⠀⠀⠀⢁⠀⠀⠰⡀⢠⢠⠀⢟⠀⡀⠰⡞⡦⠆⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⢠⠀⠈⠀⢄⢸⠘⠀⠞⠜⠀⠀⣷⡁⡀⠁⠀⢠⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡬⠀⢀⠘⣎⠀⠀⡼⠀⠀⠀⢺⣡⣤⣶⠀⠀⠀⠈⠀⠄⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠁⢀⠀⠀⡔⠐⠀⠀⠰⠃⠀⢸⡀⢻⠶⠀⠀⡀⠀⠀⠂⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠁⡀⠀⠀⠞⠀⢀⠊⠀⢸⡇⠀⠀⢁⠀⠀⠀⢂⢀⠀⠰⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡆⠆⠀⠀⠁⢸⠀⠘⣇⠀⠁⢸⠀⠀⠀⠀⠇⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠋⠃⠀⠀⠀⡼⠀⠀⢿⠀⠀⠀⡆⠀⠀⠂⠟⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠁⠀⠀⠀⠇⠀⠀⠘⡧⠰⠀⢸⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⠀⠄⠀⠀⠀⡄⠀⠀⢸⣷⡀⠀⠘⠀⠀⠀⠄⡀⠀⠀⠁⠀⣠⣄⣠⢦⠊⠁
⠀⠀⠀⡠⢄⢀⠈⠀⠀⠀⠀⡀⠀⠀⠄⠃⠀⢠⠀⣿⡆⠀⠀⠃⠀⠀⠀⡃⠀⠠⠂⢀⡹⠿⠇⢠⣀
⠀⠀⢀⠀⠘⣧⡄⠀⠀⠀⠀⠁⠀⠠⡅⠀⠀⠀⠀⣿⠇⠀⠀⢁⠀⠀⠀⠀⠸⡄⠀⠀⠈⠑⠂⡀⣈
⠀⠀⠐⢾⡀⢻⣧⠀⠠⡀⡂⠀⠀⠀⣅⠀⢀⠀⠀⡿⡦⠀⠀⠀⠀⠀⠆⡀⠀⣼⠄⠀⠀⠀⡰⠃⠀⠀⠀⢀
⠀⠀⢀⣺⠷⠆⠛⠀⠀⠀⣇⣠⠀⢰⣿⡌⠘⠀⠀⢼⣧⡀⠀⠀⠀⠠⠀⠇⠀⢴⣦⠦⡀⠀⢀⠀⠄
⢀⠄⣸⡅⠀⢀⠲⡀⠀⡀⠘⢣⠀⠀⠨⠀⣐⠀⠀⢸⡿⠁⠀⠀⡄⠘⠃⢀⣲⣿⣿⣿⣧
⠈⠀⠸⠀⠀⣶⡀⠠⡔⠀⠁⠈⠻⠀⠀⢀⠈⠑⠀⠘⣷⣀⡤⠲⠃⠀⠀⣼⣿⣿⣿⣯⣽⠆⠠⠄
⠂⠉⠐⡀⠠⠼⢟⣀⠈⣠⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⡿⡠⠄⡀⠰⠀⢰⣿⡟⠁⠀⢻⣿⣄
⡀⣀⠀⡀⠀⠀⠀⠈⠉⠓⠂⠀⠀⠠⢤⣀⡀⠀⣴⣿⡿⠙⠀⡁⢀⢰⣿⡿⠀⡀⠀⠀⣹⣷⡆
⠧⠄⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢤⠾⠽⠫⠞⠉⠌⢙⣄⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠺⠋
⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠉⠘⠓⠄⠀⠰⢤⣤⡀⠀⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣤⣤⣀⣄⡤⢰⣤⣴⣶⣶⣤⡀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠙⢿⣿⣿⣿⡿⠀⣰⣿⣿⣿⣿⣻⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⣴⣶⣧⡉⠉⠙⠛⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠒⢠⣀⣄⣠⡄⠀⢸⣇⣿⣿⠀⢀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢸⡿⠀⠈⠛⠀⠀⠀⠸⣻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠀⡀⠀⠀⠀⠀⠀⠽⠃⢸⡃
⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⠀⠚⠂⠘⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢦⠀⠀⠉⠦⠀⠀⢀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠡⢄⠀⠸⣑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⢀⣾⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⡞⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⢀⡀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣾⠃⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⠀⠀⣀⡶⠀⠀⠈⠻⣟⠛⠀⠀⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣴⣶⠿⣿⣿⣿⣷⣶⣤⡤⠀⣿⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠲⠼⠿⢿⣿⣿⣡⣤⣠⣾⣿⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢠⡄⠀⢰⣿⡿⢿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠿⠿⠿⢿⣷⣬⣤⣤⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢸⣿⢫⣉⣟⣿⣿⣿⣷⣄⠀⠀⠀⠀⠘⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⠖⠋⠉⣿⣿⣿⠛⣿⡿⣿⣿⡉⠃⠀⠀⠀⣤⠀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⣀⣠⣼⣿⣿⣿⡟⠀⠀⠉⠀⠀⠀⠀⣠⠈⠀⠂⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⢤⣨⣭⣁⣀⡀⠀⢨⢄⢀⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠻⡌⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⡾⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠳⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⣠⣦⡄⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠀⢀⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠋⠀⠀⠀⣞⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠻⣄⣀⣀⣠⠟⠋⠁⠉⠛⠶⠶⠚⢻⡅⠀⠲⣄⣀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠀⢻⣄⠀⠀⢀⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡀⠀⠀⠀⠀⠙⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠶⠛⠛⠛⠲⣦⡀⠀⢸⡇⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⡜⠁⠀⢾⡛⣿⠆⠀⢻⡄⢸⡇⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠠⡶⢾⣟⣷⢶⡆⠀⣿⠈⣧⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠈⣧⡀⠻⠾⠉⠻⠞⠁⣰⠇⠀⢿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠻⢦⣄⣀⣀⣤⠾⠋⠀⠀⢸⣇⡟⠀⠀⠀⠀⢀⡴⠶⠶⠒⠛⠚⠛⠛⠛⠛⠒⠒⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⠇⢠⡤⣤⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⡴⠞⣋⠛⠳⣦⠀⠀⠀⠀⠀⠀⠘⣇⠘⠶⣿⡿⠳⠒⠖⠒⠲⠶⠀⠄⠀⠐⠠⠿⠿⠾⠿⠿⠿⢿⣿⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣿⠁⡞⠉⠁⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣿⠸⣏⡛⣶⠀⣿⠛⣿⠛⢻⡟⠛⢻⡟⠻⣿⠛⣿⢠⡾⢛⣻⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠶⠄⠀⠀⢰⡋⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠋⠀⠈⠳⠿⣤⣼⣧⣤⣼⣧⣴⠷⠞⠁⠀⠙⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⡾⢡⡶⠛⢦⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣤⢤⢤⠤⡤⠤⢤⠤⣤⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠹⣄⣀⣬⣉⣷⠀⢻⣦⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⠀⠸⣦⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠛⠛⠛⠳⠔⠋⡡⠤⠐⠐⠶⠒⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠋⠉⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣟⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢮⡂⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⢄⠸⡆⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡙⠒⢠⠶⣅⡀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡠⠄⠁⠀⢀⡆⠀⠀⠀⠀⠪⡉⠉⠁⠀⣾⠇⡟⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠰⡁⠀⣨⠀⢸⠁⠀⠀⠀⠈⡆⠑⡀⠀⠀⠘⣦⠝⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⠃⠛⠛⠓⠒⠚⠒⠒⠂⠀⠤⠁⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠐⠁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠉⣿⠉⠀⠀⠀⠀⢀⡐⠃⢀⣿
⠀⠀⠀⠀⠀⢀⡇⠀⢀⣤⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠸⠀⢀⣴⢪⡀⠀⠀⠀⠈⠉⢀⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠁⠀⣾⣿⠉⢻⣿⠀⠤⠤⠤⠤⠤⠤⠀⣾⣟⠙⣿⡷⠸⠖⠚⠻⠦⠽⣗⣰⡦⠖⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠉⠉⠉⠛⠳⣦⣤⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠛⠹⣷⢿⡀⠀⠀⣀⠘⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣀⣤⠖⠋⠉⠻⣦⣀⡀⠀⠀⣼⠀⠀⠀⠉⢀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣀⠀⠀⢠⠞⠉⠀⢀⣠⠴⠶⠀⠉⢉⣿⣟⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣠⠈⢳⡀⢻⡀⠀⢀⡤⠴⠚⠃⠀⣴⢿⡙⣿⣆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣅⠘⣧⣼⠇⠀⠀⠴⠿⠁⢀⣴⡟⣦⠛⠉⣿⠀⠀⠀⠀⣤⡄⠈⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⠀⠀⠀⠀⠀⠈⠛⠁⠀⢸⡇⣷⠈⠁⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣇⠀⠀⠀⠀⢀⣀⣀⠘⠛⠀⣿⡀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠓⠿⢦⣀⠀⠘⠋⣭⠉⠀⣀⡼⠋⠛⢶⡛⠉⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠓⠦⣤⡀⠙⠛⣶⡶⢶⣶⡞⡿⢦⡀⠀⠈⢳⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠙⠳⠞⠙⢿⣉⣿⠛⠁⠀⠙⢦⣤⣼⠇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⡆⠀⠘⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠸⡇⢸⡄⠀⠘⣧⡤⢤⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣧⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣷⣤⠴⠶⠶⠚⠛⡻⣆⠀⡇⠀⢸⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⣷⠀⠀⠀⡏⠀⠀⠀⠀⠀⠘⣇⢻⠸⣇⠀⠀⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡘⡇⠹⣦⡼⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠘⡇⢻⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢹⡶⠒⠋⠻⣦⡀⠀⠀⢀⣠⡿⢸⡖⠛⢳⣦⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢰⡇⠀⠀⠀⠈⢿⣞⣋⣩⡥⠴⠛⠉⠉⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣛⣻⣧⣀⣀⣤⣤⠾⠏⠁⠀⠀⠀⣠⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢹⡏⠁⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⠀⠀⣀⣀⣿⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣴⣾⣛⠉⠽⢿⣷⣤⡴⠶⠚⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⣠⠶⣿⡷⢤⣬⣁⠀⠉⡷⣦⣤⣬⣤⣬⠽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢠⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠴⠶⣴⡀⠀⠀⢠⢠⡄⣠⠐⠐⠛⠳⠶⠖⠓⢲⢶⠀⠀⠉⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠂⠀⠓⣒⠘⠓⠒⠂⣘⣤⣌⢿⣦⠤⢄⠀⠀⠒⠖⢾⡟⣗⣒⣒⢠
⠀⠀⠠⠀⠀⠘⠛⠐⣏⣷⠀⠀⠀⠀⡄⠰⠦⠭⠤⠶⢾⡆⠀⠀⢸⠀⣀⠈⠿⢸⣅⣀⡀⢀⠂⠣⠶⡾⠉⣇
⠀⠀⠀⠀⠠⣢⣖⣠⠦⠜⠿⠗⠛⠀⡇⠀⠀⠀⠀⠀⢸⡂⠀⠀⠥⠾⣏⣴⡦⠬⠛⠿⠯⢘⠀⣀⡀⠹⣆⠽⣀
⠀⠀⠀⠀⢠⢲⣂⣠⣄⣐⣶⡆⠀⡠⠃⠴⣀⠀⠰⢄⠈⣻⣦⢖⡤⢄⠀⡀⣀⠀⠀⠐⢀⠃⠚⠛⠚⠂⠭⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋⣡⣷⠈⠉⠉⠉⠋⠑⢌⣉⣳⠮⠝⠈⡇⠐⠀⠛⠛⠻⠓⠊⠜⡇
⠀⠀⠀⡀⠈⣠⢤⡀⢀⡀⢀⣈⣦⡤⢤⣀⠀⣀⣀⠀⠠⠄⠀⢀⠇⢀⣀⠀⠀⠀⠄⠀⠀⡇
⠀⠀⠀⣅⣘⣳⣏⡠⠝⢻⢼⠅⣀⠈⠀⠀⠈⣡⣬⠭⡤⡀⠤⠁⣰⡦⠚⢁⢐⣦⣤⡈⠀⢽⢳⣄⠬⠤
⠀⠀⠀⠀⠈⢸⡹⠝⠃⠿⠼⠭⣅⠩⠭⠭⠩⡭⠉⡁⠔⠒⠚⠊⡗⠂⠰⠶⠛⠁⠀⠈⠤⠽⠦⠿⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠈⠈⠀⠈⠀⠐⠀⠰⡇⠀⠀⠀⠀⠀⠈⠈
⠤⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠈⠀⠐⠘⣧
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⣯⢵⣥⢠⡀⢀⠀⠀⠀⠀⠐
⠀⣄⡀⠣⠊⢀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠘⢯⡤⠠⠴⠒⠂⠠⠲⠄
⠀⠀⡐⠀⠀⠜⡕⠫⢢⠆⠀⠀⠀⢀⢶⡀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠄⠙⣦⡁⠀⠀⠀⠀⠀⠈
⠀⢹⣤⣴⣿⣷⣟⡛⠙⠀⢀⠄⡨⣠⠿⠷⠀⠀⠈⠛⠾⠿⣕⡦⣄⡀⢀⣸⠆⠀⣤⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢤⣴⡤⠾⠓⠀⠀⠑⢄⡀⠀⠁⠀⣀⠻⡏⠙⠉⠀⠙⠋⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⣀⠐⠂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⣄⣄⢣⣠⠤⠄⠄⠀⠀⠀⢤⡄⠂⢉⡻⢒⣤⠱⠀⠀⠀⠀⠀⠀⠀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⢾⣷⠿⠉⠀⠀⠀⡀⠀⠀⠀⠀⢀⠌⡤⢫⢀⡥⡹⢸⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣜⠳⣴⢏⡪⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠁⢸⠀⠓⠠⢦⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠈⢠⠁⣾⢦⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣀⠄⢊⣆⠁⡀⠀⠀⠀⠀⠀⠀⡠⠤⣬⣶⣄⠈⠐⢁⠙⠄⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⢈⠵⣊⢼⡇⣣⠎⠀⠀⠀⠀⠀⠀⢷⣮⣶⡌⠡⡉⠈⣆⡴⣄⡀⠀⠀⠀⣾⠆
⠀⠀⠀⠀⠀⠀⣀⠰⣱⠇⠀⠀⠀⠃⢀⣢⠔⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⢓⠴⢤⠃⠀⠀⠧⠄⡁
⠀⠀⠀⠀⢸⣼⡧⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠰⢃
⠀⠀⠀⠀⠸⣤⡀⡀⣸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠐⣄
⠀⠀⠀⠀⠀⠀⠁⢸⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠈
⠀⠀⠠⠄⠒⠠⠰⠌⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡕
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⡀⣦⡦⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠠⠀⠠⠀⢳⠉⠀⠀⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠞⡏⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢡⡤⠀⠚⠃⠀⠀⠠⢄⣀⣒⣒⠉⠉⠁⡸⡄⠐⣀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⡀⢀⠉⠀⠖⠉⠇⠀⠀⡐⠂⠤⠀⣠⡤⠬⣭⣷⠾⠃⢰⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠿⠶⠜⠁⠀⠰⠋⣉⣀⡀⠠⠿⠶⠄⠱⣖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⢀⠀⠀⠀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠇⠀⠀⠀⢀⠀⢀⡀⠠⣀⡈⠹⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⠇⠰⣎⠁⠀⠘⣉⢳⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⢁⡠⣤⣾⡏⠀⠀⣛⠋⠛⣿⠛⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡀⠊⠈⠛⣡⠟⠀⠐⠛⠷⠒⠀⠀⠀⠀⠀⠀⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⣾⠁⠀⠀⠀⠈⠃⠘⠀⠀⠀⠀⠀⠙⣦⣦⣴⠶⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⣰⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠘⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⢠⣿⠂⠁⠀⠘⣷⠀⠀⠀⠀⠀⠀⠰⡾⠿⠾⠟⣩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢻⣤⡶⣶⠶⠶⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⠻⣦⠀⠀⠀⣼⡟⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⣧⡀⣴⣯⣀⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡘⢿⣈⣿⣿⣿⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⣿⢻⣏⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡈
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢾⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣠
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠯⠄⠀⠀⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠶⠶⠶⠂⠻⣿⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠛⠿⢶⣤⣤⣤⣄⣰⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠶⠆⠀⠀⠀⣾⡄
⠀⠀⠀⠸⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⣀⡀⠀⠀⠀⠀⠀⢀⣠⣿⡏⢀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠛⠿⠿⣶⣤⣤⣀⣀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣿⣏⡛⠛⠛⠿⠟⢳⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠛⣸⠟⠛⠛⠛⠛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠑⣀⠃⠀⢻⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⣠⡾⢃⣰⣷⣾⡿⣫⠤⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢆⡙⠷⣦⣀⡀⠀⠀⠀⠀⠀⠙⠆⢻⣷⣾⣿⣯⣭⣴⣭⣤⣶⣤⣀⣀⣀⣀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠪⣷⡶⠋⢩⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠋⠉⠈⠉⠉⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠶⠈⠳⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⡀⠁⠀⠀⠀⠀⢀⣀⣴⡆
⠀⠀⠀⠀⠀⠰⠦⣙⣉⣿⡇⠀⠰⠶⠶⠂⠉⠁⣀⣼⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢷⣶⣿⣶⣾⣿⣿⣿⣧⣄⣈⡁⠚⠛⠓⠂⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣌⣙⣿⣿⡿⠛⣉⡛⢛⣛⣻⣿⣶⣶⣦⠠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢤⠰⠘⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⣰⠿⠿⠿⠿⠛⢳⡀⠀⠀⠀⣨⣿⠿⣷⣄⣿⠀⠀⢀⣠⣤⣿⣤⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡇⠀⠤⠀⠠⠄⠀⡇⠀⢸⡟⠉⢉⡶⣾⣟⣿⠀⠀⢸⣾⣿⣿⣿⣾⠷⡄⠀⠀⢹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠘⢄⡀⠘⠀⣠⠞⠀⠀⢸⡇⠀⣼⠁⡠⠻⡄⠀⠀⠈⢿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠛⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠉⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⢏⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢠⡎⢩⡄⠀⢀⠙⢲⠀⠀⠀⢀⡿⣿⣿⢷⠀⠀⠀⠀⠀⡤⠤⠤⠤⣤
⠀⠀⢸⠀⢀⡞⣿⡟⣆⠀⢀⠀⢸⠁⠼⠅⣟⠼⠅⣸⠆⠀⠀⠘⢁⡿⣿⠘⠃⠀⠀⠀⠀⠣⠤⠤⠤⠟
⠀⠀⢸⠿⠿⢿⡿⣿⠿⠿⢿⠀⠀⣽⠲⢤⡤⠤⠴⠃⠀⠀⠀⠀⢸⡇⣸⠂⠀⠀⠀⠸⠰⠇⠁⠲⠥⠧⠸⠅
⠀⠀⠈⠀⠀⠈⠁⠁⠀⠀⠈⠀⠈⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠰⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⡿⣷⡄
⠀⠀⠀⣤⣤⣬⣤⣤⣤⡀⠀⠀⠀⠀⢸⣹⣿⡟⠀⠀⠀⢀⠀⠀⢲⢀⡀⣄⠀⠀⠀⠀⣠⣿⣷⣶⣶⣿⣿
⠀⠀⠀⢻⣿⢹⣿⣿⣿⣩⠆⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇⠀⠓⠸⡇⠓⠀⢀⠀⠀⠸⣿⣿⣿⣿⡿⠃
⠀⠀⠀⢀⡙⢿⣿⠟⢁⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠉⢻⣿⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡁
⠀⠀⠀⠀⠀⣤⡿⡄⠀⢰⡆⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⢀⣼⣿⣧⡀
⠀⠀⠀⣀⣰⣿⡟⢻⣷⣸⡇⠀⠀⠀⠀⢿⢛⡿⠀⠀⠀⢀⣾⣿⠿⠟⠋⠈⠻⣧⠀⠀⠀⣼⢹⣿⡏⣧
⠀⠀⠀⣿⠀⠀⣾⢻⣿⣽⠇⠀⠀⢀⣴⣶⣾⣷⣦⠀⠀⢸⣷⠀⠐⢀⡐⠂⠀⣿⡇⠀⠀⠀⣸⠛⣧
⠀⠀⠀⣟⠀⠸⠇⢠⠊⢢⠀⠀⠀⣾⠃⣿⣿⣿⠿⠇⠀⠸⠿⠷⢄⣈⣁⡤⠾⠿⠃⠀⠀⠀⢹⠄⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣨⠗⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠐⢀⡔⠁⠀⠀⠀⠈⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⠑⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⡴⠁⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠐⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣠⠎⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠂⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠂⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⡠⠋⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠐⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡞⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⡰⠋⡀⠀⡰⠋
⠀⠀⠀⠀⢀⠞⢁⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠒⠒⠒⠋⠉⠉⠉⠉⠉⠉⠓⠒⠒⠢⠤⢤⣀⡀
⠀⠀⠀⡴⠃⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠑⢦⡀
⠀⢠⠞⠀⠈⣠⠎⠀⠀⡤⢤⢤⠤⢄⣀⡀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢱
⡴⠃⠀⢀⡴⠁⠀⠀⠀⢣⠀⠠⢤⣀⠀⠈⢢⠀⢸⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢻
⠣⣀⡠⠎⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠙⠲⣼⠀⡎⠀⠂⠈⠙⠒⠢⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠒⠞⠉⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠚⠁⢙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠩⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠀⡀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢀⣀⣀⣀⡤⠤⠔⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠢⠤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣤⣶⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⢘⠠⠐⠀⠗⠂⢸⣿⡟⢡⣖⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠃⠀⠀⠀⠀⠀⠀⢐⣍⣿⣿⣷⣧⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⠺⢂⡄⠀⠀⠀⠀⠀⠉⠨⣿⣿⠋⣨⡿⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠄⠀⠀⠀⠀⠀⠀⠀⠹⢷⠇⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⢀⠔⠃⠀⠀⠂⠀⠀⠐⠠⡀⠀⠀⠀⠀⠙⠦⠀⠐⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠋⠀⣼⣿⣇⠀⠀⠀⠀⠄⠀⠀⠑⠦⡀⠀⠀⠀⠀⠐⠀⡈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡕⠒⢟⡋⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠹⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢐⠀⢀⡀⠀⠀⠈⠓⡀⠈⠂⠀⢻⡀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠈⠤⠀⠀⡸⠁⠀⠐⠀⡀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠠⠄⠁⠀⠀⢀⠀⠁⠀⠀⡄⠒⠲⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢤⡃⠀⠀⠀⠀⠀⠠⠄⠀⠀⡤⠀⠀⠠⠀⠀⠀⠘⡇⢠⡏
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡌⠀⠠⠄⠀⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠀⠀⢘⡃⠠⠆
⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⢘⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⢨⣁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡅⠀⠀⠈⠤⠁⠀⠀⠀⠀⠀⠀⠀⡀⢸⡃⢸⡀⢸⠀⢸⡁⢸⠀⢸⡇⢀⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠐⠀⠀⠀⠀⠀⠑⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠁
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢈⠀⠀⠀⠀⡇⠀⠇⠀⠇⠀⠇⠀⠇⠀⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⣴⣧⣤⣶⣤⣾⣦⣴⣷⣤⣷⣼⣿⣿⣿⡟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡏⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡏⠁⠈⠙⠻⡆⠀⠀⠀⠀⠱⡀⠀⠀⠈⢢
⠂⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡼
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠠⠄⠀⠤⣤⠶⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠉⠄⠊⡀⠄⠐⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡠⠄⠀⠗⠁⠈⡠⢎⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢁⣀⠀⠀⢀⡴⠀⠀⢀⢀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⡁⠀⠀⠀⠀⠀⠘⠋⢉⠀⠀⠀⠀⠀⡁⠒⠒⢠⠀⠀⠠⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠂⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠈⠀⠀⢈⠀⠀⢀⢀⣠⡅⠀⢰
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⡀⠀⠤⡀⠠⠂⠀⠀⣠⠁⠀⡠⠁⠁⠂⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠘⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⠂⡱⢤⠈⡡⠄⠐⠒⠛⡋⠀⠈⠁⠉⠁⠀⠀⠀⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢠⠀⠀⠀⠂⠀⠀⠈⠐⠀⠐⠈⠤⠨⠀⠰⢊⠴⢦⠀⠀⠀⠆⠀⢀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⣀⣀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⢀⠄⠀⠝⠁⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠛⡄
⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠀⠂⠀⡀⠉⠖⢚⠛⠛⠒⠒⠀⠤⠀⢀⡀⣀⣠⡤⠤⠶⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠤⠤⣀⣀⡀⣀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠀⠴⠤⠀⠒⠈⠉⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠒⠂⠀⠀⠀⠭⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⠀⠀⣐⡂⠀⠭⠀⠀⠀⠀⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠄⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠛⠶⣄⢀⡴⢞⣩⠔⠀⠀⠀⠈⠛⢆⢀⣠⠴⢚⡻⠫⠭⢙⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠐⠊⠁⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠈⠀⠙⠖⠛⠛⠛⢛⣓⠲⣄
⠀⠀⠀⠀⠀⣠⢞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢾⢧
⠀⠀⠀⠀⢸⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢞⡆
⠀⠀⠀⠀⠸⣜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢀⣼⠗⠀⠀⠀⠀⠀⢀⣠⣀⠀⢰⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⡾⠀⢸⡇⠀⠙⠛⣲⠒⠲⠆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡲⣄
⠀⠀⠀⣰⠟⠁⠀⢀⣤⠶⢶⡄⣾⠀⠈⢷⠀⢷⠀⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⡏⠀⠀⠀⣼⠃⢀⡴⠒⢦⠀⠀⠀⠀⠀⠹⡌⢧
⠀⠀⣸⠃⠀⠀⠐⠙⣆⢀⠞⠀⢹⡄⠀⢸⠀⢸⡄⠀⠀⠀⣿⠀⠀⠀⠀⣇⠀⡾⠀⠀⠀⢸⠇⠀⠀⢸⡟⠀⣾⠁⠀⢸⡇⠀⣤⠀⠀⠀⠀⢸⡄
⠀⠀⡏⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠻⣤⠎⠀⠘⠓⠚⠛⠁⠉⠉⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠐⣿⠀⢀⡟⢁⡾⣿⠀⢠⠆⠀⢸⡇
⠀⠀⠻⣄⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⢀⡾⠁⢻⣦⡟⠀⠀⣾
⠀⠀⠀⠈⠉⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⢀⣾⠏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠁
⠀⠀⠀⠀⠀⠀⣷⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣬⣀⠀⢀⣀⡤⢶⡄⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠒⠒⠶⢒⣊⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠛⢄⠀⠀⠀⠀⠀⠁⠠⠤⣆⠀⢀⡀⢀⣀⣀⡀⠀⠀⣠⡾⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣭⣭⣭⣥⣤⡶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠋⠉⠉⠠⠐⠿⠓⠘⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣡⣾⠇⠀⣀⠤⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⠤⢺⠒⠓⠘⠉⠥⠠⢢⣥⣂⣠⡀⠄⠀⢠⣉⠻⣂⠐⣙⠁⠈⠃⡀⢤⣭⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣻⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢆⠘⣽⣭⣬⣉⠿⣭⠽⠿⠛⠋⢙⣉⣻⣋⣂
⠀⠀⠀⠈⠀⠁⠆⠐⠈⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠁⠌⠁⠀⠛⠗⠠⠀⠀⠴⠨⠍⠛⠉⠉⠛

⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠃
⠀⠀⠀⢀⡴⠛⠁⠀⠀⢀⣀⣀⡀
⠀⠀⣴⠋⠀⠀⢀⡴⣾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠲⢦⣄
⠀⡼⠁⠀⢀⡴⢋⡾⠁⣀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⣸⠃⠀⠀⡾⠁⠺⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⣿⠀⠀⢸⠇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸
⣰⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠉⠘⣇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠴⠶⠖⠛⠋⠁⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⡇⠀⠀⠀⠀⢸
⣿⠀⠀⠀⣶⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣀⣤⡤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⣤⣄⡀⠀⠀⠹⣆⠀⢸⠇
⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⢾⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢳⡿⢦⡀
⣿⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⡾⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇⠀⢷
⣿⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⢰⠄⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⠀⠀⠀⢸⠇⠀⢸⠇
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡟⠀⠀⠀⢰
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠴⠋⠀⠀⠀⠀⣿
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠉⠛⠛⠛⠛⢿⠛⠛⠳⣤⣀⢰⡟⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠉⠻⠇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠘⢷⣄⣀⣠⡴⠃⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⣤⡤⠶⠛⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠀⠀⠀⠀⠀⠭⠛⠑⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠠⡤⢠⠀⠀⠀⡴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⡀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠁⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠈⠀⠀⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠠⠀⠢⣜⠀⠈⠁⠒⠞⠁⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠠⢰⠀⡀⠘⠀⢀⠀⠀⡘⡀
⠀⠀⠀⠀⢀⠜⠀⢀⠀⠈⠺⡄⠁⡠⡄⡀⠂⠰⡀⠐⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠄⢀⢉⠙⠀⠀⡠⠂
⠀⠀⠀⢀⠎⠀⠀⠤⠀⢀⠀⢣⠀⠀⠉⢀⡴⡉⢫⠕⣎⢀⠼⠀⠀⠀⠐⠂⠀⠀⠆⠀⠀⠀⠀⠈⢢⠐⣒⣓⢂⣀⡒⠂
⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⠄⠀⠁⡔⠁⡟⠠⠤⣁⡊⠉⡖⠆⠇⣱⠀⠀⠰⠐⢄⠜⠨⠴⠶⠶⠶⠜
⠀⠀⠌⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⢴⠀⠀⡇⠀⠀⠀⠹⠶⡡⠦⠜⠁⠀⠀⠀⡄⠀⠀⠠⠀⠐⠂
⠀⡜⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⠱⣄⡈⠠⠃⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠄⠊⠀⠀⠀⠀⣀⡀⠒⠀⠀⠄
⢰⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢁⡔⠊⠉⠻⣷⣤⡤⠔⠉⠉⠉⠉⠉⠉⠉
⠀⠉⠓⠤⢄⣀⠐⠀⠠⠀⠀⠄⣼
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠿⠿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠋⠉⠛⢛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⠀⠀⠀⠀⣤⠀⠐⠒⠂⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⢂⡀⢉⣍⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠰⠶⠀⣀
⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠚⠛⢤⣤⣌⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⢀⡏
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠘⠢⠄⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤
⣿⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠐⢿⠟⠁⠀⠹⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⡀⠀⠀⠀⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⠀⠒⠒⠒⠒⠒⠶⠶⢤⣤⣄⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣿⠀⠀⠀⠀⢉⣤⠀⠀⠀⠄⠀⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠘⠟⣀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡿
⣿⠀⠀⠀⠀⠀⠸⠯⠀⠂⢀⣀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠛⠁⠤⣤⣤⣤⣤⣤⡤⠤⠿⠿⠷⠼
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠛⠲⢶⡶⠖⠊⠉⠉
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⣿⠀⠀⠀⠒⠒⠐⠀⠀⠀⠀⢶⡆⠀⠀⠀⣻⣤⣤⡀⠀⠀⠀⠀⠀⠐⠉⣿⣥⣤⣤⣤⣤⣤⣤⣄⢀⡐⠂⠙⢶⣀⣠⣤⣤⠀⠀⠀⣤
⣿⠀⠀⢸⠀⠀⢄⠀⠀⠀⠀⠈⢶⡄⠠⠴⠁⠀⠠⠀⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⢠
⣿⠀⠀⢸⠀⠀⡌⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠐⠋⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⢸⠀⠀⠇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸
⣿⠀⠀⢸⠀⠀⢰⡄⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠀⠀⢸⠀⠀⠈⢁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃
⣿⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⣿⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⣿⠀⠀⠀⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⠀⠀⠀⣿⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⣿⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⢸
⣿⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇
⣿⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⣿⠀⠀⠀⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠉⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⡇⠀⠀⠀⠀⠰⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⢸
⣿⠀⠀⠀⡇⠀⢠⠜⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠀⠀⠀⣆
⣿⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⣿⠀⠀⠀⠈⠙⠛⠓⠚⠛⠒⠛⠛⠛⠛⠓⠒⠢⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠒⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢦⣤⡴⠛
⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠿⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣹⣟⣿⣿⣿⣿⡿⣏⣿⣿⣿⠀⠀⠀⠖⢠⠀⠀⠀⠀⢨⠸⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⡿⠚⠋⠉⠛⠾⣿⣿⣿⣿⠀⠀⠀⠀⣀⣭⣶⣶⣦⣅⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣽⡀⠈⢯⡽⠋⢀⣼⣿⣿⣿⠀⠀⠀⠋⢿⣿⣔⣓⣼⣿⠟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⢀⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⢰⡄⠀⠀⠐⠀⠀⠀⠀⠘⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠹⣆⣀⣧⣤⡀⠀⠀⠀⠀⠀⠀⠑⢍⣈⠿⠉⠉⠿⠁⠀⠸⠏⠁⠬⠙⠒⡀
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠬⠀⣉⠑⠀⠒⠒⠂⠐⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐⠒⠀⠀⠀⠀⢠⠤⠤
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠈⠁

⠀⠀⠀⢠⡴⡶⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡆⠀⠀⠐
⠀⠀⠀⠘⠇⠧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⠛⠓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⣤⣴⠆⠀⠀⠠⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠁
⠀⠀⢀⣀⠁⠈⠀⠑⡀

⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀
⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⣀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣯⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡤⠖⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀
⠀⠀⡞⠀⠀⠀⠀⠸⣆⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣠⣶⣶⣶⣤⣀
⠀⠀⠳⣀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠈⠛⠛⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠓⢤⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠛⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢉⡷⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⡰⠋⠀⠀⢠⠞⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣠⣾⣿⣿⡿⣫⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣧⡞⠁⠀⢀⡴⡃⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⢰⣿⣿⣿⠟⢹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢸⡇⠀⠀⣾⠀⢻⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠈⢧⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠈⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢳⡀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠹⡄⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⡟⠀⠀⠀⠀⠀⠙⣆⣀⣠⠇⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⡒⢦⠀⠀⠀⠀⠀⠀⢠⡖⠇⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⢧⡼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⡀⢀⠀⣆⣄⣣⡤⠄⠀⠂⠐⠀⠀⢤⠔⠈⢩⡝⠀⣤⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠒⠀⠀⢪⢄⣾⡿⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⣠⡋⠴⡫⢀⠦⡔⠠⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠳⣦⢫⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⣸⢀⢳⠰⢦⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠉⠈⠰⠁⣯⣦⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⡠⢔⣁⣦⠀⡀⠀⠀⠀⠀⠀⠀⡤⠀⣼⣶⣄⡈⠜⢠⠹⠄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡘⠀⠀⠀⠨⢶⠁⣼⡎⣧⠃⠀⠀⠀⠀⠀⠀⢷⣵⣷⡎⠰⡉⠈⣀⡽⣤⡀
⠀⠀⠀⠀⢀⠀⡄⠸⡰⠃⠀⠀⠀⠃⣀⠵⢖⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⢗⡶⠴⠃⠀⠘⠥⡇⡅
⠀⠀⠀⠀⢸⢼⠇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠿⠃
⠀⠀⠀⠀⠸⢤⡄⡔⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⢬⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⠈
⠀⠀⠠⠄⠒⠂⠐⠎⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡗
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⣄⣄⣀⡀⠀⠀⣀⣀⣀⡠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⡾⠤⠜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣴⡄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⢠⣶⣶⣶⣶⡄⠙⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆
⠀⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣃⣠⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⠀⠀⠀⢠⣤⣤⣾⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⢰⠎⠃⣦⠀⠀⢰⣉⡹⣶⣆⡀⠀⠀⠀⠀⠙⠘⠱⡆⠀⠀⠀⠈⠛⠘⠳⣆⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠻⣴⣦⠼⠀⢰⣋⢹⣂⣸⣿⣷⡄⠀⠀⠀⠀⣶⣦⠿⠀⠀⠀⠀⠀⣶⣦⠿⠀⠀⠀⢠⡟⠉⣆⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⣿⣿⠀⡎⠀⢙⡟⠁⠈⣿⣿⣦⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⡇⣿⠀⠀⠀⠀⠀⢹⣷⢾⡟⠀⠀⣭⣭⣿⣿⡀
⠀⠀⠀⠛⠛⠀⠈⠒⠈⠉⠂⠚⠉⠉⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠘⠃⠙⠀⠀⠀⠀⠀⣴⠟⠀⢿⣠⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠙⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣨⣿⣥⣌⣿⢁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠛⠛⠉⠁⣿⡇⠀⠰⡇⠀⠀⠀⠀⣈⢠⣄⢻⡄⠀⠹⣿⣿⣿⣆⠀⠀⠀⠀⢸⣿⣸⣷⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⢸⡏⠀⠀⠀⣠⠀⣿⡇⣀⢸⡇⠀⠀⠀⠀⠑⠾⠛⠘⠁⣀⡀⢻⣿⣿⠿⠇⠀⠀⠀⠈⢹⣿⡏⢹⣿⡏⠁
⠀⠀⠀⠀⣰⣾⣿⣾⣇⠀⠀⢀⣿⣼⣿⣷⣿⣬⣧⠀⠀⠀⠀⠀⣀⣤⣴⠀⠙⠣⠈⣯⣥⣤⣤⡄⠂⠀⠀⢸⣿⡇⢸⣿⡇
⠀⠀⠀⠀⢻⣿⣿⡽⠿⢂⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠙⣿⠃⠀⠀⢀⣸⣿⣿⣿⣿⣇
⠀⠀⠠⠤⠬⠿⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠿⠀⠀⠀⠛⠻⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⡀⢀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠂⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠒⡆⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⢀⢤⡄⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⣶⣾⣷⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠓⠀⠀⠀⠀⠀⠀⣜⣿⣷⠄⢰⣻⣇⠀⠀⡄⠀⠀⢀⡀
⠀⠀⠀⠙⠹⠼⠙⠛⠛⠙⠽⠱⠽⠙⠛⠋⠡⠟⠫⠎⠛⠀⠀⠀⠀⠀⠀⠀⢀⣹⣏⣻⣀⣈⣏⣙⣇⣀⣇⣇⣶⣸⣿⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡄⠀⣿
⠀⠀⠀⢀⡶⠲⣦⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠛⠛⢻⡆⠀⠀⢰⡟⠉⠠⠾⢧⡄
⠀⠀⢀⣬⠷⠴⠿⢿⡴⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡶⠟⣠⣼⠃⠀⠀⠸⠧⠶⠶⠶⠾⠃
⠀⢠⡟⠁⠀⠀⠀⠀⡳⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣈⣹⡇
⠀⠈⠻⣦⠤⠤⠴⣾⠓⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣉⡉⠉⢹⡇⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⠁⠰⠿⡄
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠏
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⣠⠶⣆⣀⡴⢶⡄⣀⡀⠀⠀⢀⣀⣤⣀⣀⣠⣤⠴⠶⠶⠖⡶⠛⠛⠒⠒⠶⢦⣤⣀
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⣀⣀⣀⡏⠀⢹⠋⢠⠾⠋⢉⣿⣤⣤⠏⠁⠀⠋⠉⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠉⠳⣦⡀
⠀⠀⠀⣿⠀⠀⠀⣿⠀⡼⠉⡸⠋⠙⠀⠀⠀⠀⠀⠰⠛⠻⣄⡏⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⣄
⠀⠀⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠚⣼⠁⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢰⠘⠖⠛⠓⠲⠶⣤⡀
⠀⠀⠀⣿⠀⠀⠀⣿⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⣿⠀⠀⠀⣿⠀⡠⠛⠶⠶⠶⠶⠶⠶⠲⠒⢸⠇⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠶⢶⡄
⠀⠀⠀⣿⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⡄⢸⡇
⠀⠀⠀⣿⠀⠀⠀⣿⣀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡇⠀⠠⠀⢸⡦⠼⠷⢦⡀
⠀⠀⠀⣿⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⠀⢃⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⣿⠉⢹⠛⠛⠛⠛⠛⠛⠛⠛⠙⢻⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⡟⠉⠉⡟
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢀⣀⣄⣠⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣇⣀⣀⣀⣠⣅⣠⣤⣤⡇⠀⠀⡇
⠀⠀⠀⣿⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇
⠀⠀⠀⠙⣶⣒⣶⢾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣧⣤⣤⠇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠒⠉⠉⠁⠠⠤⣀⣀⣠⡤⠶⠚⠛⠛⠒⠤⣀⠀⠀⠀⢀⣀⣤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠑⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢀⠠⢄⠠⡝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠁⠀⠉⠀⠁⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠦⣄⠈⢓⠔⠁⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡴⠟⠁⡀⠀⠀⠘⡆⠀⠀⠀⢂⠒⠀⣀⠔⠫⠊⠁
⠒⠒⠀⠉⠉⠁⠐⠒⠒⠪⠭⠵⠒⠤⠤⣀⣀⣀⣀⣠⡤⠂⠙⣠⠖⠋⠀⠀⠀⠀⠸⣄⠀⠀⠀⠁⠀⢰⠀⠀⢀⠀⣤⡴⠶⠚⠛⠋⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠤⠤⠄⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠒⠒⠁⠠⢸⠀⠐⡀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣄⣀⠀⠈⠉⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠸⠀⠀⠈⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⡀⠀⠀⠈⠣⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢀⡴⢀⠀⣄⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢣⠀⠀⠀⠀⠐⠁⠀⠀⠑⠮⣆⠀⢸⡇⠀⠀⢀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠙⠶⣭⡉⠁⠀⠐⠒⠒⠀⠈⠑⠘⠁⠀⠀⢸⣤⣼⠂⣨⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⠀⠀⠀⠈⢢⡉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠀⠐⠊⢀⠀⠘⠓⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢄⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠾⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠑⡀⠀⠐⡇⠁⠐⠛⠂⢀⠀⠀⠀⠀⣸⠃⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠑⠫⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⣀⣀⡠⠆⠉⠉⠁⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠐⠋⠉⠁⠀⢀⠚⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣘⡋⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠸⠷⠀⠀⠀⠀⣺⡇⠀⠰⠋⠉⠀⠀⠀⠀⠀⠀⠀⠠⠞⣿⠀⢠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⢘⣛⠀⠀⠀⠀⠺⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠿⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠈⢩⠀⢀⣀⠀⡄⣇⢀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡅⣠⠀⢿⠀⠀⣠⡤⢤⡤⡀
⠀⠀⠀⠀⠀⢂⣀⠀⠀⠓⠓⠀⠘⠘⠀⢿⣹⡇⠃⠉⠈⠁⠁⠈⠀⠁⠐⠀⣿⠿⠹⡉⠉⠉⠈⠉⠀⣸⢃⠟⠻⡋⢿
⠀⠀⠀⠀⡜⠋⠙⢣⡀⠀⠀⠀⠀⠀⠉⠘⠀⠁⠀⠀⣰⣿⡞⣷⠀⠂⠀⠀⠟⠓⠀⠁⠀⠀⠄⠀⠄⠀⠊⠀⠀⠂⠈⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣦⠀⠂⠀⠀⠸⠽⠁⠍⠩⠧⠐⠀⠀⠀⠀⠀⢴⣶⡄⠀⠀⠀⠠⠀⠀⣠⣴⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡔⠀⡐⠘⠀⠀⠀⠀⠀⢀⣄⣴⣶⣤⠀⠀⠈⠀⠀⣀⠃⠐⣇⠀⠀⠁⠀⠀⠀⠃⢂⠀⢀⣀
⠀⠄⠀⠉⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠠⠄⠠⠃⠊⢡⣄⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈
⠀⠀⣮⠽⠯⢳⠀⠀⠀⠀⡴⣲⣶⣠⡀⠐⠂⠀⡀⠀⠀⠀⠀⢀⣁⣀⠀⠀⠈⠁⠀⠀⢀⣡⣤⣄⠀⠀⠀⡀⠀⣠⢶⡦⡀⠀⠠
⠀⠀⠙⠀⠀⠈⣀⠀⠀⠰⠃⠀⠈⣽⠧⠀⠀⠀⠁⠀⠂⢀⠒⠚⠛⠛⠻⢆⠀⠤⠀⠀⣸⢹⠟⡗⡆⠀⠀⠀⢠⢿⢽⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠄⠈⠀⠄⠀⠀⠁⠫⠀⠃⠙⠀⠀⠀⠐⠉⠈⠁⠚⠂⠀⠈

⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠆⠀⠀⢀⣀⡤⠖⠒
⠀⠀⠀⠀⢣⠀⠀⡠⠔⠒⠒⠤⠊⠑⠄⡼⠿⠂⠀⡼⠋⠉⠉⠉⠲⡄⣣⠬⠖⠲⢤⡰⠋⠉
⠀⠀⠀⠀⠀⢃⠀⢇⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠷⠆
⠀⠀⠀⠀⠀⠘⢄⠈⠻⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠐⢆⠀⠀⠀⠀⠈⡆⠀⠣⣀⢰⠀⡆⢀⡀⠀⡀⠀⠀⠀⣿
⠀⠀⢳⠀⠀⠀⠀⠰⡀⠀⠀⠈⠀⡇⠈⠉⠉⠉⠉⠁⠀⠇⢰⠀⠀⠀⠀⠀⠀⢠⣴⣋⣀⣠⠞⠲⠶⠒
⠀⠀⠀⠑⢄⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢰⢶⠀⠑⠀⠀⠀⠀⠶⠚⠁⠀⠀⢦⡀⠀⢠⣄
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢰⣛⡆⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠛
⠀⠑⠤⣀⠀⠀⢀⡠⠴⠒⠒⢺⠉⣧⡴⢦⠀⠀⠀⠀⢸⢸⠀⡔⠛⣛⡋⢳⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⠀⣿⠃⠸⣄⡾⣷⠀⣾⢸⣿⡇⠸⡝⠃⠀⣯⠭⠭⠭⠽⡇
⠀⠀⠀⠀⡞⠀⠀⢰⠒⠒⠒⡏⠀⢹⠀⠀⠀⠀⠘⠛⣿⢸⣿⠁⠀⠀⠀⠀⢹⣒⣒⣒⣻⡇
⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡿⢸⡟⠀⠀⠀⠀⠀⠈⣯⠤⠶⢶⡇
⠀⠀⠀⠘⡇⠀⠀⢸⡀⠀⢀⡇⠀⢸⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢹⣭⣭⣭⣇
⠀⠀⠀⠘⠛⠛⠉⠉⡉⠉⠉⠉⠉⣉⠉⢉⡉⠉⣉⠉⠙⣛⠛⠛⣛⠛⠛⣛⡋⢉⣝⠋⢩⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠑⠖⠚⠓⠖⠋⠛⠚⠉⠙⠉⠉⠋⠉⠙⠋⠉⠙⠋⢉⣙⠛⠉⣉⣉⡈⠉⡉⢈⡉
⠀⠀⠀⠀⠀⠀⣀⡤⠶⢦⠀⠉⠁⠀⠀⠤⠀⠀⠈⠀⠀⠤⠤⠀⠀⠀⠀⠸⣇⣟⡨⠿⡅
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠀⠀⢀⣀⠀⠀⠀⡔⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡀⠂⠀⠀⠀⠆
⠀⡄⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⡌⠿⡿⠀⠀⠤⠿⠦⠀⡆⠠⠄⠿⢿⣤⣦⠠⠇⢀⡀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠁⠀⠒⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠲⠠⠀⠀⠀⠈⠃⠀⠤⠀⠈⠁⠉⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠉⠀⠀⠀⠀⠀⠀⠒
⠄⢤⠄⠀⠀⠀⠀⠀⠀⠈⣦⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⢀⢜⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⢸⣶⣿⠃⠀⢤⣀⣭⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢁⣉⡥⠤
⠀⠀⠀⠦⠦⠠⠀⠀⠈⡅⠀⡏⠹⠀⠀⠛⢣⡀⠀⠀⠀⢀⣀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠁⠀⠀⠀⠀⠈⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠰⠦⣀⠀⠀⣀⢀⣴⠄⠀⠀⠀⠀⠀⠀⠠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁
⠀⠀⠐⠤⠠⢰⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠒⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡀⢀⣀⡀⠀⠀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⢢⣿⡟⣿⡟
⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿⣿⣿⣿⡀⢿⣧⠔
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⢣⣿⣿⣿⣿⣦⣤⣶⠋
⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣾⣿⢏⣿⣿⣿⠇⠙⡋⡟⣿⠠
⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣾⣿⣿⣏⡼⣰⢡⡷⠉
⠀⠀⠀⠀⣿⣇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣇⣿⣿⣧⣴⣾
⠀⠀⠀⠀⣿⣿⣧⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠋⠅
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣢⣽⣿⣿⣿⣿⣿⡿⠿⣯⠉⠁⠠
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⡸⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣯⢿⣧⡀⢲⡼⣿⣿⣿⣿⣿⣷⡴⠇⡔⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⡈⣌⢿⣧⠘⣷⣿⣿⣿⣿⣿⣿⣷⠏⠀⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢡⣾⣿⣿⣿⡟⣿⣿⣿⣿⢿⣿⣿⣿⠀⠈⣿⣿⣇⠙⣌⣿⣶⣿⣿⣿⣿⣿⣿⣿⡀⣠
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣷⣾⣿⣿⣷⡀⣰⣿⣿⣿⣀⡹⠾⣿⣿⣟⣿⣿⣿⣿⣟⡴⠃⠀⣰
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⢛⣿⣿⣿⣿⠯⢠⣿⣿⡿⠀⢸⣿⣿⣿⠀⠘⣿⣿⣿⡁⠀⠀⠹⣿⡿⣿⣿⣿⣿⡏⢰⣇⠀⣿
⠀⠀⠀⠀⠟⠁⢸⣿⣿⣿⣿⣿⣿⡄⠀⢠⣾⣿⣿⠟⠓⠲⣾⣿⣿⠟⠛⢿⣿⣿⠳⠻⠄⢿⢹⣿⡄⠀⠀⠀⣹⣷⠘⣿⣿⠿⠃⡏⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠉⠁⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠙⠋⠉⠀⠀⠙⠋⠁⠀⠀⠀⠈⠀⠙⠉⠀⠀⠀⠉⠉⠁⠀⠁⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃⠐⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣢⣂⢀⣀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠠⠦⠀⠰⠶⠀⠐⠒⠊⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠠⠧⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⢂⠀⠂⠁⠀⢸⠀⠀⣠⣤⢀⣤⣀⣠⣤⠈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣀⣻⠄⠀⠀⠈⠠⠴⠰⠴⠏⠄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣿⠀⣰⣶⢀⣐⡦⢀⣰⡶⠀⠠⣖⡆⢸⠉⠭⠤⠆⠅⠉⠁⠴⠧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠄⠀⠀⠀⠀⡞⠻
⠀⠀⠀⠀⠀⠀⠈⠵⠶⠲⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠀⠀⠀⠀⣷⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠀⠀⢲⡆⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⣧⢰
⠀⠀⠀⠀⠀⠚⠓⠒⠒⠒⠒⠒⠒⠸⠢⠠⠠⠄⠄⠼⠤⠤⠤⠤⠤⠼⠒⠲⠀⠘⠟⠋⠓⠚⠛⠛⠋⠋⠉⠉⠁