Skill Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣦⣤⡄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣏⣴⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⠿⣿⡿⢋⣼⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠗⣈⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢿⣿⣷⣌⣴⣿⣿⣿⢃⣤⣤⣦⣤⣤⣀
⠀⠀⠀⢻⣿⣿⠋⣉⣭⣥⣶⣿⣿⣷⣌⡛⠿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⡿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣰⣿⢩⣿⡆⣿⣿⣿⣬⣿⣿⣿⣿⣧⣷⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣧
⠀⢸⣿⣿⡘⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣴⣶⡌⢿⣿⣿⣿⣿
⠀⣿⣿⣿⣷⣌⠻⣷⣌⠻⣿⣿⣦⣤⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣙⠛⣡⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣧⡙⣿⣷⣦⢉⣍⣉⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⠏
⠀⠸⣿⣿⣿⣿⣿⣧⢸⣿⡟⢈⠻⠟⣩⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⢸⣿⣿⡿⠟⠁
⠀⠀⠙⣿⣿⣿⣿⡿⢸⣿⠇⡿⢃⡌⢿⣿⡿⢛⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿
⠀⠀⣠⣌⡻⢿⣿⣿⣾⡟⡘⢡⣿⣿⡆⢋⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠸⠿⠿
⠀⢰⣿⣿⣿⣶⣬⣙⣛⣰⣿⡆⣭⣭⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣤⡀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣛
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⡆⣿⣿⣿⣿⡟⣡⣤⣙⣩⣴⣿⣧⣄⣀⣠⣄⠀⠀⠨⣭⣭⣭⣭⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣷⢸⣿⣿⣿⣷⠈⣿⣿⣿⠿⠛⠿⣿⣿⣿⡁⠀⠀⠀⠈⠉⠉⠉⠁⠀⢀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⡄⣿⣿⢉⣉⣼⣿⡿⢡⣾⡇⠀⠀⢻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣇⢹⣿⣘⡛⢻⣿⣧⠸⣿⡇⠀⠀⣸⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⢈⣿⣿⣷⣦⣥⣤⣾⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⡿⢟⣡⣿⣿⣧⡘⠟⣉⡛⢿⣿⡿⠛⠋⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⢋⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⡈⠛⠃⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣰⠒⣦⢰⣲⡄⠀⣖⠂⢰⣰⠂⡆⢰⠀⢰⡆⠠⣖⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣇⣻⣤⣟⣸⣽⣇⣀⣬⣟⣸⣙⣆⣇⣸⣤⣸⣧⣄⣬⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⢠⣄⠀⡀⠀⠀⠀⠀⠀⢀⡀⣠⢤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⣀⣸⡿⠛⠛⠾⣿⣀⡀⠀⠀⣀⣻⠉⢉⣈⠉⢹⣁⡀⠀⠀⠀⢐⣿⣾⠿⢷⣿⡃⠀⠀⠀⠀⠀⠀⠀⣙⣋⡀
⠀⠀⠀⢛⣿⠀⢺⣿⠀⢸⡟⠁⠀⠀⢳⡄⠀⣸⣿⡀⠀⣴⠃⠀⠀⠘⣿⣿⡧⠀⢼⣿⣿⠓⠀⠀⠀⠀⢰⡼⣿⣿⡇⡆
⠀⠀⠀⠛⢻⣄⡻⠿⣣⣾⠛⠃⠀⠀⠷⢤⠘⠿⠿⠗⢠⠾⠇⠀⠀⠸⠿⣿⣀⣀⣀⣿⠿⠗⠀⠀⠀⣩⣿⡀⠿⠿⠁⣹⣏
⠀⠀⠀⠀⠘⠋⠹⡿⠉⠛⠀⠀⠀⠀⠀⠛⠞⢳⣴⠛⠞⠁⠀⠀⠀⠀⠐⠟⠛⣿⠛⠻⠃⠀⠀⠀⠀⠀⠘⠛⠲⠖⠛⠇

⠀⠀⠀⠀⢀⠀⠀⡆⠀⢀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢠⠤⣶⣤⣶⠠⣄⠀⠀⠀⠀⢀⣠⣦⣶⣠
⠀⠀⢀⡀⠀⣷⣾⢿⣶⡇⠀⡀⠀⠀⠈⣿⢋⣤⣤⣉⢻⠋⠀⠀⠀⠀⢸⢀⣰⣤⣆⡄⣿⠀⠀⠀⠀⠰⣏⠲⢞⡷⠂
⠀⠀⠀⢉⣻⣟⣀⣀⣘⣿⡋⠁⠀⠀⠀⠀⢼⡅⢀⣿⠂⠀⠀⠀⠀⠀⢸⢸⣏⣲⣻⡖⣿⠀⠀⠀⠀⡈⠙⠟⠿⠉
⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⣠⣿⣬⠙⠛⣣⣼⣄⠀⠀⠀⠀⢸⣀⣙⣋⣃⠀⣿⠀⠀⠀⠀⠙⢦⣾⣿⣿⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠟
⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⣴⣦⠀⡗⢁⣄⠀⠀⠀⠀⠀⠀⢀⡯⡸⣆⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡤⢖⡳⠀⠀⠀⠀⠀⠀⣻⣿⡃
⠀⠀⠀⡠⣈⠙⣤⣥⠚⢉⢤⡀⠀⠀⢾⣩⠇⣴⣶⡄⣏⣹⠀⠀⠀⠀⠸⣿⡿⠀⠀⡷⠴⠆⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄
⠀⠀⠈⠣⠚⢂⣿⣿⡀⠙⠖⠃⠀⠀⠀⠀⢃⣽⣿⣅⠁⠀⠀⠀⠀⢀⣾⣿⣿⣆⠉⡧⢴⡆⠀⠀⠀⠀⣀⣈⣉⣍⣉⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠧⢤⡄⠀⠀⠀⣴⢣⢨⡇⢠⣧⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣤⡴⡶⣤⡀⠀⠀⠀⠀⠀⢀⣤⣦⣀⡀⢯⡽⠀⠀⠀⠀⠀⣀⣤⢴⡚⠉⠑⡄⠀⠀⠀⠀⢀⣤⣦⠴⡄⡀
⠀⠀⠀⠀⣾⡏⢠⣦⠈⣿⠀⠀⠀⠀⠀⣾⣿⡇⠉⢻⣇⠮⠆⠀⠀⠀⢠⣿⣄⠀⢷⢲⣤⠇⠀⠀⠀⠀⣸⣿⣟⣄⢓⢱
⠀⠀⠀⠰⣿⣷⣄⣅⣴⡟⠀⠀⠀⠀⠰⣿⣿⣿⣶⣾⡇⠀⠀⠀⠀⠀⢾⣿⣿⣶⣿⣷⠀⠀⠀⠀⠀⠰⢿⣿⣚⣳⣾⡟
⠀⠀⠀⠀⠸⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⠟⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⣿⣿⡏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠈⣿⣷⢰⣿⣿⣧⣽⣿⠁⠀⠀⠀⢀⣤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⢦⣶⣶⢿⣿⣆⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⣰⣿⡿⣿⣭⣽⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠹⣿⡟⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣸⣿⣿⣾⡿⠃⠀⠀⠀⠀⠀⣴⣿⡦⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⣿⣷⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣹⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡄
⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⢦⡀
⠀⠀⣿⣿⣿⣿⣿⣿⢟⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣇⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⣹⣼⣷
⠀⠀⣿⣿⣿⣿⣿⣯⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠿⢸⣿⣿⣿⡇⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣽⣾⣿⣿⣿
⠀⠀⠈⠻⣿⣿⣿⣿⠿⠃⠀⠀⡄⢠⠀⣀⠀⠄⠀⢸⣿⣿⣿⡇⢠⠀⣄⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠉⠂⠀⠀⠀⠀⢸⣿⡏⣿⡇⠀⠈⠁⠀⠐⠀⠂⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⣛⡛⢿⣷⡄⠀⠀⠀⠀⠐⠛⠁⠛⠆⠀⠀⠀⠀⠀⢠⣾⣿⣿⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⣿⡇⣾⣿⣿⣆⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⣛⣫⣼⣿⠏⠀⠀⠀⢠⣶⣿⣿⣿⣿⣶⡄⠀⠀⠀⠹⣿⣭⣥⣤⣭⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⢀⣿⠃⢻⣿⡁⡟⢘⣿⠀⠀⠀⠀⠉⠛⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣿⣿⣦⣷⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣴⣿⣿⣿⣿⣆⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡿⢛⣩⣭⣭⣭⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢋⣴⣿⣿⡟⡙⢿⣿⣷⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⣾⣿⣿⡏⠰⠿⠌⢻⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⣿⣿⣷⢸⣿⣿⠇⣿⣿⣿⣿⣿⣿⣯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⣿⣿⣿⢸⣿⠏⣼⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⡞⣡⣾⣿⡟⠈⠛⠛⠛⠋⠀⠀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠛⠛⠋⠀⠀⠀⠀⢀⣀⠀⠀⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠛⣡⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡌⣼⣿⡌⣰⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⠟⣰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣷⡐⣿⣿⣿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⡇⢻⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠁⠈⠉⠉⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠙⠳⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡴⠋⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢠⠟⠉⢉⣩⠽⠛⠉⠉⠉⠉⠙⢦⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣀⠀⠀⣰⠋⢠⡞⠉⠀⠀⣀⡤⠤⣄⡀⠀⠀⠙⣆⠀⠀⣀⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠉⢀⡼⠁⠀⠈⠳⠤⠶⠋⠁⠀⠀⠀⠙⠲⣄⡀⠈⠳⣄⠈⢀⡴⠃⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠲⣾⡁⣠⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠲⣄⣸⠖⠋⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡟⠁⢀⡼⠛⠲⣄⣀⠀⠀⠀⠀⠀⠙⠳⣤⣀⡼⠁⠀⣤⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠧⣤⡞⠁⢀⡼⠉⠈⣧⣄⠀⠉⠳⣤⣀⣨⡇⠀⠀⢳⡄⠉⠳⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠓⠶⣏⠀⣠⠞⠁⣠⠃⣄⣀⣸⠏⢡⣤⣤⣤⣤⣽⣤⣤⣬⣷⡄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠦⠞⠳⠤⠟⠉⠀⠀⠀⣼⠀⠀⠀⠀⣧⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣶⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠛⠛⠛⣿⠛⠛⠛⠛⢻⠛⠛⠛⠛⢻⠛⠛⠛⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢹⠀⠀⠀⠀⢸⠀⠀⠀⠀⣾⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠀⣀⡀⠹⣤⣤⡤⢼⣧⣤⣤⣤⣾⣤⣤⣤⣤⠇⠤⠠⢤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀⠀⢸⡏⠁⠀⠘⢦⡀⠀⠹⣄⠀⠀⢸⠀⠀⢀⡞⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣘⡇⠀⠀⠀⠀⠙⠳⢤⣘⣦⣀⣸⣀⣴⣋⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⣠⡀⠀⢀⠄
⠀⠀⢠⣾⠛⠉⠁⠦⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠛⣻⠆
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⡢⠐⠀⢀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠯⣿⣿⢶⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠠⣅⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠼⠀⠘⠋⠐⠀⠊⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠄⠀⠜⣸⠀⠀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠐⠀⠀⠐⠉⢀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡀⢀⡀⢀⠀⠀⢠⣄⢀⣦⣴⢷⠂
⠀⠀⠀⠀⠠⣶⣶⡞⣮⡍⠀⠀⡿⢛⠉⠨⣄⠀⠘⣿⣝⣿⣷⡌⠀⠀⠀⠈⠁⣸⣛⣚⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⢻⢛⠛⠠⠗⠀⠔⠀⠈⠒⠐⠓⠒⠀⠻⠛⠏⠀⠡⠶⠂⠀⠀⠈⠀⠄⢹⣿⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠶⠿⠛⠛⠛⠛⠿⠷⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣠⣴⠾⠟⠛⠋⠉⠉⠙⠛⠛⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣠⡴⠟⠉⢀⣤⣴⠶⠾⠟⠻⠿⠶⣶⣤⣀⠉⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⣴⠟⠁⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠙⣷⡀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢀⣾⠃⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠙⢷⡄⠈⢿⡄⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣾⠃⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠈⢿⡄⠈⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⣷⠀⢹⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣿⡀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣼⠇⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠘⣷⡀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⢀⣼⠏⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠈⢿⣄⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣴⡾⢶⣿⣻⣷⣄⡈⠙⠻⢶⣦⣤⣤⣤⣤⣤⡶⠟⠋⠁⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠙⣿⠀⠉⠻⢶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⡾⠋⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⡾⠋⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⢀⣴⡾⠋⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣠⡾⠋⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡾⠋
⣾⡋⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠋⠉⠉⠁
⠘⢿⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠤⠐⠦⠀⠀⠀⠀⠀⢀⣶⠛⣦⠀⠀⠀⠀⠀⠀⢢⣤⣤⠀⠀⠀⠀⠀⠀⠄⢠⠄⢄⠀⠀⠀⠀⠀⡴⠖⢦⣶
⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠂⠀⠀⠀⠀⠀⠀⡄⣰⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡇⠀⢸⡥
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢨⠀⠀⠀⡇⠀⠀⠀⠀⠘⠛⡻⡛⠃⠀⠀⠀⠀⠀⡏⠀⠄⡇⠀⠀⠀⠀⠀⠀⢠⡟⠂⠀⠀⠀⠀⠀⠀⢸⠟⠇
⠀⠀⢠⣭⠉⢩⡥⠀⠀⠀⠀⠀⠀⠲⠁⠄⠀⠀⠀⠀⠀⣇⣠⣀⠇⠀⠀⠀⠀⠀⠘⠿


⠀⠀⣤⡤⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⣠⠀⢤⡀⠀⠀⠀⠀⠀⠀⠠⠔⡄⠀⠀⠀⠀⠀⡠⠐⠂⡄
⠀⠀⠙⣁⣤⣀⠋⠀⠀⠀⠀⠀⢠⣠⠆⠀⠀⠀⠀⠀⠀⣅⠾⠆⠾⠀⠀⠀⠀⠀⣌⠰⢁⠃⠀⠀⠀⠀⠀⡏⠁⠈⣸
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠚⠃⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠈⠐⠀⠁

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠼⠄⡀⠀⠀⠀⠀⠀⡘⢳⠟⣀⠀⠀⠀⠀⠀⣴⣫⠛⢆
⠀⠀⠀⣇⠻⣸⠀⠀⠀⠀⠀⠐⠖⠟⠟⠇⠀⠀⠀⠀⢠⡀⢠⠀⣤⠀⠀⠀⠀⠀⠱⠟⠷⠁⠀⠀⠀⠀⠀⠓⢌⡦⠋
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣰⣾⣤⠄⠰⢄⣧⣠⣴⣾⢠⡄



⠀⠀⠀⠀⠀⠀⣤⣦⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⢀⠀⠀⠀⠀⢀⡀⠠⣶⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣘⣟⡀⠀⠀⠀⠀⠀⠀⠈⣤⣠⡄⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⠀⢀⣶⣚⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣬⣿⠀⠂⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣽⠆⠀⠀⠀⢸⣿⣾⣾⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣾⣿⣶⠀⠀⠀⠀⠀⠀⢀⣅⠈⠀⠶⠀⠀⠀⣰⡀⢀⡆⢀⣆⠀⠀⠀⠀⠀⠛⣿⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠠⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣼⣿⣿⣿⠛⠃⠀⠀⠀⠀⢀⣘⣿⣃⡀⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⠀⠀⠀⠀⣀⣻⣟⣀⠀⠀⠀⠀⣠⡀⠈⠛⠃⢀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣫⢭⣝⣷⠀⠀⠀⠀⢀⣿⣏⣂⣿⣧⠀⠀⠀⠀⢰⣿⡍⢩⣿⡇⠀⠀⠀⠙⣿⣿⠄⣿⣿⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠹⠿⠇⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠛⠀⠀⠀⠀⠘⠛⠿⠿⠟⠛⠀⠀⠀⠀⠘⠻⠿⠿⠟⠋⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣉⢰⣹⢾⢶⡂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡾⠉⠉⠙⢧⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠉⠀⣀⣀⡀⠀⠉⠉⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠶⣄⣠⠶⠛⠉⠉⠉⠛⢶⣄⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⢀⡀⠈⣷⠀⠀⠀⣀⣠⣤⣄⠙⣧⠀⠘⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡤⠶⠞⠋⠻⢦⣼⡀⣶⢾⡏⢻⠀⣿⠀⢹⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⠀⠀⠀⠀⠀⢸⣿⠇⠿⠾⠷⠾⠶⠟⢀⡾⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⣴⠟⢷⣄⠀⠀⠀⠀⢀⣠⠞⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⣇⠉⣩⡿⣦⣄⡈⠙⠓⠒⠛⠋⠁⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠘⢷⣼⣛⣿⣴⠋⠈⠙⣧⠀⠀⠀⣤⠖⠛⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠶⠶⠀⢸⡟⣧⣀⡀⠀⠀⢹⡗⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠾⢤⣿⢤⡟⠹⣧⣄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⣀⣿⣸⣇⣀⣀⣿⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⣶⢰⠰⣖⠐⡆⣶⣴⢰⣖⠰⣖⡠⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠋⠈⠉⠉⠁⠉⠈⠈⠉⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢰⢷⣸⣏⡇⡿⢾⡇⣏⢸⢶⡇⣏⢸⣯⠈⡏⢸⢽⢠⣿⡌⢹⠁⣏⢸⣉⡆⡷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⡤⡄⢀⠤⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢯⡧⠷⠺⢤⡷⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠲⡖⢺⢠⠒⠒⡄⡟⢲⠖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠒⠛⠞⠸⣄⣀⡎⠳⠚⠒⠚⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⡰⠊⠙⢹⠞⠋⠙⢢⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠃⠃⠀⠘⠊⠀⠘⠘⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠆⣄⠒⠂⠀⠆⢹⢀⣤⠀⠀⠀⣀⣀⣀⣀⣀⠿⣼⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣿⣿⣷⡒⠀⢸⢺⣿⠀⢀⣸⢑⣒⣀⣒⢸⣀⣛⣚⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⢷⡞⢶⠗⠇⢸⠉⣿⠀⢸⣾⠀⠼⠼⠿⢸⠛⠋⠛⣮⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣚⡛⡛⣛⡁⢸⠀⣿⠀⢸⣿⡐⠀⠀⠛⣸⠀⠺⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠴⠶⠶⠖⠃⢸⠀⣿⠀⣸⢧⣤⡠⠤⠤⠤⢤⣤⣤⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠧⠤⠤⠤⠴⠶⠼⠀⠀⠀⠫⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠏⣤⠘⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢠⠀⣠⠞⠀⠀⠂⣰⣄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⢶⣿⡄⡀⠀⠀⢿⡀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⣮⠁⠀⠀⢀⡿⠀⠀⠀⢀⡈⠗⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡄⠀⣇⠷⣤⠀⠀⠨⠀⠀⠀⠀⠸⠁⣀⣁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣧⣾⣟⣏⢲⣿⠐⠄⠰⠀⠀⠀⠀⢠⠠⡤⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⡟⢟⠙⠛⣅⡿⠀⠀⠀⢦⠀⠀⣰⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢸⡇⠀⠀⠀⢨⡄⣤⡿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⢸⡇⠀⠀⠀⣈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢁⣤⠬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⢰⡶⠈⡄⠪⠛⡃⣙⠀⢠⡀⠀⠀⠀⠄⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡠⡁⣶⠑⡏⠡⡘⢋⠿⢐⠙⠞⠳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠌⠀⢴⡮⠺⠇⠚⠁⠋⠠⠋⢁⡰⠦⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⡀⣴⠤⡄⠀⠁⣩⡿⠳⣦⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⠊⠀⠢⠀⠀⡀⠙⠛⠳⢶⡀⠀⠸⠟⣅⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠁⢸⡷⢴⡶⣤⠙⣶⡛⠀⢀⡄⠈⠃⠀⠀⢠⠄⣴⡆⠽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠆⠀⠰⢟⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⣿⣿⡀⠈⠀⢠⠀⠀⣰⢿⡜⢿⣀⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⠀⠀⡀⠈⠃⣀⠀⠀⠠⢠⠀⠀⠀⠀⡘⠁⠿⠛⠉⠉⠓⠴⣠⢞⣥⣾⣿⣾⣿⣷⠀⠀⠓
⠀⠀⠀⠀⢀⣔⣡⠄⠀⠀⠐⣄⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⠳⣷⢶⡾⡰⠆⠀⠀⠀⠀⠀⠀⣤⣛⡟⠀⠉⠈⠉⠉⠀⣤⠊
⠀⠀⠀⠀⠻⠿⠛⠀⠀⠰⣶⡌⠀⠰⠊⠀⠀⠼⠃⠐⠁⠀⣠⠋⠀⢀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⠀⢀⠸⠆
⠀⠀⠀⠀⣄⠀⠀⣀⢠⠶⠋⠀⠀⢴⠄⠀⠀⠀⠀⠀⢡⡶⠀⣠⠶⠉⠁⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡀⠀⠀⡿⠃
⠀⠀⠀⠀⢸⣉⣋⡹⠏⠀⠀⠀⠀⠘⢧⣀⣁⠂⠀⢠⡘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠀⠀⡀⢡⠟⠀⣀⡠⠟⠁
⠀⠀⠀⠀⠀⠙⠈⠋⠀⠀⠀⠀⠀⠀⠀⢨⠹⢳⣦⣨⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡤⠚⢩⣵⣦⢰
⠀⠀⠀⠰⠀⠚⠉⠉⠉⢠⣄⠀⠀⢀⠰⠏⢙⣁⡉⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⢀⡅⢨⣿⠼⢦⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⢤⣉⡷⠾⣯⣼⡷⣤⡜⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠶⠆⠛⣋⣙⠳⠻⠷⠄⠡⠂⠈⡙⠂
⠀⠀⠀⠀⠀⠀⢀⡖⠉⠁⠛⠲⠚⠋⡁⣐⠂⠨⠷⢴⡋⠙⠂⠙⠙⡶⠄⠀⠀⠀⠀⠀⠀⠀⣼⡆⠁⠀⠁⠉⠉⠀⠁⠀⠀⠀⠃⠀⠁⠘
⠀⠀⠀⣀⣤⠄⣎⡀⠉⢙⡂⡴⠚⣂⠐⠛⢲⠾⡀⠀⠀⢂⠤⣤⣐⣀⣼⠀⠀⠀⠀⠀⢀⡴⠂⠉⠠⠆⠀⠀⠠⠀⠠⠄⠀⣀⡀⢠⠁
⠀⠠⠂⠉⠀⡰⠀⣑⠟⢈⠁⠆⠘⠁⢶⡎⠀⠃⢁⡀⡤⠉⣧⢘⣀⡅⠸⠀⠀⠀⠀⢀⠢⠄⣸⠀⠠⠤⠤⠤⠤⠤⠤⠤
⠰⠀⡀⠀⣰⠃⣔⠉⠖⠉⠀⠀⠛⠂⠒⠠⠠⣆⠀⣄⠁⠀⢀⡜⢿⠌⢀⠦⠀⠀⠀⡜⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⢀⠀⡙⠲⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠷⠚⠂⢿⠤⠶⠌⠀⠈⠂⠰⠁
⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠡⠤⠤⠤⢀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣐⣒⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⢰⠃⠁⠉⠈⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢀⡤⠶⠦⣄
⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⠖⠦⠀⠀⠀⠀⠀⠀⢠⡿⠿⢿⣿⣿⣧⠀⠀⠀⠀⡎⠉⢩⡽⢭⠉⠉⡇⠀⠀⠀⠀⡏⣲⢦⣆⡔⡖
⠀⠀⣸⣉⡙⠉⢉⣍⣇⠀⠀⠀⢀⡤⢤⡀⠀⠀⣤⢤⡀⠀⠀⠀⢨⡿⠿⢿⡿⣿⣯⠀⠀⠀⠀⡇⠀⠸⢤⠼⠁⠀⡇⠀⠀⠀⠀⠳⣉⡀⣈⣶⡇
⠀⠀⠛⠛⠓⠒⠚⠛⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣤⠶⣛⣛⣛⠳⣦⠀⠀⠀⠀⢀⡴⠉⠈⠙⠒⢦⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⠀⠀⠀⠀⢻⣿⣿⢧⣝⢿⣿⡇⠀⠀⠀⠀⣰⣶⣿⠙⠒⡆
⠀⠀⠀⠚⣩⡭⣭⠛⠀⠀⠀⠀⠀⢯⣀⣀⣀⣀⣀⡸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢺⢟⣵⢫⡿⣧⢻⡇⠀⠀⠀⠀⢹⠛⠛⣶⣶⡇
⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠻⠏⠻⠀⠰⠋⠾⠇⠀⠀⠀⠀⠈⠳⢤⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⣠⣴⢶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⡄⠀⠀⠀⠀⠀⠀⢀⣤⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢰⣉⣏⣻⣹⣋⡇⠀⠀⠀⠀⠀⣀⣤⣳⢤⣏⡀⠀⠀⠀⠀⠀⢾⣕⢗⣸⣷⠀⠀⠀⠀⠀⣾⣻⣟⣸⡁⠑⢹⡆⠀⠀⠀⠀⣤⣶⣾⣄⠀⢱
⠀⠀⠈⣽⣦⣀⣴⣿⠁⠀⠀⠀⠀⠰⣏⠀⣹⠞⠤⠃⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⡄⠀⠀⠀⠈⢻⡿⠿⠟⠲⠏
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣴⠛⢶⠀⠀⠀⠀⠀⠀⠀⠀⠼⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⣥⣥⣥⠠⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹
⠀⠀⠀⢰⣿⠟⣿⡇⠀⠀⠀⠀⠀⠀⢰⣶⡶⣾⣦⠀⠀⠀⠀⠀⠀⠀⠠⠪⢟⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⠨⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠸⠿⠤⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣆⣿⡿⠀⠀⠀⠀⠀⠀⠀⠑⠤⠊⠀⠀⠀⠀⠀⠀⠀⠉⡛⡛⡛⡛⠈⠀⠀⠀⠀⠀⠀⣇⣰⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⣄⠀⣠⠄⠒⠂⠀⠀⠐⠒⠒⠈⠉⠀⠀⠀⢡⠀⢀⠖⢉⠤⠄⣉⡑⡄
⠀⠀⠀⠀⠀⢀⠤⠊⠀⢀⣤⠶⠒⠂⠘⡖⢂⠀⠀⠀⣀⣠⣀⠀⢀⡤⠶⠤⣄⠀⢸⠀⢸⠀⠇⠀⡠⠚⣉⢝⠠⢄
⠀⠀⡠⠔⠊⠁⠀⣀⠀⠸⠀⢀⣤⠤⠄⠘⡌⡄⠀⠚⠁⠀⠈⠁⠈⠀⣀⣀⡀⠀⢨⠔⠁⡸⠀⠀⢧⡞⣊⠥⠒⠉
⠀⢎⠀⠀⠀⣰⠟⠉⠁⠀⠀⠸⠁⠀⢀⡀⠘⣷⠀⢠⠶⠛⠓⠄⠀⠘⠉⠉⠉⠀⡖⠒⠊
⠀⠈⠢⡀⠀⠃⠀⣴⠞⠃⠀⠀⡀⣠⠎⢱⠀⢹⢃⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠈⡶⢄⠀⠁⠀⠀⣀⡭⠞⠁⠀⠸⠀⢸⠈⢆⠀⣠⢶⣒⠛⠿⠟⠻⠄⠸
⠀⠀⠀⢠⠀⠎⠑⠤⡀⠙⠤⣀⣀⣀⠔⠁⠀⡌⠀⠈⢦⡉⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠈⣼⡀⠀⠀⠈⠒⠤⠤⢀⣀⣀⡠⠔⠁⠀⠀⠀⠉⠢⣀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠈⠫⣁⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣈⠆
⠀⠀⠀⠀⢐⣉⡁⠬⠄⠒⠊
⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⣀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣶⣿⠉⣷⣄⠀⠀⠀⠀⡧⣿⢯⠷⡾⡿⢭⠂⠀⠀⠀⢾⡟⠉⠉⠉⣯⡆⠀⠀⠀⢀⣂⣂⣉⠈
⠀⠀⣿⣤⣿⣻⣏⣸⡆⠀⠀⠀⢈⣿⣯⣤⣼⣿⣁⠀⠀⠀⠀⡱⠆⢸⡷⠀⠶⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠹⢧⣼⣏⣨⠟⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠻⠀⠀⠀⠀⢾⣧⣀⣀⣠⣟⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⣰⠁⠀⢘⡄⠀⠀⠀⢸⠙⠋⠋⢳⠃⠋⠋⢻⡇⠀⠀⣼⣛⣛⣛⣻⡄⠀⠀⠀⡠⠴⠟⠙⠋⠹⢶⣄⡀
⠀⠀⡼⡌⠀⠀⠀⡿⣆⠀⠀⢸⢀⣀⡀⣰⡀⣀⣀⣸⡇⠀⠀⠀⠙⣤⡜⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠘⠚⠒⠒⠒⠒⠚⠚⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠠⠊⠀⠈⠢⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣀⣀⢀⡀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⣀⡠⠤⣲⣷⠀⠀⠀⣴⠤⣤⢦⡤⢶⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠻⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⣭⣶⣾⣿⣿⣿⠀⠀⠀⢹⠀⣿⢸⡇⣾⠀⠀⠀⠛⢷⣾⢫⡙⣶⠟⠃
⠀⣴⡇⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠋⡟⠻⢿⣿⣿⣿⠀⠀⠀⢸⡆⣿⢸⡇⡿⠀⠀⠀⠀⢀⣿⡿⢿⣧
⠀⠉⠉⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠈⠉⠀⠀⠀⠘⠧⠿⠼⠶⠇⠀⠀⠀⠀⠚⠁⠀⠀⠙

⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦
⢸⣿⡿⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⢣⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠙⢦⠀⠀⢸⣿
⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣀⣀⣞⢲⣶⣶⣶⣶⣶⣾⠟⠢⣀⠀⢸⣶⣶⣾⣿
⢸⣿⣿⡿⠋⠉⠙⢿⣿⠛⠛⠻⢿⣿⣿⠿⠛⠿⣿⣿⣿⠟⠉⠉⣹⣿⣿⠉⠉⠉⠉⣿⡟⠉⠀⠙⢻⣿⣷⣄⡈⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣧⠤⠤⠤⠼⡇⠀⠀⠀⢸⠿⠧⠤⠤⠤⠼⣿⣏⠀⠀⠘⡕⢨⡧⠤⠤⠤⠼⢿⡀⠀⠀⠀⢀⣿⣿⣛⡻⠿⠿⣿⣿⣿⣿
⣸⣿⣅⡠⠤⠤⠴⠂⠱⢤⢤⡤⠊⡔⠉⠉⠉⠉⣲⣨⣿⣿⣿⣛⣷⡊⡰⠒⠒⠒⢒⣤⣕⣀⣀⣀⣺⣩⠒⠒⠒⠒⠒⣆⣹⣿⣉
⡧⠤⣁⠉⡆⠀⢰⠊⣉⣉⣉⣙⣒⠒⠢⡀⠀⡎⢡⠤⠤⠤⠤⠤⠤⣍⠙⠆⠀⡎⠁⣀⡀⠤⠤⠤⢄⠈⢱⠀⠀⡰⠉⣠⠤⠤⡼
⡇⠀⢸⠀⡇⠀⡇⢰⠁⠀⠀⠀⠀⢹⠀⡇⠀⣇⢰⠀⠀⠀⠀⠀⠀⡜⢸⠀⠀⠇⢸⠁⠀⠀⠀⠀⠀⡇⡸⠀⠀⢇⢸⡁⠀⠀⢣
⠑⢄⣸⣀⣯⣷⣷⣈⡆⠀⠀⠀⠀⡎⢸⠀⠀⠸⡀⢇⠀⠀⠀⠀⢠⠃⣾⡆⠐⣾⣈⣆⡀⠀⠀⠀⡰⢡⠃⠀⠀⠘⡄⢣⠀⣀⠼
⣶⣶⡶⣷⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⣦⣠⡿⠻⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠈⠉⠀⠀⠀⠾⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠁⢀⣴⠞⠛⠛⠷⣦⡀⠈⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡀⠀⢠⣿⠁⠀⠀⠀⠀⠈⣿⡀⠀⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠋⠉⠉⠙⠛⠻⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠀⠸⣧⠀⠀⠀⠀⠀⠀⣿⠇⠀⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⣿⡁⠀⠀⠻⣦⣀⣀⣀⣠⣼⠋⠀⣀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠈⢿⣤⣦⣀⠈⠉⠛⠛⠉⠀⠀⣼⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢠⣤⣤⣤⣤⡄⢸⣯⣀⡀⣰⡶⣦⣀⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠈⣿⠀⠈⠉⣛⣋⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠛⠷⣦⡾⠋⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠈⠉⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣦⣄⣀⣀⣀⣀⣠⣤⡶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠉⠉⣿⡄
⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣉⣭⣭⣭⣭⣍⠁⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢀⣴⡾⠛⠉⠉⠉⠉⠉⢿⣍⠉⠉⠁⠀⣿⠉⠉⠉⠉⢹⡇⠀⠉⠉⢉⣿⠋⠉⠉⠉⠉⠙⠻⣦⡄⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠻⢶⡆⢰⣶⠿⠃⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⣾⠇⠀⠀⠀⠀⣠⣤⣤⣿⠁
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⣸⡇⠀⣿⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣰⡟⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣿⠀⠀⢻⡇⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠻⣧⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣸⡏⠀⠀⠸⣧⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣤⡿⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡾⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠉⠉⠙⠛⠻⠿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠦⣤⣤⣸⣷⣤⣀⣀⣀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠛⢛⢿⠏⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠆⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠐⠀⠺⢽⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⣀⠀⠀⠀⠀⣄⡶⡀⢠⣀⠙⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠛⠋⢉⡙⠿⣿⢿⣿⣿⣿⡀⣤⠀⣰⣿⣿⣷⣿⣿⠿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢘⢿⠁⣤⠴⠒⢋⣉⣩⣝⡳⣝⣿⢿⢷⡜⠢⣿⣿⡷⢛⣁⣘⠢⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠉⠃⣨⣤⠶⣿⣿⣿⣿⣿⣿⣿⠿⢷⠀⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠀⡿⠿⣿⣿⠿⠟⡁⠒⠀⠀⠀⠀⠈⢻⣿⡿⠟⠛⣻⡏
⠀⠀⠀⠀⠀⣴⠟⠟⠓⠄⠙⡀⠀⠈⡆⠀⠀⡀⠄⠊⠀⣠⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡛⠿⣿⣧
⠀⠀⠀⠀⠀⢿⣇⠀⠀⢠⠀⠃⠀⢀⠧⠀⠀⠀⡀⠴⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡵⡤⡸⣿⡄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⢀⣠⠔⢁⡠⠔⠂⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⢀⣀⣀⠀⠀⢸⣷⡸⣿⣿⡇
⠀⠀⠀⠀⣼⣉⠙⠧⡀⢸⡿⢁⡔⠉⡗⡀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⣿⣿⣦⣼⣿⣿⣻⣿
⠀⠀⠀⣼⡟⡟⠓⠄⠘⢌⡇⣼⣀⣠⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢿⣿⡟⣿⣿⡏
⠀⠀⠀⢰⣵⣇⠀⠀⠐⠪⡿⣛⣿⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣴⣋⣍⠳⡶⣻⣯⣠⡉⢷⣽⠟
⠀⠀⠀⠘⣿⣿⠀⠀⠀⢰⠳⣾⣷⣄⠈⢆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠿⠛⠛⠉⠛⢿⣾⠛⠿⣿⢟⡿⠁
⠀⠀⠀⠀⢹⣻⣦⠀⠀⢸⠀⠘⡿⣿⣷⠈⢢⡀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢤⣀⣀⣀⣘⣿⣤⣾⣿⣼⠃
⠀⠀⠀⠀⠘⣿⡽⣄⠀⡀⠃⢁⡇⢿⣧⢠⡀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠿⠿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢾⣷⣤⣿⡤⠞⠀⠘⣿⢦⣕⢤⡀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⢷⣿⠉⠁⠀⠀⠈⠛⠷⣦⣤⣄⣀⠀⠀⠀⠀⡀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠒

⠀⠀⣀⣤⠤⣄⡀⠀⠀⠀⠀⢠⣺⣭⠷⣠⣚⣯⡷⡄⠀⠀⠀⢀⡤⡤⠤⠤⢤⢤⡀⠀⠀⠀⠀⡀⡀⠄⠄⠀⠂
⠀⣼⠛⣿⣤⡾⠻⡆⠀⠀⠀⠘⠯⣯⠝⠉⢯⣿⠽⠁⠀⠀⠀⠠⣛⠀⢠⣀⠀⣛⡇⠀⠀⠀⢰⣶⣶⣦⣦⣦⣤
⠀⣿⠟⢻⡾⠷⣶⡇⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣴⠀⠀⠀⠀⠐⣭⠀⠘⠉⢨⣭⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠈⠳⠾⠷⠴⠋⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠉⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠛⠛⠿⠿⠿⢿⡟

⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⣬⠀⠀⠀⣣⠀⠀⠀⠀⡇⠁⠁⠁⠑⠁⠁⠁⠁⡇⠀⠀⠠⣯⣭⣭⣭⣽⠄⠀⠀⠀⡤⣶⣿⣁⠍⠠⠿⠷
⢀⣞⡇⠀⠀⠀⢨⢳⡀⠀⠀⡇⡀⡀⡀⡠⡀⡀⡀⡠⡇⠀⠀⠀⠀⢑⣴⡋⠀⠀⠀⠀⠀⢺⣿⣿⣿
⠘⠚⠒⠒⠒⠒⠚⠓⠃⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠔⠁⠀⠉⠢⠀⠀⠀⠀⠓⠒⠒⠂⠉⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣶⠀⠀⠀⢠⡬⣥⡤⣤⡬⣦⠀⠀⠀⠀⠀⠀⢰⣧
⢿⣿⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⢰⣖⣫⣥⣶⣾⣿⣿⠀⠀⠀⠘⡇⢸⡇⣸⠀⣿⠀⠀⠀⠲⣶⣶⠟⡟⢶⣶⠶
⣠⣿⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠘⠻⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⣷⢸⡇⣿⠀⡏⠀⠀⠀⠀⠀⣽⣾⣿⣾⡁
⠛⠛⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠈⠓⠒⠋⠉⠙⠻⠀⠀⠀⠀⣿⣸⣇⣿⣰⡇⠀⠀⠀⠀⠰⠟⠁⠈⠙⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠎⢀⡴⠀⣠⠏⡀⠀⢄⣀⠒⠦⠤⣄⣀⣀⣀⡤⠤⠀⠀⢀⣀⡠⠤⠀⠙⣆⠀⢢⡀⠑⢄⡀
⠀⠀⠀⠀⣀⠴⠋⣀⡴⠋⣠⡞⠁⠀⠉⠲⢤⣀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⢀⣠⠴⠊⠁⠈⠳⣄⠙⠦⣀⠙⠦⣄
⠀⠀⠀⠀⠀⠠⢚⣡⡴⢾⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣤⣤⣤⡤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠈⡗⢦⣌⣓⠆
⠀⠀⠖⠒⠚⠋⠉⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠙⠛⠒⠒
⠀⠀⠀⠀⠀⠀⢰⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆
⠀⠀⠀⠀⠀⠀⡟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢹⡀⠀⡆
⠀⠀⠀⢰⠀⡸⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢧⠀⡇
⠀⠀⠀⡘⡰⠵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢧⢡
⠀⢀⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶⡀
⢀⢛⠩⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⡍⡙
⢸⢸⠀⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⠀⡇
⢸⢸⠀⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⠀⡇
⢸⠀⠀⢁⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠸⠀⠁
⠈⠀⠀⢸⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡆
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠁⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⣦⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⣧⣄⠀⡀⠀⢀⣶⣿
⠀⠀⠀⢠⣴⣾⣿⠿⠛⠿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⣾⣿⣯⠋⠉⡁⠈⠹⣿⣿⠃⠀⠀⢀⠔⠛⠻⣆⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠀⣿
⠀⠀⠀⠘⢿⣿⡄⠐⠛⠆⢦⡿⠋⠀⠀⠀⠊⢠⡀⢠⣻⠀⠀⢠⣴⡞⠉⠙⠛⠙⠃⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠟⠁
⠀⠀⠀⠀⠀⠉⣷⢉⢦⣼⣥⠀⠀⠀⠀⠘⠀⠀⢤⠿⣸⠀⠀⢺⣿⡇⠀⠀⠀⠀⢀⣿⣿⠆⠀⣀⡤⠤⠤⢦⣄⠀⠀⠀⠀⠠⠖⣨⡿⠉⠟⠃⠀⠈
⠀⠀⠀⠀⢠⣿⣿⣿⣆⠺⣴⣶⣶⣦⡀⠀⠀⠐⠒⠁⠇⠀⠀⠀⠻⠇⢀⡀⣀⠀⠸⠿⠋⠀⣼⠏⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣬⣥⡤⠚⣞⣶
⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠝⣻⣿⣹⣵⢲⡀⠲⣶⡆⠀⢀⡀⠀⡃⠰⡀⣒⠀⡄⠀⠈⠀⡀⣿⠢⢤⠀⢠⠤⠀⠟⠁⠀⣿⡇⣿⣿⢷⡶⣿⣿
⠀⢸⣿⠟⢋⢓⣾⣿⣿⣧⣤⣿⠟⠋⠁⠸⣿⣦⣽⣏⡀⠘⣷⡀⠀⢀⠀⠭⠀⠀⠖⠀⡠⠻⣜⣷⣾⣤⡄⠀⠀⠀⠀⢰⢁⣼⣋⠿⣤⡽⠿⢾⡇
⠀⠈⠛⠳⣤⣿⣿⣿⣻⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⢷⡤⣀⡙⢸⣋⣡⣀⡔⠁⠀⠈⢿⣿⣟⡛⠀⠰⠀⠀⢸⣾⣿⣿⡒⢂⣠⣐⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠁⠀⣄⣄⣰⢿⣿⣽⣿⡿⠿⠋⠉⠀⠀⠀⠙⠁⠁⠀⠀⠀⠀⠺⢿⣻⡿⠿⠀⠀⠐⠈⠁⠀⢹⠿⣿⣿⣿⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡽⠖⠚⡎⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠹⣿⣭⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⣀⣀⣶⡄⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣾⡟⠉⢩⠀⠲⠁⠀⠀⠀⠀⠀⠠⡚⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠴⠋⢉⣽⠆⠀⠀⠀⠀⠐⠀⠀⠀⣀
⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠐⠄⠣⠄⢠⠀⠀⠀⠀⠀⣼⠏⠁⠀⠀⠀⠀⠀⠀⠀⠐⡂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⣸⣿⣶⡶⡆⠀⣯⡅⠀⠀⠈⡄⠀⠐⠀⠀⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣭⠁⣀⠇⠀⠿⠃⠄⠀⠔⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠻⠏⠀⠈⠂⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣶⣶⡄⢀⢀⣤⣴⣀⠀⣤⣀⣀⣠⣀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡔⠶⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢉⠁⠀⠙⠛⠛⠉⣰⢿⣿⣿⣾⠋⠀⡷
⠀⠀⠀⠀⠀⡀⠄⠀⢀⡰⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠒⠀⢀⠀⡀⠀⠀⠀⢠⠀⠈⡀⢁⣅⣀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠉⢉⠂⣠⠄⠀⢀⣤⣴⣄⢲⠀⠀⠀⣤⣦⣄⢠⡄⢠⠀⠀⠀⠾⣦⡴⢿⣿⢻⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢺⣿⢷⣏⡷⡾⠀⠀⠸⠟⣉⠁⠘⣦⠀⠀⢹⣧⣽⡿⣷⡎⠀⠀⠀⠀⠀⢀⣼⠿⣯⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠆
⠀⠀⠀⠀⠀⠐⣟⣓⠙⢓⠳⠲⠲⠒⠀⠉⠒⠈⠁⠀⠀⠘⠟⠋⠇⠀⠑⠲⠖⠀⠀⠂⠉⠀⠆⠘⣯⢃
⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠰⠀⠐⠀⠆⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠄⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠘⠁
⠀⠀⡈⠀⠀⠀⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⡠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠠⠂
⠀⠀⠐⠂⡀⣄⣤⡄⢠⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢴⡄⠀⠀⠀⣲⠂
⠀⠀⠀⠀⡀⠈⠐⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠉⡔⡁
⠀⠀⠀⠀⠁⠖⣀⠶⢄⡀⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠆⢸⡆⠀⡼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⢀⠉⠐⠒
⠀⠀⠀⠀⠀⠂⠀⠙⠉⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⡆⢰⠁⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠐⠒⠠⢄⠄⠀⠈
⠀⢰⠋⠃⡇⣔⠀⡀⡀⢤⢠⠀⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠆⠀⠙⠂⠀⠉⠁⠀⠻⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⠀⡀⠀⠛⠛⣤⣀⠀⣀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠠⠀⢀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⢆⠀⢀⠈⠉⡀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⣦⡀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢿⠀⡀⡞⠀⠉⢠⠸⢃⠗⢠⣀⡴⠀⠀⠀⠀⠀⢸⣷⠒⠋⢸⣧⠈⠒⣲⣿⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠘⢀⡀⠈⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⢐⣞⠠⠯⠬⠅⠡⠌⠋⢅⣲⠚⣄⡠⡴⠃⠀⠀⠀⠀⠀⢻⠙⠀⠀⠀⠀⠀⠀⠈⠀⢣⡀
⣧⣄⣀⠀⡀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠠⠠⠤⠀⠀⠀⠀⢀⠈⠉⠚⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠄⠠⠠⠀⠀⠄⠀⠱
⠙⢿⣿⣤⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⠁⢂⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣌⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⣄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⣤⡀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⢸⣧⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⢀⣼⠗
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣶⡶⠟⠁
⠀⠀⠀⠀⢠⣀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠛⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢹⡿⠋⠁⠀⣠⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡏⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢘⣷⡀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⡍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣄⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⡉⠉⢹⠻⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠁⠀⣱⠀⠀⠀⢰⠁⠀⠱
⠀⠀⠀⠀⠀⢀⡠⠔⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡠⠔⠚⠋⠉⠉⠳⠖⠄⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠐⠦⣄⡀⠀⠀⣰⠏⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⡾⠀⠀⠈⢂⣀
⠀⠀⠀⠀⠀⠠⢤⣤⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣼⣥⣤⣤⡶⠉⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠂⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⣷⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣼⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⡟⠋⠉⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣦⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡟⠛⢿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣧⣤⣼⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠀⠀⠉⠉⠉⠁⠈⠉⠁⠀⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣴⣀⡁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣽⣿⣿⠿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢟⣿⣟⣹⣿⣿⣻⣿⡤⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣿⣯⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣳⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣏⡛⣛⣽⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠈⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠤⠀⠄⠀⢀⠀⢠⡄⠤⠄⠒⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⢠⣄⣀⣠⣴⣶⣷⣞⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠋⠁⠀⠀⠀⠁⠀⠀⣠⠊
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡄⠤⠛⠉⠀⠀⠀⠀⠀⠈⢧⣄⣀⡀⠀⢀⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠶⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠭⠀⠀⠀⠶⠶⠶⠶⠴⠦⠀⠀⠀⠀⠀⣤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣦⣞⣠
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠐⠒⠒⠒⠒⠒⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⣀⣰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣖⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠁⠀⠁⠐⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠂⠐⠈⠁



⠀⠀⡀
⠀⠀⠀⠨⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠡


⠀⠀⠈⠀⠀⠂⢀
⠀⠀⠀⠀⣀⣀⡸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⣀⣀⡀
⠀⠀⠘⠋⠃⠀⠀⠀⠀⠲⠠⠖⠐⠀⠄⠄⠒⠺⢛⣿⠟⠛⠛⢿⡛⠛⠒⠠⠄⠖⠒⠐⠂⠐⠀⠀⡀⠠⠀⠋⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⡄⡀⠈⠉⠀⠀⠀⣄⣾⠁⠀⠀⠀⠈⢱⣄⠆⠀⠀⠈⠉⠀⡀⡎⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⡆⣠⡍⠀⠈⠉⠀⠈⠉⠀⠘⣤⢀⡇⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠛⠒⠒⠶⠶⠦⠤⠴⠶⠶⠒⠓⠚⠒⠤
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠟⠛⠛⠻⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⣀⣤⣴⠶⠶⠶⠶⣦⣤⣀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⣴⠟⠋⠁⣠⡶⠶⠶⢶⣄⠈⠙⠻⣦⡀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣰⠟⠁⠀⠀⢸⡏⠀⠀⠀⠀⢹⡇⠀⠀⠈⠻⣆⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣸⠏⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠹⣇⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⠀⠀⠀⠀⠀⣀⣙⣷⣦⣴⣾⣋⣀⠀⠀⠀⠀⠀⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⡇⠀⠀⠀⣠⡾⠋⠉⠉⠉⠉⠉⠉⠙⢷⣆⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠹⣇⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⣰⡏⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠹⣧⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢁⣼⠏⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠁⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⠿⣦⣄⡈⠙⠛⠲⠶⠶⠖⠛⠋⢁⣠⣴⠿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠉⠛⠳⠶⠶⣶⣶⠶⠶⠟⠛⠉⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⣀⣤⡶⠶⠶⠶⣤⣄⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠿⠿⠶⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⣴⡶⠶⠶⢶⣤⣀
⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠈⠙⢷⣄
⢰⡏⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢰⠟⢻⣆⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠚⠛⣿⠆⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣾⣧⡀
⣿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⣀⠀⢸⡇⠀⠀⠀⢀
⠘⣷⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠙⠛⠛⠛⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠙⠳⠛⠁⠀⠀⢀⣾
⠀⠈⠻⣦⣄⡀⠀⠀⠀⣀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⢀⣠⣴⠟⠁
⣤⣤⣤⣤⣭⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣽⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣭⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣏⢻⡉⠉⢿⠉⠛⠛⠻⢿⣿⣿⡏⢹⣍⡉⢉⢩⠉⠛⠿⣭⣉⠉⠉⠉⠉⠉⠉⢻⣿⣿⠉⠉⢉⠉
⡇⢹⡀⠀⠀⠀⠀⣾⣿⠙⢲⡄⢀⡠⠾⢿⡌⢧⠀⠘⡆⠀⠀⠀⠀⠈⠙⢃⣼⣿⣿⣿⣿⣤⣯⠂⠀⢉⠛⠶⣤⣀⠀⠀⢸⣿⣿⠀⠘⠓⠁
⡇⠈⣇⠀⠀⠀⠀⣿⣿⡄⡈⣧⢸⣆⡃⣾⣷⡘⡆⠀⢹⡄⠀⠀⠀⠀⢀⣼⣧⡈⠙⠻⢿⣿⣿⣿⣿⣧⣀⣐⠀⠉⠛⠶⣾⣿⣿⠀⢠⣒⠄
⡇⠀⠸⡄⠀⢀⣴⣿⣿⡇⠀⢹⣿⣿⣅⢸⣿⣇⢹⡀⠀⢳⠀⠀⠀⠴⢿⣿⣿⡿⠶⠄⠀⠈⣙⣿⣭⣖⣨⣽⣶⣷⣆⣀⣀⣉⣿⣀⣀⣁⣀
⡇⠀⠀⢳⣀⢸⣷⣿⣿⣿⠀⠀⣿⣿⣿⢸⣿⣿⡄⢣⠀⠈⣧⠀⠀⠀⠀⠉⠉⢀⣤⣤⣤⣼⣿⡟⠁⠈⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⣰⢿⣧⠍⡍⢿⣿⣿⡄⠀⣿⣿⣿⣼⣿⣿⣷⡼⠂⠀⠘⡄⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⡿⢧⣠⣶⡝⠀⣠⣶⡯⠃
⡇⠀⣿⢏⣿⡌⠐⠈⣿⣿⡇⢀⣿⣿⠿⠟⠋⣹⠋⠀⠀⠀⠀⢹⣀⣀⣾⣿⣿⣿⣿⡿⠋⠉⠉⠀⠀⠀⢸⡿⣿⡗⡠⠉⠉⠉⠉⢉⣭⡍⡙
⣇⣤⣾⣿⣿⣧⠀⣀⣼⠿⠛⠋⢁⡿⠀⠀⢀⡿⠀⣠⣴⣤⣄⠀⢿⣿⡿⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿⣷⠻⠃⠀⠀⢀⣾⠋⣼⡇
⣿⣿⣿⣿⠿⠻⣏⠉⠀⠀⠀⣰⠋⠀⣠⠞⠉⠀⠀⣿⣿⣿⣿⣅⣈⣿⠤⠶⣤⣀⠀⠀⠀⣾⡿⠶⡗⢄⢸⡇⣿⣟⣴⡁⠀⢀⡞⢁⣾⣿⡇
⡟⢯⡁⠀⠀⠀⣹⡤⣤⡀⠀⢿⠀⢰⡏⠀⠀⠀⠀⠉⠿⢿⣿⣿⣿⣧⣠⣶⣿⣿⡆⠠⣾⣭⣤⣤⣤⣼⢼⡇⣋⣷⢉⠀⠀⠈⢠⣾⣷⣾⡇
⡇⠀⠹⣄⠀⢸⡁⠀⢈⣇⡴⢋⡤⠚⠃⠀⠀⢀⣠⣴⣾⢋⣍⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣟⣀⣤⡴⢧⣸⣇⣿⡟⠽⠊⠀⣠⣿⣿⣟⣻⡇
⡇⠀⠀⠈⠳⣄⠙⠒⢋⣹⢀⠞⢀⣴⣶⣶⡞⢋⣡⣞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⡴⢚⣫⣭⣥⣴⣌⢷⣿⣯⣹⣶⡄⣰⣿⣿⣿⣿⣿⡇
⣯⡄⠀⠀⠀⠈⢢⡼⠉⣠⣧⠶⠛⠛⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠋⣠⣶⣿⡌⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡽⣼⣿⣿⣿⣧⣽⠃
⣿⣄⢤⡆⠀⠀⠀⡟⢿⣹⡇⠀⠀⠀⠀⠉⠿⠿⠿⠋⢠⠞⠁⢻⣿⣿⣿⠃⣼⣿⣿⣿⡇⠸⢙⠘⠉⠀⣸⣿⣿⣿⠿⠗⢯⡹⣿⣿⡿⢿⡄
⡿⣿⣷⣄⣤⠀⠀⡇⢠⠿⣶⣶⣦⣄⠀⠀⠀⠀⢠⠴⠚⠀⢀⣾⣿⣿⠃⣸⣿⣿⡿⢿⡇⠀⠀⠀⠀⢀⣿⣿⣿⣿⢄⣀⣠⢟⣿⣿⣿⣿⡇
⡇⠈⢻⣿⣿⣿⣿⠀⣥⡀⠈⠻⣿⣿⣇⠀⠀⢠⡏⠀⠀⢠⣿⣿⣿⠋⠐⠛⠿⢫⣾⣷⡄⠀⠀⠀⠀⠘⠃⢟⣿⣷⡄⢨⣱⣿⣿⣿⣯⣽⡇
⣷⣄⠀⠙⢿⣿⣯⣤⣌⡻⣦⠀⠙⢿⡧⣀⠤⠜⠃⠀⣠⣿⣿⣿⣿⣀⠐⢤⣀⣼⠿⠟⡷⠀⣠⣤⠤⢄⡀⠀⢈⣿⣷⡞⣿⣿⣿⣿⡿⢿⡇
⣏⠻⣷⡀⠀⠛⣡⣶⡌⣷⣽⡇⠀⠀⠙⣇⠀⠀⠀⣰⣿⠟⠛⠋⠀⠈⢻⣏⣥⣴⣶⡟⢁⣴⠿⠟⠛⠲⠌⢦⠀⢾⣿⣿⡹⣿⣿⣿⣷⣿⡇
⣿⡷⢈⠻⣦⡀⠈⠛⣰⣯⡙⣿⣄⠀⠀⠈⠳⣄⠤⠟⣛⡷⢠⣴⢦⣶⣿⣿⣿⣿⡟⠀⠛⢻⣿⡛⢻⠶⣶⣬⢆⣿⣿⣿⣷⣻⣿⣿⣏⣻⡇
⡟⣠⣿⣷⠌⠳⣄⠀⠙⢿⣿⡎⠻⣷⣄⠀⠀⠈⢲⣻⣿⣷⣾⣿⣿⡭⠭⠔⣒⡟⠀⠀⠀⢸⡇⠀⢀⡤⢻⡧⣿⣿⣿⣿⣿⢷⣻⣿⣿⣿⡇
⣿⣿⡿⢃⣾⣿⡆⠁⠀⠀⢹⣧⡀⠈⢻⣷⣀⣀⣀⠙⢿⣿⣿⣿⣿⣿⣭⣥⡟⠀⢀⣀⣤⣌⣳⣶⣾⣓⡚⠳⣍⡿⣿⡽⡟⡏⢇⢿⣷⣾⡇
⡟⠿⣴⣿⣿⠏⠀⣠⣾⣿⣿⣿⣷⠀⠸⣿⣿⣿⣿⣷⣤⣟⣫⣭⡻⢿⢟⣫⣴⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠙⢪⣳⣘⣈⣬⠾⠿⠛⢛
⡇⠀⠈⠻⣅⢀⣴⣿⡟⠁⠀⢸⣿⡇⠀⠈⠻⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⣥⡤⠶⠖⠛⠋
⡟⢦⡀⠀⠈⠻⣿⡋⠀⠀⠀⣸⣿⣧⠀⠀⠀⠸⣏⠉⠀⠀⠈⠉⠉⡟⠉⠉⠉⠉⠙⣆⠀⣆⠸⣆⢀⣠⣀⣀⣰⣿⡇⣼⠁
⡇⠀⢙⣶⣄⠀⠈⣻⠀⠀⠀⡿⡿⣿⠀⠀⠀⠈⣿⡄⠀⠀⣀⠀⢀⣧⣀⣀⣤⣤⣴⣿⣄⣽⣶⡿⢻⣯⠛⣿⣿⡿⣱⠃
⣇⣴⣿⡿⠋⠳⠞⠃⠀⠀⢸⡇⡇⣿⠀⠀⡀⠀⠛⠷⠖⠋⠉⠻⣿⡁⠈⠉⣿⡾⠶⠋⠛⠛⠩⣷⣿⣯⡞⣼⣿⣷⠃⠀⠀⠀⣠⠔
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇⣧⢸⣆⣴⡇⢠⡆⠀⠀⠀⠀⠀⠘⠿⡆⠀⡧⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⡿⠁⢀⣠⣶⠟⠁⣀⠄
⡿⠃⠀⠀⠀⠀⠀⠐⢦⣄⣸⡇⢻⡌⢿⣿⣷⣿⣡⣴⠏⠀⡀⠀⠀⠀⠀⣰⢃⣿⠆⠀⠀⠀⠀⠈⢿⣿⣿⢏⣠⣶⡿⢛⣡⣶⠟⢁⡀
⡇⠀⠀⠀⠀⠐⢶⣦⣬⣿⣿⣿⠘⣿⣄⠙⠻⣿⣿⣿⣶⣿⣁⣠⡤⣶⢞⣵⡿⠃⡔⢀⠀⠀⠀⠀⠀⠙⣿⣸⣿⣿⣿⡿⣋⣤⡶⠋
⡇⠀⠀⠀⠠⣤⣤⣼⣿⣿⣿⣿⣇⠘⣿⣿⣶⡤⠾⠛⠛⠉⠉⢀⡴⣡⣾⡿⠁⡾⢁⡎⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⣿⣿⣟⣫⣴⠞⠁
⡇⠀⠀⠀⠀⠀⠈⢙⣻⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣠⢏⣼⣿⣿⠁⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣟⣥⡶⠃
⡇⢀⣀⡤⠴⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⣿⣿⣿⠇⣸⡟⢠⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿⣿⡟⠋
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⣀⣀⣠⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⡑⠀⠘⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠀⠀⡠⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠤⢶⠊⣠⢼
⠀⠀⠀⠀⠀⠀⢠⠖⡉⠂⠄⡻⣀⠀⠀⠀⠀⠀⠀⢠⣮⡤⢤⡈⣙⣿⣯⡫⠠⣌⣷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⡟⠘⣧⠀⠉⠀⢸⠄⠀⠀⠀⠀⠀⣿⣿⡇⢀⢼⠋⢩⡇⠳⣸⣃⢀⣼⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠙⣦⡙⡒⠤⠤⣤⠇⠀⠀⠀⣀⣼⣿⣿⢷⢃⡏⡄⠀⠙⡲⣿⡻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⠠⢤⣸⠙⠈⣦⣄⣚⣻⣷⣴⣿⣿⣿⡿⠋⢸⡾⠀⠀⠀⢠⠗⣻⣷⣿⣿⠟⠛⠋⠉⠁
⠀⠀⠀⢀⣏⠿⠼⠃⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠐⠀⠀⡞⣼⣿⣿⣿⠋
⠀⠀⠀⠘⢧⣤⠀⢀⣠⠞⠙⢿⣾⣿⣿⣿⣿⣿⣯⢀⣤⣿⣿⡀⠀⠀⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣻⡿⠿⡿⢷⣤⡾⠿⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣛⡿⡟⠁⠀⠀⠀⢸⡛⠂⠀⢀⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⣿⣿⡇⠀⠀⠀⠀⢸⣧⣿⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⣿⣿⣿⣿⡿⠀⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡛⠕⠉⠈⢳⡀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢿⣷⣄⠀⠀⠀⢳⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠀⢀⡟⠉⠙⢆⠀⠈⢧⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⢀⡤⠊⠀⠀⠀⠈⣇⠀⠈⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⢀⣤⠤⠴⠊⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⣤⣾⠏⠉⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠛⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠙⣶
⢸⣧⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⣏⠀⠀⠀⢣⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⣆⣿
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⠞⠁⠀⠀⠀⢿⣧⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠁⠀⠀⠀⠀⠀⠀⠻⠳⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣛⣛⣩⣍⣛⣛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣴⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣧⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⣿⣿⣿⣿⣿⡿⢋⣌⢿⣿⣿⣿⣿⣿⣧⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡅⣿⣿⡿⠿⣛⣭⣾⣿⣿⣷⣬⣛⡻⠿⠿⢿⡌⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣏⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠹⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠰⠿⠇⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣌⠻⣿⣿⣿⣄⣛⣃⠘⣛⣛⣛⣛⡓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡗⢰⣭⣭⣭⣭⣦⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣭⣭⣭⣅⢸⣿⣿⣿⣿⡿⢨⣭⣭⣝⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⣝⣛⣛⣋⣵⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠴⠎⠿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣟⣛⣻⣟⣛⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠯⢭⠭⢭⠭⠿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⢢⠰⠀⡆⠦⠰⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡆⢰⢠⠆⡄⠴
⠀⠀⠈⠀⠀⠆⡆⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠨⠘⠀⠅⡇
⠀⠀⠠⠀⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⠁⠀⠃
⢀⣠⠂⠄⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⢄
⠀⣿⣿⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟⡻⣻
⠀⠘⡠⢂⢋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠐⡙⢾
⠀⡀⡀⠈⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠃⢀⣸
⠀⠉⠘⡂⢄⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⢐⠆⠘
⣈⣉⠉⠦⠮⣀⡀⠲⠖⠩⡉⡱⠶⠀⠠⡀⡶⠊⠉⢉⣲⡆⠀⠀⢰⣖⠈⠉⠑⢦⣀⠀⠀⠶⢆⢉⠉⠲⠤⢄⢀⠱⠶⠨⣉
⠈⠁⢲⡔⠀⠉⠁⠀⢢⡄⠉⠁⠀⠐⣤⠉⠁⠀⠀⠲⡟⠀⠀⠀⠀⢻⡄⠀⠀⠈⠙⣤⠂⠀⠈⠉⠲⡔⠀⠈⠉⠀⢢⡄⠈

⠀⠀⣄⡀⢠⡀⠀⣠⡄⠀⠀⠀⠀⠀⣶⡦⢀⠀⠀⠀⠀⠀⢀⣤⣾⣦⣄⠀⠀⠀⠀⠠⠐⠒⣄⡒⠀⠄
⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⣼⣿⡗⣲⡄⠀⠀⠀⠀⠛⠛⢻⠛⠛⠃⠀⠀⠀⢃⢠⡦⢐⣴⡆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⠀⠀⠀⠀⠀⣐⣀⣸⣀⣐⡀⠀⠀⠀⠨⢑⡨⠤⢂⡰⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⣾⡟⠛⠛⠛⣿⡆⠀⠀⠀⢸⡏⠉⠉⢹⡇⠀⠀⠀⠀⣴⡺⡽⣖⢄⠠⠀⠀⠀⢀⣤⣤⡄⣤⡤⣄
⠀⠀⡟⠀⠰⡇⠀⠙⡇⠀⠀⠀⢸⡷⠤⠤⢼⡇⠀⠀⠀⢸⢞⢎⢏⣎⣾⠀⠀⠀⠀⢸⣿⣿⣷⣿⣤⣿
⠀⠀⣿⣲⣴⢳⣴⣾⡇⠀⠀⠀⢸⣇⠀⠀⢸⡇⠀⠀⠀⠈⣯⢗⣷⣮⣎⠀⠀⠀⠀⠻⠟⠃⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉

⠀⠀⣦⣬⣀⣀⣥⣴⡄⠀⠀⠀⢠⣶⣾⣷⠂⠀⠀⠀⠀⠀⠀⢀⣒⣒⡀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠱⢆⣾⣿⠀⠀⠀⠀⢀⣿⡿⣿⣷⠄⠀⠀⠀⢸⣿⡏⣄⡌⣿⣿
⠀⠀⠿⠿⢿⡿⡿⠿⠃⠀⠀⠀⠈⠀⣾⣿⠿⠃⠀⠀⠀⠀⣼⣿⣷⣶⣶⣆⠀⠀⠀⠀⠉⢡⣟⣣⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠃⠀⠀⠀⠀⢠⠖⢻⠛⠲⡄⠀⠀⠀⠀⠾⠮⠴⠧⠷⠇
⠀⠀⠀⢸⡁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣹⠶⠀⠀⠀⠀⠀⠀⡟⠑⣾⣷⠊⣻⠀⠀⠀⢠⠀⢴⠄⢴⠄⢠
⠀⠀⣀⣀⣙⣲⣒⣋⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⠾⣅⣨⠶⠃⠀⠀⠀⠈⢶⣾⣶⣾⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠂⠀⠀⠻⠿⠶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⠿⠿⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣭⣍⣩⣥⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠿⠇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠹⡷⠀⠀⢠⣿⢿⣆⠀⠀⠀⠘⠋⠀⠀⣀⣀⠀⠀⠛⠀⠀⠀⠀⣰⡿⣿⡄⠀⠀⠘⠋
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠇⠈⣿⣄⣀⣀⡀⠀⠀⣼⡟⢻⣧⠀⠀⢀⣀⣀⣠⣿⠁⠸⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣏⠉⠉⠉⠀⠀⠈⠉⠉⣉⣁⣀⣰⡿⠀⠀⢿⣆⣀⣈⣉⡉⠉⠁⠀⠀⠉⠉⠉⣹⣿
⠀⠀⠀⠀⠀⠈⠛⢷⡆⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠛⣿⡆⠀⠀⠀⠀⢰⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣠⣄⡀⠈⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⢀⣠⣄⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠘⣿⣴⡿⠋⠙⠻⣶⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣶⠟⠋⠙⢿⣦⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⣴⣦⣄⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⠿⠋⠁⠈⠙⢿⣶⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢿⣿⡿⠿⠿⣿⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣶⣿⢰⣶⣶⢸⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣩⣭⡍⣿⢸⣿⣿⢸⡇⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠛⠛⠃⠛⠘⠛⠛⠘⠃⠘⢻⣷⡼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣋⣥⣶⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡂⠠⣶⣿⣿⣿⠁⡦⢙⣛⣛⣓⣦⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠀⢠⡄⠀⠀⣍⡛⠿⣷⣶⣾⠿⢛⣉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢸⡇⠀⠀⣿⣿⣷⣦⣭⣴⣾⣿⣿⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢣⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣿⣿⣿⣆⣠⣾⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⣷⣦⣠⡿
⠀⠀⠀⠈⠹⠿⠛⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣄⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢹⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣠⠀⠀⠀⠀⢀⡾⠂⠀⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣿⣧⢶⣀⣠⣴⠟⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠻⣆⡘⢿⣿⣿⣿⣿⣿⣯⣅⡀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠈⠛⠛⠛⠉⣻⣿⣿⣿⣿⣿⡷⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣶⠄⣴⣶⣆⢈⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢹⣿⠁
⠀⠀⠀⠀⢠⣿⠀⣻⣿⣯⣾⡟⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠟⠀⠀⢤⣄⣀⠀⣀⣻⣿⣟⣀⢾⣿⣿⠟⢐⡟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⢻⣿⣿⡿⣿⣿⣿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠟⠉⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⠻⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⡿⠀⠙⣿⣟
⠀⠀⠀⠀⢠⡼⠇⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠀⠀⠙⠈

⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄
⠀⠀⠀⠀⠀⡰⢿⡿⠤⢿⡿⢄⠀⠀⠀⠀⠀⠀⠀⠖⠐⠀⠉⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⢿⣿⣿⣶
⠀⠀⠀⢸⠭⠤⠔⢀⣀⡌⠢⠤⠯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠁⠈⠉⠙⣿⡇
⠀⠀⠀⢸⠀⠀⢻⣿⣿⣿⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠛⠋⢸⣿⠠⠁⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⣶⣷⣶⣦⣿⡇
⠀⠀⠀⢸⢤⣠⣾⣿⡿⠿⠟⠛⠀⢃⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿⠟⣿⡇
⠀⠀⠀⢉⠀⠉⠉⠀⠐⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠾⠆⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠿⣄⢤⣥⣌⣤⣤⠿⡇
⠀⠀⠀⠀⠀⠣⡀⢠⠒⡀⠀⠊⠀⠀⠀⠀⢀⣀⣰⣶⡶⠄⣀⠄⢄⣤⣶⢄⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠉⠁⠀⠀⠙⠛⠋⠉⠉⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣤⡤⠤⣾⣿⡧⠤⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢞⣷⣿⡳⣤⣄
⠀⠀⠀⠀⣿⣿⣿⡏⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⣿⣶⣬⠛⠛⣵⣶⣿
⠀⠀⠀⠀⡿⣿⡿⢀⣴⠛⢿⣿⠿⠀⠀⠀⠀⠀⠀⡠⡔⢶⢾⡣⡣⡦⢄⡇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⣿⡟⠁⣾⣯⣶⡈⣿⣿⠀⠀⠀⠀⢠⡶⣭⣵⣷⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⣿⣷⢠⣯⠈⠁⠀⣿⣿⣤⠀⠀⠀⠀⠁⠘⣿⠿⢿⣯⣿⠏⠀⠀⠀⠀⠀⠀⢹⣿⠻⣟⣻⣷⡿⠃⣿⡏
⠀⠀⠈⢿⣯⣥⣤⣶⡴⢶⣦⣭⣿⡟⠀⠀⠀⠀⠀⢀⣈⣻⣟⣉⣁⣀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣬⣥⣴⡾⠿⣿⠇
⠀⠀⠀⠀⢅⡐⠋⠛⠙⠀⠐⣀⠍⠀⠀⠀⠀⠨⠿⠛⡛⠛⠃⠑⠘⡛⠻⠯⠀⠀⠀⠀⠀⠙⠛⣏⠉⠀⣰⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡤⠦⢤⡀
⠀⠀⠀⠀⢀⣾⠳⢋⠳⢚⡆⠀⠀⠀⠀⢀⢘⣷⣾⢿⣶⣿⠀⠀⠀⠀⠀⠻⣷⠖⠀⡀⠲⣾⠟⠁⠀⠀⠀⢠⡏⢶⣷⡆⢻
⠀⠀⠀⠀⢯⡉⢤⣾⣷⡄⢩⠟⠀⠀⠀⠛⣿⣟⠋⠀⠛⣻⡟⠃⠀⠀⠀⠀⠁⢤⣾⣷⡤⠈⠀⠀⠀⠀⠀⠀⢳⣌⣉⣥⡏
⠀⠀⠀⠀⠯⣤⠸⠛⠻⠀⡬⠟⠀⠀⠀⠺⠿⣿⣀⣤⣸⣿⠿⠇⠀⠀⠀⠀⣆⠠⠿⠻⠇⣰⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠻⠚⢧⡞⠳⠃⠀⠀⠀⠀⠀⠘⠛⠹⣿⠉⠛⠀⠀⠀⠀⠀⢿⡟⠛⠀⠀⠚⠻⣿⠆⠀⠀⠀⠻⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⣠⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤
⠀⠀⠀⠀⣀⠀⠸⢦⠾⠀⢀⣄⠀⠀⠀⢀⣴⡀⠹⠤⠾⢀⣴⡄⠀⠀⠀⠠⣞⡆⣤⣥⡄⢶⡶⠀⠀⠀⠀⠀⠀⢘⣿⣟
⠀⠀⠀⢻⣩⡟⢠⣾⣶⠀⢫⣬⠇⠀⠀⠀⣋⠃⢰⣿⣶⠀⢋⡃⠀⠀⠀⣴⣦⠀⣿⣿⡏⠠⣖⡄⠀⠀⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⠀⠻⡄⣠⣿⣯⡄⣠⠇⠀⠀⠀⠈⠛⢁⣼⣿⣧⡀⠛⠃⠀⠀⠀⢸⡇⣞⡯⠭⣝⠆⢹⠀⠀⠀⢀⣠⣈⠉⣩⡉⠉⢀
⠀⠀⠀⠀⠀⠰⢿⠿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⠓⢺⡸⠿⣸⠓⠋⠀⠀⠀⠹⠤⡟⠹⡥⡝⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣆⠀⠀⠀⠀⠀⠀⣀⣠⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⡄⠀⠀⠀⣠⣄⢀⣤⣤⣤⣤⡤⣤
⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠙⡆⡀⢰⠋⣤⣧⡄⠀⠀⠀⠀⢰⠿⠿⠀⠉⠭⠀⡇⠀⠀⠀⠛⠛⠘⠛⠛⠛⠚⠓⠚
⠀⠀⠀⢰⡆⠀⡿⠛⠻⠇⠀⢰⠀⠀⠀⠈⡋⠉⢛⣀⣚⠙⢀⣤⡀⠀⠀⢸⠀⣀⣼⣧⣀⠀⡇⠀⠀⠀⣾⣷⢸⣿⣹⣉⣉⣉⣹
⠀⠀⠀⠘⢷⣷⣄⠀⢀⣴⣷⠟⠀⠀⠀⠀⠙⢶⣾⣿⣿⣷⣀⡉⠁⠀⠀⢸⠀⢹⣯⣬⡉⢀⡅⠀⠀⠀⣠⣦⣠⣶⣶⣶⣶⣶⢶
⠀⠀⠀⠀⠀⠹⠿⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠛⠒⠀⠓⠛⠓⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣴⣾⡿⣿⣷⡄⠀⠀⠀⠀⠀⣠⣤⡆⣾⠟⠙⢷⡀⠀⠀⠀⢀⣤⣶⡾⢩⣤⡍⢧⠀⠀⠀⠀⠀⣉⡌⢁⣿⣶⡶
⠀⠀⠀⠀⢸⣿⡉⠀⠈⣹⣿⠀⠀⠀⠀⢰⣯⣨⡄⠻⣤⣤⡿⠁⠀⠀⠀⣾⣏⣩⠄⡉⠟⢃⡼⠀⠀⠀⠀⢰⣿⣇⠙⢻⡿⠛⠂
⠀⠀⠀⠀⢿⣿⣧⣴⣤⣿⠃⠀⠀⠀⠀⢾⣟⢉⣴⠇⣶⠀⠀⠀⠀⠀⠰⢿⡏⢡⡾⢡⡟⠉⠀⠀⠀⠀⠀⢾⣿⣿⣷⣤⣤⢈⡋
⠀⠀⠀⠀⠘⠻⢿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠿⢿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡡⠊⢑⡄⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⢐⢑⠔⠁⠈⢐⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠣⠔⠀⠀⠀⢠⣿⣿⣿⣿⠛⠛⠛⠛⠉⠁⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠅⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢅⠕⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠠⠄⠀⠀⠀⠀⢸⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠠⢁⠓⢄
⠱⣆⠀⢎⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣤⣶⣶⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢨⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠻⠿⠿⠉⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡨⠖⠋⠑⢌⣳
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠠⠈⠒⢿⡷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡽⠚⠈⠀⠀⠀⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢄⠀⠈⠀⠠⣿⡄⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⢆⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠈⠙⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠛⠛⠿⡇⠀⠀⢸⣿⡿⠿⠛⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡄⣤⢠⢶⡀⡄⡄⢴⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠌⠀⢸⠿⢻⢸⢸⡇⣿⠇⢸⢸⡇⠸⠓⡀⠰⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠢⠔⠀⢁⠀⠀⠆⠈⠠⠀⠠⠉⠁⠒⢠
⠀⠠⣐⣦⣤⣄⡢⡌⠁⡀⠀⠘⠀⠘⠈⠛⠁⠘⠀⠚⠈⠓⠀⠀⠀⠁⠀⠀⠈⠀⠀⠃⠐⢤⠀⠈⠂⠀⠀⡠⠃⠈⠇⠀⠀⢐⡀⣥⠀⣀⠴
⠀⠌⠛⠛⠋⠉⠀⠃⠀⠈⡆⠀⠈⠄⠀⢀⠄⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠆⡠⠃⠀⣬⠆⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⢀⢐⡀⣀
⠀⠄⠀⢿⣿⡗⠀⡬⠀⢂⣀⠀⠀⡼⠰⠄⠀⡀⣀⠀⠀⠃⠀⠀⠠⠞⠀⠀⠀⢀⠑⠐⡂⠡⠄⣀⠀⠀⠣⠀⡔⠉⠢⡀⠀⡰⠊⠁⠀⠀⣄⠑⡄
⠀⠈⠂⠠⠭⠀⠊⠀⡈⣤⣴⣶⣶⣶⣶⣦⣤⠀⠀⠈⠀⠃⠀⠀⢀⢀⡀⠀⠀⠀⡄⠙⠤⢄⠀⣻⠈⢈⠁⠸⠀⠀⠀⠓⡀⣄⠀⢁⠀⠌⠠⢀⠇
⠀⢀⠄⠠⠠⡀⠀⠀⡇⣿⣿⣷⣿⣟⣿⣿⣿⠀⠀⠀⡆⢰⠀⠀⠙⠺⠹⠈⠠⠀⠰⠀⠀⢛⠒⠚⠀⠂⡃⠘⡐⠀⠀⣈⠃⠈⠐⠀⠀⠀⠀⠁
⠀⠊⠀⠀⠀⠈⢂⠀⡇⣿⣿⡟⡿⣿⡿⢿⣿⠀⠀⠀⠆⠀⣤⠆⠂⠐⠒⠒⠒⠒⡖⠀⠀⠸⠀⠀⠀⠨⠁⠀⢱⠀⠀⡄⠀⠀⢠⢴⣯⡀⠀⠈⠢
⠰⣅⣀⣀⢀⣀⠃⠀⠇⠛⠿⠿⠿⠿⠿⠟⠣⠀⠀⢀⠀⠀⠉⠀⢀⡠⠄⠐⠀⠤⡀⠀⠀⠀⠆⠀⠀⡠⠀⠀⠈⠒⠒⠁⠀⠀⠁⠀⠈⢿⣄⠐⡰
⠀⢽⠔⠛⢸⡇⠀⠀⠸⠋⠉⠉⢉⣉⡉⠉⡈⠉⠻⠅⡠⠐⡠⢤⣌⡰⢄⠀⠀⡠⠄⢹⠒⠄⡀⠀⠀⡴⠒⠒⢠⡀⠀⠐⠄⠀⠐⢀⠀⠸⠿⠋
⠀⢺⢤⡄⢾⠀⠀⠀⠀⢸⠉⠀⠀⣠⡤⠀⠃⠀⢀⡐⠀⠠⡀⠀⠀⠀⢀⠀⢸⠀⠀⣘⠢⠱⠈⢄⠀⠀⠀⠒⡆⣶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠙⠲⠧⠟⠂⠀⠀⠀⠈⡄⠀⠠⡤⠛⠀⠸⠀⠑⢄⠀⠖⡱⠆⢩⠀⢊⠐⢚⡁⢎⠠⢊⠜⢀⠒⠀⠢⠄⠤⠬⠤⠤⠤⠊
⠀⠀⠀⢀⣤⣶⣶⣦⣀⠀⢃⠀⠀⠶⠐⠂⠀⡆⠀⠀⠑⠠⢀⠀⠀⠉⠉⠀⠀⠀⠀⢁⠠⠐⠁⠀⠀⠀⠀⢀⠠⢀⠀⠀⣿⣻⠏⣿⡟⣿⡏⡉⣿
⠀⠀⣴⢟⣭⡶⣶⣭⡻⣆⠘⠒⠒⠈⠉⠉⠀⠀⠀⠀⢀⣀⡤⡀⠀⠀⠀⠀⠀⢀⢠⠶⠤⠤⠀⢄⠀⠀⢠⠁⠀⠀⢑⠀⠟⠛⠛⠃⡀⠈⠛⠛⠛
⠀⠰⠿⠟⠁⠀⠈⠙⢳⢿⠀⢠⣴⣶⣶⣶⣶⣶⣶⡄⠈⢥⠜⠁⣀⡤⡄⠀⠼⢁⣄⡀⠀⠘⠃⠃⠀⠀⢸⠀⠀⠐⠘⡄⠀⠀⢘⠀⠉⠈⠈
⠀⢃⣀⡀⠀⠀⠀⠀⣄⣠⠀⢸⡏⠁⠸⠀⡏⠠⢸⡇⢠⣀⢶⡄⢈⣀⠀⠀⠄⠈⠛⢀⠔⠒⠤⣀⡸⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄
⠀⠀⢹⣷⣄⠀⢀⣼⣿⠀⠀⢸⣇⠀⡀⡄⠃⠰⢺⡇⠈⠁⠀⢠⣼⣙⠃⠀⠈⠒⠒⠁⠀⣀⠀⠀⢀⡀⠀⠑⢰⡶⠒⠁⢐⠀⠀⠀⠐⠈⠀⠐
⠐⠂⠘⠋⡻⠀⠘⢋⡙⠀⠀⢸⣿⢾⣷⣿⣾⣤⣾⣿⠀⠀⢀⠀⠈⡉⠀⠀⠀⠀⠀⠀⢺⠁⠋⠙⠛⠛⣿⣆⠘⠷⠇⠀⠺⣶⣿⣛⣓⣚⣻⣷⣾
⠀⠈⣠⣉⠤⡀⠀⠀⣈⡄⠀⣰⣿⠀⡇⡆⢀⠘⢸⣿⠀⢰⠓⠲⠾⠶⢾⠄⢄⠀⠀⠀⠇⠀⣘⢀⡀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢀⠿⠏⢀⡠⣿⣿⠿⠷⠀⠉⢿⣶⡷⠶⠿⠶⠾⠟⠀⢸⠀⠀⠄⠠⠀⠀⠀⠀⢀⡞⠋⡧⠰⠎⠀⡤⠢⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⠅⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⡀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⢠⠆⠌⠀⠸⡄⠀⡀⠀⠀⠀⠃⠀⢸⠀⠂⠀⠞⣀⣀⡀⠙⠀⠈⠀⠁
⠀⢀⡀⠤⠤⠤⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⠀⠀⢀⣀⡀⠀⠑⠀⠀⠀⠊⠀⠀⠰⠀⠘⠒⠁⠀⠀⠀⠧⠴⠃⠐⠀⠀⢼⣿⣿⣿⡗⠀⡠⠀⠐⠄
⠀⠂⠀⠀⠀⠀⠀⣳⠀⠀⡖⠃⠀⣀⠀⠀⠑⡄⢸⠀⠀⠉⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠄⠀⠀⡄⠀⠀⠙⡍⠉⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠇⠀⠈⠿⢿⠀⠀⠃⠈⡀⠀⠀⠀⠀⠀⡥⠐⠭⠂⠀⠀⠀⢀⠄⠠⣀⠀⠀⣀⠀⢀⠇⠀⠀⢠⠇
⠀⠀⠀⠀⢶⠂⠀⢸⡀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⠇⠀⠀⢀⠔⡨⢔⠁⠀⢀⠀⠀⠐⣵⡿⢷⡌⡆⠀⠜⠀⠣⡀⠀⢔⠺⠂⡄⠀⢸⠀⠘⠃
⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠟⠣⠞⠃⠈⠀⢀⠀⠀⠈⠁⠀⠸⠀⠀⠀⠠⡊⠀⠙⠾⠋⠈⠀⢸⡀⠀⠀⡄⠀⠀⠀⣤⡆⠀⠈⡄
⠀⠑⠚⠚⠛⠛⠛⠋⠀⠀⠑⠆⠀⠀⠀⠀⠀⠐⠃⠀⠉⠑⠒⠒⠢⠤⠆⠀⠀⠀⠈⠢⠠⠊⠀⠀⠀⠈⠑⠀⠊⠁⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡀⡶⢶⣂⣶⣦⡆⣶⠶⣶⡰⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⢀⠈⡉⡁⡉⢁⣉⣁⣉⣀⣁⣉⡁⣉⢉⣉⢀⢀⣀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠿⠼⠻⠻⠧⠏⠟⠿⠿⠽⠗⠸⠸⠿⠧⠿⠇⠟⠸⠻⠤⠻⠟⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣄⢹⠠⣞⡂⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡷⠖⠛⡾⢸⠀⠲⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⢻⠷⠿⠶⠶⠷⠶⡄⠀⠀⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⢰⢻⡄⠀⠠⠄⠀⠀⠽⡄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⣼⢸⣗⣒⣒⣒⣒⣒⡇⣷⣲⡄⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣧⣤⣤⡄⠀⢤⣤⡈⣩⣉⠉⠉⠉⠀⠈⠉⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⣿⣿⣷⠀⣒⠀⡇⡇⡏⡇⠀⢠⠀⡏⡭⠭⠥⠤⠭⠭⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠷⠶⠶⠞⠀⠀⠀⡇⡧⡇⠇⠀⣘⣀⡇⣭⠭⢭⣭⠭⢭⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠭⠥⠥⠭⠭⠭⠀⡇⡇⡇⢠⢊⢵⣭⡳⡇⠶⠆⡇⠰⠆⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢭⣥⣤⣬⡭⠀⠀⡇⡇⡇⠸⡸⢬⡽⢣⡟⣛⠛⡟⣻⡛⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣤⡀⠀⠀⣲⣖⠀⡇⣧⡇⠀⠉⣓⡞⡇⣇⣉⣡⣇⣒⣂⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠭⠤⠤⠤⠭⠭⠤⠷⠄⠈⠙⠉⠁⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠠⣄⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⢠⠀⡂⠀⠀⠀⣀⣤⣠⡈⡆⠀⠀⠀⠀⢰⠀⡄⠀⠀⢀⣴⠿⢿⣦⠀⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⠻⠏⣿⡇⠀⠀⠀⣈⠀⠶⠀⠀⠀⣿⣿⠛⣿⡆⠀⠀⠐⣻⡏⠁⠁⠀⠀⢰⡇⢸⢀⡇⠀⠀⠘⣿⣤⣴⡿⠀⠀⠀⠀⠋⠠⠋⠀⠀⠀⣿⣫⣿⣻⡇
⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣒⢴⣲⠀⠀⠀⣠⣴⡞⢿⡆⠀⠀⠀⠉⠸⢃⡀⠀⠀⣾⢷⣶⡶⣶⠀⠀⢠⡤⡤⢠⢤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⣄⣄⠀⠀⠀⠀⠴⣷⡄
⠀⠀⢶⣿⣾⣿⠀⠀⠀⣷⣴⠿⠟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⡆⡇⢠⣾⠀⠀⠰⣷⠀⠀⣾⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⢰⣿⡇



⠀⠰⠶⠆⠰⠆⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⣀⣠⣞⣟⡷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠⠤⠤⠤⠤⢤
⠀⢀⡀⠰⢆⡰⠆⠀⠀⢠⣄⢠⣤⠀⠀⠈⢿⡾⠾⠁⠁⠀⠀⣶⣦⣶⣤⣤⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢺⣧⣼⠇⠀⠀⠀⢸⢡⡆⡟⠀⠀⢸⠹⠀⠇⠳⢸
⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠁

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣐⡿⢣⣤⡄⠀⠀⠀⠄⠛⣿⡆⠀⠀⣀⣀⣤⣄⡀⠀⠀⠠⠀⠀⢂⠀⠀⠀⠀⠀⢠⣮⠛⠀⠀⢠⠊⡉⣳⣀⡀⠀⠀⠀⢰⣦⡤⠀⠀⠀⣄⣰⡀⡄
⠀⠘⠿⢣⣾⠍⠀⠀⠘⣿⣦⡄⠈⠀⠀⠀⣿⡏⣶⢎⣿⠀⠀⠀⠱⣖⣻⡆⠀⠀⠀⡴⠟⠁⠀⠀⠀⠘⢄⣀⠜⠉⠁⠀⠀⠛⢻⣿⠟⠀⠀⠘⡘⢤⢼⡘
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠈


⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⣼⣿⣿⣧⠀⠀⢸⢾⡯⡭⡿⠇⠀⢀⣾⣶⣶⣾⣷⠀⠀⠠⠤⢤⠤⡄⠀⠀⢠⣼⣟⡷⣤⠀⠀⠀⠀⡖⠿⠀⠀⠀⠀⠀⡠⠚⠁⡀⠀⠀⣤⣶⠀⠀⡀
⠀⠀⠘⠾⠷⠃⠀⠀⠈⠀⠀⠁⠁⠀⠀⠈⠻⠟⠉⠻⠟⠀⠀⠠⠤⠦⠹⠄⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠶⠀⠚⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠛⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠐⠠⠢⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⢸⣿⡇⠈⠁⠀⠀⢰⣋⣟⣹⣏⡇⠀⠀⠘⠁⠀⠁⠀⠀⠀⠀⡏⠀⢹⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠦⢀⢀⠂
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠁


⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣿⣿⣼⣷⣶⣧⣼⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣷⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⣷⠄⣿⡆⠀⢸⡇⣴⣟⠛⠇⢸⣿⢰⣿⣦⠀⣿⠀⣿⡟⠛⠂⣾⣟⠻⠆⣴⣟⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢛⣿⡄⢿⣇⣀⣼⡇⢈⡛⢻⣷⢸⣿⢸⣿⠙⣿⣿⠀⣿⣛⣛⠃⣙⡛⣻⣦⢈⡛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠈⠙⠛⠉⠀⠈⠛⠛⠁⠈⠋⠈⠋⠀⠈⠛⠀⠙⠛⠛⠁⠉⠛⠋⠁⠈⠛⠛⠉
⠀⢀⣤⡄⠀⢠⣤⣤⣄⠀⢠⣤⡀⠀⣠⣄⢀⣤⠀⣤⡀⠀⣤⠀⣤⡀⣀⣤⣤⡀⣤⣤⣤⣄⢠⣤⣤⣄⠀⠀⣠⣤⠀⣠⣤⣤⣄⠀⣄⠀⣀⣤⣤⣀⠀⣠⣄
⢀⣾⠛⣷⡀⢸⡏⠈⢹⣷⢸⡿⣿⣼⣿⣿⢸⣿⠀⣿⢿⣆⣿⠀⣿⡁⢿⣧⣬⡅⠀⣿⡏⠀⢸⣯⣬⣿⠇⣰⡟⢻⣇⠀⢸⣿⠀⢰⣿⢰⣿⠁⠈⢿⡇⣿⡿⣧
⣼⠿⠛⠻⡧⢸⣷⣶⡾⠋⢸⡷⠸⡿⠸⡿⠸⡿⠀⣿⠀⠻⣿⠀⢿⠇⠶⣦⣽⠟⠀⢾⠇⠀⢸⡏⠹⣷⠠⡿⠟⠛⢿⡆⠸⡿⠀⠸⣿⠈⠻⣦⣴⠿⠃⢿⠇⠙


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⡆⠀⣠⣾⣧⣤⠀⢰⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⢿⣿⢷⣄⠀⠀⣿⡇⠀⣿⣅⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⢀⣼⣿⡀⣿⡇⠀⣿⡇⠀⣈⣉⣹⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠿⠿⠟⠋⠉⢻⣿⠇⠀⣿⡇⠀⠛⢻⡏⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣴⠗⠸⣿⡀⠀⠀⠀⠀⣸⡏⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣰⣶⣤⡾⠿⢛⣥⣤⣤⣽⣷⣤⣤⣤⣴⣿⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢴⣦⣴⣦⣀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡶⠿⠛⠛⠉⠉⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣦⣶⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⢻⡇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⡦
⠀⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠶⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⠀⢹⣇⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡁
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡇⢈⣿⠟⠛⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁⠘⠛⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⣀⣀⣀⣀⣙⣃⣀⣀⣀⣀⣀⣀⠀⢀⡀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃
⠀⠙⣿⡏⠛⠛⠋⠙⠛⠛⠛⠛⠋⠀⠈⠁⠈⠉⠛⠋⠙⣿⠀⢸⡟⠻⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡀⢠⡄⠀⣤⣤⣤⣤⣥⣤⣤⣤
⠀⠀⣿⡇⢰⡿⠛⠛⠛⠛⠛⠛⢻⣧⠀⠠⠶⠶⠶⠀⠀⣿⠀⢸⡇⢸⣿⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣈⣁⣀⣈⣉⣀⣀⣀⣀⣈⣁⣀⣀⣁⡀⠈⣿
⠀⠀⣿⡇⢸⣿⠿⣿⠿⠿⢿⡷⢺⣿⠀⠠⣤⣤⣤⠀⠀⣿⠀⢸⡇⢸⣿⠀⣿⡇⠀⠀⠀⢸⡿⠛⠀⣿⡇⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠀⣿
⠀⠀⣿⡇⢸⣇⣀⣀⣀⣀⣀⣀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⣿⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⡇⠸⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠀⣿
⠀⠀⣿⡇⠈⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡷⢾⣿⠀⠿⠃⠀⠀⠀⠈⠁⠀⠀⣿⡇⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⣿
⠀⠀⣿⡇⠀⢴⣶⣶⡤⠠⣤⠀⢤⣶⣶⣶⣶⣶⣶⠀⠀⣿⠀⢸⡇⢸⣿⠀⠀⠀⢀⣴⠾⠛⠛⠛⠿⢿⣇⣸⡇⠀⣤⣀⣤⡀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿
⠀⠀⣿⡇⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⠀⢸⡇⢸⣿⠀⠀⣰⡟⠁⣴⡶⣿⡷⣦⡄⠹⣿⡇⠀⣽⣿⣯⡀⢸⡇⠀⠿⠿⠿⠂⢸⣿⠀⣿
⠀⠀⣿⡇⠀⣈⣉⣈⣉⡈⢁⡈⢉⣈⣉⣁⣉⠀⣁⠀⠀⣿⠀⢸⡇⢸⣿⠀⠀⣿⠀⣼⡏⠀⣿⣧⣬⣿⡄⢹⣇⣀⣉⣀⣉⣀⣸⣇⣀⣀⣀⣀⣀⣸⣿⠀⣿
⠀⠀⣿⡇⠀⢉⠉⢉⡉⠀⠈⠁⠈⠉⠉⠉⠉⠀⠉⠀⠀⣿⠀⢸⡇⢸⣿⠀⠀⣿⡄⠹⣧⡀⠈⠉⣩⡿⠀⣾⡟⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⢻⣿⠀⣿
⠀⠀⣿⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⣿⠀⢸⡇⢸⣿⠀⠀⠘⢿⣄⡉⠛⠻⠛⠋⣀⣾⣿⡇⠀⣠⣿⣤⡀⢸⡇⠀⠻⠿⠿⠂⢸⣿⠀⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣿⠀⢸⡇⢸⣿⠀⠀⠀⠀⠙⠻⠿⠶⠾⠿⣿⡇⢸⡇⠀⠉⠿⠉⠀⢸⡇⠀⠶⠶⠶⠆⢸⣿⠀⣿
⠀⠀⣿⡇⢰⣿⠻⣷⠀⠀⠀⠀⠀⢈⣉⣉⣉⣉⣉⠀⠀⣿⠀⠸⣧⣼⡟⠀⠀⠀⠀⠀⢀⣠⣴⡟⠀⣿⡇⢸⣷⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣾⡿⠀⣿
⠀⠀⢿⡇⠈⠻⠿⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⣿⡄⠀⠈⠁⠀⠛⠿⣿⠾⠿⣿⠋⠉⠉⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⣀⣀⣈⣛⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠶⠖⠀⠀⠀⠀⢀⡾⠁⠘⣆⠀⠀⠀⠀⠶⠶⠟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⢀⣤⣤⣤⣤⣤⡼⠁⠀⠀⠘⠦⣤⣤⣤⣤⣤⠀⠈⠛⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⡀⠈⠻⣦⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⢠⠞⠁⠀⣴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣄⠀⠀⠀⡾⠁⠀⠀⢹⡆⠀⠀⢠⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠈⠳⣤⡀⠻⣄⣀⣀⡼⠃⣠⠞⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠙⢦⣬⣭⣭⣤⠞⠁⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢠⡿⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡾⠁⢻⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣸⠃⠀⠈⣧⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⡏⠀⠀⠀⢸⡆⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠙⠒⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿
⣻⣿⣿⣿⣽⣟⠾⣭⣭⡿⢟⣿⣟⠶⣿⣽⡷⣿⣿⣿⣻⣿⣿⣿⣽⣟⠾⣭⣭⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠯⣭⣯⢿⣿⣟⢮⣭⣭⣿⢟⣿⣿⢿⣿⣿⣿
⢾⣿⣿⠿⡋⢙⠿⣳⣶⣻⢿⡋⣛⢟⣿⣾⡻⠟⠉⠻⢷⣿⣿⠿⠋⠙⠿⣳⣶⣺⠿⠋⠻⢟⣾⣿⡿⠟⠉⠻⢿⣿⣿⠿⡋⢙⡿⣶⣶⡾⢟⡉⣻⢟⣶⣾⣻
⣷⢰⣿⣿⡇⣾⣿⢰⣿⢰⣿⢠⡿⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣿⠘⣿⡐⣿⡐⣿⡇⢹⣿⣷⠀⣿
⡏⢸⣿⣿⢁⣿⡇⣾⡇⣾⠇⣼⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠸⣇⢻⣇⢻⣇⢻⣿⠸⣿⣿⡄⣿
⡇⣸⣿⡟⣸⣿⢰⡿⣰⡟⣸⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢻⡌⣿⡜⣿⡘⣿⡆⣿⣿⡇⢻
⠃⣿⣿⢁⣿⢇⣿⢣⡿⢡⡟⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢿⡜⣷⡹⣧⢹⣷⠸⣿⣧⢸
⢸⣿⡏⣼⠏⣾⢣⡿⣱⠟⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⢻⡜⣷⡹⣧⢻⣇⢻⣿⡘
⣾⡟⣼⢏⡾⣣⡟⣵⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡻⣌⢷⡽⣧⢻⣎⢿⣇
⣟⣼⣫⣞⡵⣫⣾⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡻⣾⢷⡻⣎⢿
⣾⣿⡿⣫⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⣷⢿⣾⣿
⣛⡹⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣝⣛
⣭⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣽
⣿⣏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿
⣹⣽⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⢿⢿
⣇⣧⣿⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⡟⡿⣾
⣿⢸⡸⡞⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣸⢱⢇⣿
⣿⡎⡇⣷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡟⣼⢸⡿
⢿⡇⣿⢹⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢹⡇⡏⣾⡇
⢸⣿⢸⢸⡟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⢸⡇⣿⡇
⢸⣿⢸⣇⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣸⢠⣿⢰
⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⣿⡿⢛⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣌⣴⣿⣿⣿⣿⠟⢋⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⢁⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢁⣾⣿⠸⣿⡿⠻⣿⡿⠻⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢋⣅⢸⣿⣿⢠⣤⣴⣶⣤⣴⣶⣤⣤⠀⣿⣿⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⡁⣿⣿⡘⠿⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣄⠻⠟⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣌⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣶⣶⣶⣶⣶⡆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣌⡛⠛⠛⣋⣵⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠛⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⢂⣀⡘⢿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⡌⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣯⢸⣿⣿⡿⢩⣤⣴⣶⣿⠿⠛⠿⣿⣶⣤⣤⡍⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣤⣉⣡⣼⣿⣿⣧⠹⣿⣿⡿⢡⣾⣿⣷⡌⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡘⢿⣿⡿⢃⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣦⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠉⠉⠉⠛⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠗⣠⡂⠶⠀⡏⢤⠸⠰⠇⠰⢲⣦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⡀⢘⣥⠬⠗⠚⠛⠛⠛⠛⠛⠛⠳⠦⣮⣸⡕⢃⢗⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢄⣬⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣾⣇⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⣹⠞⠁⠰⣆⣀⠀⠀⠀⠐⣿⣆⣀⣤⠰⣶⡀⠀⠀⠀⠀⠀⠉⠳⣤⡋⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠏⠀⠀⠀⢇⠉⠁⠀⠀⠀⠀⠉⣁⡀⠀⠹⣷⣄⠀⢀⠀⣀⠀⠀⠈⠻⣄⡴⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣛⡇⠀⠀⢀⣤⣾⡆⠀⠀⠀⠀⠀⠀⠙⠿⠦⠐⠋⠉⠀⢘⣧⡿⣁⠀⠀⠀⠈⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡥⢸⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡀⠈⢳⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣠⣤⣀⠀⣀⣠⡄⢰⣾⣿⣿⣶⡶⠂⠀⠀⠻⠿⠃⠀⠈⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣧⣶⣿⣿⣿⣿⣷⠌⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⡿⣆⠀⠘⣟⠛⠉⢻⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢺⡟⠹⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⡄⠀⠒⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠈⢧⠀⠈⠳⠶⠟⢠⡟⠁⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣨⡅⢴⠖⠀⠸
⠀⠀⠀⣠⡴⠶⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⡾⠀⠀⠀⠸⡇⠀⠀⢸⣦⡀⠀⣀⡀⡈⠉⠀⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠻⣤⡀⠀⣴⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⠞⠁⠀⠀⠀⠀⢷⠀⢀⠸⠆⠀⠀⠉⠻⠉⠀⠀⠀⠀⠀⠀⢰⠁
⠀⢴⡞⠛⠋⠀⠀⠈⣳⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣤⣶⣄⠀⠀⠸⣄⠿⠗⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⣠⠏
⠀⠀⠙⠳⣦⠀⠀⢼⡁⠀⠀⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⡴⠃
⠀⢸⣿⠉⠉⠀⠀⠀⠉⠉⠒⢦⡙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠙⢶⣄⣀⠀⠀⠀⢀⣀⣤⠶⠋
⠻⣏⣉⠀⠀⠀⠀⠀⢀⣔⠒⠉⠀⠀⠙⢿⣿⣷⣶⣤⣀⡀⠀⠀⠈⠉⢻⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉
⢠⡏⢍⠙⠛⠲⢲⡆⠀⠉⠛⠓⠒⠠⣤⡀⠈⠙⠻⢿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣶⣦⣤⣀
⡼⠃⠉⠀⠘⠆⣾⠁⠀⠀⢈⣿⡏⠉⠁⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⡀
⢧⠺⣤⣂⣠⣶⠛⠚⠛⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠉⠉⠛⠛⠿⢿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣀
⠘⢧⣈⣉⡴⢻⣶⠏⠀⠶⠆⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⢿⣷⣦⣄
⠀⠀⠉⠉⠀⠈⣇⠀⠧⠄⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠈
⠀⠀⠀⠀⠀⠀⠻⣌⡓⣋⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⡀⣀⣀⣤⣤⣴⣶⣶⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣤⣶⣶⣶⣶⣶⣶⣶⣿⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠟⠛⠉⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡃⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⠤⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉⠹⠆⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠆⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣿⣿⣷⣶⣤⣶⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⠇⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⡼⢩⠟⣹⣿⣿⠛⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⢀⠏⢀⣿⣿⣿⡆⢠⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠀⠈⠁⡟⢸⡇⣼⡟⠃⠇⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢨⡇⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢿⠁⣿⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀⠀⢀⠤⠒⡤⡀⠀⢀⡠⠤⣀⠀⢀⡀
⠀⠀⢸⠏⢳⣝⣿⣦⣙⣷⣌⠻⣿⣕⡄⠈⣁⠤⠐⠒⠉⠁⢡⡁⡇⠀⠘⠻⠅⠀⡓⢓⣈⣅⢡⠔⠡⡄⢖⡄⠱⡈⠉
⠀⠀⡾⢄⣀⣦⣤⢀⣀⠀⠈⠉⠉⠉⠁⢂⢀⠀⡀⠀⠀⠀⠘⠞⡇⢸⠲⠤⣔⣒⠽⣼⣧⠔⠉⡇⠉⡥⡠⣒⢂⠇
⠀⢠⠁⠉⠙⠃⠶⠾⣛⣩⣵⡶⠦⣤⡄⠀⠉⡏⡝⠉⣀⡒⠒⠓⠁⠘⠚⡲⠢⢌⡉⠑⠺⣿⠒⠸⢦⣭⡡⠴⠊⢀⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠫⠃⠀⢀⠑⠃⠀⠉⣁⢀⡀⠀⠀⢠⠃⠀⠀⠀⠉⠡⠀⠀⠀⠀⠀⠈⢳⠀⠛⠃
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠽⢿⠃⡤⠂⠉⠀⢀⠈⠉⠢⣌⠒⡦⠤⣀⡀⠀⠀⢀⠀⠀⠀⢀⠎⠉⠁⡹
⠀⢀⣍⠁⠒⠠⠤⢁⣀⠀⠀⠀⠀⡇⠀⠀⢀⠎⠀⠀⢆⠀⠸⠁⢀⡠⠈⢣⠈⠁⡒⢩⡝⡒⠤⢌⡀⠀⣜⡀⠀⢰⠁
⠀⠘⠉⣀⡤⠔⠊⠠⣀⣈⠉⠒⠒⠀⠀⣠⡿⠀⣀⣀⠠⠆⣘⠂⠁⢀⡀⠈⡆⠹⣛⣀⣉⠐⠚⠓⢡⣌⠀⠈⠲⠃⣤
⠀⢠⠊⡀⠀⠐⠑⠂⠀⠌⢳⠀⠲⠖⠛⠁⢠⡀⠀⢀⣠⢴⡀⠺⡁⡡⠁⢀⠇⡴⠇⠤⢈⠓⡄⠀⡠⡔⠂⠀⢲⣄⠉
⠀⢹⣦⠤⣀⣁⣀⣀⣀⣀⣼⠀⢠⠿⠴⣄⢄⢢⡀⠉⠒⠰⠇⠀⠑⠁⢀⠜⠀⣇⢐⡒⢡⡅⡇⢔⠑⠒⠒⣂⡭⡊
⠀⠸⣿⠀⣿⠀⣿⠇⣿⢸⠇⢀⡖⢔⣒⠨⡉⠃⠙⠢⢄⣀⣀⣀⡠⠔⠃⠰⠀⣿⢨⠬⢸⡇⢱⠀⢷⢋⣴⢟⣴⢷
⠀⠀⣿⡄⣿⠀⣿⢰⡏⡞⠀⡾⢁⡨⠙⠃⡉⠀⡀⠖⠠⢀⠶⠰⣖⠀⠉⣿⠀⢫⠃⠫⠄⡧⢸⠀⠘⣟⣵⠟⣡⠾
⠀⣤⠙⠓⠻⠶⠿⠾⠓⠁⣸⠐⢌⠛⣣⡐⠀⡾⠂⡖⠄⡘⠄⠀⣿⠆⠀⢻⡆⠘⢄⡀⣓⣠⠚⢠⡄⠘⣥⢞⣥⡾
⠀⣉⣤⣀⣠⣄⣠⣒⣂⣄⡈⠐⠀⠀⡰⡠⠊⠀⠠⠛⠂⠌⣃⣄⢩⣇⣀⣸⣇⡀⠀⣍⠉⠀⠀⢠⡀⠀⠀⠉⠀⢤
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⠀⠘⢠⣤⢄⠀⠀⠀⠀⠉⢉⠾⣿⣴⣟⣻⢆⢀⣸⣧⠰⠆⢠⣀⣀⡀⠀⠀⣠
⠀⠉⠢⠠⠠⠂⠒⠤⠤⠔⠊⠘⠛⠁⠘⠠⠀⠀⠈⠑⡇⢀⠔⠁⢦⢆⠀⣤⠈⣿⠀⠀⢿⡄⢰⣽⡃⠀⠀⠀⡀⣹⠢
⠀⠐⠂⠒⠂⡆⠀⠀⠀⢀⠀⠀⠀⠐⠈⠐⠐⠉⠁⠀⠉⠁⡀⢴⠟⢢⠠⠁⠀⠶⡇⠀⢹⡇⢰⣏⢁⣀⠀⢤⣠⠀⣽
⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠐⠦⡀⠀⠀⠀⠀⠐⢿⢶⠶⠥⢐⠡⠀⠠⠱⠑⡀⢸⣥⡀⠸⠗⣀⡙⢦⠑⣷⡺⢃⡴⠁
⠀⢰⠀⠀⠐⠀⣸⣿⣷⣿⣬⣀⣵⠄⠃⠀⠀⡄⠈⢢⠣⠀⠀⠰⢐⡂⡂⠱⠝⠀⠀⣠⠚⠁⠐⡵⡌⢻⣟⢻⣟⠐⠗
⠀⢸⣄⡠⠂⠀⠸⣄⡜⢛⣻⣿⠏⠀⠈⢀⣀⡇⠀⠀⢣⠑⡈⠚⣉⠡⠌⣁⠂⠀⢸⡠⠉⢀⢀⡤⣧⠋⡹⢧⡑⢷⡀
⠀⢸⠻⣄⠀⠀⠀⠈⠉⠸⠋⠁⠀⠀⠀⣠⠟⡇⠰⣿⡇⠛⠊⠀⢈⢦⣤⣤⠚⢥⠺⠓⠭⠔⡉⠀⢡⡜⠁⠀⠳⣾
⠀⠈⠓⠚⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠒⠂⠁⠀⠈⠙⠂⠀⠘⠋⠀⠊⠉⠉⠉⠁⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠘⠃⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⣠⣼⡇⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢹⣏⣀⠈⠉⠀⠀⡀⣹⡏⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠙⢿⣄⣰⣆⣤⡿⠏⠀⠀⠀⠈⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣈⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠻⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢾⣏⣉⣉⣹⡷⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣤⠶⠿⠶⣤⠀⣿⡀⠀⢀⣤⠾⠛⠛⠻⣦⣄⠀⠀⣿⠀⢀⣉⣉⣉⣉⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢸⡏⠀⠀⠀⢹⡇⢈⣿⢀⣾⠃⠀⠀⠀⠀⠀⢻⣆⠀⣿⠀⢈⣉⣉⣉⣉⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠈⠿⣦⣤⣴⠿⠁⣾⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢈⣉⣉⣉⣉⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣦⣤⣉⣠⣴⡶⠟⠀⠸⣧⠀⠀⠀⠀⠀⠀⢠⡿⠀⣿⠀⠈⠉⠉⠉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡉⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⢀⣾⠃⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⢀⣿⠷⣦⣤⡶⢿⣇⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣦⡶⠟⠋⠁⠀⠀⠀⠀⠀⠉⠛⠷⣶⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣿⠉⠉⢹⠉⠉⢹⡏⠉⢹⡏⠉⠉⡏⠉⠉⡏⠉⠉⣿⠉⠉⣿⠉⠉⢹⠉⠉⢹⡋⠉⢹⡏⠉⢹⡏⠉⠉⡏⠉⠉⣿
⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⡇⠀⠀⣿⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⣿
⠀⠀⣿⠀⠀⢸⣀⣀⣸⡇⠀⢸⣇⣀⣀⡇⠀⠀⣇⣀⣀⣿⠀⠀⣿⣀⣀⣸⠀⠀⢸⣃⣀⣸⡇⠀⢸⣇⣀⣀⡇⠀⠀⣿
⠀⠀⢻⣄⣀⣼⠋⠉⠹⣇⣀⣸⠏⠉⠙⣧⣀⣠⡟⠉⠉⢿⣀⣀⡿⠉⠉⢻⣄⣀⣼⠋⠉⠹⣇⣀⣸⠏⠉⠙⣧⣀⣠⡟
⠀⠀⣾⠈⠉⣦⠀⠀⠀⠈⢉⡅⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢨⡉⠁⠀⠀⠀⣴⠉⠁⣷
⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⣿⠀⠀⡇⠀⡖⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢲⠀⢸⠀⠀⣿
⠀⠀⣿⠀⢸⡇⢠⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡄⢸⡇⠀⣿
⠀⠀⣿⠀⣼⠁⣼⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣧⠈⣧⠀⣿
⠀⠀⣿⢀⡟⢰⠏⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠹⡆⢻⡀⣿
⠀⠀⣿⣸⢣⣏⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣜⣇⣿
⠀⠀⣿⣉⣉⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣿
⠀⠀⣿⠀⣧⢿⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡿⣼⠀⣿
⠀⠀⣿⠀⢹⡎⢷⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡾⢱⡏⠀⣿
⠀⠀⣿⠀⠈⡇⢸⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⡇⢸⠁⠀⣿
⠀⠀⣿⠀⠀⡇⠈⠁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠁⢸⠀⠀⣿
⠀⠀⣿⣀⣀⣃⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣘⣀⣀⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⣀⣀⡀⣀⣀⠀⣀⡀⢀⣀⡀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢿⡇⣿⠒⠀⣇⣸⡇⣸⡁⣿⡀⡁⣼⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠉⠉⠀⠉⠉⠀⠉⠁⠈⠉⠁⠁⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⢸⠛⣶⢸⣧⣴⡇⢸⡇⢸⢦⣾⠈⣿⠰⣾⣙⠘⣿⠃⣾⣻⡆⢠⣿⡄⠛⡟⠁⢻⠃⣾⠛⣦⢸⢧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠓⠘⠒⠋⠘⠃⠉⠃⠚⠃⠘⠀⠛⠐⠛⠐⠒⠛⠀⠛⠀⠛⠈⠓⠛⠉⠓⠀⠃⠀⠛⠂⠙⠒⠋⠘⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⢉⣉⣉⣉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢸⣷⣤⠚⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⢙⡶⢶⡤⠶⠞⣁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢚⠉⠉⢙⢦⠀⠀⠀⡇⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠸⢸⢀⢀⠘⡻⡀⣀⡀⠷⢤⡖⣦⠴⠇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢿⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⡿⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⡤⣤⣤⡇⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠤⠥⢤⡴⠶⠶⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣚⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠒⣻⠀⠀⠀⡇⠀⠀⡏⠁⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⠀⢠⡶⠶⠄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⢀⣶⠀⣇⣀⣀⣧⢠⢤⠈⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⠿⣴⡁⣠⠧⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢼⢸⠩⡇⡿⠛⠻⣿⢸⡘⡇⡇⠀⠀⣠⠤⠤⠤⠾⣄⠘⠶⠏⠁⡷⢸⠛⣻⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡄⢸⢸⢈⡇⡇⠸⠀⣿⢸⡇⡇⣷⠀⠀⣷⠘⠛⠛⠃⣿⠉⠓⠋⠉⡷⠸⠿⠟⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠸⠶⢃⡇⢰⠀⣿⡀⡇⢇⢻⠀⠀⣙⠚⠒⠦⠚⣣⢖⡋⠙⠲⣝⠓⠶⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠃⣸⢩⢭⡉⣇⠀⢀⡏⣇⣓⣋⣸⠠⡴⠁⠀⠀⠀⢰⡇⢯⡻⣄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠭⠭⣿⠬⠭⠤⡿⠽⠽⡇⡿⠞⢒⣺⡆⠀⠀⠀⠀⠀⠀⢳⣀⠙⠶⣋⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠜⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣴⣧⣄⡠⠄⠊⢀⣠⣾⣿⣦⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠈⠉⠈⠁⠀⠀⢀⣴⣾⣿⣿⠿⠛⠉⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠘⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠈⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⣸⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⣴⣾⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠉⠉⠛⠓⠒⠶⠦⠤⠤⠤⠤⠤⠴⠖⠒⠊⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠈⣿⡟⠀⠀⢠⣶⣿⣿⡿⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⢧⣀⣀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠛⠛⠛⠻⠗⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣤⣾⣿⣷⡀
⠀⠀⠀⠀⠀⣤⣀⣀⣰⣿⠷⠶⠿⠿⢿⡏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⣿⠿⠿⢿⣿⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠈⠙⣻⣿⣿⣿⣿⣿⢶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⢿⣿⠏⠉⠈⠁⠉⠉⠛⠿⠇
⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠀⣤⣽⣿⣧⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣯
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢴⠆⢶⠆⢠⣾⣷⣶⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡀
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⣾⠇⠀⢹⡇
⠀⠀⠀⠀⠀⠈⠉⢻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠰⣦⠈⠻⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⢀⣤⠄⣠⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢤⣶⣾⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⢀⠔⢹⠑⢄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣘⡿⠿⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠂
⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⣀⣀⡀
⠀⠀⠀⠀⢹⡿⣿⣿⢿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠯⠿⠿⠽⠽⠽⠿⠿⠯⠯⠯⠿⠿⠽⠽⠽⠽⠿⠇
⠀⠀⠀⠀⢸⡁⢀⣀⢈⣿⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣼⣿⣼⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣶⣶⣶⢲⢲⢲⣶⣶⡖⡖⣶⣶⣶⣶⢲⣲⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡻⠿⠿⣛⡛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣷⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠘⣿⣿⠇⢀⣀⣀⠀⠀⢸⣿⣿⣿⣿⡉⢉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢉⣉⣉⣉⡉⣉⡁⠀⠀⢠⣿⣿⣿⣷⠀⣨⣅⢀⣾⣿⣿⣿⡄⢸⣿⣿⣿⣭⣭⣭⣭⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⠛⣯⡅⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠛⠁⠀⠀⠘⠛⠃⠘⠛⠛⠛⠛⠓⠒⠋⠘⠛⠃⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣧⢀⡖⢺⡇⡶⢲⠖⡆⢰⢠⡖⢢⠀⡄⢰⣒⠐⡗⢀⡖⢂⠔⢢⠀⡗⢀⠔⢢⠀⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠊⠓⠚⠁⠃⠘⠀⠃⠘⠘⠃⠘⠃⠃⠐⠚⠁⠓⠘⠃⠈⠓⠚⠃⠓⠈⠓⠚⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⡠⠤⠒⢶⠖⢳⡄⡠⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠑⡄⠀⠀⢨⠀⢎⠀⠀⢠⠏⢠⢊⠞⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠘⡄⠀⠘⡄⢸⠀⠀⠸⣀⡷⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣀⣠⣄⣀⠀⢹⠀⣰⠁⢸⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⠤⠒⠋⠉⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⢟⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠉⠲⢄⡠⠴⢺⠁⠀⠀⠀⠀⣠⡶⠖⠒⠀⠈⠀⠀⣠⡶⠶⠖⠀⠘⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠀⠑⢤⡚⠣⡀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣤⠞⠋⠀⠀⠀⠙⢦⡑⣄⠀⠀⠀⢠⡾⠛⠙⠁⠀⠀⢀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢸⠃⠀⠀⣴⠞⠋⠀⠀⠙⢮⡦⡀⠀⠋⠀⠀⠀⠀⠀⠒⣋⣤⠔⣛⡻⠆⠀⡇
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⠀⠀⠐⡏⠀⠀⠀⣰⢄⠀⠑⣞⢦⡀⠀⠀⠀⢀⡴⢟⡭⠖⠈⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⣠⡤⠔⠒⠒⠦⣄
⠀⠀⠀⡴⠃⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⢀⢀⡏⠀⢣⠀⠘⡄⠙⢦⡀⠀⢿⠞⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡴⠻⡀⠹⡄⠀⠀⠀⠀⠑⡄
⠀⢀⠎⠀⠀⠀⠀⡟⠀⠀⣠⡶⠒⠀⠀⢀⢄⡾⠁⠀⠀⡆⠀⠰⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⡘⠀⠀⠹⡄⠙⣄⠀⠀⠀⠀⢸
⠀⠘⢄⠀⠀⠀⠀⠇⠀⠀⣿⠁⠀⠀⠀⣡⠞⠁⠀⠀⢠⠁⠀⠀⡇⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠁⠀⠀⠀⠀⣠⣖⠉⡆⠀⠀⠘⣄⠘⢆⠀⠀⠀⡸
⠀⠀⠀⠉⠒⠤⣄⣀⡀⠀⠁⠀⠀⢠⣞⣁⣀⣀⣀⠴⠃⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠈⢆⠘⢆⠀⠀⠈⢦⠈⠳⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠏⠉⣑⠂⠤⢄⣀⣀⠈⠀⠈⣀⣀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠈⢢⣈⣳⣀⣀⣀⡳⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡀⡿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⣀⠀⠀⠀⠀⠀⣀⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⣄⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡤⢚⣥⡮⡓⢄⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠤⠤⠀⣈⣣⠀⠀⠀⠀⠀⢀⡠⢎⠀⡠⢒⢥⡺⣟⠥⣊⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢒⡩⠋⠀⠀⢀⡠⠖⠉⠀⠈⢦⠻⣅⠛⣊⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠊⠁⡠⠖⠉⠀⠀⣠⠴⠉⠀⠀⠀⢀⡠⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠋⠀⠀⣀⠴⠊⠙⢄⠱⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁


⣿⢿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⡜⡿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⣾⢟⣿⠹⡍⣿⠋⠉⠀⣀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠊⠘⠈⣹⠎⢣⣿⣏⢠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢮⣼⠥⠂⣼⡰⠒⠀⣀⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢏⣉⣛⣾⡇⠀⠀⣤⢩⣯⡙⠀⠀⠀⠀⢀⢀⣀⣸⣇⣿⣸⣇⠂⡀⢐⣤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡃⠀⠛⢿⣿⡀⢶⣧⡀⠀⠓⠀⠀⠀⠀⠸⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣤⣀⠈⠳⠞⢻⣿⣗⢲⡤⡄⠁⠀⠘⠀⠁⢸⣿⣿⣟⣿⡿⠃⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⡟⠻⣿⣷⣦⡴⣶⡟⠙⠛⠿⠿⡄⠀⡤⠤⢀⠀⣸⡉⢹⠙⡇⠀⠇⠈⠉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣄⢕⢿⣿⣿⣿⣷⣤⣤⣶⣴⡾⠶⠶⠀⢀⠤⠬⠁⠚⠾⢧⣀⣢⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⡌⢮⣿⣿⣿⣿⣿⡻⢿⡷⠶⠶⠶⠶⠾⢂⣤⣤⣤⣤⣤⠤⣉⣉⣓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢡⣬⡻⣿⣿⣿⣿⣿⣶⠖⠶⠖⣲⣾⣿⣿⣿⣿⣿⣿⣿⠇⡏⣉⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⣿⣿⣾⣿⣿⣿⣿⣿⣑⣀⣌⣡⣴⣽⣿⣿⣿⣿⣿⣿⣾⣿⡿⢻⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⣸⣿⣿⣝⢻⢻⣿⣿⣿⣿⣿⣿⣿⠟⢻⡟⠉⠉⠝⠿⠿⠿⢿⣿⣿⡿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠴⠦⠄⢸⣿⣿⣿⣎⢧⢻⣾⣿⣿⡭⣽⣿⢀⠘⣧⣤⣤⣶⣦⣲⣿⣿⣽⣿⡇⠀⡷
⠀⠀⠀⠀⠀⠐⠶⠶⠒⠂⢸⣿⣿⣿⣿⠀⢘⣿⣿⣿⡇⢸⣿⠀⠀⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣇⡄⡇
⠀⠀⠀⠀⠀⠀⢈⣩⣿⣏⢸⣿⣿⣿⡏⣷⣾⣿⣿⣿⣷⣼⣿⠀⢠⣏⣙⣿⣿⣿⣿⣿⣷⣿⣿⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⢶⣶⢸⣿⣿⣿⣇⣿⣿⣿⣻⣿⣿⣿⣿⡆⢺⣟⣛⣛⣛⣛⣟⣟⢿⣿⣿⡇⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⢾⣿⣿⣿⣧⣿⣿⣿⣿⣿⢻⣿⣿⠆⣌⣿⢿⣛⣻⡭⠭⠩⢓⣛⣹⡇⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠛⡇⣿⠨⠥⠴⠒⠾⢿⣿⣿⣿⡇⣸⣿⡗
⠀⠀⠀⠀⠀⠀⠰⠾⠋⠾⠓⠿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡷⠀⡇⢶⡦⠰⣶⣤⡴⡶⠁⡉⠉⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⢀⣌⢉⣛⠿⠛⣿⣿⣿⣿⣛⡛⠷⠛⠡⠶⠖⣶⠛⠛⠋⢤⣠⠤⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢸⡿⠋⠹⠿⢫⡿⠳⠟⠶⠾⠨⡽⠀⣴⠤⠌⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡶⢿⣶⣄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠁⠉⠫⢻⡷⡀⠺⢭⡁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⢡⢊⡭⢛⠀⠀⠀⠀⠀⠀⢽⠇⠒⢤⣈⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣽⠀⠀⠠⠎⣼⢀⡼⠀⠀⠀⠀⠀⠀⢼⣦⠃⠀⣿⣸
⢀⠦⠃⡥⣐⡶⢲⣤⣄⣀⡀⠀⣀⡤⠚⠁⠀⢕⣿⠀⠀⣤⡊⠉⠘⠑⠢⡀⠀⠀⡠⠠⠊⢹⣟⡲⢻⣹
⠈⠡⠞⡋⠁⢃⣨⠙⣰⠆⠉⠻⣅⠀⠀⠀⠀⢕⣿⠀⠀⡿⠁⣧⠀⠀⢰⡈⡦⠜⠠⠁⠀⢸⣿⢀⠿⡏
⠀⣰⢇⢅⠀⠘⣾⠆⠙⢦⡺⡢⠈⢧⠀⠀⠀⢕⣿⠀⠐⣷⣄⣿⣄⠀⡜⠛⠲⠴⠃⠀⠀⢸⠿⢾⠁⢹
⢠⡇⠈⢮⠄⠒⠉⠀⠀⠀⡏⠁⠀⠘⣆⣀⠀⢕⣿⠀⠀⣽⠉⠿⣿⣪⡁⢀⣆⣆⠀⢀⣀⢸⢀⣨⣇⣸
⠸⡇⡔⠾⠀⠀⠀⠀⠀⠀⠱⠴⠋⢦⣇⡼⢲⣕⣿⠀⢠⣿⣷⠶⣶⣦⣤⠴⡞⡏⢫⢹⣈⣾⠟⠉⠀⢻
⠀⢻⡻⢖⠒⠒⠢⣄⠀⠀⠀⠀⠀⣼⣀⣑⣒⣽⣿⡀⠸⠀⠀⡏⣇⣱⡸⠤⢓⣿⣍⠥⢺⠏⠀⢀⠀⢻
⠀⠀⠹⣄⡑⡄⠀⡜⠀⠀⠀⣠⠾⣭⡥⠤⠤⢔⣒⣛⣯⣽⠿⠏⠓⠒⠈⠉⠙⡄⠈⠋⠁⢀⡴⢏⢀⣸⠀⠀⠄⡀
⠀⠀⠀⠀⠉⡿⠒⠓⣲⠖⠫⡐⢌⡲⣝⠖⢻⣿⣿⣿⣿⣿⣿⣶⣤⣠⡀⠀⠀⠘⢄⣀⠔⠉⠀⠸⣯⠛⡿⣭⢆⠘⡥
⠀⠀⠀⠀⠀⡗⢠⣎⡁⠢⣕⢮⡳⠋⡡⢖⡉⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠖⣿⢧⠁⠀⡌⡁
⠀⠀⠀⠀⠀⡇⢸⣿⣿⣳⣔⣉⠴⣪⣶⣿⡇⠄⠀⠄⠀⠐⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢤⣡⠁
⠀⠀⠀⠀⠀⡇⢸⣿⣿⣺⠀⡇⣿⣿⣿⣿⡇⠀⠀⣯⠑⡀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡷⠃
⠀⠀⠀⠀⠀⡇⠈⠻⣿⢿⠀⡇⣧⣿⡿⢟⣵⣀⡀⠻⣤⣘⢆⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⣇⠀⠀⠈⢿⠀⡇⠟⣫⡾⠛⠉⢿⣷⣦⡼⡋⠙⠉⢠⣤⣤⣷⡒⠒⠒⠒⠒⠒⠶⣶⣶⡾⠁
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠢⣷⠟⠫⣴⣶⠄⢀⠴⠛⠏⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⣀⣭⣿⡿⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠯⣑⣢⣌⣅⡸⣇⡀⠀⠀⠀⠀⠀⠀⢸⠟⡩⠟⠛⣲⢤⠀⢒⣾⣿⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⢚⠁⠉⠉⢧⡘⡏⠀⢀⡶⠀⠀⢰⠎⡜⠁⠀⠀⢸⣧⢻⣽⣿⠃
⠀⠀⠀⠀⠀⡇⠀⠀⢀⡤⢊⡔⡡⣢⢂⢄⠀⢳⡀⠀⣼⡷⣠⣠⣿⢸⠁⠀⠀⠀⣸⣿⣏⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣍⡛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣛⣛⣛⣛⠻⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣭⣭⣭⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣼⣿⣿⣿⣿⣧⣛⢸⣿⣿⣿⣿⣿⡟⣭⣭⣭⢻⡇⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⡟⣛⣛⣛⢿⡇⣿⣿⣿⢸⡇⠿⢿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⣾⡇⣿⣿⣿⢸⡇⣿⣿⡟⣠⣶⠟⢷⣮⡘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠟⣡⣿⡇⣿⣿⣿⢸⡇⣏⢿⢸⣿⣧⣛⠳⣿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⢸⣿⣿⡇⣿⣿⣿⢸⡇⣿⠸⡎⢿⣯⣙⣡⣿⢏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⣿⣿⣿⣿⢦⣝⡻⠷⣶⣶⣶⣿⣿⣶⡆⣶⣶⣭⣭⣭⣵⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⡎⣙⣋⢵⣿⣿⣿⣿⣶⡙⣿⣿⣿⣿⡷⠹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣿⣿⢸⣟⣛⣛⣿⣿⡇⣿⣿⡿⣫⣾⣿⡆⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣿⢸⣿⣿⡎⣿⣿⡏⣿⣿⣧⠿⣫⣾⣿⡿⢫⡘⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣇⢻⣿⣿⢇⣿⣿⣧⢿⣿⣯⣾⣿⡿⠋⠀⠈⡇
⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣷⣙⣣⣾⣿⣿⣿⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠘⠛⠋⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠤⠤⠀⠒⠒⠒⠀⠀⠀⠀⠐⠒⠒⠒⠂⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⢁⣀⣤⠤⢴⣖⣿⣿⣟⠛⠙⠛⠛⠛⠻⠿⣻⣷⣶⣶⣦⣤⣤⣀⣈⠁⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⢁⣤⣶⣿⠭⠥⠒⣋⣴⢡⣿⣿⣿⣟⡒⠒⠒⠒⠊⠉⠭⣿⣿⡙⠛⡇⠙⣿⡻⠿⠷⢦⣤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⣠⣾⣿⣿⣏⡙⠛⣛⡫⠝⡡⠻⣿⣯⡫⣈⠉⠑⠒⠂⠈⢑⡿⢟⣿⣷⣄⠙⠦⣈⠉⠙⠛⢋⣹⣿⡄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⣰⣿⢿⣷⣼⣯⣬⣭⣥⡔⠋⠀⠀⠙⢿⣷⠄⣈⣉⠉⠂⠈⡡⠴⣿⠟⠁⠀⠙⠦⢄⣉⠉⠉⢈⣁⣽⣿⣆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡯⣿⢺⠇⠀⣄⠐⢸⠟⠀⠀⠀⠀⠀⠀⠈⠒⠀⠤⠀⠀⠰⠒⠊⠀⠀⠀⠀⠀⠀⢸⣿⣟⢻⣯⣿⠛⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡟⢻⠏⣿⠀⠻⢻
⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⡗⢄⠀⢀⡿⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣿⡇⢸⠀⠈⡇⠀⡇⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠃⢸⠀⡿⠀⠂⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠇⣿⠀⢸⡇⢀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢠⠀⢠⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢸⢿⠀⣿⠀⣼⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⢸⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⣿⢀⡿⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠀⡄⠈⡀⢸⣸
⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⢠⡟⢸⡇⢰⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡇⢠⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⢸⠁⣼⢀⡏⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢱⠘⡇⣦⢻⡄
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣾⢀⡇⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⡀⣇⢹⣼⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣷⣾⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠈⠀⢺⣟⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣝⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢿⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢸⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣸⣷⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⡀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠋⠉⠁⠈⠁⠈⠉⠉⠋⠉⠉⠙⠉⠁⠉⠁⠉⠉⠉⠸⠀⠀⣸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⢠⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠤⠠⢦⠼⠿⠘⠛⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠙⢻⣤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠑⡀⢐⠀⠀⠀⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠠⠄⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡀⢴⡠⢖⠒⢦⣄⡆⡄⠉⠐⠠⠄⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠃⠒⢸⠀⠀⠉⠀⠀⠙⠀⢀⠀⡀⠀⠀⠀⠀⠀⢸
⠤⠄⠒⠒⠂⠀⠀⠒⠒⠬⡢⣀⠀⡼⠐⠒⠠⡀⠀⠜⠀⠓⠒⢲⠄⠴⠭⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠹⣀⠤⠀⠤⠜⡼⠤⠤⣀⠠⠊⠀⠀⠀⠀⠀⠋⠒⢄⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢀⡔⠀⠡⡀⠇⠑⠂⠤⠄⡲⠀⢀⠆⠀⠀⠈⠑⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢡⠈⠢⣀⡠⠔⢡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⡇⣿⠀⠀⠀⠀⢾⣿⣿⣦⡄⠀⠀⢰⣾⣷⣿⣶⠀⠀⠐⡀⠐⢆⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣴⣤⣤⣄⣤⡀⠀⠀⡀⠠⠤⢀⡀⠀⠀⠀⠀⣠⣶⡄⠀⠀⢰⡖⣒⣒⢺⣶
⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠆⠀⠀⣠⣾⣿⣿⠿⠀⠀⢸⡇⠧⡼⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠓⠀⠀⠘⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠋⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⡀⣄
⠀⠀⠀⠀⠀⠀⢸⣼⣶⣦⣦⡆⠀⠀⠀⠶⠾⠄⠀⠀⠀⣔⣻⢾⣳⡀⠀⠀⠀⢸⡛⠟⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣻⠞⠁⠀⠀⠀⣰⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠙⢿⣿⣿⣷
⢀⣿⡟⣹⣿⣿⣿⣿⣿⠿⠿⠛⠻⠿⠿⢿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⣸⣿⣰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢹⣿⣷
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣄
⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠉⠙⠂⠈⠋⢹⡇
⠘⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⣠⣄⡀
⠀⠘⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣾⠉⠈⣻⡄
⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠟⠀⠀⠀⢠⡄⠀⠀⢸⡇⢀⡏⠀⢀⣿⣀
⠀⠀⠀⣼⢋⣩⣉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢸⣿⠋⠉⠱⡟⠀⢹⠷⣦
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⣤⣤⣤⠴⠶⢾⠋⠀⠀⠀⠀⠀⣸⣿⠀⠶⢾⣇⣀⡟⠀⣼⠁
⠀⠀⠀⠘⠷⢦⢾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⣈⡆⠀⠀⠀⠀⢠⡟⠹⣦⡀⠀⠉⠉⠙⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠈⠉⠀⠀⠀⢀⣴⠟⠀⠀⣸⠇⠀⠀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠛⣶⣦⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⠴⢞⣻⡟⠁⠀⢄⡀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠀⢿⠉⠉⠉⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠈⠻⣷⣤⣄⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⣤⣼⠷⣶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠀⠀⢠⡾⠁⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠻⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢹⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧
⠀⠀⠀⠀⠀⠀⠙⠶⣆⠀⢀⠀⠀⣦⡾⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠛⠛⠉⠙⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⠋⠀⠀⠀⠀⠸⣧⣠⣤⡴⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠒⣶⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣴⠶⠶⠛⠛⠉⠀⠀⠀⢀⡟⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⡀⠀⠹⣧⡜⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⠷⢦⣍⣑⠂⠤⢨⡟⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⣏⣹⣌⡟⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣍⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⠤⠤⠧⠤⠤⠤⠤⠤⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣄⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⡆⠀⣀⣴⡿⣟⡉⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⡿⢫⣾⣿⣿⣦⡀⢰⣿⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⢿⣿⣿⣿⡟⣠⣶⣮⣿⣿⠋⠀⣾⣿⣿⣿⣿⣿⣿⣭⣄⣀⣀⣀⣰⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢀⣠⣤⣶⣶⡽⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢹⣿⣇⣤⢸⣿⣿⣦⣤⣴⣿⣿⣿⡿⣿⣿⣿⣿⣿⠏
⠀⢀⣴⣾⠟⠛⠉⠁⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⢀⣠⣿⣿⡾⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣉⣻⣿⡟
⢰⣿⠋⠀⠀⠀⠀⢀⣀⣀⣀⠀⠙⠿⣿⣿⣶⣾⣿⣿⢟⣾⠀⠀⠈⣿⣟⣁⡀⠀⠀⠈⢻⣿⣿⣿⣿⢿⣤⣄⣀
⠘⠿⣷⣶⣶⠾⠿⠛⠛⠛⠻⢿⣦⡀⠀⠉⠙⠛⠉⠁⠈⢿⣧⡀⠀⠘⣿⣿⣿⣷⣦⣄⡀⣿⣿⠿⠃⠀⠉⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠈⠻⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⣠⣴⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠀⠀⠀⠀⠀⣾⣦⡀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣤⣤⣴⡖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢐⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣽⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⢿⣷⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣯⡀⠀⠈⠙⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠛⠻⠿⢿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⢻⣿⡿⠛⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠉⢀⡀⢿⠟⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⣀⣴⣾⡇⣿⠃⠀⣼⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣿⣿⣄⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠅⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇⠀⣈⣉⣙⠛⢷⣌⢿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣷⣦⡉⠀⢻⣿⣿⣿⣿⡿⢸⡇⠀⠀⣿⣉⠛⣿⣿⣿⣷⣶⣿⣿⣽⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⡿⢟⣽⠇⠀⢸⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣿⣿⡇⢸⣏⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠉⠛⠛⠛⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠋⢠⣾⣿⣷⣾⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣶⣿⣿⣿⠿⠿⠃⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡟⠛⢻⣏⣨⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡗⠶⠀⠐⠞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣃⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣯⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣴⠾⠛⣉⣀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣦⠘⣷⡙⠿⣿⣿⠿⠟⣋⣙⣛⡛⢿⣿⣿⣿⡿⢋⣴⠟⢋⣤⣶⣿⣿⣿⡆⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡌⠻⠷⡶⠶⠞⠛⣉⣉⡙⠛⠷⣦⣭⣥⡶⠟⢁⣴⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣤⣤⣴⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⢀⣤⣶⣴⣾⣿⠟⠋⠛⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠔⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟
⠀⢻⣿⠃⠀⠀⠀⠀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠸⡏⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⠙⠛⠛⠻⠿⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠉⠛⠳⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⠶⣦⣤⣀⣀⠀⣿⣧⣀⠀⠀⠀⠀⣨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣿⣿⠿⢿⣿⢟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠊⢏⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢈⣀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠈⢿⡿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠘⠲⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣤⣤⣄⣀⠀⣀⣠⠀⠀⠀⠀⣤⡲⢆⠀⣄⡀⠉⢻⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠾⠛⠋⢉⠙⠿⣿⢿⣿⣿⣿⣧⠠⡤⠀⣠⣿⣿⣿⣾⣿⣿⢿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠘⢿⠋⢀⣠⡴⠖⠛⢛⣚⣛⠳⣎⣿⣿⣿⣦⠾⢆⣿⣿⣷⠿⠗⠀⠝⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠚⠓⢁⣈⣥⣴⣶⣿⣿⣿⣿⣿⣾⣿⣷⡅⠉⠓⠑⣿⣿⣿⣾⣿⣿⣷⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⢻⠃⠀⣿⣿⣿⣿⣿⣿⠟⢡⠄⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣭⡉⠳⡀⠀⠀⢳⠀⠀⠀⠀⠀⡠⠐⠁⢀⠜⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣥⣴⣶⣾⡇
⠀⠀⠀⠀⠀⢸⣿⡀⠁⠀⠁⠀⢡⠀⠀⢸⠀⠀⠀⠒⠉⠀⣀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣗⡨⠚⢿⣿
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢠⠀⠸⠀⢀⠜⠋⠀⢀⡠⠐⠚⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣂⢽⣿⡄
⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⢸⣄⣸⠖⢁⣠⡔⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⢀⣠⣤⣤⣀⠀⢀⣿⣯⣹⢿⣿⠃
⠀⠀⠀⠀⣠⣯⡈⠙⠣⡀⠸⣿⠁⡴⠋⢠⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠙⠿⠿⠛⠿⢿⣿⣿⣼⣿⣿⣯⣻⣿
⠀⠀⠀⢠⣿⢻⠏⠳⠀⠘⢆⡏⢸⣇⣀⣼⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠿⣿⣿⢻⣿⣿⠇
⠀⠀⠀⢸⣎⣼⡄⠀⠀⠐⢮⣿⢟⣿⣿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⢶⢈⣙⢶⡶⢾⣿⣀⡍⢻⣦⣿⠏
⠀⠀⠀⠀⣿⣏⡇⠀⠀⠀⠀⣷⣬⣿⣤⡀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠶⣿⠿⠿⠛⠛⠿⣷⣶⠿⢿⣿⡿⢶⡿⠁
⠀⠀⠀⠀⠸⡿⢷⡀⠀⠀⠸⠀⠙⡿⣿⡿⡄⠱⡄⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣿⣀⣴⣿⣇⡿
⠀⠀⠀⠀⠀⢿⡿⢷⠀⠀⠀⡇⠀⢹⢸⣿⡟⠀⠘⢦⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣶⣾⣿⣿⣿⣷⡟⠁
⠀⠀⠀⠀⠀⠸⣿⣪⣓⠀⣴⠄⢠⡟⠈⣿⣇⠘⢄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠙⠛⠏⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠞⠉⠀⠀⠹⣯⠻⣦⣷⡦⠀⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣶⣤⣄⣀⣀⣀⣀⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠈

⣿⣿⡿⣿⢿⡿⣿⣿⣿⣿⣿⣿⢿⡿⣿⢿⣿⣿⡿⣿⢿⡿⣿⣿⡿⣿⡿⣿⢿⣿⢿⣿⢿⣿⣿⣿⢿⣿⡿⣿⡿⣿⣿⣿⣿⣿⡿
⣿⣿⣷⣿⣾⣷⣿⣿⣿⣿⣾⣿⣾⣿⣿⣾⣿⣿⣷⣿⣾⣷⣿⣿⣷⣿⣷⣿⣶⣿⣾⣿⣾⣿⣿⣿⣾⣿⣷⣿⣷⣿⣷⣿⣷⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠬⠽⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣛⣛⣛⣻⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣓⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣮⣤⣤⣴⣶⣶⣦⣭⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⢻⣿⣿
⣿⡿⣿⣸⣾⢸⢫⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡏⣿⢸⣾⢸⡿⣿
⣿⣧⡇⣯⣇⡟⣼⡿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠻⡏⠟⡏⠿⢻⢻⢹⡟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡆⢷⢸⡇⡟⡏⡇⣿
⣿⣿⢷⣹⣹⢱⣿⣱⣿⣿⣿⣿⣿⣿⡿⣿⢻⠃⡇⢸⠀⡇⠀⡇⠀⢸⠸⠘⡇⠃⣿⢸⢿⣿⣿⣿⣿⣿⣿⣿⡜⣇⣧⣇⣇⣿⣿
⣿⣿⡿⣿⢧⣿⣷⣿⣿⣿⣿⣿⣿⡿⠇⢻⢸⠀⠃⠘⠀⠁⠀⠀⠀⠘⠀⠀⠃⠀⠀⢸⢸⡏⣿⣿⣿⣿⣿⣿⣿⣽⣽⣹⢿⢹⣿
⣿⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣾⣼⣿
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⡇⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⣿⣿⡿⣏⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⣿
⣿⣿⣇⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿
⣿⣿⣿⣧⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠃⠚⠘⠐⠃⠘⠚⠃⠃⠃⠘⠘⠐⠚⠂⠃⠛⠘⠀⠃⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣼⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣫⡄⣿⣿⡇⣍⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣇⣿⡇⣿⣿⡇⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⢻⣷⣶⣶⣶⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⡏⣾⡏⣭⣛⣛⣱⡍⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢛⣛⣵⡿⢟⣛⠿⣷⣍⣭⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣮⢹⣿⢸⣿⣿⣿⢸⣿⣵⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢸⣿⣿⢫⣾⣿⣜⡻⠿⢟⣼⡿⠷⢹⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡸⣿⣿⡷⣶⡌⣿⡿⠿⢿⡏⣴⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣮⣭⣾⣿⣷⣮⣾⣿⣧⣶⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠻⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⡏⠩⣍⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢓⡿⠋⠁⠀⠀⠈⢳⠜⣧⡄⣀⢤⡔⠶⡖⠊⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠁⠐⠀⠀⠀⠀⠀⢧⠾⠷⠟⠓⠛⠉⠉⠻⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣾⣿⡿⠿⠔⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣷⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾⠺⢳⣿⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠐⣿⣿⠀⡄⠹⢶⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡇⠀⠘⠣⢤⣁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡇⠀⠀⣀⣀⠈⠳⣜
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⡀⠆⠀⠀⡍⡇⡇⡀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠂⠇⠀⠀⠀⠀⠁⠇⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⡀⡀⣀⣠⣀⣀⣤⢀⣦⣴⢤⣦⣶⣶⣾⢾⣳⣖⣙⣿⠡⢍⡢⠀⠀⠸⠀⢰⢮⠰⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣄⣤⣦⣴⣶⣷⣻⣶⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣍⡓⢤⣀⠅⡘⠹⠸⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠉⠙⠻⠛⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣑⠾⣢⡄⠰⠇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠛⠟⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⣿⣿⣿⣶⣽⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠃⠰⠶⠠⠖⠀⠀⠒⣚⣘⣂⣉⣀⣋⡿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠁⡈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠂⡀⠀⢀
⠀⠀⠀⠀⠀⢠⠠⠂⡄⠀⠀⢀⠀⠀⠀⠠⠉⢁⡶⠂⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠰⠈⠆⠀⠀⠐⢀⠂⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⡰⠂⢃⠀⠀⠈⣀⠀⠀⠐⢄⠘⠋⠀⠀⠀⠀⠀⠂⠀⠀⢱⠀⠒⠀⠀⠸⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠞⠀⣴⠁⠀⠀⠀⠈⠓⢄⠀⠀⢀⡤⠂⠀⠀⠀⠀⡀⢁⢠⡀⠄⢀⠀⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠁⠠⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠉⠈⠀⠠⠀⠂⢰⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠘

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣩⣿⣿⣿⣿⠀⠘⢲⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⡇⣿⣿⣿⣿⣿⠀⠀⠈⡇⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢈⠇⣿⣿⣿⣿⣿⠀⠀⠀⣇⢿⡿
⣿⣏⠛⠿⣿⣿⣟⣩⢍⣙⣛⣛⠛⠃⠈⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣷⣶⣤⠤⠶⣶⡲⢄⠀⢀⣀⣤⠆
⣿⣿⣿⣷⣶⣤⣌⡙⠀⠀⠉⠛⠷⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣥⣴⣾⣿⣿⡴⠾⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⢠⣾⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣶⣤⣀⠀⢹⣿⣿⣿⠀⠰⡿⠿⠛⠁⢻⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣀⣤⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡶⠟⠛⠉⠁⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⡿⣿⣯⡻⢿⠿⠿⠿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣎
⠀⠀⣠⣴⣾⠿⠟⠛⠛⠻⣿⣿⣿⠿⢟⣱⣿⣿⣿⣶⠶⢿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣷⣄
⠀⣾⡟⠉⠀⠀⠀⠀⠀⠀⢹⣿⣶⠛⠉⠛⣿⣿⣿⡁⣀⣀⡈⢻⣧⠀⢠⡶⠛⠉⠀⢈⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣇⡀⠀⢠⣿⣿⣷⣮⡝⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠙⠻⠶⡶⠶⠶⠶⠖⠶⠦⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢛⣳⣾⣟⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣴⣿⣿⣿⣿⣷⣦⡸⣿⣿⣿⣿⣿⣷⣶⣝⣿⣿⣿⣿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣽⣵⣾⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⡟⠛⠉⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠛⠛⠋⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⢸⣿⠟⢁⣀⣠⣤⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠻⣿⣇⠀⢀⣤⣴⣶⣦⣤⣿⣿⣿⡏⢹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⣾⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⢀⣠⣤⣤⣜⣿⣶⣿⣯⣶⣶⣿⣿⣿⢸⣿⡇⠀⠻⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠉⠀⠀⠉⣿⠻⣿⣿⣿⣿⣿⣿⠏⢸⣿⡇⠀⠀⠈⠛⠿⠿⠿⠟⢛⣿⣿⡿⠋⠀⠻⣧⣀⣀⣠⡴⠶⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠈⠛⠛⠛⠛⠁⠀⣼⣿⣿⡀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣤⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉