Writ Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣄⠁⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠃⣧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠸⢓⠉⠃⠀⣀
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢛⠒⠀⢈⠈⠃⠛⠋⠙⠛⠛⠃⠈⠉⠉⠋⠉⠁⠉⠉⠈⠙⠋⠉⠈⠀⢀⠂⠀⠀⠀⠀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⠀⠀⠀⢢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⣦⡀⠉⠀⠀⠀⠈⠓⠃⠉⠀⠀⠀⡠⣷⠄⠀⠀⠀⠆⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⣟⡂⡤⣀⣦⡶⣀⢀⣁⣤⠔⢻⠿⠁⠀⠀⠀⠀⠀⡀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣇⣉⣉⣀⣨⣬⠼⠞⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⣤⠀⢠⠄⢠⡤⠀⡤⠀⣤⠀⣤⡄⢠⣤⢀⣤⠀⠤⠤⠤⠄⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⣤⡄⠠⠤⢠⣤⢠⣤⡅⡤⠄⣤⡄⢠⣤⢀⣤⡄⣤⣄⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⡄⣠⣄⠠⢤⠀⠴⠆⢶⡄⠠⠤⡠⣧⠄⠤⠄⢠⡤⠠⣴⠀⢠⡌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠀⢤⡄⢠⡄⠠⢤⣆⢰⡧⢠⣤⠀⢤⠄⠠⣴⠤⠤⡦⠤⢤⠤⣄⡤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⣀⣠⠤⢤⣤⠄⠙⠤⣤⣤⡤⠼⠀⢤⣤⢤⣥⣇⣲⣴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠙⢉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠟⠉⠻⣷⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⢿⣇⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⣾⡿⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⡀⣿⣽⡇
⠀⠀⠀⠀⠀⣼⣿⡇⣿⠶⠶⠶⢶⣿⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⢾⣿⠶⢾⣷⠘⣿⠁
⠀⠀⠀⠀⠀⢻⣿⡇⣿⠀⣠⣤⣤⣿⡀⣰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄⢀⣤⣿⣤⣼⣿⠛⣿
⠀⠀⠀⠀⠀⠀⢿⣧⡿⢸⡏⠁⠈⠉⢻⣿⣿⣦⣿⡿⠛⣻⡿⣿⠟⣿⡟⢿⡟⢿⣟⠛⣿⣷⣼⣿⣿⠟⠉⠀⠀⢹⣿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡍⠛⠉⣠⡾⠋⣰⡟⠀⣿⡇⠘⣿⡀⠙⣷⡌⠙⠛⣹⡏⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣷⣤⡾⠋⠀⢠⡿⠀⠀⣿⡇⠀⠸⣧⠀⠈⠻⣦⣰⡟⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣧⡀⢀⣾⠃⠀⠀⣿⡇⠀⠀⢻⣆⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣯⣀⣀⠀⣻⡇⢀⣀⣠⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢠⡴⣄⠀⡤⢦⠀⣠⢦⡄⠀⣠⢦⡀⢠⠴⣄⠀⠀⡴⢤⠀⠀⣠⠦⡀⢀⡴⣤⠀⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⢓⣋⡀⠙⣚⡀⢘⣚⠁⠀⢙⣚⠁⠈⣛⡃⠀⢀⣙⡋⠀⢀⣘⡚⢁⣀⡓⠋⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠸⠤⠇⠘⠧⠇⠸⠤⠃⠀⠻⠼⠃⠘⠤⠇⠀⠸⠤⠇⠀⠘⠤⠇⠸⠤⠇⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢼⣹⠆⢾⣹⠄⣎⡳⠀⢾⣹⠆⢰⣋⡦⠀⣞⣳⠀⠰⣏⡷⠰⣋⡦⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢁⡴⣦⢡⢶⡄⡴⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⢦⢠⠶⣄⡴⣦⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣉⣁⣈⣉⣀⣉⣛⣛⣛⣛⣛⣛⣉⣛⣛⣛⣋⣈⣉⣀⣉⣡⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡀⠀⠈⠉⠚⠠⠤⡿⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣰⡄⢸⣴⣾⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠸⡟⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣐⣷⠀⡀⢻⣿⣿⡿⠟⠛⣿⠅
⠀⠀⠀⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠢⢟⡉⠹⠗⠛⠉⠉⠈⠙⠃⠘⠛⠚⠋⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣩⣿⣿⣷⣄⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣼⢿⣿⣿⣟
⠀⠀⠀⢀⣿⣿⣻⣻⣿⣿⡶⠒⠀⠀⠸⣀⠔⠊⠁⠀⠀⠀⠠⠄⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢿⠆⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⠀⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠀⠀⠰⣶⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⡀⠀⠀⣙⡛⠛⠿⡏
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠻⠿⠻⣿⠿⠿⠿⠻⠿⠿⠿⣿⢿⡛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⢻⡇⢀⣤⠀⣤⡄⠀⢻⣧⣤⡀⢀⣤⣤⣤⡀⠀⣠⣤⡀⠀⣠⣤⣤⡀⣠⣤⣤⣄⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⢻⡇⠀⢿⠀⢹⡇⠀⢸⣏⠀⣿⠀⢹⡏⠈⣿⢸⡏⠉⢹⠀⣿⠗⢻⡃⠈⣿⠀⢸⡆⢈⣭⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠞⠁⠀⠘⠷⠟⠷⠀⠘⠿⠶⠋⠀⣸⣿⠶⠋⠈⠻⠶⠟⠀⠙⠷⠟⠁⠾⠿⠶⠾⠗⠘⠷⠞⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠉⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠉⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣧
⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠋⠀⠀⠀⣿⡇⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣇⣀⣸⣿⣿⡟⠛⠛⢛⣿⣟⣀⣀⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠉
⣿⢿⠿⠿⡿⠿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣟⣿⣿⣿⣻⣿⣟⣻⣿

⠀⠀⠀⠀⠤⠀⠀⠤⠤⠤⠤⠀⠠⠤⠤⠤⠄⠀⠤⠤⠤⠤⠀⠠⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠀⠤⠤⠤⠤⠄⠀⠤⠤⠤⠤⠀⠠⠤⠤
⠀⢀⠀⣀⢀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀
⠀⠀⢤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⡴⡄
⠀⠀⠈⠟⡿⣻⠀⠀⣴⣶⣶⣶⣶⣶⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⣶⣶⣶⣶⣶⡆⠀⠘⡟⢛⡿
⠀⠀⠀⡇⡇⣿⠀⠀⢻⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇⠀⠀⡇⢸
⠀⠀⠀⡇⡇⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇⠀⠀⡟⢸
⠀⠐⠀⡇⡇⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⡇⠀⠀⡃⢸
⠀⠀⠀⡅⡇⣿⠀⠀⠘⠛⠛⠛⠛⠋⠀⠙⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠃⠀⠀⡇⢸
⠀⠀⠀⡆⡇⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⡉⠙⠂⠀⠀⡇⢸
⠀⠀⠀⡇⡇⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡅⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⡇⢸
⠀⢀⠀⡇⡇⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⣧⣸
⠀⠈⠀⡇⡇⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠚⣿⣿⣿⣿⣿⣿⣿⣿⠋⠽⠿⢀⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⡧⢹
⠀⠀⠀⡧⡇⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠷⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠘⠂⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⡗⢸
⠀⠀⠀⡇⡇⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⡷⢸⢰
⠀⠀⠀⡅⡇⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⡇⢸⢀
⠀⠰⠀⡀⡷⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⡇⢸⢸
⠀⠀⠀⢀⡟⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⠀⠀⠀⠀⠐⡀⠜⠊⠉⠉⠉⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⣟⢸⠸
⠀⠐⠀⢘⡇⣷⠀⠀⢸⣿⣿⣿⣿⠿⠟⠛⢋⣁⡀⢂⠀⠀⠂⡜⠁⠀⠐⠀⠐⠂⠀⠀⠈⢄⠀⠀⠀⢐⠈⣍⢘⠛⠿⠿⣿⣿⣿⣿⠀⡇⠀⠀⣿⢸⢘
⠀⠀⠀⢈⡏⣿⠀⠀⢸⣿⣿⣿⡇⢰⣿⣿⡿⠟⠛⠈⡀⠀⢸⡃⠀⠀⠀⣄⠀⠀⠀⠀⠀⢸⠀⠀⢀⡟⠛⣿⣿⣿⣿⣮⢰⣿⣿⣿⠀⡇⠀⠀⣿⢸⢸
⠀⢀⠀⣸⢇⣷⡀⠀⢸⣿⣿⣿⣟⡈⡁⠘⠋⠀⠀⠀⣥⠀⠈⢇⠠⠀⠀⣤⢤⠆⠀⠀⠀⡌⠀⠀⢸⠆⢠⠛⠓⣟⣠⣅⣾⣿⣿⣿⠀⡇⠀⠀⠿⠸⢸
⠀⢠⣸⢷⣿⠶⡧⣀⣈⡉⠉⢉⡅⢠⡀⠀⠀⣠⡷⠀⠉⡒⠄⠈⠣⢀⠀⠁⠀⠀⠀⡠⠊⠐⠁⢁⠛⠃⠀⠡⠬⠦⠛⠈⢹⣿⣿⣿⠀⡆⠀⠞⠺⠻⠺⠤
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡸⡇⠀⠀⣀⡀⢤⡤⣯⠀⣀⣀⠀⡉⠁⠈⣈⢁⡀⣀⣀⢀⡀⣬⣛⡓⠶⠤⣤⠀⣇⠘⠿⠿⠿⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠋⠔⠑⠒⠉⠁⠀⠀⠀⠀⠔⠰⠁⠀⠀⠀⠀⠀⠀⠀⠘⢌⢂⡃⡃⠀⠀⠀⠀⠒⢶⠑⣆⠃⠀⠀⠀⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣼⣃⣃⣀⣀⣀⣀⣀⣀⣑⣌⡀⣀⣀⣀⣇⡨⠀⠀⠀⠀⠀⣀
⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⠄⢶⣶⣶⣶⣶⣶⣶⣶⡞⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⣶⣤⡶⢶⣆⣤⣤⠤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⣴⣧⣀⣤⡀⢸⣿⢿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡻⠃⠀⠀⠀⠀⠀⠀⢀⡏⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠔⠛⠛⠿⣭⢿⣷⣌⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⡟⡃⠀⠀⠀⠀⠀⠐⡸⠁⢾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠋⢉⣿⣧⡿⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠃⠀⠀⠀⠀⠀⠀⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⣠⣶⠿⣾⣿⣿⡀⡲⠦⠤⠌⠙⠿⢿⢿⣿⡉⢈⣹⠿⣿⡻⡧⠀⠀⠀⠀⠀⠀⠅⠀⠘⣏⠙⠛⠻⣿⠿⠿⠃⢰
⠀⠀⠀⠀⠠⠛⠿⠋⠙⣾⣿⣿⣿⣿⣶⣶⡖⠂⠤⢄⡺⣷⣿⣄⣁⡽⠡⣿⣧⠀⠀⠀⠀⠀⠀⢁⣀⣈⣝⣃⡀⢸⡿⠆⠄⠀⢠
⠀⠀⠀⠀⢀⠟⠀⠀⠈⠹⠿⣿⣙⣿⣿⣿⣿⠀⠀⠈⢻⡉⣵⡎⢙⠂⠰⠃⡅⠀⠀⠀⠀⣠⣾⣿⣿⠿⣩⣿⣿⣿⣷⣤⡠⢧⣲
⠀⠀⠀⠀⢘⣇⣀⠀⠀⡀⠠⢘⡏⠻⣿⣿⣿⡅⢠⡄⠀⠙⠿⠟⠦⠠⣴⠤⠀⠀⢰⣿⠙⢻⣿⣿⣿⢾⣼⣿⣿⣿⣿⣿⣿⡏⢸
⠀⠀⠀⠀⠻⢿⣿⣾⣿⣷⠖⢿⣧⣽⣿⡿⠿⡇⣾⡿⢷⣀⠀⡴⢶⠖⠻⣞⡀⠀⣿⣿⣄⣸⣿⡿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠠⣴⣿⣿⣿⠏⣰⣿⣿⣿⠟⠁⠀⢹⣿⢁⣶⡴⠷⢀⣾⠠⠖⣴⣢⣼⣿⣿⣿⣿⡿⠠⠀⣿⣿⣿⣿⣿⣿⣿⣿⠈⠇
⠀⠀⠀⠀⠐⣿⡿⣿⣧⠼⣻⣿⣿⣿⣷⣧⣴⣿⣿⣶⣶⣷⣶⣾⡋⠄⣒⣀⣿⣿⣿⣿⣿⣿⡧⣤⠠⣿⣿⣿⣿⣿⣿⣿⠇⠚
⠀⠀⠀⠐⠀⠙⣀⣤⢄⠀⠀⣹⢆⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⡆⣩⣉⣍⣭⣙⢿⣿⣻⣼⣿⠷⠊⢠⣿⣿⣿⣿⣿⣿⠏⣀⣤
⠀⠀⠀⠀⠀⠠⢿⡿⢙⡿⠰⣯⣸⣿⡏⢤⣽⣿⣿⣿⣟⣽⣿⣇⣠⠌⢉⠙⠿⣼⣿⡿⣿⡿⠒⠊⢻⣿⣿⣿⣿⣿⣏⢀⢼⡯⢆
⠀⠀⠀⢄⣀⣀⣦⣤⣾⣷⣶⣽⣿⣿⣿⣿⣿⣿⠋⠉⠉⢀⣿⣿⠿⡶⣯⣡⠚⠌⠉⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡎⣼⣿⣏
⠀⠀⠀⢬⣟⣿⠿⢻⣿⡿⢿⣿⣿⠿⣿⠟⣛⣛⣠⡿⠛⣿⣿⣿⣀⣀⡀⢸⣇⣤⣤⣴⣶⣄⣀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠴⣻⡏
⠀⠀⠀⠀⡏⠙⠻⣿⣿⣿⣿⣿⣿⣟⢿⣷⡿⣿⣿⣷⣾⣿⣿⣿⣧⣼⣇⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢪⠜⠃
⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣧⣍⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⣺⣧
⠀⠀⠀⠐⡆⠋⡀⣻⣿⣯⣍⣉⣁⣀⣠⣬⣤⣤⣿⣿⣿⡿⠿⣿⣿⡻⢿⣟⣿⣿⣿⣿⣿⡏⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡎⢹⡷
⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣧⣀⣿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⡇
⠀⠀⠀⢠⡏⠀⠉⢿⣿⡿⣻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡛⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣙⠇
⠀⠀⠀⣸⠀⠀⢀⡷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⣴⡏⠉⡉⠻⣿⡿⠟⠛⠑⠿⠿⣿⣇
⠀⠀⠀⣿⠀⣠⣿⡄⠙⡿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠂⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠛⡏⢀⡔⣿⣿⡿⠶⢻⠀⣤⡾⣷⣿⡆
⠀⠀⠀⣿⠰⣿⣿⡿⣦⣧⡄⢚⣻⡟⠉⠁⢤⡀⣠⣤⣸⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⠁⢙⡀⣿⣿⠀⢀⣿⡘⠛⠿⣿⣿⡄
⠀⠀⠀⢹⠐⣿⣿⣷⣀⣿⢡⣿⡿⣿⠶⢶⣻⡟⣿⣿⣿⣿⣷⢺⣿⣿⣿⣿⣿⡻⣿⣿⣿⡏⠀⠀⠁⢻⣿⣿⣿⠉⢻⣦⠀⠀⠃
⠀⠀⠀⠸⣼⣿⣿⢿⣿⣿⣿⡿⠿⠧⣴⣾⡟⠛⢛⣿⣿⣩⣿⣟⣿⣿⣿⣿⣿⡿⣿⣿⣿⠁⠀⠀⠀⣾⣿⣣⡏⠀⠀⠑⠂⠀⠠
⠀⠀⠀⠀⢿⣿⣿⣾⣽⢟⠙⣳⣶⣿⠿⠿⣿⣿⠛⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⡇⢿⣿⣿⣀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⣼⣾⣿⡿⢿⣿⣾⠀⣅⣿⠉⠚⠋⣻⣿⣉⣿⣿⢿⣿⣿⣿⣿⠛⢿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡄
⠀⠀⠀⠹⠿⣿⣿⣾⣿⣟⣧⢸⣷⠀⢰⡆⡘⢨⣿⣏⣿⣿⠟⢿⣿⣿⣷⣄⣿⣿⣿⡿⠋⠀⠈⠉⠻⢿⣿⣿⣿⡄
⠀⠀⠀⠢⣶⣿⣿⣿⣿⣿⣿⣾⣯⣴⣾⢀⣻⣾⣯⣿⣿⣿⢄⠀⠉⠙⠉⠁⠉⠉⠡⠀⠀⠀⠀⢀⡀⠀⠙⠛⠋⠁
⠀⠀⠀⠀⣤⠉⠻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⢀⣠⣴⡶⢶⡄⠀⣀⣤⠤⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠘⠋⠀⠐⠚⠁⠘⠳⢦⠤
⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⣾⣿⡽⠿⢗⣏⠙⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⢿⣷⢜⠀⠀⢀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣬⣶⣿⣴⣇⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢩⡿⠿⠟⢛⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⣃⠐⠢⢀⣀⠀⠈⠛⠛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣍⣿⡄⠹⣦⡄⠀⠰⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⡴⠂⠀⢀⣴⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⢢⣤⣴⣿⠟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠉⠼⠸⠃⠀⠀⠠⠀⠀⠀⠀⠠⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⢬⠜⠃⢰⡇⠀⠀⠇⠀⠄⠀⠀⠀⠀⠀⢁⡔⠂⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠋⠘⠁⣼⣧⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡩⠗⠡⢐⣿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⣠⡿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣓⢀⣩⡾⡏⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⢰⣹⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠾⣽⣿⡿⢷⣾⡃⠀⠀⡀⢠⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⠿⠿⣷⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣼⣿⣛⢡⣾⣷⢱⡶⣤⣇⣺⠃⢘⣀⡀
⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡜⠀⠀⠀⠀⠁⠨⣿⣿⣷⣶⣶⣶⣤⣶⣾⣿⠳⣞⣿⣁⢄⣿⣏⢸⠃⠹⣟⢿⣴⠟⠀⢈⡢
⠀⠀⠻⢷⣄⣀⣀⣠⠀⠀⠀⠀⠄⣤⡀⠀⠀⡀⢨⠀⠀⣍⣹⣿⠛⡿⠿⣿⡿⠛⢿⣴⣿⣯⣵⣾⣿⣿⣿⣾⣾⠵⠛⠋⡔⠈⡈⣿⡆
⠀⠀⢀⡰⣾⣿⠤⣿⡶⠤⠄⢁⡐⠛⠓⠀⠒⠂⠚⠓⠀⠚⢢⣬⠖⠄⠂⠀⠀⠀⠀⠈⠉⠀⠉⠟⠛⠛⠿⣋⣁⠀⠤⠤⠤⢀⠈⠛⢿⣀
⠀⠉⠉⠙⠽⠿⠿⠛⠚⠛⠀⠉⠅⠀⠠⠄⠀⠀⠀⠴⠒⠢⠾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠁⢀⣠⡀⠹⡷⠄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣶⣶⣤⣶⣶⠀⠀⢀⠀⣠⠀⠁⠀⠀⠀⠐⢁⣀⣡⣶⣿⣿⣇⣀⣙⣈
⠀⠀⠀⠀⠉⠀⠀⠀⠶⠢⠤⠤⠄⠄⣀⠀⠀⡀⢀⣀⣀⠀⣀⠤⠒⠉⡇⠉⠈⢿⣧⣽⠷⣶⣤⣤⠴⠟⠖⠒⠒⠒⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠐⠲⠦⢀⣄⣤⣀⡀⢀⣡⡄⣀⣼⠇⠀⠀⠀⠀⠀⠈⠉⠙⢃⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣘⣰⠻⠶⠶⠶⠶⢶⠶⠰⢾⢤⡬⣤⡝⢠⡤⠴⠒⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⢍⣠⠄⠈⠉⠉⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠴⠖⠛⠉⠉⠉⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡴⠋⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣀⣤⣤⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠸⠃⠀⡏⠀⣇⠘⣇⠀⠀⠀⠀⠀⠀⣠⠖⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⡀⢹⡀⠀⠀⠀⠀⠀⠁⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢧⣠⠥⣄⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠙⡞⠁⠀⠀⢹⣀⠀⠀⠀⠘⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢙⠶⠤⢒⠉⠈⠙⠲⢤⡀⢳⠘⠀⠀⣴⣄⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢅⠐⡆⠘⡆⠀⠀⠀⠀⠀⠈⢧⡀⠀⠻⣿⡆⠀⠙⠟⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢻⠀⢻⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣊⣉⡽⢂⠴⠉⢶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣠⡤⠒⠋⢁⣀⣠⠴⠚⠁⠀⠀⠈⢣⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡆⠈⠀⢹⡘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢷⠀⢧⠈⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠲⠜⡙⠀⣀⣠⠤⠖⠋⠉⠉⠛⠒⠲⠦⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠠⠤⠤⠤⠤⣄⣀⣤⠤⠴⠶⠶⠶⠶⠶⠎⠳⣄⡗⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⣒⣒⣤⣠⣤⠀⠀⠀⠀⠙⢀⣀⣠⡤⠤⠶⠒⠋⠉
⠀⠀⣠⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠀⢈⠟⢛⠟⠭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣀
⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠄⠒⣡⠔⠁
⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠤⠴⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣄⡀⠀⠀⠀⠀⢀⠤⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⠞⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠒⠶⣄⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠔⢚⡳⠖⠒⠋⠉⠁⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡴⠁⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡞⠄⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣶⡄⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠃⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⢴⡶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠔⣩⣴⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⢠⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢵⡄⢙⡏⠀⡀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠖⢻⠒⠊⠀⢰⠃⠀⠀⠀⠈⠉⠉⠛⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢸⠁⠀⠀⠀⠀⣀⡤⠖⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠏⠀⣀⠤⠖⠉⠁⠀⢀⡇⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠖⠊⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⠇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⣲⠊⠁⠙⣦⠘
⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠃⠀⣿⣿⣻⠀⠀⣀⠀⠀⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠑⠀⠀⠀⠄⠀⠀⠀⣸⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢿⣿⣿⠂⠀⠃⡻⠀⠀⠀⣾⣿⡟⠁⠀⣴⡆⠀⠀⠀⠀⠁⡀⠀⢠⡀⠀⠣⡀⠀⠀⠀⠀⠁⣥⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠘⠿⠁⠀⢰⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠈⠆⠀⠀⠀⠈⠐⠀⠻
⠁⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠄⣦
⠈⠀⠀⠉⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢺⢿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣹⡈⣿⣿⡄
⠀⠀⡀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⡄⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⣿⣧⢸⣿⣷
⠀⠀⠀⠀⠀⠀⢿⠋⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⠃⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⣰⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢿⣿⡄⢻⣿⣇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠠⢸⠇⢠⣿⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⣿⣷⠈⢿⣿
⡆⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢀⠛⠻⠟⠀⠀⣀⠀⠀⠀⠀⠀⣸⣯⣸⠄⣤⡄⢻⣿⡇⠸⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡱⠀⠀⢀⣌⡛⢦⡀⠀⠀⠙⠃⣿⠐⠝⠀⠈⢿⣿⣿⣿
⣿⣥⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡜⠠⡄⠁⠀⠀⠉⠀⠉⢦⠁⠀⠀⠼⠀⡿⣿⠀⠀⠀⠘⣿⣿⣿
⣿⣿⡆⠀⣿⠀⠀⠀⣄⢰⠇⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠉⠉⠁⠀⢸⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠰⣿⡿⠀⠀⠀⠀⢻⣿⣿
⡘⣾⣧⡄⠫⠀⠀⠀⠎⠃⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠃⠀⠀⠀⠀⠈⣿⣿
⠁⠙⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠹⣿
⠀⢸⣷⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⠈⣀⡀⠀⠀⠀⠀⠘
⣰⣿⣿⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠀⢀⠻⠟⣱⣿⣧⠉
⣿⣿⡟⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣷⠙⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⢀⣾
⣿⣿⠇⠀⠀⠀⢀⠠⠀⠀⣠⣤⣴⣶⣶⣶⠆⠀⠀⠀⣼⣿⣿⡏⠀⣴⡦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⢀⠀⣿⣟⡟⠀⠀⠀⠀⢠⣿⣿
⣿⣷⠀⠀⠀⠚⠁⠀⠐⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⢠⣿⣿⡿⠀⣸⣿⣿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠸⢱⡖⠀⠀⠐⢿⣿⠋⠀⢠⣿⣤⣾⣿⠁⠀⠀⠀⠀⠟⢿⠿
⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠰⠿⣷⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸⠋⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠛⠀⣿⡿
⣿⡇⢀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢰⠁⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⣿⠀⠀⢀⣴⠆⠀⢀⣴⡇
⣿⡇⠘⠁⠀⠀⠐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢡⠇⣐⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⣠⣿⠋⠀⠀⠈⢿⡇
⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢃⡟⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠏⠀⠠⠀⢤⡀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⢀⣿⣿⣟⣾⡇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢸⣿⣿⡇⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠃⢸⣿⣿⡇⠀⠀⠀⠀⠠⠃⠀⠀⠀⣠⣠⡄⠀⠀⡀⠑⢄⠀⠀⠠⡀⠈⠂⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠙⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠀⠀⠘⠿⠿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⠀⠀⠀⠀⢰⠛⠋⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡨⠢⠀⢸⠂⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⡀⣤⡀⠀⢀⡞⢀⠂⠀⢀⡉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⢇⣿⠁⠀⠈⠀⢸⢀⡴⠃⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⢨⣾⡇⠀⠀⢰⠂⠉⡟⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⢸⣿⠇⠀⡠⠀⠀⣰⡿⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢴⣷⣸⡏⢠⠞⠀⣤⣦⠈⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⠇⠁⠈⠙⣦⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠉⢿⣿⡿⠟⠋⠀⠀⠀⠈⠈⠉⠛⠿⢿⣇⡀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢀⡠⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠙⡉⠂⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⣄⡐⠬⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠔⠁

⠀⠀⠀⠀⠀⠀⢠⠶⣤⠀⠀⠀⠀⢠⣄⡀⠀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⠓⢯⠈⣿⢸⡇⠀⣟⢙⡆⢻⡋⢳⢼⠋⣳⢸⡿⢿⠘⡟⢹⡆⣸⣽⣇
⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠙⠛⠋⠀⠛⠋⠀⣼⡟⠁⠈⠛⠃⠈⠛⠋⠘⠛⠛⠃⠘⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠈⢿⣿⣆
⣀⣀⣀⣀⣠⣤⣾⣿⣿⣤⣤⣤⣤⣤⣾⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣶⣤⣤⣤⣤⣼⣿⣿⣦⣤⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠉⣿⡏⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣤⣤⣤⣤⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠉⠙⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠏⠤⠤⠀⠠⡬⠁⠀⠀⠀⠀⠀⠀⠀⠇⢤⢌⣁⣤⣤⡑⠠⡀⠀⠀⠀⠈⢿⣭⢭⣍⠙⠏⢻⡦
⠀⠀⠀⠀⠀⢎⢵⣈⢀⣀⠀⣗⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣯⢳⠃⢂⣀⠀⠃⠀⠀⠀⠀⠉⠑⡿⠗⠂⡀⠉
⠀⠀⠀⠀⠀⠙⠆⢑⡛⠚⡗⢡⠃⠀⠀⠀⠀⠀⠀⣀⢉⠀⠛⠛⠓⣈⢀⣢⡄⠀⠀⠀⠀⠘⠞⣿⠾⠿⠯⣀
⠀⠀⠀⠀⠀⠀⢉⣞⣶⣣⣸⣿⠄⠀⠀⠀⠀⠀⠀⠉⠉⢰⠰⠻⠤⢶⡤⠀⠀⠀⠀⠀⠀⣛⠂⠿⠿⠛⠘⣡⡄
⠀⠀⠀⠀⠀⠈⠀⠙⡿⢿⠿⣏⠏⠀⠀⠀⠀⠀⠀⠀⠠⠘⢠⢢⣆⡡⡃⡀⠀⠀⠀⠀⠀⠙⣾⡛⠶⠶⣊⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢴⡂⢠⣀⣀⣀
⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⣼⡀⠠⣀⣠⣦⣡⠀⠀⠀⠀⣀⠠⠟⠝⠶⠺⠌⠃⡀
⠀⠀⢄⠀⠸⠆⣀⠜⠀⠠⠒⠀⠼⡀⣀⠔⠀⠀⠘⡆⢸⠘⢾⠿⠐⢆⠰⠹⠀⠀⠀⠀⢻⢤⠀⣿⡃⣥⠠⡔⠃
⠀⠀⠈⢃⢲⣤⠈⣶⣒⣂⣋⡒⠶⠆⡌⠀⠀⠀⢶⢬⣥⠴⠄⠶⠶⠶⠲⠲⠋⠀⠀⠀⠈⠸⠤⠛⠀⠿⠰⠇
⠀⠀⠀⠀⠻⠞⠛⠙⠉⠉⠉⠛⠛⠋⠀⠀⠀⠀⠰⡄⣆⣀⣀⣈⣌⣬⢭⠉⢽⠀⠀⠀⢰⠰⣀⡤⣥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠿⠓⠛⠚⠛⠛⠛⠛⠁⠀⠀⠘⠰⡿⠿⠔⠻⠇⢸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡇⠀⢸⣿⣿⠃⠀⣀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠂⠀⠀⠂⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⣿⣿⡇⠀⠛⣹⠃⠀⠀⣼⣿⡿⠃⠀⢠⣆⠀⠀⠀⠀⠈⢀⠀⠁⣀⠀⠐⢄⠀⠀⠀⠀⠈⣨⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠃⠀⠀⠀⠈⠿⠃⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠳⠀⠀⠀⠀⠁⠂⠼⠇
⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⠀⢐⡀
⠈⠁⠀⠉⠀⠀⠐⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠐⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢉⡄⢻⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣄⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⠃⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⢀⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣧⠸⣿⣷
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠄⠀⠀⠀⠍⡟⢀⣼⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢻⣿⡆⠹⣿
⡆⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠼⠛⠿⠃⠀⢀⠀⠀⠀⠀⠀⢀⣿⢅⡅⢠⣤⠘⣿⣷⠀⢿
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢈⠆⠀⠀⣈⡓⢦⡀⠀⠀⠘⠛⢸⡇⠸⠏⠀⠹⣿⣿⣿
⣿⣇⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃⣥⠈⠀⠀⠈⠁⠉⠢⡉⠀⠀⢠⠆⢸⢿⡆⠀⠀⠀⢿⣿⣿
⣿⡿⡄⠀⣾⠀⠀⠀⣄⢠⡆⠀⢀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡄⢀⠏⠀⠀⠀⠀⠀⠂⠤⠁⠀⢀⣦⠀⢿⣾⡇⠀⠀⠀⠸⣿⣿
⠸⣾⣧⡀⢯⠀⠀⠀⠈⠋⠀⠀⠈⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠻⠀⠀⠀⠀⠀⢻⣿
⠃⠙⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⢰⣷⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⡀⢁
⣠⣿⣿⠏⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⠿⢋⣼⣷⡈⠁
⣿⣿⡿⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡜⠃⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⣰
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡤⠀⠀⠀⢸⣿⣿⡿⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃⠀⠀⢹⣯⢿⠇⠀⠀⠀⠀⣴⣿
⣿⣯⠀⠀⠀⠲⠋⠀⠠⠶⠿⠿⠿⠿⠛⠃⠀⠀⠀⠀⣾⣿⣿⠃⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣶⠀⠀⠀⢾⣿⡟⠁⠀⣼⣇⣼⣿⡏⠀⠀⠀⠀⢴⠿⣿
⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢠⣾⣿⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠟⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀⠸⠆⣴⣿
⣿⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢁⠂⠐⣉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⡇⠀⠀⢠⣤⠀⠀⣠⣼
⣿⡇⠘⠃⠀⠀⠠⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⡞⠠⠰⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁⠀⣴⡿⠁⠀⠀⠻⣿
⠍⠑⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⡼⠃⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡴⣀⣀⡀⠀⡀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⣾⣿⡿⣸⣟⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⡇⠀⠀⠀⠀⢀⠐⠁⠀⠀⢀⡀⣀⠀⠀⠀⠐⢄⠀⠀⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣿⣀⣸⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣷⣁⣀⣀⣘⣄⣈⣠⣀⣀⣀⣄⣀⣀⣤⣀⣀⣀⣀⣺⣿⣇⣀⣀⣀⣀⣠
⣿⣿⡟⠛⠿⠛⠻⠻⣟⠟⣿⠿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠈⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠣⡀⠈⢢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢌⠂⣀⡸⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⣀⡸⠈⢃
⠀⠑⠦⢀⣠⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠧⢀⣠⠚⡄
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠤⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠤⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠦⡈⠢⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠦⡈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠁⢀⠜⡢⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠐⠊⢀⠜⡢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠋⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠰⠉⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠮⡔⠋⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡔⠉⢀⠜⡡⠊
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠑⢲⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠐⢒⡡⠊
⠀⠀⠀⠀⠀⠀⠸⠤⠓⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠓⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⢉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡔⢉⢇
⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⢁⡔⠉⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⢁⡔⠉⠢
⠠⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢌⠲⣀⠙⠦⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢌⠲⡄
⡀⠉⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡄⠙⠆⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌
⢌⠲⣀⡸⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠳⣀⠸⠉⢃
⠀⠑⠧⢀⣠⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠧⢀⣠⠚⡄
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠤⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠤⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠦⡘⠢⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠦⡘⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡀⠃⢀⠜⡢⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠃⢀⠜⡢⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡔⠃⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠉⢠⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠮⡔⠊⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⡤⠃⢀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠑⢲⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠐⢲⡡⠊
⠀⠀⠀⠀⠀⠀⠸⠤⠓⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠓⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⢉⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⢉⢇
⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⢁⡔⠉⠢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⢁⡔⠉⠤
⢠⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⠢⡀⠈⢢⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢯⣳⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢠⡿⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢠⠏⢠⠞
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡰⠁
⢠⠞⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠳⣤⠞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⡴⠃⠀⣀⠞
⢸⠂⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠉⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢸⢉⡄⢸⡾⠁⠀⣰⠋
⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⢀⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⢤⢼⠟⣷⠋⠀⢠⡞⣁⡞
⢸⡇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡙⡛⠛⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⡁⢸⡾⠁⠀⣴⢏⠞⠕
⢸⡇⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠤⠤⠤⠤⣤⠏⠀⢀⣼⠁⠘⠃
⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⡶⠃⠀⢠⠟⠽
⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⢀⠞⠀⠀⣠⢻⠀⢀
⢸⡇⢸⣿⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⡃⠀⢀⠞⢀⣿⠀⠈
⢸⡇⢈⢻⠀⠀⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⣠⠊⠀⢈⣁⠀⣤
⢸⡇⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⣠⠞⠁⠀⠀⢿⣨⠀⣿
⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠁⠀⠀⠀⠀⢼⡏⠀⠉
⢸⡇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿
⢸⡇⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣿
⠀⡇⠀⠻⡦⠀⠀⠠⠤⠤⠴⠦⠰⠶⠖⠒⠲⠒⠒⠒⠒⣾⡴⠒⠂⠐⠶⠂⠀⠰⠦⠀⠤⠤⠤⠤⠤⠤⠤⠾⠇⠀⡃
⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞⠁⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠞⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠽⢷⣷⡹⣶⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣉⣽⣦⣀⡀⠈⠙⠻⢷⣶⡄⠀⠳⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠤⠄⠀⠉⠠⣾⠛⠛⠻⢿⡀⣀⠀⠀⠀⠀⠉⠳⢤⡈⠂⡀
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⡀⣀⣼⣯⡀⠀⠀⠀⠀⠛⢿⣶⣰⢄⡄⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠂⠀⠀⠀⠀⠀⠙⠿⣿⠦⢥⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣿⠇⠀⠀⠀⣲⣤⣄⠀⠀⠀⠈⠑⠶⢽⣷⣦⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣴⢶⡀⢀⠋⠻⢷⣦⣰⡠⠀⠀⠀⠘⡿⢿⡷⠖⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣼⡆⡴⣶⣶⠀⠈⠈⠻⣾⣧⣄⣼⠀⣙⣿⣷⣶⣶⣬⣥⣀⣿⣦⡀
⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣶⣽⣻⡿⣦⣤⢘⠉⢻⢿⢿⣿⣾⣿⣿⠿⠛⠛⠋⠁⠀⠈⠻⢦⣀
⠀⠀⠸⠿⠿⢿⢟⣿⣓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⢾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡀
⠀⠀⠀⠈⠉⠉⠁⠉⠀⠉⠛⠻⠿⠿⠻⠿⠿⠿⠟⠿⠓

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡏⠸⣗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣌⠣⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠂⢑⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⣦⡀⠸⢷⡄⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉⠀⠀⠘⢿⣦⡀⠠⡀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⠀⠢⣄⠀⠀⠀⠈⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣹⣷⣄⠙⢷⣦⡄⣀⣀⡀⠹⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢻⣸⣿⣷⡀⠙⢿⣿⣿⣧⠀⠀⢣⡀
⠀⠀⠀⠀⢀⣀⣤⡴⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠿⣦⡀⠹⣿⣮⢷⠀⠀⢳⡀
⠀⠀⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢢⡆⠉⢐⣸⢿⣷⣿⡖⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣹⣶⣬⣳⣤⡀⠀⠀⢀⢀⣀⠀⣀⡀⠀⣀⣀⣀⣐⣘⣉⣠⣉⡻⠦⠀⢏⠀⠀⢙⣿⣿⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣴⣿⣿⠿⠿⣿⡿⠿⢿⣿⣛⣿⣿⣷⣶⡾⠷⠿⣜⢿⣇⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢠⣤⣐⢷⡶⢛⣫⡵⠟⠉⠁⠙⠛⠛⠻⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠭⠟⠦⣤⠞⠳⣌⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣽⠟⣉⣥⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢝⣀⠈⠳⢾⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣡⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁




⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣀⢐⡀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄
⠀⠀⠀⠐⠒⢀⠀⠀⠀⠀⠆⠠⠂⠴⠀⠀⠀⠀⠊⢀⠀⠉⠂⠀⠀⠂⠖⡀⢠⠧⠉⠀⠀⣞⢐⠠⠄⢻⠀⠀⠀⠀⠐⠆⠂⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠰⠄⠀⠀⠠⠄⠀⠈⠀⠄⠁⠄⠀⠉⠀⣤⡿⠿⠿⠿⣶⡄⠀⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⠀⢠⡈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠀⡀⡀⠀⠀⢀⠡⠂⠄⠂⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⢀⢠⡄⣀⣀⠀⠀⢈⣈⡀⠁⠄⠥⠀⠀⠈⠀⠃⠂⠀⠀⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠒⠤⠄⠀⠀⠀⠀⠲⠀⡀⠀⠀⠀⠀⠒⠀⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠴⠤⠴⠦⠴⠀⠀⠀⠰⠖⠠⢠⠀⠀⠀⠀⠤⠈⠤⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠁⠉⠉
⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠸⠨⠍⠙⠀⠀⠀⠀⠸⠋⠒⠀⠀⠀⠀⢠⠂⠀⠀⠶⠀⠀⠀⢀⣤⠀⣄⡀
⠀⠀⠀⠀⣐⢢⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⢀⢳⠞⠀⠀⠀⠀⠀⢠⠀⠂⠆⠀⠀⠀⣶⠉⠛⠓⠖⠀⠀⠘⣽⣿⣿⣟⣭⡇⠀⣀⣾⣠⣆⣴⣶⠄
⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠊⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠉⠓⠓⠒⠁⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠈⠛⠉⠉⠉

⠀⠀⢀⠤⠀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡠⠀⠀⢤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠰⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄
⠀⠀⢶⠄⠀⠈⠶⠆⠀⠀⠀⣙⠈⠀⠀⠀⠀⠀⠉⠃⠈⠋⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣼⣴⣄⠀⠀⠀⠀⠀⢈⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠠⠁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⢴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢲⣀⣀⠀⠠⡤⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠠⠀⠀⠀⠘⢿⠟⠳⣿⣷⣤⠐⠀⠀⠀⠀⠀⠀⢆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢀⣤⣶⣶⣶⣶⣶⣶⣦⣿⣿⣿⣧⣴⡦⠀⠀⠀⢃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠶⡿⣯⠿⠿⠟⠙⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⢂⡴⠀⠀⠀⠀⠀⠀⣀⣤⡦⠀⠉⠀⠁⠀⠀⠀⠀⢠⡘⠀⣲⣀⠂⣳⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠳⠀⠠⠀⣂⣠⠾⢩⣽⣏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⠛⠋⣠⡶⣯⠿⠊⠀⠀⠀⡀⠀⠒⠀⠀⠀⢠⣀⣿⣾⢟⢀⣨⡈⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣷⡾⠁⣤⣞⣿⡏⠁⠀⣀⠀⠚⠀⠀⣠⣼⣶⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⠛⣥⣾⣿⢾⣿⣿⣶⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠺⠿⠃⣴⣿⣿⣁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⠒⢀⣼⣿⣿⡿⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⢖⠐⢦⡄⠀⣼⣿⡿⢿⡶⣻⣚⣿⣿⣿⣯⡙⢿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢣⣾⢿⣧⣾⣇⣿⣇⣿⣿⣿⣿⣟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣾⣿⡏⣿⡁⢹⣿⡟⠈⣿⣿⣿⣿⣯⠙⣻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣯⣽⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣟⣁⣙⣃⣘⣛⣻⣜⣛⣿⣿⣿⣿⣿⣛⣿⣿⣟⣛⣻⣿⣿⣛⣛⣩⣤⣾⣿⣿⣿⣿⣿⣟⣛⣙⣿⣿⣿⣿⣿⣟⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣦⡤⠶⠿⠛⠃⠀⠀⠐⠁
⠀⠀⠁⠊⠙⠲⣄⠀⠀⠀⠀⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠀⢀⡤⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢙⠲⣶⡾⣟⠇⠀⠄⡰⣄⠀⠀⠀⠀⠠⡦⠒⠒⠂⠀⠈⠁⢀⡀⠈⣿⢱⡿⣤⡀⠱⠄⠀⠀⠀⠀⠀⠀⠲⠦⠀⠀⠀⠀⠘⢧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠉⠀⠀⠀⠀⠚⠟⠒⠤⠄⢀⡞⢠⠀⠀⠀⠀⢀⣴⠛⣿⢩⣯⠻⡗⠸⠇⠉⠓⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠉⠁⠀⠀⢀⣴⡶⠀⠉⠛⠛⠚⠉⠓⡇⠀⠀⠀⠀⠀⠀⠄⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⠰⠆⠀⠀⠀⠀⠀⠐⠋⠁⠴⡤⣤⢾⣟⡿⠃⠀⠄⠀⠀⣠⣂⣠⣇⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠇⠈⡳⢦⣠⣔⣀⣀⡴⠀⠚⢉⣯⣤⡗⡼⠋⠀⠀⢀⡔⢂⠀⣯⣵⣿⠿⡀⠀⠀⠀⠀⠀⠈⠀⢠⣄
⠀⠀⠀⠀⠠⢀⠀⠀⣀⣠⠉⠒⠲⠒⠉⠉⢀⡴⣂⣝⠉⠉⠁⠀⠀⠀⣘⣛⠙⠋⣱⣿⣭⣤⣴⡟⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠛⠈⠛⢻⡛⠓⠀⢀⣀⣤⣞⡉⠂⢹⣽⠦⢄⣀⠐⠀⡀⠘⠛⣁⣴⣛⠃⠒⠻⡿⠛⠉⠳⠄⠀⠠⠄⠀⠀⠀⠀⠰⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠛⠉⠀⠀⠀⠀⠀⠀⠛⠶⠿⠷⠤⣄⣠⠞⠉⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠦⠉⠢⠀⠀⠀⠀⢌⡉⠀⢁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠦⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡤⠞
⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡤⠤⠄⠀⠂
⠀⠀⠀⠠⢄⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⡉⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠃⠠⠾⢋⣄⠀⠀⠀⠀⠀⠀⠀⠀⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠘⣤⠘⠁⢶⡀⠀⠀⠀⠀⠀⣴⣶⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠉⠐⠀⠀⢀⡀⠀⠀⠠⡀⠀⠀⠀⣀⠀⠀⠀⠁
⠀⠀⠉⠀⠀⠀⢋⣦⠀⠀⠀⣸⣿⣿⣿⣿⡟⡛⣄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣤⡀⢀⣾⣿⣷⠀⠀⠀⠀⢰⣿⣷⡄
⠀⠀⠉⠀⠀⢠⣾⣿⣇⠀⢠⣿⣿⣿⣿⣿⢁⠷⠋⠑⢀⠀⣠⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡟⠁
⠀⢠⣤⣴⣾⣿⣿⣿⣿⣦⣾⡿⠋⢸⣿⡇⡤⠼⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⢿⣿⣿⠀⠀⢀⣿⣿⣿⣧⡀⠀⠀⠀⡄
⠀⢸⣿⣿⣿⣿⣟⡟⠛⠿⠋⠀⢠⣿⣿⠁⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⠉⠀⠀⣼⣿⣿⣿⣿⡷⠀⢀⣾⡇
⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠿⢿⡏⢄⣐⡇⠀⢀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠉⠁⢀⡾⣾⡇
⠀⠀⠈⠉⠉⠉⠉⠀⠀⠠⠀⡂⠐⠂⠀⠈⠁⠒⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⢀⢎⢷⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⢐⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⢠⠋⡘⣾⣿⡇
⠀⠀⠀⠀⣻⠄⠠⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⣠⠯⡀⣹⣿⣿⡇
⠀⠀⠀⠁⠀⠀⢀⠴⠀⠈⠄⠀⠀⠀⠀⠀⠘⠁⠀⠚⠀⠀⠀⠀⠈⠹⣲⣤⣀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⣰⡓⢆⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢽⠀⢀⢠⡘⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⣿⣿⣿⣿⣶⣤⣀⢀⣿⣿⡟⠀⡰⡟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠀⠉⠈⢁⠉⣦⠙⠂⢀⠘⠛⠀⠤⠚⠛⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣰⢁⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠀⢸⣶⣿⣦⣍⣀⠀⠈⠀⠀⠀⠀⠀⠀⣗⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⢿⣰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢈⠀⣼⣿⣿⣿⣿⠛⣿⣿⣿⣶⣶⣿⣿⣧⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣴⡶⡿⠏⢉⣹⢻⣿⣿⢸⣿⣿⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠙⠛⠓⠓⠀⣽⣿⣿⣿⣿⢸⣻⣿⠀⢺⣿⣿⣿⣿⡇⠀⠉⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⢸⠰⠋⠀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇⠘⠰⠃⠀⠻⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠛⠿⠿⠿⠷⠶⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⣿⠟⠋⠉⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⢸⣿⣿
⠀⠀⢰⣶⣶⠀⣿⣿⡇⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⠀⢸⣿⣿
⠀⠀⠘⠛⠋⠀⣿⣿⡇⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⢸⣿⣿
⠀⠀⢈⣉⣉⠀⣿⣿⡇⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⢸⣿⣿
⠀⠀⢠⣀⣄⠀⣿⣿⡇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⢿⡿⠀⣿⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢠⣤⣤⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠸⠿⠿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣀⣀⣀⣀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠈⢿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⣀⠀⠀⠀⣀⣤⣾⣿⡿⠁
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⣶⣶⡶⠶⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣵⣶⣯⣶⣶⣤⣤⣀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢠⣄⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢿⢿⣶⠄⠐⠒⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠈⢀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠸⢇⠈⠁⠀⠐⠒⢲⠿⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⠨⠅⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣆
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢆
⠀⠀⠀⢠⡏⣀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢳⠄
⠀⠀⠀⠀⢹⠟⠛⠛⠶⣦⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡶⢶⠊⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⡸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠎⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠈⠀⣐⡀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⢼⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠨⠆⠀⠀⠄⠀⠀⠀⠀⠀⠏⠀⠸⠄
⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠈⠒⠂⠤⣤⠄⠘⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣀⣀⣀⠀⠀⠀⠀⠈⠐⠂⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⡀⠀⠀⠀⠘⠓⠒⣂⣉⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠀⣠⡩⠏⢌⠙⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠒⠒⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡩⠛⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⢤⡈⠂⠀⠀⠀⠀⠀⠀⠀⢀⣞⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡁⢸⠀⢧⠀⠀⠀⠀⠀⢀⡴⢋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢇⠘⡆⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡘⡤⠒⠢⣄⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⠐⠲⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⣀⠀⢀⡼⠦⣄⡀⠀⢻⠀⠎⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠋⡍⢱⡀⠀⠀⠉⠓⠘⣧⠀⠀⣿⣧⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⡠⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢳⠀⢧⠀⠀⠀⠀⠀⠈⠳⣄⠈⠛⠀⠀⠀⠀⠀⠀⠀⢀⠴⠾⣤⢀⡴⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣆⠸⡄⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⣀⡠⠔⠊⠉⣉⣩⠕⠋⠀⠈⢫⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⠀⢧⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⠀⢇⠘⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠲⠎⡇⢀⣀⡤⠖⠋⠉⠉⠛⠒⠲⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢙⣦
⠀⠀⠀⠲⠶⠶⠶⣄⡠⠴⠖⠒⠒⠒⠒⠒⠂⠳⣤⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⠭⠤⢤⣶⠤⠀⢀⣀⣘⣀⠤⠤⠖⠒⠚⠉⠁
⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⢀⠜⡩⠋⠙⠒⠶⠶⠶⠤⠦⠤⠤⠤⠤⠄
⠀⠀⠠⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢉⡥⠊
⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠊⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠾⠃⠀⠀⠐⠆⣀⣀⠶⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡎⠀⠀⠀⡄⢀⡴⠀⠀⢀⠀⢠⡲⢦⣌⣶⠛⢖⠀⡖⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⣿⡁⠀⠀⣥⣟⣶⣿⣆⠀⠀⠸⡧⣀⣿⣛⡿⣟⠴⠛⣥⠧⡽⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢹⣿⣧⠀⢡⡜⣿⢿⣿⡏⠀⠀⠀⣷⡼⠎⠉⠣⢰⣂⡈⠀⠘⠧⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠋⠰⠚⣽⣷⣄⠙⠈⠙⠛⠛⠀⣀⠴⠟⠟⠻⡛⠷⣌⢈⠀⡀⠘⡧⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠞⡡⠟⠏⠑⡶⠦⠘⠃⠉⠀⠀⠀⠀⠀⠘⠛⠻⣌⢣⠩⠒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠟⠿⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠻⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠃⠠⠀⠴⠀⠀⠀⠀⢀⠀⢠⠀⠀⢿⡀⠀⠀⣀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠣⠀⠐⠋⠀⠀⠀⢞⠛⠬⠽⡎⠁⠀⠀⠀⠀⢺⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠠⠆⠀⠈⠁⢀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢤⣤⣤⢤⣤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠁⠀⠀⡰⠛⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⣁⠀⠃⢸⢦⡀⠀⠀⠀⠀⠀⢦⣄⣀⠀⠀⠀⣀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢶⠹⡟⣭⣿⠀⠉⠀⠈⠲⣤⡀⠀⡀⠀⠛⠟⠉⠉⠉⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣧⣀⣆⢻⣮⠁⠘⣷⠄⠠⠈⠇⠘⡿⡿⣶⣦⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉⢹⠈⠙⠗⣛⣑⠎⢰⠶⠀⠀⠀⠀⠀⠉⠉⠓⠂⠰⢶⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠐⠄⡁⠈⢱⡍⢀⡀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠘⠁⡞⣤⣇⣁⣅⣀⠁⣀⠈⠡⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⣈⡅⢱⠅⣨⠀⠓⠄⠉⣐⠆⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣁⡻⠀⣤⠠⠖⢁⠽⣏⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠶⠶⠶⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠟⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⡇⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⣴⣶⣶⣶⣶⣶⣶⣶⣦⢹⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⡇⠙⢿⣿⣿⡿⢠⣾⡇⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣧⣴⣿⣿⣿⣟⢈⡛⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡟⠀⠀⠀⢶⣦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣉⣉⣉⣉⣉⣉⣉⣉⣁⣼⣿⣶⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇⠀⠀⣴⠇⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡾⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡟⣡⡟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠏⣴⠏⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⢃⡾⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⡟⣡⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⣰⠏⣴⠏⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⡼⢃⡼⢃⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⣤⣀⠛⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡈⠛⣡⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⢀⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⣤⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⣿⡟⣉⣻⠟⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠉⠉⢬⣻⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣮⣀⣤⣺⣿⣿⣹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠙⠻⠿⠿⢛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠗⠒⢄⠀⢀⣤⣤⡀⠘⠿⠋⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠈⠛⠋⠀⠠⣤⡤⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠤⠠⠀⠀⠀⠀⠀⣼⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⠲⠶⠖⠂⠀⠀⣠⢞⡉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡟⣿⣿⣧⣀⣀⣤⡶⠛⠁⠘⠁⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⢡⠁⠁⠈⡿⠍⠛⢿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⠇⠀⠀⠃⢀⣴⠋⠀⠀⠀⢸⡆⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡾⠁⠀⠐⠂⠀⠀⢹⠀⠰⠃⢀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣸⡇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⣿⡠⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡄⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣛⠳⠣⣤⣾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡆⠣⠀⠀⡶⢦⣦⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠦⣽⡏⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣧⠐⠂⠒⠂⠬⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠂⠐⠲⠄⡀⡀⢀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡠⠃⠀⣤⡈⢠⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣣⡀⠀⣾⠃⠀⠀⠀⢀⠆⠰⢠⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧
⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⠈⠉⠲⠛⠇⠐⠀⠀⣬⡀⠀⠂⣫⡌⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢳⠀⠀⢘⣤⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠸⡏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⣃⣄⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣁⡴⠺⣭⣁⣀⣤⣤⣬⣭⣽⣿⣯⣽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣾⠻⠟⠛⠛⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣴⣮⣤⣤⡴⠶⠶⠿⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠐⠒⠂⠀⠉⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠿⠛⠛⠛⠛⠻⠿⢶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣦⣀⠀⠀⠀⠀⠀⠋⠙⠒⠆⠤⡀
⠀⠀⠀⠀⠀⠀⡠⠎⠁⠀⡆⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠙⠿⣷⣄⠀⠀⠀⠈⢦⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⣿⠋⠀⡀⠀⠀⠀⠀⣀⡀⠤⠀⠠⠐⠶⠦⠤⠀⠤⢄⣀⠀⠀⠀⠉⢿⣦⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠠⠔⢊⣡⣤⣴⣶⣶⣾⡷⠶⣶⣶⣦⡤⣀⡈⠐⠢⣄⠈⢻⣧
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⡠⠞⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⣠⣾⣿⡇⠀⢹⡟⠲⢬⡙⠮⣿⡆
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣠⠔⣾⡀⠀⠙⢿⣷⣶⣶⣾⣶⣶⣾⣿⣟⣿⠞⠀⠀⣿⠃⠀⠀⠈⠓⣿⣧
⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡠⠊⠀⠀⠈⢷⡀⠀⠀⠛⠻⡿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣤⠋⠀⠀⠀⠀⠀⢀⣽
⠀⠀⠉⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠈⠶⣤⣤⡀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠐⠁⢀⠔⣢⠞⢹⡏
⠀⠀⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠀⢸⣧⠙⠲⠦⣄⣀⠀⠀⠀⠈⠙⠋⠾⠷⠶⠶⠶⠖⠚⠋⠁⢠⣀⠠⢊⣠⠞⠁⢀⣿⠃⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠢⠄⠀⠀⠀⢿⣦⠀⠀⠀⠉⠙⠓⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠚⠛⠋⠉⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⠦⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣤⣤⣀⣀⣀⣠⣀⣠⣤⣴⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠔⢒⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠆⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠿⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠠⠤⠤⠤⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⣀⡀⠠⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡣⠔⢣⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠈⠁⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢠⣯⠖⠛⢳⡀⢀⣠⣀⣸
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⣾⡕⠒⠙⠲⣷⡎⠰⢾⡇
⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⠀⠀⠀⡟⣇⠀⣶⡤⠼⠷⠀⢹
⠀⠀⠀⠀⠀⠀⡇⠘⡆⠀⡀⠀⠀⢧⢈⡖⠉⠀⢠⠤⡗⣸⣀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠓⠋⢹⡶⠉⠢⢤⣬⣿⠤⠆⠀⠈⠞⡡⣇⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠟⠓⠦⣤⣤⣤⠤⣴⠒⣋⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⡠⠴⢲⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡀⠀⠀⣀⡤⠶⠚⠛⠋⠁⠀⠀⢸⣤⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠀⡸⢷⠀⢸⡁⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇⠱⡄⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠓⠷⠸⠦⢤⣽⣄⠀⠸⡁⣀⡤⠴⠒⠚⠁⠀⣸⡆⢀⠀⠀⢹⠀⠀⠀⣀⠤⡄⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠠⠤⠤⠴⠶⠤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠳⡹⡖⠒⠒⠒⠋⢉⡽⢫⡟⠃⠀⠈⡇⢉⡽⠇⠀⣿⠉⠉⠉⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢐⣻⣼⡄⠀⢀⡴⠋⠀⠈⠀⠀⠀⣠⠇⢿⣽⡀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⣤⢤⡧⢤⣀⣀⢋⡻⢿⣀⡼⠤⠤⣄⡀⠀⢀⡴⠃⠀⢈⢿⣧⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⠉⠀⠀⠘⣵⠀⠀⠀⠀⢈⡳⠋⠀⠀⢾⣙⣚⣛⡓⠉⡇
⠀⠀⠀⡠⠔⠚⠉⠉⠉⢛⡶⠀⠀⠀⠠⠏⢤⣀⣀⣠⣤⡤⠤⠤⠾⣄⣇⣠⢀⣻⠇⠀⠀⠀⠸⡄⠀⠀⣹⠀⡇
⠀⠐⠛⠒⣲⠖⠂⠀⠤⠞⠓⠦⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢯⣉⣉⣁⣀⡇
⠀⠀⠀⠀⠛⠒⠒⠒⠶⠤⠤⢤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠒⠒⠒⠒⠒⠂⠒⢲⣤⣶⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠀⠀⢀⠔⠉⠀⣠⣀⣀⣀⣄⣤⡶⢿⣻⡽⠾⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢀⠀⠈⠲⢄⣰⠋⣀⡠⠞⢣⣭⡿⣟⣫⡵⠞⣻⠁⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⣀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠉⠠⠞⠁⣀⠄⠀⢰⠏⢠⣾⣶⣿⣿⠿⡛⡹⣇⢨⢎⡄⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀
⠀⠰⣤⡐⠊⠁⠀⠀⠀⠀⢀⣀⡄⠛⠁⣀⠄⠋⢁⡤⠂⠁⢀⠤⠈⠁⢀⣠⣼⡿⢛⡛⠛⠓⠤⢴⣁⣠⣣⡎⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢸⣽⣽⣷
⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠘⠉⠀⠐⠃⠀⡠⠀⠁⠀⠤⠂⠀⡀⠐⠞⠛⠋⣏⠧⣤⣘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠠⠄⠈⢀⡴⠛⠀⡀⠐⠊⠀⠀⠀⠀⠀⠈⠉⠉⠚⠛⠛⠒⠒⠶⠦⢤⣤⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠒⠉⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣖⠒⠚⠛⠒⠲⣦⣿⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⢻⣾⣀⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠗⠓⠐⠐⠁⠃⠛⠑⠘⠘⠀⠘⠋⠚⠂⠛⠀⠞⠀⠺⠊⠘⠊⠛⠁⠂⠀⠚⠙

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣾⠟⠛⣷⡀⠀⠀⢰⡿⠛⢻⣆⠀⠀⠀⣾⠟⠛⣷⡀
⠀⠀⠀⣀⣀⣀⣿⣀⣀⣿⣇⣀⣀⣸⣇⣀⣸⣿⣀⣀⣠⣿⣀⣀⣿⣇⣀⣀
⠀⠀⣾⠟⠉⠉⠉⠉⠉⣿⡏⠉⠉⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⠉⣿⡏⠉⠻⣷
⠀⠀⣿⠆⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿
⠀⠀⣿⡀⠀⠀⠀⠘⠟⠋⠀⠀⠀⠀⠀⠻⠛⠁⠀⠀⠀⠀⠘⠟⠋⠀⠀⠀⣿
⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣾⢷⣦⡀
⠀⠀⣿⠄⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⣿⠀⢀⣴⣿⣅⠀⠈⢻⣷
⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡾⠋⢠⣿⢿⣶⡿⠁
⠀⠀⣿⠁⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⢀⣿⠟⠁⣴⡟⢡⣾⠏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⣠⣾⢋⣴⡟⠁
⠀⠀⣿⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣼⠟⢀⣴⠟⣡⣾⠋
⠀⠀⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣰⡿⠃⣠⡾⢋⣴⠟⠁
⠀⠀⣿⠃⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣏⢀⣼⠟⣡⣿⠋
⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡏⠛⢿⣧⣼⠟⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣡⣴⡾⠟⠁⠀⣿
⠀⠀⣿⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠀⣿
⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣨⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⢞⡵⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣄⣤⣤⡤⠐⣡⣶⡋⢀⣾⢃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠂⣡⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⣉⣴⣿⣿⣿⣿⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣤⣾⡿⢋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣉⣖⣫⣾⣿⡿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣠⢻⡤⠛⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣀⢟⡽⢷⣤⣤⡆⢀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣤⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⢀⢀⠛⠋⠀⠀⠙⠛⠋⠹⠿⢻⠿⠿⠛⠛⠛⠛⣛⠛⠻⢿⣿⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣇⠀⢻⣿⣿⣿⣿⣷
⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⢿⡿⠟⠛⠛⠛⠋⠛⠛⠉⠁⠀⠀⠀⠀⠀⢄⣸⣇⢿⣿⣟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠁⠀⠙⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⣻⣶⠽⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠙⣿⣉⢡⣗⢉⣾⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢆⢛⠇⡾⡳⠸⠝⡟⡋⢭⣝⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⠳⠴⡏⢴⡴⣝⡃⢰⠰⠧⢾⡿⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢛⠉⣨⡔⢋⡗⠩⢁⡴⠧⡰⠫⠌⣵⠀⣼⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠝⡘⠝⠸⠃⠩⣠⠀⠃⢈⠐⣀⠀⠁⡠⠁⠸⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢗⡃⠂⠀⠚⠀⠉⠀⠀⠀⠀⢈⡠⠁⠀⠊⢁⣀⠀⡝⢛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢣⡈⢙⡚⠁⠈⠋⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡌⠀⠐⢀⡉⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⡰⠄⢢⠉⢀⠰⠈⠆⠂⠀⠀⣀⠀⠀⠀⠁⠈⠀⠀⠄⠀⠀⠀⠀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠈⡲⣀⠐⠀⠀⠈⢐⡤⣶⠶⠖⠀⠊⠠⡤⣠⡄⠀⠀⠀⠀⠈⠉⠘⠘⣄⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⠀⠂⠀⠀⠰⢆⣐⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⡷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠏⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠲⣆⠁⢀⠐⣾⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⡡⠀⠨⠀⠀⠀⠀⡀⠘⠛⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⣠⠞⣿⠷⢦⣤⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⣄⠀⠀⢀⡀⣤⡀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢁⡾⠃⠀⠈⠓⣄⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⢸⣶⣵⠴⠛⠉⠀⠈⠙⠳⣤⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣤⠏⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠻⣦⣄⣀⣠⠖⠋⠀⠈⠙⢾⣟⡉⠁⠀⠀⠀⢤⡴⠶⣄⡀⠈⠻⣶⣶⣤⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡰⣣⠞⠁⠀⠰⠂⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⢿⠛⠶⣤⣖⠋⠁⠀⠀⠀⠑⠄⢄⠀⠀⣀⡼⠛⣆
⠀⠀⠀⠀⠀⠀⣸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠆⠤⣄⠙⠻⠷⢦⣄⠀⠀⠀⠀⠰⠌⠻⠷⠶⠟⠦⣄⠀⠀⠀⢉⠛⠁⠀⠀⠈⣧
⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠈⠙⠶⣤⡀⠈⠙⢶⣤⣤⣤⡀⠀⠀⠀⠀⣀⠈⠓⢤⡴⠁⠀⠀⠀⠀⠀⠸⢦
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣈⣽⠶⠶⠛⠉⠀⠀⢽⣶⣦⣤⣠⣼⣷⣶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⣸⠁⡀⣀⣤⠾⠟⠁⠀⠀⠀⠀⠀⠀⠄⠀⠹⠮⣍⠙⠛⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣷⡄
⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠉⠻⢦⣀⠀⠀⠀⠠⠀⠀⠀⠀⠙⣆
⢠⣃⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⢀⣀⣁⣠⡜⠉⣀⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣳⣀⡀⣴⣤⣈⣙⢲⣤⣀⣀⣀⡀⢀⣀⣽
⣿⣿⡿⢿⣷⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣿⣷⣾⣿⣷⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⢠⣀⣰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠰⠈⠁⠀⣐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⢠⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠇⠈⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡠⠋⠔⠀⠀⠀⠀⠀⠸⢀⣀⠀⠀⠀⠀⠀⠀⡲⣆⠀⠀⠀⠄⡄
⠀⠀⣤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠀⠉⢀⢀
⠀⡎⠀⢡⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣃⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠃⠀⠸⣸⠀⠈⡀⡄⠀⠀⠸⢀⡀⠀⠀⠦⣄⠀⠀⠛⠒⠵⣄⢀
⠀⢸⠀⠀⡏⡀⠀⠀⠹⣆⠀⠀⠹⡀⢀⠔⠠⠀⠑⠲⠠⠄⡠⣹⣬⠀⠈⠀⠐⡢⠄⡀⣂⣀⣂
⠀⠸⡀⠀⠁⠃⠀⢀⠀⠈⠂⠀⠀⡼⠏⢀⣀⠀⡀⠠⠤⠊⠀⠀⠌⠂⠀⠀⠀⠀⠠⠄⠀⢀⠨⠉⠉⠂
⠀⠀⢃⠀⢸⢸⢀⠏⠀⠁⠀⠱⢄⠃⠀⢀⣀⣀⣦⠠⠒⠂⠁⠌⠀⠀⣀⡠⠄⣀⠤⠒⠈⠀⠀⠀⢰
⠀⠀⠸⡄⠈⠸⡞⠠⠂⠀⠀⠀⠀⡏⠉⠉⠀⠀⠈⠐⠂⢀⠠⠀⠀⣑⠯⠒⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣷⣀⡀⡇⠀⠀⠒⠈⠒⠤⠁⠀⠉⠉⠓⠒⠲⠤⣌⣁⠀⠀⠀⠂⠀⢀⣀⣠⣤⠤⠴⠖⢒⡀
⠀⠀⠀⢡⡏⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠏⠉⠉⠀⠀⠀⠀⠀⠀⡌⠡
⠀⠀⠀⢸⠗⢻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⢠⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⡸⣇⠹⢸⠦⣬⣄⣒⣒⣒⣒⣀⡠⢄⡀⢀⣀⠀⣀⠔⢡⠂⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡇⠋⠇⣨⣀⣀⠀⠈⠉⠉⠉⠉⠉⠀⠸⡤⠔⠋⢀⠔⠁⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢣⠀⠆⠀⠐⠁⠀⠀⡄⡇⡂⠀⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⠤⣤⣤⠤⠴⠆⡇⠃⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣀⣠⢎⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣰⠫⢡⠃⢀⠀⠀⠈⢡⠀⠀⠀⢀⡤⠂⠁⠀⠈⠉⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄
⠀⠀⠀⠀⠀⡇⠀⠈⠀⢸⠀⡄⠀⢸⠀⠀⣰⠋⠀⠀⠄⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⠔⠊⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⢞⠝⡩⠉⡈⢛⠢⡀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡇⢸⢸⠀⢰⠋⠀⠠⠀⠀⠀⠀⠀⠀⠂⢹⠄⠀⠀⡠⠒⠁⢀⡴⠛⠢⡀⠱⢄⠀⠀⠀⢠⠞⢡⠃⢠⠁⠀⡇⠀⢣⠘⢦
⠀⠀⠀⠀⠸⡀⠀⢠⠀⢰⠀⠇⢸⢸⠀⢸⡁⠠⠀⢀⡀⢀⡀⠀⠀⠂⢸⠀⠀⠨⡀⠀⢞⠁⠀⠀⠀⢈⡆⠈⡢⠀⢠⠏⠀⡇⠀⢸⠀⠀⠷⠀⠸⡀⠀⢧
⠀⠀⠀⠀⠀⡇⠀⢨⠀⠀⠀⠀⠈⢸⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠄⣀⠏⠄⠀⠀⠘⢆⠀⠀⠀⠀⡠⠋⢀⠄⠀⠀⢸⠀⠀⡰⠀⢠⠀⠀⠀⠀⢰⠁⠀⣸
⠀⠀⠀⠀⠀⢣⢀⠈⡀⠀⠀⠀⠀⡼⠀⠀⠀⠙⠦⣀⠀⠀⢀⣀⠜⠡⠊⠀⠀⠀⠀⠀⠑⢄⠈⠛⢀⡔⠁⠀⠀⠀⠀⠳⣄⠡⡀⠸⡀⠀⠀⠀⡘⣀⠔⠁
⠀⠀⠀⠀⠀⠈⠒⠒⢶⡒⣲⠖⠛⠁⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣒⠓⠒⠚⣺⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠐⠉⠉⢢⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠁⠀⡐⠊⠁⠀⡀⠀⠀⢘⠀⠀⠀⠀⣤⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⢄⠃⠡⠀⢇⠆⣠⣄⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠎⠀⠀⠀⡼⠘⡉⠀⠀⢱
⠀⠀⠀⠀⠀⣀⡠⠤⢀⡀⠀⠀⠶⢋⠀⠀⢀⡨⠒⠀⠀⠀⠀⠣⠠⠆⠈⣺⠆⢴⡇⠀⠠⠞⠀⠀⠀⠀⠸⣄⣀⠤⠄⠤⠃⠀⠀⠀⢇⠀⠙⣄⠀⢸
⠀⠀⠀⢠⠎⠀⠀⡠⠔⡋⠀⠘⠢⢄⠀⢉⣀⠀⠀⠀⠀⠀⠀⢀⡴⡚⠉⠁⠀⠀⠉⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠑⢯⡄
⠀⠀⠀⢸⠀⢠⠎⠀⠈⠀⠀⠀⠀⠀⢻⠀⠀⠈⠢⠀⠀⠀⢠⠏⠈⢀⡴⠊⠉⠙⠢⡀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡠⠔⠲⠶⢤⡀⠀⠀⠈⠢⣀⣀⡤⠁
⠀⠀⠀⠈⠧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⢸⠀⠀⣾⠀⢍⠍⡤⠀⡇⠀⠀⢸⠀⠀⠀⠰⡟⠽⢀⣀⠀⣀⣀⣳⣐⠶⢶
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⠜⠀⠀⠀⠸⡄⠀⠘⢄⡀⠀⣐⡵⢃⠀⢠⠇⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠒⡠⠃
⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⣀⡠⠔⠋⠐⠐⠈⠁⠀⠀⠀⠀⠀⠘⠦⣀⠀⠈⠉⠀⢀⣩⠔⠁⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⣀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⣤⣾⣴⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⠛⠻⠟⠛⡟⢛⠛⠃⠻⠀⠠⠤⠤⢸⣿⣶⣶⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠈⠉⠉⠻⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠛⠻⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣿⡤⠖⠚⠛⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠛⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⢃⣀⣀⡤⠤⠤⠤⣤⣀⠀⠀⠠⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣭⠄⠀⠀⣞⡿⢻⣿⣦⠀⢠⠎⢿⢿⣷⡄⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡾⡟⠀⠀⢉⡟⠀⢼⣀⣀⡼⠁⢧⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠄⠀⢀⡞⠀⠳⠤⠔⠋⠀⠀⠀⠀⠉⢳⡄⢸⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢤⣤⣀⣤⣴⠟⠁⢸⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠷⣦⠀⣾⠀⠀⣠⡄⠀⠀⠀⠀⠉⠉⠀⣰⠶⠸⡟⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣬⣷⡀⠀⣨⢷⣿⠀⠈⠁⠈⠓⢦⣄⠀⠀⠀⡴⠃⠀⠀⡞⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠻⡇⠀⢿⠤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡤⠐⠒⠒⢾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣛⣻⣇⠀⠘⣦⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠒⣲⠶⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⡇⡏⣿⠀⣀⣨⠍⠭⡿⠶⣀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠶⠆⢸⠖⠒⠒⠶⠶⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⢠⠇⠀⠀⠀⠀⢀⡀⠹⣦⡄⠀⠀⣴⣟⢯⣤⡶⠂⢸⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⣿⢸⠀⠀⠀⠀⡴⠋⠀⠐⢿⡀⠀⢀⣿⡟⣿⠁⣠⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣸⡴⠞⠉⠙⠃⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⣿⢋⣠⠿⣆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠦⣿⡇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠀⠘⣆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⢿⣸⡟⢳⡀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠷⡄⠀⠈⣿⠁⣸⠇⣼⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣠⣤⡞⣽⠀⠙⢦⣾⢿⣿⠋⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢿⠀⠀⢰⡇
⠘⠛⠛⠒⠒⢾⡏⢸⡅⠘⣆⠀⣸⣿⣚⣿⠷⠿⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠳⢶⣦⣤⡶⠿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠶⢛⣿⠟⠙⣿⡏⠙⠓⠒⠶⠤⠤⢤⣤⣤⣤⣉⣉⣉⣉⣙⣛⣒⣒⡒⠚⠛⠓⠒⠒⠒⠒⠾⠷⠆
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠠⠶⠾⣭⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠉⠛⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠓⠆⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⠒⣬⣁⣀
⡀⠀⠤⠄⠀⠀⢂⡀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣤⣀⢀⠀⠀⠀⣀⠀⠀⠈⠀⢰⠶⣤⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣷⠧⢝⡀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⠶⢦⣄⣀⣰⠋⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢆⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⠟⣳⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⣷⣿⣧⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣄⠀⠙⠛⣹⣿⣿⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⠀⠀⠀⠘⣿⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀⠔⠛⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⢀⣀⣀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠁⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⢀⡀⠀⠀⢀⣄⡖⢤⠀⠀⠀⠀⢤⣈⢃⠀⠀⠀⢿⡿⠀⠀⠀⢾⡿⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣷⡼⢦⠶⠒⠒⠿⠛⠋⢀⠀⣀⢸⡅⢀⠀⢠⣿⠃⠀⢾⣀⠈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣒⡒⠐⠂⠀⠲⠤⠌⠁⠀⠀⠀⣨⠀⣯⠁⠉⠉⠡⠀⢀⣼⣿⣿⣷⣭⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣀⣨⣿⣿⣤⣀⣄⠘⣀⣾⣿⡿⣿⣿⣿⣷⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⢶⠰⢶⡾⠟⠋⢽⣭⡄⣱⣾⣿⣿⠟⠁⠀⠉⠉⠉⠉⠉⠉⠐⠙⠈⠻⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠺⢼⢯⣴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠠⠂⠀⣼⣶⣾⣶⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣷⣦⣾⣶⣾⣷⠎⠙⠢⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢾⣿⢻⣿⣿⢉⡁⠀⠀⠀⠀⠈⠐⠃⠰⠤⠒⠀⠀⠀⠀⠀⢾⣷⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⢘⣿⣿⣾⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⣋⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⣹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⢈⣧⣾⠁⠙⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⣿⣿⣿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠶⢷⣷⣶⣶⠀⠾⣤⣤⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⣿⣿⢲⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡿⢀⣼⣯⣿⡿⢰⠇⣿⡏⠀⠹⢄⠀⠀⠀⡃⠀⠀⠀⢀⢸⠺⣿⣧⣾⣿⣰⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣿⡿⣿⣿⠗⠋⣸⣿⣶⣤⣶⣦⣡⣦⣸⠁⠀⠀⢰⣿⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⣀⣰⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⣾⣿⣿⣿⣿⣼⣿⣿⠿⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠆⠐⠐⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢰⡆⠀⡐⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣧⡤⢧⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣇⢠⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⡷⢺⡄⠀⠠⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠟⠛⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠓⠒⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠐⠛⠛⢩⠉⠁⢀⣀⣠⣤⣴⣾⠏⡄⠀⠀⣤⢠⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡀⠀⢠⠀⠀⠀⣆⠀⠰⠀⡀⠀⠐⣀⠰⢶⣾⣾⣿⣿⣿⣿⣿⡿⠋⢸⣷⠀⢠⣿⠀⢚⠛⠃
⠀⠀⠀⠀⠀⠀⣸⡇⢰⣼⠧⠊⢙⠯⢆⡄⢘⡓⠊⡉⠄⢛⠸⠛⠋⠉⠀⢁⠈⠥⠦⣀⡈⢉⠒⠼⠿⡆⠋
⠀⠀⠀⠀⠀⠀⣿⠓⢈⠀⠀⠔⠃⢠⠀⠀⢀⡀⠀⠋⠉⠁⠀⠄⠂⢉⡀⣿⠀⠠⠄⠀⠈⠿⠲⠀⠀⢈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡜⠂⢀⠀⠀⠀⢰⡆⣀⠄⠁⣸⣧⡿⠧⠠⠖⠀⠀⠀⠀⠆⠀⠸
⠀⠐⠀⠤⢀⠀⠀⠀⠐⠀⠺⠿⠯⠿⠆⠀⠉⠁⠐⠃⠸⠧⠈⡉⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠂⠈⠰⣶⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠲⢢⢤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠉⠉⠉⠉⠉⠙⠒⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡘⢡⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⠀⠀⠡⡀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠈⠄⠀⠀⠀⠀⢂⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢘⠦⠀⠀⠀⡑⡄⠀⠠⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢹⢄⡃⠀⢠⣶⣶⣦⠘⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄⢱⠈⣆⡎⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠀⢆⠀⠐⠜⣿⣏⡿⠀⠀⠀⢀⠀⠀⠐⢿⣹⣿⠃⠀⠀⡘⠀⠐⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠁⠈⢆⡀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡰⠉⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠜⠒⠤⣀⠀⠁⠂⠀⠀⠀⠀⠀⠀⢀⠠⠐⠣⢄⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⡞⠁⠀⠀⠀⠀⠈⠁⠄⣀⣀⡀⠀⠈⢀⠎⠀⠀⠀⠀⠹⣄⠀⢳
⠀⠀⠀⠀⠀⠀⢀⠜⠀⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠣⡀
⠀⠀⠀⠀⠀⠒⠓⠒⠉⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠒⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⡆
⠤⠤⠤⠤⠤⠤⠤⠤⡎⠀⠀⠀⠀⠀⠀⠳⠒⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢷⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⢰⠀⠀⡀
⠀⠀⠀⠀⢠⣮⡉⠉⠁⠈⠣⡀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠸⠁
⠀⠀⠀⠀⠠⡷⠄⠀⠀⠀⠀⠐⢄⡀⢀⠖⢎⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⡟⠂⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠢⠞⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡄⣄⢂⠑
⠀⠀⠀⠀⠘⣟⡀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣨⠤⠾⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡧⢸⠂
⠀⠀⠀⠀⠀⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈
⠀⠀⠀⠀⠀⡯⠄
⠀⠀⠀⠀⠀⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⢠⣀
⠀⠀⠀⠀⠀⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⢀⣀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⠀⠀⠹⠉
⠀⠀⠀⠀⠀⢗⡂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠃⠀⠙⠁⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣷⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣯⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠈⠙⠿⡿⠿⢿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣥⣾⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠉⠉⠙⠛⠓⠛⠻⠿⠿⣿⣿⣯⣶⣄⡀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣗⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠤⠄⠀⠀⠀⠀⠀⠉⠛⠿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⡏⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠐⠶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠋⠀⠀⡀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣦⣤⣴⣿⣶⣤⣀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⢠⢀⢈⡈⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣾⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣿⣻⣻⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣽⣭⣽⣿⣯⣿⣯⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠼⠯⠿⠷⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠀⠀⢸⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢠⣀⠀⠀⡷⠚⡇⢀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣸⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠘⠉⠀⠀⡷⢤⡇⠈⠉⠉⠉⠉⠉⠁⠀⠀⢸⡿⢠⢯⡙⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠐⠒⠒⠒⠒⠒⠂⠀⠀⢸⡷⣣⢏⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⡜⡱⢣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡜⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡱⢣⢎⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡞⡱⢣⠎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡞⢺⡵⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⣶⣶⣆⣀⡄⠘


⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠓⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⢰⡂⡖⠲⣰⠐⡖⢰⡄⢰⡢⠀⠀⣖⡂⠠⡂⠐⡖⢰⡢⡴⠒⡄⣤⠂⣖
⣤⣤⣤⣤⣤⣤⣬⣤⣷⣴⣼⣤⣧⣯⣽⣼⣼⣴⣤⣧⣤⣤⣼⣤⣧⣼⣼⣽⣴⣥⣭⣦⣧⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠤⣀⡄⠴⠄⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣌⡧⠐⣍⠁⠀⢀⠠⢴⡶⠂⠀⠁⠀⢠⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⡘⠋⣧⡄⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠙⠟⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋⠀⠀⠀⠀⠀⢂⠀⠀⠨⠇⠀⠠⣤⣀⠀⣦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠞⠁⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠀⠀⠈⢷⣀⠀⠈⠈⠀⠛⡉⢠⣿⠀⠀⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢤⡈⠀⡀⣻⠀⠀⠀⠀⠘⠁⣴
⠀⠀⠀⠀⠀⡠⠀⠀⠀⢘⠁⠠⣀⢀⣈⢱⠃⠀⠀⢐⣹⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠙⡆⠀⠠⠀⠀⠰⠀⠀⠀⢻⣌⣺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⡀⠀⠀⠁⠀⠀⠀⠳⣄⣤⡲⠂⠀⠺⠷⠶⢈⣿⡍⠛
⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⢀⠀⠀⠐⠿⠃⠀⠀⠀⠠⠀⡠⠀⡀⠀⠀⠀⠙⠽⠄⠀⠀⠀⠉⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠍⢀⠀⠀⠘⠽⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⣈⣨⢿⣶⣶⣶⣄⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⠈⡀⠁⠂⠀⠀⠀⠀⠀⠀⠩⣦⡆⠀⠀⠀⠘⣄⢬⣿⣿⣿⡿⢷⠁⠾⠀⠀⠀⠐⠤⣀⡀⠀⠠⡄⠀⠀⣄⣀
⠀⠀⠀⠀⠰⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠆⠀⠀⠀⠐⠁⠈⠀⠩⠟⡇⠒⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣸⡿⠀⢠⡄⠀⠀⠀⠀⠀⠐⠜⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣶⢴⡂
⠀⠀⠀⠀⠀⢻⡁⠀⠀⠀⠀⠘⠛⠀⠀⢨⣿⣿⡄⠀⠁⠀⢀⡤⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠾⡁⠀⠀⠈
⠀⠀⠀⠀⠀⠈⡠⠄⢀⠀⠀⠀⠀⠀⠀⢼⣭⣽⡅⠀⠀⠀⠹⠹⡄⠀⠀⠀⠀⣠⣿⣏⡤⠂⠀⠀⠀⠀⣀⢤⠀⠀⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⡀⠀⡀⠀⠀⢰⣟⣿⣻⡇⠰⠶⣄⠀⠑⠁⠀⣀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠉⠀⢀⡐⠶⣮⢿⣿⠅⠀⠀⠈⠀⠀⠀⠀⠋⠁⠀⠀⠀⢀⢀⣠⣄⣀⠀⠐⠳⠌⠀⠀⢠⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠄⣽⣶⢠⣶⣿⣿⣦⡀⠀⠈⣧⠀⠀⠰⠀⠀⠀⠀⠈⠁⢀⣠⣤⡻⠒⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡋⠁⠀⢰⠮⠏⠺⡟⣿⣯⣬⣿⣆⠀⠺⢿⠀⠀⢰⡞⠁⡀⠰⠂⠙⠇⠁⠀⢹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡇⠀⠀⠘⠀⠀⠁⠈⠹⣽⣷⣟⡛⠀⠀⠛⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠀⠀⠰⠦⡀⠀⠀⠀⠀⠈⠀⠐⠀⢀⣈⠠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⣤⠀⠀⠈⠉⣠⠀⠀⠀⠀⠉⠀⠀⠀⠀⢒⣦⠀⢰⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠋⠀⠠⣴⣦⠀⠀⣠⣠⠀⠀⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠚⠋⠉⠉⠉⠉⠉⠙⠒⠲⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠋⢂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠃⠀⠀⠢⡀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⡎⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢡⠀⠀⢠⠀⠀⠀⠀⠀⣀⠀⠀⠁⠂⠀⠀⠀⠑⢤⣀⠀⠀⠀⠱⡄⠀⠀⢠⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢆⠀⠘⢀⠄⠀⣀⣀⡀⢱⠄⠀⠀⠀⠀⠀⠀⠎⢀⣀⣉⠀⠲⠘⢦⠀⡨⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠸⠃⠦⢨⠀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠇⠀⣧⠃⢄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢆⡀⢣⡀⠀⠑⠼⠿⣧⠾⠃⠀⠀⠀⣤⠀⠀⠀⠙⢧⣼⠿⠇⠂⠀⢀⡍⠀⠘⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠈⠷⡈⠀⠂⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠘⠀⣀⠞⠁⠀⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⣬⠷⠦⡄⣀⠀⠈⠒⠀⠤⠀⠀⠀⠀⠀⠀⣀⠀⠴⠞⢅⡀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡴⠋⠀⠀⠀⠀⠈⠈⠳⣀⠀⠀⠀⠀⠀⠀⠚⢀⡝⠀⠀⠀⠀⠉⢧⡀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠤⠔⠋⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⡴⣋⣀⡠⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣀⡳⣄⡀
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⡇
⣤⠤⢤⣤⣤⣤⣤⠤⠤⢼⠁⠀⠀⠀⠀⠀⠀⠈⢦⠤⠤⠄⠀⠠⠤⠤⠄⠀⠀⠀⠀⠠⠤⠴⠀⠀⠀⠀⠀⠀⢹⣤⣠⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠀⠒⠂⠀⠀⠀⠀⠀⠐⠀⠚⢃⠀⠀⠀⠀⠀⠀⠇⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠘⡟⠃⠀⠀⠀⠀⠈⢦⠀⠀⠀⠄⠀⠠⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣠⠀⠀⠐⢆⠁
⠀⠀⠀⠀⠀⠙⣟⡁⠀⠀⠀⠀⠀⠀⠳⣤⣁⡔⠉⠰⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢫⠐⠀⠀⢁⠾⡙⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⠀⡄⢰⡙⡄⠓
⠀⠀⠀⠀⠀⠙⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣈⡡⠶⠇⡀⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⠈⡗⠆
⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠃
⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢈⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢈⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠒⠒⠒⠂⠚⠤⢄
⠀⠀⠀⠀⠀⢠⣧⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣈⣄⠀⠓⠢⠤⠄⣀⣀⣀⡀⠕
⠀⠀⠀⠀⠀⠠⣧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠰⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠋⠛⠉⠛⠉⠁⠀⠤⠀⠀⠒⠚⠉⠛⠛⠯⠭⠭⠉⠙⠛⠋⠥⠀⠂⠤⠶⠒⠋⠻⡖⢀⠳⠲⠶⠶⠶⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣸⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⠂⠀⢰⠃⠀⠉⠙⠓⠶⢤⣄⣀
⠀⠀⠀⠀⠀⣠⠴⠒⠉⠁⠀⠌⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣤⣀⣀⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡘⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠉⠉⠛⠛⠶⠶⠶⠶⠖
⠀⠀⠀⢸⠀⠀⠀⠐⢄⡰⢀⣀⣼⣠⣤⣼⡧⠴⠚⠁
⠀⠀⠀⠸⠄⠀⣠⡤⠾⠟⠉
⠀⠀⠀⠀⡴⡫⠁⠀⡆
⠀⠀⠀⢠⣧⠣⣀⣀⠄⠀⠀⠀⣠⣤⠤⢤⣄⣀⣀⣀⣀
⠀⠀⠀⢸⡀⠙⣲⠒⠒⡖⠋⠉⠓⠢⣵⠯⣀⣀⠈⡍⠙⠒⠦⣤⣤⣀⣀⣀⣀⠀⠄
⠀⠀⡀⠀⠙⠚⠑⠠⡜⠳⢄⡤⠤⡀⢸⠀⠀⠈⠻⡓⠒⠠⢤⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠴⠶⠶⠶⠶
⠀⠀⠀⠄⠀⠀⠀⠌⠀⠀⠀⠉⠑⠚⠚⠼⠁⣐⣠⠇⡊⠘⣸⠃⠀⠈⠉⠓⠒⠒⠒⠒⠛⠉⠉⠁
⠀⠀⠈⠁⢩⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⢿⡿⠿⢒⡈⠉⢩⡻⢟⡝⢋⣯⡛⠙⡷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠏⠿⣿⣿⡸⣆⢀⣿⣦⣠⡜⣿⣋⣀⣴⢿⣗⢀⣡⣤⣬⡙⠿⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣷⠞⠛⠷⠈⣻⣧⢻⣮⣗⠋⠀⠀⣿⠉⠁⠙⣪⣿⡿⠟⠋⠨⠉⢒⣦⡿⣟⢦⡀
⠀⠀⠀⠀⣰⢿⢟⠻⣯⣴⠛⠛⢳⡟⢻⣏⣿⣿⣧⡀⣼⣿⢀⣼⠟⢁⣿⢡⡟⢳⡆⣠⣾⠁⠰⢻⣶⠻⣆
⠀⠀⠀⣰⣿⣞⣿⠀⠙⢿⣷⠖⠊⣷⢠⣿⠾⠟⢛⣛⣋⣉⣉⣉⣙⠛⠻⢮⣿⣯⣵⠟⣁⡀⢀⡠⠛⠈⡝⢷⡀
⠀⠀⢠⣿⣿⣼⣿⡆⡀⠀⢈⠳⣾⡿⢋⣠⣶⣿⣿⣭⢿⣷⣿⢿⣿⣿⣷⡦⣍⠳⣿⣉⣀⠟⠺⣔⢃⣤⣶⣮⢷⡀
⠀⠀⣾⣿⣧⣻⣽⣿⣤⣄⣤⣼⠟⣱⣯⣛⣵⣋⡍⢉⢈⣿⡟⡴⢶⡍⢹⡏⣿⣷⡌⢯⣤⣓⣾⡵⠟⠋⣁⣼⣿⣷
⠀⢰⣿⣿⣷⣿⣏⢛⠛⣿⣿⠃⣾⣳⣷⠛⢫⡿⣓⠿⢛⣷⣚⠳⠾⢃⣀⣉⠣⢽⡿⣎⢻⣋⣷⣶⣾⢿⣶⡀⠙⣿⣧
⠀⢸⡟⣮⣿⡿⣿⣿⣾⣿⡇⣸⢳⡟⣥⡀⢟⡔⡡⢊⠕⣒⣮⣿⣝⢮⡱⣯⣤⠌⣿⣿⡘⣿⡾⠋⠀⡼⠻⣿⠀⣿⣿
⠀⣿⣷⡿⣤⡄⠱⡧⠙⢿⡁⣿⣼⠸⠿⠛⠋⡼⣱⢻⣎⢷⣟⡵⣌⢣⠳⣀⣤⣀⢻⣿⡇⣿⣧⣀⠀⢧⣠⠏⠀⣿⣿
⠀⣿⢸⣿⢿⣯⡴⢋⣀⢺⡇⡏⣿⡜⢿⡇⠀⣇⠇⣿⢟⣥⣝⢷⡹⣼⢀⣿⡛⠋⢸⣿⡇⣿⠿⠛⠻⠋⡥⢶⣶⣼⣿
⠀⣿⣸⣿⡟⣄⣠⣿⠿⠾⣿⣧⢸⡟⣆⠁⣀⢹⣞⠵⣫⡆⠿⢳⣱⠈⢳⣧⣽⡆⣼⣸⢁⣿⣆⠀⠀⠀⣇⠀⠈⣿⣿
⠀⢹⣿⡛⠛⠉⢁⣴⣖⢠⡿⣿⣇⢻⡽⣆⠟⢛⣱⠞⣵⡾⣿⡡⡍⠛⠛⠲⣦⡰⣱⢃⣼⣟⡻⢿⣦⣤⡿⣟⠂⣿⡇
⠀⠀⢿⣿⣶⣆⡈⢙⢛⣿⣿⡯⠿⣦⡙⢯⣻⣞⣁⠾⠟⠁⣿⠳⠆⢃⣼⢞⣡⠟⣡⣾⠳⢮⣝⡻⢦⣄⠀⠈⠀⣻⠁
⠀⠀⠈⢿⣿⠿⣝⣴⣿⣿⣿⣡⣴⢋⠹⣦⠙⠾⣿⣿⣟⣶⣟⣲⣾⡿⠟⠋⣡⡞⠋⢻⣦⢤⡟⡟⢶⣮⡛⠒⣰⠃
⠀⠀⠀⠈⢿⣦⣿⣿⣟⣁⠘⣷⠻⢮⡵⠟⣹⡶⣤⣤⣭⣭⣭⣭⣤⠴⢶⡟⣿⠀⣠⡞⠁⣼⠟⠁⠀⠉⢁⡴⠁
⠀⠀⠀⠀⠀⠻⣿⡁⣬⡈⠀⠙⣆⣾⣲⣶⠏⣠⣴⡞⢛⣿⣛⠛⣽⣶⡄⢿⡹⣮⢹⣷⠟⠁⣠⡤⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣮⣻⣷⣟⠙⣇⢉⣿⣾⠁⢈⣿⣟⣁⣿⣾⠛⢹⣿⣾⣽⣿⣏⠁⢀⠼⣛⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣽⣷⣌⡺⣿⣧⡀⠛⢿⣍⢥⣬⡿⠃⠘⢿⣿⡿⢛⣿⢂⣭⠾⠋⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣭⣛⠒⢦⡿⠧⣯⣩⣧⣴⣿⢟⣥⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠓⠲⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠉⠉⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠒⠀⠀⠄⣀⢀⡀⠀⣀⢀⠤⠂⠀⠀⢹⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠴⠋⠀⠁⠀⠀⠀⠁⠬⡀⠀⢐⡀⠀⠈⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠰⠋⠀⠁⠄⢀⠄⠀⠀⡀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢇⠀⠀⢠⣔⣊⣤⠴⠋⠀⠀⠀⠀⠀⢠⡄⠀⡀⠂⣀⠀⠀⠉⠀⣸⠃⠀⢸⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠹⡀⢠⡄⠀⠀⢀⡀⠐⠤⠠⠔⠀⠂⠉⠀⠠⠀⠀⠀⡀⢠⠏
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⣿⠀⠀⠀⠨⡑⢄⠀⢦⡀⠀⣀⠀⠀⠀⠀⠀⠀⡟⢋
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢸⡄⠀⠀⠀⢿⣶⣅⡂⠈⠉⠀⠤⠂⠀⠀⠀⠀⡗⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⡈⢳⡀⠀⠀⠈⠫⠉⠛⠟⠂⡈⠁⠀⠀⠀⠀⣰⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⣤⣀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣠⠞⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠨⢇⡉⠉⠛⠓⠒⢲⠶⠟⡋⠥⣤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢩⠁⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡆
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢆
⠀⠀⠀⢠⡏⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠹⠉⠉⠉⠑⠢⣄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠶⢶⠋
⠀⠀⠀⢀⠃⠀⠀⠀⠀⡰⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸
⠀⠀⠀⡎⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠈⠀⠰⡀⠀⠀⠀⠀⠀⠀⣿⠀⡆
⠀⠀⢸⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠘⠆⠀⠑⡄⠀⠀⠀⠀⠀⠏⠀⠹⠄
⠀⠀⠈⠲⠤⣀⠀⠀⠀⠀⠀⠉⠒⠢⠤⣠⠀⠘⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣄⣀⣀⠀⠀⠀⠀⠉⠓⠒⠉⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⡀⠀⠀⠀⠈⠓⠒⣊⣡⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⡀⣠⠩⠏⢬⠛⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠓⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠄⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠋⣽⠁⠉⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠈⠐⠀⠀⢀⡏⠀⢹⠇
⠀⠀⠀⠀⠀⢀⠀⣿⠃⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⡆⠀⠀⠀⠀⠀⠀⠀⡄⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⣀⣠⣟
⠀⠀⠀⠀⠀⣮⣱⠏⠀⠀⠀⢀⡀⡀⠀⠠⠀⠀⠀⠀⢳⠖⣊⣻⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡐⠈⠁⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠈⠧⠞⠚⠁
⠀⠀⠀⠀⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⠠⠀⢤⠠⠠⣤⠀⢤⠠⢠⢤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⠀⠀⣿⠀⢸⠀⢸⢠⠇⠀⠀⢀⢄⠈⠉⠷⡦⡀
⠀⠀⠀⠀⠀⠐⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⠀⠀⣿⠀⢸⠀⢸⡉⡆⠀⢰⠷⠁⠠⠔⣀⠀⢹⠀⠀⠀⣰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠠⠀⠠⢠⠀⠠⠀⠀⠘⡀⠀⠀⠁⠀⢠⠞⠀⠀⣼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⠲⠂⠄⠉⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢠⠈⠁
⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠐
⠀⠀⠀⣠⣀⡀⠀⠀⠂⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠐⠀⠀⠴⢦⡀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⢠⠒⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠰⠁⡅⠀⠀⠀⠿⢿⣿⣿⣶⠂
⠀⠀⠠⠀⠀⠀⠁⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠡⠐⠘⠛⠲⣾⡀⠀⠀⠀⡤⠐⠊⠉⠀⠹⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠉⠙⠧⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠀⠈⠀⠀⠀⠀⠰⡀⠛⠀⠀⢀⡇⠀⠿⠀⢈⠄⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣴⣶⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⡿⠟⠋⠉⠉⠉⠁⠀⠀⠀⡼⠁
⠀⠀⠀⢀⣿⣿⣿⣿⡿⣿⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠛⠛⠿⠛⠋⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡂⠀⠀⠀⠛⠂⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣠⠆⠀⠀⠀⠀⠀⠞⠋
⠀⠀⠀⠚⠲⢶⣤⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠲⠶⠶⠶⠶⠶⠶⣤⣤
⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢠⣶⣶⡤⠄⠀⠀⣠⣤⠀⠀⠀⢀⣠⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠛⠛⣡⣴⣾⣿⣿⣿⡿⠋
⠀⠀⢀⣠⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠲⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⣄⢠⠀⠀⠀⠀⠀⠀⠀⢠⡿⠓⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠞⢾⡿⠿⣛⡃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡋⠀⠀⠀⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⣆⠄⠠⠠⢀⠒⠟⠛⠫⢛⠾⠉⠀⠀⠀⢀⣤⣺⣷⣶⣾⣿⣶⣾⡆⠀⠀⣿⡿⠠⠆⢲⣾⣿⣿⣇⠀⠘⢿⣇⣌⢣⣌⢻⣿⡇⠀⢠⣿⣧⣾⡇⣠⣾⣿⠇
⠀⠀⠀⡠⠴⠀⠠⠆⠔⠓⠀⠀⠀⠀⠀⠀⢠⣿⣿⡗⠙⣿⣿⣿⣿⣿⠃⠀⠀⣿⡿⢰⠲⣾⣿⣿⣿⣿⣷⣄⠀⠈⣉⡉⡉⠙⣿⡇⢠⣾⡿⣿⢿⣿⣿⠟⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⣿⣿⣿⣿⡷⠀⠀⠀⣿⣿⡀⣹⣿⣿⣿⣿⡟⠛⠛⠷⣄⠙⠋⠁⣰⣿⡇⢸⣿⣷⣿⣾⣿⣥⣤⣶⡆
⣶⣶⣶⠶⣶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⠶⠛⢛⣓⣿⡿⢿⣿⣿⣿⣷⣶⣶⠹⣷⣖⠚⣹⣿⡧⠀⢻⣿⠟⠁⠉⠉⠉⠩
⣿⠟⠁⢤⣿⣿⣧⣀⠈⠻⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⠰⠀⠀⢸⣿⣿⣷⠀⣼⣿⡙⡟⢿⣿⣤⣿⣿⢾⣿⣿⣏⠁⠀⢉⠀⠀⠀⠀⢰⣦⣴
⠁⣀⣿⣿⢻⣿⣿⣿⣷⣄⠹⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢸⡄⢇⢀⣿⠟⣿⠟⡀⣿⣿⣿⣧⠈⠻⠿⠿⠋⣸⣿⣿⡧⠉⠀⣾⣧⣄⣀⣤⣾⠟
⢨⣷⡜⠃⢠⣿⡿⣿⣿⠿⠤⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣐⣆⣧⡸⠈⢡⡤⠿⠷⣿⣿⣿⣿⣿⠛⣶⣤⣴⣾⡇⢻⣿⡯⠅⢸⣿⢸⡟⠟⣿⠁
⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⣿⣿⢀⣿⣿⠀⠓⠖⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠉⠉⠉⠉⠀⢀⠀⢹⡗⣸⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣁⠛⣭⣭⣉⠍⠏⣭⡉⣍⡋⠍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢰⣤⣿⣽⡆⠀⢳⣿⣯⠛⠛⠛⣻⠿⠁
⣿⣿⣿⣿⠛⡟⡿⠿⠿⢿⠿⣿⣿⠿⠷⠶⢿⠟⢿⣿⢻⠻⢟⠻⢿⠟⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠷⠝⠿⣿⣿⡿⢻⡍⠁⠀⣠⣾⣁⣄⣀⣀⡀
⣿⣿⣿⣿⣤⣧⣦⣵⣮⣿⣤⣿⣿⣮⣴⣾⣴⣥⣾⣿⣧⣦⣾⣼⣴⣶⣵⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⠃⢸⣷⠀⠀⠀⠀⠀⠀⠈⠛⠃
⣿⣿⣿⣿⠁⠉⡹⢛⠉⡇⠏⣝⠍⣏⡽⡋⢝⣫⣹⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣇⢸⠏⠀⠀⠀⠀⠀⠀⠴⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶
⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣽⣽
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⣼⣆⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡿⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠇⣐⠼⠃⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡆⢀⣂⠒⠀⠀⠀⢸⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠀⣀⠀⠈⢥⣬⡧⠧⠀⠀⠀⠀⢺⠃
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⣢⣤⠴⢡⡄⠀⠀⠀⠀⠀⣝⠁
⠀⠀⠀⠀⠀⣀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠋⠂⢽⡦⠀⠠⣀⢨⣟⡀⠄
⢤⠔⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⡰⢞⡖⢢⡖⢋⠁⠀⠑⣲⢿⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠰⠧⠼⠥⢬⡥⣆⠀⠀⢶⢏⠁⣠⠄
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡔⠠⠀⠀⠠⢠⠌⡉⢠⠄⠀⢸⡇⠀⣸⠀⠀⠀⠀⣷⡀⠀⠀⢸⠀⠀⠀⠈⠫⠴⢶⡂⡐⡋⠛⠠⠄⠀⠰⣼⣩⡌⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢜⠃⠀⠀⠀⠀⢀⣹⠤⠠⠄⠀⢺⡇⠀⢽⠅⠀⢠⠠⠭⡄⠀⠀⠸⠀⠘⢆⠀⢠⣀⣉⠱⢗⠈⠘⡆⠁⠀⠤⣌⡏⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⡅⠀⠨⠀⠀⠀⠀⠠⠐⠟⡓⠀⠀⠀⠸⠇⠀⠠⣀⠀⠀⠀⠸⣭⠄⠀⣲⠀⠠⢈⡄⢀⣀⣉⠞⠉⠑⡯⠀⠘⠆⠠⢨⠟⠣⠉⠁
⠀⠀⠀⠢⠀⠀⠶⠇⠀⠀⡻⠀⠀⡠⠀⡄⠀⠰⠖⠒⣛⠷⠀⠀⠌⠛⠀⠀⡳⠀⠀⠀⠀⠐⣦⠀⠀⢿⡐⠐⠂⠀⠀⠘⠿⠲⠌⡆⡕⡂⠀⠦⣔⠫⠈⢡⠌⢀
⠀⠀⠀⠀⠀⠀⣿⠁⠛⢀⡿⠀⠀⡁⠀⡄⠀⠀⠀⢀⣙⠁⠀⠐⠀⠀⠈⠀⣟⡃⠀⠀⢬⢀⣁⠀⠴⡏⣁⢀⣮⣇⠄⠀⠄⠆⠠⢄⢀⢠⡸⠦⠀⠘⠑⠀⠈⣤⠠
⠀⠃⠀⠀⠀⠀⠋⣅⡀⣩⣏⠀⠀⠑⠀⠐⠀⠀⠀⠂⡞⠀⠀⠉⠀⡳⠀⠋⠀⡤⠄⠀⠀⠀⠀⠀⠨⠅⠏⡀⠳⠟⠑⠀⠀⢂⢌⠑⠙⠀⢀⡀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠸⢿⢊⠁⠀⠀⠑⠀⠀⠀⠀⠀⠃⠀⢀⢠⠄⠀⢀⠠⠤⢀⠊⠀⠀⠀⢰⠀⠙⡇⠎⢁⣮⢶⡁⣠⡜⠁⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠠⠀⠀⠀⠀⠀⡖⠀⠀⠠⠄⠀⠂⠀⠀⠘⠀⠀⠐⠂⠁⠀⠙⢐⠀⠤⣋⠢⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⡀⠀⠀⠷⠀⠀⠀⠀⠀⠁⠀⠀⠈⡄⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢔⡫⠽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⢀⡔⠋⣒⢝⡩⠭⢑⠷⣄⠀⢀⣀⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣮⣯⢷⣦⣴⢏⠈⣉⠕⡥⠒⣒⠠⡢⡸⢟⠭⠒⣒⠐⢢⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣾⡾⣟⣿⣦⣍⣥⢮⠞⣁⠀⠙⢌⡢⠜⣻⠭⢤⡵⡸⢸⣧⡤⠴⠶⠶⠶⠶⠦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣹⢻⠟⡟⠻⠦⣤⠴⣡⣎⣁⣩⣄⠀⠈⣩⠤⡄⠀⡇⡇⣸⠗⢒⣈⠭⠭⠬⡉⠒⡄⠘
⠀⠀⠀⠀⢀⣀⣀⣠⣧⣋⣾⢠⢇⣶⠶⠛⠋⠉⠉⠉⠉⠉⠉⠛⠳⢶⣏⠀⠹⣱⣼⡯⠵⠚⠓⢢⠀⡰⠀⡄⢀
⠀⠀⠀⣴⠉⣀⣀⣜⣡⢃⣾⣪⠟⠁⣀⣤⠖⣋⣭⣭⣭⣭⠻⢶⣦⣄⡈⠓⢤⣤⣤⠤⠒⢲⠀⢰⡵⢁⠜⣀⡞
⠀⠀⠀⣹⢸⢰⡫⠛⣒⣩⡞⠁⡠⣺⡟⣁⣴⣿⣿⣿⡋⣉⣩⡀⢹⣷⠈⠓⢤⡈⠳⢤⣀⣸⠀⣼⢡⢃⣼⣿
⢀⡴⣪⠟⣿⡘⣇⠞⣱⠋⢠⠞⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⠀⠀⠀⠙⢶⣄⠈⠉⠉⠛⣮⣿⢣⠸⡄
⡏⡜⡰⢋⡼⣷⣿⡞⢁⡔⢙⡆⠀⣿⣇⠀⠻⣿⣿⣿⣿⣿⡿⠀⣸⡿⠀⠀⠀⣠⠞⠁⠀⢀⣠⠞⠻⣿⠀⡆⢱
⢱⢡⠇⢈⣰⡿⠿⢧⣄⠙⠛⠷⠤⣽⣿⣧⣄⠈⠛⠛⠛⠋⠀⣰⣿⠃⣀⡤⠞⠁⢀⣤⠶⣫⣥⣤⡾⠃⡠⠃⡰
⠀⡆⣽⣼⢿⣇⠑⢒⣛⡓⢦⣀⡀⠀⠈⠉⠉⠛⠛⠒⠾⠶⠾⠟⠛⠋⠁⣠⣴⣾⠟⢡⢮⠏⡼⢿⡗⢋⡴⠊
⠀⣸⣿⢋⠏⢸⠙⢅⣈⣉⡳⣬⡉⠳⣶⠤⢤⣤⣤⣤⣤⣤⡤⠤⠶⣾⡋⢉⡴⠃⣠⠃⡎⠀⡇⢸⡿⠋
⡰⠁⡇⠈⠦⠤⠬⠭⠭⠽⠿⣷⣽⣶⣌⡙⠉⣁⣀⣀⣤⡤⢤⣤⣤⣄⣉⣁⣤⣾⢏⠞⣀⠴⠁⢀⡇
⠣⠤⠬⣭⣉⣭⣭⣭⣭⡿⢿⠶⠿⣿⠧⣉⣉⠵⢊⠭⢒⡩⠁⠒⣉⣓⡫⢋⣼⢃⣾⣫⠥⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠙⡆⡄⢣⠑⠄⠨⠅⡒⠤⢚⡡⠚⣁⠤⣴⣒⡶⠤⠖⢋⠎⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠡⡀⢑⣒⣒⠢⣌⣁⡩⢔⣫⠶⡿⢛⠿⣧⣍⡉⠁⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣈⣥⠶⠶⠭⣶⣿⠾⠟⠥⢚⡔⣡⠞⠉⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠦⡀⠁⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⢞⡀⢠⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢤⣰⣶⠶⠦⠐⠶
⠀⠀⠀⠀⠀⠀⡀⢠⠿⡹⡵⠰⣋⠛⠀⠀⠀⠀⠀⠀⠧⠖⠒⠻⠶⠦⠀⠀⠀⠀⠀⠀⠘⣋⡉⢀⡦⡂⣀⠀⡆
⠀⠀⠀⠀⠀⠸⡯⡐⢀⣲⠂⣂⡀⣄⡾⠀⠀⠀⠀⢠⢤⣀⣃⣈⡃⣴⡄⠀⠀⠀⠀⠀⠘⠽⠛⠛⢫⢗⠿⠯⠷⠧
⠀⠀⠀⠀⠀⠀⡇⣇⣈⣿⠁⣿⡇⡇⡀⠀⠀⠀⠀⠈⠂⠉⠰⠆⡒⠈⠁⠀⠀⠀⠀⠀⠀⢰⠂⠠⠤⢥⠐⡆⣀
⠀⠀⠀⠀⠀⠀⡃⣉⠉⠐⠂⠀⠑⠁⠁⠀⠀⠀⠀⣀⠀⠙⠒⢒⠃⠀⢀⡀⠀⠀⠀⠀⠺⠺⣶⣄⡖⢲⡀⡇⠉
⠀⠀⠀⠀⠀⠀⡇⢙⣶⡮⠛⣥⡆⠀⣥⠀⠀⠀⠀⠀⠿⠛⠂⠙⠛⠻⠎⠁⠀⠀⠀⠀⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠁⠑⠛⠛⠉⠉⠃⠀⠛
⠀⠀⠀⠀⢀⠀⣄⠀⠐⠴⠶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠤⡄⢠
⠀⠀⠀⠀⠀⢁⢸⢟⡶⠆⢗⣤⣇⠀⠀⠀⠀⠠⡤⢤⡄⣨⠇⠛⢓⡠⢶⠂⣰⠇⠀⠀⢀⡄⠀⠓⠩⠬⠈⢀⣀⡀
⠀⠀⠀⠀⠀⠲⠅⢰⡒⢒⢀⠀⠒⠀⠀⠀⠀⠈⠻⣨⣿⠶⢤⠤⢤⣖⠿⠴⠃⠀⠀⠀⣿⡏⢰⣛⡤⣤⣰⠊⢻⣿
⠀⠀⠀⠀⠀⢠⣬⣽⣿⣿⣶⣧⣄⡀⠀⠀⠀⠀⠀⡀⡅⢉⠛⡋⡛⢉⢤⠰⣀⡤⠀⠀⠛⠟⠀⣧⠉⢨⠆⠁⠘⠋
⠀⠀⠀⠀⣾⣿⠿⠿⠿⠿⠿⠿⢿⣿⡅⠀⠀⠀⠈⠉⠇⠘⠧⠧⠣⠀⠿⠤⠊⠀⠀⠀⠀⠀⠘⣁⣤⣇⣖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠖⠖⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠈⠙⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⠀⠀⠴⠢⠤⣤⣀⠉⣁⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⡿⠀⠀⠀⠀⠀⠀⠈⠳⣶⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢻⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣿⠇⠀⠀⠀⢠⣄⢨⣳⡄⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢾⡟⠀⢘⠀⠀⠀⠀⠈⠉⠋⣀⠀⡐⣦⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠖⠛⠛⠋⢙⠋⢸⠀⠀⠀⠃⠀⠀⣄⡀⢀⡀⢹⠟⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⢠⠋⠁⠀⣆⠀⠀⠐⣄⡀⣷⠍⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠰⠁⠠⠶⠶⢼⡄⢰⣿⣥⡉⠀⡀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠡⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⢀⡶⠋⠉⠙⢄⠀⣀⡀⣀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢧⠉⡀⠀⠀⠀⢤⠙⠓⠋⠉⠀⠀⠀⠀⠀⠑⢤⠖⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢠⡶⠶⠚⠛⠛⠻⠿⣦⣀⣀⠀⢀⣀⡠⠔⠁⠐⠀⣀⣀⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢈⡙⣏⡦⣤⣴⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⠀⢠⣽⣑⠈⠉⠉⠉⠉⠛⠛⠛⠓⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⡀⠀⠛⠛⠓⠶⠀⠠⠤⠀⢀⣀⣀⣠⠀⠄⢿⡟⠛⠉⠉⠉⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢷⣄⡠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠭⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠾⢿⣶⣶⣤⣄⣀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠉


⡿⠿⠿⠿⡿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⢤⡤
⠀⠀⠀⠀⢠⣿⠲⠆
⠀⠀⠀⠀⢸⣏⣙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⡀⡀
⠀⠀⠀⠀⣸⣯⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡰⠁⠀⠇⠀⠹⣀⠀⠠
⠀⠀⠀⠀⣽⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣒⣒⡒⠒⠒⠐⠲⠰⠐⠀⠐⠒⠀⠆⠂
⠀⠀⠀⠀⣿⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⢰⣟⣛⡋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣯⣍⡁⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⢀⣤⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣸⣧⣤⠄⠀⠀⠀⠀⡀⠐⠺⢠⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤
⠀⠀⠀⣿⠶⠶⠀⠄⠠⠀⡠⠂⢢⡄⢸⠁⠀⠀⠀⠀⠀⠀⣠⡿⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣆⠠⢁⠀⢀
⠀⠀⠠⡿⠲⠶⠀⠀⠀⠀⠢⠔⠀⡆⢸⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠠⠀⡑
⠀⠀⢰⡟⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⢸⣟⡛⠃⠀⠀⠀⠀⠀⠀⠀⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⢾⣉⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⣿⣉⣉⠁⠀⠀⠀⠀⠀⠀⢸⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⠀⡇
⠀⢀⣯⣭⣉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠤⠤⢤⢤⠤⠤⠤⠤⡬⠭⠭⣭⡉⢉⣉⡉⣉⣉⣀⡇
⠀⢸⣯⣬⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡴⢢⡅⠀⢄⠤⠰⡁⠶⠀⡄
⠀⢸⣧⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠔⠱⠄⠉⠠⠄⠣⠈⠔
⠀⣿⣤⣭⡁
⠀⣿⣤⣤
⠘⢏⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠲⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠐⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣑⠮⠃⠀⠤⢤⣀⠀⠀⡤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⡀⠼⠕⠀⠀⠒⠚⠁⠀⠚⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡕⢄⠀⠀⡐⠀⡀⡦⠀⠀⠀⠀⠤⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⢪⢢⡐⠀⠀⢧⠱⡀⠀⠀⠐⠶⠖⢠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⡀⠌⢿⣿⢳⡑⠀⠀⠈⠣⡈⠓⠒⠦⠤⡄⠘⡀⠀⠃⠂⠀⠤⠈⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠀⠀⠀⠀⠻⣿⣿⣌⢢⠀⠡⠈⠲⢄⠀⠈⢀⠞⡅⠀⠈⡀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣿⣦⡱⡀⠡⡀⢘⠀⡇⠁⠀⠁⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡄⠀⣄⡤⠾⢿⣮⣳⣌⠄⠑⠌⠂⢀⠀⠀⠀⠦⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣗⠂⠁⠀⠀⠀⠻⣯⠙⣦⢦⠈⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⣀⡀⢀⡀⢀⠑⠤⣀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠀⠈⠢⢀⠀⠀⠀⠹⢷⣤⠻⣷⡀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠸⠪⣉⠁⢀⠤⠚⢚⠒⠌⡒⠄⠱⡀
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠑⢤⡀⠀⢈⡻⠳⠿⠿⠀⠀⠀⠀⠀⠐⠧⢀⠔⠉⠀⡀⠷⠴⢦⡀⠀⠀⠁⢈⠢⡀⢱⠀⠠⠤
⠀⠀⠀⠆⢀⠀⠀⠀⠀⠠⠀⣃⠀⢀⡀⠀⠉⢀⣨⠖⢉⠁⠄⡀⠀⢄⠀⠀⠀⣀⠤⣴⢁⡞⢠⠀⠀⠀⠀⠀⢹⣀⣀⣀⠬⠤⠼⠎⠀⠂
⠀⠀⢠⠀⠀⠐⢁⠂⠄⠀⠄⡡⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠂⢨⣑⣬⢳⣆⠈⠀⠐⠛⠮⠷⠦⡶⠤⠛⠒⠀⠛⠉⠉⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⠀⢰⡴⡶⠶⠒⠛⠋⠈⠁⠐⠒⠬⠳⡀⠐⠀⠀⢀⣀⡠⠄⠀⠀⠠⠀⡀⠀⠀⠀⠀⠂⠉
⠀⠀⠈⠓⢄⠀⠀⠀⡐⠀⣉⠀⠀⣠⠤⠤⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠉⠀⠀⢀⣀⡀⠴⠁⠤⠁
⠀⠀⠒⠊⠉⠁⠀⠀⠉⠑⠂⠀⢱⣿⡷⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⣀⠠⠔⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣟⣻⣎⣄⠀⠀⠀⠄⡀⠀⠀⠤⣀⠠⠅⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⢛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣯⡿⢿⣿⣿⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢛⢿⣿⣿⣾⣿⠟⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣸⠀⠛⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡌⠀⠀⠀⠀⠀⠐⢨⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠀⠀⠛⣄⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⠀⠐⠀⠀⠘⣷⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠐⠤⣈⣋⠩⣍⡀
⠀⠀⠀⠀⠀⠀⣏⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡗⠄⡉⠄
⠀⠀⠀⠀⣠⡾⣋⣩⡄⠀⢌⠂⠀⠀⠀⠀⠀⠈⢹⣷⣄⠳⣄
⠀⠀⠀⢸⢿⡿⠟⠉⠁⠈⠒⡅⠀⡄⢠⠀⠀⠀⠈⣿⣿⡆⠙⣆⠀⠀⠀⠀⠘⣦
⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠹⣿⣿⡄⠸⡀⠀⠀⠀⠀⠈⠥⠀⡀
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⢿⣿⣷⡀⢱⠀⠀⠀⠀⠀⠈⣑⠺⡄⠀⠀⡤⢠
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⣸⠀⠀⠀⠀⠘⣿⣿⣧⠈⡦⣄⠀⠀⠀⢠⣬⠑⠁⠀⠘⠀⣸⠂
⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠇⠀⠀⠀⠀⠀⢹⣿⠟⠋⠀⠈⢦⠠⠚⠁⠀⡁⢂⠀⠖⠀⠏
⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡈⠂⠀⣴⠇⠦⠈⣃⣴⠟⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠖⢁⠴⠋⠀⠀⠀⢀⠀⢀⡜⠁⠀⠀⢿⡶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢠⡤⠞⠁⠀⠀⠀⠀⠀⠀⣴⣟⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣣⠀⠂⢰⡇⡇⠀⠀⠀⠀⠀⢀⡜⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣥⣿⣆⡀⠘⠃⠀⠀⠀⠀⢀⡴⠋⠀⠈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠉⠛⢻⡷⣄⣀⡀⠀⢠⡞⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠀⠙⠉⠙⠛⠛⠋⠀⠀⠀⣀⣀⣤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢷⣶⣶⣶⣶⣶⡆⣾⣿⡿⠿⠟⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠛⠀⣀⣾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣷⣶⣶⣽⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠉⠉⠉⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠱⠦⡕⠀⢘⠋⡙⠀⠀⠁⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠉⠁⠀⢀⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⡤⠀⠀⠀⣠⠞⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣾⣿⣧⣀⣀⣠⡴⠞⠁⠛⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⡈⠂⠀⠿⢧⠛⢿⠛⠉⠀⠀⠀⠀⠀⠘⠙⠲⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡿⠀⠀⠀⠀⣴⠊⠀⠀⠀⢲⡄⠀⢠⠇⠀⠀⠈⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠾⠁⠀⠘⠃⠀⠀⠅⡀⠋⡀⡀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡀⠀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠰⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣥⣭⣑⠶⣯⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠈⠀⢀⠛⠻⢷⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⠆⠀⠀⠀⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⢀⡁⠢⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡦⠀⠀⣼⠃⠀⠀⠁⢠⠁⠄⣉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠓⠤⠿⠀⠄⠀⢀⡁⠀⠀⢹⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠘⡄⠀⠈⣥⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠁
⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣁⣄⠁⠀⠀⡀⠀⠀⣀⣀⣀⡀⢀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣡⣖⣫⣍⣀⣤⣴⣤⣤⣼⣿⣯⣽⠷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣦⠾⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢾⣶⠷⠾⠷⠖⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣯⡽⢶⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀
⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠦⣦⠞⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠡⠤⠤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠐⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠂⠄⠀⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠉⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣿⠃
⠀⠀⠀⠀⡀⢴⡿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢍⠀⠀⢸⡿⠋⡐⠀⠀⠀⠀⠀⢠⣄⣀⣀⠀⠀⠀⠾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠑⠂⠉⠛⠛⠉⠁⠀⠀⠀⠀⢤⡀⠀⠈⣇⠈⠁⠀⠁⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⣀⢹⣁⡀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡏⠐⠦⢽⣄⣀⣠⠤⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⠀⠀⠀⠀⠀⡇⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣜⢂⡀⣄
⠀⢀⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠑⠪⣐⠠⡄⠀⣨⡥⠮⠀⠂⠁
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠁⠀⠀⠀⠀⠀⠀⠁⠒⠉
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠻⢿⣿⠟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢖⡛⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠁⠀⠈⠉⠳⣦⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣆⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀⠀⣀⣤⠔⠁⠀⠈⠛⠿⣶⠦⣄⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡠⠞⣿⣟⡋⠀⠀⠀⠀⠀⣀⣀⡿⠀⠘⠀⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⡀⠀⢠⡏⠀⠰⠘⠋⠀⠀⠀⠀⠀⠀⠈⢛⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⠛⠩⡇⠘⠀⠀⠀⠀⠀⠀⣄⡉⠳⠄⢀⡽⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠻⡞⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠼⣿⡁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠘⣧⠀⠀⠀⠹⡄⠀⠀⠀⣦⠀⠀⠀⠀⢀⡶⣤⡼⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⠀⠀⠀⢻⡀⠀⠀⠀⢹⡄⠀⣠⠞⢷⣄⣀⢀⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⢸⡇⠀⠀⠀⢀⣷⡎⢸⣆⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣰⡟⢹⢻⣄⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢠⣟⠀⢀⡼⢿⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠏⠉⠛⠋⢀⠈⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⣿⠀⠀⣠⣀⡀⠀⢹⡆⠀⢀⠀⠀⠀⢠⣔⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⢸⡇⠀⠀⢀⡼⣩⠛⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠀⠀⣀⣈⣲⣕⣤⡀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣤⣴⣾⣾⣧⣤⣤⠤⠤⠾⠟⠻⠛⠛⢷⡖⠚⠉⠀⠀⠐⠲⠷⢽⣦⡀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠖⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠣⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠰⣤⣙⠢⣄⠀⠀⠀⠈⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⣿⡀⠀⠀⠀⠀⢀⣀⠓⣲⡾⠓⠾⣧⡂⡀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣧⣤⡴⠞⠉⠛⠛⠒⠶⠶⠦⠼⠛⠉⠀⠀⠀⠸⠅⣊⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢪⣷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⡼⠋⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⠜⠀⠞⠁
⠀⣀⠤⠀⠀⠀⠀⠀⠠⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⣄⠀⣠⠤⠤⠤⠤⠤⠤⠤⠀⠴⠶⠶⠶⠶⠶⠦⠤⠄⢤⡀⠀⢀⡼⠁⠀⣠⠊
⢸⠁⣀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠤⠤⠄⠄⠬⣿⠥⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⡤⠀⢹⣠⡞⠀⢀⡴⠁
⢸⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡇⡼⠋⠀⣠⠟⣠
⢸⡀⡟⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠀⠤⣤⣤⠤⠊⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠭⠭⢹⣇⡟⠁⢀⡼⣣⠞⠃
⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠔⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣰⠋⠀⢠⠟⢱⡅
⢸⡇⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⣈⠁⠀⠈⠉⠉⠉⢡⡽⠁⠀⣴⢿
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⣼⠁⢸
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡴⠁⠀⣠⠊⣿⠀⢸
⢸⡇⠘⠷⠀⠀⢉⣉⣉⣁⣀⣀⡀⠀⠀⠀⠀⠈⠁⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁⠀⡰⠁⠀⠛⠀⢸
⢸⡇⠘⢤⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣠⠟⠁⠀⢸⣉⠀⢹
⢸⡇⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠁⠀⠀⠀⢸⣟⠀⠹
⢸⡇⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣾
⢸⡇⢠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿
⠘⡇⠀⢿⡄⠀⠀⠠⠤⠤⠤⠄⠴⠶⠀⠶⠶⠀⠐⠲⣾⡴⠶⠀⠰⠶⠄⠀⠤⠤⠠⠤⠤⠤⡤⠤⠤⠼⠗⠀⠈
⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞⠁⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠐⠐⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⣤⣀⠀⠀⠀⠀⠀⠨⠆⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⡀⣈⡉⠻⣄⣰⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠆
⠀⠀⠀⣸⣧⠀⠻⣿⢋⣉⣩⠽⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠮⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⣿⣿⡇⣶⣫⢘⣟⢒⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢩⠟⠁⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢿⠛⣷⣿⡿⣆⣻⣏⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠄⠀⠀⠀⠠⠀⣀⠀⠠⠈
⠀⠀⠀⢈⣧⡀⠘⡗⠊⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣗⡀⠀⠀⠀⣁⡌⢱⡄⠩⣐
⠀⠀⣠⣿⡗⢿⡳⢿⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠂⠤⢼⣿⣿⡛⣡⠗⡣
⠀⠀⣿⣿⣿⣄⠸⣿⠢⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠴⠖⢀⠀⠉⣉⣨⠅
⠀⠀⣿⣿⣿⣿⣦⣿⣧⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠲⣶⣯⣍⡙⠿⠃⠂⠀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⠛⠋⠉⠉⠀⠀⠀⠁
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣾⠟⠛⢿⣦⣄
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣰⣿⣿⣤⡀⠀⣸⣿
⠀⠀⣿⡇⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⢸⣿⠀⠀⢀⣼⡿⢁⣼⡿⢻⣿⡿⠁
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⢠⣾⠟⢠⣾⠟⢠⣾⠟
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⡿⠋⣰⣿⠋⣰⣿⠋
⠀⠀⣿⡇⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢸⣿⡟⢁⣼⡿⢁⣼⡿⠁
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⢠⣾⠏⢠⣾⠟
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⣰⡿⠃⣰⡿⠃
⠀⠀⣿⡇⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢀⣾⡟⢁⣼⡟⢁⣼⡟⠁
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⣠⣾⠏⢠⣾⠏
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣃⣴⡿⠃⣴⣿⠃
⠀⠀⣿⡇⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡟⠻⣿⣧⣾⡿⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⣤⣾⠟⠋⢸⣿⡿⠿⠿⠿⠿⢿⣶
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⣼⡿
⠀⠀⢿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣷⣦⣴⣾⠟⠁
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣛⣛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡇⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⣀⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣴⣾⣿⣿⣷⣌⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣮⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⢿⡄⠀⠀⠀⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⢹⡍⠛⠻⠈⠄⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⡀
⠀⠀⠀⠀⠐⠀⢠⠗⠀⠀⠀⠀⠀⠉⠓⠤⠀⡀⠀⠀⠈⣃⣀⣷⡋⠀⢀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⠁⡙⣂⣴⠄⠀⠶⠦⠶⣤⡤⣴⣞⣿⣣⣄⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢚⣛⣰⣬⣀⣀⣁⣀⣀⣀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠂⠀⠂⠩
⠀⠀⠀⠀⠀⢰⠂⠄⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣯
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⡝⠓⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⠀⢀⣀⢠⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⢀⠀⢀⠓⠂⠈⠟⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠠⠞⢻⠄⠀⠀⡇⠁⠆⠠⠀⠸⠀⠈⠀⠀⣋⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⣀⣄⠁⠀⠀⡃⠀⠀⠠⠈⠘⣖⠣⠉⣀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⣼⠿⠿⠦⢄⠀⠙⠛⠛⠛⠻⡆⠉⠀⠀⠁⠈⠐⠊⠐⠀⠐⠀⠒⠀⠀⠐⠀⠀⠛⠻⠗⠒⠂⠈⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⡏⠀⠀⠀⢸⣷⡆⠀⠀⠀⠀⣿⠉⠙⠁⠀⠈⣿⡿⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣯⡍⠀⢹⠁⠈⠄⠁⠀⠉⠀⠈⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⢻⡇⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⢸⠀⠐⠁⠠⠀⠄⠀⠀⣿⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠖⣌⠐⠁⠀⠀⠀⡹⠇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⠀⠀⠰⣦⣤⣤⣼⣀⣐⢀⣠⣥⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⢸⡁⠀⠀⠨⠀⠀⠀⢠⡶⡗⠀⡄⠀⠀⣈⠀⠀⠀⢠⣿⡟⠟⠁⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⣦⠀⠀⢠⠀⠀⠇⠀⠀⠀⠀⡆⠀⠀⠇⠀⠀⠀⠐⡿⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⣸⡄⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠆⠀⠀⣀⣴⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠈⠉⢉⡀⠑⠀⢀⠀⢀⠁⠀⠈⠁⠀⠀⠀⠄⠀⢀⠁⠀⢨⠁⠀⠉⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⢿⣷⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⢀⠐⠛⠋⠀⠉⠈⠉⠉⠙⠻⠯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢢⡀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⢀⡄⠀⡀⢀⡈⠀⠀⠤⠛⠈⠁⠀⣄⠀⠀⣾⣿⣿⢿⣻⠀⠉⠃⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⡄⠀⠀⠀⠘⠛⠀⡘⠛⠃⠀⠀⠘⠁⠶⠀⠀⢀⠀⠀⠛⠛⠛⠋⠁⡀⠀⠀⠘⢡⣿⡦
⠀⠀⠀⢀⢀⡀⠀⠀⠠⠀⢰⣿⣆⠀⠀⠀⠀⠉⠄⠀⠀⢀⡀⠀⠴⠀⠀⠂⠀⠀⡉⠀⠀⠀⠠⢀⠕⠛⣈⡃
⠀⠀⠀⠙⠉⠁⠠⠄⢠⣤⠘⠋⠉⠀⢠⣶⡖⠀⣤⣄⠀⠿⠟⠁⠀⠀⠀⢰⠀⠄⢀⠄
⠀⠀⢈⠂⠀⠀⢀⡀⠈⠙⠀⠶⠿⠀⠿⣿⠀⢠⣿⡇⠀⠾⠀⠀⠁⠈⠊⠄⡠⠀⠠⠄⠀⡀⠀⠁⠀⠀⡂⠀⠀⠈⠁
⠀⢀⡠⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠀⢨⡍⠉⣀⠀⠀⠀⠁⠀⣦⠀⠀⠀⠀⢀⠂⢀⡄
⠀⠀⠀⠁⠀⠀⢀⠀⠀⢀⠀⠄⠄⠀⠃⠀⢸⡇⠀⢨⠗⠰⠄⣦⠀⣿⠀⠹⢢⡇⠀⣰⣾⣧⠀⠀⠀⠀⠀⣠⣶⡈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠁⠀⠀⠀⢸⡇⢀⠸⣀⠀⡆⣿⠀⣻⡄⠀⣾⠁⢀⠀⠉⠛⠀⠰⢦⣤⠀⠻⠟⠁
⠀⣀⠤⡀⢴⡆⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠙⣁⠀⠙⠋⠈⠉⠉⡀⣉⠁⠴⠟⠀⠘⣿⣶⠂⠰⣿⣷⣷⠀⡾⠂
⠀⠄⠄⠇⠈⠙⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠚⠁⠛⠃⠀⢀⠀⠀⠀⠀⢀⣟⣾⢾⣮⠁⢀⣬⣄
⠀⡀⠅⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⠄⠀⠠⠊⠀⠀⠠⠆⠀⠀⠘⠣⠄⠀⠸⣭⣿⣿⡗⠀⠈⠛⠋
⠀⠀⠀⠒⠒⢒⠂⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⠀⠀⠀⢀⣤⣶⣷⣦⠀⠀⠀⠰⠂⠓⠀⠀⣛⣳⠶⣶⠅⠀⠀⠀⡾⠖
⠰⠶⠓⠒⡓⠒⠓⢈⠀⠀⠂⠠⠤⠀⠘⠀⠀⡤⢠⡠⣄⠛⠿⠟⠉⠀⠀⠀⠠⠀⣴⡆⠠⢼⣭⣭⣟⠀⠀⢠⠒⣀⡀
⢈⡙⡒⠘⠀⠀⠠⠀⠀⠈⠉⣁⡳⡀⠀⠀⠀⠀⠄⠈⠃⠀⠀⠀⠀⠀⠀⠋⠀⠀⡀⠀⠼⠒⠱⠹⡛⠀⠀⠘⠀⠡⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣷⡟⣩⣥⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⠉⠈⠉⠉⠙⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⡤⠀⣀⠒⠒⠲⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠁⢠⡟⠊⠋⠃⢘⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠬⠿⠀⠀⢀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢛⣒⡶⠀⢀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⠉⠂⣠⢾⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣿⡿⣹⣏⠳⣞⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠀⣰⣿⣻⣿⣿⣿⣿⣿⡄⠀⢹⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣾⢸⣿⣿⣿⣿⣿⣿⣿⣷⣠⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢸⡏⣷⠸⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢰⠂⠀⢿⣄⣻⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣾⡟⠃⠀⠛⠦⡝⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⢺⠃⠀⡄⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣄⠀⠀⠀⠀⠐⣝⢦⣀
⠀⠀⠀⠀⠀⢀⣞⠄⣸⠀⣸⠀⢹⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⡆⠸⣦⠀⠳⡄⠀⠈⠡⠾⢷⣶⢦⡀
⠀⠀⠀⠀⠀⡸⡌⢀⡇⠀⡇⢠⣄⣿⣶⠲⢾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠈⠁⣀⠀⢀⣴⠟⠋⠉⠁⠈⢷⣀
⠀⠀⠀⠀⢰⢡⠁⣸⠀⣸⠀⠀⠉⣿⣿⣶⣤⣌⡑⠢⢍⣉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⢦⣬⣼⣿⡟⠴⢋⣥⣶⠀⢀⣐⠳⣤⢤⣿
⠀⠀⠀⠀⡆⡘⣠⡏⢠⡇⠀⠀⠀⣧⣉⠉⠉⠉⠛⠳⢶⣤⣉⣳⣶⣤⣤⣉⣙⠛⠿⢿⣿⣿⠿⠤⢤⣄⣀⣉⣉⣽⣋⠵⠐⠊⠋⣿⣿⣷⣝⢻⣿
⠀⠀⠀⢸⠁⡗⣿⡁⣈⣠⣴⣒⡡⠞⠋⠀⠀⠀⠒⠒⠀⠼⠟⠻⢿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣤⣄⡀⠀⠀⠀⠈⠙⢻⠒⠂⢙⣀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⣾⢰⣷⣫⡜⣫⡿⠋⠁⠀⠀⠀⢀⣒⣶⣶⣶⣶⣬⣥⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠈⢯⡍⠥⠩⣾⣿⣿⣿⠿⠁
⠀⠀⠀⡇⢸⣿⠏⢰⡏⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣉⣉⠉⠙⠢⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣴⣾⣿⣾⣾⣿⣿⣿⠃
⠀⠀⠀⡇⢸⠛⠀⣈⣧⡠⠴⠾⠿⠛⠛⠋⠭⠭⠬⠿⠛⠻⣀⣀⡹⠿⠿⢟⣛⣿⡾⢿⣿⣿⣿⣿⣿⣿⣿⡛⣿⡿⣽⣿⣿⣿⣿⡿⠿⠛
⠀⠀⢴⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠚⠛⠻⠿⠿⢿⣿⣇⡿⠧⠽⠛⠉⠈
⠀⠀⠀⠙⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣄⣠⣀⣀
⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠖⠚⢿⠿⠇⣠⠙⠻⠿⢿⢶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠼⡸⠟⢩⣈⣨⣯⡽⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⡾⣉⠺⣿⠟⠉⠑⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠋⠉⢀⡴⢋⣼⣿⣷⡖⣠⠆⡀⠀⠀⠀⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⢧⡴⢋⣴⣿⣿⣿⣿⠿⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡼⢋⣴⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⡰⠁⣠⣠⠋⡴⠟⠁⠰⡰⡄⠀⠀⠀⠀⠀⡄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠂⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⢠⠃⢀⡽⠀⠻⡀⠀⠀⠀⢈⣥⣤⣶⣶⣾⣿⣿⠿⣵⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠎⣠⢾⣅⣀⣐⣡⣤⣶⣿⣿⣿⣿⣿⣿⠙⣿⣿⣷⣄⣀⠙⠓⢦⣤⡀⠀⠀⠀⣠⣆⣀⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⣿⣷⡿⠉⣿⠛⠋⠛⢿⠿⣿⣿⣿⣿⣿⣆⣿⡿⣿⣿⣿⣿⣷⣶⣾⣷⣾⣴⡾⢻⣿⡇⠀⣸⣿⣿⣿⠀⢻⡏⣿⣿⢹⢸
⠀⠀⢀⣾⠿⠋⠀⠀⠙⠻⣄⠑⠌⡿⠛⠻⣿⡛⠒⠤⣍⠻⢦⣀⠘⠿⣿⣿⢿⣿⡛⠻⠍⢿⣿⠀⠀⣿⣿⣿⣿⣷⣾⣸⣿⣿⠘⠈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⢧⡄⠀⠘⣧⣤⣤⠼⠀⠀⠉⠙⠛⠲⠖⠛⠛⠉⠛⠛⠚⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣀⡀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠉⢢⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠹⡵⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠆⠦⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⢦⢄⣀⣀⠀⠀⠀⠀⠀⢳⢸⡼⢀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⠠⢀⣀⡀⠉⠉⠒⠒⠤⢤⠤⠔⠒⠊⠉⣁⣀⡠⠤⠂⠀⠈⠙⣿⠀⠀⠀⠀⠀⣗⠭⢹⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⠀⢀⣈⠉⠒⠂⠤⠄⢸⠀⠔⠒⠈⣉⣀⠤⠤⠀⠀⢠⠀⠀⣿⠀⠀⠀⠀⢸⡎⡍⡹
⠀⠀⠀⠀⠀⠀⢸⡇⢀⢸⠀⠠⢀⣀⠀⠈⠉⠒⠒⠀⠰⠀⠐⠉⠉⢀⣀⡀⠤⠤⠄⢸⠀⠀⢻⠀⠀⠀⠀⡿⢠⢁⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⠀⢀⣀⣈⠉⠉⠁⠒⠀⠀⠀⠈⠉⢉⣀⣀⣀⠤⠤⠀⢸⢰⠀⢸⠀⠀⠀⣸⡏⡌⡌
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠘⠀⠀⠀⠀⠀⢈⡉⠁⠒⠀⢸⠀⠈⠉⣀⡠⠄⠀⠀⠀⠀⠘⢸⠀⢸⠀⠀⠀⡿⢰⢠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠒⠂⠤⠤⣈⣉⣁⠀⢸⠀⢈⣁⠤⠤⠐⠒⠒⠉⠁⠀⢸⠀⢸⠀⠀⣸⡇⡎⡌
⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⡆⠀⡿⣰⣰
⠀⠀⠀⠀⠀⠀⢸⢰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡆⡰⠁
⠀⠀⠀⠀⠀⠀⢸⢸⣐⣚⣳⣶⣶⣤⣤⣀⣀⡀⠀⠀⢸⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣛⣒⣠⢸⡇⠀⠋
⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⠚⠒⠛⠛⠛⠛⠋⠛⠉⠉⠉⠉⠛⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⢸⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠋⠉⠙⠻⣿⣿⣿⣿⣿⣿⢸⣿⣯⣥⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠉⠛⢿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠈⠙⠻⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⢿⣿⣿⣷⣦⣀⠀⠀⠀⠈⠙⠚⢿⣿⣿⣿⣿⣿⡯⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠈⠙⠻⠋⠁
⠀⠀⠀⠀⠀⢀⠀⡂⠂⢀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠠⢤⡢⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠈⠄⢀⡀⣀⣀⣀⠀⣀⠀⠠⣌⠈⢣⠈⢃⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠒⠒⠒⠂⠀⠀⠀⢸⠀⠀⠀⠂⠊⠀⠀⠀⠀⠀⠈⠂⢠⣌⣥⡀⠱⠤⣂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣱⡀⠀⠀⠀⠀⠀⠀⠖⢤⢸⠛⠀⠀⠈⠉⠑⠀⠀⡈⠁⠀⠱⠀⠀⠀⠈⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠸⣮⡀⠀⠐⢾⡆⠀⠀⠀⠙⠿⠠⠊⠀⠀⠀⠈⢩⠀⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢲⠒⠂⠀⠙⠉⢁⠑⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⣡⠜⠀⠀⠀⠀⠀⠀⠀⠦⢀⡀
⠀⠀⠀⡀⠄⠤⠠⠤⠀⠚⢄⡀⠐⡀⡂⠀⠀⠀⠀⠀⠑⠂⠓⡀⠀⠀⢀⠀⠀⠀⠀⣀⠠⠔⠋⡅⠰⠁⠀⢀⣀⣀⣀⡔⣈⡄⢀⣀⣀⡈⢂⡀
⠀⡰⠉⠀⠀⠀⠀⠀⠀⠀⡀⠡⡀⠐⡘⡄⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠈⠛⠲⡦⢄⠀⣀⠀⡤⠂⣀⠡⠌⠢⢄⠀⠀⠀⠀⢀⣤⣊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠣⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠠⠁⠀⠀⠈⠓⠒⠒⠂⠒⠄⡉⠁⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⢒⡢⠤⠤⠄⣀⣀⣀⣀⢀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠞⢟⠃⠀⠀⠀⠀⠀⢀⢀⣀⣀⡀⠠⠌⠐⠠⢀⠀⠀⠀⠀⠂⠀⠄⠐⠒⠀⠀⣀⣀⣒⣂⣉⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠁⠀⠀⠛⠴⠒⠒⠀⠛⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠛⠓⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣁⠢⠄⠀⠀⠀⠀⠂⠀⢉⣁⣀⢀⣠⣀⡤⠤⠤⠤⠒⠒⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠧⠔⠒⠒⠊⠉⠉⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠻⡍⢉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠏⢳⡄⢹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⠏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣼⡃⢸⠁⡇⠠⠤⠤⠀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠀⢸⠀⠀⢀⡟⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢸⡀⠇⠀⠶⠶⠶⠶⠶⠶⠆⠀⠶⠶⠦⠀⠀⢸⠀⢀⡞⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠶⠶⠶⠶⠀⠰⠶⠶⠶⠶⠖⠀⠀⢸⢀⡞⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⢸⡞⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠒⠒⠀⠀⡾⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠂⠐⠂⠐⠂⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡼⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣀⣐⣂⣐⣂⣐⣂⣘⣃⡀⠀⠀⢀⣀⣀⣀⠀⣾⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡃⠀⢈⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢰⠋⠀⢹⡄⠀⠀⠀⠀⠠⠤⠀⠤⠤⠤⠤⠤⠤⠤⠌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⢦⡼⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣤⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡝⢿⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⠻⢶⣤⣀⣠⣴⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣀⣤⣠⣾⣿⣿⡿⠋⠉⠛⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⣠⡴⢟⣽⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣤⣀⡀⢰⡶⠶⠶⠶⢶⡆⠀⠀⠈⣿⣿⡀
⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣠⡾⢋⣴⣿⣿⣿⠟⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⣷⠀⠀⠀⠀⢿⠀⠀⠀⢻⣿⡇
⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⢠⣾⣫⣴⣿⣿⣿⠟⠁⠀⣠⡾⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣧
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⡇⠀⢀⣾⣿⣿⣿⣿⣯⣥⣤⣤⣘⣫⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿
⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡞⣁⣠⣾⠟⠛⠛⢿⡍⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡟
⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣧⣀⡀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⣄⠀⣸⡇⠀⠀⣼⣿⡇
⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⡻⣿⠟⠉⠉⠛⠛⠛⠷⠶⠶⠶⢶⠶⠶⠶⠶⠿⠶⢶⣤⣤⣤⣤⣶⡶⠶⠶⠿⠆⢰⡏⠺⣧⡽⢃⣿⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⢀⡘⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠃⠀⠀⣼⣿⡟
⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⢀⡀⠈⠁⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⢻⣿⣧⠀⢰⣟⠁⢀⣠⣴⠾⠛⠉⠉⠙⠻⣦⣄⣀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣧⡈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣶⣶⣆⡀⣠⣄⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡿⣷⣍⠙⠶⣿⣿⣿⣙⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣮⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠓⠶⠖⢶⣯⣝⣻⣟⢿⣿⣿⣿⣿⣿⣿⣿⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⡿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠦⣙⠛⢿⡿⣿⣇⣻⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣮⢉⠲⣿⡜⣿⣿⣿⣿⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠂⠘⢿⡜⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣻⣿⣿⣆⡙⣿⡜⣿⣿⢸⣿⣿⠟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⡻⢿⡜⣿⣿⡟⠋⣼⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⣿⣿⣿⣿⣶⣿⡝⣿⣵⡶⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⢿⣷⡼⣿⡇⠸⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡻⣷⣤⣤⣬⣤⣈⣷⡹⣿⡇⢸⡟⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣛⣛⠛⢻⣿⣷⣹⣿⡿⠁⠀⣸⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡿⣿⣿⣿⣿⣿⠿⣿⣧⢹⣿⡗⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣶⣾⠟⠛⢶⣿⣧⢻⣷⣿⠏⠼⢿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣯⣅⣀⠻⢿⣧⢿⣯⡤⣶⣿⣿⡿⠃
⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⣠⡎⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣯⣙⢿⣯⣿⣷⠺⣿⡿⠛
⠀⠀⠀⣰⣿⣿⣷⣆⣄⢀⣠⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⣙⣿⣿⣾⡻⠁
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣟⣴⣦⣤⡀⠀⣸⡿⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠉⠙⢻⣏⣻⣟⣿⡅
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⡟⠁⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡙⣿⠃
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⢰⣿⣿⣿⣿⠟⡛⢻⣿⣿⣿⣿⣿⣿⣾⣿⢇⢀⡟⠀⠀⠴⣄⡀⠀⠀⠀⠀⠀⢀⡀⠠⢄⡀
⠘⠿⢿⣿⣏⠀⢒⣠⡟⠡⣀⠙⣿⣿⠿⠉⢞⣼⠷⠀⠀⠀⠈⠙⠷⠆⠀⠀⠀⠪⡵⠐⠋⠀⠀⢀⡙⠦⢄⡀⠀⠀⠀⠁⡀
⠀⠀⠀⠠⠬⣙⣛⡻⠷⣤⣤⣼⣿⣿⣶⠞⠉⠀⠹⠓⢤⢀⠀⠀⠀⠀⠀⠀⠀⡴⠤⠄⠀⣀⠀⠀⠘⠣⣶⡀⠀⠀⢠⠀⠃
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⡝⣀⠃⠀⠀⠀⠀⠀⠀⠈⠑⠐⠢⠄⠀⠀⠈⠇⠀⠀⢀⡙⠁⠀⠀⠀⠈⠉⠶⣄⡀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠰⠀⠀⢀⣴⠀⠀⡸⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⢰⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣬⣧⣴⣧⣴⣦⣴⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⢤⡤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡋⠉⠙⡄⠀⠄⣸⠀⠀⠀⠀⠀⣈⠝⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⣀⠑⢄⢠⢱⢨⢀⠇⢠⠆⢀⡤⠊⠁⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠒⠢⠤⣀⣈⠑⠬⣳⠗⠛⠺⢴⡡⠖⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠤⠤⠤⠭⢵⡇⠀⠀⠀⠈⡯⠭⠄⠒⠒⠒⠒⠒⠉⠉⠉⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⣀⣀⠠⠤⠔⠒⠒⣈⠭⢳⣄⣀⣀⣴⡫⢍⣁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⡜⠉⠉⠀⠀⠀⠀⡠⠔⠊⠁⠀⠐⢡⢻⢹⢹⠱⡝⠂⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⣀⠏⠀⠘⡀⡇⠘⢆⠀⠀⢀⣀⡀⠉⠒⠤⣤⢖⢤⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⠀⣴⣦⠀⢣⠀⡝⠀⠀⠀⡇⠀⠀⠈⠣⣀⣞⢹⣿⡄⠀⠀⢸⡎⡴⠸⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣏⢸⠀⣿⠘⡄⠨⠀⡇⠀⠀⠀⣧⠲⢦⡀⠀⢙⡈⠈⠇⣧⠀⠀⠀⡇⡀⣶⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠾⣦⡇⠀⣇⣸⣴⠃⠀⠀⠀⣿⡆⢈⡃⠀⡀⡇⡆⠀⡋⡀⠀⣰⣇⡗⡶⠐⠒⠒⠀⢸⠇
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⣃⠀⣿⠛⠁⠀⠀⠀⠀⢺⣁⣨⡃⠀⡇⠃⡇⠀⠁⣃⡽⢛⣾⠝⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠹⡉⠀⠒⠒⠛⠡⠽⢴⣇⠀⠉⠁⠐⠒⠀⠬⢷⣿⣷⣊⡃⡇⡇⣆⠀⣗⡾⠟⠛⠒⠂⣤⡄⠤⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠤⠤⠐⠒⠂⠉⠉⠉⠉⠁⠉⠁⠈⠉⠛⠻⢷⡄⡀⣿⢠⠤⠤⠤⠒⠊⠉⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠉⠉⠁⠀⠐⠒⠠⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠠⣧⡇⣿⡘⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⠤⠀⣿⣧⣧⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⣠⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⣀⣀⡀⠀⣀⣀⣀⣀⡤⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⣶⣟⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⣡⣾⣿⣿⣿⣿⣿⣾⣯⣌⣻⡿⠻⣿⠟⠿⣿⡿⢿⡿⠛⢿⣟⣙⣿⣥⣭⣿⣿⣦⡀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⠉⡙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢨⣐⠎⡁
⠀⠀⠀⠀⠀⢀⣄⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣤⣅⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠊
⠀⠀⠀⠀⡴⢌⠲⠄⢠⣾⣿⣿⣿⣁⣹⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢺⣿⣿⣿⣫⣥⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠉⠒⠙⢢⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⢯⣹⣿⣿⠛⣻⣿⣿⣛⡷⣶⣶⣦⡾⢿⣿⣿⣿⡟⣻⡿⡿⠿⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣏⡼⠋⠀⣀⣠⣼⣿⡇⠀⠹⣿⣿⡾⠋⠀⣹⣿⠀⠈⣿⣿⠃⠀⣿⣿⣿⣿⡟⠀⢀⣠⣤⣤⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀⠘⢿⣿⣿⡟⠁⠀⠀⢻⠏⠀⠀⢀⣿⡟⠀⢸⣿⡿⠀⢰⣿⠋⢻⣿⡇⠀⠸⠿⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣽⣦⠀⠙⣿⡇⠀⣸⡀⠀⢀⡖⠀⣠⣿⠇⠀⣾⣿⠃⠀⣾⣿⣿⣿⡿⠀⢠⣤⣤⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡈⠛⠿⠛⠁⢠⣿⡁⢀⣿⣿⣶⣿⡇⢀⣿⣿⠀⢠⣿⣿⠀⠘⠛⠛⠋⣻⡇⠀⠘⠛⠛⢉⣿⣿⣿⣿⣿⣿⠐⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡶⠶⣶⣾⣿⣿⣷⣾⣿⣧⣿⣿⣷⣾⣿⣿⣷⣾⣿⣭⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣟⣩⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣶⣾⣿⣿⣧⡟⢡⡌⢻⡈⣿⢀⡿⠉⡤⢹⡇⢉⣴⡏⢨⡟⢈⡟⢁⡌⢠⣿⠋⣠⠈⣿⠁⣽⠃⣩⣿⣿⣷⣷⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠶⠞⣿⣇⢀⣼⡇⠲⠖⣻⡃⠸⣿⡀⠋⠀⣼⡁⠛⡁⣼⣇⠘⢃⢠⣇⠘⠁⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⢿⠿⣰⣿⣿⡿⣿⣾⣿⣿⣿⣿⡛⠿⢁⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣤⣄⠈⢿⣿⣿⣿⣿⣿⣿⡻⣿⣿⢿⣿⡟⣿⢫⣶⣶⣿⣿⣷⣼⣿⡟⠉⠀⠈⠁⠘⠛⠻⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠉⠛⠳⠈⢿⣿⣿⣿⣿⣿⣾⣧⣶⣾⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⠠⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⢿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣦⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣌⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠈⠙⠛⠻⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣷⣤⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠈⠙⠓⠶⢤⣴⣿⣿⣴⠶⠾⠿⠛⡍⠀⠀⠀⣠⣶⣿⠿⠿⠿⠿⠿⠿⠿⠦⣄⠀⠀⠈⠿⠿⠿⠟⠛⠛⠛⠛⠻⠛⠛⠛⠻⠿⠿⠿⠂
⠀⠀⠀⠀⢀⣀⣠⣤⣤⣼⣿⣿⣿⣆⣀⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁
⠀⠀⠀⠀⠈⠉⠉⣝⣻⣿⣿⣿⣟⢻⠏⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⣿⡇⠀⠙⡄⠠⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠈⠙⣿⣿⣿⣦
⠀⠀⠀⠀⠈⠈⠐⠀⠒⠚⠛⠛⠀⠂⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠐⠀⠀⠉⠛⠛⠋⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣴⢦⠀⠘⠶⣦⣤⣤⣴⣶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢀⣀⣽⠽⠿⣿⣿⣧⣴⣾⠀⠀⠀⠀⠀⣴⠟⢣⠃
⠀⠀⠀⠀⠀⢻⣿⡛⡗⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢄⠀⢸⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣾⢟⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⢡⣮⣿⡧⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠊⠡⣼⠔⠅⢸⣿⣿⣿⡿⠟⠋⠀⠀⣴⣿⣷⠉⠀⠀⠀⠀⠀⠀⠄⠀⢀⢀⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠰⠶⣿⣧⠀⠀⠀⠀⠀⠉⠀⠀⠸⠛⠉⠁⠀⠀⠀⠀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢶⠷⠿⠿⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀⣿⣷⠀⠀⠀⠀⠀⢀⣤⣬⣄⠀⡀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢘⣿⡇
⠀⠀⢀⡀⠀⠀⡖⢹⣷⣄⠀⡀⠀⢸⠀⠛⣿⣿⠀⠀⠀⠀⢰⠿⠯⠹⠆⢁⠈⡄⠀⢀⡾⠀⠀⠀⠀⠀⢀⠤⠒⣉⣄⡶⠞⠁⠀⠀⠀⠘⣿⡇
⠀⠀⠁⠈⢦⡠⠛⣻⣿⣷⠂⡰⠀⢸⠀⠀⢻⣿⡀⠀⠀⢀⠸⣷⡖⣴⣷⠞⠀⠁⠀⣼⠃⠀⢀⠠⢐⣊⣤⣶⠿⠛⠉⠀⡰⢦⣄⠀⠀⠀⠙⠁
⠀⢀⣀⡀⠈⠧⣄⣾⣿⡿⠋⠀⠀⠸⠀⠀⣸⢿⡅⠀⠀⠀⠳⠝⠛⠞⡋⠀⠊⠀⠰⠃⠀⠀⠰⣾⡿⠛⠉⠀⠀⠀⠀⠀⠂⠸⠗
⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢱⠁⠀⠀⠀⠀⠀⠀⠀⠘⠆⢀⡔⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠂⠉⠉⢖⠄⠀⠸⠀⠀⠀⠐⠀⠐⠒⠂⢠⠔⠁⡼⠢⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢀⡀⢀⠀⢘⠀⢸⠀⣈⣀⣀⣀⡈⢁⠔⠁⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠄⠓⠋⡠⠂⠀⢈⠀⠀⠀⠀⡀⢀⠂⢤⠊⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢾⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠄⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠉⠀⠀⠀⠀⠀⠀⠀⠞⠀⠱⠂⠀⠀⠀⠀⠀⠀⠈⠂




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⠲⠶⠳⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠙⠃⣠⠾⡾⣋⢨⢈⣤⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣘⣡⣄⣠⡄⠉⠉⠻⡅⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⢃⠀⠈⠙⠳⠃⠦⡀⠄⣀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠰⠛⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡤⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠤⠒⠁⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠙⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠉⡀⠸⢄⣀⡴⠋⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⡴⠃⠀⠀⠐⠀⠀⠘⠁⠀⠀⠀⡠⠜⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⣀⡴⣶⠟⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢁⠃⠀⠀⠀⠀⢀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⢀⣠⠆⠑⣻⡿⠡⡀⠀⣜⠳⠒⠶⠶⠦⠤⠀⢀⣰⠇⢰⠀⠀⠀⠂⠀⠀⠈⠀⠀⠁⢠⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⣶⢡⡁⠀⠀⠁⠀⠀⠀⠙⢣⡤⣀⠀⠀⠁⠀⠀⠀⠉⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠟⢦⡀⠄⠀⠠⠀⢳
⣀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢠⡉⠢⡀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠲⢤⡀⠄⠁⣠⡾⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⢀⡏⠀⠀⠑⢄
⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠊⠉⢦⡈⢂⡀⠀⠀⠀⣀⠐⡏⠀⠀⠀⢈⣵⠿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠈⡇⠀⠀⠐
⠀⢸⡟⠿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠱⡄⠀⠱⣄⣳⣄⠴⣋⡤⠖⠁⣀⣤⠾⠋⣁⣴⠋⠈⠁⠀⠈⠀⠀⠈⠀⠀⠤⡴⠀⠀⠀⠀⠀⡼⠃
⠀⢸⡇⠀⠈⠛⢷⣄⠀⠘⡄⡆⠀⠀⠘⣆⠀⣀⠤⠒⠉⠁⣠⣴⠞⠛⠛⠉⠉⠉⠈⠳⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⢀⠀⢈⡼
⠀⢸⡇⠀⠀⠀⠀⠙⠳⣦⡁⠸⠆⠀⠀⠈⠋⠁⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠙⠓⠦⣄⣀⡀⠀⠀⢀⡇⠀⡀⠠⢏⣀⠀⠀⣠⠞
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣀⣴⠾⠋⠁⠀⠀⡀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣸⠟⠳⣏⢰⠆⠀⡸⠖⠋⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠟⠋⢹⠀⠀⠀⠀⠲⣼⣶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠁⠀⢀⣿⠁⢀⡴⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⢀⣀⣤⣶⢮⣭⣤⣤⣤⣤⡤⢶⣿⠈⠉⠉
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠼⡟⠋⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠸⡄⠀⣿⡇⠀⠀⠀⠀⠀⣿⠃⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡏⠹⡄⠀⠀⠀⠀⠀⢳⠀⣿⡇⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⣸⡇⠀⠀⠹⡄⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢀⣿⡀⠀⠘⣿
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⣿⣀⣤⡴⠿⠛⠉⠉⢻⡏⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣀⣀⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⣠⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⢀⡼⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠋⠁⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠛⠛⠛⠛⠛⠉⠁⠀⢀⡀⠀⢠⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣅⡤⠜⠛⠒⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣉⡉⠉⠽⠦⠠⠄⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⡎⢡⣹⡷⠆⠀⠀⠀⠀⠀⠙⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣒⡊⠉⠁⠀⣀⣀⠀⠀⠀⠠⣶⡞⠁⠀⠀⠀⠀⠀⠸⣍⡳⡄⠀⣠⠤⠴⢶⣦⡄⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢀⠘⠉⠉⠉⠉⠉⠁⠀⢠⠋⠀⠀⠱⠀⠀⠀⠀⠈⠉⠻⠤⠤⠤⠐⠒⠋⡿⡀⠘⠻⠤⣤⡘⠓⢠⣤⡄⠀⠈⠳⢤⣄
⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⠢⢄⡠⠜⠀⠀⠀⠀⠀⠀⠀⣠⢤⠤⠤⠤⢼⣏⡷⠞⠟⠛⠋⠀⠚⠭⣥⡀
⠀⠀⠀⢻⣇⣀⣀⣀⣀⣄⣤⡀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⣶⣀⣀⣘⣿⡗⠀⠀⠀⠀⠀⣴⡛⠉⠁
⠀⠀⠀⣀⡼⠧⠶⠾⠶⠗⠛⠛⠛⠉⠉⠀⠀⠀⠀⢠⣴⣶⡖⠒⠒⠘⣉⡿⠝⠿⣿⣶⢶⣦⣁⠀⠀⠀⠈⣉⣻⠿⠇⠀⠀⠀⠀⢴⠖
⠀⠀⠀⣛⣶⡶⠄⠀⠶⢾⣭⡍⣉⣭⣭⣭⡽⢦⣤⠴⠲⠞⣋⣠⠴⠞⠃⠀⠀⠀⠉⠉⠳⣬⣉⣑⡒⠤⣬⣧⣤⣄⣠⣤⣀⠀⠀⠀⠑⠢⠄
⢀⡖⠋⠉⠀⢠⡤⠴⠒⠒⠲⠖⠲⠚⠛⠛⠛⢋⣷⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠈⠉⠉⠋⠛⠙⠋⠁⣠⣿⠉⠩⢿⣄⡀
⠈⠙⣷⡶⠶⠛⠷⠶⠒⠚⠛⢻⣿⣟⡀⠀⠀⠀⠳⢦⣤⣤⣤⠤⣀⣀⣠⣤⣶⡾⠛⠙⠻⢶⣦⣤⣄⣀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠈⣧⡠⣖
⠀⠀⠀⠙⠁⠉⠙⠑⠛⠓⠶⣤⠀⢾⣁⣀⣀⣰⣖⠒⠛⠛⠻⠶⣾⣿⣿⠿⠋⠀⠁⠀⠀⠀⠈⠙⠻⠿⣿⣿⣁⣀⣀⣀⣤⡄⠀⠀⠀⠉⠝
⢤⠤⠤⠤⠤⠤⠤⠤⠴⠴⠚⠁⣠⣾⠿⡿⢿⢯⣿⣶⠶⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣷⣦⣤⣤⣀
⠰⠶⠶⢦⣄⡀⠀⠰⣖⢶⢆⡴⣟⠙⠐⠁⠺⠆⠻⢿⢷⣶⡤⠤⢤⡤⠄⠐⢂⣶⣧⣄⣾⣷⣋⣉⠓⠒⠤⢤⣶⣚⣿⠟⠛⢣⣈⠙⠃⠉⠈
⠈⣩⣭⡉⣉⣠⣀⡾⢁⣽⣿⠉⡖⠀⠀⠀⠀⠀⠈⠀⠂⠉⢙⣿⣿⣧⣴⠞⠉⠉⠉⠙⠋⠙⠛⠟⠻⠛⠿⠿⠿⠿⠷⠶⠚⢋⠏
⠀⠁⠀⠙⠉⣽⡉⢀⣼⡟⠒⢴⡆⠀⠀⠀⠀⠰⣞⠙⡛⠋⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠏⠂
⠀⠀⠀⠀⣤⣿⣼⣻⢹⣧⠁⢸⡗⠀⠀⠀⠀⠀⠈⠿⠓⠤⠀⣀⣠⣤⠤⠄⢒⣶⣉⣤⣼⣤⡉⠒⠒⠀⠤⢤⠤⠒⠈⠁
⠀⠀⠀⣠⣾⣿⡝⣛⣻⡏⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣛⣶⠾⠋⠜⠋⠛⠻⠿⡿⣟⠿⡶⣶⣿⠤⣤⢤⠤⠤⢤⡄
⠀⠀⠀⠙⣿⣿⣿⡊⣿⣿⡃⣸⠇⠀⠀⠀⠲⠿⣿⣷⠲⠶⠖⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠈⢿⡇⠀⣾⣿⣅⠾⣧⠀⠀⠀⠀⠀⠘⠉⠀⠒⢢⣤⣀⣀⡀⠀⢀⣠⡤⣴⣀⡀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠁
⠀⠀⠀⠀⠀⠈⠻⠋⢻⣿⣶⣶⣧⣀⡀⠀⠀⠀⠀⠀⠀⣀⣼⢿⣽⣉⣉⣉⣉⣌⡁⢸⡟⠉⠛⠉⠉⠉⠙⣦⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣼⣿⣻⣫⣤⣤⣤⣶⢚⣛⣉⢀⣉⣡⡭⠽⠋⠛⠁⠉⠛⢯⣄⣀⠐⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⡇⠀⠀⠛⠚⠷⣎⡉⠉⠉⣿⡅⠀⠀⠀⠀⣠⣤⣴⣶⣄⠉⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣇⠀⣠⡼⠧⣶⢹⣿⡏⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠈⠉⠒⠲⢤⣄⣀⠀⠀⠀⢀⣀⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⠚⠉⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢀⡀⠻⢬⢝⡞⠿⣧⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⢉⣤⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠻⠷⢶⣾⡿⢷⡶⠶⢟⣥⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣤⠖⠋⣡⣦⡶⠖⠋⠁⢀⣠⣴⣾⡿⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣿⡶⠟⠋⠁⠀⣀⣤⠶⠛⢩⡟⢱⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⢀⣠⡴⠟⣻⣀⣀⣠⡾⠀⠈⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡠⢴⡚⠁⠀⠀⣀⣴⠾⠛⢻⡏⠉⠙⣛⡿⠟⠃⠀⣀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣀⠤⠺⡙⣆⠀⢻⣤⣶⡟⠛⠛⠃⢀⡼⣡⠔⠋⠁⠀⠀⠀⠀⠈⣽⠃
⠀⠀⠀⠀⠀⠀⠀⡟⣴⠛⠙⣆⠀⣹⣽⠶⠛⠁⠈⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⢀⣴⠾⠃
⠀⠀⠀⠀⠀⠀⣼⠃⢧⣀⣠⣿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠀⠀⢀⣶⠀⠀⣴⣿⠽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣺⠯⠿⠧⠵⣿⣋⣛⢯⢳⡀⢸⣿⣆⠀⠹⠇⠢⠝⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠈⣷⡨⢪⢎⡇⠘⣧⡿⣻⡷⡖⢟⣛⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡑⡘⡼⠁⠀⠀⠉⣉⡩⡭⡍⢴⣶⠦⢤⣤⣤⣀⡀
⠀⠀⠀⢀⡠⢲⠟⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁⡰⠟⠁⢀⣠⣶⡭⠵⠟⠿⢾⡟⡳⢷⣶⣀⣩⣍⢻⠓⡦⣄
⠀⢀⣴⡟⣳⠈⠀⠀⠀⠀⠀⣀⡴⠾⠷⡿⠯⣤⠤⣴⣿⣻⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠹⣎⠘⢰⠷⣝⢷⡄
⢀⣎⠅⢒⡍⠀⠀⠀⠀⠀⠘⠓⠒⠒⠛⠷⢭⣀⠠⢤⢹⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣌⡓⠈⠩⢻
⣾⡇⠺⢑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠩⠃⣰⣿⣦⣤⣀⡀⠀⠐⠧⠖⠀⠀⠀⠀⢿⠀⣀⡀⠑⢦
⣧⠉⠀⢢⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠣⠀⡏⢯⣷⣶⣯⣝⢆⡀⠀⠀⠀⢀⠴⠞⢻⡆⠄⡸⠇⢠
⢻⣤⠹⠀⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠀⢁⢹⣾⠉⠀⠈⠹⣿⡇⠀⠀⢀⠇⠀⠀⢸⠗⡁⠀⠀⡀
⠀⠻⣼⢠⢰⡋⠻⣦⣤⣄⣀⣀⣀⣠⣤⡶⠋⠛⣒⣁⣾⣾⣿⡷⠶⢄⣠⣿⡇⠀⠀⠘⣀⠀⢀⣿⠏⢀⠔⢊⣤
⠀⠀⠈⠛⢶⣐⡋⢍⡣⡙⢌⣋⢑⠑⡡⢖⣒⣒⡶⠾⢿⣷⣶⣾⠛⠓⠸⣿⡇⠀⠀⠀⠛⠉⢹⠟⡆⠸⠰⣣⡟
⠀⠀⠀⠀⠀⢈⣩⣽⠷⡾⢷⣶⢷⡚⠛⠉⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠿⠞⡞⣆⣡⠟
⠀⠀⠀⣠⢾⣭⣝⡭⢊⠜⣩⣭⡳⠳⠕⠦⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⣀⠴⠫⠬⠭⣴⠿⠃
⠀⠀⣼⠛⡭⢠⡤⠚⠉⠉⠉⠉⠉⠙⠓⠶⢶⣌⣉⢷⣶⢦⡄⣄⣀⣤⣿⡽⢷⡾⠻⠁⡔⣩⠥⠛⠁
⠀⢸⣯⠀⢣⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠚⠛⠣⠴⠖⠦⠶⠖⠓⠋⠉
⠀⢸⣧⣷⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠻⣟⡈⡷⡀⠀⠀⠀⠀⠀⠀⢠⣿⠄
⠀⠀⠀⠙⠿⣯⣝⠳⣦⣤⣤⣤⡴⣫⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢀⡴⠾⠶⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢠⡟⠁⠀⠀⣽⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣠⡟⠙⠳⣦⣼⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣰⠟⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⣰⠏⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣼⠏⠀⠀⢀⡾⠁⢤⣤⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢀⡾⠃⠀⠀⢠⡟⠀⠶⠶⠶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠓⠓⢠⡞⠁⠀⠀⣰⠏⠀⠚⠒⠒⠒⠒⠛⠒⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠋⢰⣟⠀⠀⠀⣼⠃⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⢸⡟⠻⣦⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢸⡷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⣼⣹⣿⣿⣻⣿⡵⣶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣯⣽⣫⡵⠉⠉⠉⠛⡽⢿⢿⣿⡷⢳⣿⣟⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣺⢿⡻⣏⠙⠀⠀⠀⠀⠀⠀⠀⠸⠵⣴⣿⣿⣽⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣷⠇⠸⠧⠀⠠⠲⠄⠀⠀⠀⠴⠒⠘⠠⢟⣯⢿⣭⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣿⠟⠁⠀⠀⠨⣥⡆⠀⠀⠄⣪⣱⠈⣷⣿⣯⡏⣹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣌⠳⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢓⠉⢿⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣷⣤⠀⠀⠀⢤⣉⣀⣁⡄⠀⢠⠇⠀⣈⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢤⡀⠀⠉⠋⠁⢀⡔⠁⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠉⠉⢙⠐⠃⠀⠈⢣⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⡜⠀⠀⢠⡔⠁⠐⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠠⠞⠀⠀⢀⠟⠀⠀⠀⠈⠂⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠳⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠱⣄⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠔⠛⠓⠂⠠⠄⢼⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠶⢤⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⢀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠓⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢄⡀⠀⠀⠀⠁⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⠀⠀⠠⠄⡸⠗⠄⡀⢀⠠⠋⠦⡅⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠘⠇⠀⢀⡀⡀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠠⠄⠄⠀⠀⠠⠄⠀⠀⠤⠤⠄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠂⠀⠄⠰⠀⠂⠄⠀⡰⠢⠲⠖⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠀⠒⠒⠒⠒⠒⠒⠀⠀⠀⠐⠒⠒⠀⠒⠒⠚

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠞⠋⠀⠀⠈⠉⠉⠛⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠠⢤⣦⣀⡀⠀⠀⠀⠀⢘⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢀⡴⠛⢻⣿⡀⠀⠀⠀⢀⡷⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⣠⡤⠹⣿⣀⡠⠞⠉⠀⠀⠀⢀⣨⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⣾⠉⠀⣠⣭⣃⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢠⡞⠃⠀⠀⣿⠀⠀⠀⠐⣤⠞⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⡴⠋⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠔⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⠤⠤⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠉⠉⣭⣽⣿⠀⠀⣀⣀⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣀⣀⣀⣐⣒⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠻⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠷⠶⠶⠶⡾⢿⠿⠿⠿⠿⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣀⠉⢣⡀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠚⠈⣷⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡞⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠧⠄⡇⢸⠀⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⡇⠀⢀⡞⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⢉⣉⣉⣉⠉⠉⣁⣀⣉⣉⡁⠀⠀⡇⠀⡼⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣉⡉⠉⣉⣀⣈⣉⣉⣉⣉⡁⠀⠀⣇⡼⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⠀⠀⡿⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⡍⠉⡍⠉⡍⠉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠤⠀⠤⠠⠄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠙⡟⠉⠉⠉⠉⠁⠀⠀⠀⠋⠉⠉⠀⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡞⠉⠻⡍⠉⠉⠉⠉⢉⣍⠉⣭⣭⣭⣭⣭⣭⡍⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢧⣰⢆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣠⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢀⣤⡶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠾⠟⠛⠛⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢀⣾⠋⠀⠀⠀⠈⣹⣿⣆⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⢠⡿⠉⠻⣷⣄⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⠇⠀⠀⠀⠀⣼⡇⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⣿⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣷⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠘⢿⣦⡀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⠁⣴⠏⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡉⠙⠛⠛⠛⠛⠉⠁⣀⣤⠾⠃⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣶⣶⣶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣤⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠃⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣀⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣶⣤⣤⣤⣶⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⣶⣶⣷⣷⣶⣴⣶⣶⣾⣶⣷⣶⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢻⣿⢿⡿⣿⢿⠿⣿⣧⣿⡿⠿⢿⠁⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠤⡀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠉⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠄⠀⢠⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠈⠁⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇