Fry Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢦⡀⠀⠀⠀⣤⠶⠖⠛⢻⡆⣠⠴⠦⢤⣤⡄⠀⣴⠛⠛⠛⠛⣻
⠀⠀⠀⠀⠀⠀⠀⢾⡏⠁⠀⠈⢧⠀⠀⠀⡇⠀⠀⠀⢸⡇⡟⠀⠀⠀⠀⣧⣠⡏⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠘⣧⠀⠀⣿⣠⣤⣄⢸⣿⡇⣠⠴⠞⠋⠉⢿⡃⠀⠀⠀⢰⡇
⠀⠀⣠⡴⢶⣄⠀⠀⠀⢘⣧⣀⣀⣼⣇⢠⡟⠁⠀⠙⣾⣿⡇⠘⣧⠀⠀⠀⠘⣧⠀⠀⠀⢸⠇
⣴⠛⠁⠀⠀⠙⢧⡀⢐⡏⠉⠉⠀⢠⡿⠚⠛⢷⡀⠀⠘⢿⡇⠀⠘⣧⠀⠀⠀⠘⣇⠀⠀⣼
⠈⠳⣄⠀⠀⠀⠀⠻⣄⡇⠀⠀⠀⢸⡇⠀⠀⢸⠷⣄⠀⢈⣷⣤⠤⠾⣆⠀⠀⠀⢹⡄⠀⣿
⠀⠀⠈⣷⠶⢷⡀⠀⠙⣇⢀⣤⢤⣸⡇⠀⠀⢸⡇⠹⣆⢸⠇⠀⠀⠀⣿⡆⠀⠀⠀⢷⠀⣿
⠀⠀⢾⣅⠀⠀⠹⣦⠀⣿⣟⠁⠀⠙⠳⣄⠀⢸⡇⠀⠹⣿⠀⠀⠀⠀⣇⢻⡄⠀⠀⠘⣇⣿⢀
⠀⠀⠀⠙⢷⣄⠀⠈⠳⣼⡝⢷⣄⠀⠀⠈⢷⣜⡇⠀⠀⣿⠀⠀⠀⠀⣿⠈⢷⠀⢀⡾⠉⠉⠹⡇
⠀⠀⠀⠀⠀⠙⢷⡀⠀⠙⢧⡀⠙⢧⡀⠀⠀⠙⣿⠀⠀⢿⠀⠀⠀⠀⢹⡀⠘⣧⠞⠁⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⣈⢳⡄⠀⢹⣦⠀⠀⠘⢧⡀⢸⡀⠀⠀⠀⢘⣧⠞⠁⠀⢀⣒⡂⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠛⠿⣤⣘⣏⢳⡄⠀⠈⢷⣸⣇⣠⡴⠞⠋⠁⠀⠀⠀⠀⢀⣠⡴⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠖⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⢀⡠⠖⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⠵⠒⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠛⠉⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⠶⠶⠶⠒⠚⠛⠋⠉⠁⠀⠀⣀⣀⡤⠴⠒⠋⠉⠀⠀⠀⢄⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠶⠶⠶⠖⠒⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠬⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⡤⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⣁⡴⠞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠛⠉⠁⣀⣠⡤⠶⠛⠁⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⡤⠶⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠑⠣⠂⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠖⠛⠛⠛⠛⢿⡏⠉⠋⠉⠛⠛⠓⠶⠶⣤⣄⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣤⠴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠹⣇⠀⠀⢀⣀⣠⣤⡤⠶⠶⠶⠞⠛⠛⠛⠻⠛⠛⠛⠛⠳⠶⠶⢦⣤⣀⡀⠀⠀⠈⣿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣿⠶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠒⠂⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣦⣿⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⣠⡶⠛⠉⠁⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣠⣤⠶⠶⠶⢦⣄⠀⠀⠀⠰⠶⢤⣄⠀⠙⠻⢦⣄⠈⢳⣄
⠀⠀⠀⠀⠀⠀⢀⡾⢁⣴⠟⠁⠀⠀⠀⣤⠞⠁⢀⣤⣤⣴⠾⠛⠋⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠘⠷⣦⣤⣤⣄⣀⠈⠑⠀⠀⠀⠉⢳⣄⢻⡆
⠀⠀⠀⠀⠀⠀⣸⣇⡾⠁⠀⠀⢀⡴⠛⢁⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣄⠀⠀⠀⠀⠀⠹⣦⣷
⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⡀⠀⠀⠀⠀⠀⢀⡀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢀⡞⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⢻⡘⣧⡀⠀⠀⠀⠘⣧⠀⣄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⣴⡾⠟⢋⣴⠞⠁⠀⠀⠀⣴⠟⣿
⠀⠀⠀⠀⠀⠀⠸⣇⠈⠳⣦⣀⠀⠀⠙⠳⠶⠮⣭⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠔⠋⠀⠀⠀⣀⣤⠞⠁⣸⠏
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠙⠳⢦⣤⣀⣀⠐⠢⠌⠻⠦⣤⣤⡤⠼⠿⠿⢭⣤⣄⣀⣤⣤⣄⣤⡴⠞⠋⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢀⣀⣀⣠⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⢛⡛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⠿⠿⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⠀⠀⠘⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣤⣤⣤⣀⣨⡇⠀⠀⠀⠀⣏⣀⣀⣀⣤⣤⡴⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣇⣀⣀⣀⣀⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⣉⣉⣉⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠀⠀⠈⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⣤⣤⣀⠀⠀⣿
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣏⣀⣀⣹⠆⠀⣿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠛⠛⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠶⠛⠛⠛⠳⠶⣤⣿
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⣨⡟
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⠶⠞⠋
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⢀⡶⠛⠛⠛⠻⡷⠖⠛⠛⠛⣶⡴⠶⢶⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠁⠸⡆⠀⡶⠚⠛⢶⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⣻⠀⠀⠀⣿⠋⠉⠛⠳⡦⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⣶⠟⠛⢳⡄⢷⠀⠀⠀⢹⡀⣧⠀⠀⠸⣿⠀⠀⠀⠀⡿⠖⠒⣶⠦⠾⢶⣤⡶⠿⠶⠦⠀⠀⡇⠀⡼⠋⠀⢸⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⢻⣼⣇⠀⠀⠀⢷⣿⠀⠀⠀⡿⠶⠒⣆⠐⡄⠀⠀⢹⠀⠀⠘⣻⠀⠀⠀⢨⢠⠶⠿⢿⡇⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠞⠛⠉⠹⡄⠀⣈⣿⣄⠀⠀⡇⠀⠀⢻⠉⠉⠉⣷⠘⠀⠀⠀⣿⠀⠀⠀⠘⢸⠀⠀⠀⠁⠀⡿⠉⠛⢶⡄
⠀⠀⠀⠀⢀⣀⢻⡀⢨⠃⠀⠀⠀⣇⡟⠉⠉⢹⡇⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⡄⠀⠀⣿⠀⠀⠀⠀⣜⠀⠀⠀⠀⢰⡇⠀⠀⣸⠃
⠀⠀⢠⡾⠋⠹⣎⣧⢸⠀⠀⠀⠀⣿⠁⠀⠀⢸⣇⡴⠦⠴⡄⢸⡄⠀⠀⡇⠘⡇⠀⠀⢸⡄⠀⠀⠀⣿⠀⠀⠀⠀⢰⠀⠀⢠⡇
⠀⠀⠘⣇⠀⠀⢿⣿⣼⣦⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⣇⣸⣧⡄⠀⡇⠀⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⠀⡟⠀⠀⣾
⠀⠀⠀⢻⡀⠀⢸⡆⠀⠈⡇⠀⡖⠛⠻⠀⠀⢸⡏⠀⠀⠀⡏⠀⢀⡇⠀⡇⠀⣇⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⢸⢰⡇⠀⢰⣇⣀⣀⣀
⠀⠀⠀⠘⣇⠀⠀⣧⠀⠀⢁⠀⡇⠀⠀⡀⠀⢸⡇⠀⠀⠀⠁⠀⢀⠀⠀⡇⠀⣿⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⢸⣼⠀⠀⣞⣉⣩⠝⠛
⠀⠀⠀⠀⢹⡄⠀⢹⡄⠀⢸⠀⡇⠀⠀⡇⠀⢸⡇⠀⠀⢸⠀⠀⠘⠀⠀⡇⢀⣹⣀⣀⣀⣿⣀⣠⣤⣧⣤⣤⣼⡧⠶⠚⠉⠁⠀⠀⢀
⣤⠶⠶⠶⠾⠗⠒⠚⠓⡆⠈⡄⡇⠀⠀⡇⠀⢸⡇⠀⠀⢸⠀⠀⠀⡗⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢷⡇⠀⠀⡇⠀⢸⡇⠀⠀⢸⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡜
⠈⣧⠀⠀⠀⠄⠀⠀⠀⢹⠀⢸⡇⠀⠀⡇⠀⢸⡇⠀⠀⢸⡇⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡇
⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢳⣈⡇⠀⠀⣧⠀⢸⡅⠀⠀⢸⡇⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣄⣿⡀⢸⡇⢀⣀⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡞⣿
⠀⠀⣿⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⡴⠋⠀⡿
⠀⠀⢻⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠞⠀⠀⣠⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢰⣇⠀⠀⡴⢻⡇
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠸⡟⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡸⠁⢀⡾⠁⢸⠇
⠀⠀⠀⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣿⠀⠀⠀⠈⠉⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⢈⡇⠀⣸
⠀⠀⠀⢸⡆⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⣀⠀⠀⠀⠀⠀⣂⠈⠀⠀⢠⠟⠀⠀⣿
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⡏⠀⠀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⠀⢠⠂⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣀⠀⢻⠈⠉⠛⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠏⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠇⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣀⠀⠀⠀⠀⡆⢠⠋⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣷⠁⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⠶⠶⠶⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣾⠾⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠐⠒⡄⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⢇⠀⣸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠔⠊⠉⢣⠀⠰⠊⠉⠀⠈⡄⠀⢠⠼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠁⠘⡄⠱⠀⠀⠀⠈⢆⠀⡇⠀⠀⠀⢃⢀⠇⠀⠀⠉⡑⠀⠖⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠘⣄⢧⠀⠀⠀⠘⡄⠅⠀⠀⠀⢸⡜⠀⠀⠀⢠⠁⡎⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⡁⠀⠈⡆⠀⠀⠀⢡⢰⠀⠀⠀⢸⠃⠀⠀⠀⡎⠘⠢⡄⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠱⠀⠀⢰⠀⠀⠀⠀⣾⠀⠀⠀⡌⠀⠀⠀⢸⠀⠀⢠⠁⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢃⠀⠀⠇⠀⠀⠀⢻⠀⠀⢀⠃⠀⠀⠀⡌⠀⠀⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡄⠀⢡⠀⠀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠃⠀⠰⠀⡰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠢⣁⠀⢸⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⢸⠀⠀⣧⠞⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢋⠑⠲⠤⣀⢀⡇⠀⠀⣈⣀⣀⣀⠼⠒⢺⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⡄⠀⠀⢱⠀⠀⠀⢹⠀⠀⠀⡆⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢃⠀⠀⢈⡀⠀⠀⠸⠀⠀⠀⡇⠀⢠⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⣀⠤⠒⠒⠂⠉⠐⠒⠒⠤⡀⢸⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⠑⠢⣤⢀⣀⣀⣀⡀⡤⠔⠚⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⣀⣇⠀⠀⡇⠀⠀⣇⣀⠤⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⡀⠀⡴⠋⠑⢢⠀⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠑⢢⡰⠁⠔⠰⠛⠑⢦⠀⡇⢸⡏⢀⣈⡦⠤⣠⠊⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⢸⠃⠀⠀⡇⠈⠁⡘⠀⡇⢸⠇⠀⢸⠐⠄⢸⡇⢨⠤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⠉⢆⡇⢀⠀⢠⠠⠤⣼⠇⠈⠲⡟⠒⢄⢸⠠⢄⡆⠀⠀⢸⠉⠁⠤⠈⣥⠂⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢇⢰⠀⡛⠐⠄⡾⠀⠈⢠⠃⠂⢸⡇⠠⠀⡙⠀⠀⡸⠚⠲⣄⠀⡇⠀⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢦⠀⠇⠠⢰⢃⡴⢄⡌⠀⠀⢸⠀⠈⠀⠗⠊⠓⡇⠐⠀⡾⡸⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣀⣄⣘⣼⠀⣀⡟⡇⢀⠀⠁⢨⠤⠾⣀⢀⡸⡀⠈⠀⠃⠁⠰⢱⠁⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣄⣀⣀⡉⢻⠁⡇⠠⠀⡆⡇⢐⠂⣸⠡⡀⠘⠇⢸⣀⣀⣇⣆⣀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠳⡅⠀⠀⣇⠁⠘⠀⡏⢀⠀⠸⢀⡜⠀⣄⣀⣈⣉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⣄⠈⠲⢄⣿⡁⠀⢰⠀⢀⡠⠗⠋⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠈⠉⠉⠉⠉⠁⣀⠴⠃⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠦⠤⠤⢤⣄⣤⣀⣀⣀⣄⣠⠤⠤⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⡏⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠙⣦⡄⠀⠀⠀⢀⡦⣤⣀⠀⠀⢸⠇⢸⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢸⡄⢹⣷⠀⠀⠀⢸⠃⠀⡟⡇⠀⢸⠀⣾⠀⢸⠁⢠⠤⢶⣆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢧⡈⠉⠳⣄⠀⣧⠈⣿⣆⠀⠀⢸⠀⢀⣿⡇⠀⡟⢀⡇⠀⡟⢀⡟⢀⡏⠘⣆⠀⣠⠞⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀⠈⢦⡸⡄⠸⣿⡄⠀⢸⠀⢸⣿⡇⢰⡇⢸⠃⢸⠇⡼⠀⡾⠀⢠⣧⠞⠁⣀⢴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⠀⠹⣷⠀⢹⣷⡀⢸⠀⢸⣿⡇⣼⠀⡟⠀⡞⣸⠁⡼⠁⣠⠟⠁⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠒⣌⢣⡀⠳⣄⠈⣧⠀⢻⢷⣸⠀⢸⣿⣢⠇⢸⠃⣸⣵⠃⣸⣣⠞⠁⣠⢾⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠳⣌⠳⣝⣆⠘⢦⡘⡆⠈⢧⢻⡆⠘⣿⣟⢀⡏⢠⡿⠁⣼⡷⠁⣠⢞⡵⢋⣠⠴⠚⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠈⢦⡈⠻⣷⡀⠳⣽⡄⠈⢷⡗⠋⠉⠉⠉⠉⠉⠓⣾⠏⢀⡴⣷⣫⠖⠋⠀⣀⡤⠖⠋⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⣆⠈⢿⣄⠙⣿⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣯⡾⠋⢁⣠⠖⠋⣁⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣈⣷⠤⠽⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠯⢤⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⡤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⢤⣀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠞⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠲⡆⢀⣠⢄⣾⠉⠉⠉⡏⠉⠉⢹⡟⠛⢳⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⢦⢻⠀⠀⢡⢸⠀⠀⢿⠀⠀⢀⡧⠀⢤⠼⠧⣠⠼⠤⡄⠀⡇⣠⠎⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢘⣾⣧⡀⠀⢿⡇⠀⠸⠓⢹⣤⡄⣀⠀⠀⠀⡏⠀⠀⢀⡞⠛⠟⠀⡴⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⠀⠀⢡⡴⠚⢳⡄⠀⠀⠀⡇⠀⠘⢀⠀⠀⣧⠀⠀⢘⠁⠀⠀⢀⠃⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⢷⣇⠈⠀⠀⢸⠃⠀⠘⣧⠶⠴⡀⡇⠀⠀⠘⠀⠀⢻⠀⠀⢸⠀⠀⠀⠘⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⡛⠛⡆⢀⡼⢤⠀⠀⡇⠀⠀⠗⠳⡆⢰⠀⡆⠀⢸⠀⠀⢸⠀⠀⡆⡏⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣇⠀⠠⢸⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡆⠀⢸⠀⠀⣧⠃⢠⣛⡚⠿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣱⣀⣹⣀⠸⠐⠀⠀⠀⢰⡇⠀⢨⠀⠀⣀⣼⠤⠧⠤⠼⠧⠰⠾⠲⠶⡟⠒⠉⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠉⠁⢸⠀⣶⠀⠀⠀⢸⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⡆⢹⠀⠀⡆⢸⠃⠀⢸⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠘⠾⣀⠀⡇⢸⡄⠀⢸⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠓⠦⣀⠀⠀⠀⠀⠈⠉⠛⠚⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠞⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡴⠃⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠉⠢⢄⡀⠠⠦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢘⠃⢠⠎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⣀⠀⠀⠀⠀⠀⠀⠈⠂⠇⠀⠀⠈⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⢸⠠⠂⠀⡹⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀⠀⣸⠀⠀⠐⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠇⠀⡀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣘⡄⠈⠉⠒⠢⢄⣀⠀⠀⠀⠀⢀⠀⡴⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⣸⡌⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣼⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠁

⠀⠀⠀⢀⡤⢀⡀⢀⡀⠀⠀⠀⠀⠀⡀⢶⠸⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⣦⡙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠋⠁⠚⠁⠿⠁⡾⠃⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⠀⠀⠀⠀⠀⢀⣤⠸⠗⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠐⠶⠝⠓⠉⠀⠀⠀⣤⢸⡇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠺
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠵⠖⣤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡹⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣄⠀⣤⠐⣦⠰⣶⢰⡎⠷⠀⠀⠀⠀⠀⠈⠉⠺⠋⣰⡖⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣆⠹⠇⠉
⡌⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⡀⣀⡀⠀⠀⠀⠀⢀⡤⣦⠹⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⡀⠀⠀⠀⠀⡀⣴⡈⠿⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠐⠛⠰⠟⠁⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⡀⢤⡄⠿⠌⠛⠈⠁⠀⠀⣰⡘⡷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠏⢸⡆⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢶⡘⠷⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠾⣃⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢰⣄⢰⣆⢰⣆⢷⡸⠆⠀⠀⠀⠀⠀⠈⠁⠚⢁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⡈⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⢠⡤⢠⡄⢀⡀⠀⠀⠀⠀⠀⡄⢾⠹⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⡐⣶⡘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⠀⠋⠁⠛⠡⠿⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠀⠀⠀⢀⠠⣦⡘⠇⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⡘⠷⠜⠃⠀⠀⠀⠀⣶⠸⠇⠉⠀⠀⠀⠀⠀⠀⠀⠄⠺
⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠴⢏⣴⠄⣀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠙⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⣤⡐⣶⠀⣶⠸⣧⠹⠎⠷⠀⠀⠀⠀⠀⠀⠀⠘⠋⣼⡖⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶
⢀⡀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⣦⠹⠃⠉
⡄⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⣤⠄⣀⠀⠀⠀⠀⠀⢠⡔⡧⠙⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡀⠀⠀⠀⠀⡀⢼⡌⠟⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠐⠛⠰⠟⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⢀⡀⢴⣆⠻⠆⠉⠀⠀⠀⠀⣴⡈⠿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢼⡇⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠐⢶⡌⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠾⢡⡶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣆⢸⣆⠹⠇⠿⠜⠇⠀⠀⠀⠀⠀⠀⠀⠛⠡⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢾⠌⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⡖⢠⡤⣠⡄⢀⡀⠀⠀⠀⣄⢻⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡘⠷⠌⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠁⠀⠉⠁⠛⠀⠟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡰⣤⠀⠀⠀⠀⢀⡠⣶⠘⠋⠈⠀⠀⠀⠀⠀⢠
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⡈⠿⠈⠁⠀⠀⠀⠀⣶⠸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠚⢫⣶⢂⣤⠀⠀⠀⠀⠀⠀⠀⠐⢦⠜⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢦⡀⣶⠈⣷⠸⠷⠸⠇⠻⠀⠀⠀⠀⠀⠀⠀⠙⠃⠼⢏⣤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣶
⢀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⣧⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠢
⠀⠀⠀⠀⠀⠀⡠⠄⠐⢤⡇⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠱⡀⢰⠒⠉
⠀⠀⠀⠀⠀⠘⡀⠠⠀⣄⠀⢣⡏⢀⠄⢠
⠀⠀⠀⠀⠀⠀⠈⠒⠈⠀⢣⠘⡇⢸⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⢠⠇
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠀⠀⠎
⠀⠀⠀⠀⠀⠡⣄⠀⠀⠀⡎⠀⡜⠀⠀⠀⠀⡄⠀⠀⠀⠔⡆
⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⢡⢠⠁⠐⠀⠀⡎⠀⠀⠀⡎⠀⢡⠀⠀⠀⣀
⠀⠀⠠⡃⠀⠈⠅⢢⠀⣀⢸⣸⠤⠤⠤⠤⠤⠤⣤⠔⠃⣰⣚⣉⠥⠒⠁
⠀⠠⠐⠃⠀⠀⠀⡀⠓⠠⠄⠻⢀⡄⢀⠀⠀⠊⠀⠀⠜⠁⠒⣤⣀⡀⢀⣀⡀⠀⠀⠀⠤⠤⠤⠒⠒⠒⠒⠠⠄
⠣⠀⠀⣀⠰⠤⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⢀⠰⠊⠀⠄⠊⠐⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠠
⠀⢣⠀⠀⠉⠀⠀⠈⠀⠀⠀⠖⠀⠀⠀⠒⠄⠁⠀⠀⠀⠀⢀⠜⠉
⠀⠈⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊
⣤⣤⣤⣤⣤⣬⣤⣶⣶⣶⣤⣤⣤⣤⣶⣶⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠒⢲⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢸⠀⠀⠀⢀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠓⡄⠀⠀⡠⠶⠶⠆⠀⠀⢸⠀⠀⢠⠁⠀⠀⠈⡆
⠀⠀⠀⡠⠒⠓⠘⢦⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣇⣀⣸⠀⠀⠀⠃⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⡇⢀⠴⠶⠦⣄
⠀⠀⠀⢇⠀⠀⠀⠸⠆⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⡄⠀⠀⠀⢸⠆
⠀⠀⠀⠘⡀⠀⠀⠀⢳⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢰⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⡀⠀⠄⠀⠤⢄
⠀⠀⠀⠀⢃⠀⠀⠀⠘⡆⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⡀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⣷⠀⠀⠀⠀⣏⠼⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠲⣸⠀⠀⢸⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⡸⠆⠀⠀⠀⡘
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿⠀⠀⠀⢰⡇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠸⠃⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⣚⠀⠀⣸⠀⠀⠀⢸⠀⠀⣂⠀⠀⢘⡀⠀⢀⡃⠀⠀⣸⠀⠀⠀⢀⡃⠀⠀⢀⡇⠀⠀⠀⣐⠁
⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡄⠀⠠⠀⠄⠠⠠⠐⠀⠀⠤⠀⠀⠀⠠⠄⠀⠀⠠⣄⠀⠀⠀⠀⠄⠄⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠲⣾⣿⡶⢂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠀⠟⠛⠃⠨⠿⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠰⣷⣄⣸⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠻⢿⢿⡿⣿⢿⣿⠟⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣧⣤⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣿⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡁⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⣴⣯⣿⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣽⡿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣧⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠈



⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⢿⡏⠏⡿⠉⣿⠹⢻⡟⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⡻⡝⢎⣣⣶⡵⠷⠶⠿⠶⠷⣾⣵⡏⣴⡿⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⣿⣮⣣⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣾⣿⣿⡿⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠀⠀⣰⣾⠿⣷⣄⠀⣰⣾⢿⣷⡄⠀⣴⡿⢿⣆⠀⢠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠀⠀⣿⡏⠀⢻⣿⠀⣿⡏⠀⣿⣿⠀⣿⡅⢸⣿⢀⣾⠃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⣸⣿⠀⣿⡇⠀⣿⣿⠀⠙⠿⠿⠃⣾⠇⣼⡟⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣤⡄⢿⣷⣤⣿⡟⠀⢿⣷⣤⣿⡟⠀⠀⠀⠀⣼⠏⠀⢻⣧⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡃⢀⢷⡈⣻⠃⡖⢾⣓⢰⣟⡃⣸⣆⢰⠛⠒⠘⡟⢰⣦⡞⢳⡔⣧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⠟⠼⠛⠧⠸⠀⠇⠢⡽⠳⠏⠠⠟⠻⠜⠦⠖⠀⠇⠸⠟⠧⠼⠹⠏⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⣿⣿⣿⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⣭⢻⣧⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣥⣿⢃⣿⣻⡟⡷⠶⡶⠶⡿⣛⠛⣥⢷⣹⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣿⣴⣼⣷⣰⣯⣤⣽⣷⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣰⢣⠀⢀⢶⠀⠀⣴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⡀⢰⠈⢣⣠⠃⠬⢧⠎⠌⢧⡞⠸⡆⢔⢽⠀⢀⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢄⡀⢸⠀⠉⠾⠀⠀⠉⠀⣈⣀⣀⣈⣀⣀⠀⠋⢐⠝⡶⣫⡪⡇⣠⠔⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢂⠀⠉⠛⠀⣀⠤⣔⣦⠽⠼⠧⠗⠓⠚⠓⠣⠿⠿⣶⣢⢮⣊⠂⢙⢕⠡⣇⡠⢖⡛
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠉⠑⠚⠀⢀⣔⣻⠔⠋⠁⠀⢀⣠⠄⠀⡄⠀⡠⢤⠀⠤⣀⡈⠙⠲⢯⣖⡭⣈⠫⢞⢕⣧⠔⣺⠃
⠀⠀⠀⠀⠀⠰⡒⠢⠼⠇⠀⢀⢔⡳⠊⠀⠀⢴⣂⠀⠈⣤⠄⢰⠾⢆⢇⠀⠀⠀⠇⠀⠀⡄⠀⠈⠙⠷⣯⠲⣥⠪⢪⣌⡤⢴⠆
⠀⠀⠀⣀⣀⣀⣑⡄⠀⠀⣴⡻⠊⢀⡀⠈⢦⠀⠀⣙⠄⠐⡀⠈⠀⠈⣈⣋⠀⠘⠀⠀⠘⢠⠊⢹⠆⢀⠀⠙⢮⡵⣴⠑⣰⣎⣠⠤⡤
⠀⠀⠀⠘⢮⠢⠀⠀⢀⣞⡞⠁⠀⠋⠢⡀⠀⠑⠀⢀⣠⡤⣖⣟⣯⣿⠭⠽⠯⢿⣶⣤⣀⠈⠓⢁⡔⢻⠀⣀⠀⠙⢿⣻⠐⡕⣠⡞⠁
⠀⢤⠒⡒⠒⠓⠀⢠⣳⠎⠀⠘⢗⡢⠤⡈⠁⡐⡼⣎⠷⠋⠩⠄⠄⡔⡔⠰⠐⠰⢠⡉⠻⢗⣦⡀⠀⠖⠊⠀⠀⠀⠒⢳⡻⡞⢌⡉⣉⡽⠃
⠀⠀⠙⠦⡠⠀⢀⣷⠏⠠⡃⡀⠈⣱⠄⠀⢠⣾⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠆⠑⣽⣆⠀⠀⠀⠀⡀⠀⢂⢻⡵⡪⢺⠯⠤⠤⣤
⠠⣖⢊⠉⠁⠀⠸⡍⠀⠀⠀⠀⠀⠜⠀⠐⣼⡿⠀⣀⠤⢄⠀⢠⢖⠒⢄⠀⢀⣎⢏⢶⡀⠀⠀⠈⢿⣇⢀⣀⡰⠱⣀⣀⡺⣎⣧⠹⣢⡾⠋
⠀⠈⠑⣮⡀⠀⣗⡇⠀⠀⠀⠀⠀⠀⠀⣀⣷⠇⡔⢁⢖⣿⠀⡏⣽⢷⡽⣇⠸⢠⡟⢇⣣⠀⣠⡀⠘⣿⡈⢣⣀⣴⣷⠋⢀⣷⣺⠘⢟⣭⢏⣲⠄
⠐⠯⣩⢁⠀⠀⣻⠃⠀⠀⠀⣤⠀⠀⠈⢸⣿⠀⠙⠙⡯⣺⠀⡇⢽⠈⡗⣿⠀⡖⡇⠨⣾⡄⠓⡏⠀⣟⡇⢸⡧⠛⠾⠀⠠⢸⢸⠐⡱⢿⣉⡁
⠀⢀⣈⡲⠄⠀⣿⡄⢀⡀⡴⠀⠦⢤⡄⠺⣻⡀⠀⠀⣯⢜⡄⢷⢝⣆⣕⢾⠀⢥⢝⣾⢜⡇⢰⠻⠕⣿⡇⠀⠁⠀⠀⠀⠀⢸⢽⠞⣴⢟⣴⡭⠗
⠛⠫⢔⣒⠠⠀⡿⡇⠀⠱⠄⣰⣻⡋⠀⢁⢣⣇⠀⠀⠗⣧⣇⠘⢯⣚⣫⡾⠀⠈⠢⠭⠾⠁⠀⠀⢰⡾⠀⠀⣠⢄⡀⠀⠃⣎⣿⢞⢎⠟⢅
⠀⠀⣠⠜⠃⠀⢹⢥⠀⠀⠔⠉⠑⠇⠀⠀⢌⣯⢦⠀⠐⠚⠋⠀⠀⠉⠁⠀⠀⠄⢄⠀⢀⢀⠐⣠⣿⠃⠀⠘⠲⢄⡹⢀⣰⣶⡡⢣⣎⡮⠮⠷⠂
⠀⠚⠓⠒⢢⡍⠀⢟⣧⠀⠀⠀⠀⠀⠀⠀⠀⢊⠫⣷⣄⡀⠀⠀⠀⢀⡀⡈⠎⡈⡌⠐⠈⣠⢾⡷⠁⠀⠾⣠⠂⣀⠀⢎⣏⡯⣰⡫⡙⡦⡀
⠀⠀⠀⡰⠋⠀⡄⠈⢶⢣⡀⠀⠀⠀⠀⡠⠂⠂⠀⠈⠛⠻⢯⣖⣶⣢⠤⡤⡤⣤⣶⣶⡻⠚⠉⠀⢠⠂⡀⠈⠛⠁⢈⡞⡞⡐⡑⢟⠋⠑⠉
⠀⠀⠘⠁⠉⢩⠏⠀⠀⠻⡕⣄⠀⠀⢰⣁⠆⢠⠃⡰⠀⢀⠀⠀⡈⠉⠉⠉⠉⠉⠉⠁⠀⢀⢀⠂⠀⠙⢥⠚⣠⣾⣟⡿⣼⣯⣾⣿⣷⡄
⠀⠀⠀⠀⢰⡧⠼⠤⣷⠀⠈⠣⣗⡤⡀⠀⠀⢇⠴⠁⡴⣯⠀⢨⢭⠇⠀⠸⡆⠀⡶⢄⡧⠀⠈⢦⠀⠀⣠⢖⡿⣧⠏⣾⢿⡾⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠣⣀⢦⡄⠈⠑⠯⣷⢦⣄⠀⠈⠀⠀⠀⠂⠘⡄⢀⠏⠹⡀⢠⠀⠙⠀⠀⡀⠀⣢⣔⣻⢋⢼⣎⢻⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢸⠁⢀⣤⠀⠀⠉⠚⠽⣓⡶⡤⢄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣂⣤⢤⣖⡯⢗⠩⢢⢪⠶⠧⣅⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠚⠁⣼⢀⡮⢶⠀⡪⢒⢎⢍⡝⡚⡳⠿⠿⠽⢽⢯⠿⠞⡗⡋⡩⡡⣣⢊⢺⠓⢤⣚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠀⢸⢈⡵⠛⡤⢪⡮⢾⠐⡕⣱⡣⢪⢪⡎⢜⠔⠿⣔⢵⠋⠳⣍⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⡧⠊⠀⠈⣏⡴⠁⢳⣁⠎⠈⢮⡸⠀⠘⠶⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠀⠀⠈⠁
⠀⠀⠀⠀⠔⢔⠕⡢⢔⠤⢂⠤⢂⠔⡉⠔⣁⠔⡠⠂⡀⠄⡠⢀⡠⢀⡄⢀⠄⢀⠄⡠⠀⡠⠂⡠⢀⠔⡠⢀⠔⡠⢀⠄⣠⢀⡄⡠⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠊⠀⠊⠀⠊⠰⠊⡠⢊⠴⢋⠔⣁⠔⣁⠔⡡⢊⠵⢊⡤⢊⡴⢊⠔⡡⢊⠔⡡⢊⠔⠡⠊⡴⢪⠊⠔⠱⠋⠊⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣖⠛⠭⠉⠒⠀⠀⠀⠀⠚⠉⠍⠛⣲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠯⠂⠉⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠈⠐⠙⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠊⠁⣀⣀⣾⣦⣀⡀⠀⠈⢿⣿⣿⡿⠃⠀⢀⣀⣼⣧⣀⡀⠀⠐⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠫⠊⠀⠀⠀⠈⢻⣿⣿⡏⠀⠀⠀⠼⠛⠛⠇⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⠝⣆
⠀⠀⠀⠀⠀⢀⡜⡑⠀⠀⣀⣶⡀⠀⠋⠁⠉⠃⢀⡀⠄⠀⠀⠀⠀⠠⠀⠀⠘⠉⠈⠙⠀⣀⣶⡀⠀⠈⢈⢧⡀
⠀⠀⠀⠀⠀⡞⡌⠀⠀⠙⣿⣿⣿⠋⠀⠀⡠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠙⣿⣿⣿⠋⠀⠀⠡⢳
⠀⠀⠀⠀⣼⠑⠀⠀⠀⠐⠛⠉⠛⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠋⠉⠛⠀⠀⠀⠀⠋⣧
⠀⠀⠀⢀⡇⠂⠀⠀⣠⠤⣄⢠⡄⠀⣤⢀⡤⢤⡀⠤⣤⠤⠀⣠⢤⡄⢠⣄⠀⣠⡄⣤⠤⠤⠀⣤⠤⣄⠀⠀⠐⢸⡄
⠀⠀⠀⢸⠹⠀⠀⠘⡇⠀⠀⢸⡇⠀⣿⠈⠓⠲⡄⠀⣿⠀⢸⡇⠀⣿⢸⡟⣦⢿⡇⢿⠒⠀⠀⣿⠶⣞⠀⠀⠀⠆⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠙⠛⠉⠀⠙⠛⠁⠈⠛⠛⠁⠀⠙⠀⠀⠙⠛⠁⠈⠁⠉⠈⠃⠙⠛⠛⠀⠋⠀⠙⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣰⠀⣞⡓⠀⢠⢶⡀⠐⢻⠃⢰⠀⣞⡒⠀⣞⠛⠂⢀⣶⡀⢠⡞⠓⠀⢻⡟⠀⡆⣴⠛⣦⢰⣆⢰⠀⣄⡇
⠀⠀⠀⠈⡇⠀⠤⠼⠂⠟⠛⠷⠀⠸⠀⠸⠀⠤⠽⠂⠏⠀⠀⠞⠛⠳⠘⠦⠤⠀⠸⠇⠀⠇⠻⠤⠟⠸⠈⠻⠠⢸⠃
⠀⠀⠀⠀⢹⡠⠀⠀⠀⠀⣤⣀⣤⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⣀⣤⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢧⢂⠀⠀⣠⣿⣿⣿⣄⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣠⣿⣿⣿⣄⠀⠀⡐⡼
⠀⠀⠀⠀⠀⠈⢳⡡⡀⠀⠉⠿⠉⠀⣀⡀⣀⡄⠈⠁⠀⠀⠀⠀⠀⠒⠈⠀⢀⣀⢀⣠⠀⠉⠿⠋⠀⢀⢜⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣔⢄⠀⠀⠀⢀⣼⣿⣿⣇⠀⠀⠀⢰⣤⣤⡆⠀⠀⠀⣸⣿⣿⣧⡀⠀⠀⠀⡠⣢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⠄⠀⠉⠉⢿⠟⠉⠁⠀⢠⣾⣿⣿⣷⡄⠀⠈⠉⢻⡿⠉⠉⢀⠠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠄⡀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⣀⠠⣢⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠮⢤⣐⣀⣠⠤⠀⠀⠠⢄⣀⣀⡠⠵⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⢻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⣿⣿⣿⣶⡄⠀⣀⣉⠛⠛⣉⣀⠀⢠⣴⣿⣿⣿⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣿⣷⣦⡶⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢶⣶⣾⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣇⠰⣶⣶⣿⣿⣶⡶⠆⠀⢰⡆⠀⠰⢶⣶⣿⣿⣶⣶⠆⣹⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠙⠀⠀⣽⣿⣿⣍⠰⣶⣶⣿⣿⣶⣶⠆⣩⣿⣿⣯⠀⠀⠋⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠈⠙⠂⠈⣽⣿⣿⣿⠁⠐⠋⠁⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠈⠛⠆

⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⢀⣤⣶⣦⣤⡀⠀⠀⣀⣤⣶⣦⣄⡀⠀⠀⣠⣤⣶⣦⣄⠀⠀⠀⣠⣴⣶⣦⣄
⠀⢰⣿⣉⣿⣿⣉⣿⡆⢰⣿⣙⣿⣿⣉⣿⡄⣼⣏⣹⣿⣿⣉⣿⡄⣼⣏⣹⣿⣟⣹⣷⠀⣾⣏⣹⣿⣏⣹⣷
⠀⢸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⠇⢿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣯⣿
⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⠻⢿⣿⣿⠿⠋⠀⠈⠻⢿⣿⣿⠿⠋⠀⠈⠻⢿⣿⣿⠿⠃⠀⠘⠻⣿⣿⣿⠟⠁



⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠂⠉⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠀⠀⣀⡠⠠⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⡎⠁⠀⠈⠄⠀⡅⠀⠀⠈⡀⢀⠎⠓⠠⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢂⠰⡀⠀⠀⠘⡀⢃⠀⠀⠀⠃⡈⠀⠀⠀⡜⢠⠃⠈⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢠⠔⠋⢡⠀⠀⠀⢣⢸⠀⠀⠀⢸⠀⠀⠀⢠⠧⣇⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢆⠀⠈⡆⠀⠀⠀⣎⠀⠀⠀⡌⠀⠀⠀⡆⠀⢀⠇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⡄⠀⢡⠀⠀⠀⢹⠀⠀⠠⠁⠀⠀⢠⠀⠀⡘⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠰⠀⠀⡄⠀⠀⢸⠀⠀⢸⠀⠀⠀⠨⠀⠠⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⡆⠀⡇⠀⠀⠸⠀⠀⡈⠀⠀⠀⡄⠀⣸⠔⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠍⠒⠢⢄⣀⣆⣀⣀⣇⣀⢀⡤⠗⠚⠇⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢸⠀⠀⠘⡀⠀⠀⠈⠀⠀⢠⠀⠀⠸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡆⠀⠀⡇⠀⠀⣀⠀⠀⠸⠀⠀⡅⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢂⠠⠒⠉⠉⠁⠀⠈⠈⠉⠒⢤⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡦⢀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡇⠀⠈⠁⠉⡆⠉⠉⡇⠀⢀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢇⡀⡄⠀⠀⡇⠀⠀⡇⢀⡸⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠛⠛⣿⡇⠀⠀⠀⣠⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀⠀⠀⠈⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⣿⡟⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⠿⠟⠛⣿⣧⠀⣀⣠⣿⣧⠀⠀⠀⢹⣿⣶⣶⣶⣿⣷⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⣾⣿⠿⠟⠛⢻⣿⡀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⡄⠀⠀⢸⣿⡄⠀⠀⠈⣿⣇⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠛⠉⢿⣧⠀⠀⠀⣿⣧⠀⠀⠀⢹⣿⠀⠀⢸⣿⠀⠀⠀⣸⣿⣀⣀⣀⣼⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠸⣿⡄⠀⢀⣸⣿⣤⡀⠀⣼⣿⣷⣶⡿⣿⣷⠀⢸⣿⡟⠛⠻⠿⢿⣿⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣿⣿⡿⠿⠛⠛⣿⡇⠀⢿⣿⠀⠀⠀⢸⣿⠀⢸⣿⡇⠀⠀⠀⢸⣿⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢻⣿⠀⠀⠀⠀⢹⣿⡀⢸⣿⡀⠀⠀⠸⣿⡇⢸⣿⡇⠀⠀⠀⢸⣿⣀⣀⣸⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠘⣿⡇⠀⠀⠀⠈⣿⣇⠘⣿⡇⠀⠀⠀⣿⡇⢸⣿⡇⠀⠀⠀⢸⣿⡿⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣷⣴⣶⣿⣿⠀⠀⠀⠀⢹⣿⠀⣿⡇⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⣠⣿⠟⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠉⠉⠁⠙⢿⣷⣄⠀⠀⠘⣿⡇⢿⣿⠀⠀⠀⢸⣿⠘⣿⣇⣠⣾⡿⠋⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣿⣿⣸⣿⠀⠀⠀⣸⣿⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣾⣿⣶⣶⣾⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣤⣴⣶⣾⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣍⠀⠈⣩⣿⣿⣿⣯⡉⠀⢉⣽⣿⣿⣿⣍⠀⠀⣩⣿⣿⣿⣯⡉⠀⢈⣽⣿⣿⣿⣍⠁⠀⣩⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⣴⣦⣹⣿⣿⣿⣿⣤⣶⣌⣿⣿⣿⣿⣏⣴⣦⣹⣿⣿⣿⣿⣡⣶⣤⣿⣿⣿⣿⣏⣴⣦⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣓⣀⡀⢀⡉⣉⢉⡉⣀⣀⣀⣚⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⡤⠤⠄⢀⣀⠀⡸⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⡉⠉⠉⠁⠀⠀⠒⠂⠐⠈⠉⠉⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠢⠤⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠃⠘⠉⠭⠽⠟⠛⠛⠛⠋⠈⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠛⠻⠛⠛⠓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⠤⣄
⠀⠀⠀⠀⠀⢀⡴⢚⠁⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢦⡀⠀⢀⣠⡤⠖⠛⣉⠥⠄⠠⠜⠻⣦⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠉⠀⠀⠀⠁⠀⠔⠊⠉⠴⠊⠩⡝⢳⣄⡀
⠠⢶⠤⣰⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⢣⡺⡤⣿⣌⠙⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣠⢼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡲⠤⣄⣀
⡠⢀⠙⠺⠻⣷⣶⡤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣩⡾⡃⣀⣚⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣤⣽⣦⣀⠤⢶⢺
⠀⠐⠣⡅⣙⢄⣋⣭⣛⣓⣶⠶⢶⣤⡤⣤⡴⠶⠶⠶⠶⠚⡟⠛⠶⠿⢱⣚⡏⠩⠈⠓⢄⡀⠀⠀⠀⠀⣀⣤⣤⣶⡶⢿⢿⠿⠿⠟⠛⠉⠉⢀⣀⣼⠿
⠀⠀⣾⣷⡔⠀⢨⠃⠂⠘⠑⠷⢶⣧⣲⡖⠃⢂⡵⠷⠒⠛⣓⣲⡒⠄⡽⣿⡀⠀⠀⠀⠀⠙⠢⣀⣴⡟⠛⢿⡟⠀⣠⣲⣷⣲⠶⢶⣒⠚⠿⢍⡳⢤⣤
⠀⠀⡟⠬⡙⠗⠦⣤⣀⠀⠀⣀⣀⠀⠀⣙⡉⠀⢀⠭⠒⣈⠡⠖⠴⠼⢂⣿⡁⢠⢠⡔⠀⠀⢠⣾⡇⠘⡳⡖⠀⡴⣱⡁⢸⡌⠉⢳⢦⣍⡒⠤⣉⣲⠖
⠀⢀⠛⣦⡘⠀⢀⡈⠉⠛⠖⠀⣀⣀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣠⠎⢱⠙⢦⡸⢃⠄⠀⣾⣿⡇⠀⣷⡉⢙⡴⣝⢧⡀⣷⠀⠈⣶⣄⣹⣲⣇⡰
⠀⠐⡀⠈⠀⠩⠴⠿⠧⠤⠤⢤⣁⣉⣥⣌⡱⣶⡴⠭⢄⣒⣂⠭⠔⠋⠀⡸⠀⠀⠙⢼⠃⢠⡟⣿⣿⠀⠸⣿⡋⠵⢾⣭⣏⣻⡖⠐⡧⣄⣈⣉⣏⣹
⠀⠀⠑⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠙⢮⡇⠈⢻⠇⠀⢳⡉⠲⣤⣄⠤⣞⠛⢟⠿⣓⠊
⠀⠀⠀⠀⠈⠉⠒⠛⠛⠉⠁⠲⠶⠶⠶⠦⠤⠤⠄⠀⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⢿⠭⡛⡘⣤⡉⢉⠇⠈⠑⢤⣙⡚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣈⣆⠱⣸⣌⣄⠉⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⡀⡀⠀⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⢢⡈⠃⠁⢤⡨⣫⠀⣘⡂⠀⠈⢰⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣿⣆⠴⣼⠹⢻⣖⢛⡗⣎⡷⠏⣶⠃⡜⠃⠁⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠳⠘⢸⣿⢠⡸⠀⢸⡹⢋⠠⠆⠉⠀⡼⣄⡇⣧⡀⠑⢃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⢱⠘⣿⢼⡃⠀⢸⣁⢢⠃⡴⣩⠄⣟⢙⣇⢚⣪⣶⡜⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣡⣧⣿⣯⣅⣄⣸⣇⣘⣞⣠⣋⢈⣿⠰⠏⣶⣏⠾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠉⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⠄⢲⠀⢐⢰⡀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠘⢴⢸⡀⠅⠀⠀⠀⠀⢸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣃⢸⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠓⡏⠀⠀⠀⠀⢀⡠⢄⢴⠊⠉⢱⡀⡏⠉⠀⣁⢀⠖⠤⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠰⡁⠀⠈⢎⡆⠀⠀⢣⠡⠀⠀⢠⡌⠀⠀⡰⡰⠁⠉⢢
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣼⠀⣧⣤⣤⣤⣀⣱⡀⢺⠁⠘⡀⠀⠀⣾⠀⠀⢸⠁⠀⢠⠛⠳⡄⢀⠎
⠀⠀⠀⢰⡉⢖⠉⠉⠀⠀⠀⠘⠤⠃⠀⠀⠀⠉⢁⡣⢉⣇⠀⢇⠀⠀⢹⠀⠀⡍⠀⠀⠸⠀⡸⠀⡎
⠀⠀⠀⢸⠪⠭⢉⣛⣒⣛⠛⠚⠶⠚⠛⣛⣚⣛⡉⠭⠑⡟⡀⢸⠀⠀⢸⠀⠀⠇⠀⠀⡇⠀⠇⣼
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠧⣀⡇⠀⠸⠀⢸⠀⠀⠀⣃⡴⠊⠸
⠀⠀⠀⠀⡇⠀⠀⠀⣼⢷⡀⠀⠀⠀⢠⡾⣦⠀⠀⠀⢸⡀⠀⠈⠉⠒⠓⠒⠚⠒⠒⠉⠁⠀⠀⡇
⠀⠀⠀⠀⢡⠀⠀⠈⠁⠀⢁⣴⣶⣦⡉⠀⠈⠁⠀⠀⡏⡇⠀⠞⠋⠳⠀⠀⠀⠀⠞⠉⠳⠀⢰
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⣿⡟⣻⠁⠀⠀⠀⠀⢠⠄⢻⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠛⠁⢀⣀⠀⣀⠀⢼⢀⣸⡀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⡴⠋⠒⡝⡤⠖⢀⣉⣛⠫⠽⠿⣓⠒⠶⣶⣶⣶⣶⣤⣬⡿⣍⡓⠛⣉⣐⡶⣚⠭⠭⠓⢄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡃⠀⢀⡴⠛⢒⡒⢛⡒⠢⠤⠤⠒⢬⠲⣦⣄⣢⣶⠖⠥⠦⠶⠶⠛⠛⠛⠚⠲⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠉⢲⠋⠀⠀⠰⠃⠈⠁⠺⡄⠘⣀⠘⢣⡄⠰⠂⣰⠂⡤⢸⡇⣨⡃⠘⠁⠀⠀⠀⠱⡖⠁
⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠆⠤⠤⠤⢀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⡀⠤⠤⠄⠒⠒⠊⠁


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

⠀⢠⣤⡀⠀⢀⣤⠀⠀⢠⣤⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄
⢰⣟⠀⣛⣛⣛⢈⣛⣛⣋⢈⣛⣛⣋⢈⣛⡆⠀⠀⣀⡤⠖⢾⡙⣾⠻⣍⣹⣯⠭⠿⡗⠦⣄⠀⠀⠀⠀⠀⠀⢀⡠⢔⣭⣾⡏⡆⠑⡄
⠀⢸⣾⡇⠀⢸⢸⡇⠀⢹⢸⠁⠀⣿⢸⠀⠀⠀⡴⠓⠻⣟⢉⣻⣍⣵⣜⣦⡤⠷⣖⠛⢿⣉⣳⠀⠀⢀⡠⢖⣭⡤⡛⢟⡡⢊⣴⡖⡌⢢
⢾⣏⠀⣻⣿⣏⢈⣻⣿⣋⢈⣻⣿⣋⢈⣿⠆⠀⡟⢯⣉⣙⣅⣀⡼⢦⠬⠷⡙⢶⡛⢻⣉⣹⠾⣇⠖⡁⢶⣿⠟⢳⠼⣣⡄⡻⠭⢺⣴⠢⡱⡀
⠀⢸⣿⡇⠀⢸⢸⡇⠀⢸⢸⠀⠀⣿⢸⠀⠀⠀⣷⢄⡈⠉⠙⠓⠒⠚⠓⠒⠛⣦⡟⠉⢁⣠⢼⣿⢀⠈⠒⠭⣒⡁⠾⠿⠱⢪⣶⡝⠯⠱⠇⠑⡀
⢾⣏⠀⣩⣭⣍⢈⣽⣭⣍⢈⣭⣿⣍⢈⣽⡇⠀⢹⢦⣉⠙⠒⠒⠶⠴⠤⠤⠴⠓⠓⠚⢉⡠⢾⠘⢮⡓⠤⣀⠀⠉⠉⠉⠐⠒⠛⠓⠂⠀⢉⡀⡆
⠀⢸⢽⡇⠀⢸⢸⡇⠀⢸⢸⡀⠀⣿⢸⠀⠀⠀⠘⢦⡈⠉⠓⠒⠲⠶⠶⠶⠶⠒⠒⠛⢉⣠⠟⠀⠀⠉⠒⠤⢍⣉⡁⠒⠒⠒⠒⠒⠂⠈⣡⠜
⢿⣭⠀⣭⣭⣭⢀⣭⣭⣍⢈⣭⡭⣍⢨⡽⠆⠀⠀⠀⠈⠑⠒⠦⠤⠤⠤⠤⠤⠴⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠉⠉
⠀⠘⠛⠁⠀⠸⢼⣃⣀⣈⣛⡀⠀⠹⠞⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⣛⠑⡒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡴⢚⣩⠥⣦⣤⢭⣍⡓⢦⡀⠀⠀⠀⠀⢀⡔⠉⠀⡉⠀⠀⠸⠀⣀⢏⡁⡬⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡁⠠⠁⠲⠄⣀
⠀⠀⣴⢋⣴⠯⢿⠤⣟⣿⠴⡷⢽⣦⡙⣆⠀⠀⢠⠋⠂⠈⠀⠈⡇⠀⢹⠀⠈⡇⠀⢑⠙⢾⢦⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠂⠄⠂⠨⡂⢨⠕⠂
⠀⢸⢃⡿⢾⠶⢾⠶⣷⣿⠶⡷⢾⡷⢷⡘⡆⠀⠇⠈⠀⠈⠀⠃⡇⠛⢫⠛⠛⡝⠛⢫⠟⠛⡘⡆⠀⠀⠀⠀⢀⡼⠂⣈⣁⠠⠀⠂⠁⡀⠁⡀
⠀⣿⢘⣓⣚⣓⣛⣒⡋⣛⣒⣓⣚⣓⣚⡃⣷⢸⠀⠀⠀⢠⠠⢄⡇⠤⣜⠤⠤⣳⠤⢔⡦⠤⣄⡇⠀⠀⠀⣰⠆⢨⠒⠮⣀⠠⠒⢈⠘⠀⠐
⠀⢹⡘⣇⣼⣅⣽⣄⡿⣿⣤⣯⣸⣧⣼⢃⡇⠈⡄⢀⣀⢀⢀⡀⣇⣀⡸⣀⣀⣇⣀⡸⣀⣀⢎⡇⠀⢀⠮⢕⠂⢙⡁⣵⠄⢀⠠⠋
⠀⠈⢧⡙⣾⡤⣼⡤⡿⣿⣤⣧⣼⡷⢋⡞⠀⠀⠱⡈⠀⠀⠀⠀⡇⠀⢰⠀⠀⡏⠀⠀⠀⣜⡾⠀⡰⠯⢄⢼⠈⢱⡭⢕⠢⠞⠁
⠀⠀⠀⠙⢮⣙⠺⠤⡟⢿⠤⠗⣋⡴⠋⠀⠀⠀⠀⠙⢄⠈⠀⠈⠁⠐⢩⠀⠀⠝⠂⢊⣹⠟⠀⠀⠐⠠⢀⠉⠓⠽⣄⡴⠋⠠⠊
⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢥⣀⣀⣚⣀⣄⡬⠴⠊⠁⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⣀⣀⡀⠀⣀⢀⡀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣧⣿⢼⡧⣿⢸⡧⠻⡅⢸⠘⢌⢳⡜⡏⡅⠠⠄⢸⡽⣿⣼⡇⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠈⠙⠉⠈⠁⠙⠊⠓⠙⠃⠈⠈⠋⠘⠁⠙⠁⠀⠀⠈⠀⠁⠉⠙⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠀⠉⠀⠀⠀⣠⠤⠀⠀⠐⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠐⠒⠤⡄⠴⡴⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠞⠣⠒⠗⠒⠀⠀⠀⢄⢠⠤⠄⠤⣤⠲⠈⠀⡠⠶⠈⠑⠀⠤⣥⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣁⣂⣀⠁⠀⠈⠀⠀⠀⠀⠀⢂⠤⣎⣁⠠⠄
⠀⠣⠀⠐⠠⠄⠀⠀⠀⠤⠀⠀⠀⢠⠞⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠐⠁⠀⢠⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠄⠊⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠓⠦⢀⡀⠀⡠⠖⡩⠄⠀⠀⠀⢸⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⠈⠀⠈⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠭⠔⠉⠀⠀⠀⠀⠀⠘⡇⡞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠄⢠⡄⠀⠒⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢀⡀⠀⠀⠀⠄⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠈⠀⢠⠘⣠⠀⠅⠤⠀⠃
⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⢀⡌⠀⠈⠀⠸⠁⠐⠀⠀⢀⠁⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠠⠴⠄⠊⠀⢀⠘⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡜⠚⠀⠁⠂⠀⠀⢄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⣴⠄⢀⡀⠙⠂⡀⢀⡀⠀⠀⠐⠧⡄⠣⡀⠀⠀⠀⠀⠀⠛⡠⠄⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⢠⣄⣀⠀⠀⠀⢀⠀⠐⠠⠤⠀⠉⠈⠒⠈⠁⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠈⠠⢌⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠀⡐⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠈⠀⢀⠀⠒⠂⠉
⠀⠀⠀⠀⠀⢆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠨⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀
⠰⢦⡀⠴⠄⠦⠔⠃⠀⠀⣀⠤⢠⡄⡶⠠⢄⠀⡆⠀⠀⠀⡀⠀⡝⠤⢀⡀⢠⡄⢤⣤⠀⣀⡠⠄⠀⡢⠄⠠⠄⠀⠀⣀⠤⠀⢨⣅⡰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⢠⠀⡥⠀⣶⠀⠀⠀⠀⠀⠀⠀⠁⢀⢈⣛⡀⠀⢠⠀⠀⠘⠡⡤⠀⠁⠀⠕⠀⢀⡀⣤⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠀⠐⠈⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢩⠁⠀⣠⡀⠑⠓⠐⠆⠀⠀⠀⠀⠈⠉⠀⠉⠀⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠀⣀⣀⣈⡀⠴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠁⠀⠈⠁⠉⠀⠦⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠷⠀⠀⠐⡶⠀⡀⠀⠀⣀⡀⠀⠁⠀⠄⠀⠺⢷⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠔⠀⠀⠠⠤⠀⡀⠠
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠌⠓⠀⠀⠁⠀⠒⠀⠀⠀⠑⠠⠌⠂⠉⠐⠂⠉⠀⠀⠀⠀⠔⠀⠑⠒⠂⢀⣁⡀⠀⠈⠁⡠⠀⠀⠀⠄
⠀⣤⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠆⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⡀⠀⠀⠀⢈⠄⠠⠐
⡇⠀⠀⠁⢄⠘⠦⣄⢠⡄⠔⠀⠘⣰⠀⠀⠀⣀⣦⠄⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠡⠀⠀⣤⠀⠀⠀⢠⠁⠀⢠⡏⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠄⡀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡀⠘⢇⣀⠀⣀⠀⠔⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢋⡤⠀⠀⠀⠀⠐⢀⠁⠀⠀⠠⠒⠈⠉⠉⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠇⠀⠀⠀⠀⢀⠄⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠤⡀⠀⢀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣦⠀⠀⠀⢰⠀⠀⠃⢄⣀⡀⢀⣀⡰⠞⠀⡈⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠊⠔⣈⠉⠋⣀⠲⠀⠀⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣊⠤⠖⠶⠤⡑⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⣀⡀⣀⣀⠈⠁⠈⠣⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡧⠼⣀⠀⠀⠀⠀⠀⠆⠀⠴⠀⠀⢀⠤
⠀⠀⠀⠀⠓⠈⠜⢉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣰⠁⠀⠀⠀⠀⢠⠈⠁⠜⠂⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⠀⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⣈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣼⣿⣿⣿⣿⣿⢻⠀⠀⠀⠀⢀⡴⢺⣿⣤⣴⣿⣿⣿⠁⣧
⠀⠀⠀⠀⠀⠀⢰⣯⣴⣿⣿⠿⠻⠀⢸⣿⢸⠀⠀⠀⠀⣇⢄⡢⠀⠀⢹⣿⡿⠿⠀⢸
⠀⠀⠀⠀⠀⠀⣿⣿⡛⠛⢃⣴⢠⠀⠈⠋⢸⠀⠀⠀⠀⠘⢦⣐⣂⡤⡞⠿⠁⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⠁⡏⠘⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠁⢠⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡄⠀⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠰⣿⠟⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣇⠀⠀⠀⠀⢸⡀⠀⡀⠀⠀⣀⣀⣀⣀⡇⠆⠀⠸⣿⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⣼⠙⡧⡿⢿⣿⣶⠀⠀⠁⠈⠉⠉⠉⣧⡎⡿⢾⣿⣿⡗⠀⠀⠀⢸⣿⡏⠉⠏⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⠀⣿⡇⡇⠀⢸⣿⠁⠀⠀⠀⠘⠟⠁⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠏⠳⠤⠤⠭⣥⣤⣤⠤⠤⠤⠤⠼⠶⠒⠁

⢀⡀⠀⠠⠅⠀⠀⡠⠂⠈⠢⡈⠁⠠⠄⠄⠀⠂⠀⢀⠴⠀⠐⠛⠻⠀⠸⠇⠄⠄⢄⠀⠀⠊⠅⠀⠀⡠⠊⠉⠠⢀⠇⠘⡄⢀⠀⠀⠀⢀⠴⠀⠐⠛⠳⠁⠸⠔⠠
⠀⠀⠈⠀⠐⠀⠂⠁⠀⠀⢀⠜⠀⠀⠖⠀⠨⠀⠠⡀⠀⠀⡠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠂⠑⠄⠀⢀⠘⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠠⠊⠀⡀
⠺⠂⠳⣀⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠠⠀⠠⠃⠀⠈⠠⠈⠀⡔⠓⡜⢂⠀⣾⠃⠺⢆⠽⣠⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠂⠔⢂⠀⣺
⠀⠘⠂⠁⠀⠀⡠⠐⠀⠁⠐⠠⠀⠀⠀⠀⠈⠀⠀⠀⡀⢀⠈⠲⢪⡖⡛⠀⢙⠒⠂⠛⠛⠏⠀⠀⠠⠐⠀⠀⠐⠢⠀⠀⠀⠀⠌⠀⠀⠀⢀⠀⠈⠶⢢⠔⡛⠀⢛
⠀⠀⠀⠀⠀⠀⡶⠤⣂⢴⡤⢴⡆⠀⡏⠀⠄⢩⠂⠀⠀⠆⠐⠈⡧⢾⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠄⣂⢠⡆⢶⡆⠀⡍⠀⠈⢩⠂⢰⠀⠀⠀⠈⡳⢮⠃⠀⠈
⠀⠀⠉⠀⠀⠀⠉⠉⠷⠘⣣⠍⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠆⠀⠐⠂⣀⢀⡀⠁⠀⠀⠉⠀⠀⠀⠉⠉⠳⡜⣫⠝⠁⠀⢠⠀⠀⡀⠀⠀⠑⠀⠆⠀⠐⠂⢀⢀⡀
⠀⢀⠊⠀⠈⢢⠀⠀⠀⠈⠁⠀⠀⠀⠸⠄⠀⠄⠀⠀⢀⣲⠦⠀⠀⡁⠀⠀⡀⣥⠀⠠⠈⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⣀⣶⠦⡀⠰⡉⠀⢀⡀
⠀⢨⠀⠀⡠⠂⢀⢀⣑⠄⢀⣴⠤⠂⠀⠀⠀⣦⣡⡾⠟⠁⠀⡀⠀⠀⠂⠀⠀⡀⠀⢀⡀⠀⡀⠂⠀⢂⡐⠄⢀⡕⠴⠠⠀⠀⠀⣤⣱⡾⠟⠋⠀⠀⠀⠁
⡄⢢⠘⠈⠀⠀⢈⠠⡔⠲⠵⠷⣄⠒⠀⠀⠈⢜⠁⠀⠀⠠⠊⠀⠀⠀⠒⠐⠀⠀⠄⠡⠚⠈⠀⠀⠄⠀⡴⠲⠾⠶⣄⠂⠀⠀⠀⢎⠀⠀⠀⠠⠊⠀⠀⠈⠂
⢀⡁⠀⣀⠀⠀⠈⢐⠂⠠⠄⢂⠄⢀⠀⢠⣂⠀⠀⠀⠀⠀⠘⠀⠇⡈⣄⠀⠀⠀⢀⡀⠀⣀⠀⠀⠐⢐⠂⠠⠄⢂⠄⠀⠀⣘⣀⢀⠀⠀⠀⠀⠸⠁⠇⡘⡀
⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡯⠇⠀⠀⢀⠀⡰⠂⠺⠀⠀⠠⠀⠌⠢⠰⠁⠄⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⣠⡿⠇⠀⠀⢀⠀⡯⠒⠲⡀
⠀⠃⢀⠀⠀⠀⣠⠄⠒⠀⠲⠀⠐⠂⠁⠂⠈⠁⠐⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢀⠀⠀⢀⠀⠃⠀⢠⢄⡒⣀⢶⣄⠐⠂⠑⠐⠈⠁⠐
⠀⠂⠁⠐⠊⠀⠃⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⡀⡀⡀⠀⠄⢀⡀⠀⠀⢀⠀⠀⠀⠒⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠀⡀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⡆⣀⠠⠀⠒⣀⠁⠀⠦⢷⠀⠘⡉⠁⡙⠀⠀⠀⠉⠀⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⣂⣀⠠⠀⠒⡀⠁⠀⢨⠷⠄⠘⠃⠉⠛⠀⠀⠀⠉⠀⠀⠉
⠉⠀⠉⢉⡄⠀⠀⠀⠀⢀⠨⠀⠁⢀⣀⡀⠄⠀⠠⠀⠀⡀⠴⣀⢦⠀⢀⡁⠉⠀⠉⠀⠋⢉⡆⠀⠀⠀⠀⠀⡤⠀⡃⠀⢌⡄⣄⠀⠃⠀⠂⠠⠴⣐⢦⠀⢀⡀⠉
⢉⣑⠈⢛⠄⠀⠀⡠⠊⠈⠐⡈⠃⠀⠖⠀⢀⠂⠀⣀⠖⠐⠒⠋⠺⠀⠰⡃⣀⣄⢢⣄⠨⢋⠖⠀⠀⡀⠊⠈⠀⢀⠃⠐⡔⠀⠀⠀⠀⠀⠔⠐⠒⠉⡳⠁⠘⡏⣑
⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⢀⠘⠀⠀⠔⠀⠰⠀⠨⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢋⠐⠀⠀⢀⠜⠀⠀⠆⠂⢀⠀⠈⡀⠀⠀⡠⠈
⠿⣆⡵⣠⠀⠀⠈⠃⠈⠃⠀⠀⠀⠀⠂⠀⠀⠃⠀⠈⠂⠊⠀⡐⡁⣜⢂⠀⣷⢃⡿⢂⠵⣠⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠂⠂⢀⡜⡁⣄⢂⠀⣾
⠒⠛⠈⠁⠀⠀⡀⠂⠁⠀⠁⠢⠀⠀⢀⣀⣃⣀⠀⠀⠤⠀⡀⠒⠢⢔⡚⠁⠙⠐⠓⠛⠋⠇⠀⠀⠀⠀⠉⠉⠁⠢⠀⠀⢀⣀⣂⣀⠀⠀⠤⠄⡈⠖⠢⢔⡺⠀⢋
⠀⠀⢀⠀⠀⠀⡷⢒⣒⢒⡒⢶⡇⠐⡅⠀⠀⢨⠂⢀⠀⠆⠀⠈⡗⠺⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣶⢒⣂⢒⡒⢶⡆⠐⠍⠀⠀⢨⠂⢰⠀⠀⢐⠈⡷⢮⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠌⠶⡜⣣⠍⠀⠀⢀⠀⠀⠈⠀⠀⠁⠒⠒⠀⠐⢂⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠙⠌⠷⠜⣯⠈⠁⠀⠀⠀⠀⡀⠀⠀⠁⠒⠂⠀⠐⠂⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠀⠰⠀⠀⠆⠀⠀⣐⣶⠶⡄⢠⣁⡀⣀⡀⡤⠀⠠⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣶⠶⡀⠰⣁⠀⣀⣀
⠀⢠⠀⠀⡀⠂⢀⢀⡠⠄⠠⡔⠴⠂⠀⠀⠀⣤⣦⡿⠟⠁⠀⠈⠀⠀⠂⠀⠀⡀⠀⢠⠀⠀⡠⠂⠀⢀⡈⠄⠨⡔⢤⠢⠀⠀⢠⣔⣤⡿⠟⠁⠀⠄⠀⠁
⡔⢢⠊⠀⠀⠀⠀⠀⠌⠒⣋⠶⡴⠒⠀⠀⠈⢊⠀⠀⢀⠄⠊⢀⡀⢈⠁⠂⠉⠀⣔⠠⠈⠁⠀⠀⠁⠀⡌⠒⣟⠧⢴⠒⠀⠀⠀⢎⠀⠀⠀⠠⠊⣀⡀⢈⠁⠂⠉
⢀⡁⠀⣀⠀⠀⠈⠜⠂⠰⠄⢀⠄⠀⠀⢠⡇⠀⠉⠁⠀⠀⠘⠒⠃⡌⣄⠀⠀⠀⢀⡁⠀⣀⠀⠀⠀⠈⠊⠡⠤⢈⠄⠀⠀⢘⡄⠀⠉⠉⠀⠀⠘⠒⠂⡬⣄
⠁⠈⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣨⠯⠇⠀⠀⢀⠀⡸⠂⠸⠀⠀⠠⠁⡀⠃⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⡀⠀⠀⣨⡿⠅⠀⠀⢠⠀⡯⠂⠒
⠀⠁⠀⠀⠀⠄⣠⢀⡔⠭⣳⠄⠘⠄⣉⣀⢈⡁⠜⠀⠀⠀⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⢀⠀⠃⠄⣠⢆⡔⢭⣳⢄⠘⠆⣉⣀⣈⡁⠜⠀⠀⠀⠁
⠀⠂⠀⠐⠀⠀⠁⠂⠀⠀⠀⠐⠇⠀⠀⡀⠂⠀⣀⣀⡀⠀⠄⢠⡀⠀⠀⢰⠀⠀⠀⠂⠀⠑⠀⠀⠡⠀⠀⠀⠀⠈⡆⠀⠀⠠⠂⠀⣀⣀⣀⠀⢀⢠⡀
⠀⣀⠀⠀⠀⠀⣎⠀⠄⠀⠀⢈⠀⠀⠲⢿⠀⠘⠅⠀⠜⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⣀⠀⠀⠀⠀⣂⠀⠄⠐⠀⣀⠀⠀⢰⡷⠀⠸⠁⠀⠘⠀⠀⠀⠀⠀⠈⢀
⣁⣀⣉⣈⡀⠀⠀⠀⠀⠀⡈⣀⣁⢀⣀⣀⣄⠀⠠⠠⠀⣀⡒⣑⢢⡀⢀⣀⣁⣀⣉⣀⣃⣀⡂⠀⠀⠀⠀⢀⡤⣀⣃⢀⣌⣄⣀⠀⠃⠀⠂⣀⠐⣑⣦⡀⢀⣀⣁
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠛⠛⣿⡇⠀⠀⠀⣠⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀⠀⠀⠈⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⣿⡟⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⠿⠟⠛⣿⣧⠀⣀⣠⣿⣧⠀⠀⠀⢹⣿⣶⣶⣶⣿⣷⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⣾⣿⠿⠟⠛⢻⣿⡀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⡄⠀⠀⢸⣿⡄⠀⠀⠈⣿⣇⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠟⠛⠉⢿⣧⠀⠀⠀⣿⣧⠀⠀⠀⢹⣿⠀⠀⢸⣿⠀⠀⠀⣸⣿⣀⣀⣀⣼⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠸⣿⡄⠀⢀⣸⣿⣤⠀⠀⣼⣿⣷⣶⡿⣿⣷⠀⢸⣿⡟⠻⠟⠿⢿⣿⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣿⣿⡿⠿⠛⠛⣿⡇⠀⢿⣿⠀⠀⠀⢸⣿⠀⢸⣿⡅⠀⠀⠀⢸⣿⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢻⣷⠀⠀⠀⠀⢹⣿⡀⢸⣿⡀⠀⠀⠸⣿⡇⢸⣿⡇⠀⠀⠀⢸⣿⣀⣀⣸⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠘⣿⡇⠀⠀⠀⠈⣿⣇⠘⣿⡇⠀⠀⠀⣿⡇⢸⣿⡇⠀⠀⠀⢸⣿⡿⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣷⣴⣶⣿⣿⠀⠀⠀⠀⢹⣿⠀⣿⡇⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⣠⣿⠟⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠉⠉⠉⠙⢿⣷⣄⠀⠀⠘⣿⡇⢿⣿⠀⠀⠀⢸⣿⠰⣿⣇⣠⣾⡿⠃⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣿⣿⣸⣿⠀⠀⠀⣸⣿⣶⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣾⣿⣶⣶⣾⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣴⣶⣾⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣫⣭⣝⡻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢻⣽⣩⣴⡆⡇⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣾⣿⣿⣿⣷⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡹⠿⠸⣿⣧⡇⣱⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣙⢿⣿⡿⢏⣾⡏⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⢻⡜⣶⣭⣣⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⠾⡫⣶⣶⢰⣾⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣥⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣞⡃⣿⣿⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣝⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃⠐⣦⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠿⢷⢍⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠿⠥⠤⠄⠘⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⣰⣟⠍⣻⣟⠉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢀⣤⣤⢰⣿⣕⣸⣿⣅⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢣⣿⣿⣇⣿⡿⣛⣭⣭⡻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣚⣇⢿⣞⣿⡸⣯⣰⣶⣶⣶⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠞⠃⠙⠲⠛⠘⢛⢻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠀⣩⠏⣿⣭⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⡴⠋⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⣿⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣷⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁






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

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⡆⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠠⢄⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠈⠙⠤⠄⢠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠡⠂⠀⠀⠀⠀⣔⠃⠋⠓⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠠⠀⠂⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⢈⠉⠈⠀⠀⠀⠀⡌⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⡀⠀⠀⠀⠀⠈⠂⡀⠀⠂⡀⠈⠁⠁⠄⠀⡁⠀⢀⠠⡄⢀⠈⠀⠀⠠⠄⡀⣀⢀⠀⣼⣇⢄⠀⠀⠀⠈⠀⠀⢀⡀⠠⠲⠀⢀⡀⢀⠀⠀⠐⠁⢁
⠀⢰⡀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠀⠀⠁⠐⢁⡄⠀⡀⠈⠑⡽⠋⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⣤⡠⡤⣤⡴⡀⠈⢤⡀⠔⢀⠠
⠀⠀⣦⣄⡠⠆⠀⠒⠁⠉⠀⠀⠀⠀⠡⠄⠠⡁⠀⠀⠁⠀⠀⠀⢀⠌⠀⠀⢨⡕⠀⠀⡀⢠⠀⠀⠀⠠⠀⠀⠠⠀⠦⠄⠘⣄⢋⠀⠤⠂⠀⠀⠀⠀⠆
⠀⠀⠀⡀⠂⠀⠀⢀⡀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠠⠄⡀⠀⠀⢘⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠰
⠀⠀⠀⠈⠀⠀⠀⠀⠀⡌⠡⠒⢈⠀⠂⠀⠀⠀⠀⠐⠀⢀⣀⠈⠀⠀⠣⠊⠁⠆⠀⠀⠀⡀⠀⠀⠀⠔⠁⠀⠀⠀⠰⠀⠅⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠂⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⣄⣄⣀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠠⠄⠀⠊⢀⡐⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢠⣶⣿⣿⣿⠛⢛⡛⡻⡟⣿⣿⣿⣶⡄⠀⠁⠀⠉⠀⠠⠸⡈⣈⠀⠀⠀⠀⠀⠡⠀⠀⠀⠠⡐⠂⠠⠁⠀⠀⠀⠀⠀⠑⠄
⠀⠘⡒⠐⠈⠀⠀⢀⠀⢀⣿⢛⡟⣿⢻⡛⢛⣿⢛⡛⠻⠟⠟⢻⣿⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⢀⠀⣂⢀⣀⣀⣀⠀⠀⡀⡂
⠀⠀⢡⠀⡄⠀⠀⠀⠀⠘⣿⢨⠁⢹⡜⣿⢸⣿⢈⡁⡇⠀⠀⡆⣿⠃⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⡂⠀⠀⡁⠀⠁⠀⠀⠇⠀⠍⠛⠋⠀⠛⠀⠐⠈⠁
⠀⠀⠀⠆⠃⠀⠀⢀⠀⠀⠹⣼⣼⣮⣭⣿⣼⣿⣼⣧⣡⣬⣤⣴⠏⠀⠀⠀⠀⠰⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠠⠀⠐⠀⠀⠀⠉⠛⠛⠛⠻⠛⠛⠛⠛⠛⠉⠀⠀⠀⠂⠀⠀⡀⠀⠠⠄⠁⠀⠀⠀⠀⠈⠀⠀⠀⢐
⠀⠀⢅⠀⠀⠀⠂⠀⠒⠠⠀⠀⢠⠐⢀⠀⠀⠠⢀⠀⠠⡀⠀⠀⠀⠀⠄⠀⠀⠠⡈⠀⡀⠔⠀⠄⠀⠀⣀⣀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢉⠀⠸⠀⣕⠐⠘⠂⠀⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⢀⠗⠁⠀⠀⠉⠄⠀⠀⠂⠈⠂⠀⣈⠀⠀⠀⠀⢙
⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⡀⠀⠐⢀⠀⠀⠘⠕⠀⡀⠀⠀⢘⡂⠀⠀⠆⠀⠀⣀⠀⠀⠀⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠑⠂⠀⠘⠀⠰⣀⣀⠬⠃⢈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠆⠀⠁⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠐⠬⠄⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⡄⠀⠀⠀⠀⠀⠀⢀⠰⠸⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠁⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠖⠃⢀⠄⢀⠀⠒⠑⢀⠀⠀⠠⠀⠀⠀⠃⠁⠦⠀⠐⠀⠅⠀⢸⠀⠁⠈⠀⠂⠀⠐⠀⢴⣊⠖⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠂⠠⠤⠆⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⢀⠀⠀⠨⠂⠀⠀⠠⠊⠀⠀⢄⠀⠀⢀⡈⠀⠂⢰⣀⡠⡰⠱⢡⠄⠀⠀⠀⠀⡆⠀⠠⠀⠠⠀⠁⠒⠈⠀⠤⠀⠀⠀⠀⣠⠀⢀⠠
⠀⠈⠀⠀⠀⠄⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠀⠀⠠⢇⡀⠀⢸⢆⡀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠐
⠀⠢⠀⠠⠄⠂⠁⠀⠀⡀⠀⠀⡠⠔⠊⠁⠁⠀⠀⠈⠍⠀⡄⠀⠀⠈⠚⠚⠁⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⡄⠀⠀⡀⠀⠀⡄⠀⠄
⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠑⠂⢀⠉⠀⠐⠀⠄⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠁⠀⠠⠤⠄⠀⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠓⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⠉⠢⡀⡜⠀⢣⣀⡴⠉⠃⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠆⠀⠀⠈⢀⣀⣀⠉⠀⠀⠠⠔⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠐⠒⠀⢀⣠⣴⣶⡿⡿⡿⢿⢿⣿⡿⣶⣦⣄⡀⠀⠗⠂⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⡄⠀⣠⣶⣿⣿⡿⣪⣧⣾⠷⠾⠶⢾⣽⣣⡿⣛⣿⣶⣄⠀⠰⠄⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⣼⣿⣿⣛⡽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢯⣞⣽⣿⣷⡀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠃⠀⠈⠉⣉⠉⠉⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⢀⣀⡀⠉⢉⡉⠉⠁⠀⠘⠂⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠰⢾⣿⠀⠀⣾⡟⠹⣿⠀⣾⡏⢻⣷⢠⣿⠉⣷⠀⣾⠃⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣿⠃⣿⡁⢸⣿⠀⠿⠾⠏⣼⠃⣾⠛⣧⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⢠⣴⣿⣦⡄⢻⣧⣼⡿⠀⢿⣧⣼⡟⠀⠀⠀⣸⠏⠀⣿⣤⡿⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⣀⡀⢀⠀⣀⣈⡁⣀⡀⣀⡈⢁⠀⣀⡀⣀⣀⡀⢀⣀⢉⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⡤⠀⠀⠀⣓⡆⡼⣧⢸⠀⡏⣓⡄⡗⢀⡿⡆⣇⣀⢸⡁⡷⣇⣸⢸⢷⡀⠀⢠⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢠⣬⣤⣤⣤⡀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠁⠀⢀⣤⣬⣥⣤⡄⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⢐⠂⠀⠹⣿⣿⢷⣽⢶⣤⣀⡀⠀⠀⠀⣀⣀⣤⢖⣫⣲⣽⣿⠟⠀⠐⡂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠀⣀⠈⠛⢿⣧⣾⣡⡏⣿⡻⣟⢻⡟⢹⣅⣷⣿⡿⠛⠁⢀⠀⢀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⣀⠈⠙⠻⠿⠿⢿⣿⡿⠿⠿⠟⠋⠁⢀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠊⠁⢸⠀⢀⠤⡄⠀⠀⣀⠀⠀⢀⠤⡀⠀⡀⠈⠑⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠰⣀⠞⠀⠱⣀⠌⠀⠈⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢰⡁⣼⠀⡇⢸⡜⠀⢸⠁⣸⣱⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⡀⠀⠙⢦⠈⠧⠟⠂⠁⠘⠀⠀⠘⠀⠿⢠⣷⠇⡴⡇⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⢆⠱⡷⠄⣀⡥⠴⠒⠛⠋⠉⠉⠛⠦⠴⠶⣞⠳⢬⡘⠰⠞⣠⠞⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡆⢃⡴⠚⠁⢀⣠⠴⢶⡖⠒⠒⡆⢀⡤⣄⣼⠀⠀⠉⠳⡌⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣠⣞⣟⡀⠀⣸⠀⠀⢰⠇⣼⠀⢀⡇⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠎⠀⢸⠏⠉⠉⡟⠒⠲⣾⢀⣇⣀⣸⠁⠀⣀⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⠋⠙⠓⡾⠶⠤⣼⣇⣀⣠⡇⣸⠀⠀⡟⠀⣸⠁⡇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠙⠶⠤⢤⣇⣀⣀⡞⠀⠈⣹⠀⡟⠒⠲⣷⢴⣏⣸⠃⠀⣀⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠛⠓⠒⣶⠤⢤⣍⣉⣛⠒⠋⠘⠧⣤⣠⣇⡀⠀⣿⠀⣸⠉⡗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⡦⢤⣤⣏⣀⠀⡼⠀⠉⢹⠀⡴⠲⠦⣤⣭⣉⣙⠛⠛⠷⢴⠇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⣇⠀⢸⠃⠉⢹⡟⠒⠲⡾⢰⣧⣀⣀⡏⠀⠀⣽⠉⠙⠓⡾⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⣟⡿⠲⠦⣼⣄⣀⣰⠇⣼⠀⠀⢸⠋⠙⢓⡗⠶⠤⡼⢧⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⢣⡀⠀⡏⠀⠀⣽⢠⡟⠓⠲⡿⠤⣤⣼⣁⣠⠞⢁⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⠺⢧⣀⢀⡇⣸⠀⠀⢰⠇⠀⣀⡿⠟⠁⢀⡏⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠖⣠⠙⠲⣄⡀⠈⠉⠀⠘⠒⠖⠚⠛⠋⠁⣀⡤⠞⠉⠙⣛⢥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢪⠊⣤⠞⡲⢀⠉⠙⠋⠙⠦⠤⠤⠴⠶⠒⠋⠉⠀⢀⡀⠊⡏⠺⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⡡⠋⠐⡏⢰⠀⣦⠆⢀⠄⠀⢰⠂⠘⣬⠈⡗⢆⠧⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡞⢸⣿⠀⠀⢱⠄⢸⣠⠂⠹⠂⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠛⠛⠛⠷⢶⣤⡀
⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⡀⠀⠀⠀⣀⣀⣠⣤⣤⡶⠶⠶⠶⠶⢦⣤⡀
⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠛⠉⠛⠛⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣤⣤⣴⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⣀⣠⣴⠾⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⢿⢿⠿⡿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⡉⡄⢀⡈⠀⠀⠠⠌⢋⠙⠛⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠄⠆⢠⠀⠈⠠⠀⠀⠁⡄⢈⡀⣈⠀⠀⠌⡛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⣷⡄⠀⠀⠋⠀⠐⠀⠀⠠⠀⠀⠄⠈⠀⠁⠀⡄⠀⠂⡉⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⢀⣿⡏⠀⠀⠀⠄⠀⠀⠄⠀⣀⠠⠄⠒⠈⠁⠘⠀⠈⣀⢠⢀⠋⠻⣿⣗⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠁⠀⠀⠀⢀⣾⡟⠀⡄⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠉⠀⠆⠁⠀⣀⠀⠀⣀⠈⡘⣿⡆⠈⠠
⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⣾⡿⠀⠀⢀⠀⠀⢀⠀⠂⢀⡀⠀⠀⠀⠐⠀⠀⡄⠁⠀⠠⠀⣢⠀⠄⠲⢿⣿⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⣼⡿⠁⠈⠁⠀⠀⠀⠀⠘⠀⠀⠂⠀⢀⠀⢀⠀⠀⠀⠀⠀⣌⠈⠅⠠⢤⣁⣽⡿⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⣀⣼⣿⠁⠒⠀⣀⠀⠀⠀⠈⠀⠒⠐⠀⠀⠀⢰⠀⠌⠄⠤⠠⠀⠊⠁⠉⠒⣶⣿⡿⠁⠀⠀⠀⠀⠀⢡
⠀⠀⠀⢀⠃⠀⢀⣠⣴⣶⡿⣿⣿⠃⠀⠃⢀⠀⠀⢀⠉⠄⠀⠤⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠠⢀⡀⠀⠀⠀⢂
⠀⠀⠀⣂⣤⣶⠿⠋⠍⡁⢰⣿⠇⠈⠀⠀⠀⠀⠐⠀⠈⠀⠠⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢂⠀⠀⠀⠄
⠀⢀⣴⡿⡏⠁⠀⢈⠀⣠⣿⠏⡀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠁⠀⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢩
⢠⣿⡟⠁⢠⡀⠁⠀⠀⣿⡟⠠⠀⠘⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠐
⣾⡿⠠⠂⠀⠀⠈⠠⠀⠻⣷⣤⣀⡁⠠⠀⢀⠘⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠐⠤
⣿⣇⡀⠂⢁⠚⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠂⢠⠀⠐⠀⠈⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢡
⠹⣿⣧⣤⣀⠀⠀⠈⠁⠀⠀⠠⠀⠐⠨⠉⠛⠿⣿⣶⣤⣄⣠⠀⠀⢂⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡐⢀
⠀⠈⠩⠛⣿⣆⠀⣀⠐⠂⠀⠄⠰⠀⠀⢠⠀⠀⠂⠈⠛⠻⢿⣷⣶⠊⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠯⠁⠀⠀⠀⠀⠀⠤⠂⠀⠠
⠀⠀⠀⢃⢹⣿⡀⠀⠀⠄⠀⠀⠀⠂⠈⠀⡀⠀⢀⠀⠀⠄⢀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠟⠛⠉⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢿⣧⠀⠂⠀⠀⠀⠀⠄⠠⠀⠀⡀⠀⠀⠁⠐⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡐
⠀⠀⠀⠀⠨⠜⣿⣆⢀⠀⠰⠀⠀⠐⠀⠃⠀⠆⠀⠈⠁⠀⠀⠁⠀⠀⠐⠁⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠠
⠀⠀⠀⠀⠀⠉⢿⣿⡀⠰⠀⠰⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠰⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠊⠁⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠴⠐⢠⣖⣤⡤⠄⠄⠠⠠⠀⠆⠂⠈⠈⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠘⠀⠀⠀⠠⠀⠀⠈⡁⠈⠀⢠⣀⣀⣬⣤⣶⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡐⠂⠈⣁⠀⡀⠐⠆⢈⣀⣤⣤⣶⣾⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡘⢀⣀⣤⣤⣶⣾⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠛⠉⠉⠐⠠⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠠⠐⠂⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣾⣶⣶⣶⣶⣶⣾⣦⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⢀⣼⣿⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢺⣿⣿⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠚⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣆⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⢠⣿⣧⡀⠀⠀⣰⣿⣿⣄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣛⣛⣉⣭⣼⣿⣿⣿⣦⣤⣀⡀⠀⣼⣿⣿⣧⠀⢀⣀⣠⣤⣿⣿⣿⣧⣭⣉⣛⣛⢻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⢩⣭⣴⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⣭⠝⣿⣿⣿⣿⣿⡿⢋⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠻⢶⠙⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⡋⡴⠟⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣷⠙⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠀⠀⠈⠙⠻⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠟⠋⠁⠀⠀⠉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠟⠋⠀⠈⠙⠻⢿⡇


⣤⣤⣤⣤⣤
⣿⣽⠭⣭⣿
⣿⣿⠤⣿⣿
⣿⣿⣿⣿⣿

⠀⢄
⠀⠀⠑⠦⣀
⣀⠀⠀⠀⠈⠉⠒⠀⢀
⣿⣷⣦⣄⣀⣤⡀⠀⠀⢱⣄
⠉⠛⠛⠛⢿⣯⠔⠀⠀⠀⠙⠛⢦⡀
⠀⠀⠀⠀⠀⢷⣾⠀⠀⢠⠀⠢⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢻⣿⢁⡀⠀⣱⠀⠙⣦⢄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣿⣿⡄⢠⡏⠀⣶⡏⠀⣿⠈⣇
⠀⠀⠀⠀⠀⢸⣿⡿⢀⣿⡁⢠⣿⣇⠀⣿⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠛⠿⠾⠿⣧⣾⣿⣿⡀⠙⣷⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠻⣍⠓⢆⣻⡀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣹⡇⠈⡇⠀⠀⠀⠀⠀⠠⠂⠀⠄⠀⠸⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠚⠛⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠅⠀⠀⠀⠙⠲⢌⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠙⢴⡠⠀⢰⠁⠈⠔⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠠⠀⣀⣀⠛⠁⠀⠀⠦⠀⠸⡀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⣆⠀⢰⣶⣄⣀⢤⣤⣤⣦⣴⣥⣀⣀⣀⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣷⣾⣿⣿⣮⡻⠛⠻⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠓⠒⠛⠛⠛⣛⣿⣿⣿⣿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠦⠤⣤⠤⠤⠤⠴⠾⠿⠛⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠁⠀⠀⢀⠠⠤⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠤⡀⢤⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠦⠤⠐⠀⠈⠀⠠⠀⡀⣤⠀⠤⢤⡌⠛⠂⠀⠤⠆⠉⠉⠁⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⣀⣡⡒⠦⠤
⠀⠂⠀⠀⠢⠄⠀⠀⠀⠄⠀⠀⠀⡀⠞⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠠⠄⠊⠁⠀⠀⠀⠀⠀⠁⠒⠦⢀⡀⢀⡤⢊⠡⠀⠀⠀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠔⠈⠀⠀⠀⠂⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠥⠖⠁⠀⠀⠀⠀⠀⠸⡄⠃⠀⠄⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠄⠤⠀⢤⠄⠀⢂⠀⡰
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡆⠀⠀⠄⠂⢀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠄⠃⡤⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⢀⠀⠘⠀⠐⠄⠀⠀⠁⠀⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⠀⠀⠀⡐⠁⢠⡕⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠇
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣀⠂⢀⠀⠙⠂⠀⡀⠀⠀⠀⠑⢧⠘⠀⠀⠀⠀⠀⠀⠐⢁⠤⠀⠀⠘⠀⠠⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⣀⠀⠈⠀⠀⠀⠀⠢⠤⠀⠈⠀⠂⠈⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠈⠠⢌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠁⠀⡄⠀⠊⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠦⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⢀⠀⣀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠘⠲⡀⠶⠄⠦⠐⠃⠀⠀⡀⠀⢠⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠁⠤⠄⠀⠀⠀⠤⠄⠀⢀⡀⠀⢀⠠⠐⠠⠀⠀⠀⠠⠄⠀⠀⢀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣁⡀⠠⠄⠄⠀⠈⠠⠄⠈⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠠⢤⠄⠙⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠀⠀⠠⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⠀⠀⠨⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠉⠉⠀⠦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠐⠖⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠐⠈⠁⠀⠂⠀⠀⠀⠀⠐⠒⠈⣀⣁⠀⠀⠀⡁⠀⠀⠀⠀⠃
⢀⠤⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⢀⠀⠀⢀⠀⠀⠐⠢⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠈⢂⠘⠢⠄⠀⠠⠒⠀⠈⡠⠀⠀⢀⡄⠆⠆⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠰⠀⠂⠀⠀⠀⠀⠁⠂
⠀⢀⠀⠀⠀⠱⡀⠀⠄⠒⠀⠀⢰⠁⠀⢰⠑⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢂⠈⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠊
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⢆⣀⣀⡀⠀⠂⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠥⠄⠀⠀⠀⠀⡀⠀⠀⠀⠄⠐⠈⠉⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⢀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠓⢠⠄⠀⠀⣤⠚⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠢⠄⡈⠣⡙⠃⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠔⠒⣉⡐⠒⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⣤
⠀⠂⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠶⠾⡄⠀⠁⠀⢂⠐⠃⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠑⠀⠈⣉⠲⠀⠀⠀⠀⠀⠀⢀⡉⠀⠭⠭⣝⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣑⣤⣤⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⣠⣎⣉⣳⡀⠀⠀⠀⠀⡤⡤⠒⡶⣠⠒⡦⡀
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⢠⠮⠭⠭⠭⠽⠄⠀⠀⠀⣸⠥⠄⠬⠦⠨⣆⠀⠀⠀⣜⠁⣼⡙⣧⢎⢫⣾
⠀⠀⠀⢠⡗⡒⡒⢖⠠⣒⡲⣶⠾⡀⠀⠀⠀⠘⡎⠉⠉⠉⢹⠁⠀⠀⠀⠙⣟⢻⡭⡛⢹⠁⠀⠀⠀⢹⠙⠓⠓⠚⠚⢉⡇
⠀⠀⠀⠀⢹⠒⠾⡉⣘⢿⠓⠞⡟⠀⠀⠀⠀⠀⢣⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡟⢤⣺⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠉⠁⠘⠳⣰⠞⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡁⡝⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡤⢒⣂⡉⣐⠲⡄⣠⢤⠀⠀⠀⠀⠀⣀⠤⢤⠒⠢⡄⠤⡀⠀⠀⢀⡠⠔⠒⠒⠠⢄⠀⠀⠀⠀⠀⢀⠤⠒⠢⢄
⠀⠀⠀⢠⢋⠖⠧⠎⢀⡠⡫⠞⠹⡆⠇⠀⠀⢠⠞⠁⠀⠀⠣⡀⢣⣀⣈⠆⠀⣎⣀⣀⣀⣀⣀⣠⡕⠀⠀⠀⢠⠋⠀⠀⠀⠀⢣
⠀⠀⠀⢸⠈⠆⠏⠋⠻⠒⡿⢭⢅⠙⠂⠀⠀⡜⠣⡀⠀⠀⢀⡝⠁⠀⠀⠀⠀⠈⡄⢠⠈⡄⠰⢠⠁⠀⠀⠀⠸⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⢆⠳⢖⢐⢢⣀⡰⢋⡞⠀⠀⠀⠀⡗⢄⣈⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⢡⣘⠀⢃⣃⠎⠀⠀⠀⠀⡼⢑⣤⠴⠊
⠀⠀⠀⠀⠀⠑⠢⠤⠭⠤⠒⠋⠀⠀⠀⠀⠀⢇⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸⢭⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⣔⣒⣢⢠⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣴⢖⣂⠴⡒⣒⣄⠀⠀⠀⠀⠀⡤⠤⠤⠧⠧⠤⣄⣀⡀⠀⠀⠀⢸⠀⣇⠾⠼⠄⠀⠀⠀⢀⡔⡢⣄⡠⢚⠦⡀
⠀⠀⠀⠀⡎⡡⠒⠉⠒⠒⠒⢝⢦⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⣯⣗⠅⠀⠀⠀⡸⠀⡧⠋⠈⠣⡄⠀⢠⠊⠀⢱⠈⢳⠸⡄⠈⠆
⠀⠀⠀⢸⡴⠁⠀⠀⠀⠀⠀⠈⢿⠂⠀⠠⡆⠀⠷⠤⠄⠖⠋⠃⡲⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠸⡀⢀⠇⠀⢸⠀⡇⠀⡄
⠀⠀⠀⠸⠤⠤⠤⠀⠒⠀⠀⠤⠊⠀⠀⠀⠈⠑⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠀⠑⢤⣃⠴⠛⢄⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠣⠤⠤⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⡀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣀
⠀⠀⠀⠀⠀⠀⣀⠀⣞⣵⣵⠗⠺⡄⠀⠀⠀⢠⠃⠘⡀⠀⠀⠀⠀⠀⠀⣠⠊⠱⢼⢌⢷⠀⠀⠀⣠⣀⡠⠤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⢰⣊⠭⣔⡺⣽⠝⠁⠀⠀⡇⠀⠀⠀⡞⣚⠓⠙⡆⠀⠀⠀⠀⣴⠥⠔⠚⠉⢉⡏⠀⠀⢸⠀⠘⠀⣀⠈⠀⠀⠋⠀⢹
⠀⠀⠀⢎⣪⠛⠊⠉⠀⠀⠀⣠⠎⠀⠀⠀⠀⠹⠀⡇⢰⠃⠀⠀⡠⠊⠀⠀⠀⠀⡠⠋⠀⠀⠀⣞⡋⢛⡓⠒⡞⠉⣖⡛⢳⣋⡢
⠀⠀⠀⠈⠡⡀⠀⢀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠸⠋⠁⠸⡄⠀⠀⠣⢄⡀⠀⠀⡔⠁⠀⠀⠀⠀⠘⠬⠅⠈⠫⠬⠍⠤⠬⠥⠜
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⡄⠠⠃⠀⠀⠀⠀⠈⠢⠞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠲⠦⠤⣄⡀
⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠋⡿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⢯⣁⠀⠀⠀⠀⠀⢀⡠⡿⠀⠀⣠⠞⠓⢦⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⡼⠁⠀⠈⠳⣄⠀⠀⠀⠀⢀⣴⠇⣀⡄⢉⣑⠒⢲⠖⠉⢰⠇⢀⡞⡟⠀⠀⠀⠙⢦
⠀⠀⠀⡰⠃⢀⣠⡤⠖⠚⠣⡀⠀⠀⠀⠈⠳⡄⠀⠀⡟⣻⠀⡁⠀⠀⠉⠂⡼⠀⠀⡾⢀⡞⢰⣅⠀⠀⠀⠀⠈⡇
⠀⠀⠰⣗⠛⠉⠀⠀⠀⠠⠶⠾⢦⡀⠀⠀⠀⠹⡄⠀⡇⣿⠸⠇⠀⢾⠗⢀⠇⠀⢰⠇⡞⢠⠃⠈⠳⢄⡀⠀⢰⠃
⠀⠀⠀⠈⠳⣄⢠⠞⠀⠀⠀⣴⡄⠙⣄⠀⠀⠀⠘⣆⡇⣿⢜⠀⠀⠀⠀⢸⠀⠀⣼⡼⢡⠇⣤⠉⠀⠢⠉⣲⠏
⠀⠀⠀⠀⠀⠈⠻⣄⠸⠷⠀⠈⠁⠀⠈⢦⠀⠀⠀⠘⣟⣿⢌⣁⠀⠀⠀⡏⠀⢀⡇⠳⡞⢠⠈⠀⠛⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠠⠠⠜⠀⠀⠈⢧⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⠹⡝⠤⠂⢠⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⡀⠀⠃⠃⠀⠀⠈⣆⠀⠀⠀⢸⡆⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⢻⡀⢠⠇⢠⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢱⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣇⠀⠀⠀⠀⡆⠀⢸⠘⡄⠀⠈⣧⣏⡴⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠓⠷⢤⣀⣀⠀⠀⠀⢱⠀⠀⠀⢻⠀⠀⠀⠀⡇⠀⢸⡀⣧⣠⠴⠛⠉⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠉⠉⠙⠓⠒⠲⠶⠤⠼⠧⣤⣤⣴⣷⣚⣋⣉⣁⣀⣀⣀⣀⣄⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠶⠶⠤⠤⠴⠶⠖⠒⠚⠋⠉
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣻⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢠⣆⠀⠐⠒⠒⠒⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⣠⣴⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠉⠋⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠀⠀⠀⢶⣿⣿⣿⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣆⣀⣀⡀⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣦⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⣠⠄⢠⢠⣠⡄⢠⡄⡤⢤⢠⣠⡄⣤⢠⣤⠄⢠⠄⣠⡄⣤⢠⢠⡄⢠⡄⣤⣀⡤⠀⣤⢠⣠⠤⣄⣄⡄
⠀⠀⠀⠀⠙⠒⠘⠚⠐⠛⠘⠃⠓⠊⠛⠛⠓⠛⠚⠛⠂⠐⠃⠋⠓⠛⠘⠒⠛⠛⠐⠋⠋⠓⠂⠛⠘⠘⠒⠉⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⡄⠀⢸⠋⢻⠀⢰⠖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⣾⠀⣿⣀⣸⠀⣿⣤⣤⡄⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣷⠀⢸⠛⠛⡆⣿⠉⠙⣧⣿⡀⠀⣇⣸⠀⣸⠧⣄⠀⠀⣰⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢾⡀⢸⡆⢸⡆⠀⡿⢿⠀⢸⡇⠉⣿⢸⡏⢹⣇⡇⢠⡏⡞⠚⣧⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣇⠈⣇⠸⡇⠀⡇⢸⡆⢸⣅⡀⡟⢸⠀⣼⢙⡇⣼⠻⡇⢠⠏⢠⠟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢻⠀⢻⠀⣇⠀⡇⢸⡇⣸⢸⡇⡇⣿⢠⡇⣸⢡⡇⢰⢣⡏⣠⢯⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣘⣇⠸⡇⢿⠀⣇⢸⡇⡿⢸⢱⡇⣯⣼⢀⡏⣸⢁⡏⡾⣰⣿⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠙⠛⠓⠻⠾⢧⡇⣿⢸⢸⠇⣏⣸⢃⡇⣼⣼⢡⡿⢡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣼⣾⡼⠏⠉⠛⠳⠿⢧⣿⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠈⠉⠛⠓⠲⠶⠤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠛⠲⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⠤⣤⣤⣴⠏





⠀⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣤⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠐⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠝⢱⠀⠼⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠍⠱⠀⠼⢉⡀
⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡀⠀⠀⠀⠀⡴⠋⡏⡩⠦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡴⠫⡎
⠙⢣⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠆⠅⠜⢃⡶⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠰⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠰⠤⠒
⡄⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠖⡼⣿⡆⢊⠀⠀⠀⠌⠄⢉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⣖⣾⣻⡆⠊⠀⠀⠀⠌⠆⢙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠀⠀⠀⠀⠶⢋⠘⠃⢦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⠀⠀⠀⠀⠶⠛⠘⠇⢦⡴⠂
⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠨⣁⠀⡤⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣀⠂⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠀⡠⠦⡴⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡁⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢀⡠⠢⡤⠀⠀⠀⠀⠆
⠀⠀⠈⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃
⠀⠀⠀⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⢀⠉⡰⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡰⠉⠆
⠀⠀⠀⠀⡶⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠃⠠⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠜⠹⠀⠼⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⠀⠍⠹⠀⠨⢡⡀
⠦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠐⠀⠀⠀⣴⠁⡏⠩⡦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡰⠉⡏
⠙⡃⡶⠀⠀⠔⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠔⠁⠜⠃⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠐⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠐⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠒
⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆
⣗⠺⣻⡆⢂⠀⠀⠀⠌⠂⢉⠿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⣊⣖⢾⣹⡖⠊⠀⠀⠀⠈⠂⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⠀⠀⠀⠖⢋⠘⠃⢦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡴⠀⠀⠀⠀⠷⢛⠘⠃⢢⠴
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠈⣁⠐⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣁⠂⠤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡰⠦⠶⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡤⠲⡴⠐⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠀⠉⠛⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃⠀⠀⢀
⠀⠀⠀⠀⢀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠅⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠈⠉⠰⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠃⠀⠊⠀⠀⠈⠀⠐⠀⠓⠀⠀⠀⠂⠐⠀⠀⠀⠐⠈⠒⠒⠀⠁⠁⠘⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠠⠎⠉⠀⣀⡀⠠⠄⢀⡀⠁⠒⠒⠀⠐⠀⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⡤⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠄⢹
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋
⠀⠀⠀⠀⠉⠒⠠⠤⢄⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣀⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠒⠒⠒⠈⠁⠐⠀⠊⠉⠉⠉⠉⠉⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⣠⠤⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⣀⠀⢀⣀⠀⢡⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠾⠏⠀⠾⠿⠇⣼⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠣⠦⠄⣠⡟⠀⠀⣰⡏⢻⣡⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣤⣶⣏⠀⠀⢸⣟⣳⣎⡠⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠃⡍⠉⠁⠛⢻⣶⣤⣿⣡⣿⡗⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⠋⠁⢽⣶⣤⣼⣷⣤⣤⢾⣿⠋⣿⣃⠀⠑⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠃⠀⣀⡘⢿⣷⣾⣿⣿⣛⣿⡆⠀⣿⣿⣿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢴⣾⣿⠋⠀⢀⣴⣿⣿⡞⣿⣻⡟⠛⣿⡿⣷⠀⠹⣿⡟⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⣷⣾⡏⠀⠈⠻⢿⣿⣿⠟⠻⠿⣿⣿⣯⡿⠲⠟⡁⢶⢋⠁⣸⣿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣘⣿⣴⣿⠉⠉⠙⠶⣤⣄⠀⠉⠁⠀⠀⠀⠀⠀⣠⠁⠀⠀⣸⠉⣛⣡⣿⣿⣷⣦⣤⣾⣿⣷⣾⣿⡿⣀⠀⣾⣿⣿⣿
⡄⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣄⠀⠀⠀⠀⣟⣿⣶⣤⠌⠁⣄⢀⠔⡡⢠⢁⣴⠁⡼⢋⣽⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠈⡋⠀⠈⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⢐⡾⣹⠏⠙⢦⣤⣼⣞⣀⡀⣠⡟⠈⣡⡴⠟⠻⣦⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿
⣧⣄⣞⠠⣤⣀⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣯⣬⡙⣿⣷⣯⣴⣥⣄⣀⣸⣿⡟⠈⠈⢙⣋⣻⣏⣙⡦⢀⡄⠀⠐⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠁⣈⠻⠛⠻⡿⣿⠿⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢧⡢⡀⣰⣿⣿⣿⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢶⣿⣿⡃⠀⠀⠀⠶⠂⣤⣴⠿⠿⠇⠻⠛⠛⢉⣿⣿⣿⡇⠈⠳⢽⣿⣿⣿⣿⠁⠈⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢈⠉⠣⠀⢀⣀⣀⣰⠁⠀⠀⠀⠲⠀⠉⣾⣿⣿⣿⡟⠀⠀⢠⣾⣿⣿⡿⠃⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠿⢋⣀⠀⠈⠒⠒⢀⠸⠿⠟⠻⠛⠋⢀⡀⠀⠀⠘⠈⢹⣿⡿⠁⠀⣰⣿⣿⣷⣀⠀⢀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣿⣿⣿⣿⣿⡟⠹⠁⠀⠒⠋⠈⢡⣠⡀⠀⠀⠀⣀⠀⠀⠀⠙⠋⡴⠀⢹⣿⣿⣿⠿⠃⢀⣾⣿⠃⠀⢹⣿⣶⡞⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣿⣿
⣿⣡⣼⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⠀⠀⠛⠛⠛⠄⡀⠨⣿⡶⠂⠀⡀⠐⠀⠂⠙⠙⣿⣿⡀⢫⣸⣿⡅⠀⠀⠀⠀⠈⠃⣰⠄⠀⠀⠀⠀⠀⠟⠿⣿⣿⣿
⣟⢳⣿⣿⣿⣿⡟⠛⢿⣶⣤⠀⠀⠈⣀⣤⣤⣤⣄⣸⡥⠂⠉⠈⠀⠈⠠⢦⡁⢈⡀⠀⣼⣿⠀⢸⡿⣿⣿⡟⠀⠀⠀⠀⠀⠙⢿⣶⣶⣶⠂⠐⠈⠀⣿⣿⣿
⣟⣞⣿⠟⠁⠀⠀⠀⠘⠿⠒⠂⠀⠀⠀⢀⡀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣤⡆⢻⣿⣄⠀⠀⢀⣿⡿⢷⣤⠀⣀⣤⣤⣿⠟⠛⠙⣿⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣄⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⡆⠱⠈⠂⠀⢀⡀⢀⡾⠛⠁⠀⣤⠉⠉⠙⣿⣿⠟⠉⠁⠀⠑⢉⡩⠁⢠⠰⠆⠀⠀⠉⠉⠉⠉⠙⠿⣿
⣿⣿⠟⠛⠛⠃⠀⠀⢀⠀⠀⠀⠤⢤⡄⠀⠀⠸⣿⣷⣄⠀⣤⣶⣾⠀⠈⢀⠀⠀⠚⢿⣤⣀⠀⠈⢁⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣠⣿
⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠘⠉⠉⠉⠀⠉⠛⠀⠀⣠⣤⣾⣦⣀⠙⠟⠀⠀⠀⠀⣀⡴⣶⣆⠀⠉⠉⠀⠀⠀⠀⠉⠁⠀⠀⣶⣿⣿⣿
⣿⣿⡿⠃⠀⠤⠀⠀⠀⠤⣤⡀⠀⠀⠀⠀⢰⣾⠃⠀⠐⣋⣁⠀⠀⠂⣀⡀⡙⠿⣿⡧⠀⡄⠀⠀⠐⠊⠁⢀⡉⠉⠀⠄⠂⠀⠀⠄⠈⠉⢠⣴⠄⠉⠉⢻⣿
⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠘⠷⣆⠀⠀⠢⠀⠙⢀⣐⠂⠀⠀⠀⠀⣴⣿⡟⠀⠀⣄⠈⣣⡀⠀⠀⣠⣤⡘⠀⠁⢰⣦⣤⡴⢷⣄⣠⣀⢀⠀⢀⣶⣀⣀⣠⣿
⣿⠏⡤⠾⠓⠆⠄⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⠟⠛⠀⠀⢡⠠⠌⠑⣤⡀⠈⠃⠄⠀⠀⢄⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠉⠛⠋⠉⠀⠉⠙⠻⢿⣿⣿
⡇⠀⠀⠀⠀⠠⠤⠶⠀⠠⣦⠀⣐⣀⡀⠐⠀⠀⠈⠹⠆⠀⠀⠠⠄⠀⠀⣀⣸⣿⡶⢄⠀⠀⠀⠀⠳⠶⠖⠀⢱⣄⠀⠞⣡⠄⠄⠐⠁⠀⠠⠀⠀⠀⠘⢿⣿
⣿⠀⠀⠀⠀⢠⣴⣀⣤⣴⢿⣿⢿⢿⣷⠀⠄⠀⣶⠀⠀⠀⢦⣄⠖⠀⠀⠙⠁⠉⢀⣀⣀⠀⢀⠀⣰⣀⣤⣴⣾⣿⣶⣶⣄⢀⣠⠀⠀⡀⠀⠀⠀⠀⠀⢀⣿
⡿⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠐⠦⠄⠀⠁⢀⣾⠿⠦⠤⡀⢀⠀⠀⠀⠀⠀⣸⣦⣌⠉⠉⠁⠀⢀⠟⠛⠻⠻⠛⢛⡿⠛⠻⠿⠉⠀⢚⡁⠠⠀⠸⢦⣆⢸⣿
⣿⣶⣶⣤⣀⠀⢀⣰⣄⠀⠀⠐⠒⠠⣶⣶⣿⣇⠀⠀⠀⠘⠂⠐⠀⠑⠌⠹⠉⠙⠛⠄⣀⠀⠑⠈⢀⠀⠁⠀⠉⠉⠉⠀⠀⠀⢈⠢⠙⠂⠀⢠⣆⠀⠙⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠐⠀⠀⠀⠀⠈⣉⡁⠀⠀⠀⣀⠀⣀⠀⠐⠻⠷⣤⣀⠠⡈⠀⣶⡖⠋⠀⢈⣉⡀⠂⠀⠐⢻⡆⠀⠐⠉⠀⠀⢀⣹⣧⣀⣹⣶
⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣄⣀⣀⣬⣆⡀⠈⠉⠀⠀⠀⠈⠀⠨⠛⠀⠀⠀⠈⣻⣷⡅⠀⢹⣧⣤⣀⠀⠈⠀⠠⣄⢀⣀⣼⣶⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠾⠿⠿⠛⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣶⣶⣾⣶⣤⣀⣤⣤⣤⣤⣼⣿⣷⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⢉⠙⢛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⢀⣠⠴⠒⠛⠛⠛⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⢀⣴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢠⡞⠋⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠤⣄⠑⢄⠘⣿⡛⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠔⠢⠀⠱⡌⠀⠘⣿⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⠀⣸⡏⣿⣯⠃⠀⠀⠀⠀⡇⠘⠀⣿⣂⣀⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⣯⠀⢸⠀⠀⠀⠀⠀⠸⣧⡈⣿⣷⡤⠾⠀⢀⠇⠀⢠⡿⠛⠛⠛⠓⠲⢤⣥⡀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠘⣯⡀⠈⢂⠀⠀⠀⠀⠀⠙⠻⣿⣦⣤⣤⠴⠋⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⡠⠤⠤⣀⡀⠀⠙⢷⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣾⣿⡷⠖⠀⡀⠙⣦⠀⠈⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠸⡟⠷⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋⠀⠀⢰⣿⣿⣿⠇⠀⠀⠈⠀⢸⡆⠀⣸⡇⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠙⠻⡿⣿⣷⣿⠾⠒⠛⠉⠀⠀⠀⠀⠸⣿⡟⢿⣷⣧⣀⠂⠀⣼⠀⠀⣿⠃⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢀⣿⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠳⣦⣌⣉⣩⣤⠾⠋⠀⢠⡟⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢸⡟⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠂⠀⠀⢀⣾⠁⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⠲⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⠟⠋⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢬⣤⣀⣤⣤⣤⣦⡴⠞⠛⠋⠉⠉⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣖⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⡀⠀⠀⠀⡀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⠀⠀⠈⢀⠈⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠈⠈⡀⠀⠀⠀⠀⢀⡠⠔⣒⣭⡉⣭⢵⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡴⡿⢶⣼⠁⢠⡇⡇⠀⣸⣿⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⣠⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡞⡽⠷⣞⢨⠿⢭⣀⠟⠻⢯⣸⠛⢳⣱
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢂⢔⠋⠀⠀⠀⠈⠁⠀⠀⡴⠁⠀⠀⢰⠻⣥⣤⡏⢺⣤⣼⠚⣦⣤⡇⢺⣤⣼⢃⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀⠉⠀⠌⠉⣀⡢⠔⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠘⣱⡏⠉⡇⢸⠁⢈⣷⡅⠈⡏⣿⠀⢸⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⣀⣤⣟⣓⡚⣛⣛⡿⣿⣎⡟⣻⣷⣾⡿⢿⣾⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⢳⣼⠁⢸⣯⡇⠀⡟⡟⢯⡻⣄⡏⠁⣿⣿⠈⢹⡾⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣚⣶⡟⣸⠶⢾⡉⡿⣾⣋⢓⣺⣇⡙⣿⣚⣋⢹⣶⣋⠙⣚⣋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣼⠁⢈⣿⡏⠀⣸⢿⡀⢀⡏⢯⣠⣼⡇⢹⠀⢸⣾⠀⢸⣿⠇⢘⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣨⠭⠯⣅⡽⠿⢿⡄⡟⠛⢶⣺⠋⠉⡿⢸⡗⡚⢙⣒⣛⡉⣛⣛⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡸⣆⣀⠿⣧⣤⣼⠻⢧⣤⡞⢸⡶⣞⢡⡟⠉⢱⣿⠀⢨⣷⡇⣸⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⢏⢳⣾⠋⢻⣦⠏⠉⢽⣿⡴⣫⣿⢳⣤⡞⢹⠶⣟⡼⣿⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣯⣝⣲⣞⣘⣶⣶⣯⣴⣿⣟⡵⡏⠀⣼⣿⣀⠞⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠛⠿⠷⣦⣭⣭⣥⡴⠾⠛⠁


⠀⠀⢀⡀⡀⢀⢀⡀⠀⠀⣀⡀⢀⢀⢀⣀⠀⠀⢀⣀⣀⡀⢀⡀⠀⠀⢀⡀⣀⡀⣀⡀⠀⠀⠀⢀⣀⢀⣀⢀⡀⠀⠀⢀⡀⢀⠀⣀⣀⠀⠀⢀⠀⡀
⠀⠀⢹⣾⢳⡟⣏⣯⠠⠆⢿⠟⣿⢾⢿⣹⠀⠆⢸⣇⣿⠿⣿⣯⠰⠀⣿⠿⣏⡿⡿⠀⠲⢀⣸⣿⠿⢻⣷⣯⣿⠀⠆⢸⣹⣾⢯⣿⠆⠰⠀⣾⣧⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠟⢿⡄⣾⠟⠛⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⠀⠀⢸⣧⠀⠸⣧⣿⡇⠀⣿⣿⠀⠀⣿⡶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⣤⣤⣴⣿⡀⠀⣿⣿⡇⠀⣿⣿⠀⠀⣿⠇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣅⡀⢿⡇⢀⣼⣿⣇⠀⢸⣿⡿⠻⢿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⠋⠉⣿⢸⣷⣿⡇⠀⣿⠀⢸⣿⡇⠀⣼⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣿⠀⣿⣿⡇⠀⣿⠀⢸⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⠀⠀⣿⠀⢹⣿⡇⠀⣿⠀⢸⣿⠀⠀⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢻⣿⠀⠀⣿⠀⠘⣿⡇⠀⣿⡄⢸⡟⠀⢸⣿⡟⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⡀⠀⣿⡄⠀⢻⡇⠀⣿⡇⢸⡇⠀⢸⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⢷⣄⣿⣇⠀⢸⣿⠀⣿⣿⣾⠇⣠⡾⠃⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠿⣿⣄⣀⣿⣆⣿⣿⣿⡿⠋⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿

⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢻⡉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠉⣓⠢⢤⣀⣀⣀⡴⢁⣤⡀⠳⣄⠀⠀⠀⠀⢳⡀⠀⠀⣶⢬⡙⠲⣄⡀⠀⠀⠀⡰⠋⠀⠀⢸⠀⠀⠀⢀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠐⠢⢀⠀⠠⠚⠉⠙⠣⡈⠳⣄⡀⠀⢀⣷⠀⠀⡿⠀⠙⣆⠢⣙⠦⣄⠜⠀⣠⢆⡴⢁⣀⡤⠞⠉⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠡⡀⠀⢀⣀⠀⠠⢙⠢⡀⢀⣀⡤⢌⣀⠄⠉⠛⠛⢹⡆⠰⣅⢦⠀⠘⡆⠈⢻⡎⠀⣴⢃⣞⣉⠉⠀⠀⠀⠀⢰⠁⠀⢀⡠⠖⢊⡅
⠀⠀⠀⠀⠐⢧⡁⠒⠲⢬⣇⣳⡀⣏⠀⠙⠢⣄⠙⢮⣿⣇⢀⠀⠈⠻⣆⣤⠤⠴⠓⠒⠚⢚⠓⠒⠋⢉⣉⣉⣉⡀⠉⠉⠦⢀⡆⠀⠀⣰⡸⠄⠊⠁⡠⠀⠎
⠀⠀⠀⠀⠀⠀⠻⡄⢣⣀⡈⠙⠳⣌⡳⡕⢄⠈⢦⢑⣽⠿⠷⡕⢤⡀⠘⢦⣤⠤⠦⡔⠉⠉⢱⣒⣊⣭⠭⢿⡏⡤⠴⠊⠁⡼⣡⡤⢚⢁⣠⡤⠚⣉⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠢⣍⠒⠤⣄⡈⠻⡎⢇⠀⢫⡐⠦⠶⣚⢧⣙⣆⣀⣹⡛⢹⠁⢠⢅⠿⣹⣤⠼⠖⠚⠛⠓⠂⡄⣰⡷⢋⡴⣛⢯⡴⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢳⣦⠽⠷⣿⡜⣆⠀⠻⠋⠉⠉⠉⣉⣁⣈⠉⢉⠉⠒⣲⠾⠛⠉⢀⣤⠶⢋⣁⡴⢋⡼⢋⢔⣥⡾⠑⢈⠉⣉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡄⢀⢢⡄⠉⠛⢦⢄⣀⣉⣭⣭⡿⠷⣯⣥⣖⡿⠞⠁⢀⣠⣖⡵⠒⠋⠛⠓⠾⢯⣴⣣⠞⢁⡴⢂⣵⣺⣏⣇⣀
⠀⠀⠀⠀⠀⣀⠤⠤⠤⠒⠒⠓⠚⠒⠠⠷⢤⡐⠤⡀⠑⢤⠽⠒⠒⢂⡀⠼⠒⢉⣠⡤⢔⡭⢾⣽⣒⡂⠄⣤⣤⠤⠀⠈⠉⡿⢋⠴⠥⠤⠤⣤⠀⠈⠉⣙⠶
⠀⠀⠀⠀⠹⡠⣬⡒⠤⠀⢀⣀⡤⠽⣿⡿⠟⠛⠢⣌⠷⣤⣑⡾⢛⡡⠴⠒⣀⣩⣤⡖⢻⡴⠯⠤⠽⢽⣏⣲⡶⠲⠶⡶⢒⣷⣭⣗⡈⠙⠶⡶⡤⠒⠉⠁
⠀⠀⠀⠀⠀⠑⠤⠄⠒⢩⢟⣼⠴⠋⢁⣀⠴⣿⠿⢷⡟⢸⠋⠐⢉⣠⡖⠉⠜⣻⡸⡷⠀⢿⠶⠴⠤⣤⣄⣀⠉⠉⠛⠳⢯⣌⡙⠓⢍⠑⠢⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣃⠺⠁⢀⣀⣫⠔⣉⣤⣤⣀⣿⡘⠀⣔⣿⡀⠑⢮⠞⣩⢷⣳⡄⠈⢯⠛⢉⠿⠓⢯⣉⠐⠓⣢⠀⠨⠉⠛⢆⠈⣦⣉⠛⠀⠳⢄
⠀⠀⠀⠀⠀⢀⠔⡡⠊⠀⠀⠀⠀⣀⣠⣎⣁⣒⡒⠶⠤⠤⣭⡽⠻⣿⠀⠈⢿⡗⠚⣳⡙⢄⠀⠙⢁⣠⠖⢉⣼⠳⢔⠮⠥⣄⣁⣀⣀⣱⡏⡼⠙⠓⠒⠒⠃
⠀⠀⠀⠀⡠⠋⠈⠀⠀⠀⠈⠉⠁⠋⠉⠽⢿⡿⠟⠉⣉⣽⠏⠀⠰⢻⡌⡄⠈⢯⡉⠋⡛⢄⠂⠀⡰⢋⣴⢿⣀⠑⠢⢷⠒⠲⠒⠶⠠⠤⠷⠷⠤⣀⣀
⠀⠀⠀⡐⠁⠀⠀⠀⠀⢀⣀⣀⣀⡰⠖⠋⢁⡴⠲⢟⣻⠃⠠⠄⣠⡻⠓⣈⠂⣈⠑⢶⢦⣐⠛⣣⠾⢥⣤⣙⣯⣕⠢⡀⠙⠶⣒⣒⡯⠭⠤⢤⣄⣀⠀⠈⠁⡀
⠀⠀⡐⠀⣀⣤⠴⠚⠛⠉⠉⠩⠯⠤⣄⣚⣁⣤⡾⢯⣽⣲⢦⣋⠟⠁⡰⢫⠷⢬⡤⠚⠓⠪⢽⣓⠤⡀⠈⢹⣤⣈⠳⣌⠢⡀⠉⠫⣍⠉⠻⢲⡖⠀⠉⠀⠉
⠀⠠⡷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠻⢶⣆⣍⣿⠋⣠⠞⣰⠃⠤⠤⠤⠤⠴⢖⣻⣭⠷⣮⠴⠚⢿⣿⣿⡾⣷⣤⢥⣤⡬⠷⠄⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠁⡰⡃⣰⣛⣋⣓⡞⠚⠚⡛⢃⣒⣀⣢⣤⡷⠾⠟⠛⠉⠉⠲⢍⡓⢤⡑⠂⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠐⡹⣡⠃⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⢥⣀⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠑⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓
⣄⣀⣄⣀⣀⣠⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠄⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠄⠄⠀⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠉⠙⠓⠒⠒⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠛⠉⠙⠒⠲⢤⡀⠀⠀⠀⢀⣸⠤⠔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠖⠋⠁⠀⠠⠖⠡⠟⡷⣄⡀
⠀⠀⠀⠀⠀⠐⡒⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠰⡄⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⡈⠹⠿⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⢟⣹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣉⣲⣄⣀⢤⠄
⠀⠀⠀⠀⠀⠀⢁⣮⠆⢽⡍⢿⣖⢶⠶⢶⠒⣒⣚⣹⠉⠞⣳⡽⠉⠈⠢⣀⠀⣀⣤⣴⡶⢛⣟⢛⡛⣉⣁⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠸⢛⠦⢄⣀⠀⠈⠉⠙⡓⠀⣉⠀⠘⠋⡥⢎⡟⢠⢀⠀⠀⣽⠉⢤⡏⠀⡎⢻⠛⢯⣄⡈⢙⣭⠗
⠀⠀⠀⠀⠀⠀⠘⠦⢀⣠⡌⠛⠁⠤⢄⢠⣈⣁⡀⠀⡀⣠⠎⠁⠣⣌⠄⢸⣿⡄⢸⣬⣖⡽⠄⣇⢀⠦⠽⢾⡜
⠀⠀⠀⠀⠀⠀⠐⢄⢀⠈⠉⠁⠈⠀⠈⠁⠉⠉⠉⠁⠈⢀⠄⠁⠀⠈⠢⣥⠙⣧⠈⣟⠬⣉⣁⡟⢴⣟⠓⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠐⠒⠒⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠙⡓⡁⢶⠨⠇⠈⠂⠭⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠱⠘⠏⠇

⣿⣿⠻⢿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠶⡶⠶⠶⡦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣉⠀⠳⣤⡞⠁⢙⣶⣇⠀⠀⠉⢻⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢘⡾⣇⣀⡴⠋⠙⢲⣾⡁⠈⠓⢦⣰⠏⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⢳⣴⡋⠀⣠⠟⠷⣄⣰⠋⠀⠙⠦⣄⡼⠉⠳⣤⡞⠁⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢤⣠⠞⠁⢙⡶⣇⠀⢀⡾⠛⠦⣄⠀⣠⠟⠙⢦⣴⠏⠙⠳⣄⣠⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡼⠋⠳⣤⡞⠀⠈⣹⢿⣀⠀⠀⠈⡿⢧⣀⢀⡼⠉⠓⢦⣀⡞⠙⠂⢷
⠀⠀⠀⠀⠀⠀⠀⢸⠓⠿⣄⣀⡼⠁⠙⢳⣼⡃⠀⠈⠳⣦⡞⠁⠀⣨⠟⢦⡀⠀⣰⠋⠳⢤⣠⢼⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣨⠟⢦⣄⣠⠞⠁⠙⠶⣄⣰⠋⠉⢳⣴⡃⠀⠀⢩⡿⢥⡀⠀⣴⠛⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡈⠿⢧⡀⠀⣼⠛⢦⡀⠀⢀⡼⠛⠦⣤⡞⠁⠙⢶⣴⠋⠀⠀⢙⡿⣅⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢉⡿⢧⡀⠀⠉⣳⣯⡀⠀⣴⠋⠛⢦⣠⡞⠉⠙⢦⣤⠏⠀⠀⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣔⣶⣏⡀⠀⠉⢳⣴⡃⠀⢙⡾⣅⠀⠀⡴⠋⠳⢦⣀⡴⠋⠳⢦⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠙⠲⣤⡞⠁⠙⣳⣏⡀⠈⣻⢾⣁⠀⠀⢀⡟⠳⣤⡀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣰⠋⠙⢲⣾⡁⠀⢙⡶⣏⡀⠈⠳⣴⡏⠀⠀⠀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠛⠀⠉⣳⣾⡁⠀⠙⠀⡼⠃⢉⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠧⠤⠤⠤⠤⠶⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠀⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠒⠉⠁⠒⠀⢇⠀⠀⠀⠁⠀⢀⠠⠔⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠐⠂⠄⢀⠀⡐⠈⠀⠀⡇⠀⢁⠀⠁⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⡄⠂⠙⡇⠀⠀⠀⠁⠀⢸⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄
⠀⢀⠠⡄⠀⠤⠀⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢠⠤⢚⠀⠑⠀⠤⡀⠀⠀⢠⠀⠀⣀⡄⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢐⠀⠐⠀⠄
⠀⠁⠀⠁⠀⠀⠀⠀⢠⠄⠀⠀⠀⣤⠀⡄⠀⠀⠀⠀⢸⠀⠀⠈⠀⠆⠁⠀⠀⠀⠠⠘⠣⠀⡀⠠⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⡖⠈
⠀⠃⠀⠀⠀⠀⠀⠘⠃⠀⠀⢤⠊⠀⠀⠇⢠⢀⡀⠀⣈⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠀⠀⢸⠀⠀⠀⠀⣀⢀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⠸⠀⠀⠀⠀⡀⠀⢡⠘⠑⢠⠠⠄⠐⠚⠂⠀⢰⠀⣠⠅⠠⠤⠴⡄⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠉⠰⠀⠀⠀⢸
⡄⣠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠄⠀⠒⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠐⠒⠺⡀⠀⠀⢸⠀⡠
⢀⠀⠁⠐⠆⣀⠀⠀⣇⠀⠤⠀⡀⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠒⠘⠆⢠⠘⠁
⠈⠀⠀⠀⠀⢘⠁⣀⠉⠀⠂⠄⡀⠙⠂⠃⠀⠸⠀⠀⠀⡠⠄⠀⡄⠐⠀⢀⠀⠀⠀⠀⠁⢘⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠀⠀⠀⢠⠃
⠀⠃⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠂⡀⠀⠰⠀⠊⠄⡀⠠⠐⠈⠁⠀⠘⠀⠀⠀⢀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠄⠀⠀⡠⠁⠀⠄
⠀⠰⠀⠀⠀⠀⠀⠀⠈⠢⡀⢠⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⡘⠃⠤⠀⠀⢁⠳⡄⠀⠀⠈⠀⢀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⡀⠁⠀⡀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠃⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠂⠀⠀⠆⠀⣈⠄⠁
⠀⠀⢁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⡅⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⢸⡀⠜⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠼⢀⡀⠀⠀⡄⠀⠀⠀⢘⠀⠀⠀⠠⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸
⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠈⠉⠀⠀⠈⠉⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠀⣀⡀⢀⣀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣤⣀⣒⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠊⠁⢢⢀⠤⢀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠃⠀⡨⠖⠸⡄⠀⠈⡇⠀⢠⠃⠀⢺⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠠⠈⡏⠀⠀⡆⠀⠀⡇⠀⠀⡇⠀⢸⠀⠁⢸⡄⠋⡈⠁⡄⠂⠒⡄
⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⡁⣀⢠⡗⠊⠒⡏⠑⠄⡗⠊⠚⡄⠀⠸⡷⠦⣀⠸⠇⠀⠀⠃
⠀⠀⠀⠀⠀⠸⠀⠆⠑⣆⠀⠀⡏⠠⢰⡇⠀⠀⠃⠃⢰⡇⠀⠀⡧⠤⢴⠀⠂⢸⠐⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠱⡀⠠⠈⢆⠀⡃⠀⢸⢀⠒⠼⠀⠀⢸⠁⠀⢀⡀⠀⠀⠀⠀⡆⠃⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⡈⢣⡁⣀⡟⠇⠠⠀⡇⠚⢉⠑⡶⠊⠑⠆⢀⣀⣠⣥⣁⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⣁⣈⣀⠈⢇⠁⠀⠀⠁⠀⢠⠀⡇⠘⠀⠀⡸⠁⠄⠠⠤⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠳⣄⠀⢸⠀⠈⠀⠃⠀⣠⠞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠢⣀⠉⠚⠓⠒⠚⠓⠉⣀⠔⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠀⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⡤⠖⠚⢉⡠⠤⠒⠲⠆⢹⠶⠚⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡤⠖⠋⣁⣤⠖⠊⠁⢀⣠⠴⠚⠀⢰⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⢉⣠⠤⠒⠉⠀⢀⣠⣤⣄⡈⠀⣀⣠⡴⠷⠶⠦⣄⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⣤⠤⠖⠒⢉⣉⣠⠤⠖⠛⠁⠀⣤⠔⠒⠀⠈⠀⣀⡤⠜⠋⠉⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡠⠖⠋⣁⠠⠔⠒⠋⠉⢀⠀⠀⣠⠴⠤⢀⠀⢀⣀⠤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⠎⣠⠔⠋⠠⠀⠀⠠⠶⠚⠋⠀⣠⣁⡠⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡥⠶⠒⠉⠢⢄⣀⡤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠖⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣤⣄⡤⠤⠴⠒⠚⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡩⢣⠀⣤⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠓⣀⢦⠠⠤⡏⠇⠰⠰⡅⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⢆⡰⢌⡄⠹⡤⠿⠲⠁⣎⡹⠆⠡⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣄⢀⡸⠒⠎⠘⡀⡠⠞⢢⠀⠨⠁⠈⡀⢂⠈⠎⠖⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢹⠇⣤⠋⢄⠀⠑⢻⠲⠉⢇⠀⠁⠡⠐⠀⣃⣸⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⣶⠀⠀⠀⠀⠀⣀⡇⠀⠈⢆⠈⡆⠀⠈⠓⠋⢇⢃⠘⡆⠀⠀⣣⡶⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣰⠃⠀⠀⠀⠀⢸⡗⠶⠦⢤⣬⣤⣽⣄⣦⣤⣥⣸⣬⣤⡴⣶⠞⠉⠀⠀⠀⠀⠀⡇
⠀⠀⣠⡤⠔⠒⠂⠀⣈⡧⡤⠓⠒⠤⢤⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⣤⣤⣤⣤⣧⣤⣀⣀
⠀⣼⣏⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⣀⢟⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠁⢀⠀⠀⠉⠉⠉⠛⠻⢷⣦⣤⡀
⠀⡷⠬⣍⣛⣒⠚⠛⠒⠒⠒⣒⣚⡉⠥⠚⡇⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢉⡁⡄⠀⠀⠠⠀⠂⠀⠉⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⡏⠀⠀⠈⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⠀⡇⠀⠸⡆⠀⠀⣀⣴⠾⠋⠁⡠⢰⠄⠉⠀⠂⢂⣠⠤⠖⠒⡈⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⢿⣷
⢠⡇⠀⠀⠀⠀⠀⠠⠀⠀⠄⢀⡀⠀⠀⢀⡇⠀⠀⣇⣠⡾⠋⠁⠀⠀⢀⣈⣀⣤⠤⣤⠴⠿⠤⠤⣤⣉⣀⠀⢴⡴⠀⠉⠉⠠⡠⠆⠀⠀⠀⠈⣿
⢸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⢀⣴⠟⠁⠀⢀⠀⠐⠊⠉⠁⣠⠔⠋⠀⠀⡄⠀⣀⠀⠈⠉⠛⠲⣄⠀⣆⠜⠀⠀⠀⠀⠀⠀⣸⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡀⢸⣷⠟⠁⠀⠀⠺⠈⢀⡠⠂⣤⠞⠁⠀⢀⡀⠉⠀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃
⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠄⡆⠀⣠⠞⠁⠐⠁⠐⠒⠁⢠⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢻⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣰⡟⠀⠀⠀⠀⠀⠀⠋⠀⡴⠃⠀⠀⠸⠥⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⢀⣤⡞
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⣀⣤⠞⠋⣠⡇
⢸⡇⠀⠀⠁⠀⠀⠀⠀⡜⠀⠀⢨⣿⣦⣄⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠉⣀⣤⠶⠛⠉⢀⣤⠞⠉⡷⢤⡀
⣿⡇⠀⠀⠀⠀⠀⢠⠎⢀⡀⠔⠉⠳⣍⡛⠿⢿⣶⣶⣴⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡿⠿⠟⠛⢉⣁⣤⠴⠞⠋⠉⠀⣀⣤⠾⠋⠁⣀⡼⠃⠀⣿
⣿⠇⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⣇⠙⠓⠦⢤⣈⣉⡉⠉⠉⠉⠉⠉⢉⣉⣁⣠⣤⠴⠖⠚⠋⠁⠀⢀⣠⣤⠶⠛⠉⠀⣀⣴⠞⠋⠀⠀⢠⣿
⣿⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⣄⣀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⢀⣀⣤⠶⠛⠉⠀⠀⠀⢀⣴⡿⠁
⣿⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⣌⠉⠙⠛⠓⠒⠲⠶⠶⠶⠖⠒⠛⠛⠋⠉⠁⢀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⢀⣤⡿⠋
⣿⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠓⠲⠶⠤⢤⣤⣤⣤⣤⠤⠴⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁
⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⢛⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⠁
⠀⢿⣝⡛⠻⠷⠶⠿⠿⠟⢛⣋⣩⡴⠖⠛⠁⠀⠉⠛⠷⠶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠛⠋⠁
⣤⣤⣬⣭⣿⣿⣷⣾⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣭⣏⣼⣉⣯⣩⣿⣹⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠰⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⠀⠀⢸⠉⠁⠡⢀⠀⡀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⠤⠼⡀⠀⢠⡆⠈⠉⡅⠘⠈⠉⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠘⡄⠀⠀⡇⠀⢸⠁⢀⣰⡁⣼⠉⠓⢰⡇⠈⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠋⠹⡆⢀⣀⡇⡀⠋⠓⢒⡇⠀⠈⠀⠇⠀⠁⡇⠀⠀⡞⠇⠀⣠⣇⡀⢠⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠲⡇⠀⠀⡄⠃⠈⠁⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡇⠀⢰⠁⠀⠀⠀⠀⢠⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠂⠀⣷⠀⠀⢁⠀⠀⢠⡏⠀⠀⠏⠁⠀⠀⢰⠆⠀⢸⠀⠀⡜⠀⠀⣸⠀⠀⣸⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡄⠀⢸⠀⠀⢸⣃⠠⠼⠦⠓⠒⠂⠚⠂⠀⠚⠀⠀⡇⠀⠀⠛⠀⠀⡏⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⡆⠀⠸⠀⣰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⡇⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡀⣤⣀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠇⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠔⢀⣀⠀⣀⣀⣀⡀⢀⣀⣀⢀⣀⣀⣀⢀⣀⢀⣀⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠋⠉⠛⠉⠙⠛⠋⠉⠙⠋⠉⠛⠋⠙⠉⠙⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣼⣛⡲⢤⡀
⠀⠀⠀⠀⢠⠖⠋⡍⠉⡉⢉⠙⠳⣄⠀⠀⠀⠀⠀⠀⣴⠶⠾⠷⠶⣦⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⢀⡴⢚⣩⣭⡇⡿⢟⡙⢦⣹⡄
⠀⠀⠀⠠⣏⣀⣃⣀⣁⣀⣁⣘⣀⣸⡆⠀⠀⠀⠀⢾⣉⣉⣉⣉⣉⣉⡇⠀⠀⠀⠀⠀⢠⡟⠒⠻⡆⠀⠀⠀⠀⢠⡟⣴⣋⣐⢺⡇⣷⡯⢟⣫⣍
⠀⠀⠀⠀⣯⠤⠤⣤⡤⠤⣤⡤⠤⣽⠇⠀⠀⠀⠀⠈⣯⠈⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⣾⠛⢳⠀⣿⠀⠀⠀⠀⣾⢰⡇⢯⡽⢸⣧⠵⣚⠉⡇⢹
⠀⠀⠀⠀⢳⠶⣶⡒⠒⠒⠒⣶⠶⣾⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡶⠛⠀⡟⠀⠀⠀⠀⢹⡌⣇⢰⣷⡌⠄⠺⠿⣰⢇⡿
⠀⠀⠀⠀⠘⠳⠾⠷⣦⣠⠾⠷⠖⠋⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣇⠀⠀⠀⠀⠀⠻⣌⠳⢭⣴⣏⡷⠞⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠟⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⢴⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢀⣴⠻⡌⢿⢸⡟⣸⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢘⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡅⠻⣜⣳
⠀⠀⠀⠈⡏⠛⠛⠚⠛⠒⠛⠋⣿⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⣦⣀⡀⠀⠀⠀⢠⡞⠉⠠⠈⠠⠉⠄⠈⢳⡄⠀⠀⠀⠀⢀⡞⠰⠀⠄⠠⠈⡽
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⢁⡿⠀⠀⠀⠀⢹⡏⠹⡟⠉⢹⡏⢩⡟⠁⠀⠀⠀⠀⣸⣇⠐⠀⢂⣠⠞⠁
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣠⣤⣼⣧⣤⣤⣤⣤⣿⣭⣤⠀⠀⠀⠀⠀⢷⠀⣿⠀⣸⠁⣼⠁⠀⠀⢀⡤⡼⣃⡼⠓⠛⠉
⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠾⠀⠀⠀⠀⠀⠹⢦⣤⣤⣤⣤⣤⣤⣤⣤⠞⠀⠀⠀⠀⠀⠘⠧⠼⠤⠿⠴⠃⠀⠀⠀⠈⠳⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣉⣳⣆
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⣄⠀⠀⠀⠀⣾⣁⣈⣹⣄⣙⣦
⠀⠀⠀⠀⢀⣾⣩⣥⣀⣠⣩⣧⠀⠀⠀⠀⢀⡞⠉⠉⢹⠏⠉⠉⡟⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠲⢾⡀⠀⠀⠈⢷⠖⠲⠖⣶⣶⡞
⠀⠀⠀⠀⢻⣥⣄⣠⣤⣀⣤⣬⡇⠀⠀⠀⣼⢶⠶⠶⡶⠶⢶⠶⠶⡶⠶⢶⢾⡆⠀⠀⠀⠀⠀⢀⡴⠋⠳⢤⡀⢨⡇⠀⠀⠀⠘⣷⠖⢋⣡⣼⠁
⠀⠀⠀⠀⢸⡄⠈⡇⠀⣿⠀⣸⠃⠀⠀⠀⣿⣯⣛⣛⣟⣛⣋⣛⣫⣿⣚⣉⣿⡇⠀⠀⠀⢀⡴⠛⠳⣄⠀⠀⠙⣿⠀⠀⠀⠀⠀⢹⡞⢋⣠⡇
⠀⠀⠀⠀⠀⢷⠀⣷⠀⡟⢠⡏⠀⠀⠀⠀⠘⠲⠾⠭⠭⣬⣭⣭⣬⠭⠽⠷⠚⠀⠀⠀⢿⣉⠹⡄⠀⠘⣦⣠⡾⠃⠀⠀⠀⠀⠀⠈⣿⠉⡿
⠀⠀⠀⠀⠀⠘⠓⠛⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣻⠙⠖⡦⠀⠀⠀⠀⣠⠞⠳⣄⢀⡴⠛⣦⠀⠀⠀⠀⠀⣯⠭⣽⡇⣴⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⣏⡽⠷⣤
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⣷⠛⣯⣿⣹⠆⠀⠀⣰⠋⠐⣦⠈⣯⠀⢶⠈⢳⡀⠀⠀⠀⢸⠀⢸⠀⣹⠏⢹⣇⠀⠀⠀⠀⠀⠀⣠⢞⡵⣫⡞⠁⠀⠈⡇
⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡟⠛⢻⡾⠃⠀⠀⠀⢿⠀⣰⠃⠀⣽⢠⡏⠀⢸⠇⠀⠀⢀⡾⠀⠸⣾⠁⠀⠀⢸⡇⠀⠀⢠⡖⣫⡵⣫⡴⠋⠀⠀⢀⡼⠁
⠀⠀⠀⢰⡋⠀⠀⢀⣴⠏⢀⡴⠋⠀⠀⠀⠀⠀⠈⢧⡈⢁⡴⠳⣌⠃⣰⠏⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠸⣏⡽⠛⠁⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠙⢦⣴⠟⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠘⠛⠁⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠉⢳⣄⣀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠛⠓⠒⠒⠚⠃⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡏⠓⠒⠯⠝⢻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠆⠀⠀⠹⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⢉⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣤⡀⡥⠀⠀⠀⠀⠀⠀⣐⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⠞⡃⠠⡠⢹⣦⣾⣟⣽⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣻⣿⣿⣻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣯⠙⢋⡿⠣⣮⣉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣏⡉⠙⠁⠀⠀⠒⠦⠈⠉⠉⠉⠿⣿⣿⣿⣿⡟⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠻⠛⠲⠶⠶⣄⣲⣀⡀⡒⠒⠺⠽⠛⠛⢋⣼⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣛⣉⣤⣶⣶⣮⠙⠉⠉⠉⠉⠉⠉⢉⣭⣶⣶⣤⣀⣾⣿⣿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡿⣭⣙⡻⣿⣿⣿⣷⡄⠀⠀⢀⣴⣿⢿⡟⣛⣹⣿⣿⣿⣿⢾⣾⡏⠎⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠆⣈⡴⣋⠿⡝⢁⣙⠳⢶⡟⡇⠈⠀⢽⠋⡛⠞⢻⠁⢀⣰⡗⣵⠐⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡔⢡⠗⡂⣨⣍⠳⢮⠀⣀⣚⠋⠘⠀⠀⠀⢡⣴⣿⣉⣸⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣏⡟⠳⠓⠜⠉⠉⠐⠚⠛⠿⢆⠍⣀⣎⣹⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠤⠄⠒⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡠⠤⢀⠀⠀⢀⡠⠐⠂⠈⠉⠉⠁⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠒⠤⡤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧
⠀⠀⡞⠁⠀⠀⠀⠀⠀⠠⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢰⠙⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠒⠒⠒⠢⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⣀
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡁⠀⠀⠀⠀⠀⠡⢀⢔⠌⡲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⡐⠂⡄⠀⠀⠀⠉⠢⢄
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠢⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠱⡀
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⠀⠀⠀⠀⠀⢀⠐⣰⢇⢋⣾⢣⢣⢣⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢂⡿
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠘⠁⠀⡀⡀⡔⡍⠝⠸⢃⢊⡰⠡⢠⡳⢳⡅⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⠉
⢃⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⡀⠐⠌⠌⡌⢌⢜⢠⡨⠠⢡⠣⠱⣥⣻⣤⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡏
⠘⡎⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⣧⠠⠈⡌⠈⡐⠀⠆⠌⠠⢁⢃⢀⢡⡵⣻⣙⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣻⡀
⠀⡰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠳⣜⠠⠐⠀⠀⠀⠘⠀⡆⠄⠂⠂⠰⡁⣯⡪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣇
⠀⢐⢑⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠊⠕⡢⢄⣀⣀⠀⣘⡀⣸⣨⡴⢞⠗⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⡧⣟
⠀⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠌⠘⠁⠁⠂⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡁⠈⠠⠄⠀⠀⠀⠀⡀⣡⠧
⠀⢱⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣢⣜⠋⠁
⠀⠨⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢬⣵⠉
⠀⠀⢱⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿
⠀⠀⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⡟
⠀⠀⠀⡀⣗⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠔⠂⠀⠀⠀⠀⠀⢀⢠⢋⣾⡁
⠀⠀⠀⠱⣜⡲⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⣊⣵⡏
⠀⠀⠀⠀⠈⠑⢦⣉⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣪⡴⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠆⠀⠀⠀⠀⠀⠀⠐⣠⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢈⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣌⣉⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠀⠀⠀⠀⠀⠀⣠⠋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠽⠟⢦⡈⢬⢄⠀⠀⠀⠀⠀⠀⠁⠀⠠⡀⠠⣀⠀⡀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⣉⣉⣒⣀⡠⠄⡠⣁⣒⣀⣭⣤⣬⣽⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⠶⢶⣛⠟⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠈⠉⠉⠉⠁⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢴⡇⠤⠤⡄⣀⣤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠵⠞⠒⠒⠤⠭⡉⣀⠁⠈⠉⠉⠛⠸⢶⣦⢄
⠀⢀⠤⣀⠤⠐⠉⣴⣯⡤⢤⡤⢤⡶⣲⡞⡏⡍⡺⠉⠉⠐⠒⠠⠤⢄⢀⡀⠈⠁⠃⠒⠆⠠⠄⣘⠘⠪⠆
⠀⠨⣤⠀⡠⠖⠉⠁⡠⠴⠿⠒⠒⠂⠂⠢⠜⠅⠅⠁⠒⠂⠄⠀⢀⢀⠀⠀⠉⠘⠒⠂⠄⠤⣀⣀⠈⠈⠁⢪⢢
⠀⠀⠑⠐⠁⠀⠀⡴⡿⠂⠒⠒⠆⠒⠆⠄⣶⠘⠜⠆⠢⠂⠀⢤⢀⡀⠀⠀⠁⠂⠂⠠⠀⣀⡀⠀⠀⠉⠁⠒⡆
⠀⠀⠀⠀⠀⠀⢰⡇⡒⠒⠀⠆⠤⠠⠤⠠⠹⡀⠀⠀⢕⠤⣀⠀⡀⠀⠈⠁⡐⠒⠠⢄⢀⣀⡀⠈⠀⠓⠂⡴⢡
⠀⠀⠀⠀⠀⠀⢸⡐⠐⢄⠆⠄⠀⠠⠄⠀⠀⠘⠄⠐⠀⠉⠒⠬⢓⠨⠐⠀⡂⢠⣄⡀⢐⣀⢀⡉⠀⠒⡀⠔⡹
⠀⠀⣠⠤⠴⠦⠤⠷⢄⡐⠌⡢⠤⣀⢀⣀⠀⢀⠀⡁⠢⣀⠀⠒⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠒⠀⠈⠁⠐⠟
⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣁⠒⠨⠅⠐⢈⡘⠂⠐⢚⡛⠢⠤⠄⣀⡈⠀⠀⠀⠀⣀⣀⡈⠠⠐⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠂⠀⠤⠤⠤⠤⠤⠤⠒⠒⣊⠭⠐⠉
⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡒⠂⠒⠒⠒⠒⠒⠈⠁
⠀⠀⠙⠒⠦⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢶⡚⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠚⠛⠤⢀⠈⠛⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⡸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠿⠿⠿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠈⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⢀⣤⡀⠀⠀⣠⣄⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠁⠀⠀⠸⠿⠟⠀⠘⠿⠿⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠘⢿⣷⣤⣀⣠⣴⣿⠟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠉⠛⠛⠛⠉⠁⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠿⠿⠿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣤⣶⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⡇⠀⠀⠸⣿⠿⠀⠀⢿⣿⠇⠀⠀⢸⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠁⢀⣿⡟
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠛
⠀⠀⢀⣴⣾⠿⠟⠛⠛⠛⠿⢿⣶⣄⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣶⣶⣶⣶⡿⠿⠋⠁
⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⣰⣿⠁⠀⠀⣾⣷⡄⠀⣴⣿⡆⠀⠀⢹⣿⡀
⣿⡇⠀⠀⠀⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⣿⣧⠀⠀⠀⣠⣶⣶⣶⣶⣦⡀⠀⠀⢠⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠘⣿⣆⠀⣾⠟⠉⠀⠀⠈⠙⢿⡆⢀⣾⡟
⠀⠘⢿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠈⠛⠿⣿⣷⣶⣿⡿⠟⠋
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢠⣿⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣾⡏⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡟⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠸⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⡀
⠀⢀⣤⣶⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⠿⣶⣤⣀
⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣧⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⢿⣷⣄⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣤⣤⣄⣀⡀⠀⠀⣀⣼⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⠻⢷⣾⣿⣟⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣛⣿⣷⣾⠿⣿⡟⢹⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉
⠈⣿⡆⠀⠈⠉⠛⠛⠻⠿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠿⠿⠛⠛⠉⠁⠀⢀⣿⣷⠿⠛⠁
⠀⠹⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠉⠛⠿⣶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠒⠊⠙⠒⠦⡄⠀⢀⠏⠑⠒⠒⠶⠤⠤⣄⡀⠀⢠⠶⠶⠦⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⣼⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡎⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⢠⣄⣀⣀
⠀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⠇⠀⠀⠈⠉⠉⠑⡲⢤
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠏⠀⠀⠀⠀⠀⠀⠂⠀⡜
⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠂⢠⠇
⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⣀⣀⡼⠦⠴⠒⠒⣒⣚⣋⡉⠉⠉⠙⠒⠒⠒⠒⠦⠧⢤⣄⣀⡀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣀⣼⡶
⠀⠀⠀⠀⠀⣠⣾⡿⠗⠚⠉⠁⠀⠀⠀⣠⡶⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠛⠛⠶⣦⣀⠀⠉⠙⠒⠦⢤⣀⣼⠃⠀⠀⢀⣠⠤⠒⠋⠁⢰⠇
⠀⠀⠀⠀⢾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⠋⣠⠒⠉⠀⠉⠒⢄⠀⠀⡠⠐⠈⠉⠐⠠⡀⠙⢦⠀⠀⠀⠀⠀⠈⠉⠓⠖⠋⠉⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠈⢂⡐⠁⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⡅⠀⠀⠀⡀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠔⢉⣉⣑⠂⠀⠀⠀⠀⣼⠿⡇⠀⠀⠀⡀⠀⠀⣿⢿⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠐⣻⣿⣇⠀⠀⡀⠀⠀⠈⠛⠁⠀⠀⢀⠃⠀⠀⠙⠚⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣿⣿⣿⡄⠀⠐⡀⠀⠀⠀⠀⠀⢀⠎⢡⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⣿⣿⣿⣦⡀⠈⠢⠀⠀⡀⠔⠁⠀⠀⠑⠄⢀⣀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠸⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠣⡀⠀⠀⠈⠙⡍⠁⠀⠈⠙⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠓⠒⠶⠶⠤⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢸⠀⣠⠖⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣿⣶⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⠡⡀⠀⢳⣄⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠒⠫⠵⣒⣬⣦⣄⣙⣳⣄⣤⣶⠾⠛⠉⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠤⠤⢤⣼⣷⣤⣤⣤⠾⢟⣻⡿⠟⠋⠉⠀⣀⣤⣖⣊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⣯⠭⠔⠒⠒⠒⠒⠒⣺⣿⣿⣿⠟⠛⠛⠉⠉⠉⠁⢀⣀⣤⣴⡒⠋⠚⠛⠛⠙⠛⠛⠫⢭⣗⣲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⠥⠴⠾⠟⠒⠒⠒⠛⣛⣫⣿⡿⣇⣔⣀⣠⡤⠴⠚⢿⢿⣿⣿⡿⠽⢛⡓⢲⠶⠤⠤⣤⣤⣀⣙⣋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠞⠛⠿⣿⡠⠽⠛⠉⠀⢀⣤⣶⣮⣍⣀⠀⠈⠛⠳⠤⣄⣀⣉⠉⠉⠉⠙⣦⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣧⠤⠒⠋⠁⠀⢀⣠⣶⡛⠛⠛⠶⠦⠬⠭⣿⣶⣤⣀⠀⠀⠈⠻⢭⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡟⠁⢀⣀⠤⠐⠾⣻⢏⡼⠋⠓⠲⠤⣤⣀⣀⣒⣒⣉⣻⣽⣳⢄⡀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡾⠶⠚⠉⠀⢀⠎⡰⠁⣾⡡⢒⡩⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡲⣄⣤⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠻⠭⠟⣒⣒⡲⠦⠤⠤⠠⢤⣤⠎⡰⡁⣼⠟⠋⠁⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠤⣀⣐⠀⠀⠀⠀⠈⠉⠉⠉⢉⡟⢠⠃⢰⡯⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⢸⠁⡜⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⡇⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣫⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡿⢁⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠟⣽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣣⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⢯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡃⠄⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣶⣶⣶⣮⣽⣿⣿⣿⣿⣿⣯⣭⣭⣝⣛⣘⠷⣀
⠀⠀⠀⠀⠀⠀⡀⣔⡮⠟⢚⠉⠭⠁⡄⢒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣦⡀
⠀⠀⠀⠀⢀⢮⣾⣫⠀⣠⠸⢀⣈⠁⢁⣛⠫⠭⠭⠭⠭⠭⠽⠿⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣿⣿⣟⣡⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠏⢻⣳
⠀⠀⠀⠀⠸⣟⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣔⣿
⠀⠀⠀⠀⠀⠹⣿⣿⣶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠈⢾⣿⣿⣾⣯⣶⣴⣠⣤⡀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣽⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢿⣟⡿⣻⣿⣿⠿⠿⡿⡿⣻⣿⣿⣿⣿⣟⣿⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣾⣷⣿⣿⣴⡾⠿⢶⣿⣻⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠠⠒⣄⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠈⠀⡀⠀⢺⠀⢀⠄⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠘⠄⠊⠀⠠⠀⠀⠀⠸⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⢀⠂⠀⢦⠀⠀⠀⠀⠀⣀⠀⡆⠀⠈⡇⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠒⡠⠀⢠⠇⠀⠀⢰⠈⠀⢀⢿⠐⠀⠈⣠⠊⡁⠉⠰⡇⠀⠀⢱⠀⡇⠀⠀⡀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡋⠀⠎⠁⠀⢠⠒⠀⢄⠀⠀⠁⠀⢸⠀⠀⠜⠁⠀⠁⠐⣠⠁⠀⠀⠂⠀⡇⠀⠀⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢈⡇⠀⠀⠀⠠⠀⠀⢀⠆⠀⠀⠀⡇⠘⡧⠃⠀⠀⡀⠀⢰⡿⠀⠀⢰⠀⣀⣇⠀⠀⠀⠀⠇⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⡇⠰⠀⠆⠀⠀⠀⠀⠇⡶⠁⠀⠀⡐⠡⠀⢈⠠⠀⠀⢸⠁⢀⠀⠃⠀⠀⢠⡆⠀⠀⠀⡤⠆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠃⠸⠀⠀⠀⠀⠀⠀⡄⠀⠀⡐⢰⠀⣄⠁⠀⠀⠠⠘⠀⢀⠀⡇⠀⠀⠸⠁⠀⢀⠜⠁⠔⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⢠⠇⠈⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢠⠁⠀⡠⠉⢀⠈⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠠⠀⠇⠀⡇⠀⠀⡿⠀⠀⠀⡄⠀⢸⠀⡀⠰⠀⠀⠆⢀⡔⢁⡠⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠢⠤⣧⣠⣤⣤⠴⠐⠒⠚⠛⠠⣄⠁⠀⠀⠇⠀⠀⠀⢃⠔⠋⠉⠉⠉⠛⠛⠖⠛⠉⠁⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣀⣀⣠⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠴⠦⠄⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠⠤⠀⠚⠓⠈⠁

⢀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⠖⣶⠶⢌⡑⢄⠀⠀⠀⠀⡠⠖⠊⣉⡒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠞⣽⠿⣋⠐⣝⠿⣄⢣⠀⢠⠊⡠⣎⠻⢟⣍⣵⢦⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡼⠟⡡⠒⣝⠿⣫⢂⡿⠌⢦⠏⡸⡿⣋⡐⣢⠙⢋⣄⢣⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡯⢌⠳⢾⡫⢢⡙⠛⣕⢙⣦⠜⣁⠄⣝⠿⢋⡔⢢⡙⢻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢳⡾⡣⢌⠻⢟⡱⠒⣝⠟⣡⠒⢙⠻⢋⡔⢈⡻⠟⣁⢺⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⣟⣷⡾⡡⢌⠳⡾⡡⠒⣝⢻⣏⣠⠮⠿⢾⣧⠂⠻⠧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢧⢜⣷⡾⢡⠔⢻⠞⢍⠀⣞⢩⡄⠀⢧⡴⠃⠚⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠖⠋⢷⡚⢉⡀⠘⣬⠯⣤⢌⠿⢖⠻⠟⣭⡟⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣋⣸⡿⢛⠻⣧⣼⡷⣀⠻⣶⡾⡡⣢⢋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡷⣥⣵⡌⠋⣾⣤⡿⡢⢜⡿⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣜⢯⠐⢢⡈⠛⣿⣦⢞⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡻⣟⡕⢢⢜⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠩⠕⠋

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠀⠀⠂⠄⡀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠋⠢
⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢈⠧⠔⠀⠐⠊⠉⠰
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠉⠀⠒⠠⠤⠤⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠘⠛⠀⠁⠒⠒⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠂⠠⠄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠂⠈⠁⠀⠁⠀⡠⠄⠂⠂⠀⠒⠂⢄⡀⠀⠀⠀⠀⠈⠐⠂⠀⠠⢀
⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⠇
⠀⠀⠀⠑⢄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠴⠒⠁
⠀⠀⠀⠀⠀⠈⠁⠑⠂⠠⣀⢀⠀⠀⠀⠀⢀⠀⠀⢀⠀⣀⣀⣀⣀⣀⡀⣀⠴⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⠤⠀⠀⠀⠀⣀⠈⠀⡀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡡⠊⡠⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠤⠂⠉⠑⡄⠀⠀⠀⢀⠃⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠎⠁⠀⣪⠂
⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⡠⠔⠀⠀⠸⢤⡄⠤⠂⠀⠀⢀⠔⢁⠜⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⠴⠓⢆⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠂⠁⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠀⣠⠔⠁⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠋⠀⠀⠀⠐⠀⠀⠉⡁⣀⠀⠀⠤⠤⠄⢀⣉⠘⠃⣤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠤⠐⠂⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠐⠍⠢⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⢔⠉⠀⠀⠀⠀⠆⢅⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠚⣿⣿⠟⡠⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣶⡖⠒⠀⠒⠂⣶⣶⣾⣿⡀⢀⡠⠾⠋⠁⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠐⠀⠉⠙⠛⠓⠒⠀⠠⠤⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢠⠶⠖⠒⢶⡴⠶⠶⠶⣤⡤⢤⡄⢀
⠀⠀⠀⠀⠀⢀⣀⠀⣟⠉⠁⢻⠀⡴⠚⠛⣾⠀⠀⠀⢸⡄⠀⠀⠀⡇⠀⠀⣿⠋⠉⠛⣦⠀⢀⣤⣤⡀
⠀⠀⠀⠀⣾⠉⠉⢷⢹⡄⠀⠀⢧⢻⠀⠀⢹⡇⠀⠀⢰⠓⠉⢻⠒⠛⢦⡖⠛⠚⠇⢀⣿⣠⠏⠀⢀⡇
⠀⠀⠀⠀⢸⡀⠀⢈⡷⠧⣤⠀⠘⣿⠀⠀⢸⠉⠉⣧⠼⠤⣤⢀⠀⠀⢸⡇⠀⠀⠀⡏⠉⢹⠀⢠⠾⢤⣀
⠀⠀⠀⠀⠀⢧⠀⡘⠀⠀⢸⣠⠖⠛⢳⠀⢸⠀⠀⢸⠀⠀⢸⠸⠀⠀⢸⡇⠀⠀⢸⠁⠀⠀⠀⢸⠀⠀⣸⠃
⠀⠀⣴⠞⢻⡾⡄⡇⠀⠀⠸⡟⠀⠀⢸⣠⡼⠤⡄⢸⠀⠀⢸⠀⡇⠀⠸⡇⠀⠀⢸⠀⠀⠀⠀⡆⠀⢠⡏
⠀⠀⢻⠀⠀⢷⠷⠷⡄⠀⢰⣇⠀⠀⢸⡇⠀⠀⣯⡼⢤⠀⢸⠀⡇⠀⠀⡇⠀⠀⢸⡆⠀⠀⢠⠃⠀⣼
⠀⠀⠈⡇⠀⢸⠀⠀⢃⠐⠋⠉⠃⠀⢸⡇⠀⠀⡇⠀⢸⠀⢸⠀⡇⠀⠀⣿⠀⠀⢸⡇⠀⢠⣸⠀⢠⣧⣤⣤⣤
⠀⠀⠀⢹⡀⠀⣇⠀⢸⢸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⢸⠀⣾⠀⣿⠀⠀⢻⠀⠀⢸⡁⠀⢸⣇⣠⠾⠖⠂⠉⢸
⣠⣤⣤⡬⠧⠤⠿⢤⠀⡎⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⢸⠴⠛⠒⠛⠓⠛⠛⠉⠉⠉⠉⠉⣙⠉⠀⠀⠀⠀⠀⡾
⢻⠀⠀⠀⠀⠀⠀⢸⠀⢹⡄⠀⢸⠀⢸⠁⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇
⠘⡇⠀⠀⠂⠀⠀⠘⡆⢸⡇⠀⢸⠀⢸⠀⠀⠀⡇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⢿⠀⠀⠀⠀⠀⠀⠙⢮⡇⠀⢸⠀⢸⠀⠀⠀⣧⠜⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡿
⠀⢸⡶⢄⡀⠀⠀⠀⠀⠀⠉⠙⠚⠶⠾⠦⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠁⡇
⠀⠈⡇⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠋⠀⢰⡇
⠀⠀⣷⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣴⡁⠀⡰⢻⠃
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠘⡍⠓⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⢀⠎⠀⡼⠁⢸
⠀⠀⠸⡗⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠉⠑⠲⠤⣀⠀⠀⠀⠀⠀⠀⢸⠁⠂⠀⢀⠟⠀⡾
⠀⠀⠀⣧⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⢸⠃⠀⢠⠋⠀⠀⡇
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡎⠀⢠⠀⠀⠀⢸⠃
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠘⡏⠉⠓⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠇⠀⠀⠀⣼
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⡀⠀⢸⢠⠋⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡾⠃⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠴⠃
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠐⠒⠶⠖⠲⠆⠀⠀⠰⠒⠦⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⡍⣷⠀⠀⢀⡄⣶⣤⣠⣴⠀⣶⣄⡰⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣇⠀⠀⣽⠀⠀⠘⠃⠁⡉⡉⠀⢀⡀⠁⠀⠀⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠋⢀⡇⠸⠀⡖⠀⠏⡖⡸⠂⣾⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠛⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠒⠛
⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡠⠒⡰⠄⢢⣎⣁⠀⠐⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠶⠒⣒⡒⠲⢤⡀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣠⠖⠻⡄
⡟⠓⠺⣄⠀⢀⣇⣀⣹⡄⠀⢸⣄⠀⠀⠂⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⢀⠀⠈⢑⡦⢷⡀⣌⠐⢣⡄⢀⣀⡰⠟⢫⣾⠟⠁
⠇⠀⠀⠀⠉⠉⠀⠀⠀⣹⢿⣧⡎⠀⠀⠀⠀⠀⠀⠀⢸⡂⡈⠀⠀⡐⢀⡀⣈⢀⣀⣤⣾⠁⠉⠀⠛⠓⢀⣤⠶⠛⠉
⢰⠀⠀⠀⠄⠀⠀⠀⡐⠉⠀⢧⠀⠀⢐⠀⠀⠀⠀⢴⠉⠻⠽⠿⣭⠿⣯⠽⠛⣛⡻⢟⡟⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠸⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴⠚⠀⠀⠀⠉⠉⠁⠀⠈⠲⢍⣀⣒⣐⣒⣀⡨⠭⠄⠒⠉⠀⠀⠰⠀⠀⠀⠀⠈⠀⠈⠁⡆⠀⠀⡀⠀⠀⠨⢆
⠀⠉⠙⠛⠛⠒⠚⠞⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠀⠀⠱⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡨⠓⠀⠀⠀⡈⠓⢦⡀⠀⠀⠘⣆⠀⠀⠀⠀⣼⠃⠀⠀⠀⣤⣶⠶⣤⣤
⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠰⢄⣨⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠢⡀⠙⢆⠀⠀⠁⣆⠀⠀⠀⡇⠀⠀⠀⠀⠁⢀⠠⠐⡄
⠀⠀⠀⠀⠐⠳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠤⠀⠀⠀⢀⠎⠀⠀⠐⠦⠀⠀⡀⢀⣨⡦⠠⠇⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⡁⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠄⠀⠀⢀⠀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡀⢂⣤⠾⠛⠁⠐⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠄⢙⣣⡄⠀⠀⠈⠀⠉⠀⠀⠀⠀⢸⣀⣴⠟⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠘
⠀⢀⠄⠒⠢⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡤⠒⠛⠛⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⡤⢀⢄⡀
⠀⡀⢰⠀⠀⢧⠀⠀⠀⠀⣠⣄⣴⣄⠐⢃⣀⡀⠀⠀⠀⢠⠏⠀⠀⠈⠀⠀⠀⠀⢀⣠⢼⠀⠀⠤⣜⠷⠻⣇⢱⠈⡁⠀⠀⠀⠐⠀⠀⠈⠠
⠀⠄⣸⠀⠀⢸⠀⠀⢠⣾⠏⡠⠊⠀⠀⠀⠀⠉⢶⠀⠀⡟⠀⠤⠀⠀⠀⣠⣤⣶⠿⢳⣻⡀⠀⠀⣿⠄⠀⠟⣸⣄⡿⠃
⠀⡇⣇⠀⠀⣼⠀⠀⢼⡫⠜⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠳⣤⡤⠶⠊⣙⢋⠽⣷⢦⣎⡥⢴⠋⠀⡋⠘⠚⠷⠛⢛⡇⠀⠀⠀⠀⠀⡀⠠⠤
⠀⢶⢸⠀⠀⡇⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠀⠠⠃⠀⠀⠈⣼⣄⣸⣭⣴⡶⠿⠋⠃⣱⠎⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠃⠀⠄
⠀⢸⡌⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢀⣀⣈⣉⣀⠤⠒⠊⠁⠀⠀⠀⠀⠃⠀⠀⠠⠓⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠳⣤⠟⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠂⠐⠚⠀⠀⠀⢀⠀⠐⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣠⡌⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠐⠖⠒⠲⡤⠠⢤⡶⠀⠀⢀⠤⣠⠶⠶⠚⠋⠉⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⡆⢀⣬⡦
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠂⠀⠀⠁⢨⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠘⠦⠂⣀⣒⠀⠉⠝⠃⠚⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⣽⣋⣛⣛⣛⣛⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⡿⣿⠙⢻⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⣿⡿⣟⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⡄⣀⠘⣿⣿⣿⣯⣷⣦⣿⣶⠏⠀⠈⠿⠿⠛⠋⢀⣐⣺⣿⣿⠿⢿⣿⣿⣿⣏⣾⣎⣥⣦⣤⣿⣭⣧⣷⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⠃⣿⣿⣿⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠉⢩⠉⠉⠛⠛⠲⠿⢭⣽⣟⣛⣛⣛⣛⣛⣛⣯⣵⣾⣿⣿
⣿⣿⣿⣿⣿⣿⠉⠀⣿⣿⣿⣿⣧⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣄⠀⢴⢒⢳⠓⣗⠼⠆⠀⢀⣀⣨⣭⡍⣽⣍⣁⣴⠺⢻⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⣿⢺⣯⣻⡇⢰⠀⢼⠒⢢⠶⡎⡧⢣⠸⠹⠜⠃⠙⠀⠈⠉⠐⠂⢀⣠⣶⣾⣿⣿⠿⠙⠇⣿⣿⣿⣿⣷⣼⣿⣿
⣿⣿⣿⣿⣿⣿⢀⠀⠉⢀⣿⣿⡇⠊⠑⠙⠊⠰⠂⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⢹⣿⣿⣷⣤⡴⡀⠈⠻⠿⢿⡿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⢿⡆⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡿⢻⣿⣿⣿⣿⡉⣠⣿⣷⣦⣴⣾⣇⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣟⡇⠆⣁⣘⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⠀⠈⠻⣇⢸⣷⡅⣻⣿⣿⣿⣿⡟⢹⣿⣿
⣿⣿⣿⣿⣿⣿⡟⣇⠀⣻⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣾⡏⠙⠿⣶⣾⣿⡅⣸⣿⠆⣬⡭⠿⠿⢿⣺⣿⣿
⣿⣿⣿⣿⣿⣿⣇⣛⠀⢿⣿⡇⢹⠀⢀⢀⣀⣀⣠⠠⠤⠴⡀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠷⢦⣤⣾⣾⠀⠀⠀⣐⣖⣢⡿⠥⠿⠿⠿⣿
⣿⣿⠿⠿⢿⣿⣇⣁⠀⢸⡿⠧⢸⠀⠭⠤⠑⠒⠂⠃⠉⠉⠁⠀⠀⠀⠀⠀⢳⣤⣒⣠⡭⠤⠴⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠁⠀⠀⠀⠀⠈⠉⠛⠻⢾⣿⣿⣿⠀⠷⠽⠲⢖⣛⣛⣠⡤⠤⠴⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⠉⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠈⠁⠒⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣄
⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⡀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡎⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⠒⠒⠂⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⢧⠋
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⡀⠠⠄⠤⠀⠂⠈⠁⠠⠟⠁
⠀⠈⢦⡀
⠀⠀⠀⠙⠢⠤⣀⣀
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠠⡈⢣⡀⠀⢀⣠⡤⠔⠒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣰⣤⣉⣽⣿⣷⠾⠛⠋⣁⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣒⣒⣀⣙⣾⣿⣿⠛⠉⠉⢉⣁⣤⣴⣾⣏⣋⡉⠉⠙⠛⢶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠁⠀⢰⠶⠶⣟⡹⠟⠛⢉⣭⣾⣛⠙⠻⠮⣝⣋⠉⠛⠛⢟⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠒⠈⢁⣠⡴⣮⣛⡛⠓⠛⠻⠶⣤⣀⠉⠛⢭⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠤⠒⠉⡘⡝⣽⠁⣰⢯⠉⠓⠓⠓⠛⠛⠻⣦⣄⣉⢦
⠀⠀⠀⠀⠀⠀⡠⠚⠋⠓⠲⠖⠒⠀⢶⡞⠘⣰⢛⣁⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠉
⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠀⠀⠀⢸⢠⠀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⠑⡀⠀⠀⠠⠖⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⠐⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠒⠁

⠀⠀⢀⣀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠤⠄⣀
⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠤⠄⠀⠀⠀⠒⠀⠒⠂
⡀⠀⢃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠊⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠤⠐⠒
⢋⠢⠬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣁⠴⡀
⠈⠢⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣪⠄⠈⠉⣠⡁
⢰⠀⣀⣀⠀⠖⠀⠀⠀⠐⠀⠀⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠂⢂⡉⠅⠃
⠀⠁⠰⢁⠀⠀⠠⠤⠀⠀⠐⡀⠀⠀⢀⠀⠀⠀⢀⠀⣂⣰⠦⠔⠉⠁⠕
⣀⣀⣀⣀⣈⣒⣤⣔⣂⣀⣐⣒⣒⣒⣚⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⠟⠁
⠀⠀⠀⠀⠀⢀⠔⢉⠖⠁
⠀⠀⠀⢀⠔⢉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⣠
⠀⢀⡔⢩⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⡷⠟⠛⡛⠛⠻⠶⣇⣠⡄
⣔⢁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡟⠁⢄⣨⣴⣶⣦⣬⡂⠌⠻⣶⠆
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣟⢹⡟⣉⢿⢫⠻⢫⢹⢟⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡟⠸⣇⠟⣰⡸⢰⣷⣞⣎⠆⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣯⣟⣿⣿⣿⣿⣻⣿⣿⣻⣛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣏⠉⢝⠻⠿⣿⡿⠟⡋⠉⣩⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡷⣦⣌⣀⣀⣈⣡⣴⢾⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠻⠋⠿⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠀⢠⣤⣶⣶⣿⣿⣿⣷⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠸⠟⠋⠁⠈⠛⠿⠀⠀⠀⠀⠀⣰⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣏⣾⣿⡿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢨⣵⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡼⠿⢿⣛⣻⣭⣿⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣛⣯⣭⣽⣶⣶⣷⠀⠀⠀⠀⢀⣀⣰⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⠿⢟⡋⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠉⠉⠉⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠲⠖⠢⣴⠲⣄⠐⡅⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⣀⣀⣀⡀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⢀⢞⣿⣿⣿⣿⣿⣦⡀⠙⢢⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠃
⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠻⠀⠂⠀⣠⣿⣿⣿⣿⣿⣿⣿⣯⣳⠀⠘⠒⠢⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢦⡀
⠀⠀⠀⠀⠀⣠⠔⠃⡀⠈⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⣀⡔⣺⣿⣿⣽⣶⡄⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⢟⡻⠋⠀⠀⠀⠀⢠⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠘⣣
⠀⠀⠀⠀⠀⢸⡁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠊⠉⠀⠀⠀⠀⠀⠀⠙⡆⢱⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⠘⢤⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⠿⣛⡽⠋⠀⠀⠀⠀⢪⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢀⡠⠎⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⣛⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡜⠀⠀⠀⢀⡀⠀⠀⠀⣠⠎⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠒⢤⣀⣀⠬⢦⠀⠀⠀⠀⠀⠀⣀⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠋⠉⠑⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠲⠤⠖⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣟⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣽⣿⣿⣽⣯⣯⣿⣽⣽⣿⣿

⠀⠀⠀⠀⠀⣀⣀⡤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡖⠭⠒⢀⡍⡁⣈⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠃⠢⠀⠒
⠀⠀⣀⠁⠀⠀⠀⠈⠀⠀⠀⢀⡰⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⡐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⠀⠖⠀⠀⠊
⠀⠀⠁⠀⠀⠀⠒⠒⠒⢒⣈⣩⠴⠚⡅⠀⠀⠀⠀⠀⠀⡠⠞⠍⠀⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠐⠒⠒⢀⣀⡠⠤⠖
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⡀⠠⠔⢻⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠀⢠⠊⠁⠀⠀⠀⣠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⣠⠄
⠀⠀⢸⠄⠀⡀⠀⠀⠀⠀⢀⣀⡤⢾⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⠘⠀⠀⠠⠤⠐⢊⠁⠁⡔⠊⡥⠀⠀⠀⠀⠀⠀⢩⠝⣀⢋⠼⡟⢣⠉⡄
⠀⠀⠘⠀⠄⠀⠈⠉⠉⠭⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⠞⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⢼⣤⣦⣼⣤⢧⢿⠁
⠀⠀⠀⡇⠀⠀⡀⠄⠐⠀⠀⠙⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢜⣕⢕⡵⡡⡏
⠀⠀⠀⠘⠀⠀⠀⣀⣈⣀⡠⠴⡺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡯⡪⣾⢮⡇
⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢠⡆⠀⠀⠀⠀⠀⠀⣀⠄⠘⠀⠀⡚⢀⡀⠀⠃⢦⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠞⠁⠀⠀⢀⣠⠀⠀⠀⠀⣀⢄⠈⢱⠚⠇⢸⠁⠞⠂⡜⠓⢀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠐
⠀⠀⡠⠖⠂⠒⠉⠈⠁⣄⣀⣠⠞⠁⠘⠉⠀⠀⠀⠀⠁⠈⠃⣰⠸⠀⠟⠲⠀⣼⠇⠀⠈⠀⠀⠀⠀⡸⠤⠠⠄⠤⢀⡀
⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠂⠅⠧⠼⠿⠏⠀⠀⠃⠀⠀⠀⠀⠀⣺⣁⣊⣐⣂⠐⠋⠻⠐⠠⢀⠤⠀⠈⠆
⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠌⢀⡘⡈⠀⠀⠀⠀⢄⣁⣠⠎
⠇⠀⠀⠀⠀⠀⠀⡀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢨⠆⠀⢀⡈⠉⠀⠒⠃⠐⠂⠐⠀⠜
⠀⠀⠀⠀⠀⣀⣴⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⡀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠀⠤⠌⠀⠉⢐⡋⣘⠁⢀⠀⢀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⣀
⠀⠀⢀⣠⠖⠊⠉⠄⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⢸⠁⠄⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢧⡀⠀⣀⡀
⠀⣠⠋⠀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⠂⠀⠀⡄⠀⡄⠀⠠⠀⠀⢀⡀⠀⠀⠀⢸⠁⠀⠈⠃⠪⡁⢟⣠⣄
⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⣡⠞⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡒⢠⠉⠂⡠⠀⠀⡄⠀⠘⠀⠀⠀⠀⠈⠓⠀⠤⠘⣤⣗⣲⣄
⠈⢙⡁⠶⠄⠤⣤⠤⠤⡬⢄⠒⠚⠋⠀⡰⠁⠀⠀⠀⠀⠀⠀⠠⡔⠀⠀⡁⠂⡠⠊⠀⢀⡰⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⡢⡑
⠀⠈⡀⠀⠉⠀⠐⠈⠀⠐⣁⠤⠚⢀⡁⠴⢃⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠕⠊⠀⢀⡴⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞
⠀⠘⠀⠘⠀⠤⠶⠒⠒⠒⠀⠚⠩⠉⠂⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠂⣒⣨⠔⠃