Methodology Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠒⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠶⡄⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⢠⡶⢶⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄⠀⠀⠀⠀⠀⠀⡇⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣾⠋⠓⠛⠁⠈⠓⠖⠛⣦⠀⠹⣆⠀⠀⠀⣴⠋⠓⠚⠁⠀⠓⠶⠛⣦⠀⠀⠀⡇⠀⢈⡃
⠀⠀⠀⠀⠀⠀⢰⡇⠀⣀⣼⠃⢠⠖⠛⠓⢦⡀⠸⣇⣀⠀⢹⡀⠀⣀⣸⠃⢀⡴⠛⠛⢶⡄⠐⣇⣀⠀⢼⣷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠀⢸⣇⡀⠀⣿⠀⠀⠀⢀⡇⠀⢈⣹⠀⢸⡇⠀⣏⡀⠀⢾⠀⠀⠀⠀⡿⠀⢈⣹⠇⠀⡄
⠀⠀⠀⠀⠀⢀⡞⢧⠀⠀⣹⠀⠈⠳⠤⠤⠞⠁⢰⡏⠉⠀⠈⣇⠀⠀⣹⠄⠈⠳⠤⠤⠞⠁⢰⡏⠉⠀⣸⠃
⠀⠀⠀⠀⢠⠟⠀⠈⢧⡀⠳⡤⠶⢤⠀⢠⡴⢦⣤⠟⠀⠀⠀⠘⣆⠀⠻⣤⠶⢦⡀⢀⡤⠦⣤⠟⠀⣠⠏
⠀⠀⠀⠰⠯⠤⢤⠀⠈⠳⠀⠀⠀⠘⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠘⠓⠚⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢘
⠀⠀⠀⠀⠀⠀⣾⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⠆⠀⠀⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠁⠀⠺⠛⠁⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣛⣛⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠛⠛⠛⠛⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡾⠿⠿⠶⢶⣶⣤⣤⣶⡶⠶⠿⠿⢷⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣼⣿⣦⡀⠀⠀⠀⢹⡏⠀⠀⠀⢀⣴⢿⣇⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢿⣦⣾⠇⠀⠀⠀⢸⡇⠀⠀⠀⠸⣷⣴⡿⠀⠀⠀⠀⣶⣴⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠸⣿⣀⣀⣀⡀⠀⢸⡇⠀⢀⣀⣀⣀⣿⠇⠀⠀⠀⠀⢻⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠈⠉⠙⠛⠛⠻⣿⣿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⠀⠀⠀⠀⢀⣤⡆⠀⢀⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠰⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛



⠀⠀⠀⠀⠀⠀⠀⢠⡐⠀⠀⠠⠀⡀⠀⢀⠠⠀⠀⠠⠀⠀⠠⠠⠄⠠⠀⠄⠠⢀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⢸⣿⣿⣿⣿⣿⣶⣤⠀⣠⣶⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣠⣾⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡏⢰⣿⡆⢸⣿⣿⣿⣿⣿⣿⠁⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣅⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠛⣉⣤⣦⣤⣤⣦⣈⠙⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣡⣤⣌⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠉
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⣉⠙⢿⡀⢻⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⡿⠃⣸⣿⣦⣌⣉⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣤⣤⣤⣤⣤⣄⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⠈⠻⣿⣿⣿⠏⠈⢿⣿⣿⡿⠋⠈⢿⣿⣿⡿⠃⠘⢿⣿⣿⡿⠃⣸⣇⠘⠛⠛⠛⠛⠛⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃
⢸⣿⣿⣿⣆⠠⣤⣩⣤⣾⣿⣦⣬⣩⣤⣾⣷⣦⣌⣡⣤⣾⣷⣤⣌⣡⡄⢠⣿⣿⡇⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣄⠹⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣾⣿⣿⡇⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟
⠀⠈⢿⣿⣿⣿⣦⡈⠻⣿⣤⣉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣿⣿⡟⠛⠁⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋
⠀⠀⠀⠹⣿⣿⣿⣿⣦⣌⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⣤⣾⣿⣿⣿⠇⢸⣿⣿⠟⢉⠛⣿⣿⣿⣿⣿⣿⣿⠟⢉⠛⣿⣿⣿⣿⣿⣿⣿⠟⣉⠻⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⣤⣤⣌⣉⣉⣩⣤⣤⣶⣾⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⡄⠛⢃⣸⣿⣿⣿⣿⣿⣿⡀⠛⢁⣾⣿⣿⣿⣿⣿⣿⡀⠛⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠁⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⠘⢿⣿⡿⠏⠀⠀⠀⠀⠀⠙⢿⣿⡿⠃
⣿⣿⡟⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣤
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⠶⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⢀⣠⡴⠶⠶⠦⣤⡀⠀⠈⠉⠳⢦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⣰⠟⠁⢀⣠⣄⣀⠀⠙⣦⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⢠⡾⣦
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢰⡏⠀⢰⠋⠁⠀⠙⣧⠀⠘⡇⠀⠀⠀⠀⠘⣇⠀⠀⢠⡟⠛⠛⣧
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠸⡇⠀⠹⣄⠀⠀⣀⡟⠀⢠⡇⠀⠀⠀⠀⢠⡟⠀⠠⡿⠶⢶⠶⠾⣧
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠻⣆⠀⠉⠛⠛⠋⠀⣠⡞⠀⠀⠀⠀⣠⡟⠀⠀⠘⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠈⠛⠶⠦⠤⠶⠚⠉⠀⠀⣀⣤⠞⠋⠀⠀⠀⢰⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠤⠤⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⣠⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⣿⠀⠛⠀⠀⢸⡇⠀⠀⠛⠀⠀⠘⠃⠀⠀⣿⠀⠀⠈⠃⠀⣀⣀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⢸⠀⠀⣿⣤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣹⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠉⠉⠉⠉⢹⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿⢠
⠀⠀⠀⠀⠀⠙⠳⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⢸⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠾⠷⠶⠶⠶⠶⠶⠷⠶⠶⣦⡀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣬⣿⣶⣿⣷⣾⣯⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠸⠷⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠘⢯⡀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⣠⠀⠙⢶⣼⡏⠛⠳⢦⣄⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⠞⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⢳⡄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⢀⣼⠃⠀⠀⢰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠁⠀⢀⣴⠟⠁⠐⠒⠒⠚⠃⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⠋⢀⣀⣀⣀⣀⣀⣀⡀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢹⡷⠟⠁



⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⡀⡀⠀⣀⣀⠀⣀⠀⡀⢀⣀⢀⣀⢀⠀⡀⢀⣀⢀⣀⡀⢀⣀⠀⡀⠀⣀⡀⠀⣀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡼⢷⠸⣁⡇⡇⣇⡀⣟⣂⠀⡿⣶⢿⢿⣒⠀⡇⢸⠒⡗⢏⣸⢾⣅⡿⢯⣨⠿⣇⡸⣇⡹⢺⣀⡇⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⠁⠀⡁⢀⠀⡀⢀⡀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⣀

⢀⠀⣀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡏⢳⠀⠀⠀⠀⠀⢸⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⡇⠀⠀⣠⠔⠚⠉⠁⠀⡴⢲⠀⡇⢸⠙⠲⢤⡀⠀⢸⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣶⡋⠀⠀⠀⢰⠒⡆⡿⢸⠀⡇⢸⠀⠀⠀⢈⡶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠙⢦⡀⢀⣸⣀⡗⣧⣸⠓⣇⣸⡀⠀⡠⠊⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣠⠞⠳⣄⡽⢯⡀⠀⠀⠀⠀⠀⠀⠀⡨⠻⣠⠋⠉⠙⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡏⠉⠉⢹⠀⠀⠙⢦⡀⠀⠀⠀⡠⠊⠀⠀⢸⠀⠀⠀⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡽⠋⠉⠻⡀⠀⠀⠀⢸⠲⣀⠔⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⢸⠀⠀⠀⠀⣳⣄⣀⣴⠁⠀⠀⠀⢸⠀⠀⠀⡇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢇⣀⣀⣼⠀⠀⣠⠞⠁⠀⠀⠀⠑⢄⠀⠀⢸⣄⣀⣠⠇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡳⣞⠁⢠⠞⠉⠉⠑⢦⠀⢑⣴⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⣠⠞⠁⠀⠙⡇⠀⠀⡇⠀⠀⡟⠁⠀⠑⢄⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣅⠀⠀⠀⠀⠱⣤⣤⢧⣤⡜⠀⠀⠀⠀⢈⠷⣇
⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⡇⠀⠀⠙⠢⢤⣀⡀⣗⣒⣚⣙⡇⣀⣠⠴⠚⠁⠀⢸⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠉⠫⣀⣀⡼⠉⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣿⣷⡦⠀⢰⣟⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣄⡀⢀⡶⠛⠻⣦⡀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠛⠉⠀⠀⠀⠛⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠟⠋⠘⣧⣀⣠⡿⠋⠉⠋⠉⠉⢻⡏⠙⠉⠉⣿⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠉⠉⢠⡾⠛⠻⣦⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⣷⡀⠘⣧⣀⣠⡿⠀⣸⣇⠀⠀⢲⣿⣾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⢻⡟⠃⠀⠀⢹⡏⢠⡟⠉⠉⣷⡀⠀⠛⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠈⠻⣦⣴⠟⣠⣤⣤⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠸⠷⠶⠶⠶⣾⡷⠶⠶⠶⠶⢶⣟⠀⠀⣿⠀⢠⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⣦⠀⠀⠀⣠⡤⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠷⠾⠋⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣼⠃⠀⠻⢿⣟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣭⣭⣤⣤⠀⠀⠀⠀⠀⠀⢰⣶⢠⣾⣿⣷⣤⠀⣤⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣉⣉⡉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠠⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠃⠿⠘⢿⣿⡿⠇⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⣩⣶⣬⢻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠉⠛⣛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⣋⡙⣇⠻⣿⠿⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣍⣥⣿⣷⣶⣾⣿⣿⣿⣿⣿⢀⣶⣶⣶⡆⠀⢰⣶⣶⣶⡆⠀⢰⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡈⠻⢟⣡⡙⠿⣋⣌⠻⢟⣡⡙⠿⠋⣸⡎⣩⣭⣍⠁⠀⠈⣩⣭⣍⠁⠀⠈⣭⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡹⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⡇⣿⣿⣿⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣬⡛⠶⣬⣿⣿⣿⡿⠟⣫⣾⣿⡏⣴⡶⠖⢶⣶⣶⣶⡴⠒⢶⣶⣶⣶⡶⠲⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣶⣴⣶⣶⣿⣿⠿⠋⠀⢿⠷⣭⠼⠿⠿⠿⢧⢭⠾⠿⠿⠿⠧⡥⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣦⠀⠀⠀⢾⣿⣷⠀⠀⠀⣾⣾⣷⠀⠀⠀⣾⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠋⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢿⣿⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⢻⣿⡄⠀⢀⡠⢲⣽⣯⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⢈⣄⣻⠿⢺⣥⣷⣟⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⣾⣁⣉⣻⢾⣿⡟⣾⣳⣂⣤⢾⣿⣿⡿⣛⣿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣯⢤⣾⢯⣭⣭⡤⠾⣿⡿⣿⡿⣿⡿⣟⡽⠿⢿⣏⡼⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⣿⣿⣗⣒⣚⠁⠀⠈⠙⠲⠚⠋⠀⠘⠿⠛⠛⠛⣄⢠⠒⠊⣷⣏⠉⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠺⣭⣿⡽⢿⣷⡶⣿⢢⣶⡶⣶⣶⣴⣶⡂⠀⠀⠻⢶⣼⡷⣟⢹⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⢀⣼⣇⠀⠈⣛⠋⠛⠋⠛⢃⣿⡟⣘⡛⠁⠀⠀⠀⡌⠻⣷⠶⡞⠫⡀⠀⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠹⣷⢸⣗⣿⣾⣛⣿⣾⣟⡿⣿⣿⣻⣿⡿⣷⣟⣃⠀⠀⠐⠠⡹⢖⡺⣟⢉⣤⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⠀⣬⣥⡌⠁⡈⣈⠁⣥⡈⠈⣉⠉⣠⠄⠈⠁⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⣿⢡⠿⡿⣷⣾⣟⢻⣏⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠏⢀⢀⣀⣈⣀⢀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⣠⢨⣭⣩⣭⣨⣉⣭⣉⣤⣭⣉⣩⣍⡉⣉⣍⣩⣭⣭⣄⡀⠀⠀⠀⠀⠀⢻⣿⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣿⣿⠤⠶⣤⡤⠴⠦⠤⠤⠤⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠻⠄⠳⠶⠶⠖⠶⠶⠶⠖⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠈⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠘⠒⡑⠒⠺⠖⠚⠐⠟⠒⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢘⠚⣛⢛⠛⠛⡓⠛⢛⡛⣛⢛⡛⢛⣛⣛⢛⠀⣀⣀⢀⣀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣬⣩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠉⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⠇⠀⠀⠘⣧⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢻⠿⠷

⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢟⠛⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⡇⠀⠀⢸⡆⣀⢀⣀⡀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠣⣀⠐⠀⠀⢠⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣒⠂⠀⢸
⣟⠠⠀⢸⡇⠀⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠤⠤⠤⠤⠤⠄⠀⠠⠀⠀⠀⣀⠀⠀⣀⣀⣀⡀⢀⡿⠒⠂⠀⢸
⡿⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⢀⡀⠀⢀⡀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡏⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣒⣂⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⢸
⣷⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⢠⠞⢩⣒⣤⠔⠂⢑⠲⣞⠋⡲⡖⢰⣶⡲⢖⠒⢤⢶⢐⡖⣒⠒⣒⢲⡒⢲⢶⠒⣲⣒⣶⠒⢳⠦⢐⣒⠒⣦⠀⠀⠀⠀⢸
⡧⠀⠀⠀⣼⢦⣀⠈⣻⡀⠡⣈⣿⣿⠀⡇⡇⢨⣭⣠⣼⠠⢄⠉⢸⠈⠘⠀⡇⠁⡇⢸⢸⠀⣭⡽⠨⠀⣿⡅⠸⣄⣑⣋⠀⠘⠃⠀⢸
⣷⠀⠀⠀⠘⢲⣷⣶⠉⠑⠦⠐⠈⢃⠤⠚⠃⠌⠉⠘⠣⠘⠀⠑⠊⠀⠣⠠⠓⠐⠀⠘⠃⠤⠽⠀⠘⠄⠘⠉⠢⠤⠐⠃⠀⠀⠀⠐⢻
⡿⠀⠀⠀⠀⡀⠈⠻⠒⠂⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡿⠀⠀⠀⠀⢤⠀⠀⠀⣀⠠⣀⠀⡠⢀⡤⠀⣠⢧⣀⠀⢄⠄⣀⡀⢀⣀⠄⣀⢀⡀⢀⡀⢀⣀⡀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⡷⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⠟⠀⠈⡇⠐⡟⠟⠇⣲⠀⡖⠼⡆⠸⠕⠀⣷⠁⡖⢢⠈⣮⠀⢸⢱⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠝⢸
⡇⠀⠀⠀⠀⠄⠀⠀⠀⣼⣸⣧⢼⣀⣹⣇⣈⣛⣚⠂⣜⣀⣃⢠⣃⣸⣝⣀⡏⢄⣘⣚⡠⢫⣂⣸⣜⣠⠏⠀⠀⠀⠠⠀⠠⠁⠀⠀⢸
⣟⠂⣰⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⡀⠀⠈⠉⠁⠀⠀⠀⠠⠀⠀⠀⠀⠁⢠⠤⠤⠤⠤⠤⠶⠶⠒⠒⠀⠀⠐⠂⠀⠀⢸
⡏⠘⠁⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠀⠀⠀⢁⠀⠁⡀⠙⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸
⡿⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠈⠐⢠⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⠄⠀⢀⣀⠀⠐⠒⠂⠰⠈⠀⠀⠀⠀⠀⢸
⡷⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠛⠊⠁⠤⠀⡤⠛⠀⠀⠀⠀⠀⠀⢸
⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣠⣴⡿⠆⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉⡁⣈⠁⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠠⢌⡣⠥⢠⡌⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡃⡞⠀⠊⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠇⠈⠧⣀⢀⢀⡓⠀⠰⠀⡆⠢⡀⠀⣠⠆⠀⠀⠀⣰⢦⠀⢠⢶
⠀⠀⠀⠀⠀⢁⠀⠁⢖⡛⠛⡓⠊⠁⣠⠟⢡⠀⠃⠙⠀⡨⠬⠥⣼⣆⠀⠀⠙⢾⡤⡾⠋⠀⠀⣰⠉⣇⡐⣲⠀⡀
⠀⠀⠀⠀⠀⠂⡖⣀⠀⢙⡆⠀⡀⠈⣈⠕⠂⡎⡤⠒⠢⡀⠠⠤⣄⢉⡄⠀⠀⢈⠀⠇⠀⠀⢠⠤⠀⡄⠀⠙⢠⡞⢀
⠀⠀⠀⠀⠀⠸⡇⣧⠄⠚⢉⣈⠁⢰⠃⠀⠀⠿⠁⠀⠀⠑⣠⠴⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣦⠀⢡⠚⠁
⠀⠀⠀⠀⠀⠡⠙⢀⡴⠋⠉⠁⠀⠚⠂⠀⢀⡴⠋⠉⡥⠋⠁⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠰⠄⠸⡏⠀⠀⣩⠁⢈⡅⠶
⠀⠀⠀⠀⠀⠠⠀⠈⣠⠀⠘⣇⣠⠴⠒⠒⡿⠁⡔⠉⢧⠀⠀⠀⢈⠐⠲⡅⠩⠥⠀⠬⠅⠌⠀⣄⠀⣀⠴⠏⢀⣘⢢⡄
⠀⠀⠀⠀⠀⠥⠶⠄⣏⠀⡀⠈⠃⠀⠀⠀⡇⠀⠃⣠⠄⠂⠀⠀⠀⢱⡀⠘⡇⠻⣿⠟⠀⠀⢠⠏⠈⠁⠀⣤⢌⣧
⠀⠀⠀⠀⠀⠀⢀⠴⠛⢲⡧⠖⠒⠂⠀⠀⠁⢀⡞⠀⡔⠂⠀⠀⢀⠼⠁⠀⡇⠀⠘⡆⠀⢠⠞⠀⣸⠇⠀⠙
⠀⠀⠀⠀⠀⠀⠏⠀⢠⠏⠀⡔⠀⣠⣤⣄⠀⢸⠀⠀⣇⠀⢄⡀⠀⠀⠀⠀⠀⠀⢀⡇⠐⢁⠤⢰
⠀⠀⠀⠀⠐⣴⠒⠂⢾⠀⠸⡅⠈⢀⡠⠤⣄⣘⣆⠀⠘⣆⠀⠉⠛⠂⠀⠀⢀⡠⠊⠀⣰⠁
⠀⠀⠀⠀⠀⢳⠀⣀⠘⢆⠀⠈⠁⡏⠀⠀⠀⠈⠉⠀⠀⠘⡆⠀⢸⠀⣀⡤⠤⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⢀⣥⠀⠀⠒⠂⣶⠐⠀⠀⠁⢀⣘⣛⣁⣀⣀⣀⣀⣘⠀⠀⠀⢢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉⠀⢠⡀⠀⣿⠠⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⢠⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡔⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡉⠒⠒⠒⠈⠈⠁⠁⠁⠅⠌⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⠲
⢅⠀⠂⠉⠁⠁⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠄⠀⠀⠀⠂⠉⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡈⠀⠀⠀⠀⠄⠁⠀⠐⡀⠀⠀⠀⠉⠀⢀⢀⠀⢀⢀⠀⠀⠁⠄⠀⠂⠉⠉⠐⠀⢀⠀⠀⡀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠠⠀⠀⠀⠀⠘⠀⠢⠈⠀⠀⠀⠀⠀⠐⠠⠀⠠⠐⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⡀⢸
⠝⠀⡀⠀⠀⠂⡀⠠⠀⠀⠈⠂⠀⢈⠀⠆⠀⠀⠄⠐⠀⢀⠀⠀⠀⠀⣀⣀⣀⠄⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡠⠀⠀⠆⠀⠈⠀⠁⢸
⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠀⠀⠀⠰⣤⠊⠉⠈⠅⠞⢦⠀⠂⠂⡠⠄⠀⠶⠄⠀⠀⠀⠀⢀⠠⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠚⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢁⢤⢴⠀⢀⡔⡂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⠀⠀⠉⠉⠢⡀⠑⠀⠘⠔⠉⢅⢀⠴⡋⢈⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠸⠁⠉⠀⠀⠀⡌
⡀⠀⢀⣀⡤⠤⠴⠒⠒⠙⠋⠉⠉⠉⠉⡍⠋⡇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡃⠰⠈⠀⢀⠀⠀⠀⠀⡰⠃
⡃⢸⠉⠀⠂⠀⠀⢀⣠⠤⠦⣄⣄⣤⡀⠃⠀⡇⢊⠈⠁⢐⡽⣄⠀⠀⠀⢀⠀⠌⠁⠀⠀⢠⠛⠒⠒⠋⠁
⠀⠈⡇⢠⠖⠲⠒⠉⠀⠀⠀⠀⠀⠀⠙⢲⡄⡇⠣⠀⠸⠉⠀⠀⠙⠓⠒⠉⠑⠢⠤⠤⠒⠃
⠀⠀⣣⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠏⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠂⡜⠙⢿⠃⢀⡠⠀⠀⠐⠈⠀⠀⠀⠀⠀⢹⣀⣠⠁⠀⠀⠀⠀⢰⠀⠜⡄⠸⢰⡠⢄⠀⠤⠠⡄⠀⠀⠀⠀⡠⠀⡇⡠⠀⡄⠀⠀⡇
⠅⠱⢄⣸⡀⠀⠣⡀⠀⠀⠀⠀⠀⡄⢀⡴⠃⠈⢳⠀⠀⠀⠀⠀⠈⡞⠀⠸⡆⢸⠀⠀⠀⠄⠴⡇⠛⠀⠀⠀⠦⠠⡇⢆⣀⠄⠀⠀⠅
⠀⠀⠀⠀⠙⢦⡀⠈⠐⠂⠀⠒⠈⣠⠊⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⣨⣵⣶⡲⢖⣺⡿⢇⣀⣀⣀⠴⣇⠰⠀⠦⠀⡆⠐⠉⣱⢰⠊⢢⢀⡀⠀⠀⢀⡀⠐⠉⢱⠀⠀⠰⣔⣉⡰⢰⠈⢹⢸⠖⠀⣶⠐⡄⢸
⠀⠀⢀⠴⢚⣃⣦⣎⠀⠈⠉⠀⡷⢄⣀⣀⣀⡤⠋⠀⠋⠀⠙⠀⠈⠂⠙⠘⠀⠈⠀⠀⠀⠀⠀⠁⠈⠂⠊⠀⠀⠘⠘⠀⠀⠈⠒⠉⠘⠀⠀⠁⠀⠁⢸
⠂⣴⢗⡞⠋⣀⣠⣌⣳⡀⠀⠀⣏⡒⠠⠄⢰⠇⠀⠀⣀⡀⢀⣀⠀⡠⢄⢠⣀⡀⢀⠀⡀⠀⠀⢀⡀⢀⢸⡀⢀⠀⡠⢄⠀⠀⠀⡀⢐⠀⠀⠁⢱⠀⢸
⠸⣇⡎⢠⠞⠁⠀⠀⠈⠱⡄⠀⡧⣈⠉⣉⠏⠀⠀⠀⡇⠘⡇⠘⠸⡀⢀⠘⠇⠀⢦⠀⠚⠀⠀⢨⣀⢼⢸⡀⢠⠀⡀⢀⠁⣄⡠⠇⠰⠆⠀⠜⠁⠀⢸
⠇⠈⠣⢼⡀⠀⠀⠀⠀⢀⠗⠒⠓⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠒⠒⠒⠒⠶⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⡾⢿⡿⢷⡄⠀⠀⠀⢋⣬⣭⣭⣼⣿⠀⠀⠀⢠⡾⢿⡿⢶⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⣶⣶⣿⠃⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣛⣋⡅⠀⠀⠀⠀⠀⢬⣽⣫⣥⠀⠀⠀⠀⠀⢨⣛⣫⡅⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣍⢫⣶⣶⢄⣠⣤⡘⣦⡙⢋⣵⢃⣤⣄⡀⢶⣶⡍⢩⣶⣶⣤⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣷⢸⣿⡇⣿⣿⣿⣿⣬⡐⢃⣥⣿⣿⣿⣿⢸⣿⡇⢸⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢮⡛⣿⣿⢸⣿⢃⣿⣿⣿⣿⣿⣧⡌⣿⣿⣿⣿⣿⡌⣿⡇⢸⣿⣿⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠎⠳⡐⢿⣯⢸⣿⢸⣿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⣿⡇⣿⡇⣸⠟⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⢿⣿⢸⣿⣿⣿⣿⣿⡸⠇⣼⣿⣿⣿⣿⣇⣻⢟⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣙⡻⠿⣿⣿⣿⣿⣾⣿⣿⡿⠿⣛⣩⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠾⠿⠿⠶⠖⠚⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠠⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐




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

⠀⠀⠀⠀⣀⢄⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠚⠆
⠀⠀⠀⠀⠁⢈⡆⠀⠀⠂⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠠⠀⠄⠀⠂⠤⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠆⢉⡀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⢀⡴⠀⡀⡲⠅⠀⠁⠀⠈⡁⠀⠀⠀⠀⠀⠌⠜⠀⠀⠀⠀⠀⠀⣦⠀⠁⣈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢠⣀⣠⣤⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠌
⠀⠀⠀⢔⠒⢂⢀⠀⠀⠤⠤⠌⠀⠀⠀⠀⠄⠀⠀⠠⠄⢀⠈⡀⠁⠀⡇⠀⠀⠈⠐⠀⠀⠀⠀⠀⢀⠀⣠
⠀⠀⠀⠸⠂⠄⢘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠸⣀
⠀⠀⠀⢀⡐⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠚⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⠈⠣⡼⢆⠂⠀⠂⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠚⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠁⠂⠈⠄⠶⠀⡀⠘⢄⠃⣀⠂⠀⡀⠄⠀⠀⠀⠀⡀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⡠⠀⢺⠀⠉⠀⠈⠀⠀⠀⠠⠤⢀⠁⣛⡇⠂⠀⠀⢀
⠀⠀⠘⢀⠀⠀⠀⢧⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠒⢈⠤⡀⠀⠡⠐⠀⠀⢂⠀⠘⠋⢸⠆⠀⠀⠆
⠀⠀⠀⠀⠉⠣⠀⠈⠀⠀⠀⡠⡆⠀⠀⠀⠄⠀⠀⠀⠀⠄⠉⠁⠀⠨⠀⠀⢀⠀⡆⠀⠀⣸⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠛⠂⣀⠀⠀⠦⠀⠀⢀⠀⡀⣀⢀⢄⡁⢀⠔⠒⠁⠁⠀⠈⢣⣬⠂⠰⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈

⠀⠀⠀⠀⠀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢀⣀⣀⠀⣸⣿⣿⣿⡄⠀⠀⢠⣶⣶⣾⡂⠀⠀⢀⣤⡀⠀⠀⠀⠰⢾⣿⣿⣿⠆
⠀⢀⣤⡴⠶⣶⣮⡅⠀⠀⠀⣈⡉⣻⡇⠸⠿⠿⠛⠐⢦⡀⠸⣿⣿⣿⡃⠀⠀⠘⠿⠛⠓⠲⠶⠶⢬⠟⣻⣯
⠀⠸⡇⠀⣸⣿⠏⠻⠶⠞⠊⠉⠁⢻⡇⠀⠀⢸⣿⣠⡾⠃⠀⠈⠉⢿⣿⣄⠀⠀⠀⠀⠀⠻⠷⣤⣄⣸⣿⣿⡆
⠀⢰⣧⢀⣿⣟⣀⣀⣀⡀⠀⡴⠀⢸⡇⠀⢀⣼⣿⠋⠀⠀⠀⠀⣀⠀⠻⣿⣿⡛⠛⣿⠂⠀⠀⠀⠈⠉⢸⣿⣷
⠀⠀⠀⢸⣿⠉⠉⠉⢹⡇⠈⠀⠀⢸⣇⣴⠟⣻⣿⠀⠀⢀⣴⣿⣿⣿⡄⠙⣿⡟⠛⠛⠛⠶⣤⣼⠃⠀⣴⡿⢻⡆
⠀⠀⠀⢸⣿⠀⠀⠀⠘⠳⠀⢀⣴⣿⠟⠁⠀⠙⣿⠀⠀⠀⠙⠿⣿⠿⡃⠀⠘⣷⡀⠀⠀⡀⠀⠀⣠⡾⠋⠀⠘⣿
⠀⠀⠀⠸⡿⠀⠀⣤⣶⣶⣴⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⣤⣦⡄⠀⣿⣿⡀⠀⠘⣷⠀⠀⠇⠀⠀⣿⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠘⠿⠦⠀⣿⣿⣿⡟⠙⢷⣀⣀⡀⠀⣴⡷⠀⠈⠿⠿⠛⠲⢾⣿⣇⠀⠀⠐⣷⡴⠆⠀⢀⣿⠀⠀⠀⠀⠈⣿
⠀⣠⣴⣦⡀⠀⠀⠈⠙⣯⡁⠀⠀⠉⠉⢉⡧⠀⠀⢲⣀⣀⣤⣤⣤⣌⣿⣿⡀⢀⣤⣤⣄⠀⠈⠉⠉⠀⣀⣀⠈⠉⠉
⠐⣿⣿⣿⡿⠂⠀⠀⠀⣌⡗⢀⣠⣾⣿⣦⡀⠀⠀⠈⠉⠉⣁⣀⠀⠉⠉⣿⡇⢾⣿⣿⣿⠷⠀⢀⣀⣼⣿⣿⠆
⠀⠈⣛⣯⡀⠀⢀⣀⡀⠈⠀⠈⠹⣿⣿⡿⠃⢐⣧⡀⠀⣾⣿⣿⣷⡄⠀⣿⡇⠈⢻⣟⣡⠶⠟⠋⠁⠀⢉⣡⣤⣄
⠀⠀⣿⣿⣿⡄⣿⣿⡿⠶⠶⠶⠶⠶⣶⣶⠀⠀⠀⠁⠀⠙⠿⠿⠋⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡇
⠀⠀⣿⠈⢿⣿⡆⣉⣤⣶⣄⠀⠀⠀⢹⣿⣧⠀⠀⢸⡟⠛⣿⣿⠀⠀⠀⣽⡅⠀⣿⣿⣠⣴⡶⣷⡀⠀⠀⠙⣿⡏
⠀⠀⣿⠀⠈⢻⣿⠋⠁⠀⠙⠷⣄⣰⠎⢿⣿⡆⠀⢸⡇⠀⣿⣿⠀⠐⠛⠛⠃⠀⢿⡟⠉⠀⠀⠈⢻⣄⣀⡀⣿⡇
⠀⠀⠋⠀⠀⠈⢿⡄⢸⡀⠀⠀⠈⢃⣴⠿⠻⣿⠀⠈⠁⠀⣹⣿⣤⣤⣄⠀⠀⠀⢸⡇⠀⠈⢦⠀⠀⠛⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣠⡾⠛⠁⠀⠀⢻⡇⠀⠀⠀⣿⡏⠉⢩⣿⠀⠀⡶⢟⠁⡀⠀⠀⢀⣀⣤⣤⠶⢹⡇
⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣾⡇⢸⣿⣿⣿⡌⢻⣄⠀⠀⣿⡇⠀⠨⣿⣤⠘⠟⠋⣄⠹⠿⠘⠋⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⣠⣿⠀⠘⠿⣿⡿⠃⢀⣹⣷⡀⢿⡇⠀⠀⠀⠀⠀⢀⣾⠋⠀⣀⣤⠀⠀⠀⡇⠀⢸⡇
⠀⠀⡈⢧⣀⣼⡿⣦⣄⡀⠀⠀⢀⣶⣶⠶⣶⠀⠀⠀⠀⠀⠘⢛⣦⣀⠀⠀⢠⡾⢁⣴⠾⠋⠀⠀⠀⠀⠁⢀⣼⡇
⠀⠀⣇⣈⣉⣉⣀⣈⣙⣿⣶⣶⣿⡿⠃⠀⣿⠀⠀⢸⠀⠀⢀⣿⣿⣿⣷⢰⣿⣷⣿⣷⣶⣶⣶⣶⣶⣾⢦⣿⡇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠀⠀⠀⠀⠀⠈⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣁⡀⠀⠀⠀⠀⢈⠁⠀⠈⠀⠀⠀⠀⠀⡢⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠂⠀⠀⠀⠀⠀⠀⠈⠛⠀⠛⠁⠀⠀⢠⠎⠒⠀⠠⠀⠀⠀⠀⠀⠀⠐⠕⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠦
⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠐⠀⠀⠀⠀⠀⠀⠀⣏⠀⣘⠀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⢂⠀⠀⢀
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠻⢿⡉⡏⠘⠁⢨⠟⠀⠀⠀⠀⠀⡀⡀⠀⠀⢰⣷⣶⡆⠀⠀⠀⠈⠑⠀⠉
⠀⠀⣰⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠙⠓⠓⠀⠀⠀⠀⠀⠀⠀⠰⡀⡱⠀⠀⠸⠿⠿⠇
⠀⠀⠿⠤⠶⣤⣤⠄⠄⠀⠀⠀⡈⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡘⠖⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠔⠀⠀⠀⠠⡀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠢⢥⡰⠀⠁⠀⠀⠀⠛⠃⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡖⠀⠀⠀⢀⡴⡡⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢄⠺⠑⠈⠁⠀⠀⠊⠈
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⡤⠀⠀⠀⠀⠁⠀⠀⠀⠀⢳⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢦
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⡐⣇⡀⠀⠀⠀⣃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⡞⠀⠀⠈⠃
⠀⠈⠠⠀⠀⠀⠀⠀⠀⠑⠒⠒⢶⣖⠨⠌⠁⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢰⣄⡄⣠⠀⠀⠀⠀⠀⠠⣐⡀⢀⡀⠀⠄
⠀⠀⠀⠀⠰⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢟⡘⣭⠇⠀⠀⠀⠀⠀⠈⠀⠈⠫⠐⡆
⠀⢀⠸⠦⠀⠿⠫⠭⠶⠆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠃⡀⠀⠀⠈⠉⠁⢀⠔⠁⠀⢈⢢⡀⠀⠀⠀⢠⣜⣙⣃⡛⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠜⡦
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠈⠁⠀⠀⠄⠞⠐⠀⠐⠚⠁⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠄⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⠁⠉⠉⠉⠉⠀⠀⠀⠀⠠⢀⠀⠀⠀⢰⡟⠁⠁⢀
⠀⠐⠀⠀⠀⠁⠀⠐⡄⠀⠀⠐⠐⠐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠘⢶⣈⣰⣵
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠎⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡦⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⣀⢀⠀⠀⠀⢤⠤⠄⠀⠀⠀⢰⢃⠉⢀⡠⠃⠀⠀⠀⠓⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠒⠐⠀⠰⠀⠂⠐⠀⠐⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠐⠀⠂⠘⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣀⠀⠠⠊⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⢆⢂⡀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠉⠀⠁⡠⠊⠁⣀
⠀⠀⠀⠀⠀⠀⠀⡸⠀⡠⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠰⠀⠀⠐⠠⠊⡰⠊⠁⣠
⠀⠀⠀⢠⠀⠀⠀⠀⠘⠠⠖⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠡⡀⠀⠀⠀⠀⠀⡑⡡⢊⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠂⠒⢠
⠀⠀⠀⠈⠀⢀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠄⢠⡈⢓⣐⣧⡒⠠⠀⠰⣾⡯⠂⠠⠀⠀⠄⠀⠀⠀⠘⡀⠀⠀⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⠀⠤⢁⠤⠀⠀⠏⡆⡇⠀⠘⡇⢰⠀⠀⠛⠀⠀⠘⠀⠀⠀⠀⠒⠢⣀⡂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⡀⡇⠠⢀⠑⡄⠀⠀⢗⡅⠀⠀⡀⠀⢠⢄⢀⠔⠂⠀⠂⢠⠀⠀⠸⠀⠀⠀⡀⠀⡀⠘
⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⡀⠠⠂⠘⠀⢄⠸⠨⠀⠈⠀⠀⠀⠀⠀⠀⣃⢀⠀⠀⠁⢀⠀⠀⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠁⠀⠀⠀⠸⠀⡀⠄⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⢨⠀⠀⠐⠘⠀⡀⠀⢀⠀⠈⣄⠄⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⠈⠁⠉⠀⠉⢂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢢
⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⡀⠀⠀⢐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠀⠀⠒⠄⣀⠀⠀⢀⣀⣀⠀⠀⠀⠤⠀⠀⠤⠀⠠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠒⠈⠁
⠀⠀⠠⠤⠤⠠⠀⠂⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣄⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⣰⣿⣷⡀⣀⡿⣿⣟⡀⠀⠀⢀⡻⣿⡿⡀⢠⣶⣶⡄⢀⡻⣿⣿⡀
⠀⠀⠀⢰⣿⣾⣭⣷⣿⡆⠀⠀⣠⣼⣛⣻⣤⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣬⣻⣟⣥⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡟⠿⠿⠟⠛⠀⠀⠛⠿⠿⠿⢻⣿⣿⣿⣿⡿⠿⠿⠿⠟
⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⣴⣦⡀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠀⣤⣿⣿⣤⣀⣤⣿⣿⣤⣀⣠⣿⣿⣤
⠀⠀⠀⠀⠘⢻⣿⡟⠁⠀⠀⠀⠀⣀⣶⣿⣆⣰⣿⣷⣀⠀⠀⠀⠿⠿⡿⣿⣿⣿⡿⡿⣿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢠⣿⣧⢉⣽⣿⡍⣽⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⡇⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⠙⠉⠀⠉⠉⠁⠉⠋⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⣀⣀⣠⣀⠀⠀⢀⣀⣀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢼⡁⠀⡇⠀⡇⠀⠙⠂⢸⠀⢸⠀⢠⡾⠋⠀⠀⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⣀⠔⡆
⠀⠀⠀⡞⠁⠀⠧⠀⡇⠀⡆⠀⣿⠠⠘⠀⠉⡀⠐⠀⠀⠀⠆⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠄⠀⠐⠒⠉⠁⠀⠀⠋⡖⠒⢲⡏⠂⡁
⠀⠀⠈⠁⠀⠀⠀⠊⠁⠀⠁⠐⠋⠐⠋⠀⠀⠐⠈⠉⠀⠀⠁⠀⠈⠓⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠄⠀⡠⠂⠐⠐⠒⠠⡀⠀⠀⣄⣧⣤⢼⡇⡄
⠀⠀⣀⠂⠀⢐⠂⠋⠘⡒⠀⠀⢰⠈⠉⡆⢀⡄⠀⠐⠖⠀⠠⠲⡂⠀⣀⠀⠀⣠⠄⢠⡤⢠⣄⢠⣶⠠⣴⠌⢰⡀⢠⠄⡀⢰⡀⠀⡄⠉⠀⠀⡀⠈⢁⡄
⠀⢸⡌⠀⣀⢠⠤⠀⠠⡄⣀⠀⠀⠓⠂⠀⠈⠁⠀⠀⠀⢐⡀⠐⠀⠀⠋⠁⠀⠙⠂⠈⠓⠉⠉⠋⠃⠀⢋⠀⠀⠌⠀⠀⠀⠐⠀⢀⡀⠀⠄⢀
⠀⠀⠇⠀⠰⢀⣁⠑⡈⡀⠇⠐⠀⠅⠈⠀⠀⠄⠀⠁⠤⠄⠀⠠⠂⠰⠄⠀⠀⡁⠀⠀⠀⠘⠈⠀⠈⠐⠉⣤⠀⠈⠳⠲⢞⠀⣠⡄⠁⠀⠸⡏⠀⢄⠀⢰
⠀⠀⠄⠀⠀⠅⣙⢒⡋⠰⠀⠀⠀⠁⠠⠀⠠⠀⢀⠀⢰⡆⠀⠿⠀⠀⢠⡠⠆⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠉⠠⠅⠈⠀⢈⡠⡲⠀⠀⠇⠀⠈⠉⠙
⠀⠀⠀⠠⢀⠀⠹⠈⠅⠤⠀⠂⠠⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠉⠀⠉⠀⠀⣀⣀⣈⣀⣂⣀⣁⡀⠀⠀⢄⠀⠠⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⢴⠀⠠⠄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⣿⡁⢀⣀⡀⠀⠀⠐⠉⠉⣉⣉⣉⣉⢉⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠡⠈⠀⠀⠀⣀
⠀⠀⠀⡀⠓⠌⠀⠀⠀⠓⠒⠁⠰⠀⠈⠈⠀⠀⢰⡏⠀⠉⠁⣿⠈⠉⠻⠋⠉⢳⡄⠿⡄⢠⣭⣀⢀⣥⡿⠀⢀⣀⠀⢀⠀⠀⠈⠀⠠⠁⠈⠀⠀⢀⡄⣿
⠀⠀⠠⠄⠀⢸⠀⠈⠀⠙⠀⠀⢀⡀⠈⠀⠀⣀⠘⢳⡀⠀⠀⣿⢸⣷⠀⢠⣿⠀⡇⠀⡇⢸⣿⣿⠸⡇⠀⠐⠛⠛⠒⠛⠛⠀⠀⠀⠀⠀⠂⠀⠁⠀⣤⠄
⠀⢀⡀⠀⠆⠈⢀⠀⢀⠈⠀⠀⠐⢰⠢⠤⠄⡞⠀⠈⢪⣀⠀⣿⢸⡿⠀⠸⠟⢰⡇⣀⡇⢸⣿⣿⠠⡇⠠⢠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⢀
⠀⢨⣧⠀⡀⠃⢚⠀⢐⠒⢂⡀⢀⣼⠀⣼⢀⡇⠀⠀⠀⣽⡀⢿⡤⠴⠻⣤⡴⠎⠘⡯⠤⠤⠿⣷⣴⡇⠀⠁⢈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⢀⣈⡁⣤⡁⠀⢀⡀⡀⠁⠀⠉⠉⠉⠀⠀⠀⣀⣀⣬⣥⣄⣀⣀⣀⣋⣀⣀⣀⡀⢀⣀⢀⣀⣛⣀⣀⡀⢀⣀⣈⣀⡈⠈⠈⢀⠀⠀⠀⡀⠀⠀⠅⣀⠬
⠀⠐⠈⠡⡤⣤⡄⢹⡅⠠⡤⠀⠀⡀⠠⠂⢻⡙⣿⠏⣩⠀⣤⡀⠉⢿⣿⠈⠀⣶⡌⠹⢋⣉⠻⢡⣤⣾⡇⣿⢸⡁⠶⢾⡇⠠⠀⠀⠀⠠⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣶⡀⠁⠄⠀⠈⢷⡘⢀⡏⢸⣿⡇⠀⢸⣿⠀⠀⠛⣡⣴⡸⢿⣶⠘⢛⣿⡇⣿⢸⡇⡶⠺⡇⠀⠄⠀⠀⠄⢸⡄⠲⢖⢾⡄
⠀⠠⠰⠻⢻⠁⠄⠈⠈⠀⠀⠈⠁⠀⠀⠀⠀⢸⡇⢸⣿⠘⠿⠃⡄⢸⣿⠀⠀⣷⡌⣿⢻⡆⠙⢠⣿⣿⠀⣏⣸⡇⣷⠀⡠⢆⡤⡄⠀⠀⠀⢆⢹⣘⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⠀⢠⣤⣤⡄⠀⠀⣧⣸⡟⠶⠴⠞⠳⣤⣥⣾⣤⣿⣧⣽⣦⣤⡾⠦⠤⠼⣧⠭⠽⣧⣿⠈⢋⠉⠉⠁⢠⡎⠀⠘⢛⠉⠉
⠀⠀⠈⢶⠃⠀⠀⠀⠄⠠⠀⠀⠸⠿⠿⠇⠀⠀⠀⡉⣁⣀⡤⣠⠀⠉⠀⠈⠉⠙⠀⠀⢸⠇⠀⢴⠒⠠⣀⠀⠀⠁⠀⠀⠘⣆⡇⠀⠉⠀⠀⠀⠀⠀⢢⠶⠂
⠀⠀⢀⡀⠀⠀⠀⠈⠉⠉⠁⢀⣀⢀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠠⡀⠀⠀⠂⠀⢀⠀⢀⠀⡢⣀⡠⠋⢀⣠⣄⣀⣀⠀⠙⠁⠐⣿⢿⣿⡏⠃⠀⢸⡆
⠀⠀⠈⡀⣤⠀⠀⠄⠠⠤⠄⠠⣬⠐⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠰⠂⠁⠆⠀⢠⡈⡾⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠏⡁⠀⠀⠘⠇
⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡀⠁⠇⣆⠀⠀⠈⠀⢀⣾⠀⠀⡀⠄⠀⠀⠈⠢⠤⠁⠀⠀⣀⠀⡀⠀⢀⡀⠀⠥⠁⠀⠀⢀⠄⠀⠐⠑⠃⢈⠃⠁⠀⠀⢤⡀
⠀⠀⠀⠀⠁⠀⡶⠂⠀⠐⠻⠇⠀⡇⡴⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⢶⡄⣒⡆⠀⠀⠠⠀⠰⡆⠀⠊⠂⠍⠀⠀⡀⣀⡩⠂⠘⠈
⠀⠀⠀⠀⡀⠀⠀⠅⠀⠨⠄⠀⠀⡆⡆⠀⠈⠀⠀⠀⠀⠀⣀⠈⠀⡁⢀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⠀⠂⠀⡀⠀⠨⠅⠀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠸⠀⠇⠀⠀⠀⠨⠐⡀⠀⡄⠃⡧⠀⠀⡡⠁⠀⠀⠂⠈⠐⠀⠄⠂⠀⠀⠀⠂⠀⡀⠂⢄⣐⠄⠀⠀⠀⠀⠀⠈⡅⠀⠉⠁⣠⣤⠀⠀⢠⣤⡀⠀⢹
⠀⠀⢰⠀⢀⡀⡀⠘⠓⠛⢀⣀⡅⠀⠃⠀⠀⡮⠀⠀⠀⠂⠀⠀⠂⠀⠀⠈⢀⢀⠀⠀⢀⠀⠀⡇⠀⣀⢀⡁⠀⣀⠀⡉⢀⣀⡈⡉⠉⠀⠀⠈⣉⣉⣀
⠀⠀⢀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠩⠐⠂⠀⠀⠃⠀⢠⢀⠈⠀⠘⠁⣀⠀⢀⡘⠀⠀⠠⠀⠀⠀⡇⢸⣏⡸⠇⠀⡷⠀⠿⠌⠉⠁⡇⠹⣶⠁⡏⢉⠉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠈
⠀⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠈⠁⠀⠀⠀⠀⠰⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠙⣽⠀⡰⡦⣤⣀⣀⠈⠀⠐⠤⠤⡥⣀⠀⣤⡄⠀⠀⠀⠀⠀⡯
⢀⣠⣄⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠉⠙⠗⠰⣶⣤⣀⣀⡀⠋⠉⠑⠐⠒⠀⠰⠯⡀
⠀⠀⠁⠀⠘⠃⠻⠶⢤⢤⣄⣄⡀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠙⣷⡤⠶⠤⠤⣠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡆⣁⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠘⠑⠢⠤⠤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣴
⠀⡤⣄⣀⣠⠀⣀⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⠈⠀⠉⠛⠇⣴⣆⡄⣀⣀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣵⠋⠈⠉⠑⠐⠗⠠⠷⠂
⠀⠀⠀⠈⠉⠀⠛⠲⠦⠀⡀⡀⠀⠸⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣷⣀⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢺⠀⠤⠦⣴⡦⡤⣀⠀⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⠃⠿⠦⠤⣄⣀⣀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠉⠘⠻⠗⠦⣄⡤⡠⠗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡏⠐⠒⠐⠶⠤⠆⣶⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠃⠤⠤⠤⡀⡄⠀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠚⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠁⠐⠿⠷⠒⠶⠶⢖⠚⠃⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠒⠒⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠃⠀⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠗⠦⢰⡄⢠⣀⡀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠙⠢⠖⢖⡲⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠢⢤⣀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂

⠀⠀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⢱
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⢹⡇⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢄⡄⣴⢤⣤⠀⠀⢰⡰⣰⠒⡂⠂⡆⠲⠦⠀⠀⠀⠶⡖⣲⣖⡰⠶⠆⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠑⠢⡀⠀⠀⠀⠀⢸⠙⠉⠉⠋⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⡠⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣼⡃⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⣶⢰⣦⡀⠂⠀⠀⠀⠀⠈⣩⡍⣩⢩⡉⠀⠀⠀⠀⡄⢠⡤⡤⢤⢠⣄⣤⠤⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⢈⣿⣷⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⠀⢈⣇⠀⠀⠀⠀⠀⠀⣠⣞⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢿⣧⣼⣿⠏⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠟⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⣐⢉⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠚⠛⠋⠀⠀⠀⠀⠀⣀⡀⢀⡀⢀⠀⡀⢀⡀⠀⠀⠀⢉⠀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠓⠯⠥⠤⠾⠬⠇⠒⠺⠄⠀⠀⠉⠋⠉⠁⠘⠀⠉⠀⠁⠀⠀⠀⠙⠃⠋⠁⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠇⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⢏⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠖⣀⡤⠴⠚⠃⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠋⡁⠀⠀⢀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠻⣷⣤⠴⠚⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠒⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠐⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢠⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠟⠉⠉⠙⢛⢿⣿⣿⣿⣿⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⣿⠟⠉⠙⢿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡟⠛⠛⣿⣿⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⣿⣿⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣿⣿⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠍⠉⠉⠉⠡⣿⣿⣿⣿⣿⣿⣿⣿⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠸⣷⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣎⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢻⣿⣿⣿⣿⡁⠀⢀⣿⣿⣿⣿⣉⣿⣿⣶⣄⢠⣤⣹⣿⣿⣿⣿⡿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠙⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠛⠿⢿⣿⣿⣷⣶⣾⡿⠿⠛⠉⢀⡀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⣀⠉⠉⣀⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⣤⣤⣤⣄⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠉⣀⣤⣶⡾⠟⢻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⢁⣠⣴⣾⡿⠟⠋
⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡿⠟⠋⠁⣠⣴⣾⡿⠟⠋⠁⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣚⠩⠭⢭⣽⣶⣶⡿⠟⠋⠁⠀⠀⢀⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠎⠁⣤⣶⠿⠛⠉⢀⡠⡄⠀⣀⣴⡾⠟⠋⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡉⠀⠀⠈⠁⠀⠀⠀⠀⢳⣼⡾⠟⠉⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⣄⠀⠀⠀⠀⠀⣀⣴⡾⡟⢻⠀⠀⢀⣴⡿⠋⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡜⢆⡀⣀⣴⠾⠟⢉⠼⣱⠇⣠⣾⠟⠁⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠴⠲⣮⠖⢉⣶⣿⣟⣛⣒⣊⠥⠚⣡⣾⠟⠁⠀⢀⣼⣿⣿⣿⠏
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⣠⠔⠁⠀⠈⠁⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠐⢿⣿⣿⠟⠁⣴
⡇⠀⠀⠀⠀⠀⣠⠀⡀⡀⠀⠀⢂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⢤⡀⠀⠀⠰⣠⡾⡿⠋⠀⠀⣿
⣧⠀⠀⠀⠀⠀⡠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣒⣐⣦⣴⢿⣩⠒⠐⠦⠈⠀⠀⠼⢳⠆⡀⠀⠀⠀⢹
⣿⠀⠀⠠⡄⠀⠐⠐⠀⠀⣴⠀⠀⠿⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠀⠘⢻⠇⠀⠀⠀⣠⠖⡰⠃⠀⠖⠉⢱⠘⡂⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢷⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠃⠀⠀⠀⡜⠡⠊⠀⠀⠀⢀⢔⡫⠜⠁⠀⠀⠈
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠡⠄⠀⢈⡻⡀⠀⠀⠀⣛⡋⠀⠀⠀⠀⠳⠎
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠀⠀⠘⠅⠼⠀⠀⠀⠸⠭
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡠⢀⠀⢀⡌⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⠀⢀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⡀⠐⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠃⠁⠠⠀⠀⠀⡀⠄⠈⠁⠀⢀⣠⣶⡿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢀⠀⠡⠀⠀⠋⠠⠀⠀⠈⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⠀⢀⣤⣾⠿⢛⣉⣀
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠱⡒⢠⣤⣭⣵⢾⣟⠿⠉
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠲⠟⠉⠉⠉⠌⠉⠂
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⢀⠀⢠⠀⠠⠀⠁⠀⠀⡀⢀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠐⠆⠁⠋⠘⡄⠈⠑⠆⠀⠀⠄⠐⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⣤
⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⢃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⢄⠠⢀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠵
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠊⠀⠀⠨⠂⠀⠀⠐
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠈⠂⠀⠀⠀⠀⠀⢠⠀⢁⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠐⢦⡀
⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⢦⡤⣷⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠏⠀⡀⢠⡀⡀⠀⠙⡄
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠈⣧⣴⢂⠷⠜⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⠂⠀⢀⠀⠀⠀⠀⠀⠀⠰⠉⠁⠉⠱⠀⢀⠁
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡆⠀⠀⠘⠃⠈⠀⢙⣁⣑⠾⠇⠀⠦⠀⠀⠀⠀⠀⠊⠆⠀⠀⠀⠀⢀⡀⠠⠀⠀⠄⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠠
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠀⠀⢀⡜⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰
⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠠⠀⠀⢀⠈⣦⡄⠐⣢⣄⣤⠎⡀⢠⡀⠀⠀⠁⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠴⠂⢦⡀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢙⡥⠐⠀⠈⠀⡀⠀⠀⠀⠅⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⢀⠀⠄⠀⢀⠈⠀⠂⠀⠀⠀⠀⠠⣤⡎
⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⢀⠞⠠⠀⠛⡀⠠⠄⠀⢂⡀⠀⠀⠀⠀⠀⠀⠂⠀⢂⡈⠡⠀⠂⠈⠁
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⠶⠿⣿⠛⣋⣿⣿⣿⣿⣿⣭
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⠿⠿⡿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⡀⠠⢀⠄⠀⠀⡄⠔⠂⠠⡄⠀⠀⠠⠀⠠⠠⠀⠀⡄⠠⠀⠠⠀⠀⣤⠀⡄⢠⠂⠀⠀⡆⠀⠀⠀⠀⡠⠀⢄⠀⠀⠠
⢈⣑⠘⢄⢀⠀⡇⣋⢁⠸⠈⢢⠀⠀⠀⠀⠘⠉⠀⡇⠣⡀⡀⠀⠀⠁⠊⠀⢸⡉⠀⠀⡇⠀⠋⠉⠇⢄⠀⡠⠀⡀⡠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⢲⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠉⢢⠀⠀⠀⠀⠀⠀⠀⢠⠚⡶⢺⠉⡧⠮⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢰⣾⢃⡼⠀⠀⠀⠀⠀⠀⠀⠈⢆⣼⣱⣆⣉⢆⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡐⢘⢓⡜
⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠆⠀⠀⠠⠶⠶⠤⠤⠤⠬⠭⠥⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠰⠶⠶⠒⠒⠒⠒⠒⠒⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢱⢀⡖⠉⠉⠳⡄⢼⠉⠙⠀⠉⣿⠉⠀⡏⠉⠉⠀⡏⠉⢱⢰⡉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠁⠘⡄⠀⠀⢀⠇⠀⠉⢳⡄⠀⣿⠀⠀⡏⠉⠁⠀⡟⠛⡅⠀⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠈⠑⢒⠉⠀⠈⠓⠊⠀⠀⠉⠀⠀⠓⡒⠒⠀⢃⠀⠘⠈⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠑⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢠⡀⢀⡄⠀⣄
⠀⠀⠀⠀⠀⠺⠿⠇⠀⠀⠀⠀⠀⠀⢠⣼⢱⣿⣿⡎⣧⡀⠀⠀⠀⠀⠀⠀⡆⠚⠉⠉⠀⠀⠀⠀⠀⢀⣮⣦⣴⣷⣤⣽⡄
⠀⠀⠀⣀⠀⠀⣃⠀⠀⣀⠀⠀⠀⠀⠀⢹⣆⢙⠋⣰⡏⠀⠀⠀⠀⠀⠀⠲⠶⢆⢀⠀⠀⠀⠀⠀⠀⡼⢤⠄⢠⡄⠀⣤⢹
⠀⠀⣀⣋⡀⣀⣛⡀⢀⣛⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣁⣻⣀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠷⠶⠿⠾⠄
⠀⠀⠉⠉⠁⠉⠉⠁⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢾⡆⠰⣿⠂⢼⡷⠀⠀⠀⠀⠀⠸⡿⠀⠀⠐⠘⠃⠀⠀⠀⠀⠀⠀⢀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⣰⡀
⠀⠀⣴⣶⣶⣾⣶⣷⣶⣶⡆⠀⠀⠀⢰⣾⣾⣶⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡔⠁⠀⣀
⠀⠀⠘⣿⡟⢻⣿⡟⣿⣿⠃⠀⠀⠀⢘⣿⣿⠛⠀⠉⠈⠁⠀⠀⠀⠀⢠⣶⠄⣿⠁⣶⡄⠀⠀⠀⠀⠚⠂⠀⠶⠷⠀⠀⠛
⠀⠀⠀⠿⠇⢸⢿⡇⠿⠿⠀⠀⠀⠀⠈⢻⢿⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠋⠙⢺⡆⠀⠀⠀⠀⠀⢀⡀⠀⢴⠀⣠⣧⡀
⠀⠀⣠⣍⠉⢉⣭⡉⠉⣥⡄⠀⠀⠀⠀⢀⠀⢙⡋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣃⣸⣃⣺⣿⠆⠀⠀⠀⠀⣼⣧⠸⣿⡇⢹⡿⠁
⠀⠀⠉⡅⠀⠈⡍⠀⠀⠩⠁⠀⠀⠀⠀⡄⢀⢠⡄⢀⢠⡄⠀⠀⠀⠀⠘⣿⡿⣿⡿⣿⡿⠀⠀⠀⠀⠀⢹⡏⢀⠿⠄⠘⠙⠂
⠀⠀⠲⠖⠀⠰⠶⠂⠀⠶⠆⠀⠀⠀⠐⠖⠀⠒⠖⠀⠒⠓⠀⠀⠀⠀⠀⠸⠇⣿⡇⠿⠇⠀⠀⠀⠀⠀⠭⠭⠸⠤⠄⠠⠠⠇

⠀⠀⠀⠀⣀⣤⠤⣤⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⡀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢠⡞⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣤⣾⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⡟
⠀⠀⣿⢸⡷⢰⣿⡆⢾⡇⣷⠀⠀⠀⢸⣿⣿⣗⣻⣿⣿⡇⠀⠀⠀⠀⠈⣿⣯⣭⣭⣿⠁⠀⠀⠀⠀⠀⠀⣼⡿⠿⣷
⠀⠀⠹⣶⣶⣦⣭⣴⣶⣶⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢩⡿⠿⢿⡍⠀⠀⠀⠀⠀⢘⣿⣿⣾⣷⣿⣾⡇
⠀⠀⠀⠉⠻⠿⠿⠿⠟⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠨⠿⠿⠿⠇⠀⠀⠀⠀⠀⠛⠛⠓⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠛⢿⠛⡟⣿⡟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠚⠀⠀⣠⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡰⠞⠛⠋⠛⠛⢶⣄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣰⠎⠳⣆⠀⠀⠀⣠⠞⠙⣧⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡟⠀⠀⠈⠳⣤⠞⠁⠀⠀⢸⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢻⣤⣤⣤⣤⣤⣾⠁⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣗⠲⠶⠤⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡁⠸⣏⣉⣛⣛⣺⡇⠀⢛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⡜⠁⠀⠻⣍⣉⣉⡽⠁⠀⢯⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⠀⠀⢠⡄⠈⠉⠉⠀⠀⠀⠈⣇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⢤⣜⡧⠤⣤⣤⣀⡀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠉⢧⡀⠈⠓⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠻⢤⡀⠀⠀⠀⢠⠇⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡇⠀⢀⡴⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⡀⠀⠙⠚⢉⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⣿⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⣿⠀⣿⡇⣿⣿⣿⣿⣿⢠⣶⡆⢀⣀⣴⣾⣿⣿⣯⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢸⣿⣾⣿⣶⣿⣶⣿⡇⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣼⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⢸⣿⡇⠻⠿⢿⣿⣿⣿⣿⣿⣿⣩⣩⣭⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠋⠉⠉⠉⠋⠀⠀⠀⠀⠀⢀⣍⣉⣉⣉⣉⣉⣉⣉⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⣉⣉⣉⣉⣉⣉⣉⣙⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡀⣰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣛⣙⣛⣿⣿⣿⣿⣿⣿⣷⣶⣦⢸⣿⡇⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⢸⣿⠿⣿⠿⣿⡿⣿⡇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⣻⣿⣿⡿⠟⠉⠁⠸⠿⠃⣿⣿⣿⣿⣿⢸⣿⠀⣿⠀⣿⡇⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⣿⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠁⠈

⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⢰⣿⣆⠀⣻⣥⣾⣓⢸⣴⣿⢶⡟⡗⣿⣾⢺⡇⣿⠀⢲⣸⡞⣇⣾⢷⠞⣇⠀⢻⢻⢷⣶⡿⡦
⠀⠀⠈⠀⠉⠀⢈⠁⠈⠉⠈⠀⡉⣀⢀⣁⡁⠈⠈⠁⠈⠀⠀⠉⠈⠁⡈⢉⠈⠁⠀⠈⠈⢀⠁⠉⠁
⠀⠀⢺⣇⣶⣓⢸⣴⣶⢶⠞⡖⡟⣶⢺⠣⣦⣞⡓⠀⣶⢲⠲⣴⡾⡖⣿⣿⢲⣴⣚⣶⣟⣿
⠀⠀⠈⠁⠈⠉⠈⠀⠉⠈⠀⠁⠁⠉⠈⠀⠁⠉⠁⠀⠉⠈⠀⠁⠉⠁⠉⠈⠈⠀⠉⠁⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠄⠆⠐⠤⠂⠰
⠀⠀⠀⠀⠀⠂⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠛⠛⠛⠿⠿⠿⣿⣦⡀⠀⠀⠔⠀⠤⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠁⠂⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⣴⡿⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠠⠀⠠⠀⠆⠠⠄
⠀⠀⠰⠐⠒⠐⠁⠂⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠠⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⡀⠀⠀⠀⡀⣀⠀⠀⢠⣀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢃⠀⠀⠳⣄⠀⠈⠁⠉⠀⢀⠞⠀⠀⡼⢙⡆⠀⠀⠀⡀⠀⣀⠀⣀⣀
⠀⠀⠀⠆⠀⠀⠠⠀⠀⠀⠀⠄⠄⠀⠙⠚⠦⣀⠀⠈⠑⠒⠒⠒⠊⠁⢀⣠⡴⠷⠊⠀⠀⠀⠀⠀⡄⣀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠠⠤⠄⠤⠄⠄⠠⠀⠀⠀⣰⠋⠛⠛⠲⣶⡒⡒⢾⠯⠉⢀⣷⡀⠀⠀⠀⠀⢀⣀⡀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣤⣤⣄⠸⡗⠒⡿⢀⣠⣾⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡟⠰⣇⠀⢻⣼⢃⡼⠀⢹⡇⠈⣷
⠀⠀⠄⠀⠠⠄⠤⠄⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⠃⠀⠈⠙⢾⣿⠞⠁⠀⠀⣧⠀⠘⣇⠀⠀⠀⠄⠀⠠⠤⠀⢀⠄
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⡼⠃⠀⠀⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⠸⡆⠀⠀⠀⠄
⠀⠀⠤⠠⠀⡄⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢰⠇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡇⠀⠀⢻⡄⠀⠀⠀⠄⠄⡀⠠⠀⠠
⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⡎⠀⣤⠤⢤⣄⠀⣿⢠⡖⠋⢹⡆⢹⡀⠀⠀⣷⠀⠤⠀⣀⡄⠀⠄⢠⠠⠄
⠀⠀⠀⡄⡀⡠⠄⡀⠀⠀⢜⠿⣆⡀⠀⢰⡇⠀⡇⠀⠀⣿⠀⢻⠀⡇⠀⠀⣇⠘⣯⣦⣤⠟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢛⡦⢍⣙⣻⠁⠀⡇⠀⠀⡟⠀⢸⠀⢁⠀⠀⣿⠀⣿⡤⠤⢴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠉⢠⣴⣁⣿⠀⠀⡇⠀⢸⡇⠀⢸⠀⠈⢆⢀⡟⠀⢹⡀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢾⡋⡀⢢⡀⣼⣷⣀⣷⠀⠀⠹⠶⠟⠀⠀⢸⡄⠀⠈⠉⠀⠀⢸⡗⠶⠊⠀⠐⠀⠀⠠⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡦⢄⡙⢦⡸⠋⠻⣤⡤⠤⠤⠤⠤⠤⣾⡿⣦⠤⠴⠲⠶⡏⠀⠀⠀⠀⠰⠠⠄⠀⠠⠤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠟⠀⠀⠀⣸⣇⣀⣀⣀⠀⣀⣄⣸⡇⢀⣀⣀⣀⣀⣿⣄⡀⠀⠀⠀⠤⠄⠤⠤⠄⠄⠠⠄
⠀⠀⢀⠀⠀⣀⠀⣀⠀⡀⢢⡀⠀⢠⠾⠉⠀⠀⠀⠀⠀⠀⠀⢹⡏⠁⠀⣀⣀⡀⠀⠉⢻⡄⠀⠀⠄⠠⠀⠀⠠⠤⡄
⠀⠀⢠⢀⠀⡀⠀⠀⠀⡀⠀⠳⣄⣺⡠⠐⣤⠖⠛⠉⠉⠙⠢⣼⡿⠛⠉⠉⠉⠙⢳⡄⢸⠃
⠀⠀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠙⠉⠳⡞⡁⠀⠀⠀⠀⠀⠀⣹⢠⠀⠀⠀⠀⠀⠀⣿⡋
⠀⠀⢀⠀⠀⢀⠀⡀⢀⡀⠀⠀⠀⠀⣼⠷⠤⠤⠤⢤⠤⠴⠚⢻⣛⡓⠒⠒⢒⣛⣉⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠇⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠟⠀⠐⠋⠻⠵⢢⣀⢕⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢑⣀⣠⢻⠀⠀⠀⠀⠘⠈⠈⠠⠤⠠⡐⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣢⠓⡽⣶⠂⡀⠀⠀⠠⡞⢉⣀⠠⠂⠀⢀⠀⣷⠀⡏⣠⡇
⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⣷⢄⣄⣢⠇⠀⢀⠁⠀⠀⠀⠀⠉⡀⠀⠀⢹⡄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠢⠽⡿⠀⠀⠀⠈⠀⡄⣶⡶⠈⠁⠀⢸⡀⠻⢃⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄⠀⠀⢸⡦⣄⣀⢸⢘⠀⣿⠃⡅⢠⣄⠸⣇⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡎⠀⠀⠀⠀⣧⡀⠉⠚⠚⢲⠃⠠⡄⠀⢻⣆⣫⣴⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⡀⠀⠐⢀⣿⣿⣦⣤⣠⣏⣀⣴⣷⠀⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣴⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⣀⣠⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠗⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⠀⠙⠋⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠴⠆⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⠿⣿⣿⣿⣿⣿⡷⠀⠀⠠⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡤⡔⠁⠀⠀⠭⣿⡍⣥⣆⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡾⡀⢢⠀⠖⠂⢸⡆⠀⠉⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣾⡇⢻⡌⠀⠀⠀⡈⣰⡃⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠊⣧⣷⠈⠁⠀⠀⠀⠐⠊⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡚⣋⢱⢼⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡶⠔⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⢠⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⣤⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦
⠀⠀⠀⠐⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⣤⣬⣽⣯⣭⣤⣤⣤
⠀⡤⠤⠤⣤⠼⠤⣤⠤⠤⣄⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠼⠦⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠉⠲⠂⠀⠶⠀⠀⠶⠀⠀⠶
⢠⣤⠀⢀⣤⠀⢀⣤⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢾⣷⠀⣾⡆⢰⣷⠀⣾⡆⠀⠀⠀⠀⠀⠀⢸⣿⡿⢸⣿⡇⢸⣿⡇⢸⣿
⣼⣿⡆⣴⣿⡆⢴⣿⡆⢰⣿⡦⠀⠀⠀⠀⠀⢰⣿⣿⡆⣿⣷⢸⣿⡦⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠃⠈⣿⠁⠈⣿


⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⢠
⠀⣶⡄⠀⠀⡇⢀⣤⡀⣼⣿⣄⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢰⡦⢀⡞⠠⣿⡦⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⣶⣶⢰⣶⢰
⣤⣬⣤⡄⠀⡇⣠⣿⣅⢉⣍⠁⠀⠀⠀⠀⠀⣤⣤⣴⡄⠀⠿⠿⠆⠧⠀⣾⡆⠀⠀⠀⠀⠀⣴⣶⣶⡄⢸⠁⢻⡟⢼⡏⣾
⣿⣿⣿⠇⠲⡇⠉⣉⠉⣨⣿⡄⠀⠀⠀⠀⠀⢿⣿⣿⠇⠂⠐⠅⠀⠀⠈⣻⡋⠀⠀⠀⠀⠀⢻⣿⣿⠇⢼⠀⢨⡅⢨⡁⢩
⢸⣿⡏⠀⠀⡇⢨⣿⣅⢉⣉⠉⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⢰⡦⢀⡞⠠⣿⡧⠀⠀⠀⠀⠀⠈⣿⡿⠀⢸⠀⣶⣶⢰⣶⢰
⠀⠻⠃⠀⠀⡇⠙⠛⠋⣨⣿⡄⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠿⠿⠆⢧⠀⣾⡆⠀⠀⠀⠀⠀⠀⠛⠃⠀⠈⠁⢻⡟⢼⡏⣾
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠁⠙


⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷
⠀⠀⠀⠀⣠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⡇⠀⢴⡦⠀⢸⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣆
⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⣇⢰⣶⣶⡆⣸⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⢴⡆⠈⢙⡟⠋⢰⣶
⣴⣶⣿⡇⠠⢼⡤⠄⣸⣷⣶⡦⠀⠀⠀⠀⠀⣾⣿⠎⡟⠘⣿⣿⠃⢹⠱⣿⣷⠀⠀⠀⠀⠀⠀⠰⢿⡿⠈⠉⠉⠁⠾⣿⠆
⠀⠀⣉⡁⠀⢸⠀⠀⣈⣉⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⡇⠀⢿⡟⠀⢸⠀⣴⡆⠀⠀⠀⠀⠀⠀⣚⠉⠉⣃⠀⠀⢘⠋⠉⢙
⠻⠿⣿⡇⠐⠚⠓⠂⢸⡿⠿⠗⠀⠀⠀⠀⠀⠾⠿⠆⠁⠀⠀⠀⠀⠈⠰⠿⠷⠀⠀⠀⠀⠀⢨⣿⡀⢠⣿⡄⢀⣿⣅⠀⣽
⣶⣶⠾⠿⠿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣶⣷⣶⣶⣿⣾⣿⣿⣷⣿⣿⣷⣿⣿⣿⣾⣿⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠛⣆⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢻⡀⢸⡇⢸⡴⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⡇⠸⡇⠘⣇⠀⣷⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢷⠀⣷⠀⢿⠀⢻⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡁⠀⠰⡏⠳⣸⡀⠘⠀⠈⠀⠈⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣄⠻⠃⠀⠀⢻⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠳⣌⢧⡀⠀⠀⠀⢳⡀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠈⢧⡙⢦⠀⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⠷⠖⠉⠀⠙⢲⣀⣀⣀⣤⠤⠤⠿⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⣼⠭⠤⠶⠒⠒⠛⣉⡉⡇
⠀⠀⠀⠀⠀⠀⠰⣇⣤⠞⠁⣠⠶⠀⠀⠀⠀⢀⡀⠀⡴⠳⢤⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠛⠃⢿
⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠞⠁⣠⠞⠃⣠⢶⠏⢀⡞⠁⠀⠀⣸⢳⣄⡤⠶⠟⠶⠶⠤⠤⠤⣤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠈⡽⠋⢁⣴⠟⢁⡤⠚⢱⠇⣰⠋⠀⠀⠀⢰⢃⡞⠀⠀⠀⠀⠀⠰⠶⣤⣤⠤⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠳⠞⢿⣁⡴⢯⣀⡀⠈⠉⠁⠀⣀⡀⢀⡏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠓⠦⢤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⣛⠃⣼⣰⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠲⠶⣤⣌⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠁⠈⠳⢤⣀⠀⠀⠀⠀⢀⡈⠙⠓⠶⣤⣄⣈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣀⡉⠛⠶⢦⣤⣈⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠴⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋⠻⣷
⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠻⣧⣄⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠟⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡿⠃⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⢰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⣸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢋⣴⣌⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢰⡆⣶⣶⣾⣿⣿⣿⣿⡆⣴⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⢸⣇⢿⣿⣿⣿⣿⣿⣿⠇⣿⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣄⡀⠸⣿⣮⡻⢿⣿⣿⠿⢫⣾⡿⠀⣀⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣤⠀⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣤⣄⡀
⠀⠀⠀⢀⡘⣿⡙⠷⠟⢸⣧⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡾⠋⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⣿⣙⠛⢡⡶⠶⣦⠉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⣨⡾⠛⠛⠛⢷⣄⠀⠀⢹⣆⡀
⠀⠀⠀⣩⡽⠇⠸⣧⣤⠿⢀⣙⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠻⣆
⠀⠀⠀⠹⡷⠞⣷⢀⣴⣄⢻⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠁⠀⠀⠸⣧⠀⢤⣴⡄⠀⣼⠃⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠘⠻⠾⠃⠹⠛⠁⣶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⣿⢰⡞⠁⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣦⣀⣿⠀⢀⣿⣠⡾⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢰⡾⠷⠿⠾⣷⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠉⠁⠀⠀⠉⠋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣭⡛⢿⡄⠀⠀⠀⠘⠿⣾⠿⡷⠟⠀⣼⠛⠛⠉
⠀⠀⠀⠀⠀⢠⣤⣤⣴⠟⠀⢠⡾⠛⠛⢶⡄⠈⢿⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⣨⡗⠈⠻⢶⣤⡴⣦⣀⡀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⢸⣇⣀⣀⡀⠀⢿⡄⠀⠀⢸⡇⠀⢀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠉⣩⡷⠀⠈⠛⠶⠾⠋⠀⠀⣿⡉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠻⠶⠟
⠀⠀⠀⠀⠀⠀⠀⢾⣏⠀⣠⡶⣤⠀⠀⣴⢦⣄⠈⣻⡦⣠⡶⠛⠉⠉⠉⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⣿⠀⢀⣿⠀⠙⠿⠋⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠰⠟⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣠⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢠⡞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣀⣤⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣄
⠀⠀⠀⠀⠀⢠⣄⢀⡤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢸⡇⠀⣀⣀⡀⣠⣤⡀⠀⠀⠀⣴⠶⠾⠋
⠀⠀⠀⠀⠀⢀⡭⢨⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⢸⡏⠉⠿⠋⢀⣿⠆⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⣨⡿⠃⠀⠀⠘⣧⣀⡼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠻⣦⡀⠀⠀⠀⠀⠉⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣷⠄⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠈⠻⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣧⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣄⣼⡇⠀⠀⠀⠈⠻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⣤⢠⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⠀⡇⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⡴⠋⠀⢸⢸⠀⣨⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡇⢸⢸⠀⡇⠀⠀⠀⡼⠁⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⢳⠀⠀⢰⠃⠀⠀⢸⠈⣫⠴⠊⢹
⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⠀⡇⡇⢸⢸⠀⡇⠀⠀⢠⢣⡄⠀⠀⡿⡄⢶⣄⠈⢇⠀⠀⡀⡇⠀⠘⣆⣤⡀⠘⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⡇⡇⢸⢸⠀⡇⠀⠀⢸⢸⠁⠀⢸⣷⣟⢼⣎⣗⣾⡄⠀⡇⢸⠀⠀⢿⠀⣗⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠴⠧⠾⠼⠴⠷⠧⠼⠾⠤⠧⠄⠀⢸⢸⡀⢀⣇⣘⣠⠋⢧⣘⣀⣇⠀⡇⢸⠀⣴⣚⣈⣓⡍⠉⠁⠀⠀⠀⠀⢰⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⢧⣸⣆⠀⢀⣐⣀⠀⢀⠿⡀⣇⡞⠀⠀⠀⠀⠀⠀⠀⣰⢦⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠙⡷⣄⠀⠀⠀⠀⠘⢮⡟⣨⡳⡤⣉⡡⡴⢋⣤⣱⡟⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠓⣤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣛⣀⡉⠙⡇⠀⠀⠀⠀⠀⠈⠁⣹⠇⠀⠀⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠯⠇⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⡠⠖⠒⠦⣼⠀⠶⠶⠆⠀⡇⠀⠀⣠⠖⠚⠻⣋⢷⠀⠀⠊⢱⢋⡻⠒⠲⠤⣀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⣀⣈⣉⣁⣀⡇⠀⡜⠁⠀⠀⠀⠈⠹⡖⠦⠶⡞⠉⠀⠀⠀⠀⠘⡆⠀⠀⢸⣉⣉⣉⣉⣉⣉⣿⣏⣹
⠀⠀⠀⠀⢀⡹⠀⠀⠀⡆⠀⠀⠈⣇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢃⠀⢀⠃⠀⠀⠀⣤⠀⠀⢹⠀⠀⢸⠀⠀⠀⡴⠦⡄⠀⠀⢸
⠀⠀⠀⠀⡎⠀⠀⠈⠳⡷⠊⠀⠀⠈⢧⠀⢀⡇⠀⠀⢀⠀⠀⠀⠀⠸⡄⡼⠀⠀⠀⠀⣇⠀⠀⠀⣇⠀⢸⠀⠀⠀⠳⠶⠧⠀⠀⢸
⠀⠀⠀⠀⠱⢄⣀⣀⣀⣀⣀⣀⣀⡠⠏⠀⡼⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⠃⠀⠀⠀⠀⣿⠀⠀⠀⢹⡀⢸⠀⣒⣒⣒⣒⡒⠒⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠸⣤⣤⣤⣤⣤⣤⣤⣤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠒⠦⣾⠀⠀⠀⠀⠀⠷⠇⠀⠀⠀⠀⣯⡤⠤⠶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣲⣿⠭⢬⣣⣴⠃⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣼⡝⢶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⣽⣷⡾⠋⠀⠀⠀⡵⣝⣯⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠧⠤⠤⠴⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⣠⣴⣶⡿⠿⠿⢿⣶⣦⣄⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⠉⠀⣀⡀⢀⠀⠀⠉⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢰⡶⣛⣛⢿⣥⡀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢰⣿⠁⠀⠀⢿⡗⣿⣿⢇⣿⡅⠀⠀⠀⠈⣿⡆⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⠀⠀⢀⣜⣟⣶⡶⢿⢽⠞⣓⢷⡄⠀⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠠⡏⣶⢝⡇⣠⣤⣹⣮⣋⡼⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠙⠖⠛⠙⢿⣻⠇⠈⠁⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡷⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⢾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣿⣶⣶⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⣿⣿⣿⣿⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣤⣄⣠⣤⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠉⠉⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠶⠒⠛⠻⣿⣿⠛⠛⠲⠖⠛⠛⣿⣿⠟⠛⠲⠶⠚⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣶⣶⣶⣴⣿⣿⣶⣶⣶⣶⣤⣦⣿⣿⣦⣶⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⡀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣰⡄
⠀⠀⠀⠀⠀⢠⡞⣲⡀⠀⠀⠀⢀⠀⠄⠀⠊⠈⠃⠁⠎⢱⠀⠀⢱⠁⠀⠁⠀⠁⠀⠀⠐⠀⠂⠀⠀⠰⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠁⠀⠋⣠
⠀⠀⠀⠀⠀⠁⠀⠀⣣⠀⡆⠐⠐⠆⠠⡆⡀⠠⠀⠄⠀⠈⠀⠠⠈⠀⠀⠠⡀⠀⠀⠀⠀⠃⠸⠄⡰⠶⠀⠘⠅⠴⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠇⢃⠈⠁⠀⠀⡄⠀⡀⢃⠀⠠⠁⠁⠌⠙⠕⠀⠠⠀⢈⠉⠀⠄⠀⠀⠀⠀⠀⠁⠰⠈⡄⠒⠂⡀⠀⠀⠀⠀⢀⠂⠀⡀⠀⠀⢌⠀⠁⠘⢁⡀
⠀⠀⠀⠀⠀⠀⣀⡂⢀⡀⠐⡄⠠⠁⡠⢄⠀⠀⢠⠴⠉⠃⠀⠀⠀⠀⠤⠀⠼⢠⠈⠀⠀⠀⠀⠀⠀⠀⡋⡶⠁⢤⠰⠆⠀⠀⠰⠀⠀⠀⠀⠀⠂⡀⠀⠠⡠
⠀⠀⠀⠀⠈⠀⢨⣞⠡⠠⠀⠭⠗⠜⠀⠀⠂⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠳⠤⠊⠀⠐⠀⠀⠀⠒⠀⠀⣤⡇⠀⠀⡠⡠⠁⠀⠠⠀⠄⠈⠂⡀⠀⠈⢔⠆
⠀⠀⠀⠀⠀⢀⠈⠁⠀⠰⠈⠤⠀⠤⠀⢠⠤⢠⢠⡤⣄⠀⠀⡠⢀⠀⠀⠀⡔⠀⠀⢄⠀⠀⠀⠀⠠⠀⢁⡀⠀⠐⠠⠁⠀⢠⠀⠀⠀⠀⠀⡀⠐⠀⠈
⠀⠀⠀⠀⡠⠀⣍⠏⠀⠀⠀⠀⡀⠀⠀⢸⠋⠉⠉⠉⠹⠀⠐⠁⠁⠀⢀⢀⠁⠀⠀⡄⠀⠆⠀⢰⠈⡀⠈⠀⠄⢈⠄⠐⠀⠸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣀⡆⠀⠐⢆⡦⠀⠀⠙⠉⠉⢉⠀⠀⢠⠄⠈⠁⡁⢠⡀⠂⠸⠁⠀⠐⡀⢢⠀⢀⠀⡘⠀⠀⠀⠀⢀⠂⠄⠀⠀⠈⠀⠐⠀⠠⠆⣤⠖⠁
⠀⠀⠀⢀⠕⢫⠐⡄⠀⡁⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠐⠀⠛⠓⠂⠈⠚⠀⠉⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠁⠤⠀⣻⣀⡄⠠⠄
⠀⠀⠀⠀⠀⠈⠆⠼⠂⠠⠀⠠⠀⠂⢼⠂⡄⠀⠘⠀⢀⠀⠄⠀⠠⠄⢜⠠⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢉⠄⡜⠀⡌⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢐⣛⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠀⠀⠠⢏⠄⡀⠀⠰⠅⠻⣘⠕⢮⢀⡐⠀⢐⡅⠀⠀⠀⠀⠀⠀⠀⡟⢀⠁⠀⠀⢃⡀⢐⠀⢀⢶⣷⠊⠀⠆
⠀⠀⠀⠀⠐⠚⠃⠐⠀⠂⠐⠀⠀⠀⡀⠁⣀⣀⡀⠀⠀⣴⠀⠀⠅⠀⠈⢶⡁⢣⡈⡈⠀⠦⠀⠀⠀⠀⠀⡀⠀⣋⡊⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠠⠀⢀⠄
⠀⠀⠀⠂⡀⠠⠀⠡⠁⠞⡆⠀⠀⢈⠀⠀⠁⠁⠀⠊⠀⠘⠶⠰⡆⠀⣀⠄⠱⣂⠱⡀⠐⠀⠀⢀⡠⠂⠀⠀⠀⠄⠀⠀⡀⠀⠙⠂⡁⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠂⠀⠈⠃⠐⠀⠀⠀⠀⠀⠒⣀⠄⢴⠀⠈⣹⡄⠂⡃⠀⠀⠀⠀⣀⢀⠀⠐⠃⣾⢿⡀⠅⠀⠄⠀⠠⠀⠀⠈
⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠀⢠⠀⡃⠐⠀⠠⠀⠀⠀⠈⠛⡀⣀⡼⠁⠀⠀⠀⢤⠑⡀⠀⠈⠀⡈⠉⠀⣦⡶⢹⡎⠀⠘⡀⢀⠂⠒⡆
⠀⠀⠀⠀⢀⣄⡀⢀⢀⠨⡶⠀⢀⠀⠀⠀⢲⠀⠃⡀⠀⢐⠂⡀⠀⠀⠀⠈⠀⠠⡖⢲⠀⠄⠐⡂⠁⠀⠀⡈⢠⠀⠀⠁⠀⣸⡗⠀⠀⠀⠉⠅⢀⠁⠀⢄
⠀⠀⠀⠀⠀⠀⡸⠣⡀⠀⠉⠀⠀⠠⠐⢀⢴⠀⠀⠄⠀⠠⡀⠁⠀⡄⠀⠀⢀⡀⠑⠉⠀⠀⠀⡥⢘⠀⠀⣂⣀⡇⠀⠀⠃⡞⡏⠀⠒⠂⠀⠁⠸
⠀⠀⠀⠓⠀⠀⠀⠲⡇⠀⠀⠀⠀⠐⠀⠹⠿⡦⠤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⡀⠚⠈⠀⢀⣠⡄⠧⠄⠒⠒⠻⢇⢤⡐⣦⣀⣀⠄⡠⠆
⠀⠀⠀⠀⠈⠀⠀⠅⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠁⠀⠁⠀⢌⠀⠆⠐⠒⠐⠂⠒⠈⠒⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠽⠤⢄⠈⠉⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡀⠀⡀⠀⠀⣀⠀⣠⠄⢀⠀⠀⡀⣠⠀⡀⣀⣠⠀⢀⠀⢠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠕⠺⠀⠛⠇⠋⠃⠣⠈⠋⠃⠯⠀⠐⠉⠙⠸⠁⠇⠋⠓⠟⠻⠛
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠄⠄⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠠⢄⠀⠄⡄⠄⠄⠀⠀⠀⠀⣠⠄
⡇⠀⠀⠀⠀⠀⠀⠐⠁⠀⠤⠀⠄⠄⠠⠄⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⢂⠀⠠⠄⠀⠀⠀⠁⠀⠒⠐⠂⠂⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠄⡠⠀⠤⠄⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠈⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⠠
⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢀⡤⠤⠤⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⣶⣶⣦⡀⠊
⠀⠀⠀⠀⠀⠀⣀⣀⢠⡏⠀⠀⠀⠈⡆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣼⣿⣿⣿⣿⣿⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢧⠀⠀⠀⣠⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠹⣿⣿⣿⣿⡿⠡⣄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢳⣀⣸⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡠⠂⠘⣿⣿⡟⠀⠲⡌⢳⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⣶⡖⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢒⣒⠂⠀⠀⠀⠀⢷
⠀⠀⠀⢀⠖⠛⠓⢆⠀⢀⡤⠤⠤⡀⠀⣰⠋⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣤⡀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣴⣶⣦
⠀⠀⠀⢸⡀⠀⠀⣸⠀⡟⠀⠀⠀⢸⠀⣳⣀⠀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠀⠀⣼⣿⣿⣿⡄⠀⠸⣿⣿⣿⠃
⠀⠀⡴⠋⠉⠛⠋⠉⣳⠿⢤⣀⡠⠾⢏⠀⠈⠉⠁⠀⢳⠀⠀⠀⠀⠀⠀⣠⣦⣭⣯⣵⣦⠀⡻⣿⣿⠿⡃⢰⣾⣶⣿⣶⣷⡄
⠀⠀⡇⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣾⣿⣷⣶⣾⣿⣦⢻⣿⣿⣿⣿⣿
⠀⠀⠧⣤⣀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠼⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣀⣀⣀⣠⣤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇

⢠⡄⠂⠀⠀⠤⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⢀⣼⡅⠀⠀⠀⠀⠀⢀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣮⣽⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⣾⡇⣧⡄⠀⠀⠀⠀⠀⠀⡛⠛⠛⠒⠢
⢸⡇⢄⢸⡇⢠⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⢀⣿⠠⣿⠇⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠄


⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣰⡾⠛⠙⠻⣷⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣾⣽⠖⠲⡿⢦⠀⠀⠀⠀⠀⢀⣞⣉⡇⢹⠱
⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣟⣽⣿⣿⠅⠀⠀⠀⠀⢸⣸⡅⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣲⢖⡃⡺⢄
⠘⢿⣶⣴⣾⠗⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠻⢯⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢩⣷⣻⠗


⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⣤⣤⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣼⡆⢀⢈⣿⢀⠄⠀⠀⠀⠀⣠⡀⢁⢂⣠⣶⡤⠢⠀⠀⠀⠀⠀⠀⠀⢀⠐⠢
⣿⣿⣧⣤⣾⣿⡇⠀⠀⠀⠀⠀⢻⡇⠈⠘⡿⢀⠀⠀⠀⠀⠀⠸⠱⣦⡦⢫⢔⠔⠳⠄⠀⠀⠀⣯⣱⣾⣿⣶⣶
⠛⠛⠛⠻⠿⠟⠇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠛


⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢀⣀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⡇⣠⠿⢿⡃⠀⠀⠀⠀⢨⣭⢨⣭⠈⠉⠁⠀⠀⠀⠀⠀⢀⡀⣴⢰⣿⢸
⢿⣿⣯⣽⣿⣿⠇⠀⠀⠀⠀⠀⡿⠛⢾⣿⣦⣾⡇⠀⠀⠀⠀⠰⠶⠰⠶⠰⠶⠲⠶⠆⠀⠀⠀⢸⡇⣿⠨⣿⢸
⠀⠏⢀⠻⠛⠁⠀⠀⠀⠀⠀⠀⠿⠶⠿⠿⠿⠿⠃⠀⠀⠀⠀⠘⠛⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠀⠋⠘
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣟⣯⡙⠛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠟⠻⠿⣿⣿⣧⢹⠏⢫⡹⡇⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⢰⣿⣿⣿⣿⣿⣿⣎⣧⢀⡘⢙⣯⠀⠀⢸⡃⡄⠀⢀⠀⠘⠺⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⣿⣟⣿⣿⣿⠿⡿⣿⣿⣿⣿⣴⣿⣤⣦⣾⣱⣤⣀⣮⣴⠂⠀⣻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⣡⡦⢘⠻⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡻⣿⣫⣩⡷⠎⠀⠀⠀⢹⣿⣟⠻⠭⠉⠙⠛⠛⠿⢿⣿⣿⣌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠈⠙⠻⠻⣿⣽⣷⢦⣄⠁⠀⠻⣟⠻⢶⣦⣀⠀⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠘⠻⠷⢶⣦⣌⠐⠈⢢⡌⢠⣿⣷⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⡂⠀⠀⣼⣿⠻⣦⣤⡝⣦⡀⠐⣻⣾⡟⠉⠦⣀⠀⠀⠀⠀⠐⠲⣤⣀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡛⠉⡘⣧⠀⠀⠙⢷⣄⣸⣿⣟⣿⡻⠊⣝⡟⠃⠀⠀⠈⠂⠠⣁⠀⠀⠀⠀⠈⠉⠠⡘⣿
⠀⠀⠀⠀⠀⠀⠀⣤⠃⢁⠀⢀⡘⢳⢠⡄⠀⠈⠉⠉⠉⠘⠁⠀⣹⣆⠀⠐⢄⠐⠀⠀⢙⢷⣄⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠸⡻⢷⡈⠶⣤⢤⡛⢶⠀⠀⣾⣿⣿⣦⣀⣾⠿⠻⣿⠀⠀⠈⢳⡀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⢀⡼⡿⠊⠙⠀⠈⠂⢳⣟⢠⣠⢸⣿⣿⣿⠗⢾⣧⡀⠘⣆⠁⠀⠀⠘⠦⠀⠀⢿⡿⣿⡿⠆⠀⠀⠞
⠀⠀⠀⠀⠀⢻⣋⡟⠀⠀⠀⣢⣤⣤⣜⠀⡁⠈⠀⢉⢀⡄⡉⠛⠿⠆⢈⡳⣀⠀⠀⠀⠀⠀⢸⣷⢹⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠙⠢⣤⣀⣸⣿⣿⣿⣿⣷⠀⠀⢀⣠⣾⣿⣿⣶⣶⣿⣿⣿⣏⢷⡀⢀⢀⡌⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⢯⡻⣿⣿⣿⢿⢿⣧⣾⣶⣾⣿⣿⣿⣿⣿⣛⡛⣿⢿⡿⣧⣹⣦⢣⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠸⠶⠊⠁⠀⢻⡆⠀⠀⠨⡍⠙⣻⣿⣯⣿⣿⣯⣿⣿⢸⡇⠸⠇⢻⢾⣿⠃⡸⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡄⣿⣷⣘⠄⣐⣠⣲⡿⠟⠉⢁⠿⣹⣿⡧⢸⠇⠀⠀⠀⣸⠏⠀⠃⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣵⣿⣿⣿⣷⢫⢟⡟⡷⠂⡀⠃⠀⠻⠛⢀⡿⣠⠀⠀⢸⡏⠀⠀⠀⣠⡾⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠈⡟⠛⠑⣿⢻⡿⣷⣯⢟⡞⣽⠃⠆⠀⠀⣰⣶⡿⠁⢣⣀⣴⣿⠁⠀⢀⣾⠋
⠀⠀⠀⠘⠇⠀⠀⠀⡰⣧⠀⠀⣿⣄⠁⠹⣷⣏⣼⣱⣿⣤⠀⢠⣿⡟⡱⢃⣾⠉⣸⠇⠀⠀⣼⠃
⠀⠀⠀⠀⠐⣤⣴⣿⣿⡿⡄⠀⣿⡈⡄⠀⠙⠛⣻⣟⣿⣿⡄⠷⢟⠔⣵⣿⡏⢀⡟⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠁⣴⣿⣸⡇⠀⠀⢸⠿⣯⣿⣿⣷⠀⡊⠀⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡣⠁⠀⠀⢸⣿⣿⡙⣿⣿⡄⠁⠸⠏⠿⢷⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢋⠁⢄⡤⠠⠌⠶⣦⡹⣾⣿⣿⡀⠀⠀⠀⠸⣿⠃⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠠⣶⠠⠌⠀⠰⠀⠐⡦⢷⢿⣿⣿⡦⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⡡⠈⠴⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠄⢒⢒⠎⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⡿⠓⢫⡾⠀⠀⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣀⣺⣾⡇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣷⣄⠀⠀⣼⣿⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⢿⣷
⠀⠀⠀⣿⡏⠀⠀⠀⠀⣀⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⣼⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⡏⠉⣿⣿⣿⣶⣄⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢀⣴⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⣿⡇⠀⣿⣿⠀⣿⣿⣤⣼⣿
⠀⠀⠀⣿⡇⠀⢸⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⣿⣏⠀⣿⣿⠛⢿⣿
⠀⠀⠀⣿⡇⠀⠀⠙⢻⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⣿⣿⠀⣿⣿⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⢠⣾⡿⠿⣿⣿⠀⠀⠀⠀⣀⡀⠀⢸⣿⡇⠀⣿⡇⠀⣿⣿⠀⣿⣿⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣩⣽⣿⡿⠟⠀⠀⠀⠀⣠⣾⡿⠁⠀⣿⣿⠀⠀⢰⣿⡿⢿⣷⣸⣿⡇⠀⣿⡇⠀⣿⣿⠀⣿⣿⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢁⣀⣤⣤⣶⣾⣿⠿⠛⠁⠀⠀⠘⢿⣿⣾⣿⣿⡆⠈⢿⣿⣿⡇⠀⠉⠁⠀⠉⠁⠀⠈⠁⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟
⠀⠀⠀⠀⣠⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⢀⡀⠀⠀⡀⠀⠀⡀⠀⢸⣿⣿⣷⡀⠸⣿⣿⡿⣿⣷⣄⠀⠀⠀⣠⣶⣿⡿⠿⠛⠛⠉⢁⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⣾⣿⠀⢸⣿⠀⢸⣿⡏⢿⣷⣶⣿⠇⠀⠀⣿⣿⠀⢀⣾⡿⠋⠀⠀⠀⠀⣴⣾⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⣿⣿⠀⢸⣿⠀⢸⣿⡇⠀⠉⠉⠁⠀⠀⠀⣿⣿⣶⣾⣿⠃⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⣽⣿⠀⢸⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣷⣄⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣷⣤⣿⣿⠀⢾⣿⠀⢸⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⡿⠛⣿⣿⠀⣹⣿⠀⢸⣿⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠹⢿⣿⣿⣿⡀⣸⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠋⠁⠀⠀⠀⣸⣿
⠀⠀⠀⠀⢿⣷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⡟⠀⠀⠹⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣿⠟
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⢀⠀⣀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣿⣠⣽⣄⣿⡀⠀⠀⠀⢀⣽⣅⢸⡇⠀⠀⠀⠀⣠⣫⡀⣨⣧⡀⠀⠀⠀⠀⢀⠠⠚⠂⠀⠀⠀⠀⠀⡦⠀⣰⣦⠀⠀⠀⠀⢠⡄⣰⡄⢼⡆
⠀⠘⣿⠟⣿⠟⣿⠋⠀⠀⠀⠘⣿⡟⢸⡇⠀⠀⠀⠀⣻⣿⠓⢹⣿⠁⠀⠀⠀⠀⠚⠢⣀⠀⠀⠀⠀⠀⣼⣿⣤⣼⣯⣀⠀⠀⠀⢻⠇⢸⡃⠼⠄
⠀⠀⠛⠀⠿⠀⠛⠀⠀⠀⠀⠀⠿⠇⠸⠇⠀⠀⠀⠀⠙⠿⠀⠸⠟⠀⠀⠀⠀⠀⠀⠠⢽⠄⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠈⠁

⠀⠀⣄⠀⣀⠀⢠⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⡄⠀⠀⠀⠀⢀⠀⣶⡀⠀⠀⠀⠀⠀⢠⠀⣤⠀⡄⠀⠀⠀⠀⠀⠀⣤
⠀⠈⣿⠠⣾⡄⣹⡁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⢾⡸⣿⠇⡷⠀⠀⠀⠀⢺⡟⣿⢱⡧⠀⠀⠀⠀⠾⠶⠿⠶⠷⠀⠀⠀⠀⠀⣴⡶⣦
⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⣆⠀⣲⠀⣰⡂⠀⠀⠀⢸⣷⣴⣾⡇⠀⠀⠀⠀⠚⠓⠉⡷⠧⠀⠀⠀⢰⣼⣤⣿⣤⣧⡆⠀⠀⢀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠁
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⢺⠁⠄⠱⠂⠀⠀⠀⠀⣤⢀⣄⢠⡄⠀⠀⠀⢀⣼⣧⡀⠀⠈⠀⠀⠀⠀⠀⢰⣷⠄⠀⠀⠀⠀⠀⠀⠃⢀⠘⠀⠀⠀⠀⠀⠀⢐⣻⡀
⠀⢀⣸⣷⣷⣿⡘⠀⠀⠀⢴⡷⣦⣭⣾⣷⠄⠀⠀⠘⣿⡿⠁⠃⠘⠀⠀⠀⠀⣤⠄⡿⢠⡄⠀⠀⠀⠘⠂⠠⠯⠀⠐⠂⠀⠀⢰⣶⣿⣿⣿⣶⡆
⠀⠀⠘⠓⠗⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠴⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠀⠀⠀⠀⠘⠃⠛⠀⠛⠘⠃

⠀⢠⣦⣤⣤⣤⣤⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⣤⠀⠄⠐⠀⠄⠀⠀⠀⠀⡀⣀⢀⢀
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣤⠀⢠⣄⠀⣤⡀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⠀⠀⠀⠀⢰⣶⣆⣀⣀⣀⡀⠀⠀⢰⣷⣷⣿⣾⣾⡆
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣀⡀⢀⣀⠀⣀⡀⠀⠀⢐⣆⢀⣆⢀⣖⠀⠀⠀⠀⢸⣿⣿⣿⠂⠀⠀⠀⠈⣿⠀⠀⠾⠆⠀⠀⠀⠈⠁⠃⠿⠘⠈⠁
⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠀⠀⠀⠀⠀⠉
⠀⠀⣀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⣿⣴⣷⣴⣧⠀⠀⠀⠀⡺⠋⡭⠻⣆⠀⠀⠀⢀⣼⡉⣍⢹⣇⠀⠀⠀⠀⢴⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⢀⣛⡀
⠀⠸⠿⢿⣿⡿⠿⠀⠀⠀⠈⢳⣼⣿⣦⡏⠀⠀⠀⠈⣿⣶⣷⣾⡏⠀⠀⠀⠀⠸⠋⡇⢀⡄⠀⠀⠀⠀⠀⠀⣀⡘⠃⠀⠀⠀⢠⣷⣸⣿⢡⣾⡄
⠀⠀⠀⠺⠿⠃⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠸⠃⠿⠸⠃⠀⠀⠀⠀⠶⠆⠿⠸⠇⠀⠀⠀⠀⠤⠄⠀⠄⠄⠀⠀⠀⠈⠉⠁⠛⠀⠉⠁

⠀⠀⡠⠖⠒⠦⡄⠀⠀⠀⠀⣶⡶⠐⣶⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣆⣖⣰⡂⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⢸⢴⠠⣦⢰⠼⠀⠀⠀⠀⠿⠛⠒⠻⠇⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⢹⣟⣛⣻⡇⠀⠀⠀⢰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠂
⠀⠘⢿⣷⣷⣿⠟⠀⠀⠀⠀⣢⣈⣌⣡⣔⠀⠀⠀⢀⣶⣀⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠈⠧⣼⣤⣷⢼⡇⠀⠀⢀⣷⣿⣿⣿⣾⡀
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠐⢒⡒⠒⠒⠒⠆⠀⠀⠀⢠⠒⠒⠒⠒⠒⢦⠀⠀⠀⠀⠀⡴⠲⠿⠖⢢⠀⠀⠀⠀⠀⢠⣶⣿⡷⠐⠄
⠀⠀⠀⠻⠷⠀⠀⠀⡆⠀⠀⠀⢸⠀⠤⠤⢾⠉⣛⠀⠀⠀⠀⠀⡟⠿⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣣⣄⣈
⠀⠀⠼⠿⠿⠽⠿⠿⠧⠀⠀⠀⠈⠉⠩⠿⠏⠉⠙⠂⠀⠀⠀⠀⣧⣤⣤⣶⣸⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣠⠶⡶⢦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠆
⠀⠀⡇⠀⣄⢀⣄⡀⣿⠀⠀⠀⠀⣿⠒⣿⢚⡇⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣤
⠀⠀⣇⣺⣛⣟⣟⣇⡿⠀⠀⠀⠀⠈⠓⠛⠛⢧⡀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠋⠀⠀⠀⠀⠀⠀⠺⠛⠒⠛⠻
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁

⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⣤⣀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⣀⣤⣤⣠⡀
⠀⠀⣀⣰⣋⡠⠿⣶⡶⠀⠀⠀⠀⣰⠏⠁⠀⢛⢳⡄⠀⠀⠀⠀⢘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠴⢒⣄
⠀⠀⠉⠉⠻⠥⠴⢇⠀⠀⠀⠀⠀⢿⡠⠶⠴⠿⣸⠇⠀⠀⠀⠀⠸⠴⢤⡤⠦⠄⠀⠀⠀⠀⠀⡟⢖⣛⣜
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠈⠛⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠰⠾⠷⠀⠀⠀⠀⠀⠀⠀⠳⠶⠴⠶⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⡿⠿⠿⠿⠿⠿⣟⠀⠀⠀⠀⡞⠉⣩⡿⡆⠀⠀⠀⠀⠀⠀⣿⡿⣿⢿⣿⠀⠀⠀⠀⠀⡠⡼⠩⠍⢧⢄
⠀⠀⡤⣤⣤⣔⣩⣕⣾⠀⠀⠀⠀⢷⣛⣛⣳⡃⠀⠀⠀⠀⠀⠀⣿⣁⣉⣀⣿⠀⠀⠀⠀⠀⠂⡇⠐⡂⡰⠐
⠀⠀⠦⠤⠤⠬⠵⠭⠮⠀⠀⠀⠀⠀⠈⠉⠀⠹⣦⠀⠀⠀⠀⠀⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⠭⣈⣣⠴⠃

⠠⣀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡶⢀⠀⠀⢀⡄⠀⠀⠀⠐⠀⠀⡆⠀⠀⠀⡄⠀⣀⣀⠀⠀⠂
⠀⠀⠀⠀⢠⠄⡄⠀⠁⠀⠀⠄⠀⡂⠈⠰⠃⠂⠠⠀⡌⠁⠀⠀⠀⢺⡠⠀⠁⠀⡀⠐⠂⠈⠀⠡⠄⠁
⢠⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⢃⠀⢀⡀⣠⡀⠀⠀⢲⣖⣾⠀⠈⣻⠖⠀⠐⠀⠀⢁⠈⢀⠁⠁⠐
⠈⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⠀⢸⠘⠐⠀⠀⠉⠀⠀⣠⠟⠛⠛⠀⣄⠉⠀⠘⠃⠀⠀⣤⢀⡄⣤
⠀⠰⠆⠀⠘⠀⠀⠀⠀⠀⠀⢀⠸⠐⠀⢦⠐⠄⠀⡀⠀⠰⠐⡀⠀⢄⠀⡄⠂⠀⠀⠀⠁⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠉⡐⠀⢴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⣌⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠰⠂⠀⠀⡦⡆⠀⠀⠀⡰⠀⠀⠅⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠆⣀⢈⠳⡷
⠀⠒⠀⣀⠉⠁⠀⠀⠀⠁⠀⠶⠀⠀⠄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠃⠠⠐⠀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘
⠀⠀⠀⠀⠀⠀⢆⠀⠀⢈⠀⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂
⠀⠀⡀⠀⠀⠀⢀⠀⠐⠀⠠⠄⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⠀⡀⠄⠀⠃⢘⠀⠀⠀⠀⠐⠒⠙⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠐⠟⠐⠀⢘⠃⠘⠁⠬⠁⠀⢠⠀⠀⢠⠄⠀⠀⠀⠀⠀⠐⠀⠃⠘⠐⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠘⠀⠸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⠛⠀⣿⢛⣷⠀⢻⡟⠘⢻⡟⠛⠘⣿⠃⣴⠟⠛⠃⢠⡿⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⠀⣿⠙⢧⡀⣼⣧⠀⠸⡧⠀⢠⣿⡄⠻⣧⣤⡄⣾⠓⢻⡆⢸⣧⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡶⠒⢰⣆⢀⣶⠐⢶⡖⢠⣶⣄⢰⡄⣶⣀⡴⠂⢲⣶⠂⣶⣦⢰⡆⢠⡶⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠙⣿⢀⣼⣇⢸⡇⠹⣿⡇⣿⡛⣧⡀⣸⣿⡀⣿⠘⢿⣇⠸⣧⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠁⠀⠉⠉⠀⠁⠀⠈⠀⠈⠀⠈⠁⠉⠉⠀⠈⠀⠀⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠦⣤⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠏⠀⠀⠀⣠⣤⡉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢁⡼⠃⠀⠀⠀⠀⠘⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣙⣁⣙⣛⡛⠃⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⡴⠶⠶⢦⡀⣠⠶⠶⠶⣄⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠙⠳⠶⠶⠈⠳⠶⠶⠾⠃⠙⠶⠶⠶⠋⠶⣶⣚⡁⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣞⠛⠓⠒⠒⠒⠛⠛⠛⠲⢾⠩⠍⠁⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⢯⡙⢦⣄⣈⠉⠙⢛⣶⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⡉⠉⠋⠉⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⣀⣀⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢶⣶⣤⣤⠴⠖⠋⠉⠛⣛⣃⣀⣀⣀⣀⣀⣀⣸⣃⣀⣀⣀⣀
⠀⢀⡤⠞⣋⣭⣭⣍⣛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠉⠉⠉⠉⠉⠉⣩⡉⠉⠉⠉⠉⠙
⢠⠟⣴⠋⠁⣀⣄⡀⠉⠳⡌⢧⠀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⡞⠁⣿⠀⠀⠀⠀⠀⢀⣼⠉⣹⠦
⡟⢸⠃⢠⣞⠉⠉⠙⣧⠀⢹⡜⡇⢠⡴⠶⠶⠶⢦⡀⢸⡇⠀⠀⠀⠀⠀⣾⠀⠀⣿⠀⢀⡀⠀⠲⣅⣨⠏⠁⠀⠀⣀
⣧⢸⡀⠘⣯⡀⡀⣀⡿⠀⣸⢣⡇⠈⠓⠒⠒⠒⠛⠀⢸⡇⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠺⠽⢳⣀⡘⠃⠀⣀⣰⠾⠿⠇
⠘⣆⠳⣄⠈⠉⠛⠉⢀⡴⢋⡾⠁⢙⣛⣛⡛⢛⣛⠀⢸⡇⠀⠀⠀⠀⢠⡇⠀⠀⣿⠀⡴⣦⣬⣭⡅⠀⠀⣨⣭⣤⡴⣦
⠀⠈⠳⢬⣙⣓⣒⣛⣩⠴⠋⠀⠀⠰⠶⠶⠶⠦⠀⠀⢸⡇⠀⠀⠀⢀⡞⠀⠀⠀⣿⠀⠉⡉⢠⡤⠄⠀⠀⠤⢤⠀⣉⠁
⠀⠀⠀⠀⠈⣻⣹⡏⠀⠀⢀⣙⣛⣛⣛⣛⣛⣛⣛⠀⢸⡇⠀⠀⠀⠘⠓⣶⠀⠀⣿⠀⠺⠽⠛⠁⠀⠀⠀⠀⠘⠻⠿⠇
⠀⠀⠀⠀⢸⣯⠉⣿⠀⠠⠠⠄⠆⠀⠀⠀⠀⠰⠶⠀⢸⡇⠀⠀⠀⢀⡤⣿⠀⠀⠈⠓⠒⠒⠒⠒⠒⣦⡶⠒⣲⠖⠒⠚⠛
⠀⠀⠀⠀⢸⡇⠀⣿⠠⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠟⢷⡀⣠⠞⠀⠘⢦⣀⣀⣀⡀⠀⠀⠀⠀⠈⠀⢰⡇
⠀⠀⠀⠀⢸⡧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠉⠉⣧⠀⠀⠀⠀⠀⠀⠘⣇
⣤⣤⣤⣤⣬⣷⣶⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠐⢲⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⠈⡌⢫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣖⣒⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⢔⡳⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⣄⠀⢘⡙⠛⡛⡋⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⣀⢍⠉⣡⢠⠄⡄
⠀⠀⠀⠀⠀⡯⢸⠇⠣⠅⠹⠻⠃⢱⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠟⠡⡘⠄⠏⠀⡇⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣄⣤⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⡿⣿⢻⡿⣿⢻⡿⣿⣻⣿⢿⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣽⣽⣷⣿⣸⣿⣶⣾⣷⣿⣾⣷⣽⣾⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⠟⠛⠻⠿⠿⠟⠋⠈⠉⠉
⠀⠀⠀⠀⠀⣿⣿⣿⣥⣽⣿⡇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣯⣷
⠀⠀⠀⠀⠀⣀⠈⠀⡀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢿⣷⣿⣾⣭⠟
⠀⠀⠀⠀⠀⠎⠰⠴⠳⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢶⠖⠓⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣀⣶⣙⠀⠀⠀⠀⠀⢸⡎⣿⡀⣼⣀⢺⢣⠹⡄⢹⢔⡦⣿⡀⣼⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡍⠿⠉⠀⠀⠀⠀⠀⠘⠁⠁⠉⠀⠉⠈⠈⠀⠁⠈⠀⠀⠁⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠒⢰⡄⣀⢠⡔⠐⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠢⠻⠃⠸⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢸⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠘⡇⠀⠀⠘⠀⠀⠀⡟⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⣀⡴⠖⠛⠛⠛⠓⠶⣄⡀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⡀⠀⣠⠞⠁⣀⡀⣰⡖⣶⡀⣀⡀⠙⢦⡀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⠏⠀⠸⣏⠙⣩⠤⢬⡙⢉⡿⠀⠈⢷⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⡀⣿⠀⢰⠞⠃⡞⠁⠀⠀⠙⡆⠓⢦⠀⢸⡇⣠⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⠳⣆⠳⡀⠀⠀⡰⢃⡶⠋⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠘⣧⠀⠸⣧⡴⢬⠉⢩⡴⣤⡿⠀⢠⡟⠰⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣈⢳⡀⠀⠀⠘⠓⠛⠀⠀⠀⣴⢋⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢹⡄⠀⠀⠀⠀⠀⠀⣼⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣛⣛⣛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣙⣛⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠠⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⣿⠶⠶⠶⠶⠶⢦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢛⣻⣛⣟⣻⣿⣉⣟⣛⣿⣛⣿⡛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣅⡀⠀⠀⠀⣀⣤⣀⠀⠀⠀⢀⣨⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠺⢿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡿⠗⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢰⣿⣿⣿⡆⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣏⠀⠀⠀⣹⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣈⣙⣟⣉⡀⠀⢀⣙⣛⣋⡀⠀⢀⣉⣛⣋⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠦⠤⠾⣿⣿⡿⠶⠤⠶⢿⣿⣿⠷⠤⠴⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠛⠛⠛⠛⠋⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣤⣤⣤⣼⣿⣦⣤⣤⣤⣴⣿⣧⣤⣤⣤⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠂

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣶⣆⠀⠀⠀⠀⢴⣧⡀⠀⠀⠀⢰⣧⡀
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡈⠉⠀⠀⠀⠀⠠⠋⠁
⠀⠀⠀⠀⠈⠃⠐⠀⠀⠀⠘⠋⠃⠀⠀⠀⠐⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣴⣶⣦⠀⠀⠀⠀⠀⠀⣴⣿⣅⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⠉⠁⠀⢀⣠⣀⠀⠀⠉⢉⡹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢠⣤⠏⠉⠉⣷⣄⠾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠘⠛⣧⠀⢰⡟⠋⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⡆⢀⡀⢿⣦⠿⣶⢀⠀⠘⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⢼⡇⠁⠁⠀⠀⠀⠀⠈⢐⣀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠶⠿⠿⠇⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿





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

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠈⠘⠈⠁⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⡀⠀⠁⠀⠀⢆⠀⠁⠈⠈⠀⠀⠀⠀⠰⠀⠀⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠈⠘⠉
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⡄⠁⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⠈⠁⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣠⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⠀⠈⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣠
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡁⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⢈⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⡁⠀⠈⠁
⠀⠀⠀⠀⠀⡄⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠
⡄⠀⠀⠀⣤⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⢠⠄⠀⠀⠀⠠⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⡤⡔⠀⠀⠀⢤⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠐⠒⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠲⠲⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠐⠲
⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠁⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⢐⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣀
⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠂⠀⠙⠃⠁⠀⢀⣠⣽⠃⣤⠄⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠐⠀⠈⠛⠉⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠂⠀⠙⠋
⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⣀⣀⣠⣌⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡀⠀⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠛⠟⠋⢈⣿⣿⣿⣿⣿⣆⡆⠀⠂⠈⠀⡀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠁
⡀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣶⣿⣄⣲⣤⣀⠘⠳⠚⠁⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠉⠒
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠅⠛⡋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⡀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡦⠀⠀⠀⠀⠁⠈⠀⠛⠋⠉⣀⠀⠃⠀⠐⠀⠰⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠷⠐⠖⠹⠆⠀⠤⠐⠂⠀⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣤⣤⣀⣀⢀⠀⢠⠈⠉⢹⣿⡿⠭⠤⠤⠀⠀⠀⠀⠀⠀⢀⣀⠰⢞⣁⣀⠀⢠⡄⠀⠀⡀⠀⠌⣤⣤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢠⠠
⣽⣿⣿⡿⠛⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣝⠷⠀⢀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠱⠄⠀⠀⣀⠰⠀⠀⠀⠀⠈⢢
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠂⠉⠛⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈
⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣷⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⡿
⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄⢴⣼⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢦⠀⠀⠀⠐⣶⡄⠐⠶⣶⣦⡠⠶⠶⣶⣦
⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠸⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠿⢿⡯⠘⢹⡎⠙⢿⣏⠒⠠⡄⠀⢠⠽⣆⡀⠈⠻⣯⠄
⢉⡵⢿⡀⣿⣿⣿⡿⠋⠋⠉⢩⣿⣿⣿⣿⡾⠛⠃⠀⢸⣿⠟⠉⣻⣿⣿⣏⢩⣌⢸⣿⣿⣷⣶⣶⣶⣷⡀⠀⠀⠀⠀⠹⠄⠄⠹⡄⠀⠴⣩⣿⡄⠀⠈⠙⣦⣴⣶
⢹⣶⣿⡆⢻⡿⠋⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⣾⣿⣷⣾⣿⣿⣿⣿⡟⡏⠀⠛⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣾⣷⣤⣻⣄⠀⠈⠃⠙⢧⠀⠀⠈⠛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠇⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⡆⠀⣿⠀⠀⠀⠀⣠⣾⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣠⣾⠟⣳⠀⠀⣰⡾⣛⣭⣽⡻⣷⡄
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡄⠀⣿⡇⠀⣿⠀⠀⠀⣸⣿⣿⣿⢟⣭⡻⣿⣿⡄⠀⠀⠀⣿⡿⠿⠿⠿⣿⣫⣾⠟⠁⠚⠁⠀⢰⣿⣸⣿⣿⣿⣿⡞⣿
⠀⠀⠀⢰⡆⠀⣿⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⢻⣿⣿⣿⠻⠿⢇⣼⣿⣶⠀⠀⠀⣿⡿⠿⠿⣿⣣⣬⣷⠀⠀⠀⠀⠀⠘⣿⡜⣿⣿⣿⡿⣳⡿
⠀⠀⠀⢸⠇⠀⢿⠀⠸⡇⠀⢿⠇⠀⠿⠀⠀⠀⠈⢿⢟⣵⣿⣿⣿⢟⣱⠏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⣷⠾⢛⣵⣤⡀
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⠙⠓⠲⠚⠋⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢄⠀⡀⠀⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⠿⣉
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠄⠀⠀⠀⠀⠀⢀⣴⣶⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⡠⢟⠟⠡⣄⠜
⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⡀⠀⠀⢀⣀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣀⣀⠈⠿⠿⠿⠟⠁⠀⠀⠀⠀⣀⣤⡔⠁⠀⠂⢀⡷⠁
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠰⠀⠀⠀⠀⠉⠉⠙⠛⢿⣿⣁⣠⣄⡀⠀⠀⠀⡿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠇⠀⢀⣴⠟⣡⡄
⠀⠀⠀⠀⠙⠿⠿⡿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠖⠛⢻⠛⠉⠁⠀⠀⠀⠃⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⡦⢴⣯⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠰⠀⠀⠀⠀⠀⠻⢾⠶⣆⣐⡦⠀⢘⠛⠳⠆⠀⠀⠠⠀⠈⠐⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠂⣶⣲⡆⢰⡖⠒⣶⠒⢰⡖⢰⠞⠒⢀⡶⡆⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠆⠻⠙⠇⠸⠷⠀⠻⠀⠸⠗⠘⠷⠖⠸⠛⠻⠘⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⣀⠀⣄⢠⣤⢀⣄⢀⡄⣀⢀⡄⢠⡄⢠⡀⢠⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠛⣿⢀⣇⢸⡟⢿⡇⣿⢿⡀⣸⡇⢸⠙⣾⠘⣇⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠞⠉⠀⣀⣙⠋⠉⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠁⠞⠀⠀⠀⠀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠄⠠⠤⠄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣠⠤⢭⡉⣩⠤⢤⡀⠀⠀⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠈⠓⠒⠂⢓⡒⠒⠃⠛⢒⠒⠃⣶⣞⡁⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⡤⣄⠀⠻⢬⣭⣍⠉⠉⠉⠉⡇⠂⠀⠀⠀⠈⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠙⢮⣙⠒⠒⠒⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣓⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠤⠴⠒⠋⠙⣛⣃⣀⣀⣀⣀⣀⣋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡭⠖⠒⠲⢝⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇⠀⠀⠀⠀⣠⢦⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡏⢠⠖⠛⢲⡀⢳⢹⡈⣉⣉⣉⣉⠉⣿⠀⠀⠀⠀⢰⠃⢸⡇⠀⠀⢠⡞⣳⠞⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⡇⠹⣅⣀⣸⠇⢸⢹⠃⠓⠒⠒⠚⠁⣿⠀⠀⠀⠀⣿⠀⢸⡇⠺⠷⣆⡙⠁⢀⣰⠾⠗⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢦⣈⣉⣠⠴⣣⠏⠀⠭⠭⢉⠭⠀⣿⠀⠀⠀⢀⠏⠀⢸⡃⢶⠶⠴⠄⠀⠀⠦⠴⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⢶⠖⠋⠁⠠⢤⡭⢭⡭⢤⠀⣿⠀⠀⠀⠾⣤⠀⢸⡇⢴⡦⠏⠁⠀⠈⠹⢴⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⡧⠀⠈⠉⠉⠉⠉⠉⠭⠀⣿⠀⠀⠀⣠⢸⠀⠈⠓⠒⠒⠒⠲⣤⠖⣲⠖⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠰⠒⠒⠒⠒⠒⠒⠒⠲⠟⠙⢦⡜⠁⠈⠳⠤⢤⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠐⢀⣠⡄⠀⢀⠤⣤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣼⣿⣦⣄⣀⣼⢠⠁⣠⠛⠒⠪⣕⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠸⣿⣿⣇⣿⣿⣿⣿⣿⣿⠀⠀⡆⠀⠀⢸⠹⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⠐⣾⣿⣿⡯⠿⣿⣿⣿⣿⡟⠀⢤⠃⠀⠀⠚⢴⠃⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣿⣿⣧⠤⠼⠧⠀⠀⠀⢸⠀⠈⠹⡾⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⠧⣀⣸⣿⣿⣿⠀⠀⣯⣿⡿⠛⡇⠀⠀⠀⠔⠚⠋⠉⠓⢄⠀⢷⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⠄⠠⠬⣍⣛⡟⠀⠀⣿⣿⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⢹⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠁⠙⠊⡌⠣⡀⢀⣿⣿⡇⢸⡷⠖⠒⠒⠮⠀⠀⡀⢀⡦⠴⠃⣰⠞⢳
⠀⠀⠀⠀⠀⠀⢸⣯⢁⠎⠀⠐⠄⠀⠀⠀⢘⡞⠛⢻⡇⢸⡇⠀⠀⠀⠀⠀⢀⡍⠉⠁⠀⠀⠻⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠌⠙⢀⠔⠈⠀⠀⠀⠘⣷⠀⠈⡇⢸⡇⠀⠀⠀⢰⡴⠊⠀⠀⠴⠒⠢⡄⠀⢼⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢄⠤⠐⠀⠀⠀⠹⡄⠀⠃⢸⣷⠶⠶⠶⣏⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⣽⠀⠀⠀⠀⣤⣤⡇⢀⠄⠀⠁⠀⠀⢸⡇⠀⠀⢀⠈⠙⠒⠒⠒⠋⠀⣶⠋⣸⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⣷⣷⡇⠈⠒⠀⢈⠀⠀⢸⡇⠀⠀⣸⠷⠤⠤⠾⠦⠤⠤⠻⣦⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠧⣍⡇⠀⠄⠀⢸⣴⣷⣼⡇⠀⣰⠏⠀⢰⣄⣠⠖⠒⢦⡀⢘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢝⣦⢬⡑⣴⠁⣠⠂⢀⡏⢻⣿⣿⣧⠾⠯⡤⠔⠋⠈⣷⠀⠀⢀⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡙⣧⡰⢋⣴⣾⣷⣿⣿⣿⡇⠀⠀⢤⣄⣴⠾⠋⢀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡿⣿⣿⣿⣿⠏⡷⠶⠛⡋⠉⢀⣆⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠋⠀⠘⠦⠶⠿⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣶⡄⢸⣿⣿⣿⣿⣿⣿⡇⢠⣶⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠻⣿⣿⣿⣿⠟⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠁⣀⣤⣤⣬⣥⣤⣤⣀⠈⠻⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⡶⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢶⣶⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣆⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⣸⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⢀⣠⣾⠋⠁⠀⢩⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀⠀⠀⢀⣀⠉⠛⠻⠿⠿⠟⠛⠉⣁⡀⠀⠀⢀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣴⠟⠛⠻⣶⠀⠀⠀⠀⣶⠟⠛⠻⣦⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣄⣀⣀⣠⣿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢠⣄⣀⣉⡤⣠⣄⡄⣄⣤⣀⣀⠀⠀⣄⣤⣨⣄⣤⣠⣤⣴⡄⣤⣀⣤⠀⢀⡤⣠⣤⣠⢤⣌⣠⣡⣤⢄⣠⡄
⠀⢻⣼⢧⡿⣇⢽⡆⡇⡿⣿⢿⢿⠀⠀⠛⠛⠛⠋⠿⠙⠸⠛⠇⠿⠛⠿⠀⠘⠧⠻⠏⠹⠼⠿⠛⠛⠻⠝⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⣲⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠼⡟⠉⠙⣿⡆⠀⠀⠀⠀⠀⠀⠄⣠⠶⠋⠑⢢⠠⠀⠀⠀⠀⠀⠀⡠⠷⡤⢶⠙⣇⢾⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⢰⠋⢀⣀⡀⠀⡇⠀⠀⠀⠀⠀⠀⢣⠀⡵⢫⡄⠓⣄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠂⠑⣤⣤⣤⠜⠐⠀⠀⠀⠀⠀⠀⠈⢺⠉⡉⠩⡏⢸⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠦⠴⠃
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡦⣴⠦⡆⣤⡄⢰⢤⡶⠀⠀⢠⡖⡴⣶⣤⡶⣦⣦⡶⣖⣴⣶⣦⢶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⡿⡧⣏⢬⡷⢧⡟⣧⢸⠣⣽⠆⠀⠀⠉⠙⠉⠉⠉⠙⠙⠉⡫⠍⠋⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⢴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣀⣀⣠⣼⣽⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣋⣭⢭⡴⡖⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢾⣞⣟⣿⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣯⠿⠾⠟⠓
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⣿⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠻⢿⣿⣿⣿⠀⠀⠀⠋⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣌⣉⣉⣀⣤⡴⠟⠉⢳⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣤⣤⣉⣉⡁⠀⢠⣾⣦⡙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣦⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠙⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠘⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠃⠀⠈⠛⠛⠛⠛




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡟⢷⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡟⢿⡇⢹⡇⢸⠆⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣿⣭⣭⣭⣭⣭⣭⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⠻⠃⠘⠁⠘⠁⢸⡏⠉⣉⠉⠉⠉⠉⠉⠉⢻⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⣷⠟⠛⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠉⠁⢀⡾⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣥⣤⣤⣤⣤⣤⣤⣾⣦⣄⡀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢶⣄⠀⠙⢷⡄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠉⠻⠶⡾⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡶⠶⣦⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⣤⣤⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢷⣄⠀⠙⠷⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠃⠀⠀⠈⠛⠻⡿⠛⠛⠛⠛⠛⠛⢛⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿
⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⣠⡾⠁⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⣄⣴⠿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢠⣧⣀⣀⣀⣀⣀⣀⣉⣀⣸⡇⢠⡄⢀⡄⢰⣦⢸⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣛⣛⣛⣛⣛⣛⣿⡟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡇⢸⡇⢸⣇⢸⣷⣼⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⢷⣼⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⡖⠒⢲⠀⠀⠀⠀⠀⠀⠀⢀⠤⢒⠂⢐⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⢰⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⢠⢁⠞⠁⡠⠄⠈⠢⠑⡄⠀⠀⠀⠀⠀⡀⠤⡀⡀⠿⠸⠸⢂⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠠⢤
⠀⠀⠀⠀⠀⡂⡯⠭⢽⠀⠒⠒⡆⠀⠀⠀⠀⢘⠁⡆⠀⡠⠄⡇⠀⠀⠀⠀⠀⡼⠤⠂⠘⡈⠖⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⠠⠠⠤⠠⠤
⠀⠀⠀⠀⠀⠄⣏⠉⣿⢰⣄⣠⡇⠀⠀⠐⡈⢮⡀⠅⠀⣇⡴⢠⠂⠀⠀⠀⠀⠈⣄⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠑⠂⠠⣄⠔⡱
⠀⠀⢸⠈⢹⠅⠏⠉⡿⠸⡏⢀⠇⠀⠀⠀⠈⠢⠍⣁⡉⠡⠴⡣⠑⠄⠀⠀⡔⠁⠀⠎⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⠗⠠⠀⣈⣀⠠⠊
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠈⠀⠈⠈
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠔⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⡒⠀⠀⠀⠠⠂⠀⠀⠀⡠⡀⠈⠆⠀⠀⠀⠀⠀⠀⢀⠄⣒⣂⠠⣀⠆⠸
⠀⠀⠀⠋⠀⠊⠀⡄⡂⣀⢇⠀⠀⠀⠀⠀⠀⢀⠔⣉⠢⠔⡠⠲⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⣇⣠⠃⠀⠀⠀⠀⠀⡐⡐⢁⢄⢀⠐⡌⠎⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡆⠆⠀⠀⠀⠀⠴⡡⠊⠀⠑⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠜⢌⠀⠃⠁⠗⠀⠀⠀⠀⠀⡁⢇⠔⠢⡥⠊⡀⡀
⠀⠀⠀⠀⠀⠀⠈⠊⠤⠌⣜⢀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡂⠀⠀⠀⢄⠙⠪⣂⠀⠀⣊⡔⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠀⠀⠤⠤⠧⠠⠤⠸⠤⠤⠧⠤⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠒⠁

⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⢩⢩⡍⣭⠉⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⠐⠂⠐⠒⡄⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⣔⡄⢀⡢⠈⠀⠀⠀⠀⠰⠒⠒⠒⠒⠖⠶⠲⠖⠲⡁⠀⠀⠀⠀⠀⣀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠰⠳⠣⠇⠶⢰⠐⠞⢆
⠀⠀⠀⠀⠀⠙⠀⠘⠝⠀⢀⠀⠀⠀⠀⢀⣘⣈⣃⣛⠄⠀⠠⡐⣄⠇⠀⠀⠀⢀⣀⣀⡀⠁⣀⣀⣀⠀⠀⠀⠀⠆⠤⠴⠤⢤⢔⠢⢑⠾
⠀⠀⠀⢆⣀⣀⣀⣄⣀⣄⣀⠲⠀⠀⠀⠀⢤⠠⠄⠈⠀⡀⠀⠐⡈⡆⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⠡⡴⡤⡄⣼⢀⠀⠀⡁⠁
⠀⠀⢘⠘⠒⠒⠒⠒⠒⠒⠒⠓⡆⠀⠀⠈⠀⠀⠀⠓⠉⠒⠐⠚⢝⠇⠀⠀⡏⠉⠈⢰⠁⠀⠁⠈⠀⠈⠀⠀⠀⠀⠈⠳⠥⠬⠶⠥⠄⠪⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠁⠁⠈⠁⠈
⠀⠀⠀⢀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠈⠔⠈⠉⠁⢢⠀⠀⠀⠀⠀⠀⠱⠆⠒⠀⠂⠒⠀⠐⠂⠶⠃⠀⠀⠈⡤⠤⠄⠠⠤⠤⢬⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠀⢀
⠀⠀⢠⠤⠥⠔⠠⡀⢰⠠⠖⢄⡀⠀⠀⠀⠀⣀⣀⠉⠀⠀⠔⡄⠀⠀⠀⠀⠀⠀⠀⡄⢀⢄⠀⡰⡀⠀⠀⠀⠀⠀⠀⠀⣽⢛⣙⣫
⠀⠀⠘⠒⠒⠢⠐⠁⢠⠒⠦⠊⠁⠀⠀⠀⠀⣃⣸⣀⣖⣲⣀⡇⠀⠀⠀⠀⠀⠔⡆⡇⠀⣄⡀⢌⠘⠀⠀⠀⠀⡄⠀⠀⢤⠀⠀⠠⠠⠀⠠
⠀⠀⠀⠀⠢⠠⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠤⠤⠤⠤⠎⠀⠀⠀⠀⠈⠁⠁⠘⠝⠛⠫⡞⢄⠀⠀⠀⣇⣋⣀⣸⠈⠁⢃⣉⣀⣀
⠀⠀⠀⠠⠄⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠤⠈⠤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠎⠁⠉⠱⠀⠀⠰⠉⠈⠱

⡿⢿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡿⣿⣿⣿⣿

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

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣤⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢋⣿⣿⣏⢿⡁⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠔⠶⠆⠀⠀⠀⠀⠀⠀⢀⣻⣃⢀⣛⡀⣘⣃⡀
⠀⠀⠀⠀⡀⠀⠈⣭⠁⠀⢀⡀⠀⠀⠀⠀⠀⠸⢧⠘⢿⣿⠟⢸⠿⠀⠀⠀⠀⠀⠀⠸⣈⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠿⠛⠻⠛⠛⢧
⠀⠀⠀⣠⡄⠀⠀⣤⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣺⣧⣄⣇⣴⣿⡂⠀⠀⠀⠀⠀⠀⠀⠉⠉⡱⢂⡔⢄⠀⠀⠀⠀⠀⢀⣇⣿⡃⢀⣿⡀⢘⣟⣸⡀
⠀⠀⢠⣬⣥⡄⣤⣭⣤⢀⣬⣥⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣬⣧⡬⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠈⠉⠉⠀⠈⠉⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⢠⣶⡄⢀⣶⡆⢀⣴⡄⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠄⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⡤
⠀⠀⠀⣤⣽⣅⣤⣍⣤⣠⣝⣥⡀⠀⠀⠀⠀⣠⣬⣡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⠀⡘⠛⠁
⠀⠀⠸⣿⣿⣿⣿⣿⣯⣿⣿⣯⡇⠀⠀⠀⠀⣿⣿⣿⣽⠇⠰⠆⢰⠆⠀⠀⠀⠀⠀⠀⣀⠈⣿⡏⠀⣀⠀⠀⠀⠀⠀⠀⣲⡀⠠⣀⣿⣀⠀⢀⣖
⠀⠀⠀⢸⣿⡇⢸⣿⡏⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⣯⡇⠘⠿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠉⠛⠃⠀⠀⠁
⠀⠀⠀⠸⠟⠇⠸⠿⡇⠘⠟⠇⠀⠀⠀⠀⠀⠈⡿⢿⠀⠀⠀⠆⢰⠆⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠠⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠾⠻⢦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠰⡆
⠀⠀⠀⠠⠀⠀⠀⠶⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠶⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡄⢠⣦⠀⣼⣧⡄⠀⠀⠀⠀⠀⣸⣇⠀⣼⣧⡀⢿⣿⠇
⠀⠀⠰⠿⠆⠀⠰⠿⠆⠀⠰⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣴⣵⣦⣿⣿⠁⠀⠀⠀⠀⠐⢿⡿⠂⢹⡟⠀⣸⣿
⠀⠀⠀⠐⠀⠀⠀⠒⠀⠐⠀⠂⠀⠀⠀⠀⠐⠰⠎⠠⠀⡶⠀⠄⠰⠂⠄⠀⠀⠀⠀⠘⣿⡟⢻⣿⢻⣿⡟⠀⠀⠀⠀⠀⠀⣸⣇⠀⠚⠛⠄
⠀⠀⠘⠛⠃⠀⠘⠛⠃⠀⠘⠛⠃⠀⠀⠀⠈⠛⠛⠀⠙⠛⠁⠈⠛⠋⠀⠀⠀⠀⠀⠀⠿⠇⠸⠿⠘⠿⠃⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢀⡀
⠀⠀⠀⣠⡾⠋⠉⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣻⣁⣙⣃⣸⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⢠⡏⣤⡄⢀⣤⣄⢀⣤⡹⡆⠀⠀⠀⠀⣿⣿⣿⣿⣻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⣿⡿⠿⠿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣭⣤⡀
⠀⠀⠸⣇⡻⣃⠘⢿⠟⣈⢟⣡⡏⠀⠀⠀⠀⣿⣽⢯⣿⢼⣷⣿⣾⡇⠀⠀⠀⠀⠀⠀⡿⣿⣷⣿⡿⡿⠀⠀⠀⠀⠀⠀⢠⣄⣻⣟⣿⣻⣟⣠⡄
⠀⠀⠀⠻⣿⣿⣿⣶⣾⣿⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣞⣓⣛⣿⠆⠀⠀⠀⠀⠀⢀⣼⣿⡿⣿⣿⣿⢿⣿⣇⡀
⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠛⠛⠛⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠷⣤⡤⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠤⠤⠀⠀⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠶⣄⠀⠀⠀⢠⠀⠀⠀⢀⡴⠖⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣹⠀⠀⢀⣼⣄⠀⠀⢸⡁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠶⠶⠞⠛⠳⣄⠉⠀⠈⢁⡴⠛⠛⠶⠶⠞⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⣈⢁⠈⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠎⠘⠓⠆⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶
⠀⠀⠀⢰⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉
⠀⠀⠀⠘⣷⣾⢻⢘⡢⡾⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡄⡄⣄⢤⠔⠂⡤⠰⡄
⠀⠀⠀⠀⠘⠃⠈⠈⠁⠁⡜⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣻⣮⢻⣽⡟⣟⣸⣿⣟⣽⣧⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠛⠛⠛⠻⠛⠋⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡼⢡
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠊⢐⡞⠀⡀
⠀⢸⡒⡆⢰⣄⠰⡔⡇⢙⢵⢴⡆⢠⡆⠀⠀⠀⠀⠀⠀⠸⣷⡿⠑⠂⠀⠀⠀⠀⠀⠀⠀⣄⡄⡄⡀⣠⢤⡀⡄⠲⢀⣤⣰⡀
⠀⠈⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠝⢓⠀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠃⠉⠉⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣰⢶⠐⡠⢲⠂⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠉⠁⠁⠈⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠛⠛⠛⠛⠛⠛⠀⠀⠀⣉⣉⠀⠀⠀⠙⠛⠛⠛⠛⠛⠀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣍⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣭⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠛⠃⠘⠛⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠁⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇





⣾⠉⣷⢸⣹⠆⡞⢋⡁⣸⢳⡀⡿⣆⡇⣿⠘⡿⠃⣸⣷⠘⣿⠃⣿⣴⠋⢳⢸⢷⣸⠀⠰⣏⡁⢻⡟⢸⣿⡇⣿⠀⡇⣾⠋⠉⢻⠋⣾⠀⣿⢸⣽⡆⣿
⣽⣶⣯⣼⣬⣧⣽⣶⣧⣯⣽⣧⣧⣼⣧⣿⣾⣷⣦⣯⣽⣧⣿⣤⣿⣼⣶⣯⣼⣤⣿⣤⣴⣾⣧⣼⣧⣼⣬⣧⣽⣾⣧⣽⣶⣦⣼⣤⣼⣶⣯⣼⣬⣧⣿
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⣖⢉⡆⠈⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢡⡀⠁⢀⢈⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠐⢂⠀⠐⠄⠀⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠉⢉⣹⡟⠁⠀⢈⣓⠇⠀⠀⠀⠀⠀⠌⠉⠁⠆⠀⠀⠀⠈⠐⠀⠨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠆⠀⠀⠀⠀⣤⡄⠀⠈⠈⡇⠀⠀⠈⢘⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⢘⣆⠂⢀⠂⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠋⠁⠀⢀⡠⠃⠀⠀⠀⣙⡗⠀⠀⠊⠉⠃⠀⡀⢠⠀⢸⠿⣶⠺⠀⠈⠀⠀⠀⠛⡖⠐⠄
⠀⠀⠀⠀⠀⠂⠙⣄⠁⠀⠀⠀⠀⣬⣅⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠇⠀⠀⢠⠈⠊⠙⠉⢉⣋⣛⠀⢸⠀⠀⠈⡚⡏⢁⣀⣤⡠
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⠀⡒⠂⠀⠀⠀⢰⠀⠐⠀⠂⠊⠀⠀⠐⠈⠂⠀⠄⠀⠀⢀⠏⠄⠰⣀⡀⡀⠀⣿⡂⠀⡇
⠀⠀⠀⠀⣀⢀⠀⡤⡠⠀⠀⠀⠒⡄⠀⠐⠛⠒⢀⠀⠀⠸⠂⠛⠃⡜⠰⠜⠹⢶⡝⢾⡆⣴⠀⣀⠀⡂⡀⡀⢀⢘⣺⡀⠃⠓⠀⠙⠦
⠀⠀⠀⠈⢁⠊⣨⠁⠀⢸⣄⣀⠂⠠⢰⠀⠀⠀⢸⣂⣌⠠⣏⠉⠆⠀⢀⡤⠺⡝⢨⢀⣉⣀⣉⣛⡀⢁⣂⣈⣀⢀⠟⠃⢠⣊⣈⣳⡀⠓⡄
⠀⠀⠀⠀⢆⠈⢉⡀⠢⣸⣀⡀⠀⠀⠈⠀⢒⢐⠈⠀⠈⠂⣉⣁⡀⠀⡌⣾⡿⠁⣨⠊⡉⣈⡁⡈⡁⢈⠓⣞⡿⠁⠛⡄⠀⠑⠶⠛⠂⢠⡇
⠀⠀⠀⠀⠈⢒⡿⠡⣭⣼⡄⣽⠀⠒⠀⡀⠀⠀⢀⢰⡎⣠⣏⠙⣿⣅⡭⠀⣠⣤⠈⠀⠏⠏⠇⠇⠇⠘⣉⣝⣏⣉⣸⣿⣤⣥⣭⣍⣥⡉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢒⠀⠀⠂⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠑⠚⠓⠀⠀⠀⢰⠒⠐⠶⡆⠀⠂⠀⠈⣀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠜⠃⠈⠓⠘⠛⠀⠄⠀⠀⠰⠀⠀⠘⠗⢶⢰⠒⢲⡂⠐⠶⡎⢉⣀⢉⠀⡀⢀⠀⠀⠈⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡔⠀⢴⠦⢰⡖⢒⡧⠀⢵⣦⡇⠀⠀⠂⡀⣿⢸⣀⡾⢁⠄⠘⠇⠆⠐⢒⡀⠊⠐⠃⠀⠀⠀⣠⣖⣒⣂⣲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⣙⡀⢸⣌⠋⠴⠁⢘⠿⡟⢩⣭⣭⣍⢫⠀⡿⣦⠀⠀⢠⡆⠀⠀⠀⠀⠁⠀⢨⠀⠀⠀⠓⠘⠋⠛⠙⢒⢛
⠀⠀⠀⠀⠀⠀⡃⢀⢘⠀⠀⠈⠁⠰⠛⠇⢀⠀⣼⠀⠀⠀⠀⠀⠀⠉⠁⠀⢐⠀⠥⣼⡇⠀⠀⠀⠄⢠⡀⡸⠀⠀⠀⠈⠁⠁⠁⠉⠈⠈
⠀⠀⠀⠀⢆⢀⠅⢽⠈⢂⣀⣄⣂⡀⠀⠀⣾⠀⣧⣀⣀⠀⠓⠲⠆⢸⢅⠀⠻⠰⠰⠏⣇⡀⠀⠀⡀⢘⡇⣇⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠨⠵⠀⠀⠀⠀⠈⠉⠁⠉⠒⠁⠋⠉⠀⠉⠀⠉⠀⠀⠈⠉⠘⠊⠁⠔⠂⠀⠀⠑⠚⠧⠦⠤⠤⠀⠀⠀⠀⠒⠀⠒⠂⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢰⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠁⠀⠀⠀⠰⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⡀⣀⣴⣄⠀⠀⠀⠀⡀⠀⢀⣀⣀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⣠⣤⡄⠀⠐⠀⠀⠀⣡⡾⣫⡟⠉⢉⠀⠈⠁⢁⠒⠀⡉⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠴⠆⠈⠀⠀⡇⠀⠀⠠⣤⡼⠛⠛⠓⠂⠀⠀⡴⠪⠁⠨⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⣴⠁
⠀⠀⠀⠀⠀⢈⠀⠶⠀⢠⣶⣤⠀⠤⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⣥⡄⠀⡛⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⡠⠃⠀⠀⠦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠉⣦⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠘⠀⠐⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠁⠁⠉⠀⠀⠀⠀⡄⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠄⠸⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡏⣿⣤⡉⠉⠉⠉⠉⠉⠛⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠹⢶⡄⣴⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠛⠋⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠁⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠉⠉⠉⠙⠳⢦⣄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⢦⡀⠀⠀⠀⠀⢠⡞⠉⠀⢀⣤⠴⠶⠶⠦⣤⡀⠀⠈⠳⣄⠀⠀⠀⠀⢀⣴⢟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⣻⡆⠀⠀⣰⠏⠀⢀⡾⢋⡴⠖⠛⠛⠲⢦⡙⢷⡀⠀⠹⣆⠀⠀⢰⣟⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⡏⠀⢀⡟⢰⠏⠀⢀⣤⣤⡀⠀⠹⡆⢻⡀⠀⢹⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠰⣏⣉⣉⣉⡷⠀⠀⠀⢸⡇⠀⢸⡇⣿⠀⠀⢾⡀⠀⡷⠀⠀⣿⢸⡇⠀⢸⡇⠀⠀⠀⢾⣉⣉⣉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⣇⠀⠈⣧⠘⣆⠀⠈⠛⠋⠁⠀⣰⠃⣼⠁⠀⣸⠃⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣽⠇⠀⠀⠹⣆⠀⠈⢷⣌⠓⠀⡇⢸⠀⠚⣡⡾⠁⠀⣰⠏⠀⠀⠸⣏⡹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⠟⠁⠀⠀⠀⠀⠙⢦⡀⠀⠈⠋⠀⡇⢸⠀⠙⠁⠀⢀⡴⠋⠀⠀⠀⠀⠈⠻⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⢀⣀⠀⠀⠀⠙⣆⠀⠀⠀⡇⢸⠀⠀⠀⣰⠋⠀⢀⣰⡟⣷⣀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡿⠹⣧⣄⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⢿⣭⡀⣨⣽⠇⠀⢀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⢠⡶⠟⠀⠀⣿⠶⠶⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠷⠟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⣿⠛⠛⠋⠀⠀⣾⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠻⢤⣤⣤⣤⣤⣬⣭⡭⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠶⠚⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣤⠶⣶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⠆⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟⠀⠀⣿⠀⠈⢻⡀⠀⠀⠀⣤⠀⠀⠀⠉⠙⢧⡀⠀⠀⠀⣸⣏⣹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡇⠀⣴⠟⠳⢦⣼⠇⠀⠀⠀⣿⠀⣤⣤⣤⠀⢨⡟⢳⡶⠛⠉⠈⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢾⣅⣀⣠⡴⠋⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⢀⣛⡛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⣿⠀⡿⠀⣿⠀⣿⠀⣿⠀⣾⠛⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⣤⣤⣤⡤⠠⣤⣤⣤⡄⠀⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡽⠋⠉⠉⠉⠉⠉⠉⢿⡍⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠘⠛⠘⠈⠑⠑⠛⠃⠋⠓⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠓⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠙⠚⠁⠘⠓⠓⠛⠙⠃⠃⠛
⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⣁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠉⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠂⠀⠁
⠀⠀⠐⠀⠀⠀⠀⠁⢀⠀⣀⡀⡀⠤⠀⠀⠀⠀⠀⣀⠀⢀⢌⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢤⣤⢀⣤⠀⠀⡄⠀⠀⡀
⡄⠀⢠⡄⠀⠀⠀⠀⢁⡁⠈⠤⠁⠂⠀⠀⠀⠀⠰⠋⢤⠀⠂⠀⠀⠀⠁⠄⡀⠆⠆⠀⠀⠀⠀⠀⣌⡏⠀⠀⣴⣟⠀⠀⠀⠤⠄
⠁⠀⠐⢀⠖⢳⠀⠐⠊⠂⠦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⡠⠁⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠎⠁⠒⠒⠂⠺⠀⠠⠀⠀⠠⣶⣶
⠀⠀⠀⠘⠂⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠉⠛
⠀⠀⠀⢀⢀⣀⣀⡀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⢀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⡀⡀⡀⡀⣀
⠀⠀⠀⡈⣈⠉⣀⣀⢁⢈⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢉⡈⡀⣈⣈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣈⡈⣁⣁⡀⣁⣁⣉
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠁⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒
⠀⠀⠀⠀⠀⠀⢀⠊⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠨⠁⠠⠄
⠠⠂⠌⢢⠀⡄⠄⠀⠀⠂⠀⡴⠀⠀⠀⠀⠀⠀⣀⠴⠂⡆⠤⠀⠀⠀⠂⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⡠⠃
⠐⢄⡨⠌⣠⡀⠀⠀⠀⠀⡃⠀⢘⠀⠀⠀⠀⠀⣥⠑⠂⣁⡒⣲⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠚⠀⠐⠀⠒⠀⠀⠠⠂⠀⠀⡀
⠀⢐⠁⠈⠀⠀⠄⠀⠂⠀⠁⠐⠈⠀⠀⠀⠀⠀⠀⠉⠑⣶⠐⠂⠀⠀⠠⠠⠀⡄⠄⠀⠀⠀⠀⠀⠠⠁⠀⠑⠂⢐⠀⠀⠀⢰⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣄⢄⣄⢀⠤⠤⢠⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⢤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣄⢠⣤⡄⢠⢠⡄

⠀⠀⠀⠀⠠⡦⠀⠆⠰⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠐⣐⠀⠄⠀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣙⣴⣤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃
⠀⠃⣶⡟⣷⡂⠀⢠⢿⡸⠽⠟⢘⡛⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠃⠖⡛⠆⠈⠀⢘⣽⡅⠀⠀⣬⡄⠀⠀⠀⠀⡤⠄⠂⣀⠘⠀⠀⠀⠀⢈⠑⠁⠂⠀⠀⠀⠀⠀⢀⠀⠀⡈⢂⠀⠀⠀⠀⠄⣈⠰
⠀⠀⠠⠤⠄⠀⠀⠀⠚⡶⢾⡲⢿⡧⠀⠀⠀⠀⢇⠀⢀⠏⠀⠀⠀⠄⠀⠀⠈⣠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠠⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠐⠂⠈⠉⠁⠀⠀⠘⡀⠀⠀⠈⠉⠀⠒⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡰⡆⢆⢎⠱⠈⠈⠄⠕⠀⢠⠂⠀⠀⠀⠰⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⢀⡀⠄⡂⠢⠑⠈⠈⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⠒⠁⢆⡔⡀⣀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢴⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡆⢠⠄⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡀⠤⡃⠂⠁⠀⠀⠀⠀⢀⡠⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⢀⠀⠀⠀⠀⠀⠐⠡⢊⠄⡀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠒⢍⠒⠁⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⢒⠥⠄⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠙⠢⠂⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠐⢡⠄⠀⠀⠀⠀⠀⠱
⠀⠀⠀⡦⠀⠤⠀⠒⠈⠉⠢⡀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠠⠧
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⡆⠀⠀⠀⠀⠐⠦⠤⠀⠀⠀⠉⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠁⠒⠢⠤⠀⠊
⠀⢀⠁⠀⠀⠀⠀⠠⠍⡁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⠀⠀⠀⠀⠀⠰⣂⠄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠒
⠀⠘⠀⠀⠀⠀⠀⢰⣒⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⣈⠭⠉
⠀⠀⡄⠀⠀⠀⠀⠈⠆⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠤⢅
⠀⠀⢃⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢓⠊
⠀⠀⠈⡄⠀⠀⠀⠀⠐⠡⣄⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⣐⡒⠁⠀⠀⠀⠀⢰
⠀⠀⠀⢁⠀⠀⠀⠀⠀⠸⡩⠖⠂⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡐⢤⡉⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠞⠠⡀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢒⠍⠁⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢤⠖⠁⠀⠀⠀⠀⢀⡠⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠻⡀⠀⠀⠀⠀⢔⠒⠈⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢠⠄⠒⠈⠁⠀⠀⠀⠀⠉⠐⠀⠀⠤⠤⠄⠤⠤⠤⠐⠒⠂⠁⠀⠀⠀⠇⠀⠀⠀⠃⠆⠁⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣁⢆⠤⡀⡀⠀⢀⠀⠀⠀⡀⠀⠀⠀⡀⢀⡄⠀⠀⠐⠄⡀⡀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠃⡌⢀⠝⠆⢨⠸⠰⠸⠠⠃⠧⠴⠀⠀⠀⠀⠀⠀⠁⠢⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠶⠶⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠛⠈⠛⠛⠛⠛⠀⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣤⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡏⠀⠀⠀⣬⢻⡄⠠⣤⣤⣤⣤⣤⣤⣤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⠻⣦⡾⠁⣸⠇⠠⡤⠤⠤⠤⠤⠤⣤⠄⢀⣿⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⢤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠐⠒⠶⢦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠞⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣀⣀⣀⣀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠞⢧⣀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⢀⡼⠁⠐⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠁⠒⠒⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠈⢳⡄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠛⠛⠛⠛⢸⠇⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠧⣶⡄⣶⣠⣶⢄⣶⢰⠆⡖⠂⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠉⠀⠉⠈⠉⠈⠀⠉⠁⠁⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⢿⣿⢱⣶⣶⣶⢸⣿⠿⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢋⣴⣿⣦⣥⣾⣿⣿⣿⣦⣥⣾⣷⣌⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣌⢻⣿⣿⣿⡿⠿⠻⠿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣭⣭⣾⣿⣿⢋⣶⣿⣿⣷⣎⢻⣿⣿⣦⣭⡍⢿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇⠿⢋⣭⣛⣸⣿⣄⣋⣍⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣬⣭⢹⣿⣿⣧⣙⠿⠿⠟⣡⣾⣿⣿⢫⣭⣴⠷⢈⣿⡿⢛⣙⡻⣿⣟⠰⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢹⡇⢾⣿⣿⠰⣿⣿⡷⢸⣿⣿⠆⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡙⠿⢋⣭⢹⣿⣿⣿⢩⣭⠻⠟⣡⣾⣿⡶⢈⣿⣷⣬⣭⣵⣿⣯⡰⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣌⣛⣛⣛⣼⣿⣿⣿⣿⣿⣿⣷⣌⣛⣭⢹⣿⠋⣍⣋⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⠙⠛⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠻⠿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣭⣭⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣠⠴⣶⠤⣄⣠⠞⠛⠛⠛⠨⠛⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡞⠁⠀⣿⠀⠞⢳⡀⠀⠀⠀⠈⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠛⠒⠒⢺⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⣄⡈⠳⣄⣠⠞⠀⠀⠀⣶⣶⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠛⠚⠛⠛⠂⣦⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣽⡉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿