Taste Emojis & Symbols

⠀⢠⣤⣶⣴⣤⣦⣤⡀⠀⢠⣤⣦⣴⣤⣦⣤⡀⠀⣠⣤⣦⣴⣤⣦⣤⠀⠀⣠⣤⣦⣴⣤⣦⣤⠀⠀⣠⣤⣦⣴⣴⣤⣤
⠀⠿⠁⠀⠉⠉⠀⠘⡇⠀⠿⠁⠀⠉⠁⠀⠙⡇⠀⡟⠁⠀⠉⠁⠀⠙⠇⠀⡟⠁⠀⠉⠁⠀⠹⠆⠀⡟⠁⠈⠉⠁⠀⠹



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢠⣀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⢉⣤⣌⠃⡀⠀⠀⠀⠈⢋⣬⡟⠀⠀⠀⠀⠀⢠⡶⣦⠀⠀⠀⠀⠀⢀⣼⣶⠂⠀⠀⠀⠀⢀⡬⠛⡀
⠀⠀⠁⠝⠃⠔⠀⠀⠀⠀⠀⠺⠏⠸⠇⠀⠀⠀⠀⠸⠦⠾⠀⠀⠀⠀⠀⠘⠾⠞⠀⠀⠀⠀⠀⠰⠆⠸⠽


⠀⠀⠀⣰⣦⣄⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⣶⣦⠄⠀⠀⠀⠀⢿⣗⣻⣿⠀⠀⠀⠀⠀⠀⣷⠐
⠀⠀⠸⣻⣿⠞⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣰⣧⣯⠀⠀⠀⠀⠀⠘⢸⡇⠀⠀⠀⠀⠀⠀⣷⣶⡆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⢠⣤⣠⣤⠀⠀⠀⠀⢀⣀⠠⡄⠀⠀⠀⠀⠀⣤⣄⣤⡄⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠄⣤⣤⣄
⠀⠀⣠⠉⡌⢡⠀⠀⠀⠀⢀⡁⠶⠸⠆⠀⠀⠀⠀⠈⢡⠉⠀⠀⠀⠀⠀⣀⢙⣫⡴⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠙⠀⠋⠘⠁⠀⠀⠀⠈⠁⠘⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠋⠙⠉⠀⠀⠀⠀⠀⠉⠉⠻⠋

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠁⠻⢋⠀⠀⠀⠀⠀⠈⠋⡝⠃⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠀⢰⣉⡟⢳⠄
⠀⠀⠰⡶⠷⠚⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠺⠷⠗⠁⠀⠀⠀⠀⠺⠿⠷⠋⠀⠀⠀⠀⠈⠻⡹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠑⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢙⠓⣂⠠⠐⠂⠁⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡱⠀⠀⠀⢠⠀⡈⠀⠀⠐⠒⠀⠤⠀⡀⠀⠀⠀⠀⢀⣠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⡀⢠⠏⣿⣿⣶⡀⠀⠀⠀⠀⠀⠈⠑⠠⠀⢀⠈⢷⠱⠄⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠍⠀⠈⠼⠟⠀⠛⠃⠀⠀⢠⡶⠿⠿⢦⡀⠀⢀⡁⠀⢁⣵⣄⠀⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⠈⠀⠠⠀⠀⠀⠠⠄⠀⠀⠀⢠⣿⣿⣶⣾⠄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠇⠀⠀⡜⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢀⡀⠜⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠟⡯⢩⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣰⣶⠘⠚⠛⠷⢤⣤⣬⡀⢀⠀⠀⡀⠀⠀⠈⠀⠄⢀⠆⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠠⣦⢶⡇⠀⠊⠀⡰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⠀⢀⠀⠀⣀⠠⠄⠀⢈⠄⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠄⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢄⠰⡐⠄⠀⠀⠈⠀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠸⠆⠀⠀⠀⢀⡶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠆⠈⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠴⠴⠅⠤⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣐⠤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⢀⣤⡀⢀⣀⡀⣠⣤⣀⣤⡀⠀⠀⣠⣤⡀⣰⠿⠂⠈⢹⡏⠁⠶⣆⣠⣤⣤⣿⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣽⢿⡗⣛⣾⠋⣷⣙⣶⣿⡛⡃⠀⠰⣏⣼⠟⣿⠃⠀⠀⢸⡇⢸⣿⣿⣙⣷⠸⣇⣤⣿⢚⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣩⣴⣧⣭⣁⠈⠉⠁⠀⠀⠈⠁⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣠⣴⠖⠲⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣾
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡴⠞⢛⡿⠿⢿⣷⣄⣈⠳⢶⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠉⠀⢀⣀⣀⠀⢀⣀⣉⠻⣿⣿⣇⢠⡏⠀⠀⢸⡄⠀⠀⠈⠳⣭⣉⣻⣏⢹⠇⢸⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣾
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡄⠀⠐⢿⣿⡿⠀⠹⣿⣿⠗⠀⣸⣻⢿⡀⢀⣴⠏⠙⠲⠤⣤⣤⣤⣴⣿⠵⢛⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣆⠀⢀⡀⠀⠚⡂⠀⠀⢀⡴⠋⠁⢈⣿⠋⠀⠀⣀⣤⣀⠀⢀⣤⣌⠓⠦⣼⣇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣶⣶⠋⠀⠀⠀⢻⣽⡄⠀⠘⠿⠿⠟⠀⠹⠿⠿⠃⠀⣼⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣿⣿⣿⢷⡄⠀⠀⠀⠈⠹⣆⡀⢀⣀⠀⢉⡀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣧⣈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⠙⣿⢿⣿⣿⣷⣶⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠼⣱⣿⣿⣿⣿⣿⡇⣧⠀⠀⠀⠀⠀⠀⡼⢹⣿⣿⣟⣾⢿⡏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣴⣿⣿⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠘⡷⢾⣦⣬⡽⠀⠀⢷⠤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠈⣧⠀⠀⠀⠀⢠⡇⠚⣸⠃⠀⠀⠀⢸⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠚⠃⠀⠀⠀⠀⠈⢧⣼⣇⣀⣀⣀⣀⣤⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⠏⠀⠙⠀⠘⢦⣧⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⣿⣿⣿⣿⣿⣿⣿⠲⠾⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢻⣿⣻⣿⣿⣿⣿⣻⣿⣷⡀⠀⠀⠀⠀⡀⢸⣀⢀⣀⣿⠀⢀⣄⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠿⡿⢿⠿⢼⡿⠿⣿⠿⢿⡇⠀⠀⠀⠈⠳⣾⣿⣿⣟⣿⢻⣿⣿⣿⠓⠚⢶⣠⢀⢀⣄⣀⣸⣃⣀⣀⣸⡄⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚⠓⠒⠒⠒⠚⠃⠀⠀⠀⢠⡞⢹⣯⣽⡟⢿⢻⣿⣽⣿⢳⡀⠾⠛⠋⠘⠙⠟⠙⠋⠋⠙⠙⠑⠃⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠾⠭⠭⠷⢾⠾⠯⠭⠿⢾⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠰⠖⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢠⡂⠴⠋⠈⠀⠀⠀⢁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⣀⣀⡈⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⢁⡀⠀⠔⠁⠀⠀⠤⠞⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⡛⠿⠁⢀⠀⠀⠀⠀⠤⠀⠀⠙⢄⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀⠀⠀⠀⠀⠀⠢⠆⢨⠻⠷⠀⠀⠀⠀⠀⠀⣠⣤⡠⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠢⡄⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡗⠀⠀⠀⡠⠊⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⡠⡰⠗⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣮⠆⠀⢠⡞⠀⠀⣄⣑⠀⠙⠻⠶⠤⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠚⠁⠀⠀⠀⠀⠉⠒⠄⢀⠔⠈⣒⠤⠚⢁⠴⠃⠀⠀⠀⠀⠈⠢⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⢈⠢⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⢰⠀⠈⠢⡀⠀⠀⠀⠀⠀⢣⠀⠀⢈⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠑⢄⠀⠀⠁⠀⠀⠀⣸⠁⢀⠂⠐⢄⠀⠁⠀⠀⠀⠀⠀⠀⢃⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠄⠀⠀⠀⡡⠖⠉⠀⢤⠀⡇⠀⢸⠀⠀⠀⠉⠚⠀⠀⠰⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠘⠀⠀⠀⠀⠀⠆⠐⠤⠁⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⢤⣀⣀⣘⣀⣀⣀⣀⣀⡤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠉⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⣀⣤⠤⠦⢤⣌⡉⠒⢄⡀
⠀⠀⠀⡠⠊⢀⣀⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠞⢁⡔⠋⡉⠀⠈⠻⣆⠀⠙⢆
⠀⠀⡼⠁⠴⠛⠉⠉⠛⠛⠛⠿⠿⢿⣶⣶⣄⡀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢰⠃⢠⣎⣠⣀⡘⠒⠀⢢⢹⡆⠀⠈⢣
⠀⢰⠁⠀⠀⡀⠀⣀⣠⣀⣰⣄⣀⡀⠀⠉⠛⠿⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠀⠘⠉⠀⠘⢧⡀⠀⢸⢸⠇⠀⠀⠈⡆
⠀⢸⠀⠠⣤⣽⣿⠿⠛⣿⣟⣹⣿⣿⡷⣄⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠑⠦⠀⠀⣈⠃⠀⠀⢸⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠈⠹⣄⠀⠙⢯⣟⣯⠞⠀⠈⠳⣄⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⡼⠡⠆⠰⠀⣼⠀⠀⠀⠀⡇
⠀⠀⢧⠀⠀⠀⠀⠈⠓⠶⠤⠤⠤⠶⠒⠂⠀⠉⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⢀⡾⠁⠀⠀⠀⡼
⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣀⠤⠒⠊⠉⠉⠉⠉⠑⠒⠤⣀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠉⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣡⠞⠁⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠈⠳⣌⠓⢤⣀⠀⠀⠠⠤⠴⠋⠀⢀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠖⠒⠒⠉⠁⠀⡜⠁⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢧⠀⠈⠉⠒⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡠⠀⠀⠀⠀⠈⢳⠀⠀⣄⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠒⠢⠤⣀⠀⠱⡀⠀⠸⣇⢀⣀⣀⠀⠐⣋⣀⡀⣸⠂⠀⢀⠎⠀⣀⠤⠔⠒⠖⠒⠲⠤⢄⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡘⠦⣀⠀⠈⠉⠉⠀⠈⠉⠉⠁⠀⣀⠴⣃⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠑⠢⠤⣤⣀⣀⣤⠤⠔⠊⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⢀⠎⠀⠀⠀⠀⢀⣠⣤⣀⣀⣀⣤⣀⡀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⢸⢀⣀⣀⡴⠞⢯⠁⡄⠉⠉⣡⡀⠨⡟⠶⣤⣀⣀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠈⠉⠣⡒⠒⠒⠚⠛⠶⠶⠿⢛⠓⠒⠛⡢⠛⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠸⡄⠀⠀⠙⢤⣈⡀⠀⠃⣉⠘⠀⢀⣀⠔⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⠇
⠀⠀⠱⡀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠞
⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠙⠲⢄⣀⡀⠀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⢀⣀⡤⠖⠋
⠀⠀⡀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⣉⡉⠉
⣿⣏⢹⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢞⣁⣠⠶⠿⢿⣷⣶⣶⣾⣷⠶⠶⢦⣤⣬⡿⠂⠀⠀⠀⠀⢀⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⢰⣿⡵⠁⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠴⣾⡴⠋⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟

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

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠃⠘⢦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠞⠛⠉⠀⠀⠂⠀⠀⠀⠀⠀⠉⠛⣷⡶⠶⠶⠶⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠙⠳⢤⡀⢀⡤⠞⠋⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⢀⠀⠀⠀⢳⡞⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣀⡴⠛⢧⣄⡀⠀⠀⠀⠀⢀⣼⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⣦
⠀⠀⠈⠙⣦⠞⠉⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢴⣦⣀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣾⣿⡆⠈⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠉⢉⣀⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠁⢀⣴⡄⠀⠀⠀⠀⢻⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡈⠷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⣈⣉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣄⡀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠉⠁⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢻⣆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠷⣤⡀
⠀⠀⠀⠀⠀⢠⡟⠉⠉⠻⡦⠴⢿⣆⠀⠀⠀⠙⠋⠀⠀⠀⠀⠉⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠟⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣈⡿⠂
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⣤⣿⣧⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠈⠿⣤⣤⣤⡾⠉⠀⠀⠈⢻⣦⣄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣡⡶⠿⣥⣀⠀⠀⠀⠀⠀⠀⢀⣈⣠⣄⠀⣻⠿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣩⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣴⣶⠟⠉⠀⠀⠈⠙⠛⠻⠶⠶⠞⠛⠛⠉⠁⠛⠶⠋⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠾⠿⠛⠛⠉⠈⢻⣌⠻⣿⣿⡿⠋⠉⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣼⠟⠁⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⡞⠁⠀⠈⢳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣏⢈⣷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁

⠀⠀⠀⠀⠀⣠⡔⠉⢢⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠐⢅⡀⠪⠂
⠀⠀⠀⠀⠀⡇⠇⠀⠰⢂⠀⠀⢐⡡⠂⢚⣝⠒⠨⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⡤⠬⢀⠁⢁⠨⢦⠄⠀⠀⠀⠈⡢⠀⠆
⠀⠀⠀⠀⠀⢃⠀⠀⢀⠄⠀⠀⠈⠐⠠⠉⠑⠡⠐⠁⠀⠀⠰⡀⠀⡀⢀⠀⠀⠀⠈⠢⣀⠤⢀⠀⠀⠀⠀⠀⠀⠀⡀⠬⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡼⣿⡿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⡛⡛⣿⡇⡀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⡁⢈⡀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⡇⢈⠰⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠹⣦⠟⠛⠀⠀⢂⣹⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠹⣆⢰⢤⡹⣷⡀⠀⠀⠘⠋⠙⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢙⣿⠀⠓⣄⠹⣧⠀⠀⠀⢀⢴⡿⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣄⣈⣈⠀⠹⡅⠀⠈⣦⠛⠀⠀⢲⠞⠋
⠀⠀⠀⠀⠸⠀⠒⠐⢄⠀⠀⢀⢏⡿⣠⢡⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⣄⠐⠀⠠⠈⡕⢽⡏⣴⢃⣾⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡇⡄⠀⠀⠀⠀⢚⡀
⠀⠀⠀⠀⠀⠁⠀⠘⠁⠀⠀⠐⠞⣡⢋⡼⣫⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢦⡜⢇⢱⠀⠀⠀⣒⣁⡓
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⢢⣤⡥⢯⣴⣿⣿⣿⡧⣖⣀⡀⠀⠤⢤⣤⠀⡡⡆⢲⣯⣾⠀⠀⣀⢨⡐⢥⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠮⠀⠀⣿⠁⡟⢿⣿⣿⣿⣷⣾⢒⡏⢸⠃⡇⢚⡙⣿⣿⠀⠀⠉⢸⣯⣿
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠄⡀⠀⡿⠀⣧⣻⡏⠙⠛⠟⠒⠿⢿⠧⠒⠛⣂⡝⠿⠛⠲⢾⣿⣯⢺⣿⠳⠶⠄⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⡇⠀⢹⣿⣇⠀⠄⡤⣤⡃⠰⠷⡶⣴⣼⡧⠔⠺⠷⠾⠾⣽⢿⡟⠻⠷⣄⣡
⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠈⠿⠃⠠⡴⠖⠀⠲⠠⣾⣾⣿⣿⡟⠻⠷⠖⡿⠇⠿⠾⠻⢷⣦⠀⠸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⡀⢁⡀⠼⣴⠆⠒⣄⡆⠀⣿⣽⢿⡯⣷⣤⡄⡀⢰⣶⣆⡠⠶⠤
⠀⠀⠀⢸⠘⠀⠀⠴⠀⠀⠀⢀⡛⣀⠄⠀⠀⠘⠇⠀⠣⢄⡀⠀⣀⣀⣿⣿⢸⣧⣄⠀⠉⢃⡘⢿⣿⣁⣀⠀⠁⠉⠐⠆
⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⢀⡽⠋⠀⠀⠉⢽⠋⠀⠀⠀⠹⠗⢦⡬⣿⣿⠈⠙⠛⠋⠂⢀⡄⢸⣿⡏⠛⠳⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠰⠆⣀⣀⠀⠀⠀⠠⠾⢿⣯⣤⣀⡀⠀⠀⠘⠃⠿⠍⠿⣦⣤⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣶⣶⣶⣾⣿⣷⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢄⡀
⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡯⢄⠑⣽
⠀⠀⠀⠀⢊⠔⠊⢏⣩⣻⠒⢌⡂⠀⠀⠀⠀⢠⣔⣊⡈⠑⠉⣈⣒⣤⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⢀⣿
⠀⠀⠀⠀⠈⠢⢄⣛⣉⣊⠤⠒⠁⠀⠀⠀⠀⠀⠱⣄⠈⣉⣉⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢌⡡⠊

⠀⠀⠀⠀⠀⢀⣠⢺⢱⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⢸⢸⢸⢸⠤⡄⠀⠀⠀⠀⠀⠰⡿⢶⡶⢶⠶⣶⡾⡷⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡜⠀⠀⠀⠀⠀⠀⠀⠈⠢⡇⠸⠀⢰⠊⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠘⡦⠤⠤⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡐⠢⠠⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⣶⠄⣤⣤⠄⢠⡄⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⣿⣠⣿⣿⠁⣼⠃⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣾⣿⣯⣤⣿⣛⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣠⣠⣦⣤⣦⣤⣷⣷
⠉⠉⠉⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⠶⠶⠆⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠠⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣬⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠃⣀⣀⣀⣠⣤⣤⣄⠀⠀⠀⢰⡄⠀⠀⠋⠙⢿⣿⣿⡟⡹⠏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠻⡍⠻⣿⣿⠿⠁⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⡛⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⣿⣿⠃⠈⠁
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣬⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⢸⡉⠻⠳⣄⠀⠀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠘⢧⡀⠀⠈⢷⣤⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠠⢈⣁⣠⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⣦⡀⠀⠙⣿⣿⣿⣿⣿⣿⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢦⡀⣿⣿⣿⣿⣿⣿⡀⠀⠈⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠏⣿⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⣿⣿⣿⣿⡷⠶⠾⣻⣿⣿⣿⣶⣤⣤⣤⣤⣶⣿⠟⠁⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⠿⢿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⢠⡿⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣄⣿⣿⣿⣿⣿⣿⢇⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣤⣤⣶⠶⠶⢿⣿⣿⣿⣿⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⣿⣿⣿⡇⠈⠻⣿
⠀⠈⠉⠍⠁⠀⢠⣿⣿⣿⡿⠋⣼⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⣿⣿⠇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⢸⣿⢿⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠹⣿⣿⠀⣠⣿⣿⣄⡀
⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⠀⠀⣿⡄⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠞⣿⠀⢻⣿⣾⣿⣿⣿⣟⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⢿⡿⠄⠀⠀⠉⠳⢮⣄⠀⠁⠀⢐⣂⣠⠴⠚⠉⠀⢠⡿⠃⠈⢻⣿⣿⣿⣿⡏⠳⢄⡼⠋⠛
⠀⠀⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⢀⣠⣴⣿⡇⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠚⠋⠉⢉⡿⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠃⠀⠀⠀⠀⠀⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢴⣿⡋⣵⣶⣶⠀⣶⣶⣮⢙⣿⡦⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⣴⡊⠉⠉⣿⠉⠉⠑⣦⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠻⢿⢸⣿⣿⠀⣿⣿⡏⡿⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⡀⡇⠀⠀⣿⠀⠀⢸⢁⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡄⢀⠈⠳⢦⣀⡠⠴⠒⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠟⠈⠐⢁⣠⣄⠤⠂⠠⠊⣀⠔⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⢉⡉⠩⠓⣲⠴⠞⠛⠛⠛⠛⢻⡛⠛⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⣀⣐⣛⠒⣏⠀⠀⠀⠀⠀⠀⠀⠈⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣦⠀⢩⡢⢅⣐⠈⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣈⣀⣠⣤⣤⣼⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠖⠚⠛⠋⠉⠉⠉⠛⠒⠐⣲⢿⢁⡲⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⡤⣄⣀⠀⠀⢉⡉⠀⢀⠀⠀⠐⣇⡊⢿⡹⢼⠆⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢠⡴⣖⣒⣶⠶⣤⣤⠤⣦⣄⠀⠀⠀⠀⠈⠳⣾⠛⣽⣟⣋⣠⠴⠋⠀⠀⢀⠉⢉⣇⠁⢸⠀⠀⠀⠀⣄⢠⢾⡟⣦⣇⡀⣠
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⣎⠹⢺⣷⣤⣀⣤⠤⠼⠧⢀⡀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠋⠁⠁⣿⠀⠀⠀⠀⠀⠈⣎⠓⠉⢱⣚⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢺⡉⠀⠈⠙⢿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⢀⡿⠦⡶⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠋⠹⡄⠀⠀⠀⠈⣉⡽⠛⢶⣶⢶⠒⠒⣶⠾⣟⡻⠿⠟⠋⠉⠳⢤⣄⣀⡴⠏⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⣶⠶⠒⠋⠁⢠⡾⣹⣣⠞⠛⠛⠙⣗⢞⢷⡄⠀⠀⣤⣄⣀⣀⣈⣉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠝⠦⢤⣤⡤⠶⡶⢻⣫⠶⠃⣄⠱⡄⠐⣜⢮⣑⠿⣆⠀⡇⠀⠉⢉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⣻⠁⠀⠄⠈⠳⢌⠢⠘⢆⠉⠛⠛⠃⡇⣠⣾⢿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠢⠍⠒⠤⢁⠀⠀⠀⡿⢽⣿⠿⣧⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⢤⣀⣀⣀⣀⣀⣀⣀⣤⡤⠶⠚⣿⣿⣟⡳⢎⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠀⣠⠖⠀⢸⡏⣯⠻⢝⣷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠐⠒⠒⠒⣀⡥⢖⡩⠊⠀⠀⢸⣀⣘⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠐⠒⠈⢉⡡⠖⠉⠀⠀⠀⠀⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣠⣄⡀⠈⠁⠀⠀⠀⢀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⡗⠲⣶⠶⡏⠉⢹⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⢹⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢳⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠼⠀⣿⠀⠀⠈⢧⢘⣦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⠒⠚⠋⠉⠋⠀⠀⠀⠈⠉⠉⠙⠓⠒⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠛⠛⠻⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠠⣄⡉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⡀⠐⣦⠀⠉⠓⢦⣙⠶⣌⠛⠛⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣀⠀⠀⠀⠀⢀⣀⣤⣤⣍⣉⡉⠐⠦⣤⣄⠈⠳⣤⠈⣧⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠚⠉⠉⠻⠶⠶⠟⠋⠁⠀⠀⠀⠈⠉⠓⠶⠀⠈⠻⣆⠈⠁⠸⣇⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢻⡀⠀⣠⠎⢿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⣘⣛⡛⠓⠲⠶⠶⠶⠂⠀⠀⠀⠀⢀⣴⠶⠶⢶⣤⡈⠓⠀⠘⠃⠴⢫⡞⠁⣿⡇
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠐⠋⠉⠉⠀⠀⠀⠀⠀⠀⣠⠀⣼⠀⠀⣀⣤⠤⢤⡄⠈⠻⡦⠀⠀⠀⠀⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⡇⠀⡾⠉⢠⡶⠖⠲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠳⢦⣄⠀⠀⠀⠀⠀⢸⡇⠘⣇⠸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⣧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣉⣙⣛⣛⣩⡄⠈⢧⠀⠹⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⢻⣇
⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠒⠖⠦⣭⡉⠉⠉⠀⠀⠀⠀⠀⠀⠹⠊⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⣼⢋⡙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠈⠉⣀⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠾⠙⣎⠧⠀⢤⡀⠀⠀⠀⠀⠀⠐⠋⠁⢀⣀⠀⠀⠀⠙⣦⠀⠀⡼⠁⣠⣶⣿⣿⣷⣦⡀⢸⣾⠃
⠀⠀⠀⠀⠀⠀⢻⡀⢏⡈⠃⠀⠠⣍⠓⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣦⣄⣀⣸⡆⠘⢡⣾⣿⣿⣋⣀⣠⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠸⣇⣾⣰⠆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣀⣠⣿⠟⠿⠟⣻⡟⠉⢻⣿⣈⠛⠛⠋⢁⡀⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣎⠻⣧⡄⠀⠀⠀⠀⠠⢤⣄⣀⠛⠛⢋⣉⣡⣤⠤⠞⠁⣿⠀⠈⣿⠉⠙⠛⠋⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⢀⡀⠘⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣠⣿⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠲⠂⠀⠀⠉⢙⣇⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣷⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢧⣀⣤⣀⡀⢀⣰⡟⠛⣇⠀⠀⠺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣴⠃⠀⡼⠃⠀⠀⠀⠈⠉⠉⠋⠉⠀⠀⠹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢿⠀⠸⠃⠀⠀⠀⣀⣴⣶⣶⣖⣒⣦⣀⡀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠸⣆⠀⠀⠶⠟⢻⣍⠉⠉⠛⠟⠉⢩⡍⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠙⢳⣦⣤⣤⡴⢟⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠼⠋⠁⠀⠀⠀⠈⠳⠄⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⡶⠶⠾⠛⠛⠙⠛⠛⠉

⠀⠀⠀⣠⣶⡾⠿⠿⣷⣦⡀⠀⠀⠀⣠⡴⢞⣛⡉⣩⣛⡳⢦⣀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⠀⢰⡆⠀⣾⠟⠛⠛⠛⠶⣄
⠀⠀⢰⣿⡏⠀⠀⠀⠀⠛⠃⠀⢠⡞⣡⣦⣹⣿⡧⣿⣿⣿⣢⣍⢷⡀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⢠⡟⣨⣻⠿⣷⠹⣇⢿⢟⣽⣿⣿⡧⢷⡀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠙⠻⢷⣤⣄⡀⠀⢸⢧⣿⣿⣿⣶⠅⠉⠉⠫⢭⣭⣿⣶⢸⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿⡟⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠸⡎⣋⣽⣶⣾⡟⣴⢠⡽⣿⣶⡙⡿⣸⠃⢹⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⢸⣧
⠀⠀⢀⣤⡀⠀⠀⠀⢀⣿⣿⠀⠻⡜⢿⣿⣿⢹⣿⢼⣷⢿⣿⡿⣰⠏⠀⠘⣿⣿⠀⠀⠀⠀⠀⣰⣿⠃⠀⣿⡇⠀⠀⠀⠘⣿⡀
⠀⠀⠘⢿⣿⣶⣤⣶⣿⡿⠏⠀⠀⠙⠮⣝⡱⠿⠿⠻⠿⢏⣡⠞⠁⠀⠀⠀⠘⠿⣿⣶⣶⣶⣾⠟⠁⠀⠀⣿⡇⠀⠀⠀⠀⠹⣷⣤
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⠟⠛⠓⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣽⣥⣍⣹⣍⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠌⠀⠄⠈⠄⠀⢀⢠⣴⡾⠟⠛⠉⢾⣿⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢐⢀⡀⠛⡁⠠⠀⠀⠀⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠈⠒⠂⡗⣿⢛⠖⠂⠀⠀⡀⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢁⠃⠀⠀⠀⠀⠀⠑⢤⣄⣡⣆⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢄⡀⠀⠀⣀⠔⠀⡤⠩⠲⠃⠈⠁⠀⢀
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡠⠈⠀⠀⠀⠑⠲⠀⠛⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⠀⠀⠀⢀⣶⣿⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⢀⠄⣀⠤⠒⠐⠤⡀
⠀⠀⠀⠠⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠘⠈⣱⠀⠀⠠⢶⠄⣀⠀⡀⠀⠈⢣⡔⠁⠀⠀⠀⠀⠈⢢
⠀⠀⠈⠋⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠃⢾⣿⢆⡐⠀⠀⠣⡀⠀⠀⠀⠐⠀⢣⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⣀⠞⠀⡀⣀⢀⠁⠀⠀⠀⠑⢀⣀⣀⠠⠔⢀⠑⢲⠶⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠉⠉⠀⠀⠈⠀⣁⠤⠂⠒⠒⢤⠀⠀⠀⣀⢠⠔⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣷⡄⠀⠐⢠⠊⠀⠀⠀⠀⠀⠀⣷⣈⠉⠀⡜⠀⠀⠀⣠⣤⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠙⣿⠀⠁⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⡷⠞⠀⠀⠀⠀⠈⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⡄⠘⡈⠀⠐⡀⠀⠀⠀⠀⠀⣿⣿⠏⠀⢀⠀⠀⢿⣶⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠛⠛⢀⡧⢴⣿⣇⠀⠀⠀⠀⠀⠟⠏⠀⢀⣾⣷⣄⡀⠉⠙⡿
⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⠐⣾⣿⡇⠀⠙⠿⠆⠀⠀⢀⠼⢄⠀⠀⠸⣿⣿⣯⣿⣷⠞
⠀⠀⠀⠀⠀⠀⠃⢴⣿⡇⠀⠀⠻⣿⣷⡀⠀⠀⠀⢀⡴⠃⠀⠀⠉⠒⠶⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⠋⠀⣀⠔⠉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⠤⣦⣢⢈⣤⣶⡼⠄⠁⠌⠣⣀⠃⠤⠋⢽⠅⠀⠀⠀⠀⠀⣀⣀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢛⣍⠒⠛⢱⣶⡿⠿⠷⣆⡐⣂⢢⣿⡃⣸⡁⢴⡇⠀⠀⠀⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠴⠂⡰⠋⢀⢡⠟⠛⣩⠿⠚⢻⡀⠴⠷⠀⣉⡳⢾⡚⠉⣤⢋⡄⡄⠀⠀⢀⡿⠃
⠀⠀⠂⠒⠐⠂⠀⠀⠀⠀⠀⠀⠆⠄⢦⡆⣖⠸⠄⠂⢉⠀⣦⡺⠟⠉⠢⠀⠀⠛⠷⢤⡈⠒⢄⡀⠑⢢⠜⠀⡘
⠀⠠⠤⠀⠠⠤⠀⠀⠀⠀⠀⠀⣼⠁⠙⠁⠈⠠⣶⡴⠿⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣄⠈⢢⣀⠐⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣸⠀⠀⠀⠖⠠⠀⠀⠀⢠⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣶⠉⠆⢘⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠐⠀⠀⠈⠂⠀⠰⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠶⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡤⠀⠀⣀⠀⠀⠐⠆⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠈⠀⠀⢠⠀⢀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠄⠀⠒⠀⠴⠴⠀⠀⢉⠄⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⡀⠀⠀⠀⡠⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣁⠀⡄⠀⠈⢀⡋⠶⣄⠀⠋⣤⠀⣀⠈⠀⠠⠀⠀⠀⠀⠀⠂⠰⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⣃⢜⡉⠓⢤⠀⠙⡁⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀⠲⡂⢉⡀⢠⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠉⠁

⠀⠀⠀⠀⠀⠤⠶⠤⠶⠶⠶⢶⠶⠛⣛⠉⢙⡇⠀⠠⠖⠒⠲⠄⠀⡔⠚⠛⠒⠢⠀⢖⠒⠒⠒⠶⠖⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣙⣩⡥⠤⢒⡫⣿⣷⣶⠟⠓⠄⣾⣠⣎⣛⠲⠒⠛⢳⣤⣾⠦⠭⣖⠀⠉⠉⠁⠐
⠀⡀⠀⠀⣀⠤⢀⡀⢉⠵⠊⠀⠀⠔⠉⠀⠜⠀⠀⠀⠀⠄⢿⣿⣿⡟⠀⠀⠄⠀⠉⠃⠈⠀⠀⠑⢦⡀⠈
⠀⠀⠀⠧⠀⠂⢁⣴⠣⡒⠀⠀⠀⠀⢠⢂⢤⠀⠀⠀⠀⢠⠀⠀⠁⢨⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⡶⡹⣄⠀⢀
⠀⠀⠀⠠⢤⡶⢻⠨⢉⠜⡱⠀⠀⠐⠑⣣⢠⠂⠀⠀⠀⠃⠀⠀⠀⠀⠂⠰⡀⠂⠀⠐⢀⢢⢨⡙⣿⡣⠘⡆⠀⠁
⠠⢀⠀⡴⠋⡀⠂⠀⢈⢰⢇⠁⠇⢦⠀⠀⠁⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡾⡇⠐⠋⠁⠀⢹⠀⡀
⠃⠊⣼⢧⢿⠁⠴⠡⠀⢎⡠⣾⡀⢂⣀⢀⠞⡟⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠜⠀⠇⠀⠀⠉⡠⠊⠁⠢⠀⣸⠉⡇
⠀⢸⠈⠉⠀⠠⠒⠓⢠⠀⠀⠃⡁⠊⡊⠈⠄⠐⡩⢃⢀⠀⠀⣰⠀⠀⢁⡀⢈⡬⢀⡀⠀⠀⢟⡀⠀⠲⢀⡗
⠀⢸⠀⠀⠈⡐⡀⠈⠠⠀⠠⠤⠏⠈⠡⠔⠆⠓⡉⠂⠈⣁⣇⡏⠀⣀⣾⣿⠀⠀⠎⠙⠀⠰⠤⡉⠏⠀⡼⠠⠁
⠀⠘⣆⠀⠀⠀⠉⠉⠀⢸⠰⣃⢒⠠⣇⡠⡁⠢⢇⡀⡁⠁⠉⠀⡜⣽⣿⠟⠀⢸⠽⡐⢀⠀⠄⡁⠀⢰⠃⠠⠘
⠀⠀⠈⠢⡀⠀⠀⠀⠐⠀⠀⠈⠉⠀⠈⠀⠍⠕⠀⠈⢀⡐⢀⠘⠤⣡⠒⠒⠀⠀⠀⠡⠄⠠⠔⠀⠀⡟⢀
⠀⠀⠄⠂⡈⠓⠤⣀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠘⡥⠃⠀⠀⠈⠂⠐⠀⠀⠀⠀⠀⢀⣴⢆⡜⠣⠂
⠀⠀⢀⢠⢀⠀⠀⠀⠉⠙⠒⠒⠤⠤⢡⣀⡈⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢠⠀⣔⡄⠀⢐⠴⠋⢀⠀⠀⡄
⠀⠀⠀⠈⠉⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠓⠒⠤⠤⠤⣄⣀⣈⡬⠤⠥⠤⠖⠒⠉⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠄⠠⠄⢀⠀⠠⠤⠀⠂⠈⠁⠀⠈⠀⠀⠀⠈⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠇⢳⢀⡖⠀⡄⢀⢀⣖⡀⢀⡀⡀⠀⠀⠠⣔⣋⣉⠀⣀⠀⡀⠀⢀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⡴⠀⣀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠼⠋⠘⠦⠌⠘⠿⠸⠤⠿⠜⠯⠑⠿⠘⣿⠁⢀⣀⣨⠤⠋⠸⠉⠸⠻⠸⠏⠀⠀⠘⠧⠄⠀⠾⠣⡎⠸⢯⠱⡇⠶⠿⠼⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠘⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢐⢁⡀⢘⠓⢢⣀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠘⠋⣈⣁⣉⣉⣁⣠⣤⡀⢄⠀⠀⠀⠀⠠⠀⠈⠉⢉⡉⠉⠁⢀⡀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡽⠜⠁⠋⠁⠼⠿⢲⡆⡄⠀⡀⠤⢤⢄⣠⡾⠁⠀⠀⠈⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠂⠀⠁⠒⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣟⣤⠠⠤⢼⡁⠀⢀⣀⣀⣼⠋⡀⢀⣀⡀⠈⠓⡀⠀⠀⠀⠤⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⢲⡄⡐⣂⡂⠒⢤⣀⠄⠀⠀⠀⠠⠀⢀⣤⣍⠈⠁⠑⠂⢒⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡨⠒⠉⠀⠀⠀⠀⠈⢤⣀⠀⠀⠀⢀⡄⢨⡟⠀⢰⠀⠀⢨⠸⠀⠀⠀⠀⠀⠀⠀⡀⡀⠁⠀⠈⠉⠀⠉⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡈⠤⡀⠐⠀⠀⠀⢔⠈⠉⠉⠙⢷⣼⠶⠋⡠⣺⠉⠙⡄⠈⡆⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠀⠀⠀⠈⠈⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣾⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣾⠰⠷⠦⠀⠁⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⢿⡄⠀⠀⠀⠀⠀⠂⠀⠈⠀⡸⠀⠀⠀⠸⠿⠦⣄⣠⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠿⣄⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠈⠙⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢨⠀⠈⠓⠀⠀⢈⠉⠀⠀⠀⠐⢀⠀⠀⠀⡀⠀⣸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠈⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣷⡀⠀⠐⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢡⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡿⠶⠥⠀⢀⣹⣦⣀⠀⠄⠀⠀⠀⠀⣠⣿⠏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⢠⠀⣀⡀⠀⢠⢄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣖⣀⠀⠀⠉⠛⠛⠛⠛⠉⠛⠁⣀⠨⢴⠆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⣀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢚⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⡀⡠⢀⡆⣀⡀⢡⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡘⢀⠀⠐⠀⡇⢈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣔⣷⠀⡄⠧⡸⠀⠤⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡁⡸⢠⡞⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠣⠃⡇⢰⠀⣶⠂⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⠞⠃⠀⡘⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⡄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢇⠀⢀⠆⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣁⢉⠂⠄⡦⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠂⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣐⣂⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣃⣁⣋
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⢀⡀⢀⣠⡀⠀⠀⠀⡄⠀⡀⢠⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠹⢀⡟⢸⡇⡏⠗⢰⣶⡞⠣⠋⠀⣾⢹⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠦⢼⠁⣴⢆⣧⡬⢨⡼⠧⠗⠚⠦⠿⠸⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣄⡀⠀⠀⠀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⡀⣨⡛⡅⢨⠑⡎⣇⢈⡇⢛⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠬⠵⠛⠲⠞⠁⠬⠞⠑⠞⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠠⠠⠴⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣙⠉⠓⠖⢄⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣘⣀⣀⣧⣚⠁⠀⠎
⠀⠀⠀⠀⠀⣀⣀⡄⠀⠤⢄⠆⠀⠀⠀⠀⠀⠀⠀⢼⠉⠉⠉⡉⡿⠿⠛⣿⣿⠇⠀⠀⡄
⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠤⠀⠀⠠⠀⠀⠀⠀⢀⢸⠀⢀⢠⣄⡘⠢⠁⢘⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⢀⡉⢡⠆⠈⣿⠀⠀⠋⠀⠐⠐⠘⠀⠀⠀⠟⠋⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⠤⢞⠃⢀⠄⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁⠉⠁⠀⠀⠀⠀⢀⣀⡧⠀⠰⡇⠀⢀⣠⢏⣿⡠⠀⠀⠂⠖⠀⠀⠀⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⠈⢿⣶⠖⠚⠒⠛⣱⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠘⠓⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠛⠇⠁⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠂⠉⠒⠉⠉⠉⠁⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠒
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⡠⠂⠀⠀⠀⠀⠂⠀⠉⠠⠥⠠⠀⣠⣾⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠗⢺⡿⣷⣄⡀⠄
⠁⠀⠀⠀⠀⠀⠁⠄⠄⠠⠠⢀⣾⣿⡿⣽⣿⣱⣽⣿⣿⣧⡀⠀⠀⠀⢀⡤⣮⢾⠇⠀⢸⠳⡷⡷⣝⣦⣰⣆⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣇⠊⡜⢰⠻⡟⢷⠂⢀⢻⣽⣿⣆⠀⣀⣾⣿⡛⡚⠀⠁⠀⡀⢈⡿⣟⣎⠛⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠀⠀⠀⠀⡀⡠⢰⠂⢰⣋⠸⠁⠀⢈⠀⠸⠈⣿⡿⣿⢆⢰⣾⠀⠃⠀⠀⠀⠀⡀⠀⠒⠸⠞⣶⣇⠀⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠒⠀⠀⠀⠀⢶⡿⢥⡊⠀⠌⠀⠀⢸⠀⠀⠄⢸⡇⢹⡘⠟⢸⠊⠀⠀⠀⠀⠠⠇⠀⠄⠀⢆⣵⢻
⠀⠀⠈⠀⠀⠀⠄⠀⠀⣼⡗⠂⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣾⣿⡦⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⢠⣸⠄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⢟⣿⠞⠑⠀⠀⠀⠀⡀⠀⢀⠀⢚⡯⢹⣿⣟⢿⠀⠂⠀⠁⠀⠀⠀⠀⠈⠀⢘⢐⡹⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣏⢧⠠⡂⠀⠀⠀⠀⠀⠀⠀⢸⠗⣿⡟⠻⣼⣆⠰⠀⠀⠀⠀⠀⠀⠁⢀⡇⣼⡗⠀⢄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢎⡥⠁⠀⠀⠀⢀⠀⠀⠀⣆⣴⣿⣇⣸⣽⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⣁⠿⠄⠄⠀⣀⡀
⠀⠀⠀⡀⠄⠈⠂⠀⠀⠀⠀⠀⡉⠓⠒⠀⣀⣈⠨⠄⠒⠋⠉⠉⠉⠉⢡⡈⣉⡉⠂⠄⠐⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠠⠤⠔⠋⣀⡀⢀⠀⠀⠀⠐⠁⠀⠀⠈⠠⠈⠤⠀⠀⠀⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠐⠢⠂⠦⠄⠀⠁⣀⣠⢄⠀⠀⠂⠉⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⡀
⣀⣀⣀⣀⣀⣐⣀⣀⣒⣨⣁⣀⣀⣀⣀⣐⣧⣤⣤⣒⣁⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢴⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡴
⠀⠀⠀⠀⠀⡴⢋⣀⠄⢀⡞⣷⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⡼⠁
⠀⠀⠀⠀⠈⠀⠉⣀⡴⠟⠁⣿⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋
⠀⠀⠀⠀⠤⠒⠋⠉⠀⠀⠀⣿⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⢂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢳⣶⠟⠛⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠉⠙⠋⠉⠛⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣸⠃⠀⠀⠀⡔⠘⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠉⠉⠙⢻⡄⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣸⠇⠘⠢⢤⣠⠤⠞⠁⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⡽⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠷⢂⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣰⢟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡆⠀⢀⣴⠶⠾⠲⠶⣤⡀⠀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⢀⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠈⠻⠿⣶⣶⡶⠿⠋
⠀⠀⠀⠀⠀⠀⢠⠖⣦⡰⠀⢤⢀⣤⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠘⢆⠸⣷⢁⣾⣥⣾⣤⣀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⢴⡚⢲⡤⣄
⠀⠀⠀⠀⣠⠖⠒⠚⢂⣽⠾⠋⠁⠀⠉⠻⣧⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⢘⡥⢤⡿⠛⠦⣤⠷⢤
⠀⠀⠀⠀⠈⠓⣶⡾⠋⠁⡀⠀⠀⠆⠀⠀⢹⣇⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣧⣸⠀⢀⣧⡀⠀⠘⠣⠚⢷⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠁⠀⠀⡄⠀⠠⠈⣿⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⣰⢸⡏⠙⣇⠀⠈⡷⠒⢺⠙⢻⡅
⠀⠀⠀⠀⠀⠘⣧⠀⠂⠀⠰⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⠹⣮⡳⠶⢾⡓⠛⢷⣤⡴⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠠⠀⠰⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣤⣤⣤⡶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢧⣰⣤⣬⣿⠶⠚⣹⠶⠋
⠀⠀⣀⡀⠀⠀⠀⠀⠉⠲⢤⡁⠀⠠⡀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⠶⠟⠶⠶⠋
⣴⠋⢹⣷⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠠⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠘⢆⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢻⣿⣿⠆⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠐⠶⣤⣄⡀
⠀⠈⠂⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠈⢀⡟⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⣀⠀⠉⠻⣦⡀
⠀⠀⠀⢀⠘⠛⠙⠛⠛⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣰⡟⠁⠀⢀⠀⠀⠀⠀⢀⡟⠲⣄⠈⢻⣆
⠀⠀⠀⣿⡇⠀⢄⡀⠀⠀⠀⠈⠙⠛⠷⠶⢶⣤⣤⣤⣤⣤⡤⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⣿⠀⢀⠀⠀⠑⢄⠀⠀⠌⠀⠀⠘⣆⠀⢿
⠀⠀⠀⠸⣷⡀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠀⠀⢀⣾⠏⠀⠀⠀⠀⢸⣿⠀⣼⠀⢀⣀⡤⢽⡿⠂⠐⠒⠒⢻⠀⢸
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠈⠙⠛⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠟⠋⠁⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠘⣿⡀⠸⡏⠉⠀⣠⠏⢠⠈⠢⣄⠀⣼⠀⢸
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠹⣄⡰⠃⠀⠀⣇⠀⠈⣻⠃⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡈⠛⠶⠦⠤⠾⠶⠛⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣦⣤⣤⣤⣶⠾⠟⠁

⠀⠀⠀⣀⣄⡀
⠀⢠⣾⣿⣿⣿⣦⠀⠀⢠⣿⣿⣿⣿⣷⡀⠀⢀⣀⣀⣴⣷⣦⣤⡀⠀⢠⣤⣆⣶⣿⡄⠀⠀⠀⣄⡀
⠀⣸⣿⣿⣿⣿⡿⠀⠀⠘⣿⣿⣿⣿⣿⢃⣤⣿⣿⣿⣿⣿⣿⣿⡿⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⡧⠄
⠀⢻⣿⣿⠿⠋⠁⠀⠀⠀⠈⢿⣿⡿⠁⠈⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠙⢿⣿⡿⠁⠀⠀⠀⢨⠁
⠀⠀⠈⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣄⢉
⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⠀⠀⢠⣾⣿⣿⣿⣿⣆
⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠸⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡶⠀⠀⠀⢿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠁⣀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⡀⢈⣀⣠⣤⣤⣀⠀⠀⢀⣠⣀⣾⣿⣿⣧⣠⣄⡀⠀⣾⣿⣿⣶⡀⠀⣴⠿⣿⣿⡃⠀⠰⣶⣾⣿⣿⠆
⠀⢸⣗⣾⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⠛⠛⠛⠛⠓⠀⢻⣶⣴⣿⠗⠀⠈⠉⠛
⠀⠈⠻⣿⣽⣿⣿⠿⠋⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠁⠀⢠⣤⣤⡦⠀⠀⠈⠿⠿⠇
⠀⠀⢀⣠⣤⣆⣤⡀⠀⠀⠀⠀⢀⣀⢈⡙⠛⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⣤⣤⣄⡀
⠀⠀⣽⣿⣿⣿⣿⣯⡀⠀⠀⣺⣿⣿⣿⣿⡀⠀⠀⣶⣦⣶⣶⣦⠀⢀⣾⡹⣼⣏⢿⣷⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠛⠉⠋⠋⠋⠛⠃⠀⠸⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⡇⠘⣿⣷⣽⠿⣸⣿
⠀⠀⢠⣤⣤⣤⣤⡤⠀⠀⠀⠻⣿⣿⣿⡿⠁⠀⠘⢿⣿⣿⣿⠟⠁⠀⠈⠻⠿⣿⡿⠇
⠀⠀⠀⠙⠿⠿⠟⠀⠀⠀⠀⠀⠙⠛⠋⠀⡀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⢀⡤⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⣿⣿⣧⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣁⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⢏⣼⠈⡧⣰⡃⢠⢀⠀⣴⡀⠀⡀⡀⠀⠀⠀⣔⣉⣉⠀⣀⢀⣀⠀⣀⠀⠀⢀⠐⠒⠂⢀⠀⠀⢠⠆⢀⠀⢀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠟⠁⠙⠆⠁⠋⠇⠇⠼⠒⠻⠅⠺⠄⢻⠃⢀⣤⠤⠄⠋⠸⠉⠺⠛⠸⠋⠀⠀⠘⠢⠄⠰⠏⠾⠅⠻⠅⠿⠰⠻⠱⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠁⠑⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢰⡃⣤⠠⠤⠄⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⠖⢶⢂⡴⠶⣶⡐⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠀⠠⠤⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⣿⡯⠛⠀⠀⢠⠐⠉⠉⠛⡑⣄⠄⠒⠒⠒⠲⣏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠛⣓⡈⠐⢓⠂⠀⡒⠂⠚⠓⠒⠀⠀⠀⠀⠀⠤⠠⠄⢀⣑⣢⢇⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠈⡈⠉⠉⢝⠰⠄⠒⠀⠀⠓⠎⠀⠀⠀⠈⠀⠀⠞⣶⠀⢀⠀⠀⠰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠄⠰⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠄⠀⠀⡀⠀⠀⠀⢈⠶⠦⢤⣄⠰⣃⡴⠃⣀⠤⢤⠀⡀⠇⠀⠀⠀⠀⠀⠀⠴⠀⠤⠄⠄⠠⠀⢠⠤⠀⠤⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣶⠀⠉⠠⠀⠀⠀⠈⠀⠀⠀⠀⠈⢿⣯⣐⢎⣔⣁⠀⠃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡆⠀⠀⠀⠀⠀⠡⠀⡀⠀⠀⡄⠈⠈⡉⣦⣉⠀⠀⡠⠁⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣻⣄⠀⠀⠀⠀⠀⠀⠁⡠⠂⠀⠀⠀⠀⠁⠨⠷⣾⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡟⠙⠦⠄⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠀⠆⠂⠀⠒⠁⠲⠀⠆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢰⡄⠀⠀⡤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣿⢦⣀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠸⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣙⣢⠤⠀⠀⠙⠻⠿⠶⠦⠖⠶⠆⠀⢙⣋⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠯⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣠⠼⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠴⠦⠀⠀⠉⠒⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠳⣤⣀⣀⣀⣤⣴⡖⠚⠉⠉⠉⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣏⠀⠀⠉⠉⢢⡉⠉⠓⣶⢭⣀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣹⡛⠋⠉⠩⠽⢿⣄⠀⠙⠀⠈⢻⣏⠁⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠉⠀⢲⣿⣷⡶⠉⠀⠀⠀⠘⢧⡀⠀⠀⠈⠀⠱⡦⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠐⠁⠂⠔⠋⠉⠉⠙⠀⠀⠈⠀⠀⠈⠻⣶⣄⠀⠀⠀⠙⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠰⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠲⣤⣄⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⢸⠸⡇⢀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⡞⠫⣝⠛⡿⠧⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠃⡆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⣂⡤⠞⠛⢏⡀⠘⢒⣲⡶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡰⢣⠎⡰⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠈⠑⣛⣽⢟⠀⢶⢹⢡⡑⠠⣹⣭⡓⢦⣄⡀⠀⠀⠀⠔⢁⠔⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢈⡶⣻⢇⡣⠛⡱⢸⡜⢎⢧⠳⢄⢢⣩⢟⡾⣿⠀⠀⠀⠔⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⣴⡛⡱⢺⠷⠇⠃⢘⣨⠒⠼⢛⣵⠟⢁⣿⣤⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠉⠛⠓⠛⠃⠛⠚⠚⣁⡤⠖⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠈⠉⠉⠁⢀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠠⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⡀⠠⠤⠄⠀⠠⢢
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠠⠾⠿⠶⠖⠀⠀⠈⠓⠀⠀⡼⠛⣛⣛⡛⠛⠀⡁⠀⣠⣤⡄⠶⢠⡈⠳
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶⠶⠂⢀⣤⡄⠀⠀⠀⠀⠀⠄⢇⠘⢿⣷⣴⣿⣳⣶⠤⡙⠻⠷⠚⠋⠀⢸⡆
⠀⠀⠀⢀⡥⠀⠘⠛⠉⠁⠀⠂⢸⡿⠛⢁⠀⠀⠀⢀⢄⡰⠠⡀⠀⠈⠉⠉⢉⠕⣢⣤⣤⣀⡀⠀⠀⠠⣷
⠀⠀⢀⣾⣷⣶⡶⠆⠀⠀⠜⢀⣀⠐⠒⠀⠀⠀⡤⠃⠀⠀⢣⠠⠙⠒⠐⣦⠏⠘⢿⣻⣻⣻⣿⡀⠆⠀⢢⠄
⠀⢠⣾⠋⣽⣿⣦⡀⢈⠀⣴⣿⣿⠀⠀⠀⣠⠊⠀⠀⠀⠀⠘⣆⠐⠀⠆⠹⣿⡂⠀⠙⠿⠿⢟⡰⢈⡄⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⣿⠁⠀⠸⣿⣿⣷⠈⣰⣿⢿⡟⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⢸⡄⢀⠀⢠⡈⢿⡔⢀⠈⠁⠻⢒⣟⣿⣧⡀⠄⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠁
⠀⣧⠀⠀⢠⣛⡿⠿⠀⠀⠛⠛⢠⠆⡼⠁⣤⠟⡆⠀⡆⠀⠈⠈⣷⠈⠀⣁⠱⠼⢿⣾⣶⡜⠯⠈⢿⣷⢻⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁
⠀⢹⡷⠂⢀⣯⣙⡒⠘⠃⢤⢀⠀⡼⢁⡟⠰⣄⡙⠀⠀⠀⠈⠀⣿⣇⠁⠀⠀⠂⠸⠛⠘⣰⣶⣦⡀⠳⢶⠟⠈
⠀⠸⠁⠀⣿⣿⣿⣿⣷⠀⠀⣿⢰⡇⠀⣒⢧⡊⠃⠀⠀⠀⠀⠀⣿⣇⡃⡀⢀⠈⡀⡀⡃⢻⣷⣿⣷⡀⢀⣀⠂
⠀⡼⡀⠀⠻⢿⣷⣿⡿⠁⢸⡟⣸⠀⠀⠉⠀⠁⠀⡄⠀⠀⠀⠀⣿⡏⠨⢴⢩⠤⠁⠈⢩⠈⢿⣯⣿⠇⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠁
⡜⣠⣌⠀⠞⠉⠛⠉⠀⠀⠸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢹⡇⠰⠷⡴⣶⢿⠈⠾⢰⡤⠤⠤⢀⣤⡁⠀⠀⠀⠀⠀⠐⠀⠂
⣽⣿⣿⣦⠈⠀⣼⡟⣶⡌⠃⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⠀⢀⠐⠈⣧⠀⢲⣠⣬⠃⡀⠀⣶⣿⠶⣄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠠⠄⠀⠀⠂
⠸⢿⠟⢺⡄⠀⡟⣯⠡⣿⡄⠇⣿⣇⠀⠀⠀⠀⠀⢀⣙⡀⠈⢣⠀⣻⠀⠀⣔⠊⠀⠀⢀⢹⣥⡀⢏⣤
⠐⡀⠳⠞⠑⠀⠳⣯⣟⡿⠑⢶⣿⣿⠀⠀⠀⠀⠀⢈⣁⡀⢖⣀⡀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣼⡿⠀⠠
⠀⠠⠀⢀⠠⡄⠀⠈⡍⠀⠠⠘⣿⣿⠀⠀⠀⠀⠀⠀⢡⡢⠀⡡⠁⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠀⠆⠰
⠀⢏⢰⣿⣧⠀⣰⣷⠐⢠⣦⠠⢿⡿⠀⠀⠀⠀⠀⢀⠀⠙⠊⠁⠀⠈⡇⠀⠀⠀⠀⠀⢰⣯⠆⠀⣶⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠚⢸⣿⣿⠀⣽⣿⡆⢸⣿⡇⢸⡇⠀⠀⢀⠀⢀⠃⢀⠈⠁⡘⠀⠀⣿⠀⠀⠀⠀⠀⢙⣿⡇⢈⢿⣿⠀⣁⡉⠈⠀⠀⠀⠀⠀⠁⠑⠁⠀⠀⠀⠀⠁⠀⠈
⠀⠀⢾⣿⣿⡇⢻⣿⡟⢸⡿⣇⢸⡇⠀⡠⠚⠀⠈⠀⠈⠓⠺⠁⠀⠀⣿⠀⠠⠀⠀⠀⢾⢧⡇⠘⣿⣿⡇⠻⡍⠁
⠀⠀⢉⣷⣿⡇⠘⣿⡏⠀⢿⡟⠀⣧⠀⠀⢠⡖⠶⡤⢵⠀⠀⢀⠀⠚⠁⠤⢤⠬⢄⡀⠘⢶⡏⢀⢻⣿⠇⠠⠁
⠀⠀⠈⣄⠉⢠⠁⠈⠁⠀⠈⠁⠠⠎⠀⠀⣆⡁⠒⠢⣶⠆⠀⢀⢀⣀⡀⢺⣄⣁⠰⢃⡂⡀⠀⢀⣘⢛⠀⠰⠀⠀⠀⠀⡄⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡴⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠀⠀⠐⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣷⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⣿⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠀⠀⠀⠀⠀⠸⢿⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⢺⣿⡇⠀⠀⠀⠀⢀⠀⣒⢠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠶⢶⢦⠄⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⢯⣿⡇⠀⢀⡀⠀⢰⡄⣿⡷⠖⣘
⠀⠀⢀⣤⠀⠀⠀⠀⠈⡢⣀⡀⢊⠄⠒⠰⠾⠿⢿⡿⠿⠶⠀⠐⠀⡀⠀⠀⢾⣿⡇⠀⢄⠠⡄⠄⣿⡥⠶⠄
⠀⣰⠏⠉⡇⠀⢴⡞⠟⠟⡇⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⠿⢧⠀⣀⠁⢸⡞⠁⡔⠁⢀
⠸⠯⠀⠤⡀⠀⠀⣑⣦⢜⠀⢰⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⣈⠐⠌⢳⣏⡡⣍⢉⡃⣴⠂⣠⠄
⠀⣠⠾⣆⠑⣀⡾⣋⡩⠈⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠸⢏⠑⢶⠄⣶⣠⡾⢿⣿⣀
⢸⠏⠁⢸⣤⡿⠳⣄⠐⢄⠘⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣉⠣⡄⠈⠛⠀⠊
⠀⠀⠀⠈⠛⠖⡚⠂⠉⢲⡷⠟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠐
⠀⠀⠀⠀⠀⡴⠿⢷⣤⣤⡶⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⢠⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣰⣵⠤⠠⠀⠚⠐⠃⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠉⠉⠁⠀⠀⠀⠈⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡴⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣄⠀⠀⠀⠀⠲⡆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⠀⠐⠒⠶⠀⠀⠉⠄⠛⠉⢉⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠
⠀⠀⠀⠀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣀⠀⢀⡞⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⡀⠀⠘⠀⠙⠿⢷⣶⡶⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠂

⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀⠀⢰⠂⣈⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡆⠈⠀⠈⢳⡞⢤⡴⠎⠀⠀⠹⣧⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣫⡷⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢰⠀⠠⠆⠀⠐⣮⠛⢉⣁⡛⠞⠓⠧⠾⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⠈⠢⡀⠀⠀⠀⠀⢘⡀⡄⢀⣄⢐⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⣀⣠⣤⣼⠁⠀⡈⠛⢿⢠⡎⠀⠀⠀⠀⣧⡔⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣠⡿⢳⡀⠀⡁⡁⠈
⠀⠀⠀⠀⠀⠒⠒⢀⡌⠁⠀⠉⠓⡄⡌⠋⠀⠁⠀⠲⠐⠒⠂⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠈⠇⢀⠀⠀⠀⢀⣀⡥⠞⠋⠉⠓⠋⠀⠀⣀⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠼⠃⢄⣀⠀⠐⠻⢀⡇⠀⡀⠀⠀⠀⣀⠀⠀⢠⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⡋⠀⠀⡀⠀⠀⡀⠤⡀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣠⣤⡶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠤⣀⡉⠒⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠌⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠦⢞⠖⣲⣆⠁⡈⠴⠂⢒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢰⣃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠋⠙⠁⠀⠀⠀⠈⢿⣯⠦⣨⣿⡋⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠉⠉⢿⡂⠀⢠⠶⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⣈⠀⢘⣹⣷⢯⠁⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠐⣄⣀⣐⡦⢄⠞⠉⠀⡛⢢⣠⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⣤⠀⠀⡙⠉⣲⣉⠹⠞
⠀⠀⠀⠀⠀⠈⠀⠀⠶⡏⠀⠀⠙⡋⠙⣤⡄⠏⠉⠉⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡡⠃⢤⡄⠀⡴⠿⠑⠋⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠲⠀⠀⢠⢤⠄⠞⠁⠀⠈⠉⠈⠀⠎⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠱⣆⠂⣽⢧⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠁⠘⠛⠃⢄⣀⠀⠀⠀⣁⠜⠂⠀⠀⠁⠀⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡠⠔⠒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⠒⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠑⠒⣤⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡰⣠⠄⠳⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣹⡿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀⡠⣀⠀⢨⡿⠒⠛⠁
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⠙⠗⠃⠀⢱⠀⠀⠀⠀⠸⣅⠠⣙⠢⠄⠀⠠⠤⣒⠾⢁⡜⢉⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡩⠷⠤⡜⠓⢦⡗⠻⡅
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⢜⠷⢄⣉⡐⠒⢈⣉⡤⢖⠩⣢⢞⠏⢸⠀⠀⠀⠀⠀⠀⢀⢠⠎⢉⠖⠀⠲⣏⠟⠉⠉⡖⢧
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠫⣀⣒⡤⣭⣭⡥⢄⣒⠥⠚⡡⠋⢀⠆⠀⠀⠀⠀⡔⢊⣇⢈⠖⠳⣀⣀⣠⣏⣐⠤⠚⣷⠼⡀
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢿⠀⠭⠍⣉⣉⠡⠤⠖⠋⠀⢀⠎⠀⠀⠀⠀⠀⠓⠾⡀⢩⠣⢤⠼⠤⡆⠀⠈⡷⠚⣷⡴⠃
⠀⠀⠈⠲⢄⡀⠀⠀⠀⢂⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠚⠓⠒⠋⠒⠊⠉⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠆⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠘⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡲⠖⢶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⢀⡈⣇⠙⠈⡞⡿⡆⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠘⢦⡈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⡠⠊⢠⠌⢙⠳⣄⠋⠀⡗⢀⡢⠄⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⣠⢟⣀⡀⡀⠀⠀⠀⠀⠀⢠⠏⡄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠸⡄
⠀⢀⠜⠃⠠⠄⠀⢉⡁⠘⠳⡄⠉⣙⣆⠀⠀⠐⠥⡀⠀⠀⠀⠀⠀⠀⠀⠚⢡⣾⡷⡅⠀⠙⡮⠙⣦⡀⠀⠀⢸⡀⢇⣀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇
⢀⡾⠈⠃⢀⢄⠀⠑⠁⠦⠆⠹⡑⢿⡌⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⣀⡴⢻⠥⢷⡦⢵⡪⠔⢒⢼⠹⡄⠀⠀⠳⣄⠘⠦⠤⠀⠀⠀⣠⠏⠀⣀⡴⠃
⡸⠤⠰⠆⠀⠈⠀⠘⠀⠠⣢⢀⡇⠈⠧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⢹⡤⣚⢧⡦⣝⠲⠞⡼⠀⠙⡄⠀⠀⠀⠉⠓⠒⠒⠒⠉⠉⠉⠉⠁
⡷⠂⠰⠀⠀⠃⠀⠐⠓⢀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⣸⢀⣀⢗⡔⢁⡴⠚⠁
⠙⢤⣷⣀⣛⣠⠴⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠿⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡆⣿⣶⣿⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⡄⠀⣠⠖⠒⣓⡲⢶⡶⠀⠀⠀⠀⠀⠀⠀⠛⠻⡗⠦⣿⢹⡟⡷⠟⢫⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⡘⣾⠔⠂⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⠳⡈⡧⠛⢈⣩⠽⠟⠒
⠀⠀⠀⠀⠀⢀⣤⢄⣀⢀⡦⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⢷⡹⠲⠤⠤⠤⠾⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣯⠿⣰⣶⠗⠺⣿⢦⡌⠶⡄
⠀⠀⢀⡤⢚⡩⠝⠦⢠⣙⡦⠘⠖⠲⣌⠙⠢⡀⠀⠀⠀⠀⢰⢡⢊⠀⠀⠒⠾⠗⠂⠀⠀⠀⢀⢹⠀⠀⠀⠀⠀⢀⡤⢚⣩⣉⠉⣄⣚⡞⢿⠢⢸⠚⠇
⠀⢠⠏⠰⠟⠒⢸⡣⢶⢰⠦⢬⣯⠉⠉⠀⠀⢹⠀⠀⠀⠀⠘⣆⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣰⠋⢉⠀⠀⢃⠀⢰⠀⠘⠉⡇⠸⠂
⠀⢼⢀⠀⠀⠀⠈⠀⠘⠇⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡟⠀⠀⠀⢰⢧⠀⠸⠒⠒⢺⠀⠘⠓⠒⣷⠇
⠀⠸⣜⢄⡠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠁⠀⠀⠀⣇⢠⣀⣀⣇⣀⣸⠀⢤⠆⢰⠏
⠀⠀⠈⠳⠤⣤⣅⣈⣉⣡⣠⣤⣤⡤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⠤⢄⣀⠀⢀⡴⠃⠀⠀⠀⠀⠸⡼⡄⣀⣇⠀⣘⣀⣸⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣇⣸⠥⠖⠉

⠀⠀⠀⠀⢠⣶⣤⣤⣀
⠀⠀⠀⠀⢸⣝⣿⣧⡉⠙⢄⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⠂
⠀⠀⠀⠀⠈⢳⣒⣿⡻⣶⠘⡄⠀⣠⢾⣹⣿⠟⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⡇⠀⣿⣼⠿⢃⣠⠟⠁⠀⠀⠀⣀⣀⣀⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣷⣾⡇⠒⠚⠉⠁⠀⠀⢀⡴⡟⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢓⡶⢗⢄⣀⠀⠀⠀⢰⡿⣷⣿⡿⠀⣿⠏
⠀⠀⠀⠀⠀⣠⡶⢋⣩⣽⣿⠋⠀⠈⠺⢝⢧⠀⠀⢈⣷⣿⣋⣀⠟⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣼⣿⣶⣿⣿⠿⠃⠀⠀⠀⠀⠀⠳⣝⢲⡞⢓⣂⣉⣀⣀⡴⣟⣿⣻⡿⠟⢛⣷⣶⣤⡀
⠀⠀⠀⠀⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⣦⡈⠉⠙⠻⢿⣿⣾⣿⣿⣿⡳⣶⣖⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠻⣦⡀⠀⠀⠙⠛⣿⣷⣙⢁⣨⣈⠙⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⠳⣄⠀⠀⠀⠀⠉⠉⠛⠛⠟⠛⠛⠚⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠙⣆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣘⣧⣴⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣀⣀⣀⢠⣾⣿⣫⣵⣿⡅⠀⠉⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡳⠿⢿⣛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣴⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢯⣜⣁⣛⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⣰⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣐⠤⣀⠀⠀⠀⠘⠁⡤⠞⠓⠄⢀⠠⠂⠀⠀⢀⣘⣷⣻⢆
⠀⠈⠒⠦⠘⠄⠀⢀⡰⠀⠀⠘⡆⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣄⠀⠀⠀⠀⠀⠘⠛⠮⠻⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠈⠐⠃⠀⠡⠖⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⢠⠀⢄⡀⠀⠀⠙⠄⠀⠀⠀⠀⠒⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠈⠈⡀⢁⠀⠑⠀⠀⠸⣄⠴⠋⠱⡀⠀⠀⠂⠀⠀⠀⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢁⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠤⠙⠂
⠀⠉⠃⠀⠈⠉⠒⠠⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢴⣀⣁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠉⠴⠒⠂⠀⠀⠀⠠⠀⡠⠃⠂

⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⣀⡀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠉⠁⢉⠀⠈⢸⢳⠂⠀⢀⠴⠋⢠⠄⠀⠀⠈
⠀⠀⠀⠘⠀⠀⠈⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣼⣟⣛⣻⣿⣿⣿⣶⣬⣻⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣶⣶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⣶⢶⣶⣷⣶⣶⣶⣦⣤⣤⣤⣀⣀⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣭⣿⣿⣿⣿⣿⣟⣛⣻⠿⠿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡌⠉⠙⠛⢿⣿⣿⣿⣿⣿⡀
⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⠀⠀⠙⣿⣿⣿⣿⣇
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⠀⠀⠀⠙⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣷⣶⣶⣶⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣄⡀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⡏⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠄⠀⠀⠀⡀⠀⠄⠄⠠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⡀
⠀⠀⠀⠀⠀⠀⢀⠁⡁⠸⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠠⠀⡌⡈⠀⠀⠀⠀⠀⢀⠀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢃⠑⣿⠀⠤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠒⢾⡓⠂⠒⠂⠀⠀⠀⢙⣙⢉⢉⣭⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢺⣿⠠⠤⠤⠤⠤⠤⠄⠶⢤⡧⢤⠶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠶⢦⣤⠴⠦⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⢀⣈⣀⣤⣤⣤⣼⠤⠤⠤⠤⠤⠤⠤⠄⠷⡶⢶⡶⠿⠶⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠠⠈⢀⡇⠀⠐⠂⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣴⡵⢿⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣀⠀⠤⢸⠀⡀⡀⠒⠘⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠮⠐⢼⡎⠥⠧⠤⢤⡾⠥⠤⠤⠤⠶⠦⠔⠲⠶⠚⠟⠛⠟⢛⠛⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣥⣟⣉⣁⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠀⠀⠀⠀⢠⠖⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡆⣦⢊⡯⢤⣡⠀⠘⠄⠀⠀⢠⠁⠀⠠
⠀⠀⠀⠀⠀⠀⢇⠈⠰⠄⢀⠀⠀⠀⢸⣿⣟⠔⠉⣁⠀⣚⠀⢱⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠠⡆⠐⢆⢀⠳⢀⡞⠁⠀⠉⢳⠀⣁⠀⠘⡃⠀⢃⠀⢀⠁⢐⣤⠔⣺⡟⠀⣿⠀⠀⠀⢀⡐⠁⢸⠀⢀
⠀⠀⠀⠀⡠⠶⢲⣙⡒⢮⣿⣀⢸⡇⠀⠀⢀⡟⠀⢁⠀⠐⣟⠀⠶⠔⠚⠉⣁⠠⢠⢿⣇⣠⣋⣳⣤⡤⢅⣀⡀⠲⠘⡁⡇
⠀⠀⢰⠈⡡⠏⡊⢱⡸⠒⠀⣽⢸⠃⠀⢠⣾⡿⣻⢻⣧⠀⠠⠄⠀⠀⢀⣠⣀⡴⢃⣿⠋⠋⠙⠋⢘⣷⢀⡀⣗⠀⡾⢸⠇
⠀⠀⡍⢀⡴⠲⣽⡘⠫⡷⣧⠘⡟⣀⣤⣿⡏⣸⢯⣾⡣⠤⢀⣀⣠⣴⡟⢉⢽⣷⡟⣻⣷⣀⡀⣠⡴⣷⡜⣇⣜⡟⠄⠚⠤⢀⡀
⠀⠀⢆⢇⠳⠠⡏⣮⡓⠉⢸⡀⣿⡏⡽⣿⡇⣹⣿⡟⢿⡾⢷⢟⠛⣿⠀⡾⢠⡿⢰⡿⣯⣍⣻⣧⣾⡵⢧⣷⣫⢐⡈⣨⠤⠤⡏
⠀⠀⠈⠉⠒⠘⡈⠲⢁⠀⢀⣷⣿⣷⣆⣿⣆⢸⣿⠇⣸⣁⣫⣨⣦⣵⣤⣷⣈⣷⣼⢁⡁⠚⣹⡿⠁⠀⠀⠈⠛⡇⠀⢀⢀⠐
⠀⠀⠠⣤⣤⣵⣎⠳⣄⢳⡾⢡⣶⣿⣿⣿⣿⣄⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣐⢿⣇⠀⠀⠀⠀⣰⡟⠀⣄⣾⣮⡟⣀
⠀⠀⠀⢡⢹⣗⣪⢟⣪⣻⡇⣿⠟⡭⢻⡟⢡⢦⣿⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣾⣿⣷⣿⣇⣀⣠⡴⣻⠖⠚⠛⣿⠅⠒⠋
⠀⠀⠀⠈⢆⢿⣪⣯⡳⣽⠻⣮⠨⣵⠏⢐⣷⣿⣿⣿⣿⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣙⠿⣅⣸⠃⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠈⢆⠑⢵⣟⣽⠆⠻⣶⠆⣰⣿⣿⣿⡿⠋⣿⣾⡡⣷⣿⣿⣿⠋⡿⠿⠟⣿⠟⣙⣿⣿⣿⣷⢹⡇⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⢳⡄⠀⠙⢧⣸⡟⢠⣿⣿⣨⣿⠁⣼⣿⣿⣿⣶⣿⣿⡆⢰⣶⣶⣿⡿⢠⣯⣿⢟⢻⢏⣼⣧⡴⠛⠁
⠀⠠⠖⠒⠠⠒⠂⠛⠓⠮⣼⣿⠇⣼⣿⠏⢸⠇⢼⣿⠋⣤⡏⢰⣿⣻⠀⣾⡏⣽⠁⡇⢠⡿⠉⣿⣆⣿⢛⠁⠀⠀⢦⢠⡤⠖
⠀⠀⠑⠢⡄⠰⠛⣷⣦⡤⠼⣿⣧⣽⣷⣧⣴⢀⣿⢹⣦⣽⣿⣦⡄⣻⣄⣯⣦⣭⠀⣧⠘⣇⠸⠟⣼⠇⠀⠀⠀⠐⣻⡶⠃
⠀⠀⠀⠀⠀⠂⠬⢀⠘⣇⡴⠿⠆⠉⠋⣿⡏⢸⣿⢸⡇⠈⠻⠿⠛⢋⣩⣿⣿⣯⠀⣿⣻⣻⡳⣛⣁⣀⣀⠤⠶⠻⠄⠀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠫⡟⠓⠀⠀⢀⡾⠻⣧⠸⣣⣿⡇⠀⠀⠈⢯⣟⣿⣿⣿⡏⣸⡏⠹⣌⠙⢯⣄⣈⠉⠦⢀⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠄⠈⠀⣠⠽⣦⢀⣠⣿⠁⠀⠙⠛⢻⢣⣿⡄⠀⠀⠘⠻⣿⣿⣿⡴⢿⠀⢀⢮⡧⡎⠉⢲⢤⡈⢢⠀⠀⠋⠒⠒⠒⠂
⠀⠀⢀⠜⠁⠀⠀⠀⠉⣤⡒⡒⣒⣿⠦⠄⠠⠴⡷⣿⣺⣽⢦⡀⢀⠙⡭⡏⠉⢠⠼⣞⢮⠓⢷⣅⣨⠘⠸⠦⢀⡀
⠀⠀⠀⠀⠉⠈⠀⠀⢀⠞⡤⠛⡟⡹⠀⠀⠀⠀⠀⣿⠫⠔⠊⠉⠲⣯⣾⡁⠰⢾⠆⡉⡝⡇⠹⣈⡤⢀⣦⡌⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠐⠘⠡⡀⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠈⡛⠁⠀⠑⢦⣉⡚⣃⣇⠜⠀⠀⠘⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⢀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠤⠀⠀⠀⠄⠀⠀⠒⣲⣿⣷⠌⠐⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠐⠂⣀⢔⠚⠀⠐⡚⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡀⠂⠒⡈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠁⠀⠀⠀⢠⠔⠐⣀⠀⠀⠀⠘⠀⠂⠀⢀⡀⠀⠀⠀⠈⠀⡂⡀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⢐⡀⠠⠧⠄⡊⠀⡐⠀⠀⣴⡀⠀⠀⠘⣷⠀⠀⠀⢀⠀⡰⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⢞⠓⡤⢄⣚⢑⠃⠀⢈⢶⣿⠇⢆⠄⠄⣷⠀⠺⠂⡀⡠⢤⠁⠚⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠀⢤⡘⠒⠉⢸⡿⠤⡏⢄⠼⠿⢄⡦⢀⠄⠂⣀⠠⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠰⠢⠄⠲⢄⣀⠄⢀⠒⠂⠈⡀⠉⠁⠀⠀⡀⠴⠖⠘⢁⠈⠐⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠷⠉⠉⠐⠛⠂⡚⠃⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡽⣛⣻⣮⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣽⡷⣿⣿⣿⣧⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠻⠿⠿⠿⡄⢠⠊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⡎⠙⠶⠾⠿⠟⢣⣤⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣦⣄⣀⣰⣾⡯⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡧⡄⣤⣶⣾⣧⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⡉⣠⡆⠀⠛⢻⣾⣾⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⣿⣿⣿⣿⣿⣿⣿⡝⠇⠃⠁⠀⠀⠀⠀⠈⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣇⢴⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣮⣠⢀⣠⣀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣛⣛⣛⣛⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠋⠙⠛⠙⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠤⠄⡀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡠⠒⠀⠁⠒⢌⢢⠀⡠⠔⠊⡡⠄⠀⠠⢌⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠥⠄⠤⣀⣀⠜⠀⠋⠀⠀⠺⣀⠠⢄⠀⢀⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⠔⠂⠁⠀⠀⠈⠒⢄⡀⠀⣀⠤⠐⠂⠀⠐⠂⢄⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⣀⠔⠁⢀⠴⠊⠀⠑⠤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠣⢀
⠀⠀⡔⠉⠀⢀⡔⠁⠀⠀⠀⠀⢀⡠⠖⠓⢤⠀⠠⠐⠉⠢⣀⠀⠀⣀⣠⠾⢄⠀⢈⣢⣄⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠑⣄⠀⠀⠳⡀⠀⢀⡴⠚⠁⠀⠀⠀⠀⡱⠂⣀⣠⡤⠤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠉⠛⢲⡢⠄⠀⠀⢀⡜
⠀⠀⠀⠈⢂⠀⠀⠱⠞⠁⠀⠀⡀⢤⠀⡠⣪⠔⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⢂⠀⠀⠀⢀⠴⠊⡰⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠃⠀⠀⢀⡴⠋⠀⠀⢀⠔⠁
⠀⠀⠀⠀⡰⠁⠀⠀⡠⠋⢀⠜⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡔⠁⠀⠀⣴⠟⠀⠀⠀⡠⠃
⠀⠀⠀⡘⠀⠀⠀⠜⠀⢀⠎⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⠌⠀⠀⢠⠪⠃⠀⠀⠀⡰⠁
⠀⠀⡘⠀⠀⠀⠌⠀⢀⠂⠀⣮⠃⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⠎⠀⠀⠠⠱⠁⠀⠀⠀⡰⠁
⠀⠰⠁⠀⠀⡘⠀⠀⠎⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡌⠀⠀⠠⢡⠁⠀⠀⠀⠰⠁
⠀⠆⠀⠀⢠⠁⠀⡘⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠰⠀⠀⢀⢃⠆⠀⠀⠀⢠⠃
⢀⠀⠀⠀⢈⠀⢀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠇⠀⠀⡈⡈⠀⠀⠀⢠⠃
⠘⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡘⠀⠀⠠⠱⠀⠀⠀⡠⠃
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠠⠁⠀⢀⢂⠂⠀⢀⠜⠁
⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢀⡰⠁⠀⢠⢂⡆⠄⠊⠁
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠈⠒⠤⠀⡀⠀⠀⠀⠀⢀⡤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠊⢠⠞⠀⠐⠋⠉⠴⠆⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢈⠀⠂⠀⠀⡈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡔⠂⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢀⢀⣠⠞⠀⠀⠀⣮⢀⠄⠀⠀⣠⠄⠈⠋⠀⠀⢀⢁⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⡾⠛⠁⠀⠀⠀⡈⣄⡵⠂⠀⠈⠁⠀⠀⠀⢠⣄⢼⣿⡅⢨⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠋⠀⠀⠀⠀⠀⠀⠀⠰⡙⠃⠀⠀⠀⠀⡀⢀⠀⡭⠞⠉⠀⣿⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠆⠀⠀⠀⠀⠀⠈⠀⠐⠇⠀⢠⣀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠈⣀⠸⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⣾⡿⠄⢣⡄⠀⠀⠀⠀⠀⠀⠀⠁⢀⡸⠁⡤⣠⠊⠀⠀⠀⠀⠀⣀⣤⡟⢀⣀⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠘⠁⠀⣾⢿⡇⠀⠀⠀⡄⠀⠀⣠⡴⠞⢁⡎⠐⠚⠂⠀⠀⠀⠀⢋⣁⣀⡀⢿⡝⠃
⠀⠀⠀⠀⠀⠀⠐⣈⠀⡠⢸⡋⠁⠁⢀⡴⠚⣁⣤⠾⠫⠖⣺⡷⠖⠚⠂⠀⠀⠀⠀⠀⠁⠈⢻⣯⠸⠃⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠛⠧⡀⣿⠀⡰⢀⠎⠀⠘⠁⠁⠂⠀⠉⠻⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⠛⢦⡅⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⠟⢀⡘⢀⣠⠤⢼⠀⣤⡄⠾⢧⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠀⠑⣬
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⢱⣸⣿⡏⠀⠀⠀⡇⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠸⠈⣾⣻⣷⠦⣶⣘⠀⠀⠀⠀⠀⠀⡘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢿⢻⡀⠈⠒⠠⣂⠻⡄⢈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⠀⠐⠘⠃⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠑⠀⠀⠳⣄⣠⣾⣡⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣴⣻⡢⠐⠌⢯⣾⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠲⣿⡞⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⣠⡾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⣸⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⢍⢀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⢀⣠⠻⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠺⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⢠⠀⠸⣇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠀⠘⠯⡀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠘⠀⠘⠂⡠⣹⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⣆⢰⡀⢀⡀⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣎⣿⡆⠀⠛⢷⡆⠁⠡⠘⡆⠄⠠⢷⡰⠄⣂⡘⠀⢳⡄⠹⣆⠈⢻⡄
⠀⠀⠀⣀⢀⡤⢤⠀⣴⠓⢦⡙⠿⠀⣿⠠⠐⠂⠙⢶⣀⢧⣽⢿⣟⣿⣷⣆⢌⡃⡘⣆⢻⡂⣿⠀⠈⡇
⠀⣤⢨⡀⢸⡏⠀⢀⣭⡁⠄⣿⡀⣡⡾⠷⣤⡃⠒⢸⠇⠀⠀⢻⢿⢿⣽⣷⡞⡹⢷⠈⢀⢳⡈⠂⠀⠃⠸⠆
⠀⠹⠐⡀⢸⣧⡀⠘⢿⣯⠐⢸⡟⢻⡀⠀⠀⣽⢦⡙⢄⣶⣆⠙⢌⣻⣿⣿⣇⢰⣿⢇⠈⣦⠀⠀⣤⠀⠀⢆⡸
⠀⠀⣿⣿⡌⢋⡙⢦⣈⠉⢀⣼⡃⣸⠇⢠⣶⣄⠈⣣⢀⡽⠟⠇⣾⡉⣿⣿⣿⡶⣿⣏⢱⢀⡀⢨⣿⣤⡄⠈⣿
⠀⠀⠸⣟⠃⢼⠀⠀⠻⣄⠀⢹⡇⣹⣦⡀⠻⠟⢠⡞⠛⢦⣈⣠⢾⡿⢿⣿⣿⣷⢾⡏⣹⠀⠸⢆⠙⢛⡝⢀⣽⠃⠀⠀⢰⡄
⠀⠀⠀⠹⣇⣾⢰⣧⠀⢻⡄⢸⣇⡼⢜⠻⣦⡀⠨⡇⣤⡠⣿⣥⣄⢿⣌⣻⣿⣿⣶⢇⣼⡁⣤⠸⣦⣀⠀⣸⣇⠀⠀⠀⣶⠏
⠀⠀⠀⠀⢻⣇⢸⣿⢦⣼⡇⣈⣿⠁⢸⡀⠌⠳⢤⣇⢿⡿⠸⡟⠿⡟⠉⢿⣿⣿⣿⣿⣯⠐⣿⡦⠙⡟⠀⠹⣿⣧⣴⠂⠁
⠀⠀⠀⠀⠘⣯⡴⣄⠞⢨⡇⢰⡄⢀⢸⡇⠂⣠⠀⠙⢧⣀⣀⠿⣿⡾⣛⣦⣽⣿⡟⣿⠿⡄⢿⣧⢦⡝⠀⢀⡽⢀⡟⠀⠀⣤
⠀⠀⠀⠀⢠⡿⢹⡟⢁⠀⢸⡎⣱⣦⠘⣆⠉⣿⣿⡆⠠⡏⠔⠀⠉⢶⣬⣽⣿⣿⣿⡿⢸⡏⣄⢻⣄⠐⠀⠿⠀⣸⡇⠀⢠⡿
⠀⠀⠀⠀⢸⣏⢸⣟⣥⡄⠀⢏⠸⣿⠀⢸⣦⠀⠹⠂⠘⣧⠸⣿⣷⢾⣀⣿⣿⣿⡿⣷⡈⠃⢻⡜⡟⠓⣽⠀⢸⠟⠠⠆⣰⡄
⠀⠀⠀⠀⢹⣯⢸⡿⣿⣿⠀⣿⠀⠀⣰⠞⠀⢿⡀⠀⡾⠛⠶⣌⠁⢠⡼⣯⣽⣿⣷⣶⡝⣿⣷⠘⣷⢾⣧⡄⢸⠆⢴⠀⢿⣦
⠀⠀⠀⠀⣿⣿⣯⡅⢈⠉⢀⡇⠀⣰⢃⣾⣧⢸⡟⢶⢻⣿⡠⠟⠛⠛⣧⣾⣾⣿⣿⡾⢁⢿⣟⠲⠁⢸⡏⠀⣏⠀⢰⠇⢸⡟
⠀⠀⠀⠀⣿⡲⣿⣤⣴⣮⡉⣿⠒⣾⡀⠛⢛⣼⢃⣿⠈⠿⣸⠰⣷⣤⠚⣿⣿⣽⣿⣠⡌⢸⣿⠀⡾⠀⢿⡀⠹⣇⢘⡇⠸⡇
⣰⠛⣦⠸⣧⣿⣻⠦⣟⣫⣥⣿⣄⣈⣳⡖⠹⣹⣿⣏⣻⣄⡼⣦⠙⠋⢻⣝⣻⣿⣿⠋⣴⠘⢋⣀⠁⣸⡏⢿⠀⠋⣿⠓⡀⢶⡀
⠻⣤⡟⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣡⡏⡛⠧⢙⣦⣟⢩⣿⣿⣿⡛⠙⠆⠻⢿⣀⠙⣿⡈⠁⠀⣿⠘⠇⠸⡷
⠀⠀⠀⣿⢻⣿⣿⣿⣿⣛⠻⡿⣧⣄⣬⠙⠋⠙⠛⢿⡟⠻⣶⣿⣟⢿⡿⣧⣽⣿⣿⣿⣴⣿⠁⣿⠋⠸⣿⡟⠒⠀⡟⣶
⠀⠀⢠⡟⣿⡘⣿⣿⣿⠟⣦⣑⡈⠉⢳⣴⡅⠰⠤⠀⠈⠙⢦⣻⣿⣦⣀⣱⡲⣿⣾⣿⣽⣏⠀⠸⡇⣀⢹⣻⠀⡄⢟⢻⣇
⠀⠀⣼⠏⣼⠆⣿⣿⣽⣧⣟⣏⣿⠷⣼⡁⢿⣄⠀⣾⣷⣤⠀⠈⢋⢿⣯⣄⢉⡾⠿⣿⣻⣿⡁⠶⡀⢣⣄⠹⣤⢣⣌⠀⢿⡄
⠀⠰⣿⣦⢻⠄⣿⣿⣿⡥⢈⠿⣴⢟⠟⠿⣶⣿⣦⠘⣿⣿⣷⣄⠀⢶⣽⣷⣆⡹⣻⣿⣏⣽⠇⡀⢸⡏⠻⣄⠀⢹⣧⠀⢸⡇
⠀⠰⣟⣿⣿⣄⣿⣟⡙⣧⠐⠀⠝⢫⡀⠀⢈⠛⠻⣧⡈⠙⠿⠟⠁⠀⠞⢣⡞⢿⡽⢿⣿⣏⠋⠱⣾⡿⠈⢻⡀⢿⡇⠲⠸⣇
⠀⠀⢹⡇⣿⢿⡿⣿⣿⠀⢦⡳⠄⠁⡹⣆⢬⡄⠀⡈⠙⢦⣐⠄⠘⢱⡄⢠⠹⣍⣿⣦⠻⣿⠂⠰⠃⡿⠈⣾⠁⢸⡇⠘⠀⠛
⠀⠀⢸⣟⢹⣧⢠⡘⣿⡆⣀⠙⢦⠀⠠⠈⠳⣌⠁⠀⠀⢂⠈⠛⢦⣤⡀⠂⠀⣴⠻⣿⣷⣻⠐⠀⢸⡁⠀⡏⠀⢸⡛⠀⣿
⠀⠀⢸⣿⢀⡏⢱⣄⣸⣿⣿⣿⣎⠳⣄⠀⠀⠙⢦⡀⠸⣿⡿⣧⡀⠀⡉⠒⣄⡉⢿⣿⢿⣇⡶⢠⠊⣰⢰⠃⡀⠈⡇⢸⡏⠀⢠⡆
⠀⠀⢸⡤⠜⣣⠂⠀⠉⢻⣽⣿⣿⠀⡘⢦⠀⠡⠞⠑⢦⡙⢿⣿⣷⠀⠀⠀⠈⠛⢤⡙⢿⡿⢥⠟⡍⢳⡄⠀⠿⣼⢣⡌⢣⠀⠘⠇⢠
⠀⠀⢸⢰⣆⢹⠀⣿⡞⡀⠙⢿⣍⠶⠀⠴⢇⢰⣾⣷⣦⡈⢣⠝⡋⠓⠀⠰⠀⠀⠀⢻⣬⣷⣿⣰⣿⠂⠀⡆⠀⣿⠀⢀⡜⠐⡇⠀⢰
⠀⠀⢸⡞⠿⢸⠀⢿⡧⠀⣠⢀⠙⣿⣿⣦⠞⢆⡙⢷⣽⣿⣌⠣⠈⠒⢩⠶⣰⣄⣄⣠⡁⣻⡟⢩⡿⣤⢂⡆⠀⠛⠁⠈⠁⠀⣽⠀⠸
⠀⠀⢸⠀⠄⢰⠀⠴⠀⠰⠆⢻⢡⡌⢿⣿⣷⣏⣙⠢⣉⣉⠽⠛⢠⣓⣀⣬⡉⢉⡹⣇⢼⣿⡷⢧⡀⢉⣿⣷⠀⢲⠇⢠⡇⠀⢻
⠀⠀⢸⡇⢰⡜⢳⡄⢁⠖⠀⡉⣸⢿⡌⢻⣯⣿⣣⣶⠏⠥⠰⠾⣁⠉⠩⣤⣁⣨⡴⠛⣻⣿⠃⢈⡃⠘⢿⣿⠀⢸⠄⠈⣇⡴⠋
⠀⠀⠈⣧⡎⠹⠂⣻⣎⢁⣄⠣⠛⣦⡄⢠⡟⠻⠿⣷⣿⣟⣥⣻⣾⣿⣿⣿⣤⣾⣷⣾⣿⢇⢋⡈⠁⠞⠠⢹⡆⠈⣦⠀⣼⡅⠀⣠⡄
⠀⠀⠀⣿⠀⠀⢻⡟⡀⢺⣿⡇⠀⡼⢁⠀⢡⡘⠆⣹⠉⠉⣹⣟⠛⠛⣫⣿⣿⣿⣿⡏⢻⡀⡏⢀⠋⡀⢠⡌⠃⢀⡏⠠⠃⣨⠃⠙⡇
⠀⠀⠀⢷⠀⡐⠚⡁⢡⠞⠻⠃⢨⡇⣸⣦⡀⣇⢀⡭⠶⣌⠻⠿⢚⣪⡹⣿⣿⣿⠿⢀⣼⡇⢠⣾⡄⠁⡄⢃⠀⠄⠀⣤⠀⠛⠈⣶⡏
⠀⠀⠀⢸⡇⠟⡧⠙⠌⣆⠀⠀⡆⠰⢿⡿⢀⢱⡞⡀⢀⣈⠛⠛⠋⢈⣹⣾⣿⠿⢠⢿⡋⡌⢸⣧⢹⣠⣆⠘⠃⠀⢸⡅⠀⢀⡴⢸⠇
⠀⠀⠀⠀⢷⡼⢉⣶⡄⠝⠀⠀⣵⠒⠀⢰⠉⠊⠷⠀⣿⣿⠂⣠⠓⢾⣿⣿⣿⠡⡀⠾⠁⠈⢸⡟⡏⠀⡟⠀⢾⠀⠘⣧⠀⠀⠀⣎
⠀⠀⠀⠀⢿⡡⠘⣿⠇⠘⡦⢀⡽⠒⢰⠁⣶⣶⡀⢡⣈⣀⣠⣧⡀⢶⣿⡍⣹⠄⠋⠀⠠⠫⠞⢿⠃⣼⡧⠀⠄⠀⣤⡁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠙⠃⠀⠀⣾⠠⡎⠁⡄⢸⡀⢿⣿⠇⣰⠌⠁⢸⡁⠈⣻⣯⠵⢶⣠⣿⡠⠀⠠⡘⠟⢂⡾⠁⠀⢀⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢸⠀⠀⠠⠇⣈⠀⢶⠁⠀⠰⡏⢙⣦⣽⣿⣃⣶⣤⣏⠃⠈⢰⡏⣶⠈⠁⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠆⠀⠉⠀⠻⠈⠂⠈⠳⢮⣿⣿⣟⣹⡿⢋⡏⠈⠃⠘⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣧⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣋⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⡏⣼⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣽⣿⡿⠿⠋⢩⣩⠆⢀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿⡏⣋⠄⢠⣿⣿⣷⣦⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡰⣿⠺⠟⢃⣀⣿⣌⣡⣿⣻⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠇⡀⠃⠀⢰⣾⣿⠟⡻⣿⣿⠿⠟⠻⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣎⠀⠟⡋⣻⣿⡁⣼⣷⣝⣿⣶⣿⠆⠀⠒⣶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠒⠲⣄⡀⠀⠀⠀⠈⢙⠿⠉⢠⣄⣻⣾⠛⠾⠿⣟⣿⣿⣿⡧⠒⠀⠀⠀⠀⠉⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢹⣿⣪⣟⣵⡾⠗⠀⠒⣼⣿⣿⣿⠶⣶⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⢠⣾⣿⡏⠉⠀⠀⠀⠀⠰⣿⣿⣿⣿⣾⣿⣿⣶⣶⡀⣀⡀
⠀⠀⠀⠀⡠⠀⠀⠿⣿⣿⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⠛⠛⠻⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠰⠶⠀⠀⠔⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⢤⣀⣀⣀⣠⠤⠤⠚⠑⣲⡖⢶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠉⠉⠉⠉⠉⠉⠁⠶⡖⠺⡉⠠⢨⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢳⡅⠀⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠾⣷⡄⠐⡄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠶⠤⠤⣧⣼⠈⠢⡀⠆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣋⠀⡴⠊⠂⠢⠀⣀⣀⡈⠓⢄⠈⠢⢼⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢡⠊⢳⢸⠀⠢⡀⡴⢋⠀⠀⠉⠆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⡄⠈⣋⣗⠀⣴⠄⠠⢀⢀⣠⢆⡀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡜⠁⠆⣝⠻⣼⣶⣷⣢⣊⡭⢌⣀⢠⣱⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠣⣄⡅⠂⠭⣴⡞⡙⣧⠨⡤⣭⣤⣤⠟⠀⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⢀⡂⢂⠎⡀⠳⠷⠴⠖⠤⠑⡆⠀⢰⠖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣀⣀⡀⢀⠀⣀⣈⣦⡈⠪⣤⡍⡠⠁⠂⢳⣄⡀⣀⠔⢁⡴⠉⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠪⠶⢾⣟⡀⠺⢛⣛⣛⣛⠻⠽⠢⣀⡈⠑⠂⠒⠁⠀⣁⣠⡴⣿⣃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠾⠿⠭⠽⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠘⢄⡙⣦⠀⠀⠀⠀⠀⠀⠀⠓⣤⣇⠀⠀⠀⠀⠀⠀⠫⣉⣲⡴⢊⠇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠲⣄
⠀⠀⠀⠀⠀⢠⢖⠉⠛⠓⠂⠒⢄⠀⠀⠀⢀⡎⠀⠈⢇⠀⠀⠀⠀⢠⢋⠀⠌⡉⣉⡆⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢘⡄
⠀⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⢸⠀⠀⢀⠞⠀⠀⠀⠘⢆⠀⠀⠀⠸⡠⠀⠄⠄⢠⠃⠀⠀⠀⠀⢀⡠⠞⠁⣠⠀⢨⠃
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡜⠀⠀⡎⢰⠀⠀⠀⠀⠈⡆⠀⠀⠀⠙⢄⠂⣂⠎⠀⠀⡴⠖⠍⠡⠤⠴⠚⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠘⢆⣀⡤⡠⠞⠀⠀⠀⠘⢼⡀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠓⠒⠐⠒⡒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢹
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠞⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢹⡀⠀⠀⠀⠀⠀⡠⣒⠒⠚⢖⠳⢄
⠀⠀⠀⠀⠀⠀⣴⠿⠂⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⣠⠒⢾⡽⠓⠻⣼⠒⠒⢤⠀⠀⢸⢱⠁⠀⠀⠈⡇⠀⢳
⠀⠀⠀⠀⠀⢼⠉⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡞⡳⣤⠼⣇⠀⣠⡿⠷⢖⠉⠀⠀⠸⡘⠀⠀⠀⠐⠁⠀⡜
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⣸⣛⠼⠣⡄⠀⡏⠉⣣⣀⠜⠀⠀⠀⠀⠑⠤⣀⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣀⡠⠔⠃⠀⠀⠀⠐⢇⡸⠦⠤⠚⠒⠋⠚⠁⠀⠀⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢇⠒⠀⠂⠀⡀⠁⠀⠁⠀⠢⠀⠂⡇⡬
⠀⠀⠀⠀⠀⠀⠀⣦⡀⡼⢳⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⢦⡈⢠⠄⡀⠘⠀⠈⠠⡀⢉⡜⡰⠁
⠀⠀⠀⠀⠀⠀⠀⡴⣌⢣⠔⢉⢼⡀⠀⠀⠀⠀⡠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢭⡲⠤⣁⣈⣀⣉⡠⠴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢱⡤⣿⡴⠥⣜⠀⠀⠀⠀⠮⣤⡤⢾⠟⠉⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⢀⡜⢇⢀⡼⠳⣄⡏⢦⠀⠀⠀⠀⠀⢠⢣⠂⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⢸⣤⡴⠻⢄⣠⠊⠉⡿⠀⠀⠀⠀⠀⢸⡸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡔⠉⠒⠊⠁⠒⠂⠉⢉⠆
⠀⠀⠀⠀⠀⢻⡀⢑⣤⣞⠉⢲⠾⣸⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣀⣀⠴⠃⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠈⢟⠻⢤⣈⡽⠛⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠢⢤⡀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠫⠥⠝⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⡤⠂⠐⡆⠀⠀⣠⠔⠀⠀⠀⠐⡄⠀⣴⠈⠉⢠⠀⠀⡠⠤⠄⡀⠀⡠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢁⣀⣀⠸⡇⠀⠐⠀⠀⣷⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠃⣼⠁⠀⠀⠂
⠀⠀⡀⠀⢀⠀⠀⠀⠀⣆⠀⠴⠞⠛⠉⠡⡀⣷⠀⠀⠈⠄⣿⠀⠀⠀⢸⡇⠀⠀⠁⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠃⠀⠀⢀⣿
⠀⢸⡇⠀⠸⠀⠀⠀⠀⣿⠀⠐⠀⠀⣀⣠⠄⣿⠀⢀⠀⠀⠻⠀⠀⠀⢸⡇⠀⠀⠀⢠⡆⠀⢀⠀⢸⠀⠀⠀⢀⡿⠀⠀⠀⢸⡇⠀⢰⡇⠘
⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⠀⠀⡶⠛⠉⠁⠀⢻⠀⠘⢷⡀⠀⠀⠀⠀⠸⣇⠀⠣⠀⠠⠏⢀⠀⠀⢸⡄⠀⠀⠛⠁⠀⠀⠁⣾⠀⠀⠈⠉⠀⠁
⠀⠀⣿⠀⠀⠆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠆⢸⠀⠀⡀⢻⣄⢀⣀⡀⠀⠹⢦⣄⣀⡀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠶⣦⠀⢀
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠸⡇⠀⣀⣠⣤⣴⠂⠘⠓⠛⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠆⠀⠀⠀⣼⠃⠀⠀⠀⣿⠀⠀⡀
⠀⠀⢹⡇⠀⠀⠀⠀⣀⡄⠷⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠀⠀⣿⡀⠀⠇
⠀⠀⢸⣧⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂
⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠋⠈⠀⠉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣧⢴⡟⠛⠛⣶⡤⣾⡛⠛⢳⠀⠀⢀⣀⣙⣛⣛⣢⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣂⡽⠀⠸⣗⣋⡾⠚⠿⣿⣭⣟⣛⣛⠻⠬⠝⠷⠶⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠙⠀⠀⡀⠀⢉⣵⣆⠀⠀⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠄⠀⠀⠀⠠⠀⠘⠋⠀⠐⢁⡴⠞⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣤⡔⠁⠀⠀⠀⢀⣴⠞⠉⢀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠛⠶⣄⠀⠈⠛⠷⣦⣴⡶⠓⠈⠁⣨⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢈⠂⠰⠦⢤⡙⣿⡄⠀⠀⠞⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⣷⡄⠀⠀⠀⠙⠧⠀⠔⠂⢋⡤⠦⠈⡣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠉⠛⠻⡦⠐⠿⠿⠶⠁⠀⠀⠞⠁⠚⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⣶⣤⠾⠶⣶⠮⠐⢶⡾⠂⢠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⢰⠆⣀⣁⣤⡶⠦⢶⣦⣾⡷⡄⠀⣄⡈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⣨⢉⢹⣿⣯⣀⣾⠋⠸⣿⡿⢶⣼⣿⡤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⢻⣿⠃⣿⡟⢠⠀⠈⣿⡡⠄⠸⢧⠀⠙⡎⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠸⣷⢴⣿⡇⠉⠀⠀⣹⠀⠀⣄⠀⢃⡿⠀⠉⢠⡔⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇⣤⡘⢷⠃⠙⣇⣠⠴⠚⠹⣆⠀⠉⠀⢈⢷⡶⢄⣤⠀⣄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢿⣷⠋⠀⠀⠈⠁⠐⠶⡄⠈⣷⣀⠔⠁⠀⠛⣄⢀⡄⠁⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠁⣄⡈⠻⡴⠚⡉⠙⢇⠀⠋⠀⢀⣼⠿⡠⠴⠶⠄⠘⡟⠀⠸⡇⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⢠⠀⣀⣾⡅⢻⣓⡀⢘⣧⣠⠔⠉⠀⠀⠻⣆⢀⠤⠂⢻⡂⠀⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢾⡟⣉⠉⢏⠉⠀⢉⣠⠴⢏⡒⠶⠿⠦⠄⡟⠧⢤⣄⡀⣷⡀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⡇⠧⡄⠈⢣⡴⠋⢀⡀⠀⠹⣀⣀⠤⠐⢿⣄⠘⠩⠉⠈⠉⠒⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣤⣧⡀⣤⠖⠘⢧⠲⠟⠧⢄⠠⣿⠀⢠⣆⠀⢘⣇⠀⠀⣠⡸⡇⢐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢙⢻⡇⣴⠃⠀⢀⡘⢀⣤⠄⢾⣀⠤⠟⠉⣉⠸⢷⡴⠋⠁⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡈⠻⣦⠁⠀⠈⣿⠉⠀⣀⠀⠙⣆⢠⠞⠁⠀⠀⠻⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠈⢡⣼⣿⣶⠶⠾⠟⣢⡴⠿⡄⠀⠙⣟⠀⠠⠞⠣⠀⠧⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣇⣶⣄⠀⢱⡄⢠⡶⠒⠉⠉⠳⣀⡄⠀⠘⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣆⠛⠽⢡⡼⢉⡛⠀⠠⣄⡀⠠⠥⢄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢐⣀⡀⣀⢄⡈⣈⠁⠉⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠤⠤⢤⣀⣀⣀⣀⡀⠀⠀⠀⣆⡀⠀⠀⠁⠢⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢉⡳⠤⠤⠬⣅⡀⠠⠀⠀⢳⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠊⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⡈⠄⠀⠀⠀⠀⠀⠀⠀⠙⢧⡁⠀⠀⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠘⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢻⠀⢠⠇⠸
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠈⡼
⠀⠈⠉⠈⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠊⠁⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠐⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠐
⠀⠀⠀⠀⡟⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⢳⡀⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠉⠲⣤⣀⡀⠀⢀⣀⣄⣄⡤⠤⠲⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠈⡴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠔⠒⠚⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣊⢋⠙⠁⢀⠙⠂⠀⢧⠀⠀⠰⡟⠩⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡩⠉⣉⠉⠀⠀⠀⠈⡁⠁⠀⣀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡇⠣⢠⣙⠖⣠⣸⢠⡇⢧⡶⠶⢼⠂⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢓⡀⢉⠩⡈⠉⣙⠀⡀⠊⢀⣀⡀⣬⠇⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⢶⡇⠀⠁⠈⠀⠰⠘⠀⠃⠂⢸⡜⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡰⡄⠀⣀⢀⠀⠀⠀⠀⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢳⢳⣦⣌⣿⢴⠶⣴⠀⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⠁⠀⠈⠈⠁⠀⠈⠃⢼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣥⡒⠛⠓⠭⣒⣚⣒⣒⣂⣒⣊⠥⠞⠛⠂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⣆⢹⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠨⠁⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣠⣎⢹⠒⡇⡤⢠⢰⡁⢾⡴⠂⣶⣦⣆⢶⢂⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡾⠀⠟⠹⠛⠸⠃⠘⢑⠊⠉⠁⣡⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢁⣤⣤⣠⢙⢦⠀⠰⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢯⠛⠈⠉⠁⠸⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢠⡄⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣴⣤⠀⠀⠰⠀⠀⠂⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⡇⢈⣃⠸⣿⡿⠂⠀⣤⡀
⠀⠀⠀⠀⠀⠀⢴⣀⣿⠃⠈⠛⠻⢀⣀⡀⡀⠀⡀⠀⠈⣴⣿⣿⣦⠀⠀⠀⢠⠀⠂⠀⠀⠁⢰⡿⣀⡀⠈⠀⠀⢀⠄⠹⠄
⠀⠀⠀⠀⠀⠀⢚⣯⣟⣿⠶⢠⢇⣼⣾⣧⠀⠀⠀⠔⠂⣿⣿⣿⣿⠃⠀⠀⡀⠀⣢⡈⢱⠀⠈⢀⡓⠛⠿⠠⢾⡟⠀⢠⠀⠀⣠
⠀⠀⠀⠀⠀⠘⠈⡿⠛⠉⢀⡌⠸⣿⣿⣿⠃⣌⠀⠠⠀⠈⠉⠙⡅⠇⠀⠈⠉⠣⢦⠁⠀⠀⢆⡼⠑⣧⣤⢆⣀⠦⡌⠻⠂⠈⠙⡄
⠀⠀⠀⢠⡞⢰⣿⡷⠀⣹⣿⣿⠒⠀⠁⣠⠆⠙⣦⠀⠀⠙⠀⠀⢫⣤⠴⠂⠀⠀⠀⠀⢤⣀⠈⠁⣺⣿⣿⣿⣿⣷⡄⣰⣷⣄⠀⢕
⠀⠀⠀⠈⠀⣿⡿⢱⠈⣿⣿⣿⠇⠀⠀⠛⠀⢰⡄⠀⠻⣷⣠⣶⣤⠀⠀⡤⠀⠈⠀⠀⠀⠙⠋⠀⣿⣿⣿⣿⣿⣿⣷⠘⠛⠃⡀⠈
⠀⠀⠀⠀⠸⠟⠁⢈⡁⠙⠛⠁⠀⢵⠨⡀⡖⠂⠱⠀⢾⣿⣿⣿⣿⡇⠀⠆⠀⠀⠀⠀⠀⠀⠀⡖⠘⢿⣿⣿⣿⣿⡿⠀⠀⠘⠃⠐⡄
⠀⠀⠀⢻⠀⣿⣦⠹⣷⣆⣠⡀⠘⠚⠃⠁⠀⠶⠴⠶⠈⠻⠿⠿⠋⢡⣀⣠⡀⢈⡃⠀⠀⠀⢀⣴⣶⡇⠙⠿⠛⢋⣤⠘⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠃⠚⠿⠿⢾⣿⣋⠁⡴⢋⡙⢶⠳⠄⠀⣂⠀⣀⣀⡵⠖⢈⡉⢉⣀⡈⠑⠶⠄⢀⠀⠙⠛⠁⢠⣦⠀⠋⣀⣤⣠⣄⡀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⡿⠀⣿⣿⣿⠂⠧⣌⣠⣾⣏⣀⣴⢶⣄⠺⣿⠇⠀⢘⡻⢿⠿⠁⡸⡁⢠⢺⣷⡄⢠⣶⣿⣿⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠇⢀⣠⣤⡄⠀⠀⣟⠰⠆⣻⢠⡌⣠⣾⣿⣿⣿⣷⠆⡀⠑⢌⠉⢁⡀⠘⠻⠟⠋⢀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡵⣄⠈⠛⠿⠇⣀⠀⠈⠓⠚⢁⡼⠹⣿⣿⣿⣿⣿⡟⠀⠃⣀⡀⣴⣿⣿⢀⣤⡴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣟⣲⢠⣄⣀⠀⠀⠀⢀⢠⠀⠀⠈⠙⢛⣫⣥⠀⢺⣻⣿⡷⢈⣭⣤⡄⢠⣼⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢠⡰⢢⠉⠁⣤⣤⣤⠀⠛⠿⠿⠀⠀⠙⠋⢁⡈⠻⠻⠁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢃⠀⠀⠀⠘⣿⠁⠀⠠⠑⠠⠀⠆⠀⡇⠀⠈⢰⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⢠⣴⣤⣤⡀⠀⢰⡁⠇⠀⢀⢀⡀⠓⠄⠐⣡⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠻⣿⣿⣶⡄⠀⢀⣾⣿⣿⣿⣆⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠀⠀⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠙⠲⢤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡥⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⣱⠋⠀⠀⠈⢳⣤⠖⠚⠋⠉⠉⠉⠛⡷⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢰⣾⡉⠀⠀⢰⠇⠀⠰⡆⠀⠀⢻⣤⣀⡀⠀⠀⠀⢰⠇⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠸⡇⠉⠛⣿⣿⠀⠀⠀⢷⠀⠀⠀⠻⣿⣿⣿⣷⣶⣿⣦⣤⣀⣀⠀⠀⣸⣷⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⢷⠀⠀⠈⠙⢦⣀⠀⠈⢧⡀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠉⠛⠦⣤⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣿⣿⣧⡀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⣄⣀⣈⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⠶⠤⠤⠤⠤⠤⠴⠿⠛⠛⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣠⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠊⠁⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⡀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠃⠀⠀⠀⠀⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣠⠤⠤⠤⠤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡖⠒⠛⠦⠤⣀⠀⠀⠀⢸⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣷⠀⠀⢄⡀⠈⠳⢄⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢳⡀⠀⠀⠙⢆⠀⠘⢦⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠒⠉⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠐⠒⠢⠤⢭⠦⣌⣧⠀⠀⠀⠈⡆⠀⢸⠄⠀⣠⣴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄⠀⠀⣿⣀⠼⠶⢟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⡬⣶⠖⠉⠙⠢⣄⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠁⠀⢀⠂⠀⡞⣧⡀⠀⠀⠘⣆⠀⠘⡆⠀⠀⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⡂⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⣄⠀⠀⠘⠆⠀⠉⠑⠒⠒⠀⠀⠉
⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢸⣈⠒⠤⡀⠀⡸
⠀⠀⠀⠁⠁⠒⠢⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠤⠊⠑⡼⠇⠇⠀⠀⠈⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠓⠂⠐⠉⠉⠀⠀⠀⠀⠀⠀⠘⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣸⠠⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⠀⢱⢸⢸⠁⠳⡄⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠳⣾⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡤⢤⣤⣤⣴⣶⡶⠾⠶⣶⣤⣀⠸⠃⠈⣷⡀⣀⣤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠃⠀⠀⣀⣀⣀⣀⣀⣀⠈⠙⢷⡄⢠⣿⠟⠋⠀⠀⠀⠀⠈⢉⣷⣦⣀⣠
⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠚⠉⠉⠉⠀⠀⠀⠀⠉⠙⢦⣼⠿⣿⠁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠨⠽⠧⠤⣴⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⢠⣤⡤⠖⣂⣤⠴⠖⣲⡿⣿⠇⣺⣿⠟⠉⠉⠉⠉⠉⠛⠳⠶⠤⠤⠴⠶⢺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣶⣶⣛⣭⣶⣶⠟⢁⣴⠏⣼⣿⣿⢠⣤⣂⣤⡀⠀⠀⣤⣤⣤⡤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⢉⡽⠋⢁⣴⠿⡟⢸⡏⠘⠻⣷⣭⣭⣭⣤⣤⣶⠶⠶⠾⠿⠆
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⠟⠁⠀⢠⡟⢁⣴⠟⠁⢸⡇⢸⠁⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⣠⣶⠿⠛⡉⢉⡿⠿⠿⠷⣶⠏⢠⡞⠁⠀⠀⠸⡇⢸
⠀⢠⣾⠏⠀⠀⠀⠳⢮⡶⠂⠀⢀⡏⢠⣿⡄⠀⠀⢀⣠⣧⢸⣤⣤⣤⣤⣀
⠀⣾⡇⢠⡆⠀⠀⠀⠀⠀⠀⠀⣸⢠⡏⢸⣿⣀⣶⡿⠟⠻⡄⢫⣉⠉⠛⠛⠿⣦⡀
⠀⣿⡇⠸⣇⠀⠀⠀⣀⣠⣤⣤⡇⣼⣀⣼⣿⣿⠏⠀⠀⠀⠚⠾⠋⠀⠀⠀⠀⠈⠻⣦
⠀⠘⣿⣆⠉⢀⣴⡿⠟⠉⢁⠘⡇⡏⠉⠉⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠈⠻⣷⡿⠁⠀⠀⠀⠈⠻⣷⡷⠏⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⡇
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⣿⠇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⡴⠾⠋⣠⣾⠟
⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⢰⣿⣤⣄⣀⣀⣀⣀⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠙⣷⣄⠀⠀⣀⣀⣀⣀⣤⡾⠏⢀⣴⣿⠋⠈⠉⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠈⠙⢿⣶⣦⣭⣭⣭⣭⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠓⣴⠃⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠇⠀⠀⠻⠤⣤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠼⠆⠀⠠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠡⠒⠉⢇⣏⠉⠉⠑⢪⡙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣀⠣⡀⠀⠀⠸⡟⠀⠀⢠⣪⠞⠑⣜⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡎⠀⠑⠺⣦⡀⠀⣷⠀⣰⠟⠁⠀⠀⠈⡎⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠙⣲⠟⢾⣁⣠⣤⣴⣖⣒⣃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⢸⢈⠭⠭⠛⠛⠋⠉⡽⢦⡞⢤⡀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠈⡎⢆⠀⠀⠀⣠⡞⠁⢸⡇⠀⠙⢗⠀⡀⢀⢇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠘⢌⢦⡠⢊⠎⠀⠀⢸⢷⠀⠀⠀⠑⣬⢉⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡆⠀⠀⡴⠊⠳⢄⡣⢄⣀⠀⢈⠸⣀⣀⠤⣊⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠔⠃⠀⠀⠀⠉⠓⠂⠭⠭⠤⠅⠒⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠿⠟⠛⡛⠛⣫⠉⠉⣉⡍⠙⢛⣛⡛⠛⠻⠿⢿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⣠⠀⣠⣊⣥⣮⣴⣶⣿⣷⣶⣾⣿⣦⣤⣶⣒⢒⣀⣀⠈⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣰⠵⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠳⢤⡤⠄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⠳⣖⠂
⠀⠀⠀⠀⠀⠀⠀⣠⡋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠈⢟⢶⢤⣀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⡤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⣏⢲⠲⡤⣄⣀⣀⡉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣠⠤⠶⢮⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠃⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⡿⠿⠿⠿⠿⠤⠤⣿⣷⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠯⣉⡉⠉⠉⠉⠉⠉⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⢤⣿⣿⣿⣿⣷⣶⣶⣶⣶⠒⠒⢻⣿⠉⠀⠀⠈⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⢠⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠨⠘⠸⠸⠌⠇⠙⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠛⣻⠛⠛⠛⢶⠶⠶⠿⠚⠛⠛⠛⠛⠻⢿⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢹⡀⠀⠀⣏⠀⠀⠀⠀⠀⠈⠃⠀⠂⠀⠀⠀⠀⢸⡇⠀⠀⣼⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠸⡇⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣠⠞⠀⠀⢠⡏⠀⢹⡆
⠠⠀⡠⢠⣠⢀⣤⠄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡇⠀⠀⠀⠀⠉⠓⠦⠴⢤⣤⣤⠦⠦⠚⠋⠁⠀⠀⠀⢸⠇⠀⠈⣧⠀⠀⠀⠀⠀⠠⣔⢀⠔⣄⡄⡄⡄
⠐⠊⠑⠊⠈⠚⠁⠀⠀⠀⠀⠀⠀⠀⢺⠗⠲⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠒⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢛⠾⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣉⠄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢫⠳⣖⠩⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⡌⠅⢊⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠈⠔⡌⠄⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡈⣑⠂⡃⠄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠒⢤⢃⠃⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠰⢆⡂⡄⣻⠃
⠀⠀⠀⢀⠀⢀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏⠩⢐⠀⠄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⢀⠘⠂⡤⣶⡏
⠀⠀⠀⢙⠦⠆⡇⠇⠇⢣⠂⠀⠀⠀⠀⠀⠀⠘⣆⡡⠊⠀⡉⢼⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣯⠑⠂⡄⠃⣠⡟⠀⠁⠂⠀⠰⠂⢀⣦⡀⠖⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡁⠢⠄⢂⡇⣠⠖⠋⠁⠀⠉⠙⠲⣤⡓⢉⠒⡘⢡⠟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⢌⣍⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⢿⣎⠠⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣿⠀⠀⠀⢀⠀⠀⠀⠀⢸⣭⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣀⣀⣠⡤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⢀⣠⢄⣠⢄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠋⠃⠙⠂⠙⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠄⠀⡀⠀⠌⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠡⡰⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⢀⣀⠀⠀⠠⠀⣒⠂⠒⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠉⠁⣵⠂⠻⡐⠂⠠⣀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠊⠉⠀⠆⠀⠀⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⡀⢸⠃⠀⠀⠈⢲⡄⠀⠀⠀⠀⠀⣠⣾⣿⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠭⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠘⠢⣤⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠉⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠂⠀⠀⠠⠞⠉⠀⠤⡀⠀⠀⢀⠤⠤⠤⠦⠸⢿⣿⣿⣿⡿⢀⠀⠀⠀⠀⢀⣤⣶⣦
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠓⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠯⠉⢉⠔⠉⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠤⠊⠀⠐⠆⠀⠀⠀⠀⠀⠈⠉⠀⠁⠑⠀⠀⠀⠀⠀⠴⠈⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠈⠛⠿⠟⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄⣠⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠌⢁⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⣿⠱⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠓⠂⠨⡉⠓⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢢⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢧⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠷⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⢀⠚⠃⠄⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠆⠀⠀⠀⠀⠀⠀⠢⠀⠇⠀⠀⠀⠀⠀⣸⠀⢠⠐⠿⠤⠛⢢⡔⠤⣔⣓⠴⣷⡼⠲⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡄⠀⠀⠀⠀⠀⢀⡇⠸⣄⠀⠀⢠⡀⠛⠀⣨⠐⢁⠈⢄⢙⣋⣉⣉⠙⡘⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠧⠀⠀⠀⠀⠀⠈⠉⠁⠙⠓⢶⡞⠀⣕⡤⢤⢆⣁⣒⣁⡶⠴⣷⢚⣖⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠧⣄⣠⣄⣀⠀⠀⠀⠀⠀⠀⣀⣰⡾⠓⠒⣏⣀⣼⡟⠶⠾⠶⠮⠭⠽⠿⠿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠲⠛⠛

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣖⣈⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣁⣲⠄⠀⠀⢀⡀
⠀⠀⠀⠀⣿⠈⠉⠙⠛⠛⠛⠛⠛⡛⠛⠛⣛⣋⣉⠩⣿⠀⢠⠚⣁⠨⠆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⡏⢠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢀⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡀⢸⣧⢋⡈⠰⠶⠤⡔⠊⠉⠑⠚⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠶⠀⠀⠀⢀⡤⠀⠀⠀⠀⣻⣾⣏⣀⡀⠠⠤⠰⡀⠀⠀⠀⠠⣀⠀⣀⠄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⣚⠝⢡⠆⠰⢄⡀⠒⢄⡎⠀⠀⠀⠀⠀⢈⡝⠁⠀⠀⠀⢀⡕
⠀⠀⠀⠀⠀⢸⡀⠀⠰⠆⠀⠀⠀⠀⠀⢠⠟⢁⠔⠁⠇⠀⠃⠉⠢⢄⣗⠦⣤⠤⣢⢭⢭⢺⢳⣤⠢⠴⢊⡴
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⣿⠀⠀⠀⠐⠓⢪⠀⠀⠀⢰⠀⡄⠀⠀⠀⠉⠑⠒⢘⠧⣓⠵⠇⠘⡇⠈⠈⠁
⠀⠀⠀⠀⠀⠀⡕⢄⡀⠀⠀⠀⠀⠀⠀⠀⢈⣦⠀⠀⠀⠦⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠑⡄⢀⢳
⠀⠀⠀⠀⠀⠀⠈⠢⢌⣀⣁⣀⣀⣀⣀⣀⡡⠔⠓⠤⣀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⣄⡞⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉





⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠁⠀⠉⠉⠙⠒⠒⠚⠋⠁⠀⠈⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣀⡤⠴⡶⠚⠋⠉⢹⡟⠉⠉⠓⢶⠦⢤⣀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡤⠚⠛⠁⠀⠀⢱⠀⠀⠀⢀⡇⠀⠀⠀⡏⠀⠀⢈⠛⠒⢤⡀⠀⠀⠹⣄
⠀⠀⠀⠀⢰⠃⠀⢀⡴⠛⡀⠀⢀⣧⣠⣤⣾⣷⣶⣿⣿⣿⣷⣶⣾⣷⣤⣄⣼⠀⠀⢀⠛⢦⡀⠀⠹⡆
⠀⠀⠀⢀⡏⠀⣰⠋⢆⢀⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⡀⡘⠙⣆⠀⢹⡀
⠀⠀⠀⢸⠁⣸⢣⣢⣾⣿⡿⠋⠉⠀⠀⠀⠀⠀⠈⠙⠻⣿⠋⠁⠀⠀⠀⠀⠀⠉⠙⢿⣿⣷⣜⡘⣇⠘⡇
⠀⠀⠀⢸⡄⠹⣿⠿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠿⣿⠏⢠⠇
⠀⠀⠀⠀⢧⠀⠈⠙⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⠀⠀⡼
⠀⠀⠀⠀⠈⢧⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⡼⠁
⠀⠀⠀⠀⠀⠈⠳⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠞
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢤⣤⣤⣤⣤⣤⣤⡤⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣠⢤⡀⡄⡄⢤
⠀⠀⠀⠠⠤⠤⠀⠀⠠⢤⢀⡠⡀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠀⡠⠤⢤⢀⠠⠠⠤⡄
⠀⠀⠀⠒⠐⠄⢀⣀⠀⠠⠄⠠⣁⢀⣠⣦⣾⣦⣤⣸⣀⡢⠤⢄⣀⡀⢀⡀
⠀⠀⠀⢀⢀⢀⠀⣀⡠⠀⠂⣙⣿⣿⣧⡜⣿⣿⣿⣿⣿⡛⠲⢶⠖⠤⠬⠭⠤⠠⣤⣀⠀⡀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠈⡡⢡⢿⠹⡟⠓⠛⣿⡉⠁⠁⢵⣤⠈⠓⢲
⠀⠀⠀⠀⠀⠀⠈⢀⣀⠄⠐⢁⠂⠀⠉⠀⠀⠀⠈⠂⣀⡈⠀⠈⠐⠤⢑⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠙⠁⠀⠠⠂⠀⣠⡀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠢⣤⠓⡇⠈⠃⠄
⠀⠀⠀⠀⠐⢁⠊⠄⠘⠐⠁⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⢄⣺⡶⠂⠰⠐⠐⠲⠠⠢
⠀⠀⠀⠀⠀⠃⡅⠄⡐⠁⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡄⠀⠈⠘⡇⢋⠂⠀⠀⠀⠖⠂
⠀⠀⠀⠀⡀⢀⠈⠀⢆⡀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠰⠃⣀⣄⠀⢹⠐⠦⢡
⠀⠀⠀⠀⠀⠁⠄⠀⢀⡀⠀⢆⠀⢶⡀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠂⣍⢁⠡⠀⡇⣌⠌
⠀⠀⠀⠈⠇⠀⢀⢠⡼⣧⠀⣀⠀⢸⠁⠀⠀⢀⡀⠀⢀⢀⠀⠀⡆⠀⠉⡄⢀⡇⠀⢸
⠀⠀⠀⠀⠀⡀⠈⠛⠁⠀⣄⠃⠀⣨⠀⠀⠀⣼⠇⠀⠀⠀⠀⢀⠀⠀⣃⣁⣨⠀⢤⡦⠄⠈⠚⠂⠂⠃
⠀⠀⠀⠀⠢⢆⠞⣵⣾⠟⡂⠘⢀⠝⢄⠀⠀⢹⡇⠐⠐⠒⢂⡝⠉⠈⠁⣾⠷⠊⢛⠃⠀⠈⠓⠒⠈⠃
⠀⠀⠀⠀⠀⢈⡢⠉⠋⠀⢁⢀⢁⠀⠀⢑⣀⣜⠃⣀⣀⠀⠻⢥⣶⠀⣴⡋⢰⡷⠃
⠀⠀⠀⠀⠀⠀⠑⢶⠧⠀⠠⠂⠄⢀⠀⠙⠛⠻⣇⠘⢋⠀⠀⠒⠈⢛⣿⣧⠔
⠀⠀⠊⠂⠀⠀⠂⠘⢐⣶⡿⣽⣀⠀⡀⠀⣽⡓⣿⠐⠁⠐⣀⣀⡴⠕⠚⠁⠐⠀⠐⠂
⠀⠀⠀⠀⠀⢠⣄⣐⣨⣿⠋⠈⠋⠁⠀⠈⠜⣿⣿⣿⣤⣭⣤⣀⡠
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⢶⣿⠛⢃⡚⠡⢌⠍⠁

⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣤⣿⣿⣿⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢿⣾⣿⣿⣿⣿⣿⠏⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⢿⣿⣿⣿⣿⣯⢰⣿⡿⢡⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⡻⣟⡛⠛⣷⣝⣿⡟⠅⠀⢿⢹⢰⡟⠁⢀⣼⢳⡾⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠻⠟⠻⣿⣜⣷⣎⠀⠈⠈⡿⠁⠀⣾⠟⠋⣴⣿⣿⣿⣿⣿⣿⣿⢟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣧⣌⠻⣟⢿⡀⢄⠀⠁⠀⠀⠋⠀⢠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⠿⣿⣿⡻⡛⣦⠘⢾⣃⠀⡴⠉⠀⠀⠀⢀⣿⣿⣿⡿⢛⡿⢻⣿⣿⣭⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⢉⣹⡿⢷⣾⣩⠻⠶⣍⣧⡀⠹⡄⠀⠐⠀⠀⠀⠀⠈⠛⣣⣴⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⣯⣿⣶⣍⡙⢿⣾⣆⠘⢷⠀⠘⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣬⡿⢷⣮⣙⢿⡆⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋⣠⣿⡿⠿⠟⠋⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣋⣉⠺⡉⠃⠹⣆⠀⠀⢀⠀⣀⠀⠉⠀⣉⣤⣆⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⡛⠻⣿⢐⣦⣙⡇⠀⠀⠀⢈⣤⠂⡼⠉⠛⢛⣁⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡅⠐⠿⠿⣖⣊⣻⠷⠌⢭⡛⢷⣶⠷⠞⠁⠀⠠⣠⡶⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠒⢾⡷⢏⣙⠲⣇⠀⢙⡾⠧⠤⢀⣾⡷⠃⢴⣉⣁⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡉⠀⣈⣿⡷⠠⣭⡿⠶⠶⢀⡼⠟⢠⣼⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢏⣉⠩⣝⣦⣤⡯⠓⣣⠾⠦⢴⡟⣲⢏⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠇⠐⢠⠶⡦⣾⣋⣉⣁⣄⣈⣃⣹⣦⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⢿⠿⠋⠁⠐⠃⠀⠰⣿⠓⣼⣿⣿⣿⣏⡘⢿⣧⠀⠙⣦⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇⠀⢀⠄⠐⠆⣀⠠⠛⡿⠁⠈⠹⣦⠉⠉⠛⣇⠀⠀⢃⠀⠻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⢻⡏⠀⠀⠀⠀⠁⠀⡃⠐⠢⣧⣤⡔⠂⠀⠠⣤⣰⠁⠀⠰⡾⠁⣰⣽⣷⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣲⣄⡜⠓⢶⡃⠀⠀⣏⣀⢀⡸⠃⠈⢣⣤⣤⡤⠀⡈⢿⣽⣻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁⠢⣴⠇⢬⣽⡋⠈⠉⣿⣂⣀⣼⠋⠀⢱⠖⠁⠀⠛⢼⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣭⠓⠐⣧⠀⣀⣹⣋⢀⣚⡟⠉⠉⢻⣄⣀⣼⠖⡇⠀⢖⠤⡀⠀⠀⠀⠀⠀⠈⠉⠚⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⡀⢲⣿⣤⣵⣎⣁⠉⢹⣗⠚⣀⣿⢁⡍⣿⠊⢳⠶⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⢛⣿⣿⣿⣿⣿⣿⣏⡉⠙⣿⣉⣤⣿⣂⣿⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⣿⣿⣿⣿⣿⣿⣹⣿⣿⣄⣨⣿⡧⢀⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡏⣉⣿⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣠⣴⣾⢟⣻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⣰⣾⣤⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⣼⣿⡟⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁
⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠋⢙⣿⣿⣿⣿⡿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⠋
⠀⠀⠀⢿⣿⣿⣿⣿⣯⠀⠀⠈⠙⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⢁⡼⠃
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣤⣀⣠⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣉⣁⣀⡀⢺⣇
⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⡿⠿⠷⢄⠀⠀⠀⠀⠀⢀⣶⣾⣿⣉⣙⠲⣇⣀⠉⠉⠛⣿⣳⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⢀⣾⣟⣉⡡⣤⢬⣹⠟⠛⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⣿⣿⣟⣻⠏⠀⠀⠠⠀⣱⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⣿⣧⣼⢿⣶⠖⢛⣾⣾⢁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣿⣿⣿⣿⡿⣿⣋⣿⡿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠂⠁⠀⠀⠀⠀⠉⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢰⠶⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠉⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠠⠄⡐⠂⠠⠈⠀⠈⡉⠞⠁⢐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⠒⠀⠀⠀⠀⢀⠀⣠⣤⡤⠴⠊⠉⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡔⠀⢸⡀⠰⠂⠺⣟⠁⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠁⠈⠀⣨⠀⠀⠤⡀⢤⠀⢤⣇⣤⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠀⠀⠀⠀⠀⠰⠁⠁
⠀⠀⠀⠀⠀⢰⠀⠃⠀⢹⠀⠀⠁⠀⠙⠀⠀⠘⠁⠀⠀⠀⠀⠈⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠁⠠⠀⠀⠀⠀⠠⠂
⠀⠀⠀⢀⣠⢈⠀⠰⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⢐⣦⣀⠀⠐⠂⠀⠤⠀⠠
⠀⠀⠀⠸⡕⠪⢄⡀⡆⠀⠠⠀⠀⠐⢄⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠠⠃⠀⠀⠀⠄⢃⡀⠸⡆⠈⢠⡈⢂⠐
⠀⠀⠀⠀⢀⠀⠀⠘⠐⠢⠀⡁⠀⠀⠀⠑⠂⠀⠀⣀⠠⠀⠀⠠⠄⡤⠀⢀⢠⢀⠂⠀⠑⢰⡇⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠘⣦⠄⢀⠀⠀⠐⠉⠉⠐⠀⠀⠁⠀⠀⠰⠀⠀⢐⡂⠀⠀⢰⠓⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣆⠨⡇⠀⠀⠠⣤⠀⠀⠀⠄⢦⠀⠀⠀⠲⡇⠀⠈⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠐⠄⠀⠀⠙⡅⠀⠀⠀⠐⡀⠀⠀⠀⢿⠀⠀⠀⠸⡄⠀⠀⢨⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡉⡀⠀⠀⢣⠀⠀⠀⠐⢁⠀⠀⠀⢸⡄⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠀⠀⠀⠇⡀⠀⠀⢸⠀⠀⠀⠈⣧⠀⠀⠀⠉⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠚⣆⠀⠀⠀⠃⡀⠀⠀⢯⠀⠀⠀⠈⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣯⣦⣧⣾⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⢀⣠⣴⣶⡿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣷⣶⣤⣀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣃⣴⣾⣿⣿⣿⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣾⣿⣿⣿⣶⣌⣻⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⢹⣿⡻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠈⠉⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠋⠉⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⣀⣀⣀⣀⣀⣀⣤⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⣛⣛⡉⣭⣉⣙⡒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣡⠆⠢⠌⠿⠛⠀⠉⠈⠙⠁⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣄
⠀⠀⣀⣤⣶⡶⣶⢶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣞⣠⡵⠞⠛⣛⣛⣉⣙⣛⣛⡒⣒⣶⠦⠤⣤⣀⠀⠀⠈⢧
⣴⣿⣯⣍⣘⠁⠙⡟⢹⣇⢻⣿⣷⣤⡀⠀⠀⠀⠘⣿⣛⡉⠛⠛⠻⠶⣤⣀⣈⣀⣿⣁⣉⡙⢙⣲⣮⣝⣶⣄⠈⡇
⠻⣎⠙⢦⣉⡓⢦⡀⠀⠃⠎⢿⢻⣿⣿⣦⠀⠀⠀⠀⢸⣭⣍⡙⠒⠒⢻⡟⢟⡿⠁⠈⠉⠉⠉⣻⠋⠛⢻⠛⠳⠾⠤⢤⣤⣀⣀⡀
⠀⠈⢢⡀⢻⣏⠲⣝⢦⡀⠀⠈⠀⢟⣾⣿⣧⠀⠀⠀⣿⠘⣨⠟⠛⠲⢬⣷⣈⢇⠘⣃⡤⠴⠊⢉⠶⣶⢧⠀⠀⠀⠀⠀⠉⣧⠀⣹⢧
⠀⠀⠀⠙⢮⠹⡄⠈⠶⣝⢦⡀⠀⠈⠹⣿⠙⣇⠀⠀⣿⠐⡏⠸⠦⠀⠀⠈⣹⠟⠻⢥⣀⡀⣠⠏⠀⡏⠀⠑⢤⡂⣀⠀⢙⠏⢠⠇⠈
⠀⠀⠀⠀⠀⠑⣷⢤⣰⠟⠲⣝⢆⠀⠀⠁⠀⣿⠀⠀⣿⡀⡟⣛⠛⠋⠉⠉⡹⢦⣠⣞⣉⡙⠳⠦⣀⡷⠀⠀⠀⢝⣮⠑⠋⢠⡎⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠣⣷⡤⣀⠘⢮⢳⡀⠀⢴⣿⡇⠀⣿⠰⡇⢤⡄⢀⠴⠋⠀⡜⠋⢯⢧⣽⡄⠀⠈⢿⠒⢤⣌⣠⠏⠀⣴⣏⢴⣎⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⡓⢾⣧⢳⠀⠀⣿⠁⠀⠸⣆⢽⢦⡞⢁⡄⠀⢠⠃⠀⠈⢧⠡⡃⠀⢒⡞⠀⡼⢻⢁⣠⣾⡿⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣍⠲⣌⣿⡇⠀⡟⠀⠀⠀⠹⣿⡟⣧⢫⠆⡀⣼⠀⠀⢀⠨⣷⠥⠖⠋⣠⠞⠀⡼⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⢻⣇⡼⠁⠀⠀⠀⠀⠘⠿⣾⣿⣶⣥⠧⣤⣸⡤⠖⠁⢀⣴⠞⣁⡀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣤⣀⣤⣴⣾⡿⢋⣴⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠿⠛⠉⠉⠉⠉⠉⠉⠛⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣯⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣽⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠛⠛⠛⠂
⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠹⣿⣿⣿⡿⠛⣿⣿
⠀⠀⠀⠀⠀⣿⣅⣤⠘⢿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡟⠠⡷⢸⡏
⠀⠀⠀⠀⠀⢹⣿⡁⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⣼⣿⠠⣄⢸⣿
⠀⠀⠀⠀⠀⠀⢹⣿⢊⣸⣿⣿⣷⣄⣀⣀⣠⣾⣿⣿⡌⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⣿⣿⡿⠋⠉⠉⠛⢿⣿⣿⣿⣌⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠉⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢄⠀⠀⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣴⡆⠀⠘⠂⠛⠃⠀⠺⠆⢴⡆⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠉⠓⠀⠀⠀⣀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠉⠋⠰⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠁⠀⠀⠠⠄⢴⠑⣀⡰⠆⠀⠄⠇⠀⠃⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⣤⡀⠀⡀⠀⣾⡎⠛⣳⣿⣿⣆⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠼⣦
⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢠⢠⠀⠀⠀⣿⡏⢹⣿⣿⣿⣟⡇⠀⠀⠀⠀⡀⠀⢠⣄⠀⠀⢁⣤
⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⠀⠁⠠⣄⡆⠸⠃⠸⣿⣿⣿⣿⠇⠠⠀⠀⢸⣿⣿⢈⠁⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠾⠆⠀⠀⠘⠃⠛⠁⠀⢰⣿⡆⠐⠑⠘⢿⡿⢣⣾⣦⠀⣶⡌⠉⠉⠀⡦⢰⠀⠀⠘⣁
⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡃⢠⠈⠀⠀⠸⣿⡿⢃⠈⢰⣶⠀⠀⠀⠀⠂⠀⠀⠋
⠀⠀⠀⠀⠠⣶⠀⠀⠀⠚⠓⠚⠆⠀⠀⠐⠀⣧⠀⠀⠀⠠⠀⠀⠀⠀⠰⡶⢴⣾⠀⠀⢀⣤⠀⠀⠛⠆
⠀⠀⠀⠀⠀⣥⡄⠀⠀⢁⢀⢀⠅⠀⠈⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣨⡃⠀⢰⣶
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⠀⢀⡔⠀⣴⣤⣦⢠⣤⡴⠂⠠⠄⠘⣛⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⣻⠇⢀⠈⣀⢀⣌⠛⣃⠈⣿⣷⣶⣤⠀⣘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣩⠟⠀⠙⠁⠛⠀⣶⣦⠉⠀⢿⣿⣿⡿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⢶⣴⠂⠀⡀⠀⠀⠀⠀⠷⠀⠀⠁⠀⠀⡦⢰⠲⣶⣷⠦⠀⠀⢀⠀⠰⣦⡶
⠀⠀⠀⠀⠀⠀⠀⠙⠁⢠⠮⢊⠒⢀⠀⠀⠉⠀⠀⠀⠀⠀⠗⠘⠂⠀⠀⡀⣠⠠⡈⢆⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⢤⠃⠌⢀⠃⠀⢀⠄⠀⡀⢠⢠⠀⡂⠃⠀⠄⠣⠐⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠘⡅⠘⠀⠀⠀⠠⡀⡇⠡⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠞⢉⣠⣴⣶⣦⣶⣶⢶⣶⣨⣭⣝⡒⠤⣄⡀
⠀⠀⠀⠀⠀⣴⠃⣴⣷⢿⣿⣯⣿⣿⣾⣻⣿⣿⣿⣟⣷⣞⢦⣭⡒⢤⡀
⠀⠀⠀⠀⢰⣿⡀⢿⣿⣿⣿⣿⣯⣿⣾⣿⡃⠉⢘⣻⡟⣻⣿⣿⣷⡶⡌⢢⡀
⠀⠀⠀⠀⣿⣿⣷⡄⠰⣾⣿⣷⣿⢿⣿⣿⣿⣶⣻⣿⣿⠾⢷⢾⣽⣟⣳⠆⢱
⠀⠀⠀⠀⣿⣿⣿⣿⣶⣌⠻⠿⠿⣫⣿⣿⣿⣟⡛⢃⢶⣷⡄⠂⠀⠝⠻⣗⢸
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣷⣆⣰⣾⣿⣿⠁⠀⠀⠀⠈⢻⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⠀⠀⠀⠁⣠⣶⣾⣿⣿⣛⣉⡍⠛⢷⣦⣤
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣣⡄⠰⠀⢀⣾⣿⣿⣿⣿⣿⣿⣮⠀⢸⣶⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⢟⣔⣽⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣝⠿⣿⣿⣿⣿⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠦⠀⢴⣶⠆⠆⠹⣿⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡗⠀⠀⠀⠀⠆⡀⠸⠅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⣶⣄⣤⡤⣾⢆⡄⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣙⣛⣻⣿⣿⣿⣿⣿⡟⠙⠂⠈⠙⠻⣿⣿⣿⠇⡠⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⢻⣷⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣟⠁⢁⠐⠠⠀⠀⡉⠉⠻⣿⣿⣤⣠⣄⣤⡈⢻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠄⠄⠀⢀⣴⣦⠙⢿⣿⣿⣟⣤⣉⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣴⣶⣿⣿⣧⣸⣗⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⡿⠿⠛⠉

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

⠀⠀⠀⠀⡀⣀⡀⠤⠒⠈⢈⠇⠀⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡤⠀⠐⠀⠐⡄
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠷⡤⠸⢗⡈⠁⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣠⡇⠝⠣⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠐⠈⠁⠀⠀⠈⠀⢀⠄⠀⠀⡇
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡆⠀⠀⡴
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⣼⠗⡯⣄
⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠁⠀⠀⠀⣤⣷⣖⠅⣅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢨⣿⣿⣇⠀⠀⠀⠈⠀⠀⠀⠿⡿⡔⠈⠀⠀⠀⠀⠀⠀⠠⠄⡄
⠀⠀⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣰⢧⠀⠀⠀⠀⣀
⠀⠀⠀⡀⠀⠀⠁⠇⠀⠀⠀⠀⠀⢸⣟⣿⣷⡃⠀⠀⠀⠀⠐⠈⠉⠊⠀⠀⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡅⠀⢘⡀⠀⠀⢰⡌⣽⣾⣿⣿⢫⢡⠀⠀⠀⠆⠀⠀⠀⠀⠐⠈⢠⡔⠠⢄
⠀⠀⠀⠀⠀⠀⣀⠀⠃⠀⠀⠀⠄⢸⣻⣿⢿⠏⠀⠀⠀⠰⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠸⣯⡿⡟⢠⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⡀⠹⡷⠃⢸⠀⠀⠀⠈⠀⢠⠁⠀⠀⠁⠼⣄⠀⣀⡀⣀⠖
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠁⠀⠀⠅⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠟⠁⠀⠀⠀⠏⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠐⢲⣖⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠮⠔⡶⠶⣶⠶⢶⣤⣤⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠛⣒⠶⣴⣶⣴⣖⡒⣋⠁⣩⠟⢛⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠁⠀⠀⠀⠒⠂⠈⠉⠘⢧⢰⢿⣏⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣿⡞⢻⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠱⢻⠳⣄⢻⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠐⠚⠛⠛⠧⣈⠲⣽⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠒⠐⢤⡈⠉⠉⢢⠀⠑⢌⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢈⡀⠀⠀⠀⠱⠡⠊⢸⢡⠊⢂⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡌⠁⠈⣒⠢⡑⣐⡆⣰⢯⠅⢀⡜⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡇⠀⠇⣀⡀⣀⣱⢬⡮⣿⠷⣎⠥⠒⠈⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠂⠀⠀⠀⠀⠀⣸⠀⢣⠀⢀⠊⢀⠄⡅⢨⡤⠘⡟⠀⠀⣤⡤⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⠀⢀⠀⠄⠀⠘⡄⠀⢡⡀⠢⡀⠀⣀⠜⡀⠆⠡⠀⢄⠨⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⣀⣥⠂⢀⣀⣥⣥⢹⠦⣤⣟⣦⣀⠉⠉⠢⠀⠈⠜⠒⠊⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠩⠶⢶⠤⠤⠴⠒⠈



⡟⠉⣿⣛⢻⢛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢀⠎⠀⠀⠀⡇
⢰⢧⡀⠀⠀⠀⠀⡔⠁⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠋⠀⠀⠀⣼⣁⣀⣤⣄⣀⣀⡀
⡞⠀⠙⠲⢄⡀⡜⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⡴⠇⠀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠈⠉⠉⠓⠒⠲⠶⢖⡇
⡇⠀⠀⠀⠀⢈⠃⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣞⡎⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠂⠀⢠⠎
⡇⠀⠀⠈⢄⢘⢖⡊⠀⢱⣺⠁⢹⡠⠖⠒⠂⠃⠀⢰⠊⠉⠉⣦⠏⡏⠈⠁⢠⠤⠀⠒⠀⠒⢶⡖⠛⢛⡐⡂⠀⡵⠖⠒⠛⣀⣤⠒⠈
⢹⠀⠀⠀⠘⠻⠀⡀⠀⢸⣨⠀⢸⡏⠀⠀⣀⣠⢸⢹⠀⢀⠀⢩⢿⠇⠀⠰⢿⣰⢀⠀⠀⢀⣸⠂⠀⠀⡇⡇⠀⢀⠀⠀⠀⠹⠃⠀⠰
⠀⢧⠀⠀⠀⠘⡅⠇⠀⢸⡾⠀⢸⡆⠀⠀⠋⠁⢸⣩⠀⢸⠀⢸⣼⠁⠀⠀⠉⠓⡋⠅⠀⡏⢨⠅⠀⢰⡇⡇⠀⠉⢇⠈⡄⠀⠀⠠⠁
⠀⠈⢣⡀⠀⠈⢆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠈⢨⡂⠀⠈⠀⠀⠻⠀⠀⠀⠳⠶⣇⠅⠀⡷⢾⡆⠀⠐⠓⠃⠀⠀⠈⢆⢴⠀⠀⡇
⠀⠀⠀⠹⣆⢀⢵⠀⠀⣖⡆⠀⢸⡇⠀⠀⣴⠖⢹⡇⠀⣴⡄⠀⣶⣀⠀⠀⠀⠀⢅⡂⠀⡇⢸⡂⠀⠀⣀⡀⠀⠀⠀⢸⢨⠀⠀⡇
⠀⠀⠀⠀⠈⠙⡩⠀⠀⣿⠁⠀⢸⡗⠀⠀⠇⠠⣽⠀⠀⡇⣇⣠⠿⠶⠤⠤⠭⣉⡇⡂⠀⠁⢨⠄⠀⠀⡯⠃⠀⢰⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡺⠀⠀⣿⢀⡀⢸⡧⡀⠀⢀⢀⣿⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡶⢾⢠⠀⠐⢿⣂⠀⢸⠀⢸⡸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠶⠋⠟⠸⣷⡵⠛⠷⠭⣭⠭⠟⠋⠁⠀⠀⠀⢀⣀⡀⡤⠤⠤⠤⠤⠤⠄⣷⣼⡬⠭⠋⠸⠮⠟⠻⢄⠘⠒⠳⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠉⢉⡶⠒⠺⠍⢑⣢⠤⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⣀⡀⢤⠤⠵⠒⠛⠲⠤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠿⣁⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⠉⠙⠹⡍⠉⠉⠀⢹⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠐⠋⠉⠉⠉⠓⠒⠦⢤⣀⣀⠀⠀⣀⣀⣠⠤⠞⠙⣆⠀⠀⠀⠱⡀⠀⠀⠘⡇⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⢦⡀⠀⠀⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣐⠃
⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠀⡄⠀⠔⠲⠤⠀⠁
⠀⡀⠀⠀⠀⣠⠤⢦⣨⡅⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠈⠉⠳⢶⡄
⠀⠹⡄⢠⣾⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠀⠀⠀⠿
⠀⠠⠿⣆⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠁
⣇⣀⠤⠝⢿⣈⡤
⠘⣧⣀⢠⡷
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠂⠀⠀⠁⢤⡀⢀⣀⣀⣤⠖⠚⢉⡍⣉⡉⢩⠰⢶⣄
⣴⠀⠀⠉⠋⠀⠈⠙⠛⢷⡄⠀⣌⢻⣯⣷⠊⠠⠀⠙⣷⡀⠀⠀⠀⢸⣿⣧
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣈⣉⣀⣠⠰⠒⠋⠉⠁⠀⠀⠀⠈⠛⠉⠁
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⢤⢀⡀
⣤⡔⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⡈⠄⠙⠂
⠀⠀⠀⠀⠀⢀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⢠⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣄⠀⠀⠀⠈⠗⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠶⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠒⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠦⠀⣀⡉⠀⠒⠒⠂⣶⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣠⡀
⠀⠀⡲⠂⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢄⡀⠈⠉⢈⣷⡆⠀⢠⣴⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣻⡏⠁
⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠙⠚⠶⠦⠄⠑⢲⣶⣤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⡍⠛⢶⡄
⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢿⡆⠀⣀⣀⠀⣹⣿⠿⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡈⣻⢷⣄⠀⠃
⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠠⠤⠌⠙⢻⣴⣿⠏⠀⠈⠳⣴⣿⠿⠇⠀⠀⠀⠀⣀⠀⠀⢠⣴⣦⣄⠈⢿⣧⣀⣿⣦
⠀⠀⠀⠀⠀⠀⠔⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⠄⣴⣿⣯⡷⠶⣴⡆⣀⢠⣿⠀⢀⣾⠟⢛⣻⠞⠀⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢴⣿⡋⠀⠀⢀⣿⣿⡛⣿⣿⠶⢺⣿⣟⣛⣿⡆⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢰⠀⡌⠏⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠇⠀⣸⡿⢿⣿⣿⡟⠀⠈⣿⣿⣿⣿⠃⢀⡤⠊
⠀⠠⠀⠀⠀⠀⠣⡀⠀⠀⠀⠁⢠⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠘⠿⠿⠧⠀⠀⠨⠟⠟⠁⠂⠁
⠀⠀⡀⠀⠀⠀⠀⠈⢠⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠐⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡀⠀⠀⠑⢂⠀⠀⠀⠀⢀⣤⡷⠚⠉⠁⠀⠉⣀⠁⢦⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⣱⣦⣞⣫⡅⠀⠙⣂⠀⠀⠙⠀⠉⠀⠀⣙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⣀⡴⢾⡛⡉⠴⠛⠉⠀⠀⠀⠀⠉⠀⠀⣀⠀⢰⡞⠷⠿⣆⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣶⢶⡿⠁⠀⠉⠁⠀⣤⡀⠀⠄⠀⠀⠀⠀⠐⢶⣤⢦⠤⠙⠂⠦⠼⠀⠸⢋⡷⠆
⠀⠀⠀⠀⠀⢀⢀⠈⠀⠀⠀⠀⠁⠀⠀⠀⣼⡅⠳⢦⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡀⠈⠟⠈⠀⠠⡆⢀⣠⣴⠚⢹⠆
⠀⠀⠀⠀⣠⣾⠿⣄⠀⠀⠀⠀⠀⠄⠀⠀⠛⠱⢆⠀⠉⠳⢤⣀⠀⠀⠀⠀⠙⠇⠀⠀⠁⣀⠀⠀⠀⠀⢸⠟⢋⣠⣤⡟⠃
⠀⠀⠀⢠⣻⣿⡗⣿⠀⠀⠀⠀⢀⠄⠋⠀⠀⠀⠠⠒⠄⠀⠀⠉⠳⢤⣀⠀⠀⠀⠀⠀⣀⣭⠷⠐⣶⣄⢠⠖⠊⠹⡰⠿⢇
⠀⠀⠀⠈⠉⣾⠑⠛⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠳⢾⣧⡖⠋⡉⢤⣟⠈⡟⢭⡀⠀⠀⣄⠉⠂⢾⡆
⠀⠀⠀⠀⠂⣸⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⡄⢸⡆⠀⠀⠈⠢⣄⠀⠀⠈⠉⢀⣤⠶⢞⠉⢰⢳⡞⠳⢤⣄⣈⡷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⣿⡾⣾⠃⠀⠀⠀⠀⠈⠳⣤⡴⠶⠉⠀⠀⠀⠀⠈⢾⡵⠀⠄⣹⠋⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣸⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⣠⡈⠀⣷⣈⠻⣧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣀⠆⢥⠀⠉⠧⣀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢾⣧⠀⠀⠀⠙⠋

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⢛⣛⡛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠈⠿⣿⡶⠶⣤⣤⣤⣤⣤⣤⣤⡴⠶⠛⠋⠁⠐⢒⣤⣤⣾⠿⠛⠛⠛⠲⠿⣶⣄⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⠀⠀⠀⠠⣀⢀⣈⣭⠿⢺⠏⠁⣰⠋⠀⢀⣠⡴⠚⠛⠚⠿⢷⣆⣰⠟⠂⠀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣌⡉⠙⠛⠛⢋⠝⠋⠀⠰⠋⠀⠐⢁⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣄⣀⣀⣤⣤⣴⣾⣛⣥⣤⣤⣶⣶⣶⣶⣶⣤⣤⡿⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⣭⣶⣿⣿⠿⠿⠛⠛⠛⠋⠙⠛⠛⠛⣿⠃⢠⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠾⠿⠶⣿⣄⠉⠛⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⣤⣄⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣴⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠄⠀⣀⡀⠀⢀⠀⢀⢾⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⢻⡶⣿⣿⠟⠁⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⢿⣏⡀⠀⣿⣧⠀⢸⠀⣼⠀⢠⠅⢀⣴⣶⣶⣄⠺⠿⠿⢋⣿⡆⠀⠀⠘⠄⠀⠀⠈⠃⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡹⠧⣿⡆⠀⣿⢸⠀⢺⢀⡟⢀⡞⢰⣾⣿⣿⣿⣿⣷⣤⢠⣽⡧⣴⣷⠈⠐⠀⠈⠁⠀⢠⡸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⡁⠀⠀⣿⣼⣶⣿⣾⢃⡞⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⡿⠈⡟⣡⡾⠃⠀⠀⠀⠀⢼⣿⠄⢸⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠂⢠⣿⣿⣽⣿⣿⡿⠀⠀⠘⢿⣿⣿⣿⡿⠟⠁⠀⠇⠀⠘⢉⣵⣿⣿⣶⣶⣦⡈⠁⢠⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⣿⣿⠀⢀⠀⢀⢀⠀⠀⢸⣿⣿⡿⢿⣿⣿⣿⣷⠈⠉⣡⣄⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣿⣿⢀⡀⠈⡉⠛⠛⠋⠀⢀⠇⠀⢀⠀⢉⢉⢀⡏⢠⠏⡜⠀⢀⢀⠋⢻⣷⡾⢛⣿⣿⣿⣆⣸⣿⠙⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣍⡈⣧⠀⡇⢠⠀⣧⠀⢸⠀⠄⠇⠀⠇⠸⠘⠀⠘⠀⠇⠇⠾⢸⠀⢸⣿⣷⠟⠉⠺⠿⠋⠋⠰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡗⠈⠿⠿⠷⠸⣆⢷⠘⣧⠸⣆⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⢀⠖⢀⡟⢠⣿⣿⠁⠀⠀⡀⠠⠀⠂⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣤⣤⠀⢠⡀⢹⣾⣧⠸⣇⢻⡄⠀⢠⡁⢠⠀⣠⠀⢀⢀⡏⠈⠀⢊⣤⣜⠛⠏⠀⠀⠄⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢋⣴⡆⠉⠚⣿⣿⣇⢻⣎⣷⡀⠸⡃⠻⠀⠓⠀⠳⠘⠀⠀⢰⣿⣿⣿⣿⣷⠀⢀⡀⠀⠀⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⡇⠀⣀⡘⣷⢻⡞⣿⡹⣧⠀⠁⢠⣷⡄⣠⣴⣷⠀⠀⢺⣿⣿⣿⣿⣿⠑⠿⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⢿⡟⢿⣞⣿⣿⣧⠹⣧⢀⣸⣿⣿⣿⣿⣿⢴⡆⣾⣿⣿⣿⠟⣁⣀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠇⠀⣶⡄⠸⠿⠻⠛⠛⠀⣀⣿⣿⣿⡿⣿⣿⣿⣾⡇⢿⣿⠟⣡⣾⣿⣿⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣷⣿⣿⡿⠛⣤⣶⣦⠐⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠀⠈⢿⡿⠛⠁⣀⣘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣦⡀⠞⢠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⣼⠄⠀⠀⠀⠀⠀⠀⠘⠻⣷⡀⠘⠀⠀⠘⠈⡽⠃⠀⠉⠁⠂⠀⠀⠀⠀⠜⠀⠠⠀⠀⠀⣄⠀⠄⠀⠀⠀⠀⠈⠆
⡀⠁⠉⠐⠰⠦⡀⠀⠀⠀⠀⢠⣎⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⡤⠀⠘
⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢀⣼⣮⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠛⠛⠒⠀⠠⠀⠀⠉⠁⠀⠀⠂⠐⠂⠀⠀⠀⠀⠤⠀⢷⡀⠄
⠠⠀⠀⢀⣠⠟⠑⠑⠞⠚⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠒⠃⡸⣇
⣴⣶⠶⠋⠁⠀⠀⠀⠀⠀⠀⣀⣄⣠⠀⣀⡀⠀⠀⠀⠀⠀⠜⣦⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢀⣌⡀⡖⣢⠄⠀⠀⠀⠈⠳
⡀⠀⡀⠀⠀⠄⠀⠀⠀⠠⠜⠁⠀⠀⠀⠉⣿⣄⣀⡀⠀⠀⠀⠜⠳⣥⡄⠀⢀⡰⢃⡄⠀⠀⠀⠀⠛⠐⠬⡗⠻⣖⣺⣇
⡴⣒⠟⠳⠶⣄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⠁⠀⠀⠀⠀⠐⠶⠝⠳⠏⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠈⣿⡧
⡾⡃⠀⠀⠀⠈⠳⠤⠀⠀⠀⠀⠀⠀⠀⢀⡼⣧⠙⠉⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠃
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⣤⠀⠀⢀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⣦⣠⣤⠶⠂⠁⠀⠠⣰⣤⣟⠒⡀⠒⠋⠀⠀⠀⠀⠡⢠⡟⠈⠁⠀⠀⢠⡀⠑⢤⠀⠀⠀⠀⠀⢃⣠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡶⠟⠁⠠⠃⠀⢈⡅⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⢀⡄⠀⠀⠀⠀⠀⢻⡆⠀⡤
⢀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠃⠀⠸⠂⠀⠀⠀⠀⠀⠈⠲⢦⣤⠤⠶⠖⠛⠡⣈⠀⠀⠀⣱⣶⠊⠧⠄⠀⢸⠀⢀⣀⠁⠐⠂
⠈⠀⠀⠀⠀⠀⠛⠥⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⢾⡿⠉⠃⠀⠀⠀⢸⡆⠀⠈⠉⠀⠐
⣀⣤⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠈⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠈⠀⠀⠀⠐
⠟⢠⡈⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⢷⣦⣀⠀⠀⠠⠠⠒⠦⣄⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣔⣀⣀⡤
⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⢸⡆⠀⢠⠀⢰⣡⢌⣠⣿⣷⡖⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣄⡤⣤⡀
⠀⠀⠄⠠⣤⠇⢤⡰⠂⠀⠀⠈⢿⣂⠀⢡⡴⡷⣶⠋⠀⠚⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⡖⠊⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠽⡃⢀⡀⠀⠀⠀⠀⠹⠿⣿⡶⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⣤⠿⠃⠀⠀⠀⠀⣀⣾⡇
⠀⠀⠀⠀⠀⣿⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡦⠴⠋⠋⠀⠀⠀⠀⠀⠉⠁⣼⢆⡀
⣀⢀⡀⡤⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠃⠺⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡝⠀⠀⠀⠀⠀⠀⠡⣄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⢀⠈⠁⠲⠉⠀⠀⠀⠀⠘⠃⣰⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⠴⠂
⣀⣀⣐⣀⣀⣀⣀⣀⣀⣠⣄⣠⣀⣒⣀⣉⣛⣷⣠⣬⣃⣀⣁⣀⣀⣀⣀⣀⣀⣸⣋⣀⣃⣀⣂⣐⣃⣀⣀⣀⣀⣂⣸⣋⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣴⠶⠆⣠⣄⠀⢀⡀⢠⡀⠀⠀⣀⠀⠸⠀⠃⠈⣄⠀⣦⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡜⣯⠐⠁⡙⢧⣀⠈⠈⢁⣥⣤⣿⣍⡓⢠⡄⢀⡸⣦⠘⣧⠀⢹⡆
⠀⠀⠀⠀⠀⠠⠄⢒⣼⠐⠀⠺⢤⠙⣦⠉⠀⢻⣄⠀⠀⢀⢹⡶⠲⠟⣽⣷⣿⢻⣇⣠⣼⣄⠡⠸⠆⢻⡀⢘⡃⢀⡀
⠀⠀⠀⠀⢰⡖⠂⢸⡗⠀⢀⣿⣦⠁⢛⣤⣾⠏⠉⠛⢤⡈⣴⠁⢀⡀⢈⠻⣿⣿⣿⡏⢠⢻⡄⠠⠘⠂⠀⠀⠁⢸⡇⣦
⠀⠀⠀⠀⠘⣡⣧⡘⢿⣦⡀⠛⠿⠄⢸⣋⢰⣆⠀⣀⠘⠱⣿⠀⣿⣿⡀⣱⠮⣿⣿⣷⡘⣿⡵⡀⠛⠀⢀⣟⠀⠈⢳⣄
⠀⠀⠀⠀⠀⢹⡿⠧⢸⠀⠙⣦⠀⠀⢿⡁⢴⡉⠀⣿⣷⠀⣹⣄⡎⢉⠁⢻⣄⣿⣿⣿⡜⢻⡧⣿⠐⢄⠘⣿⣔⣷⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢷⡄⣚⠠⡄⠈⣳⠁⠸⠅⣼⢱⣦⡈⠉⡰⡍⠨⠛⣦⠟⠨⣏⢱⣿⣿⣿⣟⠃⢿⡀⠈⢃⡀⠙⠇⢰⠏⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠘⣿⡄⠉⡆⠀⣷⠛⢨⡏⠻⣤⠁⡷⢾⣦⣻⢻⣷⣽⠷⣼⣿⣿⣿⣟⠿⢳⣿⡘⢿⡤⠀⣶⣷⡄⢀⡀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡧⠘⠛⣣⠞⣇⠠⡏⠀⢸⡃⠀⠈⠙⢿⣝⠛⠀⣿⣉⢫⣤⣛⣿⣿⣿⣧⣿⠀⣿⡇⠀⣷⠀⠈⡿⠓⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣷⠏⠀⢏⣰⠃⠀⠸⠀⠀⣶⣆⡀⠙⣶⠞⠉⢿⡏⠀⣿⣿⣿⣿⣿⢻⣦⠹⣷⠟⢸⠀⣴⠇⢼⠇⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣟⢠⣧⠔⠁⠜⢻⢸⣿⡄⢻⡀⠻⢿⡃⠀⣍⢁⣤⣄⣸⠟⣿⣿⣿⣿⡙⠸⠇⣷⢸⣧⡌⣀⡝⢠⣿⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣷⣾⣷⠀⣿⠈⠛⠁⣠⠛⣦⠈⠁⢀⣿⡀⠿⡿⠙⢣⡿⢿⣿⣿⠻⠆⣤⡜⢣⣇⢸⡅⠀⢸⡁⢸⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡴⡏⠹⠿⠆⠻⠀⠀⡼⢃⡄⠹⣦⣀⣼⣤⠙⣶⣄⣴⡏⢛⣿⣿⣿⢻⠆⣿⣿⢈⠟⢻⡾⠀⡾⠁⢙⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⠇⠀⢠⣤⣼⡀⣺⠀⣿⣿⢀⡇⣭⠸⣿⢎⢁⣄⠀⢻⣿⣿⣿⣿⠃⠶⣹⣷⠈⡁⠸⣡⠂⢶⡆⢸⠀⢸⡇
⠀⠀⠀⠀⣤⡀⠀⣿⣮⣽⣟⣿⡿⠆⡹⠉⠻⣄⣈⣤⣾⣀⣿⣤⠉⢹⡈⢿⡿⣏⢻⣿⣿⣿⠀⡆⠘⡿⠀⠏⢠⢧⡔⠈⠷⣸⡄⠈⣇
⠀⠀⠀⣼⡇⣿⠀⠿⣟⣿⣭⣴⣶⣾⣿⣶⣶⣼⣿⣤⣾⣿⣷⡜⣷⣌⡿⠤⡀⣸⣿⣽⣿⣿⣤⣽⠀⣤⣴⠀⢿⣇⢹⡄⠀⣿⢸⡄⢻⡄
⠀⠀⠀⠈⠓⠋⣠⣼⣿⣿⣿⣿⣿⣿⣿⡇⠛⠿⢿⡿⣿⣿⣿⣾⣤⣂⣇⣤⣽⣷⡜⢿⣿⣯⣇⣠⣁⣀⣹⣦⠨⣿⣦⠀⠀⣿⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⣿⣭⢻⣿⣿⣿⣷⡎⠹⠻⠮⣶⠀⡔⠀⠈⠙⠣⢬⡛⢿⣷⡙⠋⠟⢒⣷⢿⣟⣿⡿⠉⢻⡙⠀⢿⣇⠀⢰⣬⣿⡀
⠀⠀⠀⠀⠀⢰⣷⣿⡀⣿⣿⠿⣉⢿⣶⣧⣀⢨⠛⣧⠈⠉⢐⣀⡀⠙⢸⡿⣿⣦⠜⠣⣛⣿⣿⣿⣷⠄⡘⠉⣄⠠⡟⠀⡀⠡⠘⣷
⠀⠀⠀⠀⠀⣾⠯⢹⡃⣿⣿⣿⡿⠻⣽⣋⣹⣿⣦⣻⣷⡀⢻⣿⣿⣄⠀⠘⡼⢿⣿⡀⠯⣽⣿⡼⢿⡄⠁⢚⠹⢦⠘⠃⣸⡗⠀⢻⡆
⠀⠀⠀⠀⠀⣿⢿⣼⡃⣿⣿⢿⡟⢀⠙⠿⣾⠁⠉⠻⣿⣿⡊⠻⣿⣿⣷⠀⠘⢳⡟⢳⣜⣻⣿⣧⡿⠤⢄⣸⡇⢘⡗⠀⣼⡟⣀⢸⡁
⠀⠀⠀⠀⠀⢻⡾⢟⣷⣿⣿⣬⠿⣀⢀⠐⠋⢦⡀⠰⠀⠉⠻⢦⠀⠉⠁⠠⢄⠁⠘⢫⠽⣿⡉⣿⣷⡀⠈⢹⡇⠈⣿⠀⢻⡇⣩⠘⣧
⠀⠀⠀⠀⠀⢀⣧⢻⣏⠛⢹⣿⡀⠉⠧⡑⠀⠀⠙⣤⠂⠀⠀⠀⡙⠶⣄⡀⠘⠃⠀⠈⡿⣿⣷⡘⣿⡀⠘⠃⡴⢸⡿⠀⢻⣷⠀⣤
⠀⠀⠀⠀⠀⠸⣧⠙⡿⣄⠃⢹⣧⣴⣦⡑⢤⡀⠀⠈⠳⡀⠀⢠⣷⣤⡀⠉⠳⢆⡁⠀⠳⢾⣿⣯⡽⡁⠀⢸⠁⢨⡅⠀⢻⡄⢠⡿
⠀⠀⠀⠀⠀⢸⡟⠰⢀⡈⠓⠾⢿⣿⣿⣷⠀⠳⣄⠁⢀⡙⢧⣈⢿⣏⣿⣆⠀⠀⠈⠰⣦⡙⢿⣿⣷⠟⣠⠖⢀⠉⠀⣇⢠⠇⠸⡁⠀⢸⡁
⠀⠀⠀⠀⠀⠐⣏⠁⢸⠀⣦⣀⠈⢿⣿⠿⡀⠂⢈⡄⠀⣡⣀⡉⠂⡹⢿⡿⡀⠀⢀⠀⠀⠹⢆⠹⣟⣵⠃⣤⠉⢠⡀⢉⣿⠐⠀⠱⠄⡈⠀⢘⡷
⠀⠀⠀⠀⠀⠀⡍⣿⠸⠀⢻⣧⠀⢠⡟⢷⣼⣭⡀⢰⡈⢿⣟⢿⣦⡰⠁⠢⣄⠠⢉⡀⠀⠀⠈⠻⣿⣿⢷⣿⢀⢠⡇⠈⣿⡄⠰⠇⠀⡇⠀⢘
⠀⠀⠀⠀⠀⠀⡟⠉⠸⠀⡙⠫⠀⡼⢣⡄⠙⣿⣿⣆⡹⢆⡙⠿⠿⢣⣬⠒⣈⠓⢛⢡⣦⣶⣤⣶⣿⡇⡀⠉⠯⣸⡇⠀⣨⡧⢀⡴⠀⡿⠀⠈
⠀⠀⠀⠀⠀⠰⡗⠀⢲⣤⠈⠀⢠⠁⠈⡼⣷⡸⣿⣿⣷⠆⣉⣶⣀⣤⣠⣌⠭⢀⠋⠠⠀⠰⣶⣭⣿⠟⠡⠆⠰⣾⡿⠀⢹⠀⠸⡇⢠⡝
⠀⠀⠀⠀⠀⠀⣇⡘⢿⡀⢧⡔⢃⠁⣤⢁⡙⠃⠈⡿⣿⣶⣯⣥⣬⢤⣍⣸⣶⣾⣷⠒⢺⣎⣡⣿⡿⠄⠰⡎⠀⠙⣷⠀⢼⡀⠀⢲⠋
⠀⠀⠀⠀⠀⠀⢳⡟⠀⢦⣺⠇⣠⣶⡄⠀⢹⠖⠘⠇⢈⡉⠻⠿⠿⢿⣼⣿⣿⡿⣿⣿⣻⣿⡿⠟⠑⡴⠂⡄⠘⠀⢨⡇⠈⡗⠀⠟⠂⠀⢾⡇
⠀⠀⠀⠀⠀⠀⢻⠀⠀⢨⡇⠦⢈⣿⡿⠀⠻⢀⡀⠘⠁⠠⠒⣆⠠⣶⣿⡀⡀⢖⣻⣿⣿⡟⠀⢠⡞⠃⣸⡠⠀⠀⠈⠃⠰⠃⠈⠀⣶⢤⣈⡇
⠀⠀⠀⠀⠀⠀⠸⣄⢠⠀⣖⢰⡏⠈⡁⢰⡆⢸⣿⠀⢿⡠⠚⠉⠛⠬⢩⡄⠰⢞⣻⣿⣿⠃⣼⠟⠃⢸⡏⢧⠀⠀⢰⡄⠀⢰⠟⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢀⠷⣀⡃⢸⡀⠠⢸⡇⠘⢋⠀⣺⡇⠀⣰⣶⡄⠠⣆⢰⣾⣿⣿⠏⢸⠯⠐⠠⢸⣷⡤⢿⡇⠀⠀⠀⢸⡇⠀⠸⠋⡿
⠀⠀⠀⠀⠀⠀⠀⣸⡎⠠⣿⣇⠘⡀⠀⢸⣅⣀⠘⣀⣀⠈⣄⠙⠟⠁⣎⠀⣉⡿⠥⣿⠀⠃⠀⠀⠢⣼⣯⠠⣼⡀⢀⠈⠀⠘⡇⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠈⢁⡌⠙⠁⣰⠞⢠⠏⠀⢸⠀⣿⣿⡆⠈⠗⠆⢺⠟⠣⣿⡿⠧⠘⠀⠀⢀⠈⠈⢹⡍⠀⢸⡏⢘⢀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣻⠀⠀⠐⠦⠘⡟⢣⠾⠃⢀⡸⢦⡀⢿⣧⣄⣿⠀⢶⠋⠀⢀⡦⠀⠸⠇⠀⠸⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠁⢢⡄⢣⠁⡄⠈⣇⡰⣾⣿⣿⠶⣿⠳⡞⠃⠀⢸⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣡⠀⠀⠀⠈⠀⠀⠀⠘⠉⠛⠛⠳⠾⡿⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣼⣃⣻⣧⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⢄⠀⠀⠀⠀⠀⡔⠀⠀⢢
⠀⠀⡇⢠⠛⡄⠀⠠⢦⡀⡀⠀⠀⠃
⠞⢇⠀⢈⠀⠘⠄⡀⠁⢸⡟⢀⡠⠠⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠈⠀⠀⢳⡄⠁⠒⡺⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠔⣺⠇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠓⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠀⠀⠈⠆⠀⠀⠀⣀
⠀⠀⠀⢀⡀⡀⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⡨⠓⠀⠀⣤⠘⠆⠀⠀⠈
⠀⠀⠀⠸⡜⠁⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠃⠀⢸⠆⠀⠀⠉⠀⠙⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣈⠐⠠⠀⠀⠀⠀⠀⠀⠈⠁⠱⡀⢂⠀⠀⠀⠖⠈⢉⣳⡄⢀⣀⣀⢤⣀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠘⠐⠠⠐⡡⠆⠀⢦⠠⠀⡈
⠀⠀⠀⢰⠀⠀⠀⣀⣀⡄⠉⠂⠄⠀⠈⠁⠒⠠⢀⠀⢀⣥⠟⢂⠀⢠⢛⠒⠈⠮⢇⣀⠳⡴⠦⠠⠤⠄⠀⠚⢤⡳⠆⢍⠒⠤⡀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠉
⠀⠀⠀⠀⢫⠉⠉⠉⠉⠣⡀⠀⡆⠀⠀⠀⠀⡠⠂⠀⣿⡆⢀⠀⠉⠈⣠⡄⠀⡨⠃⠘⢰⠳⠤⠤⠄⠀⠠⠤⠤⢷⣮⠌⣸⠋⠀⠀⠀⢤⣴
⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠤⢀⠦⠂⠀⠀⠀⠀⠀⠀⠁⠀⠰⣶⠛⠢⠤⠒⡈⣑⠒⠶⢔⣀⠀⠀⠀⠀⠀⢀⠰⠋⠁⠀⠈⠁⠐⠒⠞⠁⢀⡀
⠀⠀⠀⠀⠈⠢⠀⠀⣀⢀⣀⠀⠈⠊⠀⠀⠀⣠⠄⢀⠠⠤⠒⢈⠛⠖⠠⠤⢀⡈⠓⠚⠛⠒⢾⡝⠋⠻⠤⠀⡌⠁⠀⠀⠀⠀⠀⢠⣤⢴⣶⡬⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠃⠒⠠⠄⢈⡁⠄⣤⣠⣠⣤⡠⢀⣄⢀⠿⠔⣂⣤⣐⡠⠄⠊⡉⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠟⠚⠀⠉⠓⣄⠁⠀⠀⠀⠀⡀⠓⠚⠉⠋⠙⠛⠙⠛⠘⠂⠀⠤⢈⡅⠀⠀⠁⠀⠰⠃
⠀⠀⠀⠀⠀⢀⠀⢲⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠒⠚⠋⠐⠁⠀⠀⠀⠁⠀⢈⣟⠰⣯⠠⡀⠀⠒⠠⠄⡀⣨⡀⠔⢈⠔⠉⠉⠉⠒⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠬⠁⠀⠀⠀⠦⠄⡀⠀⣍⡠⠒⠉⠐⠠⠀⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠄⡀⠀⠀⠀⣀⡄⠀⣤⡀
⠀⠀⠐⠒⠐⣶⠄⠈⠀⠀⠸⠀⠀⠆⠀⠀⣬⣀⣀⣸⡜⠂⠠⠈⠰⠰⡦⠄⠠⢴⠀⠀⠉⠆⢠⣉⣶⣾⡦⢰⡄
⠀⠀⠠⣤⠠⠂⠀⠊⠀⠀⢇⠐⠐⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠉⠰⠠⠀⢀⠀⠐⠀⠂⠁⠀⠘⢀⣌⠉⣱⠞⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⠖⠀⠀⠀⠀⢀⢠⠴⠣⠐⠀⠋⠀⣀⡄⢠⣄
⠀⠈⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⠀⠀⠀⠀⢄⠘⢋⣥⡿⠃⡄
⠀⠀⠀⠀⠈⠘⠑⠀⠀⠀⡀⠈⠐⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⣲⢢⠠⠠⠀⠀⠀⠄⠀⠀⠙⢻⣷⠖
⠀⠀⡀⠀⣤⡄⠀⠐⠀⠀⢁⠁⣉⣀⠠⢀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠈⠥⠃⠀⠀⠀⠰⠀⠀⠐⡀⠀⠀⠀⠐⡄
⠀⠀⠈⠢⠀⠀⠂⠀⣀⠔⠧⠿⠵⠂⢁⡠⠀⠀⡤⣐⣂⣤⠀⠀⠀⠒⠢⡈⠂⠀⠀⠀⠁⠀⡄⠀⠘⢄⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠄⠰⠉⠁⠀⠁⠀⠐⢀⠀⠀⠀⢿⣿⠏⠀⠐⠀⠀⠀⡀⠀⠀⣔⠄⠄⠀⠁⠄⠔⣆⠠
⠀⠀⠐⢀⠀⠀⠀⠀⠀⠐⠠⠤⠄⠂⠄⢠⠐⠡⡀⢀⣀⠀⠀⠀⢴⣶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠃⠆⠀⠀⠂⠀⠓⠀⢁⠄⡀⠀⠀⠀⠤⠄⠀⠀⠀⡀⠀⠀⢠⠚⠃⠀⠄⠀⠀⠀⡀⠀⠀⠠⡀⡠⢠⠄
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠂⡆⠻⠃⠀⠀⠐⠀⢀⡀⠀⠴⠑⠢⠀⠓⠀⠀⠀⠀⡀⠠⠀⠀⠐⠀⠈
⠀⠀⢀⠐⢀⠀⠀⠠⠈⠀⠀⠀⠑⠦⠤⠠⠭⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠦⠀⡜⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠉⠉⠛⠲⢤⡀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⠶⠶⠶⠶⠶⠶⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⣰⠃⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣤⠴⠖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⠤⣄⣀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢳⡀
⠀⠀⢰⠇⠀⠀⠀⢠⠞⠁⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⡀⠹⣆⠀⠀⠀⠈⢧
⠀⢠⡏⠀⠀⠀⣰⠋⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣼⣧⠀⠀⠀⠘⣇
⠀⣾⠀⠀⠀⢰⡷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⢹⡄
⢰⠇⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⢤⣄⣀⠀⢀⣀⣠⡤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⣇
⣼⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹
⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘
⠹⡄⠀⠀⠀⠀⢻⡀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢠⠟⠀⠀⠀⠀⡼
⠀⢻⡄⠀⠀⠀⠀⢳⡀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡴⠋⠀⠀⠀⠀⣼⠁
⠀⠀⠹⡄⠀⠀⠀⠀⠙⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣤⣀⣀⣠⣤⠴⠚⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⢠⣤⣛⣷⣾⣃⠀⠺⢿⠃⢀⢮⠇⠀⢶⣦⣼⢇⡀⠀⠀⠈⠛⠁⠀⠀⠀⠊⠀⠃⠀⠀⠀⠘⡀⠘⠛⠁
⠀⠀⠙⠋⠉⣿⣿⠆⠀⣤⡮⠗⠋⠀⢠⡶⢷⣵⣿⣿⡇⠀⠀⣰⣿⣷⣦⠀⠀⠘⣁⣀⠀⠀⠃⠐⢀⣠⣄
⠀⢀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠷⡀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠸⠿⣿⠟⠀⢀⡀⠀⠈⠁⢠⣶⣷⡜⠛⠋
⠀⠘⠿⠁⠀⠀⠀⢳⡐⡔⠎⠂⠉⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⣞⠩⠙⣾⠆⠠⠀⠛⠋
⠀⠀⡀⡀⢀⠀⠀⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠈⠓⠻⢧⡤⢼⡄⠀⠀⠀⣀⣤
⠀⠀⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣾⡇⢰⡇⠀⠀⠀⠸⣔⣣⠇⠀⠀⠀⠙⠛
⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⣾⠀⣿⡆⢀⣤⣄⣴⣶⡆⣿⡄⣿⡶⢸⣿⣷⣴⣠⡆⠀⠀⠀⠀⠀⣀⣴⡖
⠀⠀⠀⠈⠛⠛⢁⠀⠀⠀⠀⢿⡟⢻⣇⠿⡯⠁⠛⠋⠁⠀⢀⣤⠀⠈⠀⠀⠈⢹⡇⠀⠐⠛⠛⠛⠛⠉
⠀⢠⠄⠀⢀⢀⣼⡀⠀⠀⢀⡘⠋⠀⠁⣰⠶⠀⠀⠀⣤⠄⢼⡿⠀⡀⢀⠘⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠻⠙⠟⠙⠁⣁⣀⠀⠀⣿⡇⠀⠖⢰⣿⢀⣶⣿⠄⣭⡷⢸⣧⠄⠷⣿⢇⣿⢠⣾⠿⠀⠀⠀⡂⠀⠁
⠀⠀⢀⠀⠀⠀⠙⠿⠁⠀⣿⣇⣠⡇⢸⡇⠀⠛⠋⠀⠀⠀⠀⠀⢠⣤⡿⢸⣯⡌⠉⠁⠀⠀⠀⠀⠀⠀⠚
⠀⠀⣸⣧⠀⡠⠐⠀⠀⠀⠛⠛⠈⠁⠀⠀⠀⠀⢀⣀⢴⣏⡂⠀⠈⠉
⠀⣶⣿⣿⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⠠⠶⠚⠋⠉⠀⠀⠀⢀⣀⡤⠶⠂⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⣦⣀
⠀⠙⠛⠋⠀⠀⠈⠀⢁⠀⢠⡄⢠⣶⠀⠀⠀⠀⠀⠺⠿⠋⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⢀⢀⠀⠀⠉⠀⣲⣟⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣶⢆⡀⠀⠈⠛⠛⠋⠁
⠀⠀⠀⢠⠁⠀⠁⠀⠀⠀⡽⢿⠿⠟⠛⣠⣀⠀⠀⠀⣄⡖⠛⠀⠀⠀⣶⣄⠊⠀⠈⠙⢡⢂⡀
⠀⠀⠀⠀⠉⠈⠀⣠⣿⣿⠇⠀⣀⣀⣈⠁⠘⠳⡶⣠⡀⠀⠀⠀⣴⣿⣿⣧⣦⡀⠴⠀⠀⠐⠃⠀⠠⠆
⠀⠐⢿⣿⠀⠀⣼⡿⠟⠁⠀⣿⣿⣿⣿⠀⠀⠉⠁⠈⣠⣄⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡀⠘⠀⠂
⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠙⠻⠁⠀⠀⠀⠀⠈⠛⠛⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠤⠀⠀⠉⢉⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡞⠉⠉⠭⣩⣤⠈⠘⢀⡉⠀⠈⠐⡄
⠀⠀⠀⠀⠂⠀⠀⠀⣀⠔⠃⠉⠉⡩⠭⠛⢂⣉⠀⠀⠤⠀⣇⣀⢀⡄⢀⠤⠌⠖⠁⢡⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡠⠀⠀⠀⢀⡴⠎⠀⢱⡀⠀⠀⡟⠘⢤⠴⠞⠓⠼⣀⠡⠣⢤⠂⠀⡏
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠸⠁⠀⠀⠰⡋⠀⠀⠀⠀⡇⠀⠀⠘⠄⠈⡓⣒⠀⠴⠄⡤⠀⠂⢁⡠
⠀⠀⠐⠀⠀⢸⠀⠀⠐⠘⠶⣦⠴⠀⠀⠀⢀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⢸⡀⠘⠦⠁⠀⠚⠀⠀⠈⠁⠁⠀⢰⠃⠀⢀⡠⠔⠚⣩⠍⢉⠀⢉⡙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠱⣄⠐⠒⡖⡀⠀⠀⢄⠀⠀⣠⠋⢀⠔⢁⢄⠀⠈⠁⠀⠀⠉⠂⡀⠉⠀⠈⠑⢦⡀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠈⠓⠤⣄⣀⠀⣀⣀⠀⠐⠁⡠⠋⣀⡡⠎⣠⠁⠄⠉⠁⠀⠀⠀⠸⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠐⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⡏⠀⢠⡹⡀⠀⠀⠀⣠⡤⠂⠀⠐⡈⠂⠀⠣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣠⠐⢈⣭⣉⣁⡲⡄⠀⢰⠃⡤⠂⠀⠀⢀⠽⠦⠠⣤⠶⠷⢤⡀⠀⠀⠀⠀⠐⠂⠀⠈⡄
⠀⠀⠀⠀⠀⠀⡜⣡⣼⡿⣿⣿⣷⡿⠈⠄⢸⠈⢣⠀⠀⠀⢨⡀⠀⠀⢉⠀⠀⠈⣿⣆⢀⣠⠄⠄⠀⠀⡠⡧
⠀⠀⠀⠔⠃⠀⡁⠒⣻⣿⣿⣿⣽⠇⠀⡀⢸⠀⠀⠱⣄⠀⠀⠈⠂⣀⣈⠲⣀⢠⣟⡽⡅⡂⠀⠄⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠱⣸⣿⣿⠿⢿⠟⠁⡔⠀⡈⢧⠀⢠⠈⢦⡀⠀⠀⠀⠈⡝⡁⠿⢅⠊⠉⠓⠀⠴⠈⠄⡼
⠀⠀⠀⠀⠀⠀⠀⠉⠊⠡⠤⠀⢂⢊⡀⠄⡀⠈⢦⡀⠃⡀⠉⠢⣀⠀⠘⢧⠀⠚⡋⡤⠟⠀⠐⠃⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠠⠒⠑⠢⠱⠀⠱⣄⠀⠀⠀⠈⠓⠦⣌⠒⠈⢁⡤⠤⠆⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡧⠁⢰⠫⠹⢉⠀⠀⠀⠈⠙⠢⢤⣀⠀⠂⠈⠳⣄⣈⡠⠔⠊⠁⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣐⠀⠓⠀⠎⠀⠁⠀⠀⠀⠀⡀⢀⣉⣀⡀⠀⠉⢁⣀⣤⠔⠊⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠀⠀⠉⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢄⡀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠻⡀⠀⠀⠘⠀⠁⣄⠀⠀⠀⢠⠆⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠐⣦⣀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠹⣆⠀⠀⠀⠀⢻⠀⠀⠀⢁⡀⠀⢰⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠤⢄⡀⠀⠀⢸⠀⠀⠘⣦⠀⣀⡀⠀⣤⠌⣠⠛⠁⠀⡆⠀⠀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⠅⠀⠀⠈⣿⠏⠹⢦⠍⢰⠇⠀⠀⢰⠁⢀⡾⢁⠀⠀⠀⠀⣀⠤⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡀⠀⠀⠈⠳⣶⠀⠀⢀⡌⠀⠆⠈⢻⡇⠀⡆⠀⣎⣰⠏⠀⠌⠀⢠⠄⠂⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠉⠳⢤⣽⠀⠀⣼⠁⠀⠀⠀⠀⢱⡀⠇⢠⡿⠁⠀⣾⡄⠈⠁⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠰⢶⣶⣦⣄⢀⡀⠈⠳⣄⠀⠈⠳⢦⡟⠀⠀⢀⡀⠀⠈⣷⣶⠎⠀⢀⡾⠋⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢌⡋⢤⡉⠠⠙⣦⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠸⠃⠀⣰⣏⣀⣀⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡙⢦⣀⠙⢷⡀⠀⠀⠀⠻⣤⡁⠀⣰⡦⠚⠃⠉⠉⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠲⠶⠀⠀⡀⠀⠙⢀⠹⣶⠘⠛⠂⣤⣄⡀⠘⣧⠞⠁⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡀⠙⠦⡀⠳⣄⠀⠀⠈⠉⠲⣦⡀⠀⠀⠀⢀⠀⣤⣼⠀⠒⠒⠀⠀⠉⡡⠿⠟⠁
⠀⠀⠀⠀⠀⠈⠓⢂⡀⠶⠄⠘⠂⠀⠛⠶⢶⠀⠹⠆⠀⠀⠀⠀⠀⠻⡄⣀⡴⠞⠋⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⡖⠒⢀⠤⢄⣀⠳⠀⠸⠦⢄⠀⣀⠀⠤⠾⣁⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠻⠖⡄⠐⠀⠉⠛⠻⣦⣀⠀⠛⠁⠀⠀⠀⠈⠳⡆⠁⠀⢀⡄⢀⣀⣀⠠⠐⠒⠚
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠐⠦⡄⢠⣄⠢⡔⢄⠀⠀⠙⠲⣤⡀⢀⣤⠄⠒⠚⠒⠂⠰⢿⣦⠈⠉⠉⠐⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠠⠼⣿⣦⣤⣄⠀⠀⠈⠃⣀⡀⠆⠀⠀⠀⢀⣹⣧⠀⠀⠀⠀⡀⣤⣥⡴⠶⠖⠈⢉⠉⠉⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠛⠷⢶⣋⡉⠿⠶⣉⣀⠀⠀⠀⢀⣀⠴⠟⣤⠤⠞⠋⠀⣀⡀⠀⠠⢤⡤⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡒⠂⢠⠈⠳⠖⠠⢀⠈⠛⠷⡁⠀⠀⣀⣀⡈⠁⣤⠶⠋⢉⡄⠐⠀⠉⠁⠀⢠⡄⠀⠙
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣶⣶⠞⠿⠓⠚⠓⠨⢷⣄⠀⠹⠒⠀⠀⠈⠙⠛⢠⣤⢀⡈⠀⠈⠁⢴⣀⠤⠌⠃
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢈⣧⣤⣤⢠⡶⠒⠂⢠⡈⠣⣄⠘⣀⠀⠀⠀⢠⡴⠒⠻⡷⠖⠂⢠⣄⡉⠀⠀⠱⠃⡀
⠀⠀⠀⣀⠀⠀⡌⠁⠲⡀⠸⡇⠀⡤⠁⠤⠀⠀⠀⠘⠆⠙⣄⢉⡷⠉⠹⢦⣤⠈⠀⠈⠀⠁⠄⠘⡉⠀⠂⢲⠀⠑
⠀⠀⠀⠈⠓⢰⡿⠀⠀⣾⣶⠷⠦⣤⠄⣰⡆⠀⢰⡷⡤⠚⢻⡶⢀⣤⣀⠁⠀⠈⢠⡌⢱⣣⣄⠘⠗⠀⠈⠀⠶⡈
⠀⠀⠀⠀⣠⡋⠀⠀⡰⠋⠃⡀⠀⠀⠀⠙⡛⠃⠀⠛⠆⠀⠀⢳⣀⡄⢩⣐⡦⠈⠛⠀⢀⠍⠛⢳⡆⠀⠀⠀⣀⠙⠀⠀⠰⡆
⠀⠀⠀⡐⠿⠃⠀⠰⠁⠀⠀⠑⠀⠀⣀⠊⠀⣀⡤⢴⣤⣴⣞⠉⠁⡧⠈⠁⠐⣦⠄⠛⢫⡤⠀⡀⠘⣂⡀⠀⠁⠀⠀⢠⠄⠃
⠀⢰⡤⠃⢀⢀⡼⢤⡀⠀⢀⣤⠲⡖⠙⠓⢾⡋⠀⣸⡆⠀⠙⢇⠁⠀⠀⠀⡌⠁⠀⠀⢸⠀⠀⢹⡞⠉⠉⠖⠂⠀⢠⡦⠀⣤⠰
⠀⢸⠇⢀⡼⠁⠄⠀⢹⡟⠉⢀⡴⠃⠀⠀⢀⠉⠈⠉⠳⠀⠀⠺⣷⠶⠶⢤⡿⠀⠀⠀⠊⠀⠁⠰⡇⠀⠀⠋⣀⣼⡏
⡀⠘⢀⡌⠳⠖⠀⠀⠀⠛⠰⠂⢡⠀⠀⠀⠸⠂⠀⠀⠀⠀⣠⠞⠁⠀⡀⠀⠘⢣⡀⠀⠀⠀⠀⠀⢧⡀⠀⡀⠀⢸⡇⠀⠀⠰⡆
⠈⣄⡿⠁⠀⠀⠣⠄⢀⣀⠀⠀⠀⠀⢀⡼⠟⠛⡟⠓⠦⣠⡏⠀⠀⠀⠁⠀⠀⠀⣿⣠⠤⠴⣶⠴⢎⡁⠀⠀⠀⠸⣇⠠⠎⠠⡇
⠀⠉⠰⣧⠈⠳⣄⢀⡜⠋⠘⠓⣄⣠⡟⠀⠀⠀⠁⠀⠀⠹⠇⠀⠐⠀⠀⠀⠒⠂⣼⠃⠀⠀⠀⠀⠀⠙⢦⡤⡖⠢⡀⠀⠀⣠
⠀⢠⣾⣹⠆⠀⠘⣿⡀⢀⠀⠀⠈⠿⠀⣤⡴⠀⣆⠀⠀⠀⡃⠀⣀⣤⣤⣤⣄⢀⣿⠀⠀⠀⡀⢠⡤⣴⠟⠀⢀⡄⠀⣦⠐⢠⡦
⠀⢮⡄⠉⠀⢄⠀⠛⠑⠞⠀⠄⠀⠘⣇⠀⠀⠀⠈⠁⠀⠈⣡⠞⠁⠀⢄⠀⠈⠛⣏⠉⠀⠉⠁⠀⢠⡏⠀⠀⡞⠐⡿⠁⠘
⠀⠾⢫⣴⠶⢄⡀⠰⣄⣤⣄⣀⠀⠀⢙⠶⠉⠙⠋⠁⠶⣤⡏⠀⠀⠀⠈⠀⠀⠀⠈⣣⠶⠆⠀⢶⡜⠃⠀⢉⡀⠼⠇⣀⠀⡟
⠀⢠⠸⡇⠐⠀⠙⣶⠋⠀⠉⠈⠛⢶⡎⠀⠀⠀⠃⠀⠀⠈⢷⠀⠙⠛⠁⠀⠙⢠⠟⠁⠀⠀⢰⠆⢱⣆⣰⠋⢥⡀⣀⣥⣴⠇
⠀⢸⣧⠀⠛⡀⠀⠘⣷⠴⢀⡀⠀⠀⢻⡀⠢⣤⠀⠠⠄⠀⠠⣧⣠⣤⣤⣄⣰⠇⠀⠀⠀⡰⠃⠤⠋⢨⠋⠠⠜⢁⠙⠁⢼⠆
⠀⠸⠙⣧⡀⠑⢤⡀⢡⠀⠈⠑⠂⠀⢀⣧⠀⠀⠀⠀⠀⣠⡴⠋⠁⢠⠇⠈⠳⣆⠈⠁⢈⣤⠒⠶⢦⡀⠀⠀⠀⡮⠠⠂⣠
⠀⠀⣠⠀⠑⣄⠀⣨⡽⠶⣤⡀⣠⠖⠉⠉⠉⠉⠱⢦⠾⠁⠀⠀⣠⠘⡀⠀⠀⢸⣤⠖⠉⠀⠀⣷⠀⣴⣶⠶⢢⡀⢀⠾⠋
⠀⠀⠙⣶⡆⠀⢻⡁⢰⡄⠀⠹⣧⢀⣀⢀⡀⠀⠀⠈⣦⠀⠈⠉⠀⠀⢉⠁⠀⢀⣏⣀⣀⣀⡴⠛⠻⠃⠈⡇⢀⡼⣾⠃
⠀⠀⠀⠘⢷⠄⠈⠁⠙⢠⠀⠀⠙⡄⠀⠀⠑⠠⠄⠠⠛⢀⡠⠤⠛⠋⠁⠀⠙⠧⡀⠀⠀⠀⠀⠀⠇⢀⡀⠁⢨⠟⡀
⠀⠀⠀⠀⠀⠂⣰⣧⣀⡀⢑⡀⠠⣧⡤⠒⠒⠲⠒⠢⣴⠏⠀⡀⠀⠀⠀⣠⠀⣀⡿⠒⠒⠛⢶⣞⠓⠀⢀⡘⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠙⣯⠙⣇⠉⠁⠺⠏⠀⠀⠀⠀⠀⠀⠈⢣⠈⠉⠐⠀⠁⠉⠸⠋⠀⠀⠀⠀⠀⠛⠓⠫⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠂⠀⠀⠐⠄⠁⠂⠀⠶⠐⠂⠐⢷⣀⠀⠀⠀⢀⣇⠀⢀⠀⠒⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⢌⣀⣸⣆⠀⠀⠀⠀⣠⠆⠉⠉⠛⠀⠈⢙⡟⠉⣀⠤⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣾⣿⣶⣶⣦⣤⣤⣤⣯⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠿⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣪⣥⣘⣚⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠈⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢻⣿⣿⣿⣷⠀⠂⣼⡀⣀⣤⣶⣿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠘⠛⢟⣈⣽⣖⣲⡻⣻⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣷⣶⣶⢀⡿⢿⡿⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠛⣿⠂
⠀⠀⠀⠀⠀⢀⠀⢀⣠⣼⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⠟⠁⠀⠀⠠⢻⡄
⠀⠀⠀⠀⣠⣷⣶⠾⣿⣿⣿⣿⣿⣣⣴⣀⢄⡀⠀⢀⣀⣀⣀⣩⣝⣻⣿⣇⠀⠀⠀⣘⡸⣧
⠀⠀⠀⠀⣿⣿⣇⠀⠀⠉⣿⣿⣿⣿⣿⣟⣭⡶⠟⠋⠉⠉⠀⢀⡈⠉⠙⠻⠆
⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⠟⠁⠀⢀⣠⠶⠶⠦⣤⣬⠀⢀⣠⣤
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠸⠿⢿⡟⠁⠀⠀⠀⠀⣤⣠⣦⠴⡶⠥⢴⣿⢿⠎
⠀⠀⠀⠀⢀⢯⠀⠀⠀⠀⠰⠻⣆⠁⠀⢠⡆⠀⠀⠀⠀⠉⠘⠃⠀⠀⠻⠂
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠐⢀⢹⣦⠠⠂⠡⠄⠀⠀⠀⠀⢠⠀⠀⠀⠘⣤
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠀⣠⠀⠈⣙⣻⣤⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⣀⠀⠀⠀⢷⣾⣤⣬⣭⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣧⣛⢤⠀⠀⠣⠹⡎⢻⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣧⣄⠴⠃⠀⠹⣿⣿⣧⠀⠀⠀⠹⣄⠁⢀⢄⡀
⠀⠀⠀⢠⠰⠶⣾⣶⣶⣾⡷⣄⠀⠀⠀⠀⠈⢻⣿⣷⣤⣤⣤⠤⠗⠒⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⡝⢄⠀⠀⢠⣀⠰⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⠳⣦⣺⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⠿⠿⠿⣿⣦⣙⠛⠟⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠉⢿⣿⣿⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⠁⠀⠀⠀⠘⣯⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣇⢰⡔⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⣗⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣿⠾⠛
⠈⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣷⠉⣷⠀⢠
⠀⠢⡀⠀⠄⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣿⡰⢻⢁⣦
⠀⠀⠐⡞⠀⠀⠀⠑⢦⡀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⢀⣿⢷
⠀⠀⠀⠑⠀⠀⠀⠀⠈⣇⠀⣀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠣⣬⣿⡾⣫⣾
⠣⠀⠀⠀⠈⠀⣇⠀⠀⠀⠀⠀⢀⠀⠀⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠀⠀⢢⠑⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇⣷⣿
⠀⠁⠀⠀⠀⠱⣝⠆⠈⠀⠀⠐⠁⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⢠⠾⠁⣜⠅⠀⠂⠉⠳
⠀⡖⠀⠀⠀⠀⠘⠋⠘⠓⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠁⠀⣸⠐⠀⠀⠀⠀⠦
⠀⡄⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣟⠀⠀⠀⠀⠀⢠
⠀⠃⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⣾
⠀⢽⢸⠀⠀⠠⠀⠀⠀⢀⠄⠀⠘⡆⠀⠰⠀⠃⠆⡄⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠠⠌⠠⡆
⠀⠸⠀⠆⠀⠘⠄⠀⠂⠀⡃⠐⢆⡀⠀⠣⠄⠠⠊⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⠁
⠀⠈⠇⢠⠀⠀⠀⠀⣀⠒⠂⣶⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣏⠀⠀⠀⠀⡏
⢦⡀⠐⠈⡇⠀⠈⠀⠘⠗⠀⠈⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠴⠻⠀⠀⠀⡸⠇
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠠⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠤⠒⠀⠄
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠡⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠂⠀⠀⠊⠀⠀⠁⢀⠀⠀⠈⠆⠀⠂⠀⠀⠀⠁⢀⠂⠀⠀⠀⡔⠁⠀⠑⠄
⢀⠀⠀⠐⡄⠀⠀⠀⠀⠀⢁⠀⠹⠐⠀⠀⠀⠀⠀⠀⠒⠈⢀⠁⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠢⣀⡠⠃⠀⠀⠁⠄⠈⠄⠈⢀⠀⠀⠸⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠑⠀⠑⢄⡀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⢀⠂⠐⡀⠀⠀⠈⠀⠈⠁⠀⢀⠀⠀⠀⠀⠃⠄⠀⣄⠡⡄⠀⠑⠤⠤⠐
⠀⡀⡀⠀⠀⠀⢌⠀⠀⠀⠀⠊⠀⠀⠀⠈⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠅⠀⠀⠀⠀⠈⠀⠠⠀⠐⠀⠁⠀⣀⠀⠠⡀
⠀⢸⡀⠈⠀⠀⠀⠈⠐⠠⡀⠀⠀⠒⠋⠑⠀⠀⢀⣤⣀⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡄⠀⠀⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠀⠀⡌⢃⠀⠀⡀
⠀⠀⠣⡀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠂⠀⠐⠀⠀⢹⣿⣿⣿⣷⣿⡄⠈⠢⠄⠀⠀⠀⠇⠀⢠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⣀⡩⠞⠁
⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⡀⠀⠀⠈⠐⠀⠐
⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⢢⡢⠀⢀⡀⠀⠀⡠⠀⠀⣀⣀⠠⠐⠁⠀⠀⠠⠀⠠⠄⠂⡀⠀⠀⡀⠤⠀⠀⠀⡠⠂⠁⠈⠉⠙⢂⠐⠒
⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡀⠐⢄⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⠀⠐⠁⠀⢊⠠⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠠⠀⠀⠐⠀⠀⠉⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⡀⠀⠀⠀⠀⢀⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡂⡄⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠑⠄⡀⠀⠀⢀⠄⠁
⠀⠀⠀⠈⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠈⢉⡃⠀⠀⢠⠀⢂⠀⠀⠀⢨⠀⠀⠑⠠⠆⡀⠀⠀⠀⠀⠀⡠⠒⠂⠐⠠⡀⠀⠈⠁⠉
⠐⠂⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢘⠁⠀⠀⡅⠃⠀⠸⡄⠀⡄⠀⠀⠀⠁⠐⠃⣀⠠⠀⠀⠀⠄⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠁⠀⠉⠀⠀⠀⢁⠀⠀⠰⡀⠀⠆⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⡀⠀⠐⠌⠐⠡⠄
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠂⠀⢈⣉⢐⠀⠀⠐
⠀⠀⠀⠐⡁⠉⠒⠄⠐⠀⠐⢄⠀⠀⢀⡀⡠⠃⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣀⣴⣆⣀⡀⡀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠉⠉⠁⠢
⠀⠀⠀⡀⠀⠁⢂⡀⠊⢀⠀⢀⡁⠈⠁⠀⠀⢠⠊⠁⠀⠹⠓⠤⡀⠀⣠⣾⣿⣿⡟⣿⣿⣿⣿⡿⣿⢿⣟⠀⠀⠀⢇⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢘⡤⠤⢠⣀⠀⠀⠘⠄⡀⢈⡀⣀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡇⢸⣿⣿⣿⣿⡇⠉⠋⠁⠈⠁⠉⢉⠉⠀⠀⠀⠀⠁⠂⠀⠀⠀⡆
⠀⢀⣭⣦⣴⣿⣿⣿⣦⠀⠐⠀⠀⠀⠀⠀⠀⠱⡠⢄⣀⡀⠔⢋⠀⠀⠙⠿⢿⡿⠁⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠄⠐⠁
⠀⠙⣿⣿⣿⣿⣿⣯⡟⠀⠀⠀⡀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠠⠊⠀⠑⠆
⠀⠀⠀⠙⠛⠛⠛⠛⠁⠀⢀⠊⠀⠀⠳⢄⠀⠀⠉⠀⠀⡀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠀⠀⠀⠐
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠂⠀⢀⠠⠀⠁⠀⠈⠀⠉⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠑⡀⠀⠀⠀⡠⠀⠐⠒⠂⠀⡀
⠀⢀⣴⠃⠀⠉⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠚⠈⠉⠁⠢⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠰
⠀⢸⣿⣆⠀⠀⠀⢠⠆⠸⠀⠀⠀⠀⠀⠘⠀⠀⠸⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠻⣿⣦⣄⣀⡥⠀⠀⠢⡀⠀⡀⠔⡡⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡀⠀⢀⠠⠊
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐