⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡤⠞⠲⣄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠎⠀⠀⣶⠟⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣤⣤⡀⠀⠰⠟⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠳⠁⠀⣸⠇⠀⢾⡷⠘⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⠀⢀⠔⣁⣤⣤⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡸⠋⠉⠉⠙⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣠⣶⡆⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡾⠃⡀⠀⠀⠀⠀⠀⠀⠀
⢀⣴⡿⠿⠿⢷⣤⡾⠛⠛⠛⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⡡⠎⠀⠀⠀⠀⠀⠀⠀⠀
⢾⣿⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣀⡴⠚⣡⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⣠⠔⠛⠁⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠙⠳⣄⡀⠀⠀⠀⡴⠋⢀⣠⠞⠉⠀⠀⠀⣸⡇⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠹⠀⢀⡌⠀⣠⡿⠁⠀⠀⠀⠀⠀⠹⣄⠈⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠎⢀⣾⠏⠀⠀⢀⣤⣤⣤⣄⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⠀⣾⡇⠀⠀⢐⣋⣤⡀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠈⠻⠿⠋⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢁⠈⠻⣿⣶⣤⣄⣤⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣠⠒⣶⣦⠀⢸⠀⡇⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣿⣄⠈⢁⣴⠟⢀⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠙⠛⠛⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣀⣀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢾⣅⣉⣠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢷⣀⢀⣆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢣⡞⣡⡴⠖⠒⢶⣄⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡉⣿⠀⡏⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠁⢻⡄⣷⠀⠀⠀⢀⡀⡿⠆⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⠮⠻⣜⢧⡀⠀⠿⠟⠁⠀⠀⠀⠀⣠⣾⢵⣇⣤⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢯⡳⣄⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠈⢳⠛⠂⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣷⣄⠀⠀⠀⣼⠁⠀⢠⠄⠀⢈⣟⡁⠀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣒⣩⣭⣝⣒⠮⣝⣧⣄⠀⣯⠀⠀⠹⠧⢴⡿⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣎⣼⡋⠉⠈⠉⠙⠛⠶⣽⡻⣷⣿⡀⠀⠀⠀⢈⣁⣀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡗⣂⠀⠀⣀⡆⠀⠀⠀⠀⠈⠉⠙⠿⣆⠀⢠⡏⢁⢈⡆⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠭⠯⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢧⡈⠉⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠶⣄⠀⠀⢿⣆⣤⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⠀⠀⡿⠀⠀⡸⣿⠰⠆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠉⠉⠁⠀⡰⠃⣿⠀⠓⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠀⢰⢻⢀⣏⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠁⠈⢙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣾⢰⡍⠀
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣼⢇⠷⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⡘⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠙⠿⢷⣶⣤⣤⡴⠶⣫⣿⠟⡬⠁⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⣇⠛⠂⠀⠀⠀⠀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⢋⣽⡿⡫⠶⠉⠀⠀⠀⠀⠀⠀
⠀⠃⠤⢀⡀⣀⡀⣠⡄⠠⣶⡆⣀⡤⠖⣫⣾⣿⠻⠌⠋⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠀⠉⠁⠙⠁⣠⡴⠚⣁⣴⣿⠟⣥⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⣠⣾⠿⣫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡴⠋⠀⣠⣾⠟⢡⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣰⠟⠀⣠⡾⠋⠀⠀⡿⡆⠀⢀⡶⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣼⠃⠀⣴⠟⠀⠀⠀⢰⡇⡇⠀⠘⣇⣀⡔⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣸⠃⠀⣼⠃⠀⠀⠀⠀⢸⡇⠘⢦⣀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢀⡏⠀⢰⠃⠀⠀⡰⠀⠀⠸⣷⡀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣤⠚⠉⠙⢦⡀⠀
⢸⠇⠀⢸⠀⢀⠆⡇⠀⠀⠀⢣⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣷⠄⠀⠀⠹⡄
⢸⡆⠀⢸⠀⣼⠁⣷⠀⠀⠀⠈⢧⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠶⠆⠹⣷⣄⠀⠀⠈⠻⣖⢤⣀⡀⠀⠀⢀⣠⣴⡿⠀⠿⠆⣰⠇
⠀⠸⠀⠀⠀⢶⡆⠀⠀⠉⣁⠀⠀⠀⠈⠳⣄⡉⢙⢛⢛⡏⣥⠘⢗⣠⠞⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⣀⡀⠈⢦⠉⠓⠶⠶⠶⠒⠚⠉⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡞⠉⠉⠙⣦⠈⢷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠱⠴⠞⢀⡿⠂⢸⡇⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠴⠞⠁⠀⣼⡇⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢡⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀