Toilet Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡒⠖⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⢤⣤⡤⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠁⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⡔⠒⠶⠂⠒⠶⠂⠒⠳⠓⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢁⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣯⠉⢑
⠀⠀⠎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⡓⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⡎
⠀⠀⠳⡄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠀⠀⠀⠀⠀⠀⠁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⡇
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⢤⠠⠄⠄⢤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣷⣵⣦⣤⣤⣤⣤⣤⣶⣵⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠂⠤⠤⠤⠤⠤⠒⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⣀⣰⣄⣀⣀⣀⣀⣀⣀⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⢠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣬⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠷⠿⠶⠶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡰⠟⠛⠀⠉⠉⠉⠛⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢆⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠿⢷⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡴⠟
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠉⡘⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠠⠁
⠀⠀⠀⢀⢦⡄⢊⣥⣤⣬⠒⠀⠀⠀⠀⠀⠀⠀⠀⢁⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠙⡜⢛⡳⡉⣦⠀⠀⠀⠀⠀⠀⠀⠀⢊⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠳
⠀⠀⠀⠀⢀⢠⣥⠂⠘⣱⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠓⢦⣤⡀⢀⣀⠀⣀⡤⢒⡍⠚
⠀⠀⠀⠐⠂⠀⠙⠻⠿⠚⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡤⠋⠀⠀⠀⠀⠘⠓⠲⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⣴⡾⠿⠟⠛⠛⠛⠻⠿⣦⣄⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢠⣾⢟⣠⣴⠶⠚⠊⠉⠙⠛⠒⢦⣽⡆⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣤⣤⣄⡀⠀⢠⡿⠃⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣷⡶⠞⠋⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡛⠛⠿⠿⠿⠿⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⢿⡛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠈⠁⠀⠀⠀⠀⠀⠀⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣿⣿⣷⠤⠄⠀⠀⠀⠀⠀⠀⠀⣾⠀⡀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣶⠆⠀⠈⢳⡈⠀⠀⠀⠀⠀⠀⠀⠃⢠⡇⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣸⡟⡀⠀⠀⠀⠀⠀⠀⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⡿⣯⣤⣤⣤⡤⠴⠊⠀⠀⠀⠀⢢⡄⠻⠅⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠢⠤⠤⠶⠶⠖⠓

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⡻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⡛⢻⣿⡟⠿⢿⡿⠿⢿

⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠈⣽⠋⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣶⠤⠤⢄⡀⠀⣀⣀⣀⣀⡐⢳⣄⣀⣀⣀⣀⣀⣤⣄⣤⣤⡍⢱
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠈⠙⠉⠉⠀⠀⠀⠈⠛⠛⠛⡃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⣾⡤⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢹⠅⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⡄⢸⠂⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⣀
⠀⠀⠀⠀⢠⡇⢸⡄⠀⠈⠀⠀⠀⠘⠤⠴⠼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠋⢸⠀⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣿
⠀⠀⠀⠀⠘⠇⢸⡁⠀⠠⠀⠀⠀⠀⢀⣤⡀⠀⠀⢀⠠⠄⢸⠀⠀⠀⠀⠀⠀⠀⠸⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⠀⣼⠏⠙⠄⠀⢼⡺⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⢠⢸⠅⠀⢸⠀⠀⠀⡴⠁⠀⠀⣸⠀⠘⢋⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾
⠀⠀⠀⠀⠀⢸⢸⡂⠀⢸⡄⠀⠀⢳⣦⣤⣴⣟⠀⠀⠉⠠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻
⠀⠀⠀⠀⠀⠈⢸⡇⠀⣼⣇⡤⣤⡧⡻⠿⢟⣠⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⢸⠏⠀⣿⠈⠹⢿⠰⣀⣉⣀⡼⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸
⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⢿⣀⣼⠒⠫⣉⣿⡖⠒⠛⠛⠃⢹⣦⣤⣤⣤⣤⣤⣤⣤⢤⡆⢸⣧⡤⠤⠤⠤⠤⣤⣤⣤⣾⡇⢸
⠀⠀⠀⠀⠀⢘⣸⠟⢻⢻⠀⠉⠁⠐⢤⣤⡽⠇⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⢸⡇⠁⠀⠀⠀⠀⠀⠀⠉⠸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠻


⣀⣠⣠⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀
⣿⣷⣺⣯⣝⣽⣩⣙⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣼⠛⠛⠶⠶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠞⠛⢻⡆
⠙⠷⠶⣤⣤⣀⣀⠈⠉⠉⠛⠛⠓⠶⠶⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠛⠛⠋⠉⠉⢀⣀⣠⣤⡴⠶⠞⠁
⠀⠀⠀⠀⠈⠻⣯⠙⠛⠳⠶⠶⠶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠶⠶⠶⠛⠛⠋⠉⣽⠏
⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠷⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠾⠛⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢰⠟⠉⠉⠙⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠛⠋⠉⠉⠻⣆⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣋⣿⣛⣛⣛⣭⠭⠭⠽⠖⠒⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣀⣤⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⢦⣄⡀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⢹⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⠻⣝⠒⠤⠤⣀⣀⣀⣀⣠⣤⣦⣀⣀⣀⣀⡠⠤⠖⢚⡽⠋⠀⠀⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⣷⠘⡇⠀⠀⠀⠀⠀⠀⣀⣼⡤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢲⡶⠶⠶⠶⠶⣶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧⣤⠤⢶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⣟⣺⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡉⠛⠶⣭⣛⠿⠿⠿⠷⣶⠾⠿⠿⠿⠛⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠻⡍⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⣀⣀⣀⣠⠟⠀⠀⠀⠀⠀⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠕⠂⠉⠉⠐⠪⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡟⡟⠈⣻⣻⠃⠙⡟⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣴⡕⠁⠀⠀⠀⠀⠀⠀⠈⢫⣦⠤⠤⡄
⠀⠀⢀⣀⣀⣀⣀⣠⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣹⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠁
⠀⠀⠈⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡞⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠣
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⢀⣀⡀⠀⠀⠀⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣎⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡢⣀⣀⡀⠀⠀⢀⣀⣠⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣉⣉⣉⡉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⠠⠤⠄⠤⠤⠤⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⠓⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⢁⣈⣉⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⣠⠞⠋⠉⠉⠛⢦⡈⢿⣦
⠀⠀⠀⠀⠀⣰⣿⡿⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠹⡄⢻⣆
⠀⠀⠀⠀⢠⣿⡿⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⣿
⠀⠀⠀⠀⣾⣿⠃⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⡇
⠀⠀⠀⢠⣿⡟⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⣿⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣼⣿⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢸⣿⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏
⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⢛⣫⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⢿⣿⣄⣠⣤⣶⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠻⢛⣉⣥⣾⡿⠿⠛⠉⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠈⠉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠛⠋⠁⠀⠀⠀⢠⣾⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢷⣦⣄⠀⠀⠀⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⠘⡄⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣄⠀⠀⠀⠙⠛⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠟⠋⠀⠀⠀⣰⠇⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠙⢮⣟⠿⣿⣶⣦⣄⣀⡄⠀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⣀⣀⠤⠔⣋⣥⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠈⠛⠶⣭⣛⡻⠿⣿⣿⣷⣶⣶⣭⣭⣤⣤⣤⣭⣭⣽⣿⣶⣶⣾⠿⠟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠈⠉⠛⠒⠶⠭⢭⣭⣽⣛⣛⣛⣛⣛⣋⣉⡉⠉⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠻⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠉⠛⠛⠿⠿⠷⢶⣶⣶⣶⡿⠿⠿⠟⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣉⠓⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣷⣶⣦⣬⣍⣉⣉⣙⣛⣛⣉⣀⣠⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠒⠒⠒⠒⠈⢉⠟⠁⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⢸⠒⠒⠒⠀⠤⠤⠤⠄⠒⡎⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠈⡏⠉⠁⠒⠒⡖⢲⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡇⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢣⢈⠀⠸⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢸⡀⠀⢇⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢸⡇⠀⠈⢣⣤⣤⣴⣾⡿⠧⠤⠤⠤⠤⠤⠤⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⡤⠬⣶⣥⠔⠒⠘⣿⡿⣏⠀⠠⣶⠶⠟⠛⠛⠛⠋⠩⠿⠶⠶⢤⣍⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠤⣄⣉⡉⠒⠢⠵⢮⣍⣒⡪⠭⢉⣁⣒⣒⣒⡒⠒⠒⣒⣚⣁⣠⠤⣚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠉⠒⠲⡄⠀⠈⠉⠙⠛⠛⠛⠛⠛⠒⠚⠓⠒⠒⠒⠉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠑⠒⠒⠒⠚⡏⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠬⣉⡑⠒⠢⠤⣄⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠬⣑⡒⠤⣀⣇⣀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠬⢍⣙⣛⠲⠶⠶⠖⠚⣉⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⡈⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⠴⡄⠠⠙⡿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡁⠂⡀⠂⠐⣾⣿⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠾⣠⠒⢰⣿⣿⣦⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢆⡀⠄⠉⠀⠐⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠁⠠⠀⢀⢑⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣀⠠⠐⣿⣏⣽⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡠⢤⣀⣠⣤⣴⣶⠦⠤⢤⣤⣠⣤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠚⠛⠋⠉⣤⠋⢀⠀⠈⢉⣀⠄⠈⡀⠨⢿⣛⠁⠀⣸⠷⠦⠄
⠀⠀⠀⠀⠀⠀⡠⠖⢁⡴⠒⣚⡛⢶⣀⠀⠀⣿⠄⠀⣉⢫⡀⠚⠃⠀⣾⠥⠾⠛⠃⠀⠀⣲⣷⡦⢄
⠀⠀⠀⠀⢀⡶⣠⠖⢋⠀⣠⠟⠁⢀⣹⠿⠒⠶⢴⠞⢁⡜⠻⣆⢠⡟⢁⣤⠶⠖⠻⣟⠛⠃⢀⣑⣶⣷⡀
⠀⠀⠀⠀⡟⣱⠏⢀⠟⢀⡇⢀⠞⣩⠴⣏⠀⠀⢄⠀⣾⠀⠒⢈⠞⠀⢸⡇⢀⡘⠳⢻⡆⢠⡏⢹⠛⢯⡑⡄
⠀⠀⠀⢸⡖⠙⣆⠀⢀⠌⠀⢸⡇⠁⠀⠤⢼⡀⠀⣿⠃⠀⠠⠞⢀⠀⢸⣦⣿⣁⣠⣼⣆⣸⡁⢸⡆⠈⢻⣾⡄
⠀⠀⠀⠀⠁⠈⠲⠗⠤⠤⠤⠀⠙⠳⠶⠄⠈⣧⢀⡋⣀⣦⣄⣠⠾⠷⢾⣿⠀⠉⠉⠻⠟⢉⡵⠋⣠⣴⣾⣿⠳
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠲⠓⣒⠂⢲⡾⠛⠉⠀⠁⠈⠉⠁⢀⠀⠀⡿⠛⠛⣷⣶⡶⢿⣇⠐⣋⢱⣾⣷⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⣠⣤⠶⠋⠉⠙⠒⢤⡿⠶⢼⣷⠶⠙⠻⠛⢀⣰⣿⡟⣫⣼⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢸⠁⠀⠀⢀⣀⠀⢢⠀⠹⣦⡀⠀⢤⣤⡔⠛⢋⡛⢻⣯⠁⠠⣿⠿⠥⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⣿⠈⢄⠀⠱⣀⡈⣧⡤⠾⠻⣙⡟⠋⠀⢶⡈⢦⣤⣀⠼⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣤⡔⣲⣾⣷⣶⣤⣤⣬⣕⣤⣀⣘⣿⢀⣀⡌⢀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣿⣿⣿⣭⣭⣟⣿⣽⣯⣽⣟⠿⠷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⠿⢿⣿⣏⣛⣻⣯⣭⣿⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⠀⠉⠛⠚⠛⠛⠛⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀⠀⠀⠘⣶⡄⢈⠀⠰⠟⠐⠒⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⣿⠙⠀⠐⠀⠀⢸⣷⠘⠀⣄⠸⠂⠀⠀⣄⢀⣽⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣄⢸⣷⣿⣿⠟⡟⠀⠆⠀⢸⣿⡌⠐⢻⠀⠓⠀⠀⣼⣿⣿⠁⢨⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣠⣿⣿⣿⣿⡟⣠⠇⡆⢲⠀⣼⣿⣄⠀⣼⠀⡐⢰⠈⣿⣿⣿⣶⣿⠃⡿⢳⡀
⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠈⢀⠛⣿⡻⡇⠀⠃⢸⠀⣿⣿⡏⠀⢸⠀⢃⡞⠀⣿⣿⣿⣿⣿⣴⣧⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠁⣀⡤⠔⠋⠤⣀⠙⢧⣀⠨⣽⠀⣿⣿⡇⠀⢸⠃⠀⠁⠀⠃⢀⣿⡋⡛⠟⣿⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣦⡈⠀⣿⣤⣿⣿⣿⣤⣼⠀⢸⠀⠀⡠⠞⠉⠀⠙⠋⠛⠛⣿⣧
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠘⣦⡏⢿⢿⣹⠏⣿⡿⣄⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⢠⠀⠘⠂⠠⠎⠉⠒⢠⠊⠈⠁⢐⣿⡟⡋⠨⣝⣿⡯⢿⣵⣮⣿⠀⠸⠀⠀⠈⠀⠀⠀⠆⡀⢸⣿⡄
⠀⠀⠀⠀⠀⡠⠀⡁⠀⠀⡀⠀⠀⢠⠀⠀⣿⣿⠃⠠⣾⠛⠉⡛⠷⢌⠻⡿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⢹⡻⡆
⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠁⠀⠀⠈⠀⢰⣿⡿⠂⠀⢹⠇⠟⠲⡯⠂⠑⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠈⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⢀⠀⠀⣼⣿⣥⢀⣤⡆⠙⣟⠉⠲⣤⣀⣰⣷⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⢒⣷
⠀⠄⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢈⠀⢰⡿⣟⠹⢹⠟⠁⠀⡏⠀⠜⣿⡏⠙⠻⠃⠀⠀⡀⠀⡢⠀⠰⠅⠀⠀⣼⠘⣄
⠀⡁⠀⠀⠀⡀⠀⠀⠀⠀⡠⠔⢁⠴⣣⣴⠟⡠⠊⠀⢠⠀⡇⠀⠀⣌⡳⣠⣷⣤⠆⡂⠀⠈⠁⠀⠀⠀⠀⢀⡟⠀⣸
⠀⠀⠀⠀⠰⡄⠀⠔⠁⠀⠀⣀⣤⠞⠁⠀⢈⡤⠀⠀⢸⡀⣡⠀⣠⣿⣿⠿⡟⢮⠙⠳⣴⡄⠀⠀⠶⠀⣀⣿⡣⣠⡟
⠀⠀⠀⠀⠀⡇⠀⢀⣠⣴⡾⢿⣴⢲⣶⣶⡟⢧⡞⠉⡣⠀⣃⠀⠉⣷⣤⠔⠚⣦⡁⠤⣈⣛⢦⡀⠀⠐⠺⣽⣿⣿⡇
⠀⠀⠀⠸⡖⠁⣠⢿⣿⠛⠀⢈⠺⠀⠈⢈⢻⡂⢄⢫⣹⠀⠂⢀⣋⣿⡡⣀⣼⠟⠇⡀⠉⠉⡟⠙⠶⣦⣾⡿⣿⡟
⠀⠀⠀⠀⠹⣶⣿⣾⡏⠀⠂⠀⠐⢁⡀⢮⣦⣿⣮⠫⠉⠀⡅⠘⣽⢏⣼⣿⠋⣤⠠⢉⡄⡀⡄⠀⣴⢭⣽⣿⣿⠇
⠀⠀⠀⠀⠀⢹⡿⣿⣷⠀⢞⢶⡀⠂⠑⢶⣿⣿⣿⡗⠂⠀⠁⢰⣿⣼⣿⣷⣾⠗⢠⡵⠁⣣⠂⢰⡏⢾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⢿⣿⡅⢼⡦⣴⣦⣄⡀⢙⡻⣿⣿⡇⠀⣤⣿⣿⣿⣿⣯⡥⢶⠛⠀⡴⠃⠀⣾⢿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣟⠾⣿⣿⣿⣿⣿⣶⣾⣿⣦⣭⣾⣿⣿⣿⣷⡶⠯⠶⠛⢁⣰⣾⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣭⣍⣻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣯⣁⢄⣠⣠⣴⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢭⣿⣿⣿⣿⡿⠃⠈⠻⣯⣽⣿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠖⣛⣭⢭⣽⡯⠶⠶⠶⠾⢿⡭⠭⣭⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣉⣥⠾⠗⠛⠁⠀⢢⣾⡠⠀⠀⠀⢀⡼⠇⠀⠀⠐⢤⣈⣳⢬⣑⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡴⠞⢋⡭⠤⠤⠶⠒⠂⢰⡾⠋⠀⠀⢀⡞⠉⠀⠀⢀⣠⠴⠞⠳⠦⢤⣀⠉⠓⢮⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡽⠋⢁⣤⠞⣁⣠⠤⠤⡤⠂⠀⡞⠀⠀⠀⢀⡟⠀⠀⠐⣾⠋⠀⢀⣠⠤⠤⣄⣬⠷⠤⠤⣽⠶⣟⠦⣀
⠀⠀⠀⠀⠀⠀⢀⣼⣵⠋⡴⠚⢉⡴⠊⠁⢀⠀⠀⠀⠀⣸⣁⠀⠀⠀⢸⣇⠀⠀⠀⠈⣷⡞⠉⠀⠀⠀⠘⠄⠀⠀⠀⣿⠀⠈⠳⣌⢣⡀
⠀⠀⠀⠀⠀⢀⣼⠃⣺⣼⣁⡞⠁⠀⠀⠐⠘⢦⠀⠀⢀⡏⠀⠀⠀⣠⠞⠉⠑⠀⠀⠀⡼⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠘⣦⢷⣄
⠀⠀⠀⠀⢀⡞⢹⠞⠉⠀⠈⠑⠤⢤⡜⠀⠀⢀⡷⢄⣼⠁⠀⢀⣼⠃⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⡤⠤⠤⡀⠀⠀⠀⢠⡇⠀⢀⡿⠳⡌⢣⡀
⠀⠀⠀⠀⣸⠁⣼⠀⣀⠦⠀⠀⠀⠘⠀⣠⠊⠁⠀⠀⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⢻⠀⠀⠱⡄⠀⢸⠀⠀⠒⠻⡀⠀⠀⠉⢀⡠⠋⠀⠀⢹⣦⣷
⠀⠀⠀⡰⣹⠛⠉⣹⠇⠀⠀⠀⠀⠀⢀⡇⠀⠀⡄⠀⡟⠀⠀⠀⣧⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⢀⡇⠀⠀⠀⠀⠁⠀⠐⢲⡏⠀⠀⢀⣀⡞⠈⢿⢱
⠀⠀⢰⡇⡇⣀⣮⠀⠁⠀⠀⡖⠒⠒⠚⠇⠀⠀⢁⠀⢃⠀⠀⡀⠉⣀⣠⡴⠾⠛⠋⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⠀⢀⠘⣦⡇
⠀⠀⠀⣷⠋⣡⠞⠢⠄⣀⡐⠃⠀⠀⠀⠀⠀⠀⠘⣆⠈⢢⣀⡿⠟⠋⠁⡤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡠⠔⠇⠠⠀⣯⠀⠀⠀⣸⡀⡸⠐⣸⡇
⠀⠀⠀⢇⢰⡇⠀⠀⢀⡔⠓⢄⡀⠀⠀⠒⠚⠃⠀⣨⡿⠛⠉⠀⠀⢀⣴⠆⠠⣤⡤⠤⠤⠴⠥⠤⠴⠐⠀⠁⠀⠀⠀⠀⢠⡏⠀⣀⣀⠀⠙⢇⠐⢠⡇
⠀⠀⠀⢸⡇⠙⠀⠀⠀⢣⠀⠀⠀⠉⠑⣤⣤⣤⠞⠁⠀⠀⢀⡠⠖⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⠋⠀⠈⠉⠀⠁⠀⡿⠀⡜
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡼⠁⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⣀⣀⡀⠀⠀⣸⠶⠶⠶⠶⠤⠤⠴⠞⣡⠞⠁
⠀⠀⠀⠀⠹⡄⠀⠘⠦⢄⣀⢀⣀⣠⠴⣾⠁⠀⣴⠁⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⣠⠤⠖⠚⠉⠀⠈⠓⠀⠀⠀⠁⠀⢀⣀⣴⣦⣤⡴⠚⠁
⠀⠀⠀⠀⠀⠙⠶⠤⢤⣤⣬⣡⣥⠤⠖⣿⠀⠀⠸⣄⠀⠀⢀⠀⠀⠀⡟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣤⣀⡀⠉⠉⠉⠑⠲⠖⠓⠀⠀⠀⢂⣤⣤⣤⣶⣶⣿⣟⣫⣽⣻⢭⣤⣶⣶⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⣠⡤⠴⠶⣾⠻⣿⣿⣿⡿⢽⣾⣵⣟⣯⣟⣻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⠛⣿⣿⣏⣩⣭⣽⣿⣷⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⠻⣿⠛⠻⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢠⡞⠋⠙⠲⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠛⢉⠉⠉⠻⣿⣄⠀⠀⠀⠈⠉⠉⢳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⣿⣠⡶⠛⠁⠐⠻⣿⡟⠀⠀⠀⢸⡀⠀⠶⢿⡀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠃⠀⣠⡼⠿⠀⠀⡄⠀⠀⣿⡇⢀⡀⠀⠈⢧⡀⠀⠀⢻⡤⠴⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣼⢋⠀⠀⠀⣴⠋⠀⠀⠀⣠⡇⠀⠈⢿⡿⠋⠀⠀⠀⠀⣿⠂⠘⢿⡀⠀⢠⡝⠷⣄
⠀⠀⠀⠀⠀⢀⣼⢿⡇⠀⠀⠀⣯⠀⠀⠀⠈⡿⠀⠀⠀⠈⡡⠔⠀⠀⠀⠀⣿⠀⠀⢸⣷⣄⡀⠹⣆⠹⣇
⠀⠀⠀⠀⢠⡎⠀⣼⠿⠿⠋⠀⣿⣦⠆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⠀⠀⣿⡄⠀⠀⠀⠉⢻⣶⢿⡀⢻⡄
⠀⠀⠀⠀⡼⠀⢸⠃⠀⠀⢠⣾⠟⠁⢀⠴⠲⠶⣏⠀⠀⣿⡀⠀⢰⣇⡀⠀⠘⣷⡤⠶⠂⠀⠀⢻⡎⠻⣤⣿⡀
⠀⠀⠀⣠⠿⠿⢿⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⠛⠁⠀⠀⠀⣿⠀⠀⠀⠀⠲⠾⠷⠤⡈⠻⣆
⠀⠀⢸⡇⠀⠀⠈⠢⠀⠀⠸⣧⣀⠄⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⣼⣃⠀⠀⠀⠀⠀⠀⢤⣾⠁⠀⠀⢀⠴⠖⠶⠶⢾⣿⡆⢀⡤⠄⠀⠀⠀⠀⠈⣷⠀⠀⣠⠶⠛⠛⠻⣿⠈⣿
⠀⣾⢻⡿⠋⠑⠂⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠀⠀⠀⠀⠀⠀⢠⡏⠀⠾⠃⠀⠀⠀⣀⣀⣠⣿⡀
⠀⣿⡈⠁⠀⣀⣤⣶⡿⠟⠷⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⣀⡀⠀⣾⠁⠀⠀⠀⣠⡶⠟⠋⠉⠉⠀⣿
⢀⡼⠻⠿⠟⠋⢹⠃⠀⠀⠀⠈⠛⠦⠤⠤⠴⠶⠶⣤⣾⣿⡇⠀⠀⠘⠋⠉⠀⠀⠙⣷⠶⠶⠟⠉⠀⠀⠀⠀⠀⣠⣿
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠀⠀⢀⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⢸
⣿⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡶⠋⠀⠀⠀⣀⣴⠿⠟⠛⠒⠲⠒⠋⠀⠀⠀⠀⣼
⢸⡟⠓⠚⠛⠉⠉⠉⠛⢻⣷⣶⣄⡀⠀⠀⠀⠀⠀⠐⢤⣠⣿⡟⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉
⢸⣇⠀⠀⠀⣦⠀⠀⠀⠉⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣸⠃⠀⠀⠀⣀⡴⠂⠀⠀⠀⢀⣠⣿
⢰⣿⠶⠶⢤⣿⡄⠀⠀⠀⢀⡀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠁⠀⠀⣠⠞⠋⠛⠓⠒⠲⡶⠛⠋⠀⠀⢠
⢸⣟⢧⣄⡀⠈⠻⢦⣀⣀⣸⠇⠀⠀⠈⠻⢦⣤⣄⠀⠀⠀⢸⣄⠀⠀⠀⢀⡴⠞⠁⠀⠀⢶⡀⠀⠀⠁⠀⠀⠀⣠⡟
⠀⣻⣆⠉⠉⠛⠒⢰⡟⠉⠹⣆⠀⠀⠀⠀⠀⠈⠹⣦⣄⡀⣸⡿⠿⠶⠞⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣾⢿
⠀⢹⣿⠷⣾⡆⠀⠈⠁⠀⠀⠘⢷⣄⣀⣀⣀⠀⠀⠀⠙⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣤⣀⣠⣴⢾⡏⢁⡟
⠀⠀⠻⣆⠈⠷⣤⡀⠀⠀⣄⠀⠀⠈⠉⠉⠳⠄⠀⠀⠀⠀⣯⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⣄⠀⠈⠁⣠⡞⢀⡾⠁
⠀⠀⠀⠈⢳⣦⣀⠉⠛⠚⠻⣦⡀⠀⢷⣄⡀⠀⠀⠀⣀⣴⢿⡶⠀⢐⣈⠁⠀⠀⣀⠀⠀⣽⠶⠶⠟⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠙⣿⡛⠻⠶⡄⠈⠛⠛⠋⠉⠛⠛⠛⠛⣉⣤⣿⡅⠀⠉⠙⠛⠛⠛⠁⢀⣼⣏⣀⣤⡴⣾⡿
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⢶⣄⣀⣀⣀⣀⣤⠶⠟⠋⢉⣼⠟⠳⠦⠤⣤⣤⡴⢞⣿⠟⢉⣩⣤⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣶⣬⣭⣿⣿⣿⣇⠀⠐⠈⠉⠻⡆⠀⠀⠀⢀⠤⠴⠋⣡⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣭⣙⣛⠒⣋⣠⠞⠻⠶⢦⠶⠶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠑⠒⠂⠀⠐⡂⠀⠀⣻⠁⠀⠀⠀⠀⢀⠀⣀⠀⠀⠠⢤
⠀⠀⠀⡤⣀⠀⠀⠆⡆⠀⠠⣬⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⠀⠉⣉⣁⠀⠀⠤⠤⠤⠄⠒⠺⠆
⠀⢸⣏⡋⣩⣾⠇⢸⡇⢡⠄⢸⠀⠧⠰⢃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠙⠇⠛⠋⠀⢸⡇⡸⠀⢸⠀⠀⠀⢾⠀⠀⠀⠀⠐⠀⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣽⡀⠀⠀⢠⠞⣫⠾⣾⣧⠀⠀⡄⡇⠀⠅
⠀⠀⠀⠘⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⣾⠀⠀⠀⠐⢦⠄⠀⠐⠀⣤⣤⣼⣷⣴⣦⣶⣾⣿⣿⣷⣶⣷⠾⣄⠛⠁⣿⠟⠀⠀⢀⡘⠀⡇
⠀⠀⠀⢠⠀⠀⠀⢸⡗⠃⠀⢸⠀⠀⠀⣿⡀⠀⣀⣄⡀⠀⠀⠛⠰⠿⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡟⣳⠧⣉⣴⣿⣆⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⠀⠀⠀⢸⡇⠁⠀⢸⠀⠀⠀⣿⠰⠈⠉⠉⡳⢦⡦⠀⠀⡀⠀⠀⠖⠀⠒⣰⢠⣿⣿⣿⣿⡷⠿⣊⣿⣿⣿⣿⠇⠀⠘⡇⠀⡄
⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⢸⣀⢠⣤⣽⡀⠀⠀⢰⠃⠀⠀⠀⠈⠃⠀⠋⠀⠲⠀⣾⣰⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡄⠄⢰⠃⠀⡅
⠀⠀⣀⣀⣀⣀⣀⣸⣇⣶⠿⣔⣻⣿⢿⣿⡇⠀⠠⠬⠠⠀⠀⠀⠁⠀⣶⡖⠠⣄⠘⠟⠿⣿⢫⣾⣿⣿⣿⣿⣿⣿⣿⠙⠁⠀⠼⠀⠀⣧
⠀⠀⠀⠈⠉⠉⠙⠛⣢⠞⠋⣹⢳⣶⡿⣿⣇⠠⢶⠐⠐⠀⣆⣿⣾⣿⣿⠟⠂⠀⠀⠠⠀⡁⢨⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⡸⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠆⡟⣾⢈⣇⣿⡟⢰⡾⠓⠤⡀⠈⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⡝⠋⣽⣿⣿⣿⣿⠏⠂⠂⠘⣃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠃⢿⣄⠙⣿⣷⣻⣇⣼⣴⣴⠰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠈⠙⣟⣩⣤⣀⡲⠆⢠⣿⠈⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⢸⡋⢠⣿⣗⡏⣯⢿⣽⣿⣇⣿⣿⠁⠀⡆⠀⠀⠀⠀⠀⠀⠐⠀⠘⡆⣨⣿⡟⠀⠀⠀⣐⣀⡈⠂⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⢸⢡⢾⡿⣧⠁⠰⣾⡿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠅⢸⣿⣿⠉⠉⠉⢭⠉⣭⠉⠉⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣜⠴⠂⢸⣊⣤⠟⣿⠀⢤⡴⣳⣿⡻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⡶⡿⢿⣶⣤⣀⠀⠀⠀⢨⡦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⢺⠁⣻⣿⣿⣶⣾⣀⡤⣰⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠄⡄⠀⠀⠙⣿⣿⡋⣭⠢⠀⣷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠁⢀⣰⣿⣿⠯⠟⠛⠋⢩⠞⠉⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠏⣠⢾⣿⣦⡁⠚⠃⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢶⠟⠻⠽⠖⠀⠀⢀⠴⡫⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣾⣽⡿⣿⣿⠇⠀⠀⡿⢨
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠠⣄⡄⣰⣶⣄⣀⣀⠀⠐⠊⠁⠀⢀⣴⣿⣻⢿⡏⢰⣿⣿⡀⣷⣾⣿⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣼⣿⣸⣿⠟⣿⣷⣶⣶⣶⢶⣿⣿⣿⣿⣏⣠⣿⣿⣿⣿⣿⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⢼⡓⢂⣿⣿⣛⣹⣿⣼⢹⣿⣿⠏⣡⣿⣿⣷⣝⠿⢿⣯⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⣺⠞⢃⡯⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣮⣾⠿⠉⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣇⠴⠞⠋⠉⠉⠉⡯⣷⣿⣿⣿⣿⠿⠛⢋⠃⡀⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⡀⠀⠀⠀⠀⡠⠔⠂⠒⠈⠉⠀⠀⣀⢀⡀⣀⢀⣀⢀⡀
⠂⠀⠀⠀⠀⣀⠄⠄⠂⢀⣀⣨⠤⠥⡄⣐⣂⠂⢐⡐⠀⠄⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠠⡎⠀⠀⢠⠚⠉⢠⠔⣪⣥⣤⣤⠄⠀⠀⠀⠀⢢⠀⠈⠀⢀⣠⠲⠃⢈⠀⠨
⠀⠀⢠⡄⠘⠒⠀⠈⠀⠀⠈⠈⠁⠋⠲⠲⠌⠂⠀⠀⠐⠁⢐⠀⠈⠁⠀⢀⠔⠀⡸⠃
⠀⠀⠀⠉⠂⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⠂⠂⠁⢠⡢⠀⠁
⠀⠀⢀⠄⡌⠀⠈⠁⠉⠉⠉⠁⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⢀⠀⠠⠐⢁⡜
⠀⠀⠀⠂⠡⡀⠈⠠⠄⣀⣀⢀⡀⢀⣀⡀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠄⣈⠃
⠀⠀⠀⠀⠀⠈⠂⠕⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⣀⠤⠀⠀⣀⡎⡁⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣄⠀⠀⠀⠀⠀⠀⢀⣀⡠⠦⢴⠞⢋⣴⡁⠀⠀⡀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠑⢶⣤⡄⡠⠤⠤⠄⢒⡁⠀⡠⢆⢅⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣶⣚⣛⣓⠊⠴⢋⠆⡅⠀⠀⠀⡄⠂⣀⣤⠀⠀⠦⠀⠀⠖⠋⠐⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠁⠈⠹⣷⠶⠖⣷⠷⠊⡁⠀⠀⠀⠁⠀⢀⠘⠃⢀⡀⢷⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢰⣄⠀⠀⠉⠀⠐⠟⠒⢂⡺⢽⣻⡁⢀⠀⣀⠁⠀⠐⡃⠀⠠⡠⢀⠀⠂⠀⠀⠁⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠉⠀⠀⠂⢘⣾⣷⣶⡦⠝⣫⠁⠋⠐⢃⠤⢰⠁⢀⡄⢠⣦⣦⠈⠂⣈⠁⡀⠤⠀⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠀⠄⠀⡀⡀⢠⣾⣴⣶⠮⢿⣴⠃⠀⠀⡠⠁⠄⠌⠀⠈⠠⠄⠋⠉⡠⠔⠄⠀⢳⠠⠄⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠠⠀⠃⠀⠀⢳⡭⠤⠄⢰⣺⡵⠁⠁⠀⡁⠠⠁⠈⠴⠐⠁⣰⠃⠀⠁⠒⠤⠀⠀⣇⠠⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠒⢠⡀⠆⣭⠸⣿⡿⣛⣭⡿⣃⡶⠌⢀⠀⠈⠀⢁⠂⠀⠉⠉⠲⢄⠀⣀⣀⣈⠀⢀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣄⣰⣛⣿⠟⠀⠗⠉⢶⣿⣋⠙⠀⠚⠃⠀⠀⢠⠀⠰⣤⠢⠴⠦⠔⠊⠁⠀⢀⠀⠔⣸⠃⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠺⡷⠢⣤⣀⠀⠈⣩⡥⠤⠌⡀⠀⠠⡤⠤⠦⡤⠐⣬⣄⠀⢀⢄⣀⡀⣤⠠⢞⣃⠬⠒⠓⡡⠶⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣠⡴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡇⠀⡠⢤⠀⢺⢄⡀⢠⠖⠂⠀⡔⢰⠈⠧⣄⠀⣀⢀⠈⠆⠀⡴⢤⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠒⠚⢄⢀⠀⡸⠀⡄⢸⠀⡌⢠⡇⣼⠉⢁⠎⢀⠃⣼⠀⠀⠁⠀⡘⠁⡐⢀⠎⢀⡄⣠⠋⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠶⠐⢀⡘⠘⠐⠓⠊⠀⠈⠉⠉⠁⠀⠀⠉⠁⠀⠀⠉⠀⠑⠒⠊⠉⠀⠀⠓⠛⠄⠚⠶⣁⠼⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡷⠶⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣇⣀⡀⣴⡆
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠁⠀⠀⠀⠈⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡖⠤⢀⣤⣶⣶⣾⣿⠷⠶⢾⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⣠⢾⣿⣋⡁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡀⠀⠀⠀⠀⠀⠀⡿⠸⠇⠘⣸⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⣿⢸⡟⠛⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⣇⢀⠀⠀⠁⢞⣿⣿⠟
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠙⠎⢧⠀⠀⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠻⣿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠜⡄⠀⢻⣿⣿⣿⠿⣿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠀⠈⢿⣿⣷⠀⠀⠉⠻⣷⣄⠀⣼⣿⠃⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠐⡀⠀⠘⢿⣿⣆⠀⠀⠀⣾⣻⣿⣿⠋⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠈⢆⠀⠈⠻⣿⣿⣶⣶⣿⣿⡿⠻⣦⡀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠙⠆⠐⢻⣿⣿⣿⠟⠋⠀⠀⠘⢷⡄⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⣀⠀⠹⢿⣧⡀⠀⢀⣀⣤⣾⣿⣿⠟⠻⣿⣄⠀⠀⣠⢖⡈⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⣿⠿⠿⠛⠛⠋⠉⠙⢶⣤⣼⣿⣷⣄⢁⣿⣿⠟⣱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣀⣀⣹⣿⣿⣿⣿⣿⡿⣷⣿⠟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⢿⣿⡿⠿⣿⡜⡇⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠒⢄⠀⡰⠁⢀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⠻⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠜⠰⠣⠃⢆⡜⠜⢱⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⢟⣵⡿⢏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⡔⠁⣄⠀⡠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⣄⠀⠀⠼⣟⣵⣿⠏⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠞⠐⢖⡁⡜⢀⡀⠀⣀⢠⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣟⣿⣿⠏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠋⠀⠀⠰⠓⠁⠎⠸⠱⠁⠏⠸⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡆⢤⡀⠀⠠⠤⢠⠤⡤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣻⠿⠘⢯⣭⣼⣿⣾⣓⣛⣚⡃⠀⠠⠀⢽⡦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠓⠀⠉⠇⡱⢊⡁⢠⣿⣿⠋⣥⠀⠉⡈⣄⠠⢞⡁⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⢀⡴⢿⡦⠙⠃⠄⣥⢸⣿⣿⠺⡇⠀⠠⠇⣼⠠⠘⠃⠀⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠈⠃⢠⡼⢂⠀⢐⡟⢸⣿⣷⠺⠿⠓⠛⠂⢩⠁⠇⢦⠘⢀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠄⡄⠀⣃⢘⠀⣨⢀⡌⠁⠀⢿⣿⣿⣅⡀⠀⢲⠞⠃⡆⢸⣄⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣥⣤⣄⡀⠵⣆⠀⠁⠻⣀⠘⣃⣼⣿⠁⣤⡀⠘⠏⢀⡀⠹⢩⠈⠢⡄⢠⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⢀⡀⠀⠣⢪⣣⡌⠑⠠⣄⠈⠉⢹⣿⣷⡧⡞⠛⠲⣷⡏⠀⠡⠜⡠⢦⡌⢈⣭⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡻⢂⣯⢅⡤⠄⠘⡇⠀⢳⠈⠙⣷⠟⢽⡿⢻⠃⢠⠀⢹⣆⠠⣞⣀⡀⠘⠓⠀⠀⠏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣱⣼⡀⠉⠁⠘⡃⣷⣀⠈⠟⢋⠁⠠⣾⣿⢨⡟⠁⠀⢸⢋⡞⠁⣿⠀⣀⠀⡀⠀⡄⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡛⠇⢁⠀⢾⠀⢰⠏⠘⡎⠁⢠⣞⣂⢤⣿⣯⡉⢉⡷⠆⠠⠊⠑⠀⠱⠦⠇⠀⠀⠈⠁⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠿⡧⠀⠸⠂⣸⡏⠘⢟⠀⠠⠄⠤⠆⣿⣿⣿⣯⠤⣴⠆⠀⠤⣛⣖⠲⠀⢦⠀⣀⣉⣠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢦⡀⠡⢀⠀⠞⢻⡆⠀⢸⡇⠀⣀⠈⠳⣾⣏⠁⠀⠀⠐⠂⠒⠀⠈⠝⠀⠀⢈⣰⠟⢉⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢀⠟⠳⡚⢧⠀⠘⠓⠒⠀⢿⢖⠀⠀⣷⣾⣿⣶⣟⠹⡏⢰⣄⣀⡀⢄⠰⣄⣁⣀⡀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠗⠈⢢⣉⣀⣲⢦⡤⢠⣾⡅⠀⢨⣿⣿⡯⡭⣢⠆⣻⡶⣟⠀⠘⣀⠟⠃⡄⠈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢶⡦⢁⣀⠁⢹⣄⠙⠻⣵⡦⢼⣿⣿⣿⡐⠓⠈⠁⠉⣼⡛⡿⠇⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠣⠈⠁⢀⠊⠉⠠⠾⠀⠠⣶⣾⣧⠟⣽⣿⣻⠟⠒⠀⠀⠜⡿⠓⠀⡀⠀⠤⡸⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠠⢄⣼⢷⡘⠚⣡⣄⠠⣄⢀⢀⣈⣿⣿⣷⡿⢶⡀⠀⠉⣁⡄⠀⣵⣀⠀⢹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⢧⣘⣈⡝⡾⣰⣷⣮⠉⠙⠻⣿⣿⣦⡤⠤⠀⠀⡰⢿⣤⡌⠁⠈⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⢿⠙⠦⢞⠋⡃⠁⡶⣶⣿⣿⠯⡤⠀⡀⠀⣢⠔⡌⢣⣤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡘⣶⡞⠃⠈⢳⣽⣿⣿⣶⣦⡄⡠⠋⢥⡾⢇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣦⠴⠋⣸⠏⠈⠿⠾⣖⣀⣖⣈⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠹⣵⠋⠳⢴⣖⢦⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠬⠦⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠁⣟⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣠⢤⠀⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⢆⣀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠳⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠤⡄⠀⠀⠀⠝⠃
⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠀⠀⡇⠉⡵⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡠⠼⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⡋⠁⣀⢀⢀⠀⠀⢠⠇⠀⡇⠀⢀⣀⡠⠔⠐⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⢧⡀⠀⠠
⠀⠐⠀⠀⠀⠀⠉⠛⢹⣸⣠⠃⢀⠞⠀⠀⠉⠉⣉⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⡞⡳⠋⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⢠⠒⢢⣀⡄⢠⠤⠼⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡙⠁⠀⠀⢰⢇⡠⣢⠖⢠⠤⠤⣄⠀⠀⣇⠀⠀⠈⠁⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⣣⠞⣵⡵⢦⡀⣳⠀⠈⠓⠂⠀⠀⠤⠦⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡡⠚⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣰⣁⣀⣀⣀⣀⣱⣄⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⡀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣋⡹⡅⠋⠉⠙⢱⣴⡏⠋⠉⢻⡶⢰⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣍⣹⡄⢱⡔⠺⠘⣊⣳⣿⣯⡽⠦⣴⢾⣷⣞⠡⣾⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠱⠀⢺⣾⠇⢽⢧⠐⢩⠗⠶⣿⣿⠁⣤⣤⣹⡆⠿⢃⡟⠀⡏⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣾⠇⠦⢾⣼⠀⢾⣾⣤⡤⠶⠾⣿⡿⣆⣾⠛⠨⢹⢰⣏⠀⢠⣿⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⠇⢤⣄⣠⣤⡴⢋⣁⣴⡤⠀⢀⣿⣿⣾⠇⠰⢦⣿⡄⢹⣖⠀⠻⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢸⠀⣠⠞⠻⣆⣠⠴⠋⡉⠉⠙⠻⣿⣿⣿⡀⣶⠀⠀⢳⡾⠛⣧⠀⠩⢿⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣍⠓⠿⢤⣞⣙⠁⣤⡶⠧⣴⡆⠀⢸⣿⣬⣇⠹⡄⡰⢼⢧⡈⢾⣆⣠⣼⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠰⡏⠈⢼⣧⣬⣯⡍⠹⠶⣤⡛⠹⠅⠏⠰⣿⣿⣿⣯⡷⣌⠳⠀⣤⠄⠀⣹⠋⠀⣽⡏⢢
⠀⠀⠀⠀⠀⠀⢠⡶⢿⡐⠋⠴⠶⠶⠦⢴⣷⡜⢧⣀⠴⠾⢦⡜⣿⡻⢇⣀⣾⠶⢲⡷⠒⠀⠙⢀⠸⣵⠇⢸⠇
⠀⠀⠀⠀⠀⠀⢸⣷⣘⣟⢋⡉⢳⣼⢠⡶⢛⣛⣶⡠⠬⠤⣟⢷⣿⠀⠀⣏⢻⣦⣴⡇⢀⣤⠔⣛⡣⣌⣀⡇⣴
⠀⠀⠀⠀⠀⠀⠈⠿⡞⠙⡿⢿⠁⣇⠘⠁⢼⠙⠂⣷⠠⣤⣴⣾⣿⡀⡀⢻⡟⠉⠣⣝⠻⣎⡞⠡⡄⠈⣯⣼⢁
⠀⠀⠀⠀⠀⠀⢸⡦⣤⣿⣷⢀⡔⠈⣻⠿⠿⢦⣷⠟⡋⠍⢲⣿⣿⣻⠟⣳⣄⠓⣤⣤⣼⣁⠀⠘⠁⢰⠇⣽⣻⡀
⠀⠀⠀⠀⠀⠀⢻⠀⢘⣿⡎⣟⢦⡀⠁⠠⣆⠈⢍⣿⢦⡴⡚⢻⣿⠇⣬⡀⣾⢧⣻⣦⠟⢁⣴⣫⣴⣿⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠸⡗⠀⡁⢹⡨⣤⡙⠾⠁⢺⡿⣛⣿⣌⣇⠀⣼⣿⠀⠹⢄⡼⠛⢧⠻⣠⣵⡶⣿⣿⡏⢸⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣤⠙⢦⣥⣿⣽⣷⢚⣹⢾⠫⢆⠶⠁⡼⢹⣿⠸⣵⣟⠈⠀⢨⢧⠟⠲⣼⣹⣿⠃⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣠⣤⡈⠋⡿⠶⠦⠜⢧⣌⣖⢛⣵⣿⣿⠠⠿⣿⣳⡾⠛⠛⣤⣼⡿⠋⢹⡃⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢙⡞⢿⡦⠈⠙⢦⡬⠉⠉⢹⣿⣿⣷⠞⡉⢁⡠⣦⣠⣀⠛⡡⣺⠾⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠋⠹⢾⡛⣷⠶⠾⣄⣠⡉⢻⢻⣿⢛⣩⠶⠿⠒⠚⠿⡌⢦⣘⠑⡿⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠴⠷⠧⠐⠢⡤⢁⡘⠸⣿⣿⣿⡿⣶⡶⠖⠀⠀⠓⠋⠙⣻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠙⠀⣔⣉⢻⣶⣶⣷⣬⡯⠉⢤⡀⠠⠭⡲⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣥⣤⣬⣡⡼⠟⠁⠸⣏⣘⣛⣀⣥⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⢀⠴⠂⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠀⠐⢄
⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠒⠒
⠀⠀⠀⠀⠉⣉⣽⣿⠿⡏⠁⠀⢠⠊⠉⢲⣀⡾⠋⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣽⣿⡿⠏⠁
⠀⠀⠀⠀⠀⡿⠋⣁⣤⡇⠀⠀⢸⠀⠀⣰⠋⢧⡀⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢁⣠⡆
⠀⠀⠀⠀⠀⣶⣾⠿⠛⡇⠀⠀⠀⠠⡀⠂⠀⠀⠁⡐⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠟⣀⣤⣶⡇⠀⠀⠀⠀⠈⢔⢄⠀⠀⠂⠀⠀⠀⠠⡈⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⢀⠇⠀⠆⠀⠐⠀⢠⡤⠴⡆⠀⠀⠀⡟⢁⣤⣶⡇
⠀⠀⠀⠀⠀⣿⠿⠛⢁⡇⠀⠀⠀⠀⠀⠈⡀⢱⠆⠀⠀⠀⠀⠀⠙⢀⣀⣀⡐⠁⠀⠀⠀⠀⠀⡜⠀⡘⠀⠀⠀⠀⢐⠒⠿⠅⠀⠀⠀⣿⡿⠛⢉⣀
⠀⠀⠀⠀⠀⣃⣴⣾⡿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠹⠀⠀⠀⠀⠀⢰⠃⢀⠃⠀⠀⠀⠀⢒⣛⣭⡅⠀⠀⠀⣏⣤⣶⡿⠇
⠀⠀⠀⠀⠀⡿⠋⣁⣤⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢇⠀⠄⠀⠀⠀⠀⠀⠠⣬⣬⡇⠀⠀⠀⡿⠛⢉⣠⣦
⠀⠀⠀⠀⠀⣶⣾⠿⠛⡇⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠜⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠩⠥⢖⡂⠀⠀⠀⣷⣾⡿⠟⠁
⠀⠀⠀⠀⡀⠋⣁⣤⣾⢇⡀⠀⠀⠀⠀⠀⠈⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⠀⠀⠀⠀⠀⠀⠀⣘⢒⣲⠇⠀⠀⣀⠛⢁⣤⣶⠗⡀
⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠠⠠⢀⣀⣀⣀⣀⠉⠉⣉⣙⣉⣁⣈⠭⠂⠀⠀⠀⠀⠀⠂⠉⠭⠅⠀⠀⠠⠠⠤⠤⠤⠤⠂
⠀⠀⠀⠀⠈⠢⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣀⠀⢠⠀⠀⢠⠀⣀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣦⡯⠐⠁⠀⠀⠀⠒⢸⣧⡆⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⡟⣤⠀⠀⠀⠀⠀⠀⢠⠙⣅⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠓⠷⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⣀⡱⠃⠀⡀⠀⠀⠀⠀⠀⠀⣀⢘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⡀⠀⠀⠀⠀⠀⠀⡰⠁⢠⣥⣤⡦⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠨⠗⠂⠀⠀⠀⠘⠬⣿⡟⠦⡀⠀⠀⠀⠴⢪⣿⡧⠊⠀⠀⠀⠀⠀⠀⢀⡧⠄⢺⡇⠀⣹⠢⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⣦⢀⠀⠀⣌⣠⡚⣉⠀⠀⠀⠀⠀⢀⣤⠊⠍⢹⣆⡌⠛⠛⠁⠀⣶⠋⠁⢦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡆⠀⡿⠏⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠓⠒⠒⠒⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣁⣁⣀⡀⠠⠄⠐⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⠖⠒⢉⠙⠛⠉⠥⢤⣤⣤⣤⣤⣤⡤⠤⣤⣤⣤⣀⣉⣙⣛⡛⠛⠛⠓⠲⠦⠤⢄⣀
⠀⠀⢀⠤⠒⠀⡀⠄⠂⠠⠐⠊⡉⢍⣐⣀⣠⡤⠤⠶⠖⢂⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣉⣉⣙⣓⠒⠢⠤⢄⣀⠀⠉⠑
⢀⠔⠁⣠⠖⠁⠀⢈⣀⣤⣶⠿⢛⣋⣭⡤⠶⠶⠞⠛⠛⣉⣉⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠙⠓⠒⠈⠉⠓⠶⢄
⡌⠀⠌⠁⢀⣤⢞⡿⣹⣥⡶⠟⢋⣡⣴⣶⠶⢖⣛⣫⠭⠉⠉⠀⠀⠀⠀⠀⠙⠛⠓⠒⠒⠒⠲⠶⠤⠤⣤⣀⣀
⠀⠀⢀⡶⢋⣸⠿⢋⠕⣫⡴⠞⠛⠉⢁⣀⠤⠔⣒⠒⠒⠒⠒⠒⠐⠀⠉⣉⣉⣉⣉⡉⠉⠉⠋⠐⠒⠀⠀⠀⠉⠙⠓⠦⣄
⠀⢀⣟⣴⠟⢁⢔⡵⢟⡩⡲⠼⢒⠩⠁⣀⣈⣤⣤⣤⣤⠤⠶⠶⠶⠶⠦⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⠤⢄⡀⠀⠀⠀⠀⠈
⠀⠈⡟⠁⣰⣻⢏⠔⠁⢀⣤⣶⡷⠞⢋⡉⠭⠴⠒⠒⠀⠀⠀⣉⣁⡀⠀⠀⠀⠀⠀⣤⣤⣄⣀⡀⠀⠀⠀⠉⠲⡄
⠀⠈⠃⠀⢱⠇⠀⢀⣴⣿⠟⢉⡤⣐⠄⡂⠤⠀⣀⣀⣀⣤⡤⠤⣤⡤⢤⡤⠤⠤⠤⠀⡀⠀⠀⠈⠑⠢⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢠⡟⡐⣡⢖⡡⢊⣠⣤⡶⠞⢛⣩⡥⠴⠶⠒⠒⠒⠒⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠜⠁⣡⢶⠟⢋⣡⠴⠚⣛⠿⠟⠋⠉⢉⣁⣉⣉⣉⣀⣀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⠁⡠⠊⠁⣢⡼⣆⣠⣤⣶⡾⣿⣿⠿⠿⠶⠦⠤⣄⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣡⠶⣫⡵⢟⣫⣭⣽⣶⣤⣤⣤⡤⠤⠤⠤⠄⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠞⣵⡿⣫⢖⣵⣿⡿⢟⣯⠏⣩⠤⠶⢒⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡾⡏⢐⡵⢋⡕⡡⡺⠋⣠⣮⠴⠚⣛⣩⣵⣞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠡⠀⡾⠁⠎⠈⡐⣡⡞⡫⢂⣴⣾⡿⠋⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⡇⠀⠀⢠⡾⢻⢊⡴⡫⠋⣩⣴⠿⣫⠭⠉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠸⡇⢆⡟⡐⣱⡿⢋⠕⣪⣴⣾⣯⡭⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠠⢊⣾⢟⢟⢫⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡾⠐⢋⣴⠿⣋⣤⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣧⠞⡻⢛⣩⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⢸⣶⢟⣥⣶⡛⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⡿⢫⣥⡶⠚⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⣭⣤⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣥⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣵⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⠶⠟⣿⡿⠋⠁⣼⠟⠁⢀⣠⣤⣤⣉⣹⣿⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉⣰⡿⠁⠶⣾⡟⠀⠀⢸⡏⠀⣴⠟⠉⠉⠉⠙⢿⡅⠀⠀⠈⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⢀⣀⣤⣿⡁⠀⠀⠹⠇⠀⠀⠈⠁⠀⠀⣠⣶⠟⠀⠀⠸⠇⢀⡄⠀⠈⢿⣌⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⢠⡾⠛⠉⠉⠈⣿⡄⠀⠀⠀⠀⣀⣤⣴⣶⢶⣿⡁⠀⠀⠀⠀⠀⣾⠃⠀⠀⠈⢿⡄⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⣾⠃⠀⠻⣦⠀⠸⠃⠀⢀⣴⠿⠋⠉⣀⠀⠀⠈⠁⠀⠀⠀⣠⣾⠋⠀⠀⠀⣠⣾⠃⠀⢠⡾⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠛⠀⢠⣶⠾⠋⠀⠀⠀⠹⣧⠀⠀⣀⣾⠃⠀⠀⣾⠋⢀⣀⣤⣤⣴⣶⣾⠋⠁⠀⠀⢀⣾⠛⠻⣦⡀⠘⠃⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⣤⡀⠀⠀⢶⣤⣀⠀⠀⣠⣿⡷⠟⠛⠁⠀⢀⣰⣿⠟⠛⠉⠁⠀⠀⠀⠹⠇⠀⢴⡶⠟⠃⠀⠀⢸⣿⠷⣶⡀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⡇⢰⡟⠈⢿⡆⠀⠀⠈⢹⣷⠟⠋⠁⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⢺⡆⣀⣤⣤⣤⣤⣀⣀⣀⣀⣠⡴⠆⠀⣿⠀⠘⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢿⡇⠸⣧⠀⣼⠇⠀⠀⢰⡿⠁⠀⣤⣄⣀⣴⠟⠋⠀⠀⠀⢀⣠⣤⡿⠟⠋⠉⠀⠀⠉⠉⠙⠛⠉⠙⠷⣦⡀⠁⠀⢀⣠⣴⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣧⠀⢻⡆⠻⠿⠿⠻⣿⡇⠀⠀⠀⣹⡟⠁⠀⠀⣀⣴⠾⠛⠉⠁⠀⢠⡀⠀⣀⣤⡶⠶⠶⠶⢶⣦⣄⣈⠛⠀⠘⠻⣯⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢀⣴⠟⠛⢿⡇⠀⠀⢠⣿⠀⠀⠀⠚⠋⠁⠀⠀⣿⣶⠶⠿⣷⡾⣋⣥⡶⠶⠶⢶⣦⣄⡉⠙⠻⣦⣄⠀⠉⣰⡖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠿⠃⠀⠀⠈⣿⡀⠀⠈⠁⢀⣴⠶⠆⠀⠀⣠⡾⢋⣴⠀⠀⣿⠻⠿⠷⠶⠶⠶⢶⣦⣽⣿⣦⡀⠀⢻⣦⣴⠟⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⢀⣀⣘⣿⣿⠶⠟⠛⠁⠀⠀⢀⣾⣿⣤⣿⠁⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣾⣇⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠙⠻⠶⣶⣦⣴⠾⠟⠁⠀⠉⠻⣧⡀⠀⠈⢹⣿⠿⠶⠶⣶⣤⣄⣀⣀⣀⣀⣼⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣤⡾⠿⣇⡀⡀⢀⣸⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡈⠙⠛⢛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣤⡶⠶⢶⣶⣶⣶⠶⠦⣄⣄⣀⣀⣀⠄⠒⠐⠂⣄
⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠿⠿⣯⡴⠿⠿⠿⠷⢶⣤⡈⠙⣷⣆⠀⢻⣷⣸⣄⣀⣀⣠⣴⣿⣧⣀⡀
⢠⣴⣿⡿⠟⠛⠉⠁⣀⠀⠰⠚⠁⢴⣄⠐⣦⡀⣠⡙⣿⣆⠘⣿⡆⠿⢿⣿⣿⠠⠤⠧⢼⣿⣿⠻⣿⣷⣤
⢸⣿⡇⠀⠀⠀⠀⠀⠁⠐⢩⠾⣦⢐⣹⣦⡘⣿⣿⣧⡘⣿⠰⣿⠇⣠⣾⣿⣿⠀⣶⣶⢸⣿⡇⠀⠀⣿⣿
⢸⣿⡇⡠⠀⠀⢠⠂⠀⢂⠀⢼⣿⣦⡈⠉⠙⠻⢿⡧⠷⠋⠐⠙⠛⢭⣿⣧⣏⠀⠛⠋⠈⠸⠀⠀⠀⣿⣿
⠸⣿⡇⠁⠀⢀⠃⠱⢰⠃⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠃⠀⣀⣤⢠⣾⡇⠀⠀⣿⡿
⠀⣿⡇⠀⠀⠙⡄⠀⣾⠀⠀⢸⣿⠃⠶⠶⠒⣲⡀⠀⠀⠔⠶⠶⠶⡌⢿⣿⣼⡇⣿⣿⢸⠏⠁⠀⢀⢿⡇
⠀⣿⢸⠀⠀⢀⡜⡶⢻⠀⠀⢸⡏⠀⡠⢶⡗⠯⠃⠀⠀⠰⢶⡽⢎⠛⠀⣿⠟⠃⠉⠀⠀⣇⡀⠀⢸⢸⡇
⠀⢹⡏⠀⠀⠈⢤⡜⠸⡀⠈⣿⠇⠀⠀⠈⠁⠀⠀⠄⠀⠀⠀⠉⠁⠀⢀⢦⣞⠡⣴⣶⣾⣿⡇⠀⡸⣾⠁
⠀⠸⣧⠀⠀⠀⠸⡇⠀⡇⠀⠱⣇⠀⠀⠀⢀⣀⣤⣤⣤⣴⣀⠀⠀⠀⠀⡟⠁⡾⠿⠿⠛⡇⠀⠀⣧⡿
⠀⠀⢿⣷⠀⠀⠀⡇⠠⠃⠀⠈⢿⣷⣤⢖⣩⣭⣥⢶⣾⣭⣬⣙⠦⣠⣼⡟⠀⠀⠀⠀⣀⣧⡆⢸⣿⠇
⠀⠀⠸⣿⣇⠀⠃⢧⠀⢠⠀⠐⣾⣿⣾⣿⣿⡉⢹⣶⣿⡉⣹⣿⣿⣿⣟⣿⣀⢰⣿⣿⣿⡟⣧⣿⡿
⠀⠀⠀⢿⣿⡄⠈⠃⢦⢸⠀⠀⢼⣧⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡇⣷⠾⠭⡥⢤⣷⣾⣼⠇
⠀⠀⠀⠈⢿⣳⠀⠀⠀⣨⡄⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣼⣿⣤⠤⢧⣼⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣧⠀⠀⢹⠀⡂⠨⡏⠀⠈⣷⠀⡂⢸⠂⢐⠀⡇⢐⡦⣖⠀⡂⢸⡀⠀⢈⢛⡝
⠀⠀⠀⠀⠀⠘⢿⣳⡀⠻⠤⠴⠖⡴⠾⠤⠶⠤⠷⢶⠤⠶⢴⡷⠶⠦⡶⠶⠦⠼⠃⣰⣳⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⡧⠄⠘⡇⠠⠄⢸⠀⠿⢀⡇⠨⡤⠆⠀⠀⢀⣴⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠛⠒⠛⠛⠚⣷⣾⣿⣶⠛⠓⠛⠃⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣀⠀⠛⠋⢁⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣉⣁⣀⣀⡀⠀⠐⢶⣶⣶⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠒⠲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡟⠁⠀⠀⠀⠳⠀⠀⡠⠒⠉⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⡐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⡀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠒⣴⣾⣿⡷⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⣀⣀⠤⠒⠁
⠀⠀⠀⠀⢠⡈⠉⠁⠀⠠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠈⣿⡠⢀⣾⣡⠆
⠀⠀⠀⠀⠈⢧⣀⠀⠀⣀⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠓⠛⠇⠚⣁⣴⠀⠀⣤
⠀⠀⠀⠀⠀⠀⣿⢁⠘⣿⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡼⠁⣸⠆
⠀⠀⠀⠀⠀⠺⠘⢿⡀⠋⢀⣼⣿⡀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⠺⣧⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠁⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠀⠀⠀⠰⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡿⢛⣤⣍⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⣻⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⣼⣿⣿⣿⡄⡤⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠁⣀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⣶⣿⣿⠀⠀⠀⠀⠀⢤⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠻⠟⢻⠄⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡸⠛⠚⠋⣡⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣄⡠⠀⣽⣄⣀⣾⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣁⠀⠀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢍⠉⢉⣹⠟⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⠈⢀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠠⠀⠀⠠⠀⠀⠘⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠓⡀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠠⠀⠠⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠈⠀⠀⢀⠐⠀⠀⠜⠃⠁⠀⠀⠀⠀⢂⢀⠀⠀⠀⠂⡀⠐⠠⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠃⢀⠀⠀⠄⠂⠀⠀⠀⣨⠄⠰⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠘⠀⠊⠀⢀⡀⠀⠀⢄⠠⠠⠄⠀⠀⠤⠤⠜⠂⠠⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠈⠀⠀⠼⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠘⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠠⢀⠀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⣶⣤⣤⣤⣤⠀⢠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠹⣿⣿⣿⣿⡟⠀⣸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣀⣀⣈⣉⡻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⣿⣿⣿⣿⠁⡙⢿⣿⣷⣿⣿⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠏⠷⠀⣿⣿⣀⢑⣻⡛⣮⠙⣿⡏⠀⣠⣬⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠆⢸⠏⠀⠀⠀⠘⣿⣿⣿⣿⠟⠉⣡⣿⠃⠚⠁⢩⣽⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⠛⠻⣷⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣀⠉⠉⡿⢁⣔⣀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣾⡇⠀⠀⢀⠔⠉⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡿⠀⠀⡠⠂⠀⠀⢀⣩⣿⣿⣿⣿⣍⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡌⠀⢀⣴⣿⡿⣿⣿⣿⣯⠻⢿⣦⡀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢠⣎⣀⣴⠟⠋⢰⡇⣿⣽⣿⣿⣧⣠⠉⢳⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡏⠙⣿⠁⠐⣣⣾⣿⣦⣿⡹⣿⣿⡿⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢰⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣄⣾⡏⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⠋⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⢀⡄⠀⠀⣠⢾⣿⡿⠷⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡇⠀⠀⠀⠸⢻⠇⢀⣴⢶⡯⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣧⣰⡇⠀⠀⢀⣴⡿⢏⣺⣀⣼⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣷⡀⠀⠸⠟⠁⠈⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠈⠉

⡀⠀⠀⠀⣠⠤⡾⠿⠻⠿⠧⠤⠤⢤⣀⣀⡖
⢱⣖⠋⠻⣅⠐⠈⠀⢠⠔⠒⠒⠒⠦⠤⠤⣉⣉⠓⠲⠤⢬⣑⡒⠢⠄⣀
⢸⡉⠳⣄⡈⠓⠦⣄⣀⠀⠀⠀⠈⠉⠒⠒⠒⠬⠍⣛⠒⠦⣤⣉⡙⠒⠦⣄⣀⠀⠀⠀⡀
⠀⠑⢦⣀⠉⠓⠦⣄⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠈⠉⠓⠲⠤⣌⡙⠒⠤⣀⠁
⠠⣀⠎⠉⠙⠒⠤⢔⡉⠛⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣉⠒⢤⡀⠀⠀⠠
⠀⠈⢷⠒⠤⢀⡀⠀⠀⠉⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣈⡐⠒⠥⠤⣒⣂⡈⠙⣦⠈⠁⠸⡄
⠀⠀⢈⠑⢦⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠦⠤⠭⠭⠥⠤⠤⠤⢳⡀
⠀⠀⡀⢠⡆⠉⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠠⠤⢄⣀⣀⡀⠀⠀⠀⢂⣠⠾⠁
⠀⠀⠱⣧⣀⠁⠂⠄⠀⠀⠀⠉⠉⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⢤⣄⣀⣉⣉⣙⣏⣉⠄
⠀⠀⠀⠘⢯⡙⠓⠶⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⣀⣀⣀⣀⣀⣉⣭⠽⠋
⠀⠀⢈⠀⠀⠉⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⣀⣀⣀⣈⣉⣉⣩⡧⠄
⠀⠀⢺⡀⠘⣦⡃⠀⠈⠉⠛⠓⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣭⢽⡋
⠀⠀⠀⠙⢦⣘⡝⠓⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠉⠉⢉⣡⠴⣺⣃⠀⣴
⠀⠀⠀⠀⠀⠈⠙⢷⡶⠤⢤⣄⣀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠚⢛⣉⡤⠞⠁⢈⣽⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠴⠋⣠⣯⣀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⡈⠉⠉⠉⠉⠉⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠈⢁⡴⠚⠁⠀⣨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣛⣛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢚⡥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠛⢻⡛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠊⢀⡀⠂⠠⠴⢛⡷⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⢾⣧⣄⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠁⠀⠰⠚⢉⣠⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⠛⠉⠉⠀⠀⠀⠠⠄⠲⠶⠛⠋⢁⣴⣣⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣄⣀⣀⡰⠄⠀⠀⠀⢀⡀⣠⡭⠛⠁⣈⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢉⡹⠋⠉⠁⠀⠀⠀⠀⠉⢀⠠⠖⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣉⡉⠁⠀⠀⡀⠀⠀⠠⠖⠛⢧⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠉⠁⠀⠀⠀⠁⠀⠀⢀⣠⣤⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠛⠒⡀⠀⠀⠀⢈⣩⣥⠖⠋⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣹⣗⡒⠠⠁⠀⠀⠀⢀⣠⣬⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢭⣈⠉⠀⠀⠐⣀⣐⠣⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠀⠈⠙⠓⠲⠦⢼⡧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⠷⠀⠀⠀⠀⣀⢳⡖⠚⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠄⠈⠁⠀⠐⣶⠂⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠭⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣑⠶⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠶⠶⠶⠶⠶⢤⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⣛⣋⣩⣍⣉⣙⣛⠻⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷⣿⣿⣴⣶⣶⣶⣤⣤⡤⢟⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠈⠉⠒⠛⠋⠉⠛⠛⠛⢛⣡⠄⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⢿⣛⣛⣛⣛⣛⣛⣛⣛⣉⣭⣴⣶⣿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⠤⠤⠤⠤⣤⣤⠤⠤⠶⠖⠛⠉⠁⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠢⠬⢭⣭⣭⣭⣭⣭⣭⣿⠿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⣛⣓⠒⠶⠶⠒⠒⠒⠂⣀⡀⠐⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢑⠒⠶⠶⠶⠶⠶⠶⠾⢟⣻⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣉⣉⣉⣉⣉⣉⣡⠤⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢭⣛⣛⣛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠦⠭⠭⠭⠥⠄⠀⠀⠀⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣒⠒⠂⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡑⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣓⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⠶⠶⠶⢶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⣉⣁⣤⣤⣤⣶⣶⣶⣦⣤⣤⣈⡉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⣉⣤⣴⡾⠿⠛⣋⣉⣭⣥⣤⣤⣤⣤⣬⣍⡙⢿⣦⡀⠈⠃
⠀⠀⠀⠀⠀⠀⣠⡿⢋⣴⣾⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣧⠀⢀
⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣴⣿⢿⡏⢀⡾
⠀⠀⠀⠀⠀⠈⠀⠀⠻⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⣛⣋⣭⣵⣶⣿⡿⢛⣵⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠈⠻⣯⣝⣛⠻⠿⠿⠿⣿⣿⠿⠿⠿⠛⢋⣡⣴⣿⣿⠟⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣸⣻⣿⣿⣷⣶⣶⣤⣤⣤⣶⣶⣾⣿⣿⣿⡟⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡤⢤⣿⣿⣿⣿⣿⠥⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠟⠓⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣶⣿⠯⠷⠦⠤⢤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⠻⣶⣮⣽⣿⣷⣶⣶⣭⠍⠛⠛⠿⠷⣶⣤⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡈⠉⠛⠿⠿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣈⣉⣿⡦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡀⠀⠀⠀⠀⠲⢤⣤⣬⣉⣉⣉⣉⣉⣉⣻⣿⣿⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣄⡀⠀⠀⠀⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⠀⠀⠀⣻⣿⣿⣷⣾⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣉⠁⠀⠀⣹⣿⣿⣿⣿⡷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣉⡁⠀⠀⢤⣭⣭⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡍⠀⠀⣈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣗⣂⠀⠈⠭⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣥⣤⡄⠘⢻⣿⣿⣤⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠙⠿⢿⣥⠀⠸⢿⣿⣿⣆⡈⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣿⣗⡘⢿⣿⣿⣿⣤⡁⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠀⢰⣦⠈⠀⠀⠸⠿⣦⠉⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠐⠀⠀⠙⠋⠀⡀⠀⢴⡄⠉⣧⢸⣿⣿⣿⠁⠸⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠈⠀⠀⠀⢈⡀⢀⡙⠀⣿⣾⣿⣿⠏⠁⠀⠀⡀⠈⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢀⠀⠀⢤⣤⣿⣿⣷⢰⣿⣿⡿⣣⣀⠀⠀⠀⠡⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠖⠚⠛⠛⠛⠙⠋⠋⠙⠛⠛⠓⠒⠒⠒⠒⠲⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠸⠆⠀⠐⠒⠲⠒⠖⠒⠒⠐⠒⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠄⡤⠠⢀⠄⠀⣤⣠⣤⣤⠄⠤⠀⠤⡤⠤⠤⠤⣤⣤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣤⣀⣀⠀⠠⢢⠀⠀⠀⡄⠀⠒⠒⠀⡠⠒⠀⠄⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠰⡄⠀⢈⠂⠰⡨⠤⠒⡄⡄⠀⠀⠀⡆⢀⡀⠀⡠⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠟⠀⠉⠉⠁⠀⠀⠈⠁⠉⠀⠈⠉⠁⠀⠀⠀⠈⠁⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠳⠤⠤⠶⠒⠖⠠⠲⠒⢶⠒⡲⠶⠶⠶⠲⠶⠦⠦⠤⠠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣤⢤⡄⠀⢠⠞⠉⠉⠉⠙⢳⠀⠀⠀⣠⡄⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣀⠈⠉⠀⠀⠈⠳⠒⠶⠒⠚⠋⠀⠀⠀⠀⠀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠚⠒⠓⠚⠛⠛⠛⠛⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠃⠀⢀⡤⠴⣶⡶⢾⣿⣿⣿⣿⣿⠿⠛⠛⠛⢻⣟⣻⡿⢶⣶⣦⣄⠀⠀⠼⠃⣀⣀⣀
⠀⠀⠀⢀⣀⡀⠆⠀⢸⣷⣶⣤⣤⣭⣭⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣶⡿⠿⢿⣛⣫⡆⠀⠀⠀⠤⣤⡉⠛⡆
⠀⠀⠀⡏⢡⣾⡛⠀⠈⢻⣿⠿⠿⠿⠿⠭⠭⢭⣭⣭⣭⣭⣠⣤⣿⣋⣉⣉⣩⣭⣵⡶⢀⣀⣤⠤⠾⡛⠀⢤⡀
⠀⠒⡀⠀⠀⢈⡉⠛⠛⠛⢶⣶⣶⣒⣒⡒⠀⠀⠀⠀⠠⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⡷⠀⠀⠀⠀⢀⣹⡇⠐⠟
⠀⠀⠀⠀⣀⠻⢤⣤⣤⣀⣘⣿⣿⣿⠿⠍⠉⠩⠤⠤⠤⠤⠬⠿⢿⣿⣿⣿⣿⣿⡿⠤⠤⠶⠖⠚⠋⠉
⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠻⣿⣯⣭⣍⣉⣑⠶⠶⠤⠤⠤⠤⣬⣭⣵⣶⣾⡟⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠒⣤⡀⠛⣻⣿⣷⠒⢒⣲⣶⣤⣤⣤⣤⣶⣶⣿⣿⣻⣯⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣁⡉⠀⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣭⣉⣛⣿⣿⣿⣷⣖⠛⠛⠋⠀⠈⠡⠶⠿⠞⣛⡋⠀⢀⣉⣷⠘⡦⠀⠀⠀⠀⠰⡿⠉⠞⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠿⣿⣿⣯⢭⠝⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣏⡉⣐⣶⠀⠀⠀⠀⠀⠀⠙⠂⣴⠏⣠⠄⢈⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡗⠂⠈⠿⠿⢿⣿⣒⣶⣶⣖⠛⠛⡻⠿⠿⣟⣛⣿⣿⡁⠀⢀⣀⡀⠀⠀⠀⠘⠁⡴⠋⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠂⠀⠀⠬⠿⠿⠿⠷⣶⣶⣶⣾⣋⣙⡛⠟⠯⡿⣀⠒⠚⠋⢠⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⣤⣀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠉⢙⠛⠻⠿⢷⣶⣾⣿⣭⠿⣦⣤⠶⠟⠀⠀⠀⠉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠘⠉⠀⣹⣆⣤⠤⠴⠖⠀⠀⠀⠀⢤⡘⠦⠤⢤⠀⠰⠾⠿⢿⡿⢿⡧⠀⠀⠪⠅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⣤⠀⠀⠉⣥⠶⠖⠛⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣷⣿⡏⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⡀⠀⢀⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⠀⠰⠰⣇⣼⣿⣋⠀⠸⠀⠀⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠋⠉⠒⠠⡀⠀⣀⠄⠒⠉⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡔⠁⠀⠀⠀⠀⠀⠀⠘⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠒⠒⠢⢄
⠀⠀⠀⠀⠔⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⡀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠠⠤⠤⡀⠀⠀⢀⣀⡀⢀⡀⠤⠤⢼⣠⢚⠉⠙⠒⠲
⠀⠀⠀⢱⡀⠀⠀⠀⠢⢄⠞⠁⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠘⠁⠀⠀⠀⠀⠈⢂⠀⠀⠀⡸
⡠⠚⠣⠚⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⣠⡄⠀⠀⠀⠀⠃⠀⢠⠃
⢣⠀⠀⠀⠀⢰⣉⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢀⠃
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⢠⠄⠀⠀⡜
⠀⠈⢧⠀⠀⠀⢠⣤⠴⠒⠳⡀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠘⠐⠒⠒⣟⣀⣠⠼
⠀⠀⠈⢇⠀⠀⠈⡀⠀⠀⠘⣌⠂⠄⠤⠀⠀⡰⠻⡑⠀⠀⠀⠠⠀⢨⡇⠀⠀⠀⠀⠀⠀⠠⠞⠉⢱
⠀⠀⠀⠈⢆⠀⠀⣱⠀⡄⠀⠈⠑⠒⠀⠒⠊⠀⠀⠑⠢⠤⠤⠤⠒⠱⣇⠀⠀⠀⠀⠀⠀⢰⠼⠚⠁
⠀⠀⠀⠀⠈⠶⣉⡤⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠜⠁⢠⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⠠⠔⢦⡀⠀⠀⠀⣀⠔⠳⣄⠀⠀⠀⣠⡀⠀⣀⠜
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⣀⡀
⠀⠀⠐⠁⠉⢄⠀⠀⢰⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⣤⣿⣭⠁⠀⠀⠀⣀⣡
⠀⠀⠘⠦⠠⠓⠀⠀⠺⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠁⠀⠀⡄⠀⠀⠀⠀⢹⣝⠋⠀⠀⠀⠀⠉⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠈⠉⢱⣶⡾⠁⠀⠀⠀⠀⢢⣉⠀⠀⠀⠀⠀⠆⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⡷⠆⠰⢾⡄⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⠄⢳⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣥⠒⠒⣬⡞⠀⠀⠀⠀⢠⠭⡄⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣷⡆⠸⢛⠀⠀⠀⠀⠀⣴⣶⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣧⡄⠰⢞⠀⠀⠀⠀⠀⠋⣿⣏⠀⠀⠀⠀⠀⠐⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⢀⣿⣿⣿⡟⢛⡆⠀⠀⠀⠀⣇⡄⢰⠾⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡏⡅⢠⣾⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⡴⠏⠁⢨⠴⢦⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠇⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣞⠻⣿⣿⡏⡀⠀⠀⠀⠁⡇⠀⠀⢸⠈⠀⠀⠀⠀⠸⠆⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢠⣦⣄⣿⡟⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠛⣿⣿⠇⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⢉⣽⣿⣿⣷⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⣰⣾⣿⡉⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠙⠿⣿⣿⣷⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⠘⣿⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣾⣋⠀⠀⠀⠀⠀⠀⣀⣨⣍⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠿⠿⠿⠿⣷⣆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⣀⡀⢀⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⣬⣷⣵⣿⠏⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⣙⣿⠛⠛⠛⠷⠷⠟⠛⠛⠂⠀⠀⢠⣶⠾⠶⠷⠿⠿⠿⣿⣷⣄⠀⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⡀⡀⡄⢀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠶⠛⠛⠛⠛⠻⠿⡾⠾⠷⠶⠷⠴⠶⠾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⣬⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣤⣶⣶⣶⣴⣤⣀⡀⢀⠀⣀⣠⣠⣀⡀⢀⢀⢀⣀⣤⣤⣀⡀⣀⣠⣠⣤⣤⣶⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠢⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠄⢀⣀⠀⠄⠀⠢⠤⠄⠤⠄⠐⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠈⠉⠈⠉⠲⢄⠀⠐⠈
⠀⠀⠀⠀⠀⠀⡀⠀⡁⠂⠀⠑⡄
⢀⡂⠀⠀⠀⠘⠟⠀⠏⠀⠀⠀⢱
⢠⠑⡄⠢⠄⣀⠀⠀⠀⡀⠀⠀⠘
⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⡌⠑⠢⢄⣀⢀⠀⡄⠊
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠢⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠒⢄
⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠁
⣶⣶⡶⡶⠶⢶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠤⢄⡠⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢺⢀⣱⠦⣼⣤⡐⠲⡖⠚⡒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟⡴⠁⠈⡴⠚⣿⠀⢸⡄⣹⡀⡱⣄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡈⠀⢧⠀⢀⡇⠀⣿⠀⣸⠃⠈⣅⡷⠬⢻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡡⠖⢠⠞⠀⣋⡤⠶⣿⠀⠈⠙⠒⢸⠀⠐⡚⠄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡁⠒⠚⠲⡈⠁⠀⠀⢸⡄⠀⡇⠀⢆⢀⡴⠧⣜⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠤⠞⠁⠀⠀⡇⢀⡞⠀⣾⠁⠸⣤⣄⠘⡜⢀⡄⠌⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣦⣠⡟⠛⣶⡇⠈⢷⣀⣿⡴⠖⠁⠤⠴⢧⣈⠓⡖⠛⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠈⠉⠷⣄⠀⠈⠀⠀⠙⣿⡁⢀⡤⠒⣀⣀⣈⣠⠇⢸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣷⡄⠀⠈⠛⠉⠛⢦⡀⣸⠋⢉⣀⡼⠉⡄⠀⠤⢠⡟⡘⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⠁⡉⠙⢶⡞⠉⠁⠀⠃⣿⡀⠀⠤⣀⣤⣿⠚⠛⢉⣄⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣹⡆⠸⡄⠳⣌⣓⡶⣿⠁⡴⢀⡼⢀⠈⣧⣠⠏⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⣳⣄⠦⠴⢦⣡⣄⣿⣤⣤⠴⠊⠁⠴⢋⣸⠄⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡓⢤⣙⡀⠰⡁⡄⢹⡀⢹⠖⢦⣤⠴⠋⢳⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡉⠓⠦⠭⣿⡗⠛⠓⢒⣤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠁⠙⠒⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡀⠀⠀⡀⠀⠀⠀⣀⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠞⢁⠤⡈⡆
⠀⠀⠀⠀⠀⠈⠀⠀⢈⢀⠎⠀⢸⠀⡀⠂⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠅⢸⠀⠈⡄⠀⠀⠀⠀⠀⠀⢆⡸⠔⡁⠀⢀⠥⠄⢠
⠀⠀⠂⠐⠀⠀⠀⠀⢀⠠⠊⠀⠂⠀⠀⡀⠁⡆⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠰⠈⠀⠀⠈⠑⠒⠁⠀⢠⠁⠀⠀⠀⠀⠀⠀⡀⠤⠊⠀⠀⠀⡩⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⢂⠐⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠊⠀⠀⠀⠀⣀⠠⠔⠁⠀⠀⠀⠀⠒⠒⠠⣄⠀⠀⠤⠯⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⣀⠈⠠⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠽⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢨⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠁⠁⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠬⠄⡂⠨⠄⠀⢅⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠂⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢎⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡡⠝⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠁⠉⠉⠉⠀⠀⠀⠈⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠁⠢⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠉⠉⠉⠑⠀⠐⠒⠲⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢤⠴⠒⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡶⠧⠤⢤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠀⠀⠂⠰⠆⠄⠄⠀⠠⣀⡀⠤
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡀⢻⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⡦⠀⠀⠁⢰⡆⣀
⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠴⠶⠶⠖⢢⡍⠁⢰⠁⠀⠀⠀⠀⠀⠀⢀⡤⣿⠐⣸⢁⣄⠀⠀⠀⢀⠀⡇⠶⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠉⠈⠉⠹⡧⠀⠀⠀⣇⠒⠂⠈⠀⢸⠃⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⡇⠀⠀⠀⠀⡶⠉⠀⢀⠀⠀⠀⠀⠀⠘⢠⠄⠀⠀⣽⣦⣄⣀⠀⣠⠃⠀⠀⠀⠀⠀⠘⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⠀⠀⠀⠀⠀⣠⠆⠙⠓⠂⠉⠯⣤⡀⠹⠂⡀⠀⠶⣜⣁⠀⢦⣸⠆⠀⠀⡀
⠀⠀⡠⠋⠀⠀⢀⠾⠊⠋⠈⠣⡄⠀⠀⢸⠀⠀⣇⠀⠀⠀⠤⠰⠁⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠉⢦⡆⠈⠳⡎⠰⠅⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠊⠄⢠⠋⠀⠀⠀⠀⠀⠰⠀⠀⠘⡇⠀⣿⠀⠀⠀⠦⣼⠀⣀⠀⠀⠤⠀⣀⣀⢬⡹⠀⠸⢌⢫⡄⡾⢠⠓⠀⠤⠀⡀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠴
⠀⠀⠆⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡇⠀⢣⠀⠀⠀⠀⠡⠀⢳⣤⠀⠀⠲⢶⠏⡘⠀⠀⠀⠈⢦⡙⣤⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡬⠀⢀⡆
⠀⠀⠀⠀⣦⣼⡄⠀⠀⠀⠀⠀⡸⠃⠀⠀⡇⠀⣺⠀⠀⠀⠀⠀⠙⠢⡄⠐⠂⠂⡠⠖⠀⠀⠀⠀⢀⡌⡁⢬⡳⠖⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠙
⠀⠀⠀⠀⠁⠀⠉⠠⡀⠀⣀⡴⠁⠀⠀⠀⡑⣿⣫⣄⠀⠀⠀⠀⠀⡀⠏⠀⠀⠘⢦⡀⠀⠀⠀⢸⠋⢘⡿⠢⣴⡜⠁⠀⠉⠙⠛⠶⠤⣀⣠⠊⠉⠁⠐
⠀⠀⠀⠀⡈⠁⠈⡀⡏⠉⠁⢠⠀⠀⠀⠀⣧⡀⢩⡄⣰⢄⣴⠋⠉⠃⢀⣄⡰⠀⠀⠉⠉⢃⡀⢈⡟⠉⠡⣠⣼⢓⡤⢄⡀⠀⠀⠀⠀⠀⡁⠀⠉⠀⠐⠆
⣀⡐⠀⠁⣀⠀⣀⠔⠓⠰⠆⠉⠉⠑⠶⡄⣙⠙⠡⢶⣅⠛⢷⣶⡄⠀⠀⠅⠠⠀⠀⠀⠀⠁⠸⠋⠀⠀⡜⠁⠀⠀⠀⠀⠀⠋⠰⠂⡄⠈⠣⣤⠲⠶⠀⠐
⣉⣈⣁⣀⡀⣀⣡⠤⠤⠖⠘⠓⠒⠒⠒⠢⡙⢄⡀⠈⠘⡣⣀⣩⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡷⠊⠀⠻⡑⠉
⠀⠀⠀⠈⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠻⢯⡄⢀⠸⢇⠀⠀⠀⠀⠀⠀⠀⣸⢣⣤⢠⡴⠁⠀⠀⠀⠁⠴⠀⠀⠀⠀⠀⡮⠀⠀⠀⠀⢐⢹⡷
⣀⣀⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠤⠤⠤⠤⢾⠀⠀⠀⠀⠀⠀⠀⡇⢺⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠁⠀⠠⣇⡀⠀⠀⠀⠠⠘⠁
⠀⠀⡌⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡯⠄⠀⠀⠀⣺⠀⠀⠀⠘⠀⠀⠀⡃⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠈⠀⠀⣰⠆
⣀⠀⣐⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡆⢳⠒⠒⠀⠉⢹⠂⠀⠤⠤⠤⠀⠤⠁⠀⠀⠈⠉⠒⠦⣄⡀⠀⠀⠀⠀⠀⠘⠁⠈⢠⠤⠚⠁⠀⠘⠃
⠀⠠⡴⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠤⠀⣀⢀⠀⡈⠠⠉⠉⠈⠁⠂⢉⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢠⣄⠀⣀⠀⠀⠀⠀⠀⠒⠁
⣀⠀⠸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⠀⠈⡏⠀⠠⣀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⣡⣈⡐⠇⡀
⠉⠀⠀⠙⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠤⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠢⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠔
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠹⡄⣀⣀⣀⢀⡀⣀⠠⠤⠤⣤⠖⠂⠁⠠⠄⠀⢻⠀⠀⠀⢸⣷⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡄⠂⠁
⠀⠁⠀⠀⠀⠁⠀⠀⢩⠙⢤⣀⡀⠀⣠⠞⠁⡇⠀⢀⣀⠀⠀⠾⠀⠀⠀⢰⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠈⣄⠀⠀⢸⠀⢀⣀⣀⡏⠀⢀⣀⡠⢤⡇⠀⠀⠀⢺⠀⡇⠀⠀⠀⢛⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠞⠻⠀⠈⠁⠉⡟⠛⠈⠁⠀⢀⡇⠀⠀⠀⢸⠀⢰⠀⠀⠀⢸⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣤⠤⠤⢶⠒⠒⠚⢉⠭⡀⠀⠀⠀⣇⣠⠋⠁⠂⡄⠀⠙⠀⠀⠀⠀⣏⠀⠸⡆⠀⠀⠘⡇⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠠⠀⠁
⠈⠀⠀⠀⠀⠀⠑⠄⠀⣏⠀⠘⠀⠁⠀⠁⠁⠀⠀⢀⠇⠀⣿⢀⣀⣠⣰⡇⠀⠀⢿⡀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣈⣉⣀⣀⠤⠖⢁⣤⠎⠀⠀⠀⠀⢸⡇⠀⢠⡞⠉⠁⠀⠈⣳⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠁⠀⠸⠶⠒⢤⠒⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠋⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣛⣛⣟⣻⣻⣿⣿⠛⣿⣟⠛⢛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⡴⠚⠛⠛⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢀⠀⠀⠈⠉⠙⢧⠀⣀⡀⣠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠉⠙⠁⠀⠈⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠸⠅⠘⠀⠀⠁⠀⢀⠀⠀⠉⠆⠀⠀⠀⢤⠀⠀⠘⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⣤⣃⡀⠀⠤⠊⠀⠀⠀⢠
⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠉⠉⠙⠛⠿⠿⢯⣭⣍⣉⡉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠲⠶⠶⠦⠤⣤⣥⣤⣦⣤⣀⣀⣀⣀⠀⣀⠀⠀⠀⠐⢶⢺⡇⠀⠀⠐⠠⣄⠀⠀⠀⠁⠈⣂⢀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣏⡙⠂⠂⠀⣸⠘⣧⡄⢀⠀⠀⠀⡄⠀⠀⠀⠀⢻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣤⠶⠃⠀⢸⡀⠈⠀⠉⠈⠀⠀⠁⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⣤⣄⣤⡼⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⣶⠀⢰⣦⡆⠐⡶⠀⢰⠀⠀⣶⠂⢰⣦⣶⠀⣶⠀⢰⣶⠄⠐⠀⢴⡆
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠙⠛⢀⣛⣂⡈⠉⠁⢀⣋⣀⣘⠓⠀⠙⢃⣈⣉⣙⡀⠛⠂⣘⣙⣁⣀⡀⢚⣃⣀⣀⡀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡄⠀⢨⣿⣿⣧⠀⠀⢸⣿⡿⣿⣿⡆⠀⢸⣿⠟⠻⣿⡆⠀⣿⡿⠛⠻⠃⢸⣿⡿⠿⢿⣧
⠀⠀⠀⠀⠀⢸⣿⠁⠈⣿⡇⠀⢸⣿⢿⣿⠀⠀⢸⣿⠀⠀⣿⡷⠀⢸⣿⣦⣶⣿⡃⠀⣿⣷⣶⡆⠀⢸⣿⣇⣠⣼⡿
⠀⠀⠀⠀⠀⢸⣿⣶⣾⣿⡁⠀⣾⣿⢸⣿⡇⠀⢸⣿⣷⣿⡿⠃⠀⢸⣿⠋⠙⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡟⢻⣿⡁
⠀⠀⠀⠀⠀⢸⣿⠿⠿⣿⣇⠀⣿⣿⣾⣿⣇⠀⢸⣿⠋⢿⣿⡄⠀⢸⣿⣦⣶⣿⠏⠀⣿⣿⣿⡿⠇⠸⠿⠇⠀⠿⣷⠄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡿⢸⣿⡟⠛⣿⣿⠀⢸⣿⠆⠈⠿⠟⠀⢛⣋⣉⠉⢀⣴⣶⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⢸⣿⣦⣶⣿⠃⣸⣿⠅⠀⠘⠛⣁⣀⣀⢠⢤⡆⣴⡟⠋⠉⣻⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⠿⠟⠁⢀⣀⣀⣠⣤⣤⣭⣭⣭⣭⣥⣭⣥⣬⡥⠶⠞⣋⣁⣈⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⣠⡴⣾⠛⢻⣿⡆⢠⡆⠀⠀⢀⣤⡄⢠⣶⠛⣿⠀⣼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⠋⠀⠹⠦⠾⠟⠀⣿⢷⣷⣴⡏⣸⡿⣾⡇⣸⠟⠀⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⡇⠀⠀⣠⣤⣶⡄⢸⡟⣼⠏⣿⣥⡟⢱⡿⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣿⣿⣶⣿⡿⠿⣿⠃⣾⠃⠿⠃⠈⢀⣀⣉⣥⣤⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⣠⠏⠀⣋⡤⠴⠞⣛⣉⠉⠁⢀⣽⡿⠁⣺⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⡇⠀⢀⡴⠃⠀⠁⠀⠶⠶⢶⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠸⣿⣷⠞⠋⠀⠀⢀⣠⣤⠶⢞⣛⠉⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣫⣥⣒⡛⠋⣁⣠⣤⣤⣾⣿⣿⣿⣿⣯⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⠂⠟⢲
⠀⠀⠀⠀⠀⠀⢠⡾⠋⣠⠶⠶⣦⠁⡠⣾⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⠀⣻⣄⠋
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣀⣼⠋⠀⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡼⣧⣀⣠⣀⣽⣿⠂
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢁⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⣸⣿⣿⣿⣿⡟⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⢹⠁⠀⠀⠙⠦⠬⣭⣍⣉⣉⣉⣉⣉⣙⣛⣛⣛⠛⠛⠛⡻⢋⣴⣿⣿⡿⠟⠉⡇
⠀⠀⠀⠀⠀⠀⢠⣆⡽⢮⣡⠴⣄⢠⠀⠀⠀⣠⣾⣵⣧⣤⣴⣿⣽⣁⣉⣉⣙⣟⡛⠐⠾⣿⠻⠛⠁⠙⠋⠉⣧
⠀⠀⠀⠀⠀⠀⠸⡏⣽⡿⠉⣿⠟⢸⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣩⣿⣿⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⢸⢾⣀⣿⣐⡇⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢀⡇⠀⢈⡬⣉⣩⢭⡭⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⢸⠀⠀⠹⠤⠻⠟⠘⠇⠨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠾⣉⡛⠿⠿⢿⡿⠃⣸⠤⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠉⠉⠉⢉⠛⠛⠛⠛⠿⠿⠷⣿⡁⠊⠀⣿⡆⠐⠐⣈⣃⡒⠁⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⡀⠉⠁⠀⠀⣿⡇⠀⠀⠉⢻⣿⣶⣦⣄⢲⡄⠀⠀⢀⣠⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⢶⣶⣿⣯⣤⣤⣤⣭⣭⣭⣭⣥⣴⡶⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠋⠉⠉⠉⠀⠈⠉⠉⠉⠙⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⡷⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⣡⣤⣤⣤⣤⣤⣤⣡⣤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠁⢙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣉⣉⣉⣩⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢉⣿⣻⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢒⡂⠀⡀⠰⠎⠩⣉⠰⠆⠒⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠉⣉⣀⣉⣀⣀⠌⡽⠀⠀⣡⠔⠛⠐⠲⠬⠓⣄⣌⠉⠲⢆⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⡡⠄⠔⠁⠀⠄⠀⢈⡹⠴⠧⢤⡞⠁⠀⠲⣧⠔⠖⠋⠉⠉⠉⢙⠮⣭⡛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡍⠀⡜⠁⠀⢠⠀⠀⠀⣇⠀⢁⠀⠀⠀⢀⡷⠆⠀⡴⠁⠀⠀⠀⠀⠀⠀⢱⠀⠀⠙⠒⠮⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⠼⠦⡤⠂⢀⠀⠘⠦⣀⡀⠈⠖⠁⠀⢠⠎⠑⠒⠛⢺⡇⠀⠠⢄⣺⠀⠀⠀⢸⡇⠀⢂⠀⠒⠓⢍⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡠⣃⣀⣸⡀⠀⠀⡁⠀⠀⠄⠈⠁⠀⠠⣄⡾⠀⠀⠀⢀⠀⠻⣄⠀⠀⣿⡤⠀⠀⢸⠅⠀⠈⠲⠞⢀⠘⡄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⠈⡀⠤⢀⡗⢉⡀⠀⠉⠀⠀⢀⣷⡀⠀⠘⣧⠀⠈⠀⠈⢢⣀⣀⣠⠴⠃⠀⠀⠀⣸⢧⡀⠀⠀⠀⢨⠀⠀⢸⢊⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⡜⢀⣀⠀⠃⠀⠁⢀⣀⡴⠊⢹⡇⢱⡀⠀⠘⣆⠀⢀⡴⠋⠁⠀⠄⠀⠀⠀⢀⣰⠃⠀⠉⠓⠲⡖⠁⡗⠀⠨⣀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠏⡰⠉⠀⠀⠐⢤⠖⠛⠤⡀⠀⡀⢇⠀⠇⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢀⡜⢧⠀⠀⠀⠀⠘⠁⠀⣸⠀⠀⠈⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠈⡸⠀⠃⠀⠙⠒⣶⠋⠀⠀⠀⣀⠀⡇⢸⣤⡤⢤⣴⡿⠥⣤⣀⣀⣠⠼⠛⠉⠉⠁⢀⡼⠤⠤⠤⠤⢴⠋⠐⠁⠀⠀⢳⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠨⡱⠀⠀⠂⠉⠹⣅⠀⠉⠀⠀⠈⢹⡿⠋⠀⠀⢰⠃⠀⠀⠀⠈⠛⣤⣀⣀⣀⠐⠂⠙⠀⢀⣠⠤⠀⠀⡷⠤⠒⠂⢀⣞⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠄⠐⢄⡀⠈⠉⠙⢋⡡⣴⠏⠀⠀⡰⠁⠘⣆⣠⠀⠸⠀⠀⠀⠀⠀⠉⠛⠓⠚⠋⠉⠁⠀⢀⡴⠃⠀⠀⠉⠁⠈⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠉⠿⠶⠶⢶⣶⠇⠀⠀⠀⡇⠀⠢⡤⢧⣀⠀⠀⣄⣀⣠⣄⣀⡴⠦⠤⠤⣴⡶⠞⠿⠶⠶⡦⢤⣤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠉⠉⠉⠉⢻⡄⠀⠤⠒⢧⠀⠈⠀⠀⣈⣁⠔⢋⠈⠁⢀⣠⣴⡶⠶⠛⠁⣀⣉⠀⠌⢁⡰⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠐⠂⠀⠈⠳⣄⣀⠀⠀⠸⠀⢀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣌⣒⣤⣄⣁⣀⣀⣀⡤⠞⠉⠉⢻⣿⣿⣶⣂⠉⠉⣀⣤⢤⠴⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠒⠿⢍⡛⠛⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠀⠐⠂⠀⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠁⣤⠀⠀⡀⠔⠀⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⠀⡄⠄⠐⠑⠀⡃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠀⠀⠀⠀⠀⠀⡘⠀⢈⡄⠀⠀⠀⠀⠀⠸⣁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠄⠊⠀⠀⠠⠈⡀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠂⠐⠒⠠⠂⠀⠈
⠀⠀⠀⠳⠀⠀⠀⠀⢀⠎⠈⠀⠀⠉⠘⡄⠀⠠⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⠎⠑⠂⠀⠀⡀⠀⠀⠀⠀⠀⣀⠄⠄⡀
⠀⠀⠀⠀⡀⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡀⠤⣠⠊⠁⠀⠉⠦⠠⠤⡀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⢠⠊⠁⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢄⠀⡀⠀⠰⠉⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠒⠢⡀⠀⠀⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⢲⢀⡔⢦⢠⡖⡄⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠈⢄⣀⡀⡀⡀⢀⠝
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⠈⠣⠞⠘⠋⠐⠆⠀⠀⠀⠘⠀⠀⢀⠀⠄⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⣀⣀⢀⡀⡸⠀⠀⠀⠀⠀⠀⢠⠛⠀⠠⡆⠰⠗⠰⢒⠀⣂⠀⣖⠀⠠⠀⠆⠀⡜⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠤⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⡀⠀⠀⠀⠀⠀⠔⠀⠀⠘⠀⠁
⠀⠀⠀⠀⢠⣀⡀⠀⠈⠀⠀⠀⠀⠀⠈⣄⠀⠑⠂⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠒⠘⠁⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠈⠒⠀⠒⢄⣀⣀⠔⠓⠂⠂⠀⠀⠀⠀⡰⠤⠤⡀⠀⠀⠈⠂⠀⠉⠀⠉⠀⠩
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡤⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠀⢀⠂⠈⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠄⡀⠀⠀⢠⠋⠀⠀⠀⠀⠉⠠⠀⠀⠀⠐⠂⠁⠀⠠⠄⠄⡀⠀⠀⠀⣰⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠞⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⠀⠱⡀⠀⠀⠀⠀⢀⡪⠀⢀⠄⠐⠐⠀⠂⠠⡀⠀⠀⠀⠉
⠀⠀⠂⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠉⠓⠒⠢⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠑⠂⠅⠉⠀⠀⠀⣴⠀⠀⠀⠀⠈⠢⠀⠀⣀⠄⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠁⢀⠀⠀⠀⠀⡀⠈⠀⢀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⣒⠄⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢻⣛⣟⣙⣛⣟⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡠⠼⣀⠦⠄⠀⠀⢀⠀⣿⡇⠀⠀⢈⡴⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠈⠀⠉⡃⠀⠀⠀⢷⣻⡇⠀⠀⠈⠛⠋
⠁⠄⠋⠠⠄⠀⠀⠈⣄⣿⡇⠀⠀⣈⣴⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⣶⡆
⠀⢀⣀⡀⠀⠀⠀⠀⢹⣾⡇⠀⠀⠈⠛⠋⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢠⠞⣩⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡇
⠀⠛⣹⠟⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢶⡆⠀⠀⠀⠀⡶⠀⠀⠀⠀⡀⠸⣿⡇⠀⠀⢠⢞⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢚⡴⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢀⣠⠁⣠⡀⠀⠀⢁⠀⣿⡇⠀⠀⠰⢋⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢉⡴⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠂⠀⠟⢄⡈⠀⠀⠈⣄⣿⡇⠀⠀⢴⣯⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⣩⣞⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠹⣾⡇⠀⠀⠀⢈⡀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠙⠛⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠶⢻⣷⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⣶⡆⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠰⣫⡾⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⡇⠀⠀⡠⠺⣀⠔⠄⠀⠀⢠⠀⣿⡇
⠀⠀⣿⡇⠀⠀⠀⣀⠃⣀⠀⠀⠀⠄⠈⣿⡇⠀⠀⠘⣡⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠉⠢⠄⠀⠀⠀⣦⢿⡇
⠀⢰⣿⡇⠀⠀⢎⢉⡶⡉⠑⠀⠀⢰⠀⣿⡇⠀⠀⠘⣡⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠘⣿⡇
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⢧⢻⡇⠀⠀⠙⠿⠟⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢛⡽⢟⠀⠀⠀⠀⠀⠸⠁
⠀⠈⠛⠃⠀⠀⠀⣤⢿⣦⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⣤⡄⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢈⡴⢛
⠀⠀⢰⣷⠀⠀⠀⠰⢋⣵⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢿⡇⠀⠀⠀⣠⠁⢀⠀⠀⠀⠄⠀⣿⡇⠀⠀⢈⡴⣻
⠐⠀⢸⣿⡆⠀⠀⠰⢋⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢀⠷⡁⠀⠀⠀⠸⡀⣿⡇⠀⠀⣨⣞⣻
⠀⢡⠘⣿⡇⠀⠀⠰⣫⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⢻⣽⡇⠀⠀⠈⠛⠁⠀⠀⠀⠀⠲
⠀⠈⣦⣿⡇⠀⠀⠺⣷⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⢤⢿⣷⠀⠀⠀⠀⠈⢻⠇⠀⠀⠀⣶⡆⠀⠀⠀⠀⠟
⠀⠀⠹⣿⠇⠀⠀⠀⣀⡀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠰⢋⣴⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⡇⠀⠀⠠⠴⡀⠶⢄
⠀⠀⠀⢸⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠰⢋⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠁⠀⠋⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡠⠾⣀⠶⢄⠀⠀⢠⠀⣿⡇⠀⠀⠰⣩⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠁⠐⠉⠢⠀⠀⠀⠀⣆⢿⡇⠀⠀⠺⣷⠞⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠚⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠸⣾⡇⠀⠀⠀⣠⡄⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢘⡵⢿
⠀⠀⠗⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠛⣹⢟⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣿⡇⠀⠀⠀⢀⠃⠀⠀⠀⠈⠄⠈⣿⡇⠀⠀⢊⡴⢻
⠀⠀⠇⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢘⡵⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠔⠉⣦⠋⠰⠀⠀⢰⠀⣿⡇⠀⠀⢈⣴⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣾⡿⢿⣿⠿⣿⣦⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⡿⠛⠋⠉⢀⣴⠋⠀⢀⣴⠃⠀⠸⠿⠿⠛⠛⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⣽⠃⠀⢠⡾⠟⠁⢠⣴⣟⣁⣠⠀⠀⠀⠀⠀⣴⠋⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⢠⡾⢿⡀⠀⠀⢀⣤⣤⣿⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⣿⠀⠀⢷⠀⠀⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣼⠋⠁⠀⢀⡾⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠳⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⣷⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠟⠁⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠶⠶⣤⠀⠀⣿⠀⠀⢀⣴⠀⠀⠙⢷⡟⠁⣠⡼⠃⠀⢰⣧⣄⠀⠀⠀⣀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⡀⠀⠀⣾⠛⠛⠁⠀⣀⣾⡁⠀⢀⣀⣼⡷⠞⢋⡀⠀⣠⠟⠁⢹⡇⠀⠀⠉⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣇⠀⢠⡇⠀⠀⠀⣀⣽⠿⠷⠛⣋⣩⣤⠶⢾⣯⣁⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣷⣶⣶⡾⠋⣤⠶⣿⠟⠉⠁⠀⠰⡟⠁⢉⣩⣥⠶⠶⠖⠚⠓⠒⠺⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣧⡘⠃⠀⠻⠆⠀⢀⣠⡤⠶⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷⣶⣶⣶⣿⢿⣿⣤⣤⣴⣟⠛⠲⢤⣤⣤⣤⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠹⣿⡆⠀⠀⢻⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠁





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠉⠁
⡀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⠛⠋
⣇⠀⠀⠀⠠⠾⢿⢛⣩⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠋
⣿⣿⠖⠀⠀⠀⠸⡿⢻⡟⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⣿⣿⠟⠁
⣷⣦⣤⣀⡀⠀⠘⠒⢻⣧⣿⣤⡞⠋⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⡿⠋
⣿⣿⣿⣿⣿⣏⣀⣀⡙⣿⣿⣿⣄⣤⣿⣥⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡟⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣼⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣴⣤⣤⣤⣤⣤⣤⣄⣤⣾⣦⣤⣄⣀⣴⣿⣿⣿⣤⣀⣿⣀⣀⣰⣾⣿⣂⣀⣀⣤⣶⣾⣦⣰⣾⣇⣸⡃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠒⣒⣒⣒⠒⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢀⡄⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣀⡀⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣿⣿⣿⣿⣿⡸⣿⣿⣿⡿⣿⣿⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⣾⡿⠿⢿⠛⠧⠈⠻⠿⠷⠘⠋⠘⢋⣵⡿⣟⡅⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠉⠉⠙⢲⡶⠲⣶⠋⠁⠀⠀⠙⢿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⠀⠀⠀⠀⢀⡇⠀⢷⠀⠀⠀⠀⠀⣸⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡙⢦⣀⣀⡠⠊⠀⠀⠈⠳⠤⠠⠤⠞⣱⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣰⣿⣿⢁⡀⠀⠀⠀⢀⣶⣿⠶⢤⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢧⠀⠀⠀⠠⣿⣿⣷⣀⣤⣴⣿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⠋⠀⠀⠀⢀⣾⡿⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⡀⠀⠀⠘⣷⡀⠀⠀⢾⣿⣿⣿⣿⠛⠛⠋⠀⠉⠉⣽⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⡿⠁⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡓⠶⣦⣼⣷⡀⠀⢸⣿⣿⣿⣿⣷⣤⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⡄⠀⢠⣿⡟⠁⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠉⠙⣟⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⡟⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠳⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠁⠀⠈⠋⠻⠿⠻⠿⠿⠿⠟⠛⠿⠋⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠉⠩⠙⢋⡀⡈⠉⠉⠩⠰⡒⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢄⣠⠀⣠⠖⠋⠉⠀⠀⠀⠀⠉⠙⠲⢌⠀⠣⡀⠉⢙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡤⠊⢀⠞⠁⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠱⡀⠈⡗⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⠃⢁⠎⠀⠀⠀⣠⣴⠒⡶⣋⠶⢆⠀⠀⠀⠀⢱⠀⢣⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠖⠂⢸⠀⠀⠀⠘⡄⠓⠀⠓⠉⢁⠎⠀⢀⠀⠂⢈⠀⢀⠕⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣔⠁⡀⢸⠀⠀⠀⡀⢸⡀⠀⠀⠀⡎⠀⠀⠀⠀⠁⡸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⡄⠈⢆⠀⠀⠠⢀⡇⠀⠀⢸⠀⠀⠀⢠⠐⢠⠃⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠴⠎⣀⠀⠈⢦⡀⠀⠀⠛⠀⠀⡃⡀⡄⡆⠀⡰⠃⠀⠀⡔⠋⠁⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠤⠤⢄⠀⠸⠀⠀⠀⠀⠀⠀⠀⢱⣠⡄⡉⠢⢄⣸⡀⢑⣇⣠⠤⠔⠋⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠑⠀⢇⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠰⣎⡀⢘⡽⢟⡆⠈⠑⢦⡜⠁⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠀⠠⠔⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣉⣉⣤⡴⢞⡧⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡘
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⢠⣀⣴⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡌⠀⡜
⠀⠀⠀⠀⠀⠰⡁⠀⢀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠠⠀⠀⠀⠀⠀⠈⠻⣙⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠘⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠊⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠰⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡔⠉⠉⠲⡀⠀⠀⠀⢻⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢥⣌⣊⠔⠰⠀⠆⡔⡠⢠⢀⠎⡞⠠⠃⢋⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠄⠊⠀⠀⠀⠀⢸⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠢⠤⠤⠤⠤⠤⠔⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣱⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢣⡸⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⢈⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡴⠆⣨⢔⠵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣷⠆⢐⠵⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠕⡪⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣩⣂⠅⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣩⣿⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢘⣻⢹⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢕⡏⢸⠀⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡠⢀⠴⣂⢔⡢⢊⠤⣂⠔⡪⠔⡡⢐⡠⢂⠴⣂⠔⣢⢔⡠⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠥⢻⡰⣛⢕⡾⢳⣄⣿⢔⡠⢂⠄⡢⡤⠔⠔⠠⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠖⠚⠁⠘⠤⠄⠒⠢⠤⢤⣀⣀⣀
⢀⡀⠀⠀⠤⠤⠔⠒⠚⠋⠉⣡⣤⣶⣶⣶⠆⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣽⠿⠷⢦⣄
⡎⠀⠒⠤⣄⣀⠀⠀⠀⠀⣀⣉⣉⣭⡥⠴⠶⠖⠒⠛⠛⠛⠉⠉⠉⠀⢀⣀⣀⣀⣀⣤⣽
⢿⣦⣀⡀⠀⠉⢻⠋⠉⠉⠉⠈⢀⣀⣀⣀⣤⣤⣤⡶⠶⠶⠾⠛⠛⠛⠛⠋⠉⠉⠉⠁⣿
⢸⡆⢉⠙⠛⠃⠸⡶⠒⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⡀
⠘⡇⠀⠀⠈⢈⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡇
⠀⣿⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⢻⡄⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷
⠀⠸⣇⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⢻⡄⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠈⣷⡀⠆⡠⠁⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⡟
⠀⠀⠀⠈⢷⡄⠀⠄⠀⠁⠚⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡼⠃
⠀⠀⠀⠀⠈⠻⣦⡘⠁⠡⠐⡌⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣯⡶⠿⠶⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡂⠈⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠠⣀⠉⠻⢷⣦⣄⣀⣀⣴⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⣀⠂⠈⠉⢹⣿⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⠛⢸⣧⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⣿⠙⣧⡀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠈⠻⣶⣄⠀⠈⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠉⠛⢷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠨⠙⠻⣶⣦⣄⣀⠉⠛⠿⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠿⢶⣦⣄⣈⡉⠛⠻⠿⠷⣶⣶⣶⡶⠾⠿⠟⣋⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣶⣶⣶⣶⣶⣶⡶⠾⠟⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⠀⣀⣠⡴⠞⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⣤⣄⣠⣠⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣄⣀⣀⣀⣴⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣋⡴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠞⣫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡴⢋⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⡴⢋⡴⢋⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡙⢦⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⣡⠛⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠳⡌⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠏⡴⢃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠹⣄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡿⢠⠟⡸⢁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣾⢡⡞⣰⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢱⡄⢿
⠀⠀⠀⠀⠀⠀⢸⡇⣼⢃⠏⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⢇⢸
⠀⠀⠀⠀⠀⠀⣼⠁⣿⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸⠈
⠀⠀⠀⠀⠀⠀⣿⢠⡇⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⢸⡇⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣿⠈⣇⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⢻⡄⣿⢸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⢀
⠀⠀⠀⠀⠀⠀⠸⡇⢹⡌⣦⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⢻⡈⢧⠹⡄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡘⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⣦⠱⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⣧⠻⡄⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡰⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⢷⡘⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠴⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠹⣆⠳⣄⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠳⣌⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡘⢧⡀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡌⢷⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢣⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡘⣧⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠋⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣭⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⣶⣯⣥⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣤⣄⣻⣿⢿⣿⣟⣛⣛⣛⣛⣛⣛⣻⣟⣁⣀⣤⣤⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠻⠿⢿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⣻⣮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣿⣿⣿⣿⣯⣬⣥⣤⣤⣤⣤⣶⠶⠶⠶⠾⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣍⣻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣉⡛⣻⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠉⣩⣿⣄⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣛⣿⣿⠛⠓⠚⠛⠛⠛⠉⠉⠉⠉⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢶⡶⢶⠶⠶⠶⠊⢉⡹⣿⠟⢖⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⢉⡾⠁⠀⠀⠀⢻⡇⠀⠲⣤⠞⠐⢿⣟⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠂⠀⢀⡤⠂⣠⡟⠀⠀⠠⠀⠀⢀⡿⠀⠀⢸⡇⠴⠞⢹⣇⢻⡿⠿⢿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⣴⠇⠂⠀⠀⠀⠀⠤⢾⣀⡔⠀⠉⢣⠀⠀⠀⢢⡞⠁⠀⠀⣠⠞⢻⡄⠠⠞⣿⡆⠠⡀⢸⣿⣿⡿⣷⣄⡀
⠀⠀⠀⣀⠄⠚⠁⠀⠀⠀⠴⣁⠀⠀⠀⠉⢠⠂⢀⠞⠃⠀⠀⣸⠀⠀⠀⠀⡟⠀⣼⠁⠄⠀⠙⣿⣤⣿⢾⣿⡇⠀⡈⢻⣿⣆
⠀⢀⣾⠃⠀⢀⡠⠀⢠⠂⠰⠈⠓⡀⣀⠔⠁⠀⡎⠀⠀⠀⠉⠛⢄⡀⠀⢰⡅⠀⣿⠀⢀⠔⣠⣿⣿⡏⢸⠹⣷⣹⡌⡌⢻⣿⣆
⠀⣼⡅⠀⢠⢘⠁⠀⡿⠀⢀⠀⢀⣿⠃⠀⠀⢠⣇⣀⣀⣠⣤⣤⣀⣤⣤⡾⠀⠈⣧⠀⢆⠀⠁⣿⣿⣧⡸⢰⣿⣿⣷⠈⢾⣿⣿⣦
⣾⢻⡇⠀⢠⡜⠀⠀⠃⠀⠀⠁⣿⣿⠀⣀⡤⠚⠋⢩⠉⠁⠀⠀⠀⠀⠀⢀⣤⠶⢯⣄⣀⢁⣈⣿⣇⠿⠶⣿⣿⠉⠀⣠⣿⣟⢹⣿⣆
⣿⡾⠀⠙⠻⠱⣄⡀⠀⠀⠈⠒⣿⠟⠉⠁⠀⠀⣀⣿⠀⠀⠀⠒⠉⣉⠉⠉⠀⠀⠀⠉⠛⢿⠛⠉⣠⣴⣾⣿⠟⠀⢈⣿⣿⣿⣿⣿⣿
⢿⢿⠀⠀⠀⢀⠀⠁⠂⡤⠀⢨⠃⠀⣿⣄⡀⡜⠋⠀⠀⠀⠀⠀⠀⠀⢉⠛⣻⠀⠀⢀⡀⠘⣦⣿⠿⠛⠙⣦⣹⣷⣿⣿⣿⣿⣿⣿⣿
⠈⠚⠓⡦⠀⠈⢳⡖⠉⢀⣠⡟⠀⠀⢨⠉⠉⡀⢀⠀⢠⠀⠱⣦⠐⠂⣩⡾⢋⣾⣤⣤⣴⣀⡉⢀⣴⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠱⡄⢀⣶⣴⣾⣤⣴⣿⣿⡇⠀⠀⣼⠀⠊⢀⣾⡀⠈⣀⢞⣡⣾⣿⣷⣾⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⣛⣿⣿⣿⣿⣿⣟⠻⠿⠿⠃
⠀⠀⠈⠺⢿⣟⣿⣛⣿⠿⠛⠉⣄⣀⣌⠉⣉⣥⣤⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⢿⡉⠉⠉⠒⠺⢦⣄⠀⢀⣒⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣟⠿⢛⣡⣤⣿⠏⠙⠿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠂⣦⣴⢺⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠉⠉⠳⣍⡛⠿⠿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣌⣻⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⣤⣀⡀⠀⢀⣠⠴⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠙⣿⣷⠋⠀⠀⠀⠀⠀⠀⠉⣧⣀⣤⠤⣤⣀
⠀⠀⠀⢀⣤⣶⣿⠀⠀⠀⠀⠀⠀⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢙⣦⡤⠖⠒⠲⢤⣀
⠀⠀⡴⠁⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠈⢧
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡴⠟⠋⠉⠉⠙⠲⢤⡀
⠀⠸⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠙⠂⠀⢠⠴⡄⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀
⢀⣴⠿⠦⠀⠀⠀⠀⠈⠛⠶⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠰⠟⠀⢠⣤⠀⠁⠀⠈⠙⠲⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠉⠳⣄
⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠬⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣧⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠈⢳⠀⣰⠂⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠹⣄⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⢀⠀⠀⠀⠀⠈⡇⠙⠶⢶⣄⣀⣀⣀⡠⠇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⡤⠖⠲⢾⡁
⠀⠈⣳⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⣠⣼⠃⠀⠀⠀⠘⣯⠉⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠘⠧⠀⠀⠀⠙⡆
⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢸⡽⠀⢀⣀⣀⡀⠀⠀⠉⠳⠶⠤⠶⠀⢀⣀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⣄
⠀⢻⡀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⠸⠋⠀⠀⠀⠈⠙⠻⣟⡛⢶⣶⣤⣤⣤⣄⣀⣉⣛⠛⠛⠁⠀⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⣸
⠀⠀⠓⢄⣀⣀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠉⠲⣍⡛⠿⣿⣿⣿⣾⣭⣭⣭⣉⣛⠛⠛⠛⠿⢿⠧⠤⣤⣄⠤⠴⠋
⠀⠀⠀⠀⡾⠛⠉⠙⢿⡀⠀⠀⠀⠰⣄⣀⣀⣠⠋⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠙⡄⠀⠀⠙⢦⡈⠙⠲⣍⡻⢿⣶⣤⣍⣙⣿⡶⠶⠯⠄⣀⣀
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠙⢦⡀⢠⡝⠳⢮⣙⠳⢯⣿⠲⠤⣄⡀⠀⠉⠉
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⣀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠙⣾⠷⠤⠤⠽⠻⢯⡀⠉⠒⠦⣍⡃
⠀⠀⠀⠀⠀⢹⣦⣄⣀⡀⠀⠀⠀⠘⢶⣤⣤⣀⣠⣤⠶⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠙⢶⣦⠤⠤⠖⠋⠉⠳⡄⠀⠀⠀⠀⠉⠲⣄⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠻⣇⠈⠁⠀⠀⠀⠀⣠⠟⠉⠉⠁⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠙⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⣸⣿⠦⢄⣀⣀⣀⣠⡤⠞⠁⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠛⠛⠛⠀⡄⡏⠉⠉⢹⢠⠀⠛⠛⠛⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣤⣤⣤⠀⠃⣇⣀⣀⣸⠘⠀⣤⣤⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠉⠉⠀⠀⠀⠐⠒⠀⠀⠀⠉⠉⠉⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠀⠶⠀⠠⠄⠠⣤⠀⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠴⠀⠠⡄⠀⣤⠀⢠⡄⠀⣈⠀⠀⠁⠀⠘⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠶⠀⠰⠆⠀⣤⠀⣠⡄⢀⣀⠀⣈⠁⠀⠛⠀⠘⠃⠀⠖⠀⠰⠄⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠠⡄⠀⣤⠀⢠⣄⣀⣈⣁⣈⡋⠀⠛⠃⠘⠟⠀⠶⠆⢰⡦⠀⣤⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠰⠆⠀⣤⠀⣠⡄⢀⣤⠖⠉⠀⠀⠀⠀⠀⠉⠛⢿⡀⢾⡦⢠⣦⠀⣤⡄⢀⣀⠀⢈⠁⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⣤⠀⢀⡄⠀⣈⠀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⠖⠁⠀⠀⠀⠀⠘⣏⠀⠛⠁⠘⠇⠀⠶⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠤⠀⢠⡀⢀⣀⠀⢈⡁⠀⣻⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⡶⠿⠀⢶⠆⢠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⡩⠊⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠂⠀⠀⠄⣠⡀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠈⠁⠀⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠖⠀⠠
⠀⠀⠀⠀⠀⠛⠀⠰⠆⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠄⢠⡄⠀⣀
⠀⠀⠀⠀⠰⠦⠀⣤⠄⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⠀⢀⡀⠀⡀
⠀⠀⠀⠀⢠⡄⠀⣀⠀⣈⡉⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⢀⡁⠀⠈⠀⠀⠁
⠀⠀⠀⠀⣀⠀⠀⠁⠀⠙⠁⠘⠟⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢉⣉⠀⢈⠁⠀⠛⠀⠘⠂
⠀⠀⠀⠀⠉⠀⠘⠃⣴⠟⠋⠙⠣⢰⣶⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠛⠀⠛⠃⠐⠗⠀⠰
⠀⠀⠀⠀⠀⠀⠺⢸⣏⠀⠀⠀⠀⣤⣤⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠇⠐⠷⠀⢰⠄⠀⡀
⠀⠀⠀⠀⠀⠀⣦⠀⢻⣦⣤⣤⠖⣉⡁⢈⡛⠻⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⡿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⠆⢰⡆⠀⣤
⠀⠀⠀⠀⠀⣾⠁⢀⣄⠀⢀⡁⠀⠙⠁⠘⠛⠐⠿⠛⢿⡿⢿⣿⡿⣿⣿⢩⣭⡁⣙⡻⣿⣶⣤⣤⣤⣤⣴⣶⣿⡟⢠⣦⠀⣠⡀⢀⡀
⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠘⠃⠘⠛⠀⠾⠇⢰⣶⠀⣴⡄⣠⣭⢀⣭⡁⢈⣛⠀⠛⠃⠸⠿⠛⢿⠟⢻⣿⠁⣤⠀⢠⡄⠀⣈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠀⠰⠆⠀⣶⠀⣠⡄⢀⣭⠀⣈⡉⠈⡛⠁⠻⠟⠘⠿⠃⢾⡷⢠⣶⠀⣤⡄⢀⣌⠀⢈⠁⠀⠉⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣄⠀⣈⠁⢈⡋⠀⠛⠃⠘⠿⠀⠿⠇⢰⡶⠀⣴⡄⢠⣄⠀⣉⠁⠈⠉⠀⠛⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠘⠋⠐⠿⠂⠲⠖⠠⣶⠀⣤⡄⢀⣌⠀⢈⡁⠀⠉⠀⠘⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠰⡶⠀⣤⡄⢠⣄⠀⣈⠀⠀⠉⠀⠙⠁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⢶⣤⣤⣄⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠖⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡴⠁⠘⡁⣏⡆⠠⢳⠒⡄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠙⠁⠈⠀⠈⠉⣠⠃⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠈⢻⣄⠀⠀⠀⠀⠀⢹⠀⠀⠀⡰⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠻⣦⡀⠀⠀⡇⠀⢠⠁⠀⢠⡞⠁⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡤⠤⠶⠤⠼⠤⢤⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠶⠶⠶⠶⠶⠶⠶⠾⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣒⣒⣒⣊⣉⣉⣉⣉⣉⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⢿⣯⣍⣙⠫⣍⡙⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣬⣓⢮⣙⠻⢭⣓⠿⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢭⣙⠺⣿⣲⠯⣗⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⣈⠓⠦⣭⣛⠦⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢤⣈⠙⠲⠦⣬⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣬⣭⣻⣶⢴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣤⠤⠶⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠁⢀⡸⠀⠀⠀⠀⣠⠚⠉⠉⠈⠲⡀⠀⠀⣀⡀⣀
⠀⠀⢀⡀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠠⠄⠐⠉⠀⠀⢀⡸⠀⢠⠊⠉⣁⠠⢤⡀⠀⠈⠁⠉⣷⠀⠀⠀⠗⠠⡀
⠀⠈⠉⠀⠀⠀⠒⠠⠍⡑⠂⠄⣀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠂⢮⣀⣀⠜⠁⠀⢸⠀⠀⠓⠀⠀⡱⠀⠀⠀⠴⠋⢀⣀⠀⠀⠤⠘⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⢀⠉⠒⠤⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⢀⠀⢀⡀⠀⢣⡀⡀⠠⠐⠁⠀⠤⠄⠀⠀⠉⡸⠀⠀⠐⠁⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡐⠀⠀⢀⢀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠈⠉⠁⠠⢀⣤⡤⠴⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠢⠤⢠⣀⣁⡀⠒⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣒⡒⠢⠭⡂⢄⡀⠀⠀⢠⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠢⡀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠂⠤⠤⠤⠤⣀⠀⢁⣹⡀⡀⠈⡆⠈⠉⠉⠁⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠲⢄⡀⢀⠤⠤⠤⣀⡀⠀⠀⢠⡄⠀⠀⠀⡹⠉⠈⠦⠤⠔⠁⢠⣀⠀⠀⠈⡧⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⢬⠉⠒⠀⠀⠠⢿⠀⠀⠈⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⡦⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣆⠀⢀⡤⠤⠚⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠲⠄⠠⠴⠆⢀⠠⠔⠁⠈⠁







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡴⠦⣤⣄⢀⣠⠤⠀⢠⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠇⢀⠳⡖⠀⣻⣿⣼⠏⠀⣤⠀⠍⠻⢿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⡤⠊⠀⡞⠀⠈⢽⣻⣿⠄⠀⣠⠈⠒⢅⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠘⠄⠀⡄⠀⣠⠷⠒⠒⢶⣿⡇⠐⠀⢇⣠⠇⠀⠄⡃⣿⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⠋⠀⠀⠀⢀⡴⠁⠘⠂⢳⢸⣿⡧⠄⠀⠈⠁⠐⠲⣾⣀⢌⡙⠾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⠐⠀⢀⠔⠉⠀⣔⠀⣤⡿⢾⣿⡇⠀⢘⡷⠒⣤⠖⠀⢁⠀⠙⠚⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠈⡇⠠⣀⡴⠀⠉⢠⡼⠆⠀⡇⣨⣿⡧⠀⣻⠇⠀⢇⠀⢸⠉⠀⣙⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢨⡧⠀⠇⡀⠠⠤⠴⠆⢠⠃⠠⠐⠳⠄⣿⡃⠘⠉⠀⠀⠀⠀⠈⠃⠀⡞⣩⢉⡿⣿⡄
⠀⠀⠀⠀⠀⢀⡞⠀⣆⣀⣁⣄⠀⠠⢤⣟⡀⠐⠓⡄⠻⣿⣧⣀⠀⠟⠀⠣⡀⠀⠁⣀⣠⣻⢸⣷⣿⣷
⠀⠀⠀⠀⠀⢨⠇⡀⢀⡠⠤⠀⠀⠄⠀⠈⠉⠑⠀⠈⣶⣿⣧⡄⠀⠔⠀⠀⣽⠗⢉⠉⣯⣵⡿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⢨⢂⢸⠀⢠⣀⡄⣧⠈⣆⠀⠆⡤⠊⠁⢴⣿⠀⡀⠄⡠⠇⠈⠁⣴⡿⢋⢩⣶⣰⣿⡿⣿⡇
⠀⠀⠀⠀⠀⣸⡿⠀⠓⠀⠇⠀⠈⠀⠾⠚⠦⣤⠆⠁⣼⣿⠀⡄⠀⢶⣿⣶⣶⡁⠑⢘⣆⡿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣽⠈⢆⠀⡸⣀⠀⠃⢀⡴⠃⠂⠴⠂⠀⣴⣿⠷⠋⠀⡏⠀⣦⠈⠿⠻⠿⡋⣥⣴⡿⣻⣿⣿
⠀⠀⠀⠀⠀⢹⣀⢼⣆⡀⠈⣻⣄⢀⠈⠄⠲⣄⠀⢋⣿⣿⠆⢢⠞⠀⢀⣿⠗⠚⣠⣾⠿⣟⢻⣾⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣏⠀⣈⣴⠉⡈⠉⠻⣧⢄⣀⣽⢛⠛⢿⣿⠃⣈⣨⠟⠉⣩⣬⣷⠟⣱⣿⣿⣦⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣷⢳⡄⠙⢠⠇⣀⣀⡿⢤⠄⠁⣉⢴⣿⣿⠞⠉⠁⣠⠂⣠⣿⣶⣾⣿⣿⣿⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢿⣦⢹⡞⣋⡘⢛⡅⢹⣾⡙⢦⣉⣼⣿⣿⠃⣠⠞⢆⣛⣿⡟⢰⣿⣿⣿⣛⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⣮⣽⣩⢛⣣⣤⡼⠋⠁⣨⣿⢿⣿⣿⠾⢇⡚⣿⣿⣿⡇⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣵⣾⣟⣋⣴⡛⠻⣿⣿⣿⣿⣿⠰⣦⣼⣿⣿⣿⣿⣯⣙⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠶⢀⡠⢠⠤⠐⠀⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠘⠇⠀⠈⠛⠒⠚⠡⠶⠤⠤⠤⠄⠀⢈⠀⠀⠀⠱⠆⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⠀⠀⠀⠁⠀⠀⠀⡀⠒⠒⢒⣒⡉⠉⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠣⡂⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣐⡤⣴⣾⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣆⡀⠀⠀⠀⠀⠀⠈⠭⣐⣰⣽⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⢹⠿⡛⠋⠋⢀⠀⢀⣭⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⣶⣴⡤⠀⠀⢸⣻⠅⠀⠀⠠⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣏⣉⣄⠤⢠⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢀⢀⠀⠀⠠⣉⣙⣛⣏⣄⠘⡛⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠁⠐⠀⢱⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠀⠀⠀⠀⠈⠄⠈⠂⠀⡁⠇⢌⣻⣛⣷⠌⢀⣊⠧⡀⠈⠂⠠⡀⠀⠀⢀⣀⡠⢀⡀⠀⠀⠀⠀⣈⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠒⠰⠁⠀⡀⠏⠈⠈⠀⢈⣛⠡⠉⠀⣘⣿⣿⠀⠈⠀⡠⠀⡅⡁⠀⠀⠀⠂⠀⡱⠀⠄⢀⡄⢂⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠂⠀⠦⡂⢁⠀⠄⡀⠀⠀⡷⠦⣀⣀⢬⣿⡯⠡⡀⠀⠁⠀⡈⠣⠀⠀⠀⠠⠄⠁⠀⠀⣘⠈⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠂⢀⠐⠕⡂⠠⠀⢠⡀⠂⠀⡀⠁⣻⡏⠁⣜⢫⢾⣀⢥⢊⡀⢪⠁⠀⠀⠀⠀⠈⠢⠀⠀⢤⢋⣈⣁⠔⠏⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡄⠀⣠⠋⠠⠀⠘⠱⠂⠜⡴⢦⣍⡍⠹⣿⣒⡰⢝⡊⠄⣸⣄⠀⠁⠀⢁⡂⠠⠀⠀⠣⡀⠀⡞⠨⠦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠉⠀⠐⠀⠰⠀⠀⠀⠀⢁⠀⠙⠿⣣⠈⠻⣶⣟⣦⣥⠀⡅⢠⠀⠀⠀⠈⠀⠀⢐⣄⠃⡆⠘⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠘⠻⠇⠀⠀⠀⢀⠀⡆⢀⠄⠁⠈⠁⠠⠭⠿⣿⠏⡉⡴⠃⠀⠀⠀⠄⠊⠙⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠀⠀⠑⠈⠀⠁⠂⠀⠀⠀⠀⢀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣦⣶⣯⣽⣶⣴⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡄⠀⠀⢨⣍⠀⠀⢀⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣾⢿⠉⠁⠈⢙⡿⣧⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣿⠀⠀⢸⡇⠀⠀⣸⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃⢸⡇⠀⠀⢸⡇⠸⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⢹⡇⠀⢸⣷⠀⢀⡿⢻⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⡏⠀⢸⡇⠀⠀⢾⡅⠀⢻⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡟⠈⣷⠀⠘⠏⠀⢸⡇⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟⠀⢠⡟⠀⠀⠀⠈⢿⡀⠀⢿⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⢻⡄⠀⠀⠀⣿⠀⠈⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⢀⣾⠁⢠⡿⠀⠀⠀⠀⠀⠈⢷⡀⠘⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣾⠀⠀⠘⣧⠀⠀⢸⡇⠀⠀⢿⡀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠁⢀⡿⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⢻⡀⠀⣾⠁⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠇⠀⠀⠀⢸⡇⢠⡏⠀⠀⠀⠘⡷⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠺⠷⠶⠶⠶⣶⠶⠶⣶⠶⠶⠶⠾⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡆⢀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣇⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣀⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠂⣠⣾⣿⣾⠃⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⠟⠁⠀⠈⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡀⢾⣿⣿⣿⡿⠋⠀⠀⠀⢀⠄⠡⡀⠀⣴⣿⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣾⣷⡀⠹⣿⠟⠁⠀⠀⠀⠀⢸⠀⠀⣱⣾⡿⠃⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣴⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠘⣤⣾⣿⠟⠣⡀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⡀⠀⠀⠀⠀⣠⣾⣿⣟⠁⠀⠀⠈⠲⣄⡀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⢀⡴⠊⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⢿⣶⣦⣄⡻⣿⠋⠘⢦⡀⠀⠀⠀⠀⠙⠓⠶⡄⠀⠀⠙⢦⡀
⠀⠀⣴⠋⠀⠀⠀⠀⣿⣏⣈⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣠⣿⣿⣭⣿⣦⣄⠀⠀⠙⠢⣄⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠈⠳⣄⠀⠀⠀⠘⠿⠿⠿⠟⠁⠀⠀⠀⢀⡠⠖⠋⠉⠉⠃⠉⠻⠿⠛⠿⠶⣄⡀⠀⠉⠓⠒⠠⠞⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⠶⢤⣄⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⢠⠞⠉⠀⢈⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣇⠀⠀⠈⢳⡄⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣿⠀⠀⣠⣾⠋⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠹⢷⣄⣀⣠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡶⠿⠛⠁⠀⡏⢹⢹⠙⠋⡏⡇⣿⢸⢹⠉⡏⡏⠏⢹⠀⠀⠉⢉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠃⠘⠸⠀⠀⠇⠇⠿⠸⠸⠀⠅⠇⠂⠘⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣠⠄⠛⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸⡇⠀⠀⠀⠀⣠⡇⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣹⠆⠀⠀⠰⣏⠀⠰⡏
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡁⠀⠀⠀⠀⠙⣦⠀⠉⠢
⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⠋⠀⠀⠀⠀⠀⣠⢸⡆⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣦⠄
⠀⠀⠀⠀⠀⠠⠁⠀⠀⢸⠀⠀⠀⢀⣠⠞⠁⠀⠳⢤⣄⡀⠀⠀⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣀⣹⠦⢤⣄⡀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠙⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡅⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠻⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣠⢿⣀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠤⠤⠤⠶⠶⠚⠛⠉⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⢸⡆
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠄⠘⢀⣾⢁
⠀⠀⠀⠀⠀⠠⡂⢻⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⡱⢃⠞⣠⠂
⠀⠀⠀⠀⢀⡩⠒⣁⠜⠛⢳⠶⣦⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⠴⢶⠞⣻⠟⡱⢣⠎⡔⠁⠴⠁⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠐⠁⠘⠁⠈⠀⠋⠀⠉⠀⠋⠐⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠛⠓⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠏⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠈
⠀⢀⡟⠁⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣊⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠒⠈⠉⠉⠉⠙⠒⠢⣄
⢠⠟⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠈⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠈⢳⡆⠀⠀⠐⠒⠉⠉⠉⠀⠀⠀⠀⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠄⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏
⠀⠀⠉⠒⢲⡶⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢄⡀⠀⠀⠉⠉⠒⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠐⠻⣍⠉
⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠄⠀⠀⢀⡴⠒⠲⣌⠢⠀⠀⠑⠲⢄⡀⠀⠀⠀⠉⠁⠀⠀⠀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢄⣀⡼⠀⠀⠀⠀⠀⢀⣈⠑⠄⠀⠀⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢯⠈⣹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣛⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⣿⣿⣶⣿⣷⣿⣿⣿⣿⣶⣶⣯⣭⣽⣻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡟⢻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣟⣿⣿⣿⡿⠟⠿⣿⡿⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣯⡿⠿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⠷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣴⣿⣿⣶⢿⣿⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢁⣰⣟⢩⣭⢿⣷⠋⣶⣾⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⠞⣁⣴⣿⡿⠛⡯⢚⣿⣿⣿⣟⣻⣻⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣰⣿⣿⣿⣿⣿⣶⣿⡿⣬⡟⠛⠛⠟⠛⠦⠐
⠀⠀⠀⡀⠀⠻⢱⠜⣿⣿⣿⣿⣿⡿⢿⠋⣽⠛⠓⠃
⠀⠀⠀⠸⠷⢸⣿⣼⣿⣿⣿⣿⡿⢶⠖⢛⡇
⠀⠀⠀⠀⠀⠹⠿⣿⡿⢿⡛⢿⡷⣬⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠈⠁⠀⠀⠈⢻⣷⡠⢀⣀⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠤⠄⠀⠀⠚⠛⠚⠻⠿⠶⠿⠿⠿⠷⠒⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠰⣶⣶⠀⢰⣶⠀⠂⠀⠰⣶⣶⣶⣶⡆⢤⡐⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣖⣺⣿⣇⠀⠀⠀⢻⣿⣿⣿⣿⣤⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣻⠿⢛⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣭⣭⣿⣿⣿⣿⣿⣿⠿⠿⠟⢛⣛⣿⣿⣿⣿⡄⢠⣤⣾⣿⣿⣿⣿⣿⣧⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⢾⢿⣿⡿⡻⢿⣝⣿⣿⣿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡄⠁⢫⣿⣿⣦⣤⣿⣞⣿⣿⣷⡈⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⢡⣿⣿⣿⣷⣶⣟⣻⡾⠛⣻⣿⣿⣾⠿⣿⣧⢹⣿⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢻⣷⣿⣿⠋⠉⢻⣿⣿⣯⣴⢟⣫⣼⠿⢿⣧⣿⣿⡆⢻⣿⣿⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⣠⣾⣿⡿⠃⠀⠀⣠⣿⣿⣿⣿⣿⣯⡵⠋⣸⣿⣿⣿⣿⣾⣿⡿⢋⡾⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷⣾⣿⣿⠟⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣽⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣄⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡼⣿⣿⣧⣄⣰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣍⠻⢿⣿⣿⣿⡏⢉⣉⣋⣩⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣫⣶⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⡛⠿⣿⣯⢷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣹⣿⠿⣋⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠈⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣥⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣛⣷⣿⣿⣿⡿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⣶⣿⣿⣿⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡀⢻⣿⠋⠛⠁⠹⣿⣿⣿⣿⡟⢫⡛⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣕⣺⣿⣇⠀⠀⠀⢻⣿⣿⣿⣿⣤⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⠿⡛⣛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣭⣍⣿⣿⣿⣿⣿⣿⡿⠿⠟⢛⣛⣿⣿⣿⣿⡄⢠⣤⣾⣿⣿⣿⣿⣿⣧⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⢶⢿⣿⡿⠻⣿⣝⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡄⠠⢫⣽⣿⣦⣤⣿⣞⣿⣿⣷⡘⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠟⠁⠿⠿⠿⠷⠶⠟⠻⠾⠛⠻⠿⠿⠞⠻⠿⠇⠹⠿⠆⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠛⠙⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⢹⠖⠛⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⣝⣴⠶⠀⠐⠻⢷⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠺⠀⠀⠀⢈⡔⢢⠀⡴⠋⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⡤⠤⠼⣦⢸⠀⣧⡀⣨⠷⠒⠒⠀⠐⠂⠄⠀⠀⠠⠒⠋⠉⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠊⠉⢹⠀⠀⠀⠈⠧⠤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⡳⢤⣀⡐⢄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠁⠐⠒⠀⠐⠀⠀⠸⠤⠄⠀⠠⢏⠉⠉⠉⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⢶⣚⣠⣄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅
⠀⠀⠀⠉⠀⠀⠀⠂⠈⠁⢈⢧⡀⠀⠀⠉⠀⠈⠙⠒⠦⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡶⠃⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠓⡤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⣠⠞⠳⢄⣀⠀⣀⣠⠴⠋⠳⣄⠀⠀⠀⣀⡴⠋