Laundry Emojis & Symbols

⠀⠀⠀⠀⠈⡝⠉⠉⠉⠉⢫⡞⠻⣬⡟⢻⣽⣟⣛⣛⣛⣛⣻⡏⠉⡇
⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠮⠷⠾⠩⠷⠾⠹⠷⠷⠷⠾⠷⠷⠧⠤⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠠⠄⠤⠤⣤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠉⠉⠓⠢⣄⠀⠀⠀⡔⠉⠁⢀⣀⡴⠊⠨⠰⠍⢶⠂⣱⣈⣓⣤
⠀⠀⠀⠀⢸⠀⠀⠀⣠⠊⠀⡠⠒⣉⣁⣀⣁⠒⢄⠀⠙⣄⠐⠁⠀⠀⠀⣴⣭⣥⣤⣤⣬⣥⣴⡀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠀⠀⣰⠃⢠⠊⡴⠋⠁⠀⠀⠈⠙⢦⠑⡄⠘⣆⡇⠀⠀⢰⠦⠤⠤⠤⠤⠶⠤⠴⡇⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡏⠀⡄⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢹⠆⠀⢀⠷⠶⠶⠶⠶⠶⠶⠶⠶⢶⠀⠀⢠⠁
⠀⠀⠀⠀⢸⠀⠀⣇⠀⢁⢸⡀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣸⢰⠄⠼⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⢄⣈
⠀⠀⠀⠀⢸⠀⠀⠹⡄⠈⢄⠙⢤⣀⠀⠀⣀⡤⠋⡰⠁⢠⠏⠘⠐⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⢠
⠀⠀⠀⠀⢸⠀⠀⠀⠙⢦⡀⠑⠢⠌⣉⣉⠡⠔⠊⢀⡴⠋⠀⠀⡀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠲⠤⣤⣀⣀⣤⠤⠖⠋⠀⠀⠀⠀⡇⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠃⠈⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⢁
⠀⠀⠀⠀⠈⠢⡄⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣀⢠⠶⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣸



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠉⠁⣤⠔⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠈⠁⠢⢀
⠀⠀⠀⠀⠀⢀⡠⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠤⠤⡤⠔⠈⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠒⠀⣈⡉⠍⠁⠀⢢⠀⠀⢀⠄⠐⠀⠀⢖⠒⠢⡴⠁⠀⠀⣀⠄⠐⠉⠀⠀⠀⠀⠀⠀⠁⠢⢽⡀
⠀⠀⠀⠀⠀⠋⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⢃⠔⢁⣠⠄⠀⠀⠀⠉⢹⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠃⠀⠀⠀⠒⠂⠀⠀⠠⢄⡀⠀⢸⣷⠟⠉⠀⠀⠀⠀⠠⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⠐⠒⠂⠠⠝⣂⣸⣧⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠈⠒⠒⡒⠢⣄⡉⠐⠢⢔⣋⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⣇⠀⠙⢦⣉⠀⠀⡸⡏⠉⠙⡏⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⡏⠀⠉⢻⣏⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠸⣷⣤⡀⠈⢉⠿⢀⠈⠀⠀⢳⣶⣽⣷⣦⣄⣀⣑⡒⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠛⠉⠀⠀⣠⡟⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⠁⠸⠢⢄⣀⠀⠀⠇⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠲⣤⠀⠀⠀⠀⣇⣀⡤⠤⠒⢛⣿⠁⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠉⠁⠀⣀⣀⠀⠠⣄⡀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠈⣇⠀⠀⠀⡟⠀⡐⠀⡆⠈⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢃⠸⠀⠀⠀⠀⠀⢇⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⢹⠀⠀⠀⡇⢀⠁⢠⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⠀⢗⠒⠂⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⢰⠃⡌⠀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠂⠀⡧⢀⢀⠴⡃⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⠐⠁⢀⡁⠐⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡀⠀⠀⠀⠀⠣⠀⠆⠀⠀⠀⠄⠀⠄⠀⠀⠀⠇⠄⠂⠀⠀⠀⢀⡀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡅⠑⡀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢀⠊⢉⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢡⠀⢁⠀⠀⠁⠀⢸⠀⠀⠀⠉⠉⠁⠀⠀⢠⠀⠀⠀⠀⠘⠀⡌⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⡀⠈⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⠂⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠒⠁⠀⠘⡄⢀⠆⠀⠀⠀⠀⠀⠀⠀⢆⣀⠜⠀⠀⠑⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠑⠒⠃⠀⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⢩⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠉⠁⠀⣀⣠⡤⢼⣦⣄⡀⠈⠉⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠛⠉⠉⠀⣀⣠⡤⠶⠚⠛⠉⠁⠀⣼⠀⠀⠉⠓⠶⣤⣀⠀⠈⠉⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠚⠋⠉⠁⠀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣠⣤⣤⣀⣀⡈⠉⠓⠦⣤⣀⠀⠉⠛⠳⣤⣀
⠀⢀⣠⡤⠖⠛⠉⠀⠀⣀⣠⣤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠦⣬⣷⠟⠁⠀⠀⠈⠉⠻⣄⡀⠀⠀⠈⠙⠲⢤⡀⠀⠉⠙⣳⢦
⠀⣿⠣⢤⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠶⠤⣤⣤⠽⠿⠚⠋⢁⣼
⠀⢹⣦⣀⡈⠙⠲⢦⣄⡴⠒⠛⠷⠶⠛⠉⠉⠳⣄⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠒⣶⠊⠁⠀⢀⣠⡴⠞⠉⣿
⠀⠈⣧⠈⠉⠑⠦⣄⡈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠈⠢⣄⣤⡤⠤⠶⠦⣤⣤⣤⣀⡀⠙⠳⣦⡤⠔⠚⠉⠉⢧⠀⠀⠀⢸⡦⠖⠛⠉⠁⠀⠀⠀⡏
⠀⠀⢹⡄⠀⠀⠀⢠⡍⡗⠦⣄⡉⠛⠶⣤⡀⠀⠀⠀⠈⠓⠲⠶⠒⠀⠀⠀⢀⡩⠿⠚⠉⠁⠀⠀⠀⠀⠀⢸⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣧⠀⠀⠀⠘⣿⡅⠀⠈⠙⠲⢄⡈⠙⠛⠶⣦⣄⡀⠀⠀⠀⣀⡤⠶⠋⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⣷⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢻⡀⠀⠀⠀⠻⣕⢤⡀⠀⠀⠀⠙⠲⢦⣄⣀⠉⠛⢳⡖⠋⢁⣠⠴⠖⢻⡇⠀⠀⠀⠀⡄⠀⠀⠀⢸⡇⠀⣸⠀⢿⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠛⠾⢷⣶⣤⣀⣠⠟⠀⠈⠙⠲⢼⡗⠚⠉⠀⠀⠀⢸⠁⠀⠀⠀⡼⠀⡀⠀⠀⢸⡇⠀⠏⠀⢸⡆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡇⢠⠃⠀⠀⢸⡇⠀⠀⣀⣸⡇⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⠀⡌⠀⠀⠀⠸⡷⠞⠋⠉⠁⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠀⠀⣀⣴⠇⠀⠀⠀⢀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⣠⠶⠶⠖⠚⠋⠁⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧⠶⠋⠁⠀⠀⢀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣤⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⢸⡏⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣼⣧⣤⠶⠚⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁
⠀⠀⠀⢰⣰⠀⠀⢸⢾⡇⠀⠐⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠐⠀⠐⠀⠂⡰⠂
⠀⠀⠀⢸⢼⡀⠀⢸⢻⡄⠀⢸⣿⡿⣿⣿⣿⠇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⣳⣄⣀⣠⣜
⠀⠀⠀⡼⠘⡇⠀⠸⠻⠇⠀⢸⣿⣷⢾⣿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠟⠘⠇⠀⠀⠀⠀⠀⢸⣿⣿⣾⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢿⣋⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⠾⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠉⠓⢦⣀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠘⠓⠒⣄⣀⡀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠘⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠂⠒⠛⠓⠀⢰⣿⣿⢸⡀⠀⠀⠀⠀⢀⣶⠆⠀⣀⠁⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡄⠀⠘⣿⣿⣿⠁⢾⣦⠀⣀⠀⠁⠀⠸⠿⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⡀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⠀⠀⠁⠀⠻⠷⠀⠀⠀⡀⠀⠀⠀⢻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣀⣠⣤⣴⡄⠀⠀⠀⢈⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣠⠖⢉⣉⣉⠑⢦⡀⠀⠀⠀⠀⠘⠿⠿⠿⠀⠀⢀⣼⣿⣡⣤⣤⣿⣷⠀⠀⢠⠎⠈⠛⠁⠱⡄
⠀⠀⠀⠀⣼⢁⣾⡿⣿⣿⣿⠂⢱⡀⠀⠀⠀⠀⣀⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠟⠉⠁⠐⠃⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⡇⠐⣯⠀⠀⠀⢹⠀⢸⡇⠀⠀⠀⢀⣿⡟⠶⣄
⠀⠀⠀⠀⠘⣄⠙⠷⣶⡶⠟⢀⡞⠀⠀⠀⠀⢸⣿⣿⣦⣬⣷⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠈⠑⠀⠤⠀⠒⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠆⠀⠸⢿⣿⣿⣿⠟⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣾⠉⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣀⡈⠉⠉⢳
⠀⠀⠀⠀⠀⠀⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡖⢲⢰⠊⡇⢰⠉⡇⠀⠀⢰⠧⠈⠁⠘⠦⠀⢰⡍⠉⠨⢷⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠙⠁⠈⠉⠁⠈⠈⠁⠀⠀⠘⢤⣆⣐⡶⠃⠀⠳⣤⡀⣠⡞⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⠤⠤⠭⠭⠭⠭⠭⠭⠭⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠿⠷⠶⠶⠾⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠉⠉⠉⠓⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⣀⣠⣤⠤⢤⠤⢤⣀⡀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠴⡾⠉⡜⠁⣰⠃⢠⠎⠀⡜⠓⣦⡀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡴⢣⠞⣠⠎⢀⡴⠁⣰⠃⢀⡞⠀⣰⠃⢸⣆⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣼⡵⠃⡴⠃⢀⠞⠀⡴⠁⢀⠎⠀⡰⠃⣰⠋⢸⣆⠀⠀⠘⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠟⢀⡞⠁⣠⠋⢀⡞⠀⢠⠎⠀⡴⠁⣰⠃⣰⠋⣼⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⢠⠞⠀⡰⠃⢠⠎⠀⣠⠋⢀⡞⠁⡰⠃⡰⡥⠚⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠋⠀⡜⠁⡠⠃⠀⡴⠁⢠⠎⢀⡼⠁⡼⠑⣇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠀⢹⣀⠞⠀⡴⠁⢀⡞⠀⣠⠋⢠⡞⢀⡜⠀⠼⠉⡿⠲⡆⠀⢠⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⣏⢀⡜⠁⢠⠎⠀⡰⠁⢠⠏⢀⡞⠀⡼⢀⠖⠀⡶⠀⠀⠸⠁⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⢿⡀⣰⠃⠀⡼⠁⣰⠋⢀⡎⠀⠜⢁⡟⢠⡿⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠙⠧⣄⡞⠀⡴⠁⢀⠎⠀⠌⢀⣏⠴⠋⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠉⠚⠓⠒⠛⠒⠚⠋⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠃

⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠱⠤⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢈⠤⠂⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠚⠀⠀⠀⠁⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠒
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀
⠀⠀⠀⢠⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠄⠃⠀⠀⠀⠀⠐⡀⠀⠁
⠀⠀⢀⠆⠀⠀⠀⠀⠀⠣⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢡
⠁⠀⠉⠒⠒⠶⠤⠤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠈⠁⡀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠠⠐⠂
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢸⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠁⠀⠈⢆
⢘⠃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⠀⢀⣂⣰⠀⠄⠀⡂⠀⢠⠀⠀⠒⠊⠀⠀⡀⠀⠀⡀⠀⡆⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⢂⠈⡁⠀⠠⢀⣤⠤⠤⠃⠉⠀⠀⠀⠠⠀⠴⠀⠅⢤⡄⠴⠶⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠈⠀⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⠀⢀⠴⣠⣤
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⠔⢊⠉⠑⣧⠞⢁⡸
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠴⠒⠋⠹⢓⠎⠁⠀⠉⠂⠀⠀⠀⠙⠀⠋⠘⠉⠿⣧⠤⠚⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠂⠉⢀⡀⠀⠀⠠⠄⢀⠀⠐⠐⠀⠠⠀⠀⠀⠀⠐⠂⠠
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⣀⠀⠀⠀⠉⠀⠂⠀⠀⡄⠀⠀⠀⠀⡁⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⢀⡼⠢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡔⠀⠀⠀⠁⠈⠄⠀⠀⠀⡀⠀⠀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠻⠄⠀⠀⠃⠀⠀⠀⢀⠀⠃⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠈⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠄⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⡀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠
⠀⠉⠭⠩⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠠⠇⠀⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠉⠀⠀⠀⢀⣈⡙⠛⢄⡀⠀⠉⠑⠒⠲⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⣀⣈⡉⠒⠈⢉⡢⠖⠊⢩⢳⠀⢹⠙⠤⢄
⠀⠀⠀⠀⠀⠀⠀⡼⠶⠶⠶⠒⠒⠒⠒⢚⣧⣲⣤⣀⣀⣀⣀⣉⣉⣡⣖⣉⡉⠉⠛⠒⠚⠒⠒⠶⢇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣏⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣸⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠯⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡆⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡸⠓⠒⠛⠓⠒⠒⠒⠒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠳⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⠧⠤⠤⠤⠤⡤⠤⠤⠤⡤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠵⣄⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢙⠶⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠶⠒⠒⠒⠒⠒⠶⠒⠒⠲⠖⠶⠶⢶⠖⠒⠒⠒⠒⠒⠒⠒⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠋⠀⠘⠃⠂⠘⠛⠀⠛⠁⠀⠛⠀⠀⠀⠀⠘⢣⡤⠿⡼⠉⠿⣂⠉⠉⠉⣑⠳⠋⢢⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⢄⣠⠤⠤⠤⠤⠤⠤⠤⣄⠤⠤⠤⠤⠤⡴⠋⠀⠀⠑⠢⣄⡀⣩⠶⣍⢁⣀⠤⠜⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠈⠹⡀⡏⠁⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⣐⣂⣴⣐⣀⣐⣰⣖⣀⣀⣐⣦⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠺⠭⠭⠭⠭⠭⠭⠭⠭⠽⠯⠭⠭⣷⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⣟⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡞⠒⠒⠒⠂⠀⠶⠶⠶⠶⠶⢾⣦⡷⠶⠶⠶⠶⠶⠆⠐⠒⠒⠒⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⢺⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⢺⡭⣷⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡤⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⠀⠀⠐⠒⠒⠒⠒⢺⣷⡟⠒⠒⠒⠒⠒⠀⠀⠈⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⢰⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢰⡀
⠀⠈⣤⣤⣤⣤⣤⣬⣭⡍⠀⠀⠀⢸⣿⡆⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠁⠠
⠀⠀⣤⣉⡉⡉⢉⣩⣍⣀⠀⠀⠀⣼⣿⡟
⠀⠀⠉⠉⠉⠙⠉⠉⠉⠽⠆⠰⣿⠿⠿⠃⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠐⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⢸⡟⠀⢠⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠈⠉⠛⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⢰⠛⣿⣿⣷⡀
⠀⠘⠁⠀⠟⠀⣀⢀⡄⠀⠀⠀⠸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣾⣿⣿⠁
⠀⠀⠀⢀⠀⠀⠻⣤⣧⠀⠀⣀⡀⠙⠻⠿⢟⠁⠀⠀⠀⠀⠀⠀⠰⠐⠐⠇⠈⠀⠀⠉⠀⠀⠀⠔⠂⠈⢻⣿⡯
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⣤⡀
⠀⠀⣠⣤⣠⣬⣌⢁⠀⠀⢴⠀⠀⠆⠀⣠⣴⣦⣤⣄⣠⣀⣀⣄⣀⣤⣴⣶⣿⣿⣷⡄⠀⠀⢸⣿⣿⣧⡀⠀⣀⣠⣤
⠀⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠘⣿⣿⣿⠇⢶⣿⣿⠿
⠀⠀⠀⠉⠉⠁⠀⠀⠀⣤⣤⡄⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⢀⠀⠈⠉⠁⠀⢈⣉⠉
⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠂⠈⠙⠿⠦
⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣦⣤⠀⠁⠀⣀⣀⣀⣀⣀⠀⠀⢰⣿⣿⣽⡟⢀⣠⣦
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠀⠀⠀⠉⠙⠉⠉⠉⠀⠀⠸⠟⠟⠋⠁⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠘⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠃⢰⣷⣤⣤⣠⣿⠃⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠒⠋⢉⣉⡀⠀⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠒⠛⠉⠉⠀⠀⠠⣴⣶⣾⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⠿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣀⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠒⠋⠉⠉⠉⠁⠀⠒⣂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠉⠛⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣋⡤⠔⠒⠒⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢀⡞⠀⠀⢀⣤⣶⣶⣤⣀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⣿⣿⣿⡿⠛⣁⠥⠤⠤⠴⠶⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡼⠀⠀⣴⡿⢋⣴⣾⣿⣿⣧⡀⠀⠀⢿⣷⠀⠀⠀⠀⣿⣿⡏⣰⠋⠀⢀⣀⠤⠴⠒⠒⠒⠚⠛⠛⠒⠒⠲⢶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢢⡇⠀⣰⡟⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⡇⠀⠀⠀⣿⣿⢀⠇⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠤⣀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⢸⠀⠀⣿⠃⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣿⣷⠀⠀⠀⣿⣿⢈⡞⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⢸⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡇⢸⠀⣀⣤⣴⣶⣿⣿⡿⠿⠿⠿⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢸⢸⠀⠀⣿⡄⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⡿⠀⠀⠀⣿⣷⡌⢻⣿⣟⣯⣥⣶⡶⡶⠷⠿⠿⣆⠀⠀⠀⣾⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠜⡆⠀⢹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠀⠀⢰⣿⡇⠀⠀⠀⣿⣿⡇⢸⣿⢛⣭⡤⣶⡖⣶⡇⢾⣿⠰⠀⠀⢠⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢳⠀⠀⠻⣷⣈⠻⣿⣿⣿⡿⠁⠀⠀⣾⣿⠀⠀⠀⠀⣿⣿⣿⠈⣿⡟⣛⡁⣭⡄⣶⡆⣶⣖⢳⣶⣶⣾⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢧⠀⠀⠈⠻⠿⠾⠿⠋⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⣿⣿⣿⡄⢿⡟⣻⣅⣭⣤⢴⣆⢰⡆⢰⡦⣼⣿⣽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⢻⣋⢨⣥⢨⣦⢰⡦⢰⡆⣼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠈⣿⣿⣿⢈⣭⢰⣧⢰⣦⣼⡧⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⣿⣿⣿⣉⣿⢨⣯⣸⣯⣼⡇⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⡇⢸⣿⣿⣏⣿⣭⣯⣼⣯⣽⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⡄⡇⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⡉⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⣀⣀⡀⡀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⣿⣿⣿⣿⡳⡑⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠀⢿⣿⣿⣷⠹⡜⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⣇⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡎⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⠇⠛⠀⠂⠁⠀⡀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢜⢿⣿⣿⣿⣿⡿⢻⡗⠋⠀⢀⢀⡀⠀⠀⠀⡀⠠⠀⠀⠀⢀⡰⢄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠛⠛⢿⡈⠙⠓⠈⠀⠀⠀⠭⠤⠤⠿⠶⠶⠒⢡⢖⣞⡋⠁
⠀⢤⣤⣄⣠⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠁⠀⠀⠀⠀⠐⠒⠂⠐⠒⠒⠂⡀⠉⠀⠈
⠀⠀⠉⠛⠛⠿⠿⢿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠐⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣤⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡉⢩⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣥⣄⣀⣀⣀⣁⣀⣠⣤⣤⣶⣾⣿⣿⣿⣍⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡅⡎⠉⠉⠉⠉⠉⠉⠉⠉⡆⣴⠞⠻⣦⡀⣴⠞⠻⣦⡀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣀⣀⣼⠃⣷⣀⣀⣼⠃⣿⠛⣿⢛⡟⢻⡛⣻⡛⣻⢛⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡧⠻⠳⠟⠛⠛⠛⠺⠒⠛⠣⠬⠛⠛⠡⠠⠬⠟⠛⠥⠠⠿⠛⠛⠛⠛⠛⠛⠻⠛⠿⠛⠛⠰⠀⢾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠴⠶⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣀⡤⠤⠒⠒⠒⠒⠤⢤⣀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⡴⠋⠁⣀⣤⣤⣤⢤⣤⣤⣀⠈⠙⢦⡀⠀⠀⠹⣦⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠃⠀⠀⡰⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠙⢆⠀⠀⠘⣧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡏⠀⠀⣰⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⣆⠀⠀⢹⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠐⡅⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢨⠂⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⡀⠀⠀⣇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣸⠀⠀⢀⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣧⠀⠀⠘⣄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⠃⠀⠀⣼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣆⠀⠀⠈⢦⡀⠈⠳⢦⣄⣀⣀⣀⣀⣠⡴⠞⠁⢀⡴⠁⠀⠀⣰⠏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠢⣄⣀⠈⠉⠉⠉⠉⠁⣀⣠⠔⠋⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠈⠉⠙⠒⠒⠋⠉⠁⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣤⣤⣤⣠⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠣⢤⡤⠤⠤⠤⠤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⠤⠤⠤⠤⢤⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠁⠈⠉⠁⠈⠉⣄⡤⣀⡄⠈⣠⢌⡉⠉⣉⠋⡋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⡀⢸⠀⠀⠀⢊⡧⣷⠇⠈⠷⠾⠟⠀⠳⠶⠿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠉⠉⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠂⠒⢒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣆⢐⡀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⣴⠶⠟⠛⠛⠷⣦⣑⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠄⡀⠹⣀⡀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠄⢏⠈⢰⡇⠀⠀⠀⠀⠀⠀⠁⡈⣆⢸⢢⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠘⠓⠀⠀⠀⡤⣾⠿⠭⣡⠀⠎⠤⠵⠾⢇⣀⠀⠀⣀⣀⡴⠊⠀⢸⠄⠀⠀⢼⠆
⠀⠀⠀⠀⠀⡠⠀⠴⠴⠵⣤⣤⣦⡺⠿⠓⠶⠾⠿⠼⠃⣤⣀⡙⠊⣩⠍⢱⠒⠨⠆⠰⠇⠰⠠⡄⣀⣸⢠⠄⢀⢸⡀
⠀⠀⠀⠀⠈⣀⠴⠒⠒⠀⠀⠉⠛⠛⠓⠒⠉⢉⣿⣿⡡⠤⠤⠭⠥⠥⣶⡌⠫⣥⠒⠒⠺⣯⣐⣧⠚⠁⠊⠒⢺⡞⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠠⠁⡤⣤⢤⣤⣠⣀⣂⣀⡂⠀⢂⣾⠛⣛⣋⠀⢛⠛⡃⠀⡻⡇⠈⠈⢡⡶⢾⡇⢾⠶⠞⠋⢂⢴⠟⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠠⡀⣹⠈⢛⠛⠚⠛⠘⠋⠛⠛⠋⠘⠛⠛⠀⠛⠒⠃⠀⠛⢓⣦⡔⣸⣁⣀⡐⠽⣤⣤⡶⠗⠁⠀⢀⠤⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠶⠆⠳⠶⠖⠘⠿⠾⠘⠟⠒⠂⠚⠛⠟⠀⠛⠛⠛⢀⡇⢈⠉⠉⠓⠋⠁⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠻⠶⠆⠿⠿⠂⠷⠶⠆⠷⠽⠷⠸⠿⠟⠘⠿⠗⣡⣾⣶⣶⣶⣶⣶⢶⠷⠷⠶⠖⢶⢶⣶⣶⣇⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠓⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⠈⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⠛⠛⠛⠿⠶⠶⠶⣦⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⣉⣠⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀⣉⣉⡉⠛⠛⠛⠷⠶⢶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⢀⣤⡶⠟⠛⣩⣴⠟⠿⣿⡟⠛⠩⣉⠉⠉⠛⠛⠻⠟⠛⢻⣧⣭⣭⣿⡿⣿⣿⡿⣷⣶⣶⣤⣤⣄⣩⣭⣟⢿⡄
⠀⠀⠀⠀⣼⠏⠀⢴⣿⣏⣀⣙⣂⡈⠻⣶⣶⣧⡶⠶⠾⠛⠛⠷⢶⣌⣿⡄⢰⠈⢢⣀⠹⡻⣿⣍⣽⣿⡛⠋⠉⢤⡙⢿⣇
⠀⠀⠀⠀⠹⣧⣀⣀⡀⠉⠉⠉⠛⠛⠻⠶⠿⣿⣶⣶⣔⣀⣒⣒⣈⣉⣻⣿⣼⠶⣦⣄⣤⡶⠿⠿⢭⡉⠙⢦⡀⠈⠁⠀⣿
⠀⠀⠀⠀⠀⠈⠹⣿⠛⠛⠻⠶⠶⢶⣤⣤⣤⣀⣀⣀⠈⠉⠙⠛⠛⠿⠿⢿⣶⣦⣤⣽⡿⢿⡷⣦⡀⠉⠀⠀⠀⢀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠷⠶⠶⣦⣤⣤⣄⣀⣀⣀⣠⣼⣷⠘⣷⠀⠹⠄⠀⢸⡇⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⢻⡆⠀⢰⠦⡄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠀⣿⠀⢿⡄⠀⠀⡆⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣧⣨⡇⠀⢸⡀⡇⠀⢸⠉⣿⠀⢸⠛⡶⠀⣠⣤⡄⠀⠀⠀⢰⣿⡇⢸⡇⠀⠀⠃⠀⠀⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢰⡶⣦⠀⢨⣭⡁⠀⠘⠶⠿⠀⢸⣤⣇⠀⣿⠀⡇⠀⠀⠀⠸⢾⡇⣿⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣇⣹⠀⢸⡇⣿⠀⢸⡏⢻⠀⢰⠶⣤⠀⢙⡛⠃⠀⠀⠀⢀⣸⣧⢻⣿⠀⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣬⢭⡄⠈⣛⡛⠀⠸⠧⣾⠀⢸⡀⣿⠀⣿⠉⡇⠀⠀⠀⢸⣯⣿⢸⡏⠀⣧⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢸⡀⡇⠀⡏⢹⠀⢰⡶⣦⠀⢀⣉⣉⠀⠻⠶⠇⠀⠀⠀⠘⢻⡇⣸⡇⠀⢸⡆⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⠓⠇⠀⢿⣼⠀⢸⣇⣻⠀⢸⡇⢹⠀⢰⠶⣦⠀⠀⠀⢰⡟⣿⣿⣄⣤⣬⣇⣠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⠷⠼⠂⢸⣀⡏⠀⠀⠀⠸⠷⢋⣼⠟⠋⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⢶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠶⠶⣦⣤⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄⡠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣃⡀⠆⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠦⠀⠀⠀⠶⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡀⠀⠀⠛⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣛⣢⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠛⢣⣄⠁⠀⠀⡀⣀⠻⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀
⠀⠀⠀⠀⠀⠀⡶⠊⠀⣤⣀⡀⠁⠉⠴⠆⠉⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠤⡈
⠀⠀⠀⠀⠀⢰⣷⠒⠒⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⠒⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠖⠲⣶⡆⠀⠀⠦⠄⠀⠤
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡟⠁⠀⣄⠀⠀⠉⠀⠀⠀⠀⠃⠀⡀⠈
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⣤⣶⣦⣄⡀⠀⠀⠀⣿⠀⠀⠀⢀⣤⣶⣶⣦⣤⡀⠀⠀⠀⡇⠐⠀⠸⠀⠀⠀⠀⠀⠀⢀⡀⣀⣅⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣇⠀⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣿⠀⠀⢰⣿⡿⠿⣿⣿⣿⣿⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢿⣮⣿⣿⣿⡿⠅⠀⡀⠀⣿⠀⠀⠀⠻⣄⣾⣿⣿⣿⡿⠁⡀⠀⡃⠐⢂⠀⠀⠘⠁⣀⡦⠀⠀⠀⠀⣀⣀⣼⡖
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⠀⠉⠉⠉⣀⡠⠞⠁⠀⣿⠀⠀⠀⠀⠈⠙⠛⠛⠉⣠⠔⠁⠀⣧⠀⣀⣤⣀⠀⠀⠀⡀⢀⣀⣤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠐⠶
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠿⠟⠛⠀⠀⠤⠤⠄⠀⠤⠤⠤⠤⠤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⠄⠀⠀⠀⠀⡤⠀⠀⠀⠄⠀⢠⠀⠀⡄⠀⠀⠄⣄⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⢤⣼⠀⢴⠋⠌⠠⠀⠀⢠⣤⢠⠄⠸⠁⠀⠈⠀⠀⠅⠀⠀⠀⠈⡄⠀⠠⠀⠹⠆⠀⠈⠂⠢⠉⠿⠆



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣽⣯⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣷⣷⣶⣿⣿⣷⣶⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⠁⠀⠀⠀⠀⠈⠉⠉⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣒⣀⡢⢀⠀⠀⠀⠀⢘⡟⠀⠀⠀⢀⣤⣤⡭⣛⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠗⣐⣻⠿⣗⠀⠀⠀⢸⡇⠀⠀⠀⠈⣿⢿⠷⠶⠶⢿⣿⣿⣿⠇⠀⠀⠀⣠⣤⣄⣸⡆
⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⡀⠀⠈⢿⣿⠟⠉⠉⠈⠙⢃⡄⠀⠐⣾⠁⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠈⣛⡟⠀⠀⣰⣿⣿⣟⣋⣿⠁
⠀⠀⠀⠠⡀⣾⡿⢛⣻⣿⣿⡀⠀⠀⢱⣀⠀⠀⠀⠀⠀⢻⡆⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠃⠀⢠⣇⢀⡀⢹⡟⠁
⠀⠀⠀⠀⠙⢿⣷⠋⠀⠐⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⢸⠉⠑⠒⠊⣳
⠀⠀⠀⠀⠀⠸⠛⠆⠀⠀⣸⠀⠀⠀⠲⠶⣤⣄⡀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⣠⡤⠤⠶⢾⡟⢁⣀⡀⠀⠸⡀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢉⣤⣤⣤⠟⠈⠙⢦⡀⠀⠀⠀⢻⣿⡀⠀⠀⠀⡏⢰⠠⡀⠀⠙⠛⠙⠿⠴⠶⠝⡗⢠⡞⠁
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠐⣿⣶⠿⠟⣋⣤⠶⠶⠤⣄⡙⢀⠀⠀⢹⣹⣧⠀⠀⠀⠀⠸⡄⠀⠈⠀⠀⠀⠀⠀⢠⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⣿⠁⢠⠞⠉⠀⠀⣀⣀⡀⠈⠿⣆⠀⢀⡿⠈⢧⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡀⠀⠀⢀⡴⠚⣉⣠⢤⠠⡲⠁⢀⣼⠃⠀⠈⢷⡀⠀⠀⠀⠀⣉⡛⠛⣛⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢀⡋⣰⠟⠋⠉⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠉⠉⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⡀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣤⣶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⢀⠀⣠⡶⠶⠶⠶⠶⠶⠒⠒⠒⠚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣓⣒⠒⠒⠶⠶⠶⣤⡀⢀⡀
⠀⠀⠀⠀⠎⣼⠃⠀⣴⠶⠒⠛⠛⠛⣛⣛⠉⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⡉⢉⣩⣍⠉⣉⣉⡙⠛⣷⠀⠀⢹⡆⠓
⠀⠀⠀⠀⠀⣿⠀⠀⣿⢰⣟⣻⡇⢼⣏⣻⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠒⠚⠀⠻⠶⠇⠀⣿⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠹⠟⠁⠈⠙⠿⠁⠘⠷⠶⢤⣤⣤⣤⣤⣤⠾⠃⠸⠭⠝⠀⠯⠭⠇⢰⡏⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣓⣚⣒⡒⠲⠶⠶⠶⠶⠶⠶⠶⠛⠛⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠞⣛⣩⠭⠭⠭⠭⠭⠭⢭⣛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⠾⢋⠴⠒⠋⢁⣀⣠⣤⣤⣤⣤⣤⣄⡈⠙⠲⢬⡙⣿⡉⠉⠛⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃⡴⠁⢠⡴⢛⣩⠭⠒⠚⠛⠛⠛⠒⠮⣝⢷⣄⠀⠙⣎⢻⡝⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⡼⠀⢠⡟⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡹⣆⠀⠸⡀⢿⡀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠘⡇⢀⡶⢾⡇⡇⠀⢸⡅⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⢻⡀⠀⣇⢸⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⣷⠘⣇⠸⡇⣇⠀⠸⣇⢧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⢸⠸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⢻⡸⡄⠀⢻⡌⢮⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⠁⠀⢸⢰⡇⠀⢠⡏
⠀⠀⠀⠀⠀⢠⢸⡇⠘⠳⠾⢿⡜⣆⠀⠙⣦⡙⠦⢌⣀⣀⣀⣀⣀⣀⡤⢞⣱⠟⠀⢠⢏⡾⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠘⠈⣇⠀⠀⠀⠀⢻⣌⠳⣄⡈⠙⠳⠶⣤⣭⣭⣭⣭⣤⠶⠛⠁⣀⠔⢁⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠳⣦⣙⠲⠤⣀⣀⣀⠀⠀⢀⣀⡤⠖⢊⣡⡴⠋⠀⠀⠀⢀⡟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠐⠦⠉⠛⠶⠶⠤⣬⣭⣭⣥⠴⠶⠞⠋⠉⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠒⠶⠶⠶⠶⠧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟
⠀⡀
⠀⠂

⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⣰⠟⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠈⠙⣆
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⡆⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⣤⣤⣴⣶⣶⣾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠛⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡍⣦⣀⡀⢠⠀⣀⣠⡎⡍⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡄⢉⣵⡓⢿⡟⣿⠗⣳⣍⠁⣼⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⣼⣩⢻⠳⡷⠎⢣⣾⣁⣹⣦⠋⠶⡷⢻⢫⣹⣤⡈⣧
⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣠⣤⣾⢷⣴⣯⣥⣀⣶⠿⠀⣼⣥⠺⢢⢽⡄⠼⢷⣆⣠⣭⣷⣴⢿⣦⣤⣀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢸⣿⣿⣶⣶⣿⣿⣍⣿⣵⡿⣿⣿⣿⣿⣿⣷⣄⣿⡈⠀⠈⣸⣏⣴⣿⣿⣿⣿⣿⡿⣷⣽⣏⣹⣿⣿⣶⣾⣿⣿
⡞⢁⢱⢷⣿⡿⠯⡭⠿⢛⣽⠿⠛⠛⠛⠛⠲⢿⣿⣿⣿⣿⣵⢦⢶⣽⣿⣿⣿⣿⠷⠚⠛⠛⠛⠻⢿⣝⠻⠯⡭⠿⣿⣷⢷⢠⠙
⠑⢌⡁⠈⡻⣿⣷⣶⣚⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⣧⣿⡿⣿⣿⣿⡿⣿⣧⡟⠁⠀⠀⠀⠀⠀⠀⣠⣿⣟⣲⣶⣿⡿⡋⠀⣉⠔
⠀⠀⣨⠴⣿⣿⣟⣻⡿⢿⣿⣯⣷⣶⣦⣤⣤⣶⣾⣉⣿⣷⣿⣿⣿⣷⣿⣟⣹⣶⣦⣤⣤⣶⣶⣯⣿⣿⠿⣿⣛⣿⣿⡷⢬⡀
⠀⠀⢧⡚⠈⡟⢬⣿⣷⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⠛⠉⠀⠈⠙⠻⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣶⣿⣯⠜⡏⠘⣣⠇
⠀⠀⠀⠀⠀⣡⠖⣿⠟⠿⢽⣿⣿⣷⣾⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣾⣿⣿⣿⠽⠟⢿⡗⢦⡁
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠈⢉⡉⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⣉⠉⠀⠀⠀⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⣤⣤⣠⡟⠁⡴⠃⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠋⠁
⠀⠀⢀⣀⠀⠀⠓⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢶⣤⣀
⠀⠀⣏⠀⠴⢤⠦⣾⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠙⠿⠆⠀⢰⣿⢐⡇⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣸⠀⠀⢀⣤⣤⠞⠋
⠀⠀⣀⣌⣤⠆⣼⠃⢸⣷⠀⢸⡇⠀⢰⡀⠀⠀⠀⠀⣠⠄⠀⢠⣿⠀⠀⣀⡈⠁⢀⣴⠟⠉
⠀⢰⡏⠀⠁⠀⢹⡄⢹⣿⠀⢸⡇⠀⢸⣿⠀⠀⠀⣼⡿⠀⠀⢸⡏⢀⡀⣨⠟⠀⣼⡟
⠀⠀⣿⠂⣀⣀⡀⢻⡌⢻⡀⢸⡇⠀⢸⡟⠀⠀⣰⡏⣴⠀⠀⣸⡟⠈⣹⠁⠀⣼⣿⡁
⠀⠀⠈⢻⡋⠀⠁⢀⣽⠶⢿⠈⣧⠀⣿⡁⠀⠀⣿⠃⢿⠀⠀⣿⣁⡿⠋⢀⣾⠟⢉⣷
⠀⠀⠀⣾⣧⡾⠀⣾⠃⠀⡘⣧⣹⡄⢸⣇⠀⠀⢿⡀⠘⢧⣀⡟⠈⣠⡴⠛⠁⠀⠈⢿⡀
⠀⠀⠀⢿⠀⠀⠀⣿⢀⡾⢁⠀⠉⢳⣜⣿⡀⠀⢸⡇⠀⠀⢉⣴⠾⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡷⠂⠀⢻⡾⣱⠏⠀⣴⡿⠉⠛⢻⡄⠈⣿⢀⣴⡾⠋⠀⠀⠀⠀⠀⢀⣷⠏
⠀⠀⠀⣼⠋⠀⠀⠸⢷⡟⢀⣸⡟⠁⢀⡖⠀⢻⣦⠟⠋⠁⠀⠀⡄⠀⠀⢀⣷⠾⠃
⠀⠀⠀⠘⣧⠀⠀⠀⠘⠛⠛⠋⠀⠀⠻⠶⠶⠛⡉⠀⠀⠐⢦⡖⠃⠀⣀⡾⠁
⠀⠀⠀⠸⣧⡀⠀⡀⠀⠀⠀⢧⣠⠀⠀⢀⣀⣠⠷⣄⣀⣠⡾⠃⠀⠉⣿
⠀⠀⠀⠀⠸⣧⣴⠏⠀⠀⠀⠀⢙⣻⣷⡚⠉⠁⠀⠸⣏⠉⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠛⠋⣩⢙⠻⢷⣤⣀⠀⠈⠛⠒⠚⠋
⠀⠀⠀⢰⣞⣷⢸⣿⣄⡀⠀⠘⠶⠟⣠⠤⢤⡈⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠁⢸⡇⠈⠛⢶⣄⡀⠀⠙⠚⣯⣤⣶⣶⠀⠙⠻⢶⣦⣄⠀⠀⠀⠀⣏⣹⠆
⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠋⡵⢶⣻⠆⠀⠈⠙⢻⣶⡄⠀⠈⠉
⠀⠀⠀⠀⣴⣋⠙⢷⣄⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⣀⣤⣤⡶⠞⠋⠉⡇⢀⡠⢤⡀⠀⠀⠀⡶⠛⡆
⠀⠀⣀⣸⣇⠙⢷⣾⣉⠳⣤⡀⠀⠀⠀⠀⠀⠈⠙⣿⠟⠋⠉⠀⠀⠀⠀⠀⣇⠈⣀⣨⡿⠀⠀⠀⠀⠉⠁
⠀⠈⢯⡽⠙⠛⣿⠀⢈⣷⠈⠙⢶⣄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠁⣰⠛⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠈⠻⣦⡀⠀⠀⢺⠀⠀⠀⢀⣠⣴⠶⠛⠉⠀⠀⠀⠀⠻⠤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣾⡿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠉⠀⠀⠉⠉⠛⠛⠶⢶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣶⣶⣿⠿⠿⠿⢿⣿⣿⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣒⣛⣛⣛⣻⣭⣭⣭⠭⠿⠶⣶⡶⠖⣛⠛⢻⡟⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠁⠀⠀⠘⣿⠐⣿⣿⡿⠿⠿⠿⠇⢸⡇⠀⣾⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠄⠆⠂⠐⢘⠀⢸⡇⠀⠙⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⣿⠟⠿⠿⠀⣿⠀⢀⣃⣁⣁⣬⣤⣤⡼⠷⠶⠖⠺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣶⣶⣶⣿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⢀⣠⣤⡶⣶⣶⡶⢶⣤⣀⠀⠀⠀⠀⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣶⡿⣫⣷⣿⣿⣿⣭⣛⠶⣝⢷⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢷⣻⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⡿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠹⣿⣷⡽⢿⣿⣿⣿⡿⢟⣫⣾⣿⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠀⠀⠘⠿⣿⣷⣭⣽⣿⣿⣭⠿⠛⠁⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⢀⣀⣠⣤⣶⣶⣾⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣀⣤⣤⣴⣶⠾⠿⠛⠛⠉⣿⡇⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⣀⣴⣶⡷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠋⠉⠁⠀⠀⠀⣀⣀⣠⣄⣤⣤⡶⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣴⣶⣾⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠉⠉⠉⠉⠉⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠄⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣴⣿⣾⣄⣄⠀⠀⠀⢀⡀⠀⠀⠀⢠⣀⣷⣿⣦⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⣿⡿⣿⣿⣿⣾⡄⠀⠀⠀⠀⢠⣷⣿⣿⣿⢿⣿⣧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿⣿⣇⣀⡙⠛⣿⣿⣦⡀⢀⣴⣿⣿⠏⢋⣀⣸⣿⣿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠘⢉⠀⠀⣭⠙⠾⣿⣿⠇⠸⣿⣿⠿⠋⣭⠀⠀⠉⠃⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠈⢤⣤⣬⡤⠴⠋⡉⠀⠀⣉⠙⠢⢤⣥⣤⡴⠁⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠉⣡⠄⠐⠈⠀⠀⠀⠀⠁⠂⠠⣌⡉⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣀⣀⣤⣿⣇⣀⡀⠀⠀⠀⡀⠀⠀⢀⣀⣸⣿⣤⣀⣀⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣾⣩⣍⠉⠉⠀⠻⣽⣷⣼⣿⣿⣿⣿⣦⣾⣯⠟⠀⠉⠉⣩⣅⣷⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⠥⡶⠿⠀⢰⡶⡶⠿⣛⣛⣿⢿⡿⣿⣟⣛⠿⢶⢶⡆⠀⠿⢶⠬⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢿⣾⡿⠾⠻⠟⠻⠞⠻⠟⠿⠿⣿⡿⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⢮⠛⠛⠛⢊⣉⣈⣁⣉⡑⠛⠛⠛⡡⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⢇⢦⡀⠀⠙⠛⠛⠛⢿⣿⣿⡿⠛⠛⠟⠛⠁⢠⣴⡺⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠉⣳⡘⢎⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡱⢃⡞⠉⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣏⣉⡎⠳⣿⣿⣶⣶⣶⣾⡿⣜⠷⣶⣶⣶⣾⣿⠞⢱⣉⣹⡗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣝⡦⣀⡉⠉⠛⢿⣟⣤⠤⣛⡿⠋⠉⠉⣀⢴⣫⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢾⣿⣭⣭⡵⠶⠿⢿⣭⣭⣿⡷⠾⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣤⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣀⣤⡀⣹⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣼⣿⣿⣦⡈⢳⣀⣀⣤⣤⣶⣶⣶⣶⣤⣤⣀⣀⡞⢁⣴⣿⣿⣿⡙⡿⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡿⠗⠀⠉⠀⠀⠀⠙⠿⠟⠉⠀⠀⠀⠉⠀⠻⢿⣿⣿⣿⠈⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡛⠉⠀⠀⢀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⢀⡀⠀⠀⠈⢻⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⠁⠀⠀⠲⣄⡙⠓⠶⠆⢸⡟⣿⡆⠴⠶⠚⢋⣠⠖⢀⠀⠈⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠻⢦⣌⣙⠛⠿⢦⢸⠀⠘⡇⠶⠟⢛⣋⣡⡴⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⠸⡀⢠⢁⣴⡾⠛⠋⠁⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⡀⢀⠀⠀⠀⠀⠀⠀⠙⠃⣷⡞⠘⠁⠀⠀⠀⠀⠀⠀⢀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⣍⣛⠷⠲⣦⡀⠀⢦⣄⠀⠘⠃⢀⣠⡔⠀⣠⡴⠞⠞⣛⡍⡙⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⣿⣿⡿⠿⣿⣿⢧⡈⠙⠀⠀⠀⠘⠉⢀⡼⣿⣿⠿⢿⣿⣷⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⡙⢯⣁⡀⠈⠻⠆⢣⠀⠀⣀⠀⠀⠀⡞⠴⠋⠀⢀⣈⡽⢫⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣽⠸⣧⣼⣄⣈⣍⣉⣙⠛⠀⠀⠉⠈⠀⠈⠓⣉⣉⣉⣩⣇⣤⣾⠃⣯⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⡇⠮⣭⣭⡭⠖⠀⠀⠀⣶⠃⣀⣀⠸⣦⠀⠀⠀⠲⢭⣭⣭⠔⣸⡿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡻⢻⡀⠀⠀⠀⠀⠀⠀⣼⠀⢾⣄⣤⡷⠀⣧⠀⠀⠀⡀⠀⠀⢠⠏⢏⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⡾⠁⠀⠳⣄⠀⡡⣮⡦⣄⠈⠀⠀⢹⡏⠀⠀⠁⡀⢴⡵⢈⠀⣠⠏⠀⠘⢿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣷⣲⣄⠀⠙⢦⡑⢌⠳⢿⡷⢷⣈⣛⣗⣱⡶⢿⠿⠟⡡⢊⡼⠃⢀⣠⣶⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⣇⠀⠀⠙⠦⣍⠒⠉⠉⠙⠛⠛⠋⠉⠛⠚⣩⠞⠋⠀⠀⡸⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣄⠀⠀⠀⠈⠓⢦⡀⠀⣀⣀⠀⣀⡴⠛⠁⠀⠀⢀⣠⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢯⣷⡀⠀⠀⠀⠉⠻⠿⠿⠟⠁⠀⠀⠀⢀⣾⡽⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡺⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠟⡻⠛⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⠛⠟⠻⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠠⣧⠂⠀⠀⠀⠀⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⠀⠀⠱⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⣆⢰⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠘⢷⣶⣶⣤⣖⢶⣄⣶⣶⣶⣶⣤⣶⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣰⣶⡤⣶⢲⢖⡶⣲⣶⣶⣶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⡛⢷⣿⡟⡿⢿⣌⣛⣿⣿⣿⣿⣟⢳⡄⣠⢄⣠⣄⢠⡾⣫⣽⣽⣛⣷⣯⣽⡾⣷⣿⣾⣞⣫⣾⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⢿⣷⣿⡿⣿⣾⣾⣟⣻⣿⣈⣿⣿⣿⢿⣿⠮⣿⣾⡿⣮⣋⣽⢏⣟⣧⣾⣟⣛⣿⢿⣿⣻⣶⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⣏⣻⢟⢿⡉⠉⠉⠉⠛⢿⡿⣿⣿⣿⣿⢿⣿⠻⣶⣿⡿⠟⠉⠉⠉⢉⣽⣧⣾⣼⣷⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⣯⣯⣬⣷⣿⣦⣀⣀⣀⣀⣿⣷⣸⣯⢿⣪⡿⢻⣯⣾⣁⣀⣀⣀⣤⣾⠿⣻⣿⣟⢻⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠾⣿⣿⡟⠭⣭⣵⡿⣯⡶⣿⣿⣿⣵⠟⠻⣿⡿⢻⡮⣹⣯⣛⣛⣛⡻⣿⢻⢿⣟⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣯⣼⡿⣻⣿⣥⣽⣿⡟⣉⡿⠁⠀⠀⠈⣿⣿⣿⡏⢒⡛⢿⣯⣳⣽⣿⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠿⣽⠿⠟⠉⠛⠛⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⡿⠿⠋⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣤⡔⠀⢠⣤⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢱⡀⠀⠀⠀⠀⡀⢴⣧⡀⠀⠘⣏⠀⢸⡏⠀⠀⣨⡷⠄⡀⠀⠀⠀⠀⡜⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡕⣠⣴⠶⠿⢷⡄⠙⢷⠀⠀⢹⠀⢸⠀⠀⣼⠋⢀⡴⠿⠷⢶⣄⢌⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠘⣜⢧⡀⠀⠀⠘⠄⠀⢣⠀⠘⠀⠘⠀⢰⠁⠀⠊⠀⠀⠀⣰⢋⡌⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢻⡄⠀⠹⣦⡑⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⢤⠜⣵⠟⠀⠀⣾⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡸⣷⡀⠀⠈⢿⠇⠀⠃⠀⠀⠀⠀⢀⡄⡄⠀⠀⠀⠀⠀⠁⠘⢿⠋⠀⢀⣼⡟⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠓⣹⣿⡦⠐⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⢿⡄⠀⠀⠀⠀⠀⠀⠀⠑⢴⣾⣿⡜⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠢⡀⠏⠁⠸⣿⣿⣶⣤⣦⠄⠂⠀⠀⣠⠂⠞⠁⠀⠀⠻⠀⢆⠀⠀⠈⠢⣴⣤⣴⣾⣿⡇⠀⢸⠀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⣸⣶⣄⠙⣄⠀⠀⠉⠛⠋⠉⠀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⠉⠙⠛⠋⠀⠀⢠⠞⢠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠛⠻⢻⣆⠈⢦⣠⣼⣧⠀⠈⠉⠉⠉⢀⣀⣴⣄⣀⡰⠀⢣⣀⣀⣶⣄⡀⠉⠉⠉⠉⠀⣰⣷⣄⣠⢊⢰⡿⠻⡏⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣻⣿⡾⢿⡏⢿⣷⣶⣤⡬⢿⣿⣿⣿⡛⠛⠇⠀⠈⠛⠛⣿⣿⣿⡿⠭⣥⣴⣾⣿⠙⣿⢿⣾⣾⠁⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣿⣿⣇⠀⠉⠛⠻⣿⣿⣄⣠⣿⡿⢛⡓⠂⠀⠀⠀⠀⠒⡛⢿⣿⣧⣀⣼⣿⠿⠛⠋⠁⢠⣿⣿⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠹⣾⡿⣆⠀⠀⠀⠈⠀⠠⠤⠄⠈⠉⠈⠀⠀⠀⠀⠐⠁⠈⠁⠠⠤⠤⠐⠀⠀⠀⠀⢠⠿⣯⡟⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⣽⣷⡀⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⠀⠀⠀⠀⡠⢀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢁⣴⣿⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⢘⢻⡿⣿⣷⠠⣄⣀⣸⣿⠒⣶⣶⣄⠀⠀⠀⢀⣴⣶⡖⣻⣗⣀⣠⡄⢾⣿⠿⡿⠃⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠛⣾⣿⡿⠛⠻⠕⠛⠻⢿⣷⠤⢶⣿⠿⠛⠊⠿⠛⢻⣿⣷⠟⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠘⢿⡿⢂⡤⡀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢀⡠⡀⠻⣿⠏⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠊⢀⣥⣾⣏⡉⣛⠂⠀⢒⠂⠀⢛⠉⣩⣷⣮⣀⠈⠢⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢸⢿⣿⣿⣿⣿⠷⠞⠛⠛⠶⣿⣿⣿⣿⡟⡇⠀⠀⡐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣌⠘⣿⣿⠛⠑⣛⣉⠛⢉⣙⡋⠙⢿⣿⠇⣰⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣷⣼⠖⣩⠴⠖⠒⠒⠒⠒⠦⢄⡐⢯⣴⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠉⠉⠠⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠢⠉⠉⠉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡀⠀⠀⠀⣰⠿⠖⠚⠒⠾⢧⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠊⠁⠀⠀⠀⠀⠀⠀⠉⠂⠐⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠿⡏⠉⠩⡈⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠀⠀⢀⠀⠁⠀⠀⠀⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⣶⢠⡀⠑⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⢠⡱⡆⠑⢤⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠙⡅⠀⠈⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢣⠀⠀⠀⡇⠄⠀⠀⡸⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⢀⢷⠀⡠⢿⠂⠀⠂⠀⠀⡀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠍⠉⠁⠈⠻⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⡀⠑⠀⠀⢀⠀⠄⣀
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠀⠸⡄⠙⠒⠁⠀⠀⠈⠙⠐⠄⠀⢀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠀⠀⠘⠆⠈⠄⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠂⠠⢄⡀
⠀⠀⠀⠀⠀⣸⣿⠀⡀⢀⠠⠀⠀⠀⠁⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡀⠀⡆
⠀⠀⠀⠀⠀⠿⠘⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠤⠀⠀⠠⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣷⠀⠁⠀⢃⠀⠀⠈⠀⠁⠀⠀⠈⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠧⠲⣄⠶⢏⣤⠐⠛⠁
⠀⠀⠀⠀⠀⡿⠀⠀⠀⢘⡷⠦⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠶⠒⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢃⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠐⠛
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠋⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠶⠶⠶⠖⠒⠶⠦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣿⣿⣿⣷⡶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠻⢽⣓⣶⡶⣶⣶⣒⣚⣛⣿⣿⣯⣭⡿⠿⣿⠿⣶⠚⠛⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢹⡟⠉⠉⠉⠀⠘⣿⠰⣶⠿⠿⠿⠿⠃⣿⠀⢰⣿⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣀⠀⣿⠀⠐⢐⢐⢈⠨⠀⣿⠀⠀⣉⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡀⠀⠀⠀⢸⡇⠘⠛⣉⣋⣀⣿⣤⣤⣤⡶⠶⠶⠚⠛⠛⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⡇⠀⠀⢰⢸⡿⠿⠟⠛⠛⠋⠉⠉⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡇⠀⢸⢸⢸⡇⠀⠀⠀⠀⢀⡴⣚⣭⣶⣶⣖⠮⣝⢿⣦⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣷⢸⢸⢸⠸⡇⠀⠀⠀⢀⣞⣼⣿⣿⣿⣿⣿⣿⣎⢳⡹⣷⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢹⢸⢸⢸⢸⠀⣇⠀⠀⠀⣸⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣇⣿⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⢸⢸⢸⠀⣿⠀⠀⠀⢻⢻⣿⣿⣿⣿⣿⣿⣿⡿⢀⣇⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⢸⢸⠸⠘⠀⣿⡇⠀⠀⠘⣞⢿⣿⣿⣿⣿⣿⠟⣡⢞⣾⠏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⠃⣿⡇⠀⠀⠀⠈⠢⣝⡻⠿⠿⣗⣫⣷⠟⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡄⣿⣧⡄⠀⠀⠀⠀⠀⠉⠙⠛⠉⣉⣀⣠⣤⣶⣶⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠃⣿⣿⡇⠀⠀⣀⣀⣤⣤⣶⠾⠿⠛⠛⢹⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢻⣿⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⣀⣸⣧⡴⠿⠗⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⢸⠀⢀⣀⣀⣤⣤⣦⡶⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠙⠦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠖⠦⣄⡤⠴⠖⠁⠀⠀⠀⠂⡨⠏⠙⠟⠢⢤⣀
⠀⠀⠀⠀⠀⢠⡤⣤⡀⠀⠀⢀⣠⠴⠒⠛⠁⠐⠀⠸⠁⠀⠂⢁⡀⠀⠀⠰⡉⠀⡠⠴⠲⠒⠊⠨⢦⡀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⢿⢱⣎⢳⡄⡴⠋⠀⠀⠈⠉⢀⠟⠉⠀⠠⠀⠙⠉⠋⠀⠀⠀⠈⠁⠀⠘⡀⠀⠀⠀⠀⢳⣠⡟⣇⢹⠂
⠀⠀⠀⠀⠀⢸⡞⣯⣧⣿⢰⣦⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠇⠐⡀⢰⣦⡄⠉⢿⣽⢸⡇
⠀⠀⠀⠀⠀⢸⡟⢮⣿⣇⠀⠉⠀⠀⠀⠀⠠⠄⢀⠂⠀⢀⠄⠀⠀⠀⠤⠀⠀⠀⢀⠀⡄⢀⠀⠈⠀⠀⠂⣸⠼⣾⡇
⠀⠀⠀⠀⠀⠀⢷⣤⣽⣍⡙⣮⠥⠄⣀⣰⣀⣀⣁⠄⠖⠺⠐⠄⠀⠀⠀⢤⡂⠀⢸⠀⣠⣘⣋⣠⣦⠶⠾⣡⣬⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣟⠛⣿⡟⠛⢻⣶⣾⣮⣀⣀⡩⠀⠀⠀⠐⠀⠀⠀⣸⡛⣳⣂⣉⣙⣫⣭⣭⠷⠿⠖⣛⣉⡼
⠀⠀⠀⠀⠀⠀⠀⢻⡍⢫⡙⠻⠿⠯⢏⡉⠛⠛⡇⠀⠀⠀⢀⠀⢴⣿⣿⠉⠭⠭⠭⠽⠿⠷⣴⣶⡶⠟⡿⠋⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡗⢾⡛⠷⢶⣆⡚⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠉⡗⣒⣲⣂⣴⣶⣶⣬⡥⣤⣶⢿⣶⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⡿⣶⢶⡖⠿⢤⣤⣶⡇⠀⠀⠀⠀⠃⠀⠀⣯⠤⢭⣭⣤⣤⣴⣚⣉⣠⣥⢾⡗⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡳⣽⣦⣌⣹⡷⣦⣤⣽⠆⠀⠀⠀⠀⠀⠀⢠⣗⣒⣚⣛⣙⣩⣤⣟⠉⣭⣐⣿⢆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⢯⣦⣉⣙⡲⢀⣉⡟⢀⣴⣦⡄⠀⠀⠀⢸⡷⠈⠛⠛⣏⢋⣩⣷⣛⣉⣽⠿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⢧⣙⡛⠯⡬⢛⡃⠘⠻⡿⠀⠀⠂⠀⢸⡕⡬⠵⠾⠟⠋⠡⠕⣛⣫⢾⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⢦⣍⡙⢻⡍⢛⡇⠀⠀⠀⠀⠀⡀⠀⣏⣩⠷⠿⠾⠝⢛⡫⠗⢋⣴⠷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⢮⣙⡻⠿⣻⠇⠀⠈⠀⠀⢀⣿⣦⢋⡴⠖⠲⠾⢛⣫⡷⢛⣩⣶⢦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡷⣮⣛⠭⣝⠢⠦⠤⢥⣤⣖⡦⠍⣛⣛⠾⠴⠛⣛⣫⣛⣩⣵⡿⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⡦⣍⣛⣍⡛⠻⣶⣳⢄⣛⣛⣛⣻⡯⡽⠛⢛⣻⣽⣯⡥⢾⣯⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⡙⠨⣛⣦⣉⣛⡿⢿⣍⣩⣉⣩⣍⣭⡔⣛⣉⣭⣾⡯⠖⣉⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡙⠳⢿⠿⢮⣙⣛⣛⣂⣤⣭⣥⣤⣍⣉⣹⡭⠶⣗⡴⠊⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣁⠢⠬⢭⢉⢵⣦⡤⣥⡴⡽⠍⠤⠒⠉⣁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠦⣤⣤⣤⣤⣤⣤⠤⠶⠒⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⠀⠀⢸⣷⠀⠀⢰⣶⢰⠀⠀⡶⠄⠲⠶⣶⡶⠐⠀⢀⠀⠠⠄⢲⠀⠀⣻⠿⣲⣦⢸⡇⠀⠁⠀⠀⠀⡀⡆
⠀⠀⢰⠄⢸⣿⠀⠀⢸⢀⢺⠀⠀⡇⠀⢀⣿⠋⠁⢦⣴⠏⠀⢀⣠⠟⣾⡰⠙⡀⢩⠏⢸⡗⠀⠀⠀⠀⠀⠁⠇
⠀⠀⠀⠀⢸⡇⢈⠀⠘⠉⢸⠠⠈⢧⣼⡾⠁⣤⡻⠞⠁⣀⠰⠉⠀⠈⠽⠛⠃⣙⠀⠀⠈⠸
⠀⠀⠀⠀⠨⣇⢸⠀⢸⠘⢉⠀⢸⣸⡏⠁⢿⣼⠟⣁⡶⠡⠀⠀⢂⡀⢀⣾⣽⣁⣶⠀⡀
⣶⠀⠒⠀⠀⣟⣾⠈⢹⠈⠈⠀⣾⣿⠑⡜⢿⡟⠀⠀⠃⠀⡀⠙⠚⠗⠸⠻⠛⢿⠟⠰⠙⠻⣦⡄
⣿⠀⡇⠀⢀⣿⠈⡄⢸⠘⢠⣤⣿⡣⣄⢓⣚⣷⣦⡴⣤⣤⣬⡀⣀⣁⠂⠁⢀⠀⠀⠀⠐⠠⣽⣿⠆⠀⠀⠀⠀⠤
⢿⠀⠀⠀⢸⣿⠀⠀⣸⢏⠈⠿⠉⣗⡀⠀⣻⣴⠟⡷⠖⠠⣔⠑⠒⢚⢷⠄⠸⢓⣶⡀⢴⣶⢾⠏⠀⠀⠄⠀⠀⠀⢀
⣿⠀⠀⢠⣾⣿⠀⠀⣸⢬⠀⠁⣾⡑⣳⣪⡿⣫⢆⡥⣠⢴⣯⠞⢩⡅⠀⠀⠘⠈⠙⠡⠀⠙⠂⠀⢰⠀⡀⠀⠀⠐⣾
⢹⡂⠀⠈⢹⣿⠀⠠⢸⠈⠀⠺⢻⣷⡉⠩⠂⣽⣥⠠⣾⡿⠃⠀⠀⠀⠀⠈⠀⣤⡀⢀⣠⣶⠆⡁⢸⠃⠁⠀⠀⠀⣽
⢸⡆⠀⠀⢸⡯⠀⢀⢸⠸⢀⡈⢈⣽⠎⢙⠀⢳⣉⠀⠀⣠⣤⣶⣷⣦⣤⣤⡴⢫⠆⢈⡹⠁⠔⠃⠘⠀⠀⠀⠀⠀⠙
⢸⡇⠀⠀⢀⣿⠀⠍⢸⠷⠐⢠⣿⠡⠴⡟⣄⠐⠿⣷⠀⢟⣤⣤⠌⠓⠀⠀⠉⠑⠚⠈⠀⠀⠀⠀⠀⠠
⠈⣧⠀⠀⢸⣿⠠⠒⢸⡄⡀⠄⠀⠀⢀⢨⠊⢷⣆⣈⡀⠨⢸⣥⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐
⠀⢿⡄⠀⠀⢻⡀⠀⠻⠷⡑⠩⠤⠀⠋⣻⡇⠀⣿⣯⠀⠄⢀⠀⣠⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⣿⠀⠀⢸⠇⠀⠠⢤⡟⠠⠀⠒⢊⡝⠑⣳⠛⣁⠀⠀⣄⡑⠛⠋⠉⢙⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⣾⠁⠀⠐⣾⠀⠠⠌⢰⠇⡆⢠⣬⠟⡷⣿⣿⣾⠫⠀⠲⣤⠉⠛⣿⠃⡘⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠛⠁
⠀⢸⡄⠀⠀⢿⠐⠀⠀⡟⢀⣿⠪⠁⠘⣿⣦⣈⠀⠚⠁⠂⣸⠀⢠⡏⢰⠇⠀⠘⣇⠀⠀⠀⠀⠰⠄⢰⠖
⠀⠈⣷⠀⠀⢸⠀⣴⢀⠆⢩⡾⣰⠂⠂⢰⢤⣭⠤⢄⡴⠿⠋⣀⠞⣡⡎⠀⢀⢶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐
⠀⠀⣿⡆⠀⢸⠀⢻⠘⣆⠹⣴⠇⠀⢧⣀⣀⢸⡷⣬⣙⣛⣩⡵⠾⠃⠀⣠⣾⣾⠿⠀⠀⠐⠀⠀⠀⠀⣀⠀⠀⠀⠄
⠀⢠⡏⠀⠀⢸⡄⠈⢸⡉⡆⡜⣀⣤⡤⠋⠉⡸⠃⠈⠛⠋⠁⠀⣀⣤⣾⢿⣿⠷⢠⢀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠠
⣸⣿⠁⠀⠀⡈⠃⠈⠘⢃⡏⠀⢠⠶⠶⢶⣋⣡⣤⡄⠀⠰⢺⢟⣽⣥⠀⣿⠅⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢸⣇⠀⠀⡀⣧⡀⠐⠀⠨⡁⠀⠀⠀⢻⣽⣿⣞⣯⣤⣴⣿⠈⢸⡏⡂⣸⠁⠁⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠼⠋⠀⠀⠀⠙⠻⠆⠀⠀⠙⠖⠀⢲⠕⠟⡿⠻⠿⣿⡿⠏⠀⠘⠀⠁⠾⠸⠀⠈⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠈⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⢾⡁⣹⠆⠀⠀⠀⣿⠁⠉⠈⠉⠉⠁⠉⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⢛⣋⣉⣉⣛⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣥⠖⠛⠉⠉⠀⠈⠉⠙⠓⢦⣉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⢷⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⡄⢷⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⡆⢷⠸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢸⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢷⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠃⡟⢸⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣇⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢦⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣡⠟⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣍⠓⠦⢤⣤⣀⣀⣤⠤⠖⢋⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣪⢍⠕⢼⡁⠀⠀⠐⢮⡓⢄⠀⠀⣀⠤⠔⢒⣦⢤⣤⣤⡀
⠀⠀⠀⠀⠀⣀⣀⠀⣠⠎⢸⡁⡀⠼⠤⠤⡻⢶⡛⣷⣒⠬⣻⢔⠋⠀⠀⣴⣿⣿⣿⠿⠛⣇⡀
⠀⠀⠀⡎⠉⠀⠀⣈⣭⣟⡛⠛⠛⠓⠒⠒⠯⣉⠁⠀⣀⣙⣎⠁⢳⣠⣾⣿⣿⡯⣒⠒⠒⠁⠈⠉⠩⡇
⠀⠀⠀⡟⠀⣠⢮⣽⣒⣂⡀⠭⠭⢝⣲⠀⠀⣀⣻⣖⡤⡀⠉⢈⡿⡿⢛⡻⣿⣵⣛⢮⠷⣮⣤⠛⡷⣮⣦⡤⣀
⠀⢠⣾⣇⣰⢈⠟⢹⣟⣓⠻⠦⣴⡿⣣⣼⣿⣿⣿⣧⣦⣬⣽⣧⣄⢃⠀⠤⢈⡽⠛⠋⠁⢀⠶⠧⣟⣱⠿⠃⡸
⢰⠋⡇⣨⢣⠊⠀⢸⠀⠀⠉⠑⠒⠬⠭⣙⣛⠻⠿⠿⢿⣿⣿⣿⣿⣬⣶⣿⣀⠠⠷⣦⠔⠁⠀⢀⡼⠖⠒⡏
⠀⠑⠷⠤⡅⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⢆⢀⠴⠋⠀⠀⢀⡇
⠀⢀⠀⠀⠈⢑⠖⣃⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⡀⡤⣀⣤⣀⢤⠀⠀⢠⢤⢠⣤⢠⢲⢠⣷⢰⣷⠀⠀⢸
⠀⢸⠀⠀⡠⠋⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⡇⡟⣤⣿⢸⡄⠀⢸⣾⢸⢻⢸⢸⢸⢹⢸⢻⠀⠀⣸
⠀⢸⣠⠊⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⡃⢻⡃⣳⡃⣳⠋⣲⠁⠀⢘⣟⢘⣎⢨⣏⢨⢏⡰⢧⠀⠀⡇
⠀⣿⠃⠀⠀⢿⡃⠈⡆⠀⠀⠀⠀⠀⠀⡿⡇⡇⡇⡇⡇⡇⡇⣧⡇⠀⢸⣸⢸⢸⣸⢿⣷⡞⣿⡇⠀⠀⡇
⠀⠸⡄⠀⠀⣸⣧⡤⣻⡄⠀⠀⠀⠀⠀⢧⠇⢧⠇⠷⠇⠧⠇⠧⠇⠀⠸⠼⠸⠜⠙⠊⠙⠃⠙⠁⠀⢸
⠀⠀⠈⢶⡫⠗⠚⠛⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠁⠈⠷⣄⡀
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠁⠀⠀⠀⠀⠈⠷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⣾⠉⠩⠈⣧⡀⣀⣤⣤⠴⠶⣤⣤⣤⣀⣸⣄⠀⠠⠀⠀⠀⠠⠄⠀⣠⣇⣀⣤⣤⣤⠶⠦⣤⣤⣀⢀⣼⠁⠍⠉⣷
⠀⠹⣇⠀⡠⠬⠛⠉⠆⠐⠀⠁⠀⠀⠉⠉⠉⠂⠂⠄⠤⠀⠀⠤⠠⠐⠐⠉⠉⠁⠀⠀⠈⠀⠂⠰⠉⠛⠥⢄⠀⣸⠏
⠀⠀⢻⠀⢀⡃⠀⠁⠀⢄⣀⣡⣶⣦⣥⣄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠄⠀⠀⣠⣤⣴⣶⣎⣀⡠⠀⠈⠀⠘⡀⠀⡟
⠀⠀⠸⣇⠀⡀⠐⢤⣶⡞⠋⠁⠀⠀⠀⠙⠻⣦⣤⠀⠀⠄⠠⠀⠀⣤⣴⠟⠋⠀⠀⠀⠈⠙⠳⣶⡴⠂⢀⠀⣸⠇
⠀⠀⠀⢻⡄⠄⠀⠀⠉⠛⠲⠶⣤⣤⡤⠶⠟⠉⠉⠀⠀⠂⠀⠀⠀⠉⠉⠻⠶⢤⣤⣤⠶⠖⠛⠉⠀⠀⠠⢠⡟
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠙⢷⣄⡠⠀⠀⠀⠀⠀⠀⠀⢀⠠⢢⡴⠞⠋⠙⠓⢦⡔⠄⡀⠀⠀⠀⠀⠀⠀⠀⢄⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣒⡲⡂⠤⣐⣂⣭⠾⠋⠀⠀⠀⠀⠀⠀⠙⠷⣭⣐⣂⠤⢐⢖⣒⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠤⠠⡦⢰⢀⡀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠅⠰⠞⠒⠐⠒⠄⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⢠
⣀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣀⣀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⠿⠿⠛⠛⠛⠛⠛⠛⠋⡁⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠙⣿
⣀⣀⣀⣤⣤⣄⣀⢀⠀⠀⣈⡒⢀⠀⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⣴
⠉⠈⠉⠉⠉⠉⠉⠭⠤⠤⠬⠄⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠙⠛⣛⡛
⠅⠀⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻
⣛⠻⠖⠓⠒⠒⠂⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠛⠛⠉⠩⠯⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠
⣂⣀⣤⣒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠉⠉⠉⠉⠉⠁
⡀⡐⠐⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠍⢃⡀⠤⢤⣄⣀⣀⣴⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣿⣦⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠿⡏⠙⠋⠙⠋⣿⡿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠖⠂⠀⠀⠀⠀⠋⠁⠠⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣦⡘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣀⡀
⠀⠀⠀⠀⠀⣴⠚⠉⠉⠉⠉⠉⠉⣹⠆⠀⠀⠀⠀⠀⠀⠸⣧⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⢟⠛⠳⣄
⠀⠀⠀⠀⣀⣹⡶⠤⠴⠖⣶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣧⣆⣤⡴⠚⢷⣠⢞⠕⠁⠀⠀⠈⢷⡄
⠀⠀⠀⢰⢱⠛⠓⠒⠒⠒⠛⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠑⢁⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⢸⡘⢦⡤⠤⠤⢤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⣴⠏⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⣷⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⡅⠀⠀⢀⡠⢄⠀⠀⠀⠀⠀⠀⣠⡾⢁⣴⠋⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⢠⢎⣠⠼⠁⠀⠀⢀⣴⠟⠛⣦⡞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢻⡀
⡔⠒⢤⡀⠀⢀⣀⡀⠀⠀⢸⠀⠀⣇⣀⣉⠉⠀⠀⠀⣀⣀⣿⣀⣰⠟⠻⣦⡀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠙⠒⠤⢵⡄⠸⣄⡙⡄⠀⣼⠀⢀⣯⣍⠉⠙⠛⠋⠉⠉⠉⠈⣿⠁⠀⠀⠀⠙⢦⡀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣶⡏⠀⠻⣍⣉⠛⠳⠶⠶⠶⠶⠒⠛⠋⢳⣄⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠶⠖⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣷⣄⡀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠠⣏⣉⡩⠶⠀⢻⣿⠿⣛⣋⠀⠀⠀⠀⣠⡾⠛⠛⠛⠉⠉⠉⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣠⠀⠀⠀⠀⠀⡈⣿⣟⣭⣵⠞⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡔⠋⣠⠏⠀⠀⠀⡴⢚⡇⠀⠉⠈⠛⠋⠁⠀⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠾
⠈⠋⠁⠀⠀⠀⠸⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠁⠀⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⢶⡞⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠸⣇⠀⠀⠈⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢉⡁⠀⠀⠉⠙⢶⣄⠀⠀⣴⠞⠋⠁⠀⠀⠈⠙⠳⣄⠙⢦⣀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢡⡾⠋⠁⠀⠀⠀⠀⠀⠙⣦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠳⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡿⠀⠀⠀⠀⣠⣤⠀⠀⠀⢹⡇⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠀⠀⣀⣼⠃⢻⣄⠀⠀⠀⠀⠀⠀⣀⡾⠈⢷⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠉⠀⡼⠋⠀⠀⠀⠉⠻⡄⠀⠀⢠⡞⠉⠀⠀⠀⠉⢳⡀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢸⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⣇⣰⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠉⠙⢿⠻⣆⠀⠀⠀⠀⠀⠀⣻⠞⠁⠀⠀⠀⠀⣤⡴⠟⠀⠀⠹⢦⣤⡀⠀⠀⠀⠀⠻⣯⠀⠀⠀⠀⠀⠀⠀⣴⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣿⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⡇⠀⢿
⠀⠀⠀⠀⠀⢀⡾⠁⠀⢠⡟⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢸⡇⠀⠸⣇
⠀⠀⠀⠀⠀⣼⠁⠀⢀⡾⠁⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⢀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠸⣇⠀⠀⢻⡀
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⢀⡾⠉⠳⢦⡴⠛⠁⢻⡄⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠹⡆⣿⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠘⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠙⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢠⣾⣇⣹⡆⢻⣄⠀⠀⠀⠀⠀⠀⣀⡿⢠⣿⣈⣷⣄⠀⠀⠀⠀⠀⠀⠸⠟⠀⢀⣰⠏
⠀⠀⠀⠠⠶⠶⠶⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⠿⠷⠶⠶⠶⠶⠶⠿⠶⠾⠿⠶⠶⠦⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⢀⣤⣤⣤⣤⡄

⠀⠀⠀⠀⠀⠀⣠⡾⠳⡀⠀⠀⠀⢀⡠⣚⢦⡤⡀
⠀⠀⠀⠀⠀⣾⣧⠏⢑⢾⣄⣠⠔⠉⠀⣸⠄⢠⣿⢳⡶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣠⠟⡠⢚⢁⠂⠙⠧⣀⣀⡀⠁⠀⢧⡏⣸⠅⣿⣃⠄⣩⠖⠉⢙⡦⠴⣒⠉⠳⣤⣄⣀⡀
⠀⠀⠀⠸⡗⠒⠚⠋⣉⡭⣷⣺⡥⢠⣿⠢⣠⣠⠇⡯⢾⠉⡠⠊⣠⠴⢚⡩⠔⠚⣩⣄⠀⠙⢶⡉⠍⠻⡄
⢀⣀⠀⠀⡌⠙⠒⣊⣁⣨⠄⢀⢀⡞⢂⢢⢹⡿⠖⠉⡠⠊⡠⠚⠶⠚⠁⠀⡔⠉⠀⠈⠙⠓⠲⠥⢬⡀⡇⡰⠲
⣾⢉⠱⠀⡄⢠⢠⢃⢠⡄⠆⠋⢰⠡⣸⠜⠋⣀⡀⣎⣴⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠁⠳⡅⣽⢡
⠸⡜⣇⠀⢳⠃⢃⡎⢁⠀⢠⡄⠛⡙⠁⠘⢉⢠⡖⠉⢀⡇⢰⠻⣦⣀⣀⣀⡀⢃⣠⠵⠒⠒⠒⠓⠢⡄⠈⢋⣮
⠀⢡⠙⢦⡈⠁⠘⠀⠁⢴⣾⡇⠰⢁⣤⡶⣿⣾⡘⠀⠀⣠⠊⢠⠄⠀⣀⣠⡜⢋⣠⠒⠁⣀⠀⠀⣸⢒⣀⡼⠛
⠀⠀⢹⢤⡀⠀⠀⢀⡠⠿⠷⠧⠶⠥⠞⣛⡛⠃⣾⣆⠀⠁⠀⠉⠀⡟⣏⣧⢁⡻⣮⢣⠀⠉⠀⠀⡏⢉⢀⠆
⠀⠀⠘⣧⠙⠄⠐⠐⣟⠉⢩⣭⠉⠉⢙⣛⡀⠀⣿⢸⡆⠀⡔⠀⡰⠋⢹⣬⠏⠱⣜⢮⣝⣓⣓⡀⠠⡂⡜
⠀⠀⠀⠘⣦⠁⠀⠀⠤⠆⠀⠴⠆⠀⠐⠒⠂⠀⢿⢸⡇⠀⢀⡼⠁⠀⢨⣭⠁⠀⠀⣛⡒⠫⣭⠇⢝⡲
⠀⠀⠀⠀⠹⣆⢐⠀⠩⠥⠀⠩⠭⠄⠀⠭⠄⠀⠬⢿⣇⡴⠯⠭⠀⠀⢘⣒⠀⠀⢀⣒⠀⢠⠄⢠⣱⠁
⠀⠀⠀⠀⠀⢹⡄⠂⠀⠩⠄⠀⠭⠥⠀⠩⠭⠀⠨⠭⠛⠀⠭⠭⠀⠀⢀⣀⠀⠀⠒⠃⠀⠝⢐⣪⡇
⠀⠀⠀⠀⠀⠀⢻⡌⠀⠀⠯⠀⠈⠭⠄⠈⠭⠀⠨⠭⠁⠀⠭⠭⠀⠀⠭⠀⠀⢀⣚⠀⣸⠃⢪⡿
⠀⠀⠀⠀⠀⠀⠈⢿⡔⡂⠈⣧⡀⠩⡭⠀⠭⠥⠈⠭⠅⠀⠭⠭⠀⠀⢭⠂⠀⠸⠁⢀⡞⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡌⠄⠈⢥⠀⢈⡂⠐⣒⠀⢐⠀⠀⣉⣉⠀⢘⣛⠀⠀⡿⠀⣜⢁⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠈⠥⠀⣒⡀⢒⡂⢒⣂⠀⢐⣒⠀⠐⠃⠀⠨⠅⠠⠃⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠴⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠈⠀⠀⠀⠀⠀⠉⠀⠀⠠⠖⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⡠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡞⣠⠖⠉⠀⢠⡞⠁⢀⡾⠁⢀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣠⠗⠁⠀⢀⡠⣾⠁⠀⣾⠀⠀⡼⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣩⣤⡤⠴⠒⠉⠀⣿⠀⣸⠇⢀⡾⠁⢠⠞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢿⣰⣯⡶⣋⠤⠚⠛⠳⠆⠹⣄⠀⡆⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠀⡀⠀⠀⠀⠈⢦⠀⠀⠾⠋⠀⠀⠉⠉⠉⠀⠀⡠⠀⠐⢤⠀⠘⣶⡇⢪⡖⠀⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁⠂⣤⠠⠙⠒⠀⢠⢩⡆⠀⣠⣬⣌⠑⡄⠀⠀⠀⢰⣿⣿⣶⡀⠁⠀⠛⠁⠈⡛⠠⠞
⣀⡀⠤⠄⣠⠂⠀⠀⠀⠁⠀⠀⠙⢄⡀⠄⠐⠈⠀⠀⠑⠻⠀⠹⢿⣿⡇⠄⡠⠖⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⡜⢀⠀⠄
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⠀⠀⠀⠀⠙⠒⣀⢰⡄⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠉⠀⠀⠀⠀⣀⡤⢲⠙⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⡀⠸⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢲⠶⡲⠖⠒⡉⠍⠀⠃⠀⠀⠀⢡⡀
⠀⠀⠀⡆⠀⠀⡀⠀⠀⠀⡇⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⡀
⠀⡇⠀⡇⠀⠀⠧⠆⠀⠀⡇⠀⣿⣀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠈⠄⠀⠀⠀⠀⠀⣀⣠⡤
⠀⣷⠀⣧⠀⠀⠀⠐⠀⠐⠓⠂⠉⠀⠀⠀⠀⡀⠀⠠⠤⢤⣀⣀⡠⠔⠁⠀⢑⣄⡄⠀⠀⠀⠀⣀⠜⠐⠂⠠⡤⠀⠀⠀⠈⢄⣤⠴⠒⠋⠙⠢⢄
⠀⢹⠀⢸⡀⠤⠄⠀⠀⣀⠠⠄⠂⢀⡀⢀⣀⠀⠤⠀⠠⠗⣴⠂⠀⢀⠠⠖⠋⠡⢑⠦⠂⠐⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⣦⡀
⠠⠘⠉⠀⠀⠀⡠⠂⠁⡀⠀⠀⠐⠃⠀⠀⠀⢙⠁⠀⠹⢤⡙⡢⠈⠉⠒⠀⢴⡒⠈⡀⠀⢀⡀⠄⠐⠉⠀⠀⠀⠀⢀⠠⢖⠉⠄⣈⡱⠶⢖⡊⠉
⠀⠀⠀⠀⠀⠀⣄⠀⠣⡀⠀⠀⡈⠀⢠⠉⠑⠥⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⠆⣠⠊⠑⠠⣀⢀⡀⠄⠒⠈⠀⠀⠀⠂⠌⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠈⠐⢄⡈⠑⣤⠀⠀⠆⠀⠀⣠⠼⠆⠘⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡄⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠏⠲⣄⣀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢌⣂⢐⠦⡀⠁⠀⢀⡰⠶⢃⡀⠁⠒⠠⣀⠀⠀⠀⠀⢀⠠⠀⠊⢀⡤⢺⠐⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠈⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡈⠑⠤⣀⠻⣀⡉⡼⠀⠀⠀⢀⣀⠦⠐⠈⣀⡠⠔⠊⠁⠀⠸⣠⡇⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠉⠂⠤⠤⠄⠒⠈⢁⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢠⠌⠙⠤⠄⠀⠐⢒⣉⡠⠤⠛⠻⢅⡀
⣀⣤⠴⠖⠋⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠀⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡄⠀⠀⢨⣅⠀⠀⠀⠀⠀⠈⠙
⠧⣄⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡐⠳⡀⠀⠈⡄⠉⣒⣤⣤⠖⠊⠉
⠀⠀⢉⣲⠤⣔⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠗⠂⠔⠒⠈⣱⠀⣈⠢⠀⠙⠲⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⢣⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠉⠁⠈⢳⠀⠀⢀⠔⠒⠉⠁⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⢀⣀⣀⠀⠻⣉⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⠀⠀⡎⢀⠎⠁⠉⣉⠖⠸⡄
⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⢠⡾⠋⠁⠀⠀⠀⠱⡈⡄⠀⠀⣠⠶⠚⠲⠄⠀⠞⠉⠉⠙⠓⠀⢸⠀⣠⠃⡸⠀⠀⡜⢁⢔⠭⠒⣉⠵
⠀⢀⠴⠊⠁⠀⠀⣠⡤⠄⠈⠀⠀⣴⠞⠛⠁⠀⢣⢣⠀⠈⠀⢀⣀⡀⠀⠀⠀⣠⣤⣤⣄⠀⢸⠊⠁⡠⠃⠀⠀⠑⠯⠥⠒⠉
⠀⢸⡀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠃⠀⠀⢀⣄⠀⢟⠆⠀⠰⠛⠉⠉⠃⠀⠀⠁⠀⠀⠀⠀⡘⠉⠉
⠀⠀⠑⢄⠀⠘⠁⠀⣴⠟⠉⠁⠀⢀⢄⣰⠋⠸⠀⠘⠟⡄⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢑⠦⡀⠀⠃⠀⠀⢀⣨⡵⠛⠁⠀⢰⠀⢀⡇⠸⡀⠀⣠⠶⣖⣛⣛⡛⠛⠳⡄⢰⠁
⠀⠀⠀⠀⡌⢀⠎⠢⣄⠀⠚⢍⡁⠀⠀⢀⡠⠃⠀⢸⠁⠀⠱⣄⠛⠉⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⢇⠘⠀⠀⠀⠙⠢⣄⡈⠉⠉⠁⠀⠀⣠⠎⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠈⢿⡒⠤⣀⡀⠀⠀⠀⠉⠉⠁⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠉⠒⠤⣈⡉⠒⢄
⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠬⢷⡞
⠀⠀⠀⠀⠀⣸⠤⠄⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢆⠀⠀⠀⠀⣠⢖⣦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡥⠋⣳⣧⣀⡴⠊⠁⢸⡅⢠⣿⢲⠶⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢴⣫⣐⡈⠻⢤⣀⣀⠉⠀⠷⡏⣼⢠⣿⡤⣀⠽⠊⢉⡷⠶⣚⠉⠳⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⡒⠋⢁⡴⡺⠿⠵⢦⡾⠷⣤⣴⠁⡗⠯⣡⠞⣡⠴⣊⡥⠖⢋⠩⢤⣀⠙⢮⠄⡉⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠆⠍⢋⡥⠤⠶⢀⡄⡾⢙⠆⣻⠟⠋⢁⠞⣡⠞⠑⠋⠁⠀⠼⠁⠀⠀⠩⠉⠓⠒⢦⣸⢰⡋⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡡⡌⠀⣆⡐⡜⣤⠶⠚⠉⢠⣇⡴⢋⣡⠤⢀⡷⢾⡄⢀⡀⠀⠀⠀⠀⡀⠀⡀⢐⣀⣀⡀⠀⠻⣼⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠧⢿⡀⠹⠐⢸⠁⠆⢀⣶⠀⡐⠀⢀⣤⣲⢯⠀⠸⢁⡿⠙⠲⠶⠶⠤⣤⠖⢋⠍⠉⠉⠉⣵⢀⠨⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⠂⠀⠀⠀⢈⣿⣿⢀⣤⣺⣯⣿⡿⣦⡀⠠⠊⠀⠎⢠⣴⢶⠫⣤⡺⡅⠀⠆⠀⠀⡯⠖⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣝⢦⡀⡠⢾⡦⠤⢼⡥⠴⢶⣶⡂⠀⣿⢷⠀⠀⡀⠀⣼⠼⡼⣦⢦⣻⣜⠤⠤⠤⠘⢀⢂⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⢀⠀⠘⣣⠀⢸⣧⠀⠐⡶⠆⠀⣿⢸⡆⠈⢀⡼⠁⠀⢻⡇⠀⠙⣿⣿⣛⣓⡦⡴⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠤⠀⢒⡀⠐⣛⡀⠀⠘⡀⠀⣹⣾⡇⣰⣿⡄⠀⠀⠿⠆⠀⠀⠭⠀⢈⠁⢐⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠄⠀⠦⠀⠘⢓⠀⢐⣒⠀⠐⠛⠟⢐⣒⠁⠀⠨⠭⠁⠀⢰⠃⠀⠿⠠⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡈⠀⠨⠤⠀⠶⠄⠀⠒⠀⠀⠤⠀⠀⠭⠅⠀⠭⠭⠀⠀⣒⠀⣼⠃⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡠⠀⢹⡄⠘⣟⡀⠸⠧⠀⡭⠀⠈⠩⠅⠀⢨⡃⠀⢸⠁⢀⡖⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡡⠀⠭⠄⠘⠇⠀⣓⠀⣿⠀⠈⣛⡁⠀⣻⡃⠀⡿⠀⣼⢡⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠩⠄⠩⠄⠐⠀⠠⠄⠀⠤⠤⠀⠉⠀⠀⠇⠐⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠶⠶⠤⠤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠒⠛⠋⠁



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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣍⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢿⡷⠰⠾⡿⠿⠷⠾⠿⠿⠿⣿⡭⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣛⡻⠿⠿⠿⠻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡶⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠈⠀⠉⠩⠿⠟⠻⠟⠛⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠉⠠⠟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞⠿⣿⡿⠿⡿⣿⣿⣻⣿⣿⣿
⣿⣿⣿⡋⢐⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⠀⣻⣿⣻⣶⣿⣿⣿⣼⣿⣿⣿
⣿⣿⣿⡿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣦⡀⠰⠀⠀⠲⣂⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠿⢟⣿⣿⣿⣿⣿⣿
⣿⣿⣷⡷⡟⠛⠉⠁⠴⢘⣺⣿⡏⠀⣹⣿⣅⣀⡀⠀⡀⠀⠀⣴⠾⠶⣦⣿⡟⠃⡀⣢⡀⠀⢀⣀⠀⡠⠀⢑⠀⠀⠀⢶⣤⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⡀⠀⣀⣀⣸⣿⣿⣿⣷⣿⣿⣿⣷⣿⣤⣀⣬⣡⣴⣿⣷⣤⣤⣿⣿⣮⣽⣯⣿⣷⣿⣿⣷⣶⣶⣼⣶⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣷⣿⢽⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣄⠀⠀⠠⣀⠀⠙⠳⡀⠀⠀⠀⡘⣷⡄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢦⠄⠀⣩⠉⢀⣤⣶⡇⠀⠀⠶⣿⡷⠂⠀⠀⠘⠿⠀⠀⢻⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⡏⠀⠴⠋⢠⠟⠁⠛⠀⠀⠐⠟⠉⠀⠀⢀⣨⠬⠛⠂⠀⠸⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠋⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣷⡤⢷⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⢠⡤⢤⡄⢤⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣻⠇
⠀⠀⠀⣠⡴⠋⠀⠀⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠅
⠀⠀⠘⠋⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡀
⠀⠀⠀⠀⣴⡟⠈⠠⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⡿⠟
⠀⠀⠀⣸⠟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⡀⢀
⠀⠀⠀⠉⠀⠀⠀⢹⣶⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⠿⠶⠂
⠀⠀⠀⠀⠀⠀⢀⡜⠙⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠙⠛⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣀⣠⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠒⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠇⠀⠀⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠁⠀⢠⣤⠴⠂⠀⠀⢀⡀⠀⠀⠀⢠⣤⣾⠏⠀⢠⡄⠀⢘⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠲⣔⠀⠀⠀⠀⢀⢟⠋⠁⠀⢠⠋⠀⠀⠉⠐⠶⠿⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⣦⣀⠈⠉⠉⠀⠄⠀⠀⢻⡒⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣯⣭⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣫⣭⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠛⠋⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣯⠁⠀⠀⠀⠀⠀⠀⢩⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⣠⢖⣚⣩⣥⣴⣖⣦⣄⡙⠇⠀⠀⠀⠀⠀⠀⠟⣁⣤⣖⣶⣤⣭⣛⣒⢦⡀
⠀⠀⠀⠀⢦⣴⣾⣕⣻⠿⠿⠿⢿⠿⣿⣿⠗⠀⣀⠀⠀⠀⢀⡀⠐⢿⣿⡿⠿⠿⠿⠿⢿⣓⣽⣶⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠃⠉⠉⠉⠀⠀⢀⣿⠀⠀⠀⠸⣇⠀⠀⠈⠉⠉⠁⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠋⢸⡇⠀⠀⠀⠀⣿⠈⠳⣄⠀⡀
⠀⠀⠀⠰⣦⣄⣀⣀⣀⣤⠤⠶⠛⠁⠈⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠈⠀⠙⠳⠦⢤⣄⣀⣀⣀⣠⡶
⠀⠀⠀⠀⢻⣷⡙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠘⢃⣭⣛⣛⣛⣫⣅⠛⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣱⣿⠃
⠀⠀⠀⠀⠀⠹⣷⡀⠻⣿⣦⣀⣀⣀⣠⣤⣴⣿⣿⡿⠋⠻⣿⣿⣷⣤⣤⣀⣀⣀⣠⣾⡿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⠈⠛⠛⠛⠛⠛⠛⣿⣿⣋⣀⣀⣀⣈⣻⣿⡟⠛⠛⠛⠛⠛⢋⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⠖⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣭⣿⠘⢯⣤⣤⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⠁⠀⠀⠀⠀⠀⠀⢰⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠛⠛⠳⠶⣤⣀⣀⣤⠤⢿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⠋⡴⠄⠀⠀⠴⠂⠈⡏⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⣀⣀⣀⣿⠀⠀⢻⡝⢣⡟⠀⠀⣿⡀⠀⠀⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡰⢋⡴⠖⠒⠻⢦⣐⣒⠈⠭⢐⣒⡿⠟⠛⠲⠶⠾⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢄⠀⢸⡆⠀⡇⢸⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⠒⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⡃⣞⠛⣋⣉⣉⣹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡹⢋⡞⢡⠄⠀⣄⢲⡹⣝⡆⠀⠀⠀⠀⠀⠀⠀⢠⡿⢛⡻⣿⠟⣛⠻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡈⠀⠈⠀⠸⢘⡣⠘⠄⠙⠀⠀⠀⢰⣲⠀⠀⠀⠀⣿⠿⣯⡀⣇⠸⣿⢇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⠞⠁⡠⠀⢠⠀⠀⠀⠀⣄⠀⢠⡀⠀⠀⠀⠀⡠⣄⠀⣹⣧⢭⣥⠿⢷⡶⠻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⣀⡀⠀⠰⠁⢀⡾⠛⠛⠳⣄⣸⣤⣄⢱⡞⢙⡇⠈⠂⠊⠀⣿⢅⡲⣿⢘⣻⡘⠟⣻⡽⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣉⣹⢆⡴⠚⠛⣷⠶⣤⡟⠻⣄⣠⢼⡇⠙⠋⠁⠀⠀⠀⠀⠻⣮⣥⣾⣶⣾⠿⠛⠉⣾⣿⡞⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⡁⢾⠀⣰⠶⣿⡀⠘⣷⠀⠙⣷⠀⠻⡟⠓⣦⠀⠀⠀⠀⠀⠈⠉⠋⣉⠀⠀⠀⠀⠉⠉⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⠴⢿⡾⢧⣽⡄⠈⢧⠀⠈⢧⠀⠘⣇⠀⠹⣆⠘⡇⠀⢰⣷⠀⢀⡀⠀⣿⠀⠀⠀⠀⢀⣴⣶⠸⣷⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⠀⢀⡿⣞⠁⢻⡄⠈⢧⠀⠈⢷⠀⠈⠀⠀⠘⣶⠇⠀⢸⣿⠀⣼⣧⠀⣿⣀⣶⣶⣦⠘⠿⣦⡄⢿⣿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠈⠙⣩⠖⠛⣰⠶⣿⡄⠈⠧⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⣿⣀⣿⠿⣷⣿⢿⣇⣾⣿⣄⣴⡾⠏⠘⠃⠰⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⠀⠀⢻⡄⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⠿⠟⣠⡉⢁⣈⠛⣋⣈⠛⣡⠀⣶⣾⣆⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣧⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢿⣇⣼⣿⣼⡟⢻⣿⣿⣀⣿⠻⣿⠏⠙⠁⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠈⠳⢤⣤⣤⣶⣋⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠙⢛⣿⠙⠿⠿⢃⣙⠛⣿⣠⣿⡶⣶⡜⢿⣦⣍
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⢸⡁⢈⡇⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠙⣆⠀⢸⣆⠀⢿⣿⠋⣴⣶⡀⢸⣿⣧⣿⡇⣿⡆⢸⣿⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠉⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠘⣧⠸⣿⠀⢘⣿⡀⣿⣛⣿⡌⣿⡝⣿⣿⢸⣷⣾⢿⣦⣤⣼⡟
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣠⠏⠀⢿⡿⠛⢿⣇⣿⠟⠙⠿⢿⡇⠈⠛⠀⠛⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠤⠴⠚⠁⠀⠀⢸⣷⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣗⣈⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⢫
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⡜⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢱⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡾⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣎
⠀⠀⠀⠀⠀⣇⡀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⠤⠤⠤⠤⠤⠤⢤⣀⡀⠀⢀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢰⡉⠀⠈⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢉⡆
⠀⠀⠀⠹⣏⣉⣉⣉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⠏
⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣦⠀⠀⡴⠢⡀⠀⡠⠲⡄⠀⢠⠖⢄⠀⢠⠔⢦⠀⠀⡴⢢⡀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⡟⠚⡇⠀⡟⠒⡇⠀⢸⠒⣻⠀⢸⠓⢻⠀⠀⠓⠚⡀⠀⠀⡇
⠀⠀⠀⠀⢸⡇⠀⠠⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⢹⠀⠀⡧⢤⡇⠀⣧⠤⡇⠀⢸⠤⣾⠀⢸⡤⢼⠀⠀⡧⢴⠁⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸⠀⠀⡇⠀⡇⠀⡇⠀⡇⠀⢸⠀⢺⠀⢸⠀⢸⠀⠀⠆⠀⡇⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⣸⠀⠀⡟⠚⡇⠀⡟⠛⡇⠀⢸⠛⢻⠀⢸⠋⢻⠀⠀⡟⠛⡄⠀⢸
⠀⠀⠀⠀⠀⣇⠀⢠⠁⠨⠉⠉⡗⠒⡖⠊⡟⠉⡟⠑⢲⠒⢶⠒⢶⠶⢲⠒⠁⡇⢸⠀⠀⣇⣠⡇⠀⣧⣀⡇⠀⢸⣀⣼⠀⢸⣀⣸⠀⠀⣅⣠⠆⠀⣼
⠀⠀⠀⠀⠀⣸⠀⢰⠓⢚⠀⠀⡟⠚⡇⠀⣟⠒⡇⠀⢸⠒⢻⠀⢸⠓⢻⠀⠀⡗⢺⠀⠀⡇⠀⡇⠀⡇⠀⡇⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⡇⠈⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠀⢀⡦⢼⠀⠀⡧⠤⡇⠀⣿⠤⡇⠀⢸⠤⣾⠀⢸⠤⢼⠀⠀⡧⢼⠀⠀⡗⠒⡇⠀⡷⠒⡇⠀⢸⠒⢿⠀⢸⠒⢺⠀⠀⠗⠺⡁⠀⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠳⠞⠀⠀⠓⠔⠁⠀⠙⠶⠃⠀⠘⠢⠋⠀⠈⠢⠚⠀⠀⠳⠞⠀⠀⠓⠖⠁⠀⠙⠖⠃⠀⠘⠶⠋⠀⠈⠢⠚⠀⠀⠳⠞⠀⢠⠇
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⡧⠐⢿⠂⠀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⡉⠑⢾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢡⠃⠆⠠⣷⡄⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡹⠁⠀⣸⠷⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢼⠘⠿⣲⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣁⣐⣂⣁⣤⢤⠛⢷⡀⠀⠀⠀⠀⣀⣤⣶⣦⡴⢁⣎⠉⠁⠀⠀⠀⢀⣴⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⢸⠃⠈⠁⠀⠘⠊⠉⠪⡑⠦⢐⡨⠿⣿⡯⣿⣷⣼⣯⣷⡄⠀⠀⣠⠏⠈⢯⣦⣿⠋⢹⡄
⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠀⣿⣿⣞⣯⣿⡼⢏⠳⢔⣫⡟⣴⠀⣸⢹⡀⢸⡾⡇
⠀⠀⠀⣛⡭⠭⠝⠒⠒⠉⠉⠉⠛⠓⠒⠲⠤⠤⠤⠀⣀⣀⣀⠀⠯⠽⣿⣛⣿⣿⡥⠄⠀⠉⢩⢎⡾⢹⠀⢸⡄⢳⢈⣧⡙⢦⡀
⠀⠀⢰⠋⡿⠿⢿⣶⣶⣶⣦⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢹⢀⡾⠯⠝⢻⣷⠁⠀⠀⠀⣇⣾⢁⡾⠀⢈⡇⢸⠸⢸⡎⣇⣷
⠀⠀⢸⠀⡇⠀⠀⠀⣀⣈⠉⠉⠉⠙⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡟⣿⣶⣿⣇⠀⠀⠀⠈⠉⠻⢧⣀⣼⣤⡷⠤⠼⠃⠈⠁
⠀⠀⢸⠐⡇⢀⡔⣫⠭⠭⡙⢷⡄⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣽⣻⠿⢿⣿⡆
⠀⠀⢸⠐⡿⣿⢸⠃⠀⠀⠘⡌⣿⣤⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣥⣹⣷⣶⣯⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣀⣀
⠀⠀⢸⢸⡏⠹⡜⢧⣀⣀⡴⢠⡿⢭⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⣿⠉⠀⠀⠀⠀⠀⠀⢀⣜⣹⡭⢷⣿⠋⠁
⠀⠀⢸⢸⡇⠀⠈⠒⠭⠭⠔⠋⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡤⣤⠤⠿⠀⠀⣀⣀⣀⣀⣀⢎⣾⡿⢣⡿⠁
⠀⠀⢸⢸⡇⠤⠤⢀⣀⣀⣀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢇⣠⠔⠒⢉⣩⠭⠤⠒⠲⠏⢾⡷⢳⡿⠳⢤⡀
⠀⠀⢸⢸⡧⢤⣀⣀⣀⠀⠀⠉⠉⢹⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣧⡀⠔⠋⠁⢀⡀⠈⢀⡀⠤⠀⡉⠙⢂⣠⡏
⠀⠀⢸⢸⡗⠂⠤⠤⠄⣉⣉⠉⠓⢺⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢛⣏⠙⢛⣟⡲⠲⠶⠾⠗⠶⠚⠻⠿⣿⣏⠗
⠀⠀⢸⣸⡋⠙⠒⠂⠄⠤⠤⢈⣉⣹⡇⢸⠀⠀⠀⠀⢀⣤⣶⠋⠀⠀⠀⠟⠘⡆⠈⠻⢀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠛⠻⠿⢷⣶⣶⣤⣤⣄⣀⣸⡇⢸⠀⣀⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⠢⣀⣀⠀⠀⠀⠀⠀⣀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⠁

⠀⠀⢰⡐⠂⠀⠀⠀⠀⠀⢴⡀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣁⣐⠄⠀⠀⠀⠀⢰⠀⠀⢀⣠⣄⣀⡀
⠀⠀⠈⣳⡄⠀⠛⠛⠀⠀⢿⣿⡆⠀⠓⣿⣿⣷⡄⠐⣾⣆⠘⢷⡀⠈⢭⣿⣎⣶⣆⠀⠀⠰⢾⣷⣄⣀⣉⣉⣀
⠀⠀⠀⣽⣿⣄⠀⠰⠶⠆⠀⠙⣿⡄⠀⠙⠛⠛⢻⣄⠈⠋⠀⠈⢷⡄⠀⠛⠛⠛⠻⣇⠀⠠⣄⠉⠻⣏⠉⠉⠉⠀⠀⢀
⠀⠀⠀⣿⣿⣿⣦⣶⣦⣤⣤⣤⣾⣿⣶⣤⣤⣴⣶⣿⣦⢤⡤⣴⣾⣿⣖⣒⣶⣶⡖⢿⣧⢀⣿⡗⢀⣾⣿⣾⣿⣿⣿⣷
⠀⠀⠀⣼⣿⣿⣧⣿⣿⣯⣿⣿⣿⣿⡿⠿⡿⣿⣶⣾⣿⢿⡇⣿⣿⣿⣿⠿⣿⣽⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⣿⣿⣿⣿⣿⣇⣿⣿⣿⡇⡅⣉⣿⡻⢿⣷⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⢿⣿⣿⡿⠉⠹⡿⠯⢹⣿⣿⣿⣿⡘⣿⣓⣿⣿⣉⣁⣉⣽⣿⣿⣷⣾⣿⣿⣿⣟⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⡏⢷⣾⡯⠭⠼⣿⣿⣿⠟⢼⣍⢸⣿⣿⢿⣿⣿⡿⠛⡋⠛⠻⣻⣿⣿⡇⠶⢸⣿⠗⣻⣿⣶⣖⣶⣶⡇
⠀⠀⠀⢡⣙⠻⡇⠹⠿⣷⣶⣚⣿⣿⣿⡞⢷⣿⣛⣻⣿⣏⣹⣿⣿⡾⢷⣦⣤⣷⢚⡿⡇⠛⡘⢿⡼⠿⠿⠭⠯⠉⠩⠁
⠀⠀⠀⠀⠙⠓⠃⠉⠓⠋⠉⠀⠉⠛⠋⠀⠈⠁⠉⠋⠉⠁⠀⠉⠀⠀⠈⠙⠻⠿⠍⠁⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⢤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠒⢒⡒⠲⠶⠤⠤⠤⢤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠔⠀⠀⠃⠀⠀⠀⠀⠀⠁⠉⠉⠉⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠘⠚⠁⠃⠀⠀⠨⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠊⠀⠀⠀⣇⠀⠀⡅⢈⣉⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠊⢏⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠇⠃⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠆⠦⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠐⠀⠀⠀⠉⠓⠢⢄⡀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢉⠉⠀⢇⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢧⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡌⠀⢰⡘⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢱⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⠀⢨⡃⠀⠀⠀⠀⠀⠇⡆⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⣿⡀⠸⠀⠀⠀⠀⠀⠀⢀⢧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠷⡉⢀⠄⠀⠀⠀⠀⠀⠀⠺⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠏⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠈⠽⣆⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⢀⣠⣴⠟⢃⠀⠀⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠊⠃⣀⠀⠀⠀⠉⠉⠀⠚⠛⠛⠉⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡆⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠕⠦⣀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠻⡇⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠂⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠄⡀



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣲⡶⠶⠶⣶⣒⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⣤⣤⡇⠀⠀⠀⠀⠀⢠⡒⠒⠒⢲⠀⠀⠀⠀⠀⢀⡔⠒⠒⣎⠀⠙⠢⢄⣀⣠⠤⠚⠁⠘⡖⠒⠲⡄
⠀⠀⠀⠀⠀⡴⠒⠒⡟⠒⠒⠒⢳⡀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡆⠀⠀⠀⢀⠎⠀⠀⠀⠈⠙⠲⢤⢴⠋⡦⠤⠤⠒⠋⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⡇⡎⡜⠀⠀⠀⠀⠀⢳⠀⠀⠀⣠⠞⠒⠒⠀⠚⢧⣀⣀⣠⣋⣀⡀⠀⠀⠀⠀⠀⠀⠈⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⣧⠟⠀⠀⠀⠀⠀⠀⠀⢣⢠⠎⠁⠀⣀⣀⣀⠀⢀⡈⢳⠖⠤⠤⠬⠭⠭⣉⣉⠭⠶⣆⠇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢰⠋⠁⠀⠈⡆⣿⢱⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣤⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⡇
⠀⠀⠀⠀⠀⠳⠦⠤⠤⠤⠤⠤⠤⠴⢲⡏⠀⡇⠀⠀⠀⠀⡇⣿⢸⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠯⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⡇
⠀⠀⠀⠀⠀⢰⠤⠔⠊⠉⠉⠉⠒⠒⣾⡇⠀⡇⠀⠀⠀⠀⡇⢿⣸⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣶⡟⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⠙⠒⢄⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠷⡿⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠛⡆
⠀⠀⠀⠀⠀⣘⡒⠤⠤⠤⠤⠤⠤⠐⣻⡇⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠤⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⡅
⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠉⠉⠉⠁⠀⣇⠀⠈⠉⠐⠒⠒⠂⠉⠁⠀⡟⠒⠒⠲⠦⠤⠤⣧⠤⠲⣟⡿⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⠇
⠀⠀⠀⠀⠀⠳⠤⠤⢀⣀⣀⣀⠤⠤⠔⠻⠤⣀⣀⣀⣀⣀⣀⣀⣀⠴⠋⠙⠛⠓⠒⠲⠶⠾⠖⠚⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠈⠇⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠐⠀⠀⠀⠸⣿⣿⣿⣿⣏⠐⢀⡁⢀⠀⠀⠀⠄⠠⠘⠛⣛⠘⢸⡟⠀⢹⣷⣷⠘⢡⡆⣶⢤⠀⢀⣴⠗⡀⢤⣿⣷⢡⣶⢿⣿
⠀⠀⠓⠀⠀⢁⠼⣲⢛⣾⡤⣀⠠⡶⠀⢀⡀⢠⣿⣿⣿⣿⢧⣗⣦⡴⢫⣄⠀⠀⠀⠈⢲⡿⣟⣀⠼⣧⡄⣸⣿⣿⡿⢦⡅⣀⣿⢷⣶⣴⣶⣿⡟⣿⣿⣿⣷⣿⣿
⠀⠀⠒⢂⣤⣭⣼⣻⠟⠿⠽⢉⠤⣄⠀⠈⠛⠀⠉⣿⣿⣿⣾⣿⣧⣷⣧⣾⣷⡄⢀⣴⣬⠁⠟⣃⡂⣟⠄⢿⣿⣏⣴⡾⣿⣿⣽⣿⣿⣿⣻⣾⡇⢀⢀⣤⣿⣿⣿
⠀⠀⠈⠀⠻⠹⡛⢯⣖⠀⢀⠀⠤⢾⠄⠐⠃⢀⡜⠉⢿⣿⣿⣾⢿⣿⣿⡿⣿⣿⡾⣿⣿⡀⠚⣀⠀⢹⡏⣿⡿⡏⣄⣠⣷⡯⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⠀⠠⠇⢁⠂⣝⠂⣴⣆⡀⠠⣦⠐⠶⠋⠀⢀⢻⣿⣿⣟⣸⣷⣿⣿⣿⡝⠿⠻⢿⠇⣠⣿⣀⣼⢇⣷⣿⣿⣿⡿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠁⠰⠀⠐⣿⠛⣿⣿⠗⠱⠁⠂⠠⠀⣚⢹⣿⣿⣿⠻⠿⣷⣿⣿⣿⡒⠈⡤⠃⠀⣏⡿⣿⠺⡇⣿⢸⣿⡿⢋⡚⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠋⠻
⠀⠀⠀⣫⡀⠔⠺⡆⠹⡶⠋⢉⡀⠉⡈⠀⠀⣤⢠⣴⡁⠈⠙⠆⠀⠙⠛⢿⠟⠁⠀⠓⠶⠀⢸⣿⣿⢁⠿⣾⣼⣿⣶⣞⣐⢦⣾⣿⠿⣯⡿⣿⣿⠿⣿⣤⠜⠛⠛
⠀⠀⢠⡟⠣⠄⢀⠁⠀⠄⠀⠀⠐⢶⠃⠀⡠⣈⡀⠈⠁⠀⠀⠤⢃⠀⢀⠀⠁⠚⠀⠃⡀⠀⠘⡇⣿⠸⠂⢾⣿⣿⣭⣿⠃⠐⠿⠿⠷⣯⣤⣼⠛⢿⠞⠹⠆⠐⢆
⠀⠀⠸⡇⠀⠱⠂⠀⠄⠀⠀⠀⠀⢠⠄⠀⠀⠈⠙⢡⣤⣤⣄⠥⠈⠁⠀⠀⠀⠀⢠⡄⠀⡀⠀⠀⠈⠀⠀⠈⠙⠘⠉⠀⠀⠉⢠⡟⢰⣼⡍⠀⠰⣄⠠⠆⠀⠀⠈
⠀⠀⠀⠹⠀⢀⠀⢬⡀⠀⠀⠀⠀⠀⠀⠰⠀⣀⠀⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠙⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠈⢧⡁
⠀⠀⠀⠀⠠⠘⠁⠀⠁⠀⠀⠀⠀⠤⠐⠃⠈⠁⠐⠂⠀⠀⠛⢆⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠶⠀⠀⠆
⠀⠀⠈⠀⠀⠀⠀⠊⢀⣘⢟⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⣀⡀⠐⣀⠀⠀⠀⠀⠢⣀⣠⡈⣤
⠀⡀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⣶⠘⣿⣅⡾⠏⣁⠀⢠⣾⣷⣾⣿⣤⠹⢤⠀⢀⠄⠐⠀⠄
⢠⡇⠀⠶⠀⢀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠵⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢿⣿⣿⣿⣤⣟⣿⣿⣯⣿⣿⣿⠷⣦⣾⣾⣿⡯⣖
⠀⠃⡀⣿⠀⠉⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠘⣻⣯⣯⣿⣹⣯⠙⣿⡟⢹⢃⣿⣿⣿⣿⣯⣿⡴⠇
⠀⠀⣡⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⣄⢫⣿⢩⣿⡦⣿⡆⣰⠿⣿⣿⣻⣿⣏⣿⠁
⠠⣀⢿⣯⡛⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡏⣯⡛⣶⡇⠱⡛⠁⠉⣰⣿⣿⣿⣿⡟⠈

⠀⢠⡞⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡛⠛⣛⠓⢦⠰⣏⠉⣿⠀⠀⠀⠀⣠⠖⠚⠲⠏⣳
⠀⣸⢠⣧⠸⡄⠀⢠⠖⢒⡲⠖⢛⡆⠈⣿⠸⠯⢇⡸⠃⢸⠀⠛⣛⠓⢦⢠⡇⢠⠏⠙⠦⠞⢠⠖⢛⡛⠋
⢀⡇⣨⠭⡄⢻⡀⣿⠀⣏⣹⠀⣿⠀⠀⡟⠀⡖⣦⠙⡆⢸⠀⢸⣉⡇⢸⠞⡇⠘⢆⣀⠤⡀⣿⠀⣏⣙⡛
⣯⣤⣭⠿⣇⣀⣽⠘⢦⣈⡥⣦⣬⡗⢸⣥⣀⣉⣥⡴⠃⠸⣤⣤⣉⡤⠞⠀⠘⠦⣤⣠⡤⠟⠘⢦⣌⣡⡽
⠀⠀⠀⠀⠀⠀⣠⣶⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢠⡏⢁⢤⡉⣸⠀⢀⡴⠋⠉⡇⠀⠀⢾⣉⣉⠉⢳⠀⠠⣞⣉⡉⠙⡆⠀⢀⡴⠟⠳⢤⡀
⠀⠀⠀⠀⠈⠧⣈⠓⢯⡉⠀⠀⠛⡯⠀⡇⠀⠀⠀⢁⡞⢀⡜⠀⠀⠈⣵⠋⢴⡃⠀⡏⢠⡎⠙⠞⠁
⠀⠀⠀⠀⢰⠚⢮⣱⠀⢹⠀⠀⣀⡇⠀⣇⡀⠀⡴⠋⢠⣮⣄⠀⢠⡖⢮⠗⠀⡷⠀⠳⣄⡙⢋⣳⠄
⠀⠀⠀⠀⠸⠼⢶⣠⠴⠋⠀⠀⠷⠤⠤⠼⠃⠀⠳⠶⠶⠦⠾⠀⠈⠓⠦⠴⠚⠁⠀⠀⠀⠻⠏
⠀⠀⠀⠀⠀⠀⠈⠉⢠⣤⣤⡀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠀⠀⣀⣀⣀⡀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⡇⢀⡤⣄⠀⠸⣄⣳⢯⣈⡇⠀⠿⠤⠤⠗⠀⠫⠴⣦⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠼⠀⠘⠦⠞⠀⠠⣾⠏⢰⡿⠃⠀⢠⡞⢳⠀⠀⠀⣾⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢲⡄⠀⢠⣞⣍⡳⢤⡠⣦⠀⠀⠀⠀⣹⣾⠀⠀⠀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠁⠀⠈⠓⠚⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠙⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠛⠉⠁⠀⠀⡇⠀⠈⠙⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠉⠀⠀⣀⣠⣤⠴⠞⠛⡟⠓⠦⣤⣀⠀⠈⠉⠛⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠀⠀⣀⣠⣤⠶⠞⠛⠉⠉⠀⠀⠀⢰⡇⠀⠀⠀⠉⠛⠲⢤⣄⡀⠀⠈⠉⠛⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠛⠛⠉⠉⠀⠀⣀⣠⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣠⣴⠦⣤⣤⣄⣀⠈⠉⠛⠶⢤⣀⡀⠀⠉⠛⠶⣤⣀⡀
⠀⢀⣀⣤⠴⠚⠉⠁⠀⠀⣀⣀⣤⡴⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⣦⣿⡾⠋⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠈⠉⢛⡳⢦
⠀⢿⠓⢤⣀⣀⡤⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠤⢤⣄⣀⣬⠿⠶⠚⠉⢁⣼
⠀⢸⣆⡀⠈⠙⠲⢦⣄⡀⡠⠴⠖⢶⣤⡶⠟⠋⠙⢢⡀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⢶⣤⠒⠋⠁⠀⠀⣀⡤⠞⠋⣿
⠀⠀⣿⠉⠓⠲⢦⣄⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⣀⣀⣀⣤⣀⣀⣀⡀⠀⠀⠙⠶⣤⣄⣀⣠⠴⠖⠛⢿⡁⠀⠀⠀⢻⣀⣤⠴⠖⠋⠁⠀⠀⢠⡇
⠀⠀⢸⡆⠀⠀⠀⠈⢙⠶⣤⣀⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠙⠯⣍⣉⣉⣉⠉⠉⠉⠙⠛⢷⣤⠤⠖⠋⠉⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣷⠀⠀⠀⠀⢸⣠⡇⠈⠛⠦⣄⡈⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣦⠸⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢹⡆⠀⠀⠀⠈⣟⣇⠀⠀⠀⠀⠙⠲⣄⣀⠈⠉⠛⠷⣦⣄⡀⣀⡤⠖⠋⠉⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿⠀⣿⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⢮⣗⣦⣄⡀⠀⠀⠀⠉⣻⠶⢦⣄⡀⠉⣿⠁⣀⡤⠖⠚⠉⢉⡇⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⣿⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣤⡴⠋⠀⠀⠀⠉⠓⣿⠋⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠀⡄⠀⠀⠀⣿⠀⠀⠁⠀⢸⡇⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⠃⣸⠁⠀⠀⠀⣿⠀⢀⣀⣀⣼⡇⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠋⠀⠀⠀⠀⣿⠾⠛⠋⠉⠉⠀⠀⠀⣀⣴⡟
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠃⠀⠁⠀⠀⢀⣀⡛⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉
⠀⠀⠀⠀⠀⠈⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⣤⡶⠶⠟⠛⠉⠀⢀⣠⡴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣇⣠⡴⠞⠉⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⣿⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠿⠶⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡏⠉⢠⣬⣻⡛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⢿⣿⣿⣿⡇⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡝⢶⣆⠀⠙⠻⢿⡇⠀⢸⣿⣿⡇⠀⢀⣠⣴⣿⡿⠇⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣙⢿⣦⡀⠀⠀⠀⠈⢻⣿⣷⣿⣿⣿⠿⢋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣾⣿⣷⣾⣿⣿⣿⠩⠭⠝⠋⣁⢠⣭⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠈⠁⠉⢹⣿⣿⣿⣾⣿⣷⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣆⣀⣰⣧⢸⣟⠘⠿⠿⠃⠻⠿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣍⣉⣹⣿⠇⢸⣿⡄⠀⠶⠶⠖⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠉⠉⠀⢸⣿⣿⣾⣧⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⢸⣿⣿⣿⣿⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣹⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣾⠉⣙⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣗⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⡀⠀⠀⣀⠀⠀⠀⡀⡀⠀⡀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠈⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣠⣴⣾⣿⣿⣿⢿⡿⠿⠛⠛⠿⢿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠘⣇⠀⢀⠀⠀⣼⠁⠀⠉⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠉⠹⣄⠀⠀⠀⠀⣿⣤⣿⣷⣤⡟⠀⠀⠀⠀⣴⠏⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠁⠀⠀⠀⠙⣦⣠⡴⠖⠛⢛⣉⠉⠛⠛⠶⢤⣀⡾⠃⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠙⢶⣄⡀⢀⡴⠋⣡⠴⠚⠋⠉⠉⠉⠙⠓⠦⣄⠉⠳⣄⠀⠀⣀⡴⠟⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠈⣻⠏⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠈⢷⡟⠁⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣀⣀⡀⠀⢰⠏⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢷⠀⠀⠀⠀⢘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠉⠙⠛⣿⠀⡏⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⢸⡄⢸⡷⠞⠛⠛⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠶⠶⠚⠛⢿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠷⢤⣤⣄⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠘⣧⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⡟⠀⠀⠀⠀⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⣀⡴⠞⢷⣄⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⣴⠿⣤⡀⠀⠀⢀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡟⠉⠀⠀⠀⢙⣶⣄⡈⠙⠒⠲⠶⠖⠒⠛⠁⣀⣴⠟⠁⠀⠀⠙⠳⣤⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⣰⠏⠀⠈⠉⠛⡶⠶⠶⠶⣶⠞⠛⠉⠹⣆⠀⠀⠀⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣾⡃⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⠀⠀⠙⣧⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⣄⣀⣿⠀⠀⠀⠀⢸⡇⢀⣀⣤⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠦⠾⠴⠒⠒⠂⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠠⠖⢀⣀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠂⠀⠁⠢⡘⡉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⡔⠁⠀⢀⠐⠠⡐⡉⠢⡀⠀⠀⠳⠤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⡢⠄⠀⠀⠀⠉⣦⠤⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢀⠀⠀⠀⠀⠸⠤⣔⡔⠀⠒⠀⠈⢢⣄⣱⣤⡄⠀⠀⣀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⡄⠀⡀⠀⠀⡀⠀⡁⠈⠊⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢁⠤⢺⣧⡀⠀⠒⣠⣀⠀⠀⠀⣀⠋⠀⠛⢡⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠛⠚⠱⠁⡀⠓⠚⠁⠀⡸⠡⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡊⠉⢠⠁⠀⠘⠿⠛⠞⠀⠻⠿⠃⠀⠠⢋⣀⣌⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠈⢀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠙⢷⣶⡄⡒⠃⠀⠀⠀⠔⠁⡀⢩⠡⣈⠆
⠀⠀⠀⠀⠀⠀⠸⣀⠀⢀⢤⡿⡐⣂⡐⢀⡽⢢⠄⠚⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⣅⠂⢀⢤⡺⡉⢃⡀⠂⡸⣢⡀⠐⠁⣼⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠮⣑⣭⣭⣭⡭⡭⡕⠣⣊⠥⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣔⠮⣳⠯⠭⠭⢭⣭⣭⢝⡻⣆⠐⠸
⠀⠀⠀⠃⣀⠀⠀⢰⡢⡽⢽⣒⣒⣒⣒⣒⣲⣔⢭⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣢⡝⢐⡓⣓⣒⢛⣿⣟⡌⢒⡭⣆
⠀⠀⠀⠀⠀⠀⡄⠈⠙⠀⡒⠖⠒⠒⠒⠒⠖⡈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠀⠘⡶⠶⠒⠲⠶⠞⠂⠀⠛⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠆⠀⣢⢃⣸⣷⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢨⠇⠈⠭⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠐⠀⠘⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠙⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠃⠊⠛⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠈⠃⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⢣⡱⡞⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡙⠄⠀⡠⠔⠉⢉⣀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⢀⡀⠀⠀⠀⢀⡠⠊⠁⠀⣈⣆⣯⠀⣠⣶⠗⠉⠱⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠁⢠⢃⢾⠱⣀⠠⣚⢣⠀⢀⠴⠊⠀⠑⠤⡙⠓⠤⠤⠀⠠⠬⠆⠒⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡌⠰⢁⠎⠀⠳⡈⠢⣀⠡⠻⡁⠀⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠒⠂⠉⠁⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⢠⠁⠎⠀⠀⠀⠈⠲⢄⡐⠤⡘⣄⠀⠀⣀⠀⠀⠠⠠⢄⠀⠀⡠⠒⠢⢄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡡⠊⠀⣂⡸⠄⠒⠉⢃⠀⣀⡀⠬⠷⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠈⠢⣀⠷⡀
⠀⠀⠀⣠⠄⠠⠤⢒⡡⢊⠥⠔⠊⠁⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠢⡀
⠀⠀⠀⠏⠀⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⣘⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠃⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⡘⠒⠄⣀⠂⣀⠠⠤⠐⠀⠲⠀⢂⠀⠀⠀⢀⡠⠤⠦⡄⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠤⢤⡀⡀⠀⢸⠀⡠⠤⠀⠀⠒⠂⠀⠀⠠⠄⣀
⠐⡤⠄⠤⠦⢼⠀⠀⠀⠀⠀⠆⠀⢨⡄⠄⠤⠼⠵⡄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡴⠡⠤⠠⠤⢯⠀⠒⢄⠀⠀⠀⠀⠀⠀⠈⠱⢲⣄⡀
⡠⠇⠀⠀⠠⡌⠀⠀⠀⠀⠀⠰⡀⡰⠀⠀⠀⠀⠀⠈⢆⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠎⠀⠀⠀⠀⠀⠀⢂⠀⠀⢣⠀⠀⠀⠀⠀⡰⣱⣗⣝⢝⢓
⠓⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠨⠀⠆⠀⠀⠀⠀⠀⠀⠀⠓⠢⣸⠀⠀⠀⠀⡔⢰⢏⢾⣢⢃⠊
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡌⢀⣀⠀⣀⣀⠀⣀⡀⢀⣈⣄⠀⠈⢢⠀⠀⠀⠀⠀⠀⣀⣤⣧⣀⢀⣀⡀⢀⣀⠀⣀⣀⠀⠉⠢⢄⡌⣰⢣⡿⣱⢃⠂
⠀⠀⠀⡶⠒⠀⠒⠀⢀⠀⠀⡐⠁⠀⠀⡇⡇⠅⢸⠀⣀⠸⠀⠈⢦⡀⠀⠱⣄⣠⠔⡖⣋⢠⠀⡳⢾⢸⡰⡇⡇⠀⡅⡇⠘⠀⠀⠀⠀⠉⢳⢿⣼⢣⠂
⠀⠀⠀⠐⣄⠀⠀⠀⠀⠑⣴⠁⠀⠀⠀⡅⡇⢹⢸⠀⣧⢰⠀⠀⠂⠐⠤⠆⣉⢘⡩⡇⠖⠸⠀⠽⢸⢸⠀⡇⡅⠀⡃⡇⢘⠀⠀⠀⠀⠀⠘⠀⠈⠁
⠀⠀⠀⠀⠈⡗⠂⠀⠒⠉⠀⣒⡁⢈⢒⠁⢑⣋⠀⣓⡋⢈⢒⠁⢙⣊⠀⣑⡋⢈⣓⡁⢘⣊⠀⣑⡋⠈⣒⡁⢉⣚⠀⣑⡊⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠈⠀⡇⡇⠈⢸⠁⢉⢸⠐⠆⡥⢬⠀⣇⡿⠀⠀⡇⠀⠈⠀⠃⢸⢸⠀⡇⠁⠀⠁⠃⠈⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⢸⠀⠈⢸⠀⠀⠆⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⢸⢸⠀⡇⡄⠀⡀⡆⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠂⡁⢈⠂⠁⠑⡊⠈⠁⡃⢈⠐⠁⢙⢘⠀⡃⡃⠈⡀⡁⢘⢈⠀⠑⠈⠈⡂⡁⢉⢈⠁⡑⡊⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠁⠈⠀⠇⡇⠈⢸⠁⢐⠸⠀⠀⠁⠈⠀⠁⠈⠈⠀⡇⠀⢨⠀⡇⢸⢸⠀⡇⡇⠀⠁⡇⢈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⢰⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⢾⢸⠀⠇⡇⠀⠁⡇⢰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣀⡃⢈⣐⠁⢑⣊⠈⣃⡀⢈⣐⠁⢙⣈⠀⣃⡋⢈⡂⡁⢘⣈⠀⣃⡈⠈⣐⠁⢙⣈⠁⣁⡊⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠁⠈⠀⠇⡇⠈⢸⠁⠀⢸⠀⠀⠅⠈⠀⠁⠀⠈⠀⡇⠂⠨⠀⠁⢸⠸⢆⡃⠏⠀⠀⡇⠸⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⡆⡇⠀⢸⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⢸⢸⠈⡇⠃⠀⡆⡇⢠⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠈⠂⠃⠈⠒⠁⠑⠊⠈⠓⠂⠘⠒⠁⠑⠘⠀⠃⠊⠈⠂⠁⠑⠘⠀⠑⠊⠈⠒⠃⠑⠒⠁⠓⠊⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠔⠁
⣿⣿⡿⠛⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⠏⠁
⠀⠀⠀⠀⢤⣴⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠰⡇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣿⣿⣿⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⣠⣤⣧⣤⣾⡀⢀⣧⣤⣤⣤⣄⡀⠀⠈⠙⢷⣌⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣠⠖⠚⢯⣁⢀⣀⣀⡀⠉⠉⠀⢀⣀⣠⠬⣝⠳⣦⡀⠀⠛⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡖⠁⠀⢀⡸⠋⠉⠉⠉⠉⢉⣻⠟⠉⠉⠁⠀⠈⣇⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⣰⡶⠚⠛⠛⠛⢣⣀⣠⢿⣤⣀⣀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠉⢹⠄⠈⢷⣄⣀⣀⣀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣻⣿⡶⠶⠤⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠆⠘⢧⣀⣀⣤⠀⠀⠀⣠⣼⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⠴⢧⣀⡀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠲⠤⠖⠁⠈⠷⠶⠶⠶⠶⠶⠶⠶⢶⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠾⡇⠀⠀⠹⡛⢻⡁⢀⡿⠻⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⠆⢨⠿⡟⢿⠄⠀⢀⣧⣄⣉⣉⣀⣀⣀⣀⡤⠤⠔⠛⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠃⢪⡀⠀⢸⠋⢻⠛⡏⠉⡿⠛⠃⠀⠀⠀⠀⠠⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡍⠛⢿⡶⠾⠿⣶⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⣀⣤⡶⠛⠛⠛⠓⠒⠶⠶⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣁⠀⠈⢳⡒⣦⣀⣀⣀⠳⠤⠴⢶⡀⠀⢀⣖⣋⣀⣤⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠉⠁⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢽⡆⠚⢽⣍⢋⡉⣩⡯⠓⢸⡽⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡰⢮⣟⡙⣇⠸⠍⣿⣷⢾⣿⠩⢇⣸⢋⣻⠵⢎⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡡⡮⢸⣏⠀⡏⠸⣦⣓⣸⠽⠯⣇⣚⣤⠇⢩⠀⣹⡇⢵⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢻⡀⢠⡟⠻⣄⣹⣆⢻⣈⣿⡖⢒⣿⣁⡟⣰⣏⣤⠟⢻⡄⢀⡟⠾
⠀⠀⠀⠀⠀⠀⠀⠐⢉⣁⡈⠢⠹⣦⣉⣄⠳⣽⣿⣦⢏⣼⣿⣷⣧⡹⣴⣿⣯⡞⣠⣩⣴⠏⠔⢁⣈⡉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠈⠋⣁⣤⡈⢆⠈⢿⣾⣿⣿⣷⣭⣍⣩⣭⣾⣿⣿⣿⡿⠁⡰⢁⣤⣎⠙⠁⠥⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠰⡯⢄⢙⣉⣂⢺⣼⢃⠁⠸⣿⣿⡿⢿⣿⣿⠇⠈⡼⣧⡟⣐⣉⡋⡀⢽⠆⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠾⠟⠛⠉⣷⡿⡌⠱⠠⠤⢼⡟⢻⡷⠤⠄⠎⢡⢿⣾⠉⠛⠻⠷⡲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡌⣡⠈⠀⢹⠀⠉⠉⢷⡜⢆⣴⣤⣼⣇⣸⣥⣤⣤⡴⢢⡞⠉⠉⠀⡗⠀⠁⣌⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⡀⠀⣟⡃⢀⣀⣨⣿⠌⢡⣬⢭⣼⣧⡭⣥⡍⡡⣿⣅⣀⡀⢘⣻⠀⢀⡇⠈⠃
⠀⠀⠀⠀⠀⠀⠐⡚⠀⢸⣧⠀⡄⡗⢿⣿⡿⠛⠘⢣⠹⡜⣾⣶⢣⠏⡜⠓⠛⢿⣿⡿⢺⢨⠀⣼⡇⠀⢓⠂
⠀⠀⠀⠀⠀⠀⠸⢃⠐⢷⡍⠸⢿⣅⠘⠿⠃⢸⡜⠃⠓⡙⠏⠻⢋⠚⠘⢣⡇⠘⠿⠃⣈⣿⠄⢩⡾⠂⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡇⢀⡼⠁⠠⠀⣀⡧⡀⢻⡷⣜⢈⡁⣣⣾⡗⢀⢼⣄⠀⠄⠈⢧⡀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣭⣤⠀⢀⡜⡿⡓⣋⠄⣸⣅⣽⣯⣨⣇⠠⣙⢚⣿⢣⡀⠀⣤⣭⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⢿⠆⣆⡷⢁⠙⢫⠟⡽⠟⢻⢛⠿⢧⠹⡝⠊⠄⢶⣰⢰⡿⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠸⣆⢠⣤⢋⢎⠡⠠⠒⠈⠶⣭⢣⡔⢒⡶⠓⠒⡆⠎⠡⣙⣤⡄⣰⠇⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠛⠻⠷⠒⣾⡆⢠⣠⣜⣠⠠⣿⣧⠀⢰⣽⠐⠶⠟⠛⠻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠊⡤⠒⠰⣶⠿⡇⠀⢉⣏⣉⣴⣏⠛⠀⢸⠿⣶⠆⠐⢤⠑⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⡔⠀⣀⠰⠄⣷⠀⠈⢿⣻⢻⣾⠃⠀⣾⠠⠦⣀⠀⢠⠑⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠄⠀⠀⠄⢾⣟⠃⠀⠀⠙⠋⠁⠄⠚⣻⡷⠠⠈⠀⢠⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠠⢂⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⢂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢁⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠁⠁⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⢀⡊
⠀⠀⠀⢄⠀⠘⢶⣄⠀⠻⡅⠀⠀⠀⠀⠈⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⢀⠀⠀⠀⠀⡽⠁⢀⣴⠞⠀⢀⠄
⠀⠀⠀⠈⠢⠀⠀⠁⠠⠄⡠⡀⠀⡀⠈⠓⠀⣂⣀⡀⠈⠀⠀⠀⠀⠀⠈⠀⣀⣀⡀⠐⠉⠀⡀⠀⡠⠀⠤⠀⠀⠀⠠⠊
⠀⠀⠀⠀⢸⠷⣤⣈⣒⣀⣠⣨⡴⠖⠛⠋⠉⠉⠉⠉⠉⠛⠲⢤⠴⠚⠋⠉⠉⠉⠉⠉⠛⠛⠶⣬⣠⣀⣀⣊⣠⡴⢿
⠀⠀⠀⠀⢸⡆⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⣾
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠛⠲⣄⡀⠨⠿⠯⠀⣀⡴⠚⠋⠉⠉⠉⠛⠲⢤⣀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠙⠲⠤⣤⣤⡤⠶⠚⠛⠁⠀⠀⠙⠙⠳⠦⣤⣤⡤⠴⠚⠉⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠶⠶⠶⠒⠛⠉⠉⠁⠀⠀⠉⠉⠙⠓⠲⠶⠶⠶⠶⠶⠚⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠉⡁⠉⠈⠋⠿⢿⠟⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢁⠀⠘







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆


⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠳⢛⢦⣀⠀⣠⡄⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⣐
⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣻⣟⣋⣈⠙⠀⠁⠤⠀⣤⣶⣰⣦⣴⣞⠶⣯⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⢐⣦⣶⣶
⠀⠀⠀⠉⠁⠺⢿⠿⣟⣿⡿⠿⠿⢿⣿⣿⡿⣿⣿⣿⣿⣿⠿⠿⠟⠁⠞⢶⠤⢒⣴⣄⠀⠹⣿⣿⣶⣯⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣭⣿⣿⣿⣭⣭⣥⣤⣤⣴⣦⣤⣤⣤⣤⣤⣭⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠑⢤⡀⠀⠀⠀⠀⠀⢈⠄⠀⢤⠈⠀⠀⢐⠂⠀⠂⠀⠀⠈⢠⠀⠀⣌⠀⠀⠀⠀⠀⠀⡠⠞⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠲⢄⠀⠀⠀⠈⠁⢀⣈⣀⠠⢤⡾⠷⠲⠿⣦⠄⢀⣸⣀⠀⠋⠀⠀⠀⣀⠔⠋⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢀⣿⢋⠙⠻⠄⣸⠃⠀⠀⠀⢿⡀⠼⠋⢉⢻⡇⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢢⣀⢰⣦⣿⣿⠗⠀⠀⠁⠀⠀⡀⠀⠀⠁⠀⠰⢻⣿⣵⡎⢀⣤⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣛⣿⣿⣿⠃⡀⡀⢰⠀⢀⡾⠻⣄⠀⢸⠀⠠⡈⢿⣿⣿⣟⡡⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠢⠀⢴⣽⣿⡏⠁⢀⣴⠃⡜⠀⣾⠁⠀⢹⡄⠸⡈⢧⣁⠀⠉⣿⣿⣴⠀⠠⠂⠀⠀⠀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⠁⠀⠜⠋⠉⢀⣾⠁⡆⢿⠀⠀⣸⠇⠀⢻⣄⠈⠉⠓⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⣀⠀⠀⡟⢀⠀⡀⠀⠀⣾⡇⠀⢡⠈⡇⠀⠏⢰⠀⠈⣿⠀⠀⠀⠀⠀⢸⡅⠀⢀⡤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⣷⢼⡄⢻⡀⠀⠘⣧⠀⠀⠇⠀⠀⠀⠃⠀⢠⡟⠀⠀⣼⠀⣾⢸⡇⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣀⠀⢿⡎⣧⠀⠳⣦⣀⠈⢳⡀⠀⠀⠀⠀⠀⡴⠋⢀⣤⠾⠁⢸⡏⣿⠄⢀⡀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡔⠉⢈⠉⠀⣼⡇⣿⣀⣀⡀⠙⠻⡄⠁⢠⣠⣇⣀⠐⠁⡼⠋⠁⣀⣀⣼⡙⣿⠄⠉⢉⠈⠱⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣠⠀⣸⠋⢸⡇⢹⣿⣿⣿⣷⣄⠀⣠⠿⠿⠿⠿⣦⠀⣀⣴⣿⣿⣿⣿⠀⣯⠈⢻⠀⢄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠙⠸⣯⣛⣿⣿⣿⡄⠀⠀⠀⣾⣿⣿⣛⣫⠟⠘⠁⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠒⡾⡿⠀⠀⠀⠒⠺⠟⠛⠛⠳⠀⠀⠘⠛⠋⠛⠿⠒⠂⠀⠀⠼⣿⡖⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡄⠀⠘⠁⠛⢀⣄⣄⠈⠷⣦⣴⠂⠐⠀⠀⠀⠀⠐⠐⢶⣤⡶⠃⢀⣄⣠⠜⠁⠑⠀⠀⣏⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠁⢀⣈⡇⠀⠀⠀⠂⠘⣴⠞⢷⣦⣈⠻⣦⣤⡤⡀⠀⣠⣤⣤⡞⢉⣠⣾⠿⢦⠞⠐⠂⠈⠀⢀⡇⣄⠀⠁⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠠⠀⠀⠘⠒⠠⠔⠀⠀⢘⣿⢙⣷⣤⣹⣿⣯⣾⣿⣟⢡⣴⡟⣹⣝⠀⠀⠢⠄⠀⠊⠀⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⡀⡠⠂⢀⡅⢺⠉⠻⣷⣿⣓⣬⠉⡏⣩⣖⣻⣶⡿⠋⡻⠂⣅⠐⠢⡀⡀⠀⠀⠐⠒⠒⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⡌⠀⠁⠈⠩⠐⢤⠁⡴⠾⢻⡏⣹⣛⣟⣻⠉⣿⠿⠦⡄⢡⠖⠨⠉⠄⠀⠈⠄⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢢⡐⠀⠀⠈⣠⠙⣿⠿⠋⢉⣉⣉⣉⠙⠻⢿⠏⢠⡈⠀⠀⠀⣤⠂⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣡⠀⠤⣄⢀⠐⠃⠀⡇⠻⣶⠾⠋⠁⠀⠉⠻⢶⡾⠁⡇⠈⠀⢂⢀⡤⠄⢠⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠘⠒⠊⠁⠀⠈⠛⠀⠁⠈⠁⠘⠉⠀⠀⠉⠖⠚⠀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢠⠒⡀⠋⠉⠉⠙⠂⡘⢄⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⠖⠒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠚⠛⠓⠲⢦⡀⢸⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⢶⠛⢳⡆
⠀⠀⡾⠀⣦⠀⢷⠀⠀⠀⠀⣀⣤⣤⣀⣤⣤⣤⡄⠘⠛⡇⠀⡟⢳⡆⠀⡷⠈⢹⡆⠀⣧⣤⣤⣄⠀⠀⣼⠃⠀⡴⠶⣤⠀⣸⡇⠀⣠⡤⠤⣴⠶
⠀⢰⡇⢰⣿⡇⠘⣇⠀⠀⣼⠋⢀⣤⣌⠉⠀⡴⠇⠀⢰⡇⠀⠛⠋⠡⣞⠁⠀⢸⡅⠀⣡⠤⡄⠈⢷⠀⡏⠀⢸⡇⠀⠈⠉⠉⢁⡞⠁⢠⠤⣄⣀
⠀⣾⠀⣴⠶⢦⡀⢹⡄⠀⣇⠀⢿⡀⣸⠆⠈⡇⠀⠀⢸⡇⠀⣿⢛⡦⠈⢳⡀⢸⠃⠀⣏⠀⡿⠀⢸⠀⢿⠀⠈⢧⣀⣤⠶⣄⢸⡇⠀⢿⣀⣩⡉
⣿⠁⠀⢙⣷⣟⠃⠀⢻⡇⠻⣄⠈⠉⣁⣄⠀⠛⡷⢸⡏⠁⠀⠉⠉⠁⣀⡼⠀⢸⣀⣠⡈⠉⢁⣠⠟⠀⠈⠳⣄⡀⠀⠀⣀⡼⠃⠻⣄⡀⠉⠁⣙
⠉⠉⠉⠉⠁⠉⠉⠉⠉⢁⣀⠈⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⢿⣤⠤⣄⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⢰⠋⠀⣤⢤⡈⠀⡇⠀⠀⣴⠞⠁⠀⢹⠀⠀⠀⠀⢿⣁⣀⡀⠀⠹⡆⠀⠀⣾⢁⣀⣀⠈⢹⡄⠀⠀⣠⡴⠞⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠓⢮⣟⠛⠁⠀⠀⠹⠶⡷⠀⢸⠀⠀⠀⠀⠈⠉⢩⡟⠀⣰⠇⠀⠀⠈⠋⣤⠿⢀⣸⠇⠀⢰⠏⢀⡴⠒⢶⡴⠃
⠀⠀⠀⠀⠀⠀⣀⣈⣛⠦⣄⠈⠳⡄⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⢀⣀⠷⣄⠈⠹⡆⠀⢸⡄⠈⢧⣄⡤⣄
⠀⠀⠀⠀⠀⠀⣿⠀⡙⠦⠾⠀⢀⡿⠀⠀⢠⡴⠇⠀⠸⢦⡄⠀⢠⡟⠁⠀⠾⠷⠶⡆⠀⢰⡟⠙⠓⠛⠀⢠⡇⠀⠀⠙⢦⣄⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠿⠴⠗⣦⢠⠴⠞⠁⠀⠀⠸⠦⠤⠤⠤⠼⠇⠀⠘⠷⠶⠶⠶⠶⠶⠇⠀⠀⠙⠳⠦⠴⠖⠋⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⡇⠀⠀⠀⠀⠀⠀⢠⡖⠲⣄⣴⠖⢦⡀⠀⢰⡖⠒⠒⠲⡆⠀⣰⠟⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢰⡇⠀⣴⠚⢶⡀⠀⠘⠦⣄⣿⠳⢤⣸⡇⠀⠘⠛⠛⠛⠛⠃⠀⠈⠳⢋⣹⠆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡾⠀⠀⠻⠦⠾⠁⠀⠀⣶⣿⠇⢠⣾⡿⠁⠀⠀⢰⠛⠻⡆⠀⠀⠀⠀⡏⠁⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⣠⠴⠶⢤⡉⠀⢀⣌⡁⠀⠀⠀⠀⠘⠲⣦⡇⠀⠀⠀⠀⣩⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠇⠀⠀⠘⣷⠚⠛⢦⣝⠓⣋⡽⠃⠀⠀⠀⠀⠀⣶⡿⠃⠀⠀⠀⠸⣇⣀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⣉⣉⣁⡀⠀⠉⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣷⣛⣋⣉⣉⣉⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⢉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠤⣄⠀⠀⣠⠤⣄⠀⠀⣠⠤⣄⠀⠀⣠⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⠀⣿⠀⠀⡇⠀⢹⠀⠀⡏⠀⢹⠀⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣇⠀⣿⠀⠀⣇⠀⣼⠀⠀⣇⠀⣸⠀⠀⣇⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢛⣁⠀⠀⠙⢛⣃⠀⠀⠙⣛⡋⠀⠀⢙⣛⠛⢧⠞⣛⡓⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣾⠉⣷⠀⢰⡏⠹⡇⠀⣼⠉⢹⠀⠀⡏⠉⣧⠀⢰⠋⢹⡆⢻⡀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠀⣿⠀⢸⡇⢸⡇⠀⣿⠀⢸⠀⠀⡇⠀⣿⠀⢸⡀⢸⡇⠘⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠶⠋⠀⠈⠳⠞⠁⠀⠘⠲⠛⠀⠀⠛⠾⠃⠀⠈⠳⠞⠁⠀⡇⠀⠀⢰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⡴⠲⣆⠀⣰⠚⣦⠀⢠⡖⢶⡀⢀⡶⢳⡄⠀⣴⠒⣆⠀⣰⣿⠀⠀⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⣿⠀⣿⠀⣿⠀⢸⡇⢸⡇⢸⡇⢸⡇⠀⣿⠀⣿⠀⣿⠘⢶⠴⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⠴⠃⠀⠹⠦⠟⠀⠘⠧⠞⠁⠈⠳⠼⠃⠀⠻⠴⠏⠀⠘⠦⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡴⢦⡀⢀⡴⢦⡀⢀⡴⢦⡀⢀⡴⢦⡀⢀⡴⢦⡀⢀⡴⢦⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠷⠼⠃⠈⠧⠼⠃⠈⠧⠼⠁⠈⠧⠼⠁⠘⠧⠾⠁⠘⠧⠾⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣤⡄⠀⣠⣤⡀⠀⣠⣤⠀⢀⣤⣄⠀⢀⣤⣄⠀⢠⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⣿⠀⣿⠀⡇⠀⡇⢸⡇⢸⡇⢸⠀⢸⠀⣿⠀⣿⠀⡷⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠹⣤⠟⠀⠻⡴⠇⠀⠷⡼⠃⠘⢧⡾⠀⠸⢦⠟⠀⠻⣤⠏⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⢤⡤⣔⢲⣆⣲⠰⠯⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠸⢹
⠀⠀⠀⢀⠆⡃⡞⠙⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠡⠤⠠⠤⠐⠒⠒⠒⠢⠤⡀
⠀⠀⠀⠀⠀⢀⠃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⡸⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⢀⠠⠤⠐⠒⠂⠀⠀⠒⠒⠤⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠐⠒⢲
⠀⠀⠀⠀⢰⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠄⢇⡀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⢸⠀⠠⠀⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡖⠉⠉⠀⠘⠀⠁⣢⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⠠⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡁⢘⠀⠀⠀⠀⢈⠀⠈⠉⠁⠈⠉⠉⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⡀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠤⠀⡇⢰⠀⠀⠀⠀⠸⡀⢰⠁⠐⠒⠈⠂⢸⠀⠀⢀⠀⠀⠀⠈⠄⠈⡁⢀⠀⠠⠀⣊⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣧⠜⠀⠀⠀⢀⡔⠃⠈⠢⠨⠬⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⢌⠀⠠⠀⠀⠀⠊⠀⡀⠈
⠀⠀⠀⠀⠈⡗⠢⣤⠦⠒⠙⠀⠀⠀⢠⢊⡔⠊⠈⠉⠉⠁⠐⠒⢄⠀⠀⠠⠀⠀⠀⠀⠀⠀⡐⠉⠁⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠈⡄⠀⠀⠀⣀⣀⡀⠀⢘⢓⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⢬⢠⡈⡰⠂⠤⠀⡤⠠⠔⡨⡤⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⡜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠌⠡⠁⠀⢀⢂⣤⠀⠀⠀⠀⠀⠈⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠐⠒⠀⠉⠀⠀⣀⠠⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠤⠤⠤⠔⠒⠋⢉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⠋⠉⠀⠀⠀⢀⣼⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣏⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠹⣦⠀⠀⢀⣀⢀⣴⣿⡁⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢧⣤⣄⣀⠀⠹⣧⠀⢀⣽⠟⠁⠈⠁⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠉⠙⠛⠶⢾⣷⣟⡁⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠉⠛⠻⠶⣦⣤⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁⠀⠀⠀⠀⠀⠀⢸⣏⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⣨⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⠛⠷⢦⣤⣄⡀⠀⠀⠙⠷⠟⠁⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣴⠟⠛⣻⡷⢶⣤⣀⡀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡾⠋⠀⠀⠈⠉⠛⠛⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡟⠁⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠉⠀⠀⠀⢿⡄⠀⠀⠀⢀⣴⠟⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢀⣴⠟⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⠟⠁⢠⡾⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠁⠀⢀⣾⣯⣀⡀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠛⠁⠀⠀⠀⠀⠀⣼⠇⠀⠉⠙⠛⠶⢦⡾⠋
⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⣼⠇
⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⠏
⠀⡾⠃⠀⠀⠀⠀⠀⣰⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⣰⠏
⣼⠇⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⣼⠏
⠻⣦⣤⣤⡶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠃⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣤⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣶⡞⠋⠁
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⢩⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠋⢉⣀⡤⠴⠖⠋⣻⠙⠲⢤⣀⠉⠛⠲⢤⣄⡀
⠀⠀⠀⠀⢀⣠⠤⠴⠒⠚⠉⢁⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⢀⡇⣠⣤⣄⣈⡉⠓⠢⢤⣀⠉⠓⠦⣄⡀
⠀⣤⠒⠊⢁⣀⡤⠤⠖⠚⠉⠉⠀⠀⠀⠀⢀⣠⠴⠒⠚⠓⢺⠟⠁⠀⠀⠀⠙⢦⣀⣀⡈⠙⠢⣄⣈⡩⠗
⠀⣇⡉⠚⠯⣄⣀⠤⠴⣤⡴⠖⠲⣄⣠⠖⠋⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⣀⣠⢬⡭⠟⠋⠉⣀⠤⢺
⠀⢹⡈⠉⠲⢤⡙⠲⢤⣄⡀⠀⠀⠈⠣⣤⠤⠤⠤⠤⠤⣤⣈⠑⣲⠤⠖⠊⠙⡇⠀⠀⢷⠔⠚⠉⠀⠀⢸
⠀⠈⣇⠀⠀⢰⣿⠑⠢⢌⡙⠶⣤⣀⡀⠈⠉⠉⠁⠀⣀⢴⡛⠉⠀⠀⠀⠀⠀⢱⠀⢠⢸⠀⠀⠀⠀⠀⡟
⠀⠀⢹⠀⠀⠀⢿⣄⡀⠀⠉⠲⢤⣈⡙⠳⢦⡤⠒⢉⣠⠴⡇⠀⠀⠀⡀⠀⠀⢸⠀⣸⢸⡆⠀⠀⠀⢠⡇
⠀⠀⠸⡆⠀⠀⠀⠙⠛⠷⣤⣀⡴⠁⠉⠓⠲⡗⠋⠁⠀⢰⡇⠀⠀⡸⢀⠀⠀⢸⠀⠋⠀⡇⠀⠀⠀⡼
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢀⠇⡌⠀⠀⢸⣀⡤⠴⠇⠀⠀⢠⠇
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾⠀⠀⠘⠀⠀⠀⢀⡼⠁⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡿⠀⣠⠴⠒⠒⠋⢉⣀⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠛⠋⠁⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⡇⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠧⠖⠚⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠙⢿⡆⣾⡟⠙⢻⣇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣴⡿⠃⠻⣷⣴⡾⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠿⠛⠛⠛⠛⠿⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢋⣁⣤⣶⡶⠾⠿⠿⠶⣶⣦⣤⣈⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣰⡿⠋⣠⣴⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡈⠻⣷⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣼⠟⢀⣾⠟⠁⣀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠘⢿⣆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⡏⢠⣿⠋⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠈⢿⡆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡿⠀⣿⠇⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣸⣿⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⢸⡿⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋⠙⠛⠛⠛⣿⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡀⡀⠀⠀⣀⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣇⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⠛⣿⠟⠋⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡄⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣼⡟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣷⡀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣰⡿⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⢿⣄⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢁⣼⠟⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠻⣷⣄⡉⠻⠷⣶⣤⣄⣀⣀⣀⣀⣤⣴⣶⠿⠋⣡⣴⡿⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣉⣉⠛⠛⠛⢋⣉⣁⣤⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠟⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⢿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⢸⠉⡜⠛⠛⠛⠛⠛⠛⠛⠛⠛⡍⢉⣵⣿⣭⡉⠉⣩⣽⢿⣍⠉⢹⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⢫⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⠈⣿⢰⣏⠀⠀⢹⡇⢸⡷⢶⡤⢴⡴⢤⡶⢤⡤⢴⡤⢾⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⣤⣠⣤⣤⣤⣤⣤⣠⣤⡇⠙⢷⣶⡾⠋⠀⠻⣶⣶⠟⠁⢸⣧⣴⣧⣼⣧⣼⣦⣼⣧⣴⣧⣼⣼⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⠉⠉⠉⠁⠈⠉⠉⠉⠛⠳⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣀⠴⠒⠉⠁⠀⠀⠀⠀⠈⠉⠒⠦⣀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⣠⠞⠁⢀⣠⡴⠶⠛⠛⠛⠛⠶⢦⣄⡀⠈⠳⣄⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⡜⠁⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⢣⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡿⠀⠀⠀⡜⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢣⠀⠀⠀⢿⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⢠⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠘⡄⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠈⡆⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⠁⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣿⠀⠀⠀⢱⡀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢀⡎⠀⠀⠀⣿⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠳⡄⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⢠⠞⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠘⠦⣀⠀⠉⠛⠳⠶⠶⠶⠶⠾⠛⠉⠀⣀⠴⠃⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠈⠓⠢⢤⣀⣀⣀⣀⣀⣀⡤⠔⠚⠁⠀⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣄⣀⣀⣀⣀⣠⣤⣴⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣟⣉⣉⣉⣉⣉⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣉⣉⣉⣉⣉⣹⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣍⣁⣀⣀⣀⣀⡬⠍⣷⠀⣞⠹⢅⣀⣀⣈⣉⣉⣉⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠤⠤⠤⠤⠤⠤⠶⢲⡏⠀⢹⠓⠲⠤⠤⠤⠭⠥⠤⢴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣄⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⣶⡄⠰⣿⣿⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢴⣿⣿⠀⠀⢸⣷⣄⣴⣶⣷⠀⠀⣀⡙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣾⣶⣶⡄⠸⢿⣿⢀⣀⢸⣿⣿⣿⣿⣷⣤⡀⠉⣁⡁⠈⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⠀⠺⢿⣟⣫⣵⣿⠿⢏⣿⣿⣜⡿⠿⢿⣿⠶⣯⣅⣺⣿⣿⣿⣷⣆⣼
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣙⡻⠿⠿⠿⠿⣟⣫⣥⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣮⣍⣛⣛⡿⠻⣛⣛⣭⣤⡄
⠀⠀⠀⠀⠰⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠿⠛⠋⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠉⠛⠛⠁⠀⠀⠀⠀⢾⣿⣿⣿⣿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⣀⡀⠀⣾⡆⣿⢸⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢰⣆⠀⢸⡇⠀⣿⡇⣿⢸⡇⠀⣿⠇⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣘⣁⣀⣀
⠀⠀⠈⣿⡄⢸⣿⠀⣿⡇⣿⢸⡇⢠⣿⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠸⣷⠘⣿⠀⣿⡇⣿⢸⡇⢸⡿⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣬⣄⠻⠂⠻⠃⠟⠘⠇⠸⢇⣠⣥⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣣⣤⣤⣤⣤⣤⣄⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣵⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣦⡻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣷⡹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠿⠇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣲⣿⣷⣿⣿⡎⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣿⣿⣿⣿⢰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⡿⣈⣫⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣂⣘⣛⣛⣛⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡼⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠶⠶⠶⠶⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢉⣀⣀⣀⣀⣤⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⣉⠉⠉⠉⠉⡿⠁
⠀⠀⠀⠀⢰⡶⠒⠒⣲⡿⠂⠀⠸⡇⢸⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢙⡆⠀⣏⠉⢉⡇⠀⠀⡇⠀⢠⡴⢤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢻⣀⡴⠋⠀⠀⠀⠀⣇⠸⡀⠀⠀⠀⠀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣥⣤⣀⣈⣉⣉⠁⠀⢰⡇⠀⠀⢹⡄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠶⠶⠦⠉⠀⠀⢸⡇⠀⠀⠀⠹⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⡠⠒⠉⠉⠛⠦⡀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⢰⡇⠀⠀⠀⠀⠀⣠⠶⠗⠀⠠⠀⢸⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⣇⠀⠀⠀⠀⠀⠙⠛⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢦⣄⡀⠀⠀⢀⣠⠏⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢱⡀⠀⠁⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣤⣄⣤⡤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠤⠤⠤⠤⠤⢐⣒⣒⣒⣛⣋⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠙⠛⠒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⣀⣤⣤⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢳⣄⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⢠⠣⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⢣⠀⠀⠀⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠷⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠀⢀⠀⠀⠀⠴⠏⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠘⢦⣅⠀⠀⠂⠀⡆⠀⠢⡆⠀⠀⠀⠀⠀⠀⠀⠀⡬⠂⠀⡄⠀⠂⠀⠀⣥⠞⠀⢀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠤⠀⠙⠦⠀⠂⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠠⠞⠁⡠⠄⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⣝⣀⣉⣠⡤⠴⠚⠛⠛⠛⠒⠲⢤⠴⠒⠚⠛⠛⠚⠲⠤⣄⣈⣁⣘⡥⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠁⠀⢀⣀⣤⠤⣄⣀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⣀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢾⡉⠀⠀⠀⠀⠈⠑⣧⠀⢠⡟⠉⠀⠀⠀⠀⢈⡹⠆⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠓⠶⠤⠤⠖⠊⠁⠀⠀⠙⠒⠦⠤⠴⠒⠋⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣀⣀⣀⣀⡤⠤⠶⠒⠲⠦⠤⣄⣀⣀⣀⣀⡤⠖⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠀⠀⢀⢄
⢠⣄⡰⠂⣈⡥⠂⠀⠶⣟⣀⢄⡾⠛⣁⣀⣤⡀
⡀⠀⠀⡩⠟⠿⠿⠷⠶⠾⠷⠾⠟⠋⠉⠙⣋⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠏
⠈⠉⠐⠠⠤⢄⢀⣀⡀⠀⠀⠀⣀⡀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠠⠤⠀⠠⣠⠤⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢿⡏⠀⠀⠀⡠⠔⠶⠁⠐⠐⠈⠀⠀⠒⠻⣷⣬⡁⢀⣀⡉⢈⣺⡷⢂⢤⡀
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⢿⢢⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠘⠂⠌⠉⠄⠄⠀⠀⠀⠈⠙⡎⠡⠆
⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡋⠀⠀⠀⠀⠀⠲⠆⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣐⡀⣬⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣽⡆⠀⠀⠀⠀⠀⢙⡂⠲⠤⠀⠀⠀⠀⢈⣉⣉⢨⣭⠍⠬⠤⠄⣒⣚
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣿⡇⠀⠀⠀⠀⠀⠀⠭⢘⡒⠀⠀⠀⠀⠠⠥⠤⠐⢒⡂⣒⣛⢉⣭⠥
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣯⡅⠀⠀⠀⠀⠀⠀⢑⡠⢭⠀⠀⠀⠀⢘⣛⣃⢨⠭⠄⠤⠶⠐⣒⡂
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠰⠶⠆⠐⠒⠀⠛⠁⠈⠉
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣭⣿⣯
⠀⠀⠈⠀⠠⠀⠒⠀⢤⠴⠶⢾⣟⠹⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠠⠜⠉⢆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠚⠒⠈⡐⡆⢄⡀⠀⡐⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⡤⠔⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡠⠔⢀⡀⠀⠀⠛⢶⢇⠀⠀⠀⠀⠈⠄⠀⠈⠂⠔⠉⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⣀⡠⠤⠤⠤⠇⠚⠁⠀⢠⡞⠉⠉⠁⠀⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⣄⡐⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣔⠩⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⡰⢨⠀⠀⠀⠀⠀⠀⢠⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠚⠋⠴⠒⠲⠦⠤⣄⣀⣀⣐⣹⠄⣀⠀⠀⠀⡀⠎⠂⠉⡄⠀⠀⠀⠀⠀⣬⣹⣀⣀⡀⠤⠤⠤⠀⠒⠂⠈⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠈⠉⠙⠛⠋⠋⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢉⢑⠒⠢⠤⠤⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠒⠈⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⠬⠀⠀⡃⠀⠆⠀⢸⠈⠉⠉⠉⠷⢒⡒⠀⠀⣀⠐⠪⡏⠀⢘⠁⠈⠇⠀⢰⠊⠀⠀⠀⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⠈⠉⠁⠀⠘⠀⠀⠀⠀⠆⠀⠉⠀⠀⠀⠑⠤⠆⠀⠘⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⠀⠀⣏⣰⡆⠀⢠⠤⠤⠀⠀⣀⣀⣀⠀⠀⣀⣀⢀⡀⠀⢠⠤⠀⡄⠀⢰⣉⠽⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠸⠀⠀⠇⠈⠗⠀⢨⠀⠀⠀⠀⡀⠀⠘⠀⠀⡁⠀⠀⠀⠀⢀⠀⠀⠁⠀⢰⠀⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢⠀⠀⠃⠀⠈⠀⠀⠁⠁⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢠⠤⢤⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⡀⠀⢠⠤⠤⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⡁⠀⢸⠀⠘⠀⠀⠗⠀⢠⠀⠀⡅⠤⠤⡄⠀⠸⠒⠒⠃⠀⢠⠤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠃⠀⠈⠀⠀⠀⠀⠃⠀⠠⠀⠀⠇⠀⠠⠆⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠂⠀⢠⠤⣄⠀⠀⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⢠⠤⠤⠄⠀⢰⠈⢹⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⡇⠀⢨⠀⠈⠀⠀⡁⠀⠈⠀⠀⠗⠀⠠⡄⠀⠘⠀⠀⠀⠀⢸⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠐⠃⠀⠈⠀⠀⠀⠀⠓⠠⠨⠀⠀⢤⡤⡤⠄⠀⠘⠀⠀⠁⠀⠈⠉⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢰⡒⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠒⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡝⠆⠀⠸⠀⠘⠀⠀⠆⠀⢴⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠃⠀⢨⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠴⠀⠀⢤⣀⣰⠀⠀⡍⠀⠰⠇⠀⠰⠤⠤⠃⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠠⣄⠀⠀⠀⢀⡀⢀⠀⠀⣀⣀⣀⡀⠀⠀⠤⠤⡄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠈⠉⠀⠀⠈⠀⣻⠀⠀⡏⠀⠈⡁⠀⠠⠄⠀⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣄⡀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣶⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋
⠀⠀⢰⣿⣿⣿⡛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⣿⡟⠛⠛⠉⠁
⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠻⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠈⣍⣍⣍⡁⠀⣈⣹⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⠀⣉⣁⣉⣉⣉
⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣃⣾⡿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠉⠉⠉⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠄⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠐⠂⡈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀
⠀⠀⠘⢉⠉⠙⠀⠀⠘⠃⡀⡀⠊⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠀⠒⠠⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠽⠤⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠬⡀⡨⠀⠀⠀⠊⠀⡖⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢱⠀⠊⠁⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⢀⠙⠐⢁⠤⢊⠂⠀⠀⣈⠂⠤⠊⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⣉⡉⠙⠀⡇⠀⠀⠀⠀⠀⠀⠈⡠⠊⠀⠒⠓⠊⠀⠀⠒⣄⠀⡂⡂
⠀⠀⠀⠀⠀⠀⠠⠐⠊⠁⠉⣉⢀⡐⡲⠛⣈⠝⢀⠀⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣞⠠⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⢄⣀⡠⠴⠮⠤⠤⠜⠀⡇⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠠⠤⢤⠐⠊⠥⣂⡈⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠃


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠙⠀⠉⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡁⠀⠀⠙⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠊⠀⠈⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠄⠀⠈⠀⠈⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⣉⣀⡀⠀⠀⠀⠀⠀⠂⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠒⠠⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠒⠚⠉⡉⡉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢲⢙⠠⢙⠃⠄⢡⡐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠰⠀⠐⣤⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠆⣀⠀⢳⡨⡀⠌⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠢⣁⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠉⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⠚⠿⣭⡉⠛⢶⣄⡀⠈⠹⠋⣥⣬⠙⠏⠁⢀⣠⡶⠛⣉⣽⠿⠓⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠛⠛⠶⣤⡈⠙⢦⣄⠙⢿⣷⣷⣤⠿⠿⣤⣼⣾⡿⠋⣠⡾⠟⢁⣤⠶⠛⠛⠻⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠁⠀⠀⠀⠀⠀⠀⠙⢶⣄⠙⢷⣤⡉⠛⠿⠷⠾⠿⠛⢉⣴⡾⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠴⠾⠿⢿⣶⣤⡀⠀⠙⢷⣎⠉⠛⠷⢶⣶⣶⡶⠾⠛⠉⣡⡾⠋⠀⢀⣠⣶⣾⠿⠿⠦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣴⣾⣿⣷⣮⡻⣿⣯⣄⠀⠙⠻⢶⣤⣀⣀⣀⣀⣤⣶⠟⠋⠀⣠⣴⣿⢟⣿⣿⣿⣷⣦⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣼⣿⣿⣿⣿⣿⣿⣾⡿⢿⣷⣤⡀⠀⠈⠉⢻⡟⠋⠁⠀⢀⣠⣾⣿⢟⣿⣿⣿⣿⣿⣿⣿⣇⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢺⡔⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠙⢿⣿⣷⣦⡖⠁⠈⢲⣴⣶⣿⡿⠟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠹⠋⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠭⢿⢻⡆⠀⠀⢰⡟⣿⠭⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠙⠏⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠯⠭⡑⢄⠻⡇⠀⠀⢸⠏⡠⠊⠭⠽⠿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠌⣻⠛⢛⠋⣡⣤⡶⣆⡀⠀⠃⠀⠀⠀⠀⡜⠀⣀⣴⠶⣤⣄⡉⡛⠛⣟⠡⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣇⣄⡀⠀⠀⠀⣀⡄⣤⠛⣹⢟⣵⣿⡋⠉⠁⠀⠀⠀⠀⠀⠀⠘⠉⠙⣿⣮⡙⢟⠛⣠⣤⣀⡀⠀⠀⢀⣠⢼
⠀⠀⠀⠀⠀⠀⠀⠸⡸⣿⣿⣿⣿⣿⣷⠛⣋⣴⡿⢻⠽⣳⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⣞⠿⡿⢿⣶⣍⠛⣿⣿⣿⣿⣿⣿⢏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣝⠻⠟⣛⣡⣴⣿⣿⣫⡆⠀⢢⡉⠻⣝⢷⣦⣤⣤⣴⡾⢫⠟⢋⣴⠁⣷⣝⣿⣿⣦⣌⣙⠛⠛⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⣿⣿⡷⠝⢳⡀⠀⠹⣿⣿⣧⣍⡙⢛⣩⣼⣿⣿⡿⠁⢠⠟⠡⢾⣿⣿⣿⣿⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⡟⠀⣼⣦⡀⠀⠀⠈⠙⠿⠿⠿⠿⠿⠿⠟⠋⠀⠀⢀⣼⣇⠀⢻⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⠸⠋⣠⣾⠟⣫⡟⠒⣤⣀⡀⠀⠀⠀⠀⠀⠀⡀⣀⣤⠀⢛⡛⠛⢳⣄⠻⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠘⢉⣴⢞⣫⣵⣾⣿⣧⣠⣦⣄⣈⣏⡉⣙⣋⣉⣩⣁⣀⣾⣄⣹⣿⣿⣭⣉⣃⣨⣉⣋⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡝⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⢏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⣇⢹⠻⣿⣿⣿⣿⠻⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⠟⣿⣿⣿⣿⡏⡟⢸⡿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⡄⠀⠹⣿⣿⣿⣄⢩⠀⢀⣃⣬⣤⣤⣭⣘⠀⠀⡇⣰⣿⣿⣿⠏⠀⢠⣿⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⣿⣄⠀⠨⠽⠟⣋⣥⢞⣫⣭⣶⣶⣶⣶⣬⣝⣲⣬⡙⠻⠛⠁⠀⣠⣾⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡝⣿⣿⣶⣿⣿⣿⣿⠿⠛⢉⣡⣤⣤⣌⡉⠛⠿⣿⣿⣿⣶⣶⣾⡿⢋⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣉⠉⠉⠉⠁⢀⣠⣾⡿⠛⠉⠉⠙⢿⣷⣤⡀⠈⠉⠉⠉⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠀⠒⠋⠁

⠀⠀⠀⠈⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠿⣿⣿⣿⣿⣿⡍⣁⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣰⣤⠀⠀⠀⠀⠀⠠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠒⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⣾⡿⠿⡿⠀⣠⣤⣤⣤⣄⣀⢈⠻⢿⣷⣶⣄⠀⠀⠀⡄⠀⢀⣠⣴⣶⣶⣶⣾⣿⣷⣄⠀⠀⠀⠀⠀⢶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠃⣀⡀⢠⡾⣿⢿⣿⣿⣿⠋⠻⠋⠙⠛⠛⣷⣦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣤⣦⠀⠀⠈⠐⠀⠀⢠⠀⢀⢀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠋⠉⣸⣿⠟⠀⠀⠘⠛⣿⣿⣷⣶⣾⣿⣆⠀⠉⠛⠂
⠀⠀⠀⠀⣀⣔⣿⣿⣆⡀⠀⠀⠀⠀⠈⠀⠀⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣏⣿⡋⠂⠀⠰⠟⠃⠀⢀⣤⣴⣷⣾⠟⠿⢿⣽⣿⣿⣷⣶⣶⣾⣷⡆
⠀⠀⠀⠀⠓⠃⣼⣿⣿⠃⠀⠙⠂⠀⠀⠐⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣯⣴⡶⠾⠟⠋⠛⠛⠛⠛⠋⠀⢀⣿⣿⣏⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⡜⢿⡟⠁⠀⠀⠀⠐⣾⣿⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣄⠀⠅⠀⠀⠀⢠⣤⣤⣾⣿⣿⣿⣿⣿⡀⣙⣿⣿⣿⡇
⠀⠀⠀⢀⣻⣇⣬⡷⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣵⣿⣿⣿⣿⣿⠀⠘⠛⠀⠀⠀⠀⠿⠋⠻⠉⠁⠉⢻⣛⣩⣿⣿⣿⣿⡿⡇
⡄⠀⠰⠸⢿⠿⠙⢥⣤⠀⠀⠀⣰⣿⡿⢻⣟⠻⣿⠿⣥⣿⣿⣿⡿⠁⠰⢿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣿⣿⣿⣿⣿⣿⣟⠂
⡁⠀⠀⠀⠀⠀⠐⠿⣿⢧⠄⠓⣾⣿⡀⠀⠁⠀⠀⠋⠉⣳⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣷⣿⣿⣿⠄
⠃⠀⠀⠀⠀⠀⢰⠆⠉⠀⠀⢶⣿⣟⢣⡤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣸⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿
⡆⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢨⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣀⣤⣤⣀⣀⠄⣠⣾⣿⠿⠿⣿⠟
⡀⠀⠀⠀⢰⣶⣤⣴⣆⣠⣤⣬⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄⠁⣀⣄
⠃⠀⠀⠀⠈⠿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠠⢴⣶⡄⣣⣶⠟⠂⠀⠻⣿⣿⣿⣿⣿⣿⣿⠃⠘⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠿⠋⠉⠉⠉⠀⠀⠀⠀⠀⠁⠠⠐⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠛⠛⠀⠀⠀⣴⣟⠋⠉⠀⠁⡸⠏⠀⠀⠀⠉
⠀⠀⠀⠀⠁⠀⠀⣄⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⣤⣀⠀⡀
⠀⠠⣼⡇⠀⠰⠀⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠏⠹⣿⣿⠿⠃⠀⠰
⠀⠀⠀⠀⣤⣄⣤⡀⠀⠀⠀⠈⢀⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠈⠁
⣶⡶⢲⣾⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣷⣶⣶⣾⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣿⣷⣶⣷
⣿⣷⣥⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⡛⠋⠽⠋⢀⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⡟⢁⡀⠀⠀⠉⣛⡿⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⠀⢩⡭⠀⠀⠀⣭⡍⠘⣿⣇⠀⠀⠀⠀⠀⠀⣸⡟⠀⢽⡆⠀⠀⠀⣾⠅⠀⣿⡀
⠀⠀⠀⠀⠀⢸⡀⢉⠦⣖⠿⠍⠉⠭⢟⣲⢼⡁⢸⠂⠀⠀⠀⠀⢸⡁⢨⢶⡺⠭⠍⠉⠭⠽⣒⢾⠀⣸
⠀⠀⠀⠀⠀⠀⠑⢾⡄⠤⠄⠀⠀⠀⠠⠤⢸⡷⠋⠀⠀⠀⠀⠀⠀⢱⣾⡀⠤⠄⠀⠀⠀⠠⠄⣸⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⢶⣢⠄⠀⠠⣔⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⢷⣒⡀⠀⣐⣲⠵⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⢀⣴⡶⢶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣾⠁⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⣰⡟⠋⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠃⣶⡀
⣠⣤⣮⣤⣤⣤⣤⣘⣃⣀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣸⡇
⣽⣿⣿⣿⣿⡏⠉⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡟⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⣿⣿⣿⡇
⣽⣿⣿⣺⣿⣿⠘⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⡇
⠻⠿⠿⠾⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⣴⠻⠛⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣄
⠀⠀⣿⠀⠀⠀⡀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠈⢿⡆
⠀⣠⢅⣴⣿⠿⠿⣷⣧⠀⠀⠀⠀⠀⠀⠀⣫⣴⣾⣷⣶⣽⠁⠀⠀⠀⠀⠀⠀⠀⣨⣴⣾⣿⣶⣬⡃⠀⠀⠀⠀⠀⠀⠀⠸⢇⣀⣀⣀⠸⡇
⣼⡯⣾⣿⠁⣴⣶⡌⣿⡇⠀⠀⠀⠀⠀⣸⣿⠋⣤⣤⡉⢿⣷⠀⠀⠀⠀⠀⠀⢰⣿⠋⣤⣤⡌⢻⣷⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠿⣿⣿⣦
⢿⣷⢻⣿⡄⠻⠿⢃⣾⡇⠀⠀⠀⠀⠀⢹⣿⡀⠿⠿⢃⣼⣿⠀⠀⠀⠀⠀⠀⢸⣿⡀⠿⠿⠃⣸⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠠⣿⣿⣿⡇
⠈⠻⣮⡻⣿⣷⣶⡿⠟⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣶⣾⠿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⣰⡟⠋⠙⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠛⠿⣦⡀
⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⢹⡇
⣰⣂⣿⠚⠛⠛⠃⣿⣓⣦⠀⠀⠀⠀⢀⣌⣋⣠⣠⣤⣠⣘⣫⡀⠀⠀⠀⠀⠀⡦⠽⢥⣭⣭⢥⡽⠵⡆⠀⠀⠀⠀⠀⣾⣿⣾⣿⣿⣿⣿⣾⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠨⢻⣿⣿⡟⠛⣿⣿⣟⠇⠀⠀⠀⠀⠀⢳⢸⣿⡿⠿⣿⣿⢰⠃⠀⠀⠀⠀⠀⢰⣶⣶⣶⠒⣶⣶⣶
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢼⣿⣿⡇⢰⣿⣿⣯⠂⠀⠀⠀⠀⠀⢸⢸⣿⣿⢸⣿⣿⢸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣼⣿⣿⣿
⠿⠿⣿⣿⢿⣿⣿⣿⠿⠿⠀⠀⠀⠀⠰⣾⣿⣿⣷⣾⣿⣿⣷⡆⠀⠀⠀⠀⠀⠈⠳⣽⣿⣿⣿⡵⠋⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⢀⣤⡶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⣶⣤⡀
⠀⠀⣾⠃⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠙⣷⡀
⠀⣀⣿⣀⣀⣀⣀⣿⣀⡀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇
⢨⣿⣷⣶⣶⣶⣶⣾⣿⡇⠀⠀⠀⠀⠀⢰⡷⢾⠿⡿⢿⠿⡿⡆⠀⠀⠀⠀⠀⢸⡿⢿⠿⡿⢿⠿⡿⡇⠀⠀⠀⠀⢠⣶⣶⣾⣿⣿⣿⣷⣶⣦
⠀⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⢸⣿⡏⠩⡏⠈⡏⠍⣿⣿⢸
⠰⣿⣿⣿⣷⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣟⣹⣏⣟⣹⣏⣿⡇⠀⠀⠀⠀⠀⢸⣿⣻⣟⣿⣻⣟⣿⡇⠀⠀⠀⠈⢸⣿⣿⣶⣷⣶⣷⣶⣿⣿⠈

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⡏⡽⠛⠛⠛⠛⠛⠛⠛⠛⢯⢉⣭⡿⣿⣍⠉⣩⣶⢿⣭⠉⡽⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⢿⠉⠉⢹⡂
⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏⠀⠀⣿⢶⣿⠀⠀⢹⡇⣯⡷⢶⠶⣶⠶⣶⠶⣶⠶⣶⠾⣼⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⣷⣤⣤⣤⣤⣤⣤⣤⣤⣼⣈⣻⣷⣾⣋⣀⣙⣷⣾⣟⣁⣿⣷⣾⣶⣿⣶⣿⣶⣿⣶⣿⣶⣿⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠟⠛⠛⠻⠶⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⢀⡠⠴⠒⠚⠉⠉⠒⠒⠢⢤⣀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⣠⠞⠁⢀⣠⣤⡴⠶⠶⠶⣤⣄⡀⠈⠳⣄⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠏⠀⠀⢀⠞⠁⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠈⠳⡄⠀⠀⠹⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡟⠀⠀⢀⠎⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢻⡄⠀⠀⢻⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⡇⠀⠀⢸⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣧⠀⠀⠘⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡄⠀⠀⣏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⠀⠀⣸⣃⢀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⢸⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⡟⠀⠀⢰⡟⠀⢸⠉⠉⠉⠉⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧⠀⠀⠈⢆⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣸⠃⠀⠀⣼⡇⣀⣸⡟⠿⠿⠛⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣇⠀⠀⠈⢦⡀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⢀⡴⠃⠀⠀⣰⡏⢁⢀⡽⠉⠉⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠙⢦⣀⠈⠙⠛⠷⠶⠶⠾⠛⠋⠁⢀⡴⠋⠀⠀⢀⣴⢿⠀⡎⡹⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠀⠀⠉⠛⠲⠤⣤⣤⣤⣤⠤⠶⠚⠉⠀⠀⢀⣠⠞⠁⢸⣀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⣶⣶⣶⣶⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠉⣉⣩⠭⠭⣉⣉⣉⡁⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠒⠒⠒⠒⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⢉⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠤⠤⠤⢤⣞⠳⠤⠤⠤⠤⠤⠤⠤⠔⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡤⠤⠴⠤⠤⠤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡄⢶⠁⠉⠙⠛⠋⠛⠛⠛⠛⠋⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉

⠀⠀⢀⣠⣄⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠿⠻⢿⣿⣿⠿⠛⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⡋⠙⣿⡏⠀⠀⠹⡏⠀⠀⠈⢿⡇⠀⠀⠀⣿⡇⠀⠀⠀⣿⠃⠀⠀⠈⣿⠁⠀⠈⣿⠋⠉⣿⣿⣿⣿⣿
⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣷⣤⣤⣤⣾⣧⣤⣤⣤⣿⣧⣤⣤⣤⣿⣦⣤⣤⣴⣿⣶⣶⣶⣿⣿⣿⣿⡿⡟⠋⠁
⠀⠀⠀⠀⠈⠉⢿⠁⠈⠹⣿⡿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⡟⠉⠉⡟
⠀⠀⠀⠀⠀⠀⠘⣆⠀⣰⣿⣧⣤⣼⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣇⠀⣠⠇
⠀⠀⠀⠀⠀⠀⣤⣿⡿⠻⣿⡿⠉⠉⢻⡏⣿⡇⢹⣿⠀⠀⠘⣿⡇⠀⠀⢸⡟⢹⣿⠉⣿⠋⠉⢻⣿⠿⢿⣿⣤⠆
⠀⠀⠀⠀⠀⠀⠋⢹⡇⠀⠸⣿⠀⠀⢸⡇⢸⡇⢸⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⢸⣿⢠⡿⠀⠀⣼⡇⠀⢸⣏⡹
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣶⣿⣧⣀⢈⣿⣾⣷⣤⣿⣀⠀⣸⣿⣇⠀⢀⣸⣧⣼⣿⣼⡇⣀⣴⣿⣷⣴⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⣿⣿⡛⢻⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠛⣿⣿⠁⠀⡝
⠀⠀⠀⠀⠀⠀⠀⢠⣈⣷⡾⣿⣿⠛⠻⣿⠻⣿⠏⣿⠁⠀⠈⣿⠃⠀⠀⣿⠟⣿⡟⢻⡿⠛⢻⣿⠷⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⠀⢻⠀⠀⢿⠀⣿⡄⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⣿⠀⢸⠇⠀⠸⠁⠀⢹⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠀⠀⢸⠀⠀⢸⡄⣿⡇⣸⡇⠀⠀⣿⡀⠀⠀⣿⠀⣿⠀⣼⠀⠀⡄⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠖⣾⣧⣀⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣄⣴⡿⠒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠀⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠑⣳⣶⣶⣾⣿⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⡟⠉⠀⠈⠻⣿⠗⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣶⠀⠀⢀⡄⠀⠠⠶⣿⣦⣼⡟⠛⣿⠃⢠⣿⣷⣦⣄⡀⠀⠀⠀⠀⠻⡛⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣏⠀⢈⣁⣤⣄⡈⠁⠀⠈⠀⠀⡄⠈⠘⢷⣿⠀⢼⣿⣿⣿⣿⣷⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣥⣾⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠻⠀⠀⠾⠃⠀⢸⣿⣿⣿⠿⠛⠛⠀⠅⠀⢠⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⢿⡿⣾⡉⠀⢰⠀⠀⠀⢀⠀⠀⣼⣿⣆⢀⣴⡆⠀⠀⠀⣻⣿⡇⠐⢚⣿⡟⠀⠀⠈⠙⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣫⣩⣿⣷⣶⣆⠀⠀⠀⠈⠁⠀⡀⠈⣿⣿⣿⡅⠀⠀⠀⠻⠿⣿⣤⢿⣿⣇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⠿⠿⠋⠀⠀⠀⣀⣴⣿⣿⣷⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⢠⠟⠿⡟⠋⠀⠀⣠⡖⠀⣤⣺⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣀⠰⠀⡔⢹⣿⣿⠟⡩⠗⠻⢿⡛
⠀⠀⣤⣶⣄⠀⠈⠀⠀⣀⣀⠀⢀⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠐⠛⠋⠐⢠⢳⣿⣿⡟⠀⠀⣾⡗⢹⣧
⠀⠀⢿⣿⣿⣷⡄⠀⠀⠛⣿⣷⣌⠉⠙⣛⠟⠙⠉⣸⡿⠛⠻⠟⠛⠂⠀⠀⠀⠀⠀⠀⢸⣿⠙⢿⡇⠀⢰⣿⠗⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⣷⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⣀⣀⡀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣀⣸⣿⣿⣿⣅⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⡀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣛⣛⣘⣓⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡧⡤⣾⢤⣅⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡓⣛⣚⣓⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡦⠥⠼⠬⠴⠁⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣼⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠁⠛⠈⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⠠⠄⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠤⠤⠤⠤⠖⠒⠒⠒⠒⠛⠃⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠢⠤⣀⡀
⠀⠀⠀⡟⠓⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⢄⣀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠉⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠄⠒⢒⠒⠒⠂⡈⠉⠭⠭⠍⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠄⠒⠖⠒⠒⠒⠂⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠀⠀⠀⠐
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⣠⠒⠀⢶⡄⠀⠀⡔⠢⠀⡖⠐⠀⢏⠈⠀⡇
⠀⠀⠀⡏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡅⠀⠀⠀⠛⠀⠀⠧⠀⠀⠳⠄⠀⠙⠂⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠴⠋⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠤⣤⠴⠶⠶⠶⠚⠛⠒⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢠⣤⠠⠤⠠⠤⠔⠒⠒⠒⠊⠉⠉⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠉⠈⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡒⠷⠚⠛⠛⠛⠓⠒⠤⢀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⠊⠀⠀⠀⠀⣠⡤⢒⠂⠀⠀⠒⠤⣀⠀⠀⠀⠑⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⠀⡠⠊⠀⠀⠀⠈⠑⠢⡀⠀⠈⢳⣄⠀⠀⠈⢆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠈⢺⠁⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠹⠂⠀⠀⠘⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⡆⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⢡
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⡀⠀⠀⠀⠘⣕⠢⢀⣀⣀⡀⠄⠂⠀⠀⠀⡠⠁⠀⠀⠀⢀⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣣⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠀⠀⢀⠎
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠉⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠤⠤⠤⠄⠒⠊⠁
⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠐⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠄⠒⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⢀⣀⣀⣤⠤⠤⠔⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠁