Shoreline Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣿⡟⡻⠱⣿⣷⣿⣟⠿⠅⠠⣿⣿⣟⠣⣼⣿⣿⡟⠀⠀⠀⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣶⣿⣿⡿⡹⠱⣷⣾⣿⣿⣿⡓⠀⢫⣿⣿⣟⠐⢿⣿⣿⠃⠀⠀⠠⣿⡏⣼⣿⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⣵⠁⢬⣺⣿⣿⣿⣿⠃⠀⢽⣿⣿⣏⠼⣿⣿⡟⠀⠀⠀⢸⣿⢧⣿⡏⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣿⣿⣿⢷⠃⡘⣾⣿⣿⣿⣿⡏⠀⠠⢿⣿⣿⠇⣭⣿⣿⠁⠀⠀⠀⢘⣧⣾⠏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣒⣿⣿⣿⣯⡎⢙⣾⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣟⢀⣶⣿⡏⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⣿⣟⡝⠈⢳⣿⣿⣿⣿⠇⠀⠀⠀⠐⢿⣿⣿⠈⣷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢍⣿⣿⣿⢳⠁⣙⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⢸⣿⡟⣴⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢪⣿⣿⣿⣿⠂⢙⣳⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⢻⠷⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣭⣿⣿⣿⢋⢀⣹⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢾⣿⣿⣯⢀⣴⣻⡿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⢽⣿⣟⠃
⠀⠀⢀⣠⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⡿⢡⣰⢿⣿⠃
⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠨⠛⢿⣧⢸⣿⡷⠈
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠋⠁⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣷⣶⠭⠀⠀⠟⠩⣭⣙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠖⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣤⡶⢖⣶⡶⣶⢶⣦⣝⠃⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⣿⣿⣿⣿⡿⣿⢿⣿⠿⣷⣿⡟⣴⡿⣿⡇⢹⡆⢻⡅⠀⠀⠀⠀⠙⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣴⣿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣥⡘⠎⢐⣫⣭⣽⣾⡟⣴⡿⢡⣾⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⣿⣿⣷⠖⣢⢂⣶⡐⠦⣭⣻⡿⣸⠏⠁⠀⠁⠀⠀⠀⢀⣀⣤⣤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣶⡤⠀⢀⣀⣴⣿⣿⣿⣿⣿
⠋⠁⢁⡾⠋⢸⡟⢿⣦⡀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣠⠟⠋⣑⣑⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠙⠁⠀⠀⠀⠘⣿⣿⣿⣿⡿⣿
⠀⠀⠈⠀⠀⠸⠁⠘⣧⠁⠀⣸⠀⠀⠀⠀⠀⢀⣾⣁⣴⣿⣿⣛⣿⡟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡯⠀⠀⠀⠀⢀⡞⡿⢋⠵⢋⣩⣙⣢⣬⣅⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡇⠀⠀⠀⠀⠀⠘⢱⣿⠚⠛⠉⣉⣟⣋⣙⣛⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣀⣀⣀⣤⣤⣤⠈⠘⢷⣶⣶⣶⣿⣦⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀⡀
⣾⡟⠁⢛⣿⣿⣿⣿⣿⣴⠶⠾⠿⠿⠿⠿⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⣿⣷⠷⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣎⢷⣚⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢛⣉⣭⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⣀⣀⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣬⣭⣉⣉⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣧⣤⣤⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡟⠸⠈⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡴⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⡀⣀⣀⣀⣀⠀⣀⣀⣀⣠⣠⣄⣁⣾⣤⣤⣤⣴⣶⣶⣷⣖⣶⠂
⠛⠛⠛⠻⠶⠿⢿⣿⣏⣉⣻⣟⣻⣿⠿⠍⠉⣡⣶⡖⣒⣛⠛⢛⡧⣭⢉⡀⠀⠈⠹⠁⡉⠀⠁⠈⠈⠉⠋⠀⠀⠀⠁
⢦⣴⣶⣶⡯⣿⣽⣿⣿⣿⣿⣿⣿⠗⢀⣰⡀⠁⠀⠀⣿⠍⠁⠀⠁⠀⠂⠀⠀⠀⠁
⠀⠶⣾⡿⠿⣟⣿⣿⡿⠛⠋⠉⠀⠀⠈⠙⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡰⠂⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣴⠀⠄⠀⡀⡴⢀⣴⣄⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⣇⣀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠠⢠⡁⠁⠚⠉⠋⠘⠁
⠤⠀⠤⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠀⢀⣀⠀⠀⠐⠂⠀⠀⠈⠁⠀⠀⠒⠒⠂

⣀⣀⠠⠤⣤⣖⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈
⠚⠛⠲⣄⣀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠈⠛⠀⠀⠀⠀⠀⠁⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠀⠁⠀⠒⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠘⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠁

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣿⣿⡿⠛
⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠉⠉⠙⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠶⠀⠀⠀⠀⠀⠛⠉⠉⠙⠆
⠀⠀⠀⠀⠀⡟⢠⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠚⠇⠀⡃⠀⠀⠈⣇⠀⠁⠀⣤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠒⣨⠄⢋⣉⠠⠈⢀⣀⣦⣀⣀⣀
⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠈⠥⠤⢬⠿⣉⡭⠍⠛⢉⣁⠈⠉⠒⣤⣀⡀⣀⣀⣀⠤⠤⠄⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⠤
⠀⠀⠀⠀⡇⠈⠀⠘⢳⠀⠀⠠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⠳⢦⣤⣀⡀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⡰⢄⠀⠀⠀⠀⠁⣀⠄⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠐⡄⠀⠂
⠀⠀⢀⣟⠈⠀⠀⠀⠀⠀⠉⠁⢀⠌⠀⠱⡄⠀⠀⠀⠀⠀⣀⣀⣀⢀⡀⠘⡒⠀⠀⠀⠄⠀⠠
⠀⠀⠸⡈⠁⠀⠀⣀⡠⠖⠂⠀⠀⠀⠀⠀⡙⢦⡀⠀⢀⡀⠀⠀⠀⠘⠣⠉
⠀⠀⢠⠇⢆⣘⡏⠁⠄⠈⢁⠀⠀⢄⠀⠀⠈⢦⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣁⡴⠶⠮
⠀⠠⢼⢤⡶⢟⡀⠀⢠⠀⠉⠀⠀⠀⠢⡀⠀⠀⠈⠙⠲⢤⡤⡄⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠂
⠀⠀⠀⢤⡶⢶⡦⠖⠊⠀⠀⠀⠀⠀⠠⡈⠒⢤⣀⡀⣀⠀⣀⡀⢠⣤⠤
⠀⠀⠀⠐⠋⠀⠹⣦⡀⠀⠀⠀⠀⠐⢄⠈⠲⣟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀
⠀⠀⠈⢧⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠠⣌⡙⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⡆
⠀⠀⠀⠈⠀⠈⢦⡀⠠⣿⣿⡶⣄⠀⠂⠠⠀⠀⠉⠛⠶⣦⠤⠤⢄⣠⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠
⠀⠀⢠⣄⠀⠀⠀⠙⢦⡱⢝⠿⣮⣀⠀⠐⠲⢤⡄⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⣄
⠀⠀⠈⢿⡇⠀⠀⠀⠀⠙⣿⣷⣌⠻⢇⣤⣀⠠⠷⠄⡄⠀⠀⠀⠀⠀⠀⢀⠔⢀⠂⠉⠃⣠⠀⠀⠀⠉⢻⣿
⠀⠀⢰⣾⣿⠄⡠⢤⣀⠀⠈⠹⣷⣕⡦⡉⠊⠙⠲⠦⠤⠀⢀⣑⣤⡀⠀⠛⣐⠚⠓⠂⠀⠀⠸⠍⠀⠀⠠⠿⠷
⠀⣠⣾⣿⣷⣶⡄⠀⠙⠛⠀⠀⠀⠙⢿⣟⠷⢄⡀⠐⠀⣹⠇⠁⣀⠠⠀⠀⡀⠀⠐⠆⠀⠤⠤⠤⠤⠤⣴⠆⠖
⠀⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣀⣹⣷⣾⢿⣂⣡⣶⡀⣈⣙⣶⣾⣦⣤⡴⠿⢄⡾⠁⠀⠀⠀⠀⠂⡰⠆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣁⣠⣤⣴⡒⠀⠀⡉⠉⠁⠤⢼⣇⣀⣀⣠⣤⣤⣶⡁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠉⠙⠛⠳⠬⣍⣉⣉⠉⠈⡁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠲⢶⣚⣻⣿⣭⣿⣶⣦⢀⣄⡀
⠀⢿⣿⣿⣿⣿⣿⣿⡿⢷⣤⣆⠀⠀⠀⠀⠐⠿⠛⠀⢀⣈⣉⣭⣉⣱⣶⠀⠒⣒⡒⢒⣂⠈⠉⣉⣥⣬⣴⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣔⣂⡀⠀⣀⠀⠀⠴⣭⠩⣯⣽⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠛⠿⠻⠿⠿⠿⠉⠉⠿⠯⠄⠀⠉⠁⠀⠉⠿⠿⠯⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣄⣠⣄⣠⡀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣤⣤⣤⡴⠉⢿⣷⣤⣶⣶⣴⣤⢸⣇⣴⣄⣸⣿⣿⣬⣅⠩⣤⠺⢛⣅⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣥⡠⢶⣶⣶⣤⠀⠈⣿⡟⠛⠛⣿⢻⠿⠈⢷⠬⠿⡲⠛⠞⠙⢃⠌⢳⣿⡝⠿⠃⢠⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠁⠀⠉⠩⠗⠀⠀⠙⠁⢦⣀⠛⠷⢡⣂⡈⣽⣿⣾⡶⠿⠆⠾⡡⠌⡻⠥⠀⠀⠷⢬⠀⢣⠀⠀⠀⠀⠠⠄
⠀⠀⠀⡀⠀⠀⠀⢀⡀⠤⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⢀⣀⢡⠆⠹⠷⠾⡹⢻⣿⣟⣧⣬⡀⠀⠀⢾⡧⢀⣴⠒⠂⢸⣏⠀⠀⠐⠈⠹⠄⢴⡺⣶
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠀⠁⠀⠀⠀⠀⠈⠉⠀⡄⠊⢛⢀⣤⢶⢶⠶⠿⠣⠉⠭⠁⣷⡶⢖⠁⠀⠀⠀⣿⣏⡷⣤⠀⠉⠀⣦⠀⠀⠀⠀⠸⢳⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠈⠀⠀⠀⠀⠀⣴⣉⣠⡾⡟⣿⡵⣆⢙⣥⣦⣶⠀⣠⠴⢼⣇⠀⢠⣶⣖⣦⡁⣿⣹⠿⠯⣿⡒⠈⠠⣴⡦⠀⠀⠉⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠉⠀⠀⠀⠀⠀⢀⣴⠾⠟⠉⢁⡼⠏⠉⠰⡾⡟⠋⠛⣷⠾⣠⢏⢣⣿⣦⠉⠙⠿⠚⠁⢿⣿⡀⣴⡏⠓⠐⠀⠚⣁⣀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠠⠔⠊⠄⠀⠀⠀⠀⠀⡀⢀⣴⠛⠁⣠⢤⡖⠋⢘⠳⠒⠢⠋⠐⡚⣲⣕⣖⢮⢠⣡⣿⡿⠁⠀⠀⠀⠀⠀⢻⡅⠀⠇⠴⢖⠶⣤⣈⡇
⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⡑⠟⠩⠐⣰⠿⠋⠀⢁⣯⠿⡏⠷⣒⢄⢠⠉⠉⠈⠓⢃⣾⣿⡅⠀⠀⠀⠀⠀⠀⠈⢻⡲⡌⣀⠨⣤⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣮⡡⠞⠥⢰⡖⣻⣯⡥⡆⠠⠋⢉⡔⣁⣉⣤⠰⠾⠛⢶⠀⠀⠠⢾⣟⢯⡁⡀⠀⠀⠀⠃⠀⠀⠀⠙⠁⠉⠀⣸⠃⠈⣇
⠀⠀⠀⠀⢀⠀⢀⣠⡤⢖⣠⡾⠉⣱⣾⠛⠉⠰⠀⠁⢈⢀⡀⠀⠨⠍⢙⢞⣣⠐⠀⡄⠤⠉⠻⠉⡻⣡⣋⣶⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠊⣠⣄⣞⣿⠶⠋⠁⣶⡖⣁⠈⡭⠛⠮⡥⠲⠉⠁⡀⠀⠀⡛⠝⠌⠘⠓⠀⡄⣠⠋⢣⠔⠑⣿⣿⡇⣿
⠀⠀⠀⡘⠩⠞⠉⠀⣀⠐⣲⠙⠂⡈⠀⠙⠋⢀⢀⡀⠀⣠⠠⡄⠀⢐⠀⣄⢤⡀⠺⡤⠜⠣⡇⠀⠈⢿⣿⡓⣾⣦⢤⡀
⠀⠀⠀⠀⠀⢀⢬⠏⠑⡛⠑⠐⠏⠙⠄⠘⠎⠀⢀⠙⢤⡁⣤⡠⢨⣎⣵⢄⢩⣐⠆⢭⠆⠂⠀⠀⠀⠀⠁⠀⠘⠻⢆⣅
⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠒⠀⠀⠠⠐⠤⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢡⡤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣬⡝⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣆⠠⡴⠀⠰⠀⠀⠀⠀⠀⠀⣆⣶⢌⠁⣤⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⢫⢃⠀⣄⠑⠶⠊⣀⢄⣨⡀⠀⠀⠀⠈⡽⠋⠀⠐⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠉⣘⣀⢻⢀⢀⢉⡁⠞⣡⠶⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⠅⡴⢩⣿⡷⡒⢠⣤⠆⠐⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⡫⠃⢬⣿⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠷
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢀⢀⣬⣽⡂⠌⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣠⣄⣄
⠀⠉⠀⠀⠀⠀⠉⡈⠉⠉⠈⠩⣿⣉⠙⣀⣶⣶⣖⣾⡿⠿⠿⠙⣉⣀⠀⣬⣭⣼⣭⣯⡀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠉⠀⠈⠉⠉⠉⠁⠹⣷⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠖⠒⣶⣴⣶⣭⣄⣈⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠰⠶⠄⠂⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⡤⠤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡠⢾⡿⣿⡛⠳⢦⡀⠠⣽⡿⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠸⠃⠀⠀⠃⠠⠋⢰⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢰⠀⠠⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⣿⠿⡋⠁⠀⠀⠀⠀⠀⠆⠀⠀⠀⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣸⣄⢼⣤⣀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⣈⣳⣆⣶⣶⣶⣼⣿⣷⣏⣽⣿⣿⣿⣿⣿⢿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠿⠟⠚⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⣛⠿⠿⠿⠿⠿⠿⠿⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠁⠀⠐⣒⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠐⡒⢒⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⣉⠉⣉⣀⣀⣠⣤⡴⠖⠋⡉⠁⠀⠀⢀⠒⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠶⠶⠶⠶⠿⠛⠿⠛⠛⠛⠋⠉⠙⠉⠙⠛⠉⠉⠉⠀⠀⠀⠈⠁⠲⣶⣤⣄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠄⠀⣀⡠⠦⠲⠖⠶⠞⠛⠉⠀⣼⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠾⠛⠛⣉⣉⣉⣠⣤⣤⣤⣀⣤⣄⡀⠀⠀⠈⠀⠀⠀⠀⣀⣀⣤⠴⢚⡁⠀⠀⠀⠀⡀⠐⠋
⠀⠀⠀⠀⠀⢀⣈⣉⣙⣂⣉⣩⠭⠤⠶⠞⠛⠛⠉⠉⠉⠙⠙⠋⡉⣀⣉⣩⡝⠟⠛⠛⠛⢛⡩⠍⠙⠀⠉⠒⠚⠯⠀⣒⠊⠁⠀⠀⠀⣡
⠀⠈⠁⠉⠈⠈⠉⠉⠉⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠶⠒⠊⠉⠉⠀⠚⠉⠀⠀⢀⡠⠴⠂⠁⠀⠀⠀⠀⠀⢀⠀⡀⠀⢸⣷⣶⣳⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠂⠀⠀⢀⣀⠤⠄⠒⠀⠄⠈⠂⠀⠀⠀⠀⠀⠀⡠⠢⣴⣆⡿⠟⣡⣾⠁⠀⣨⡌⡁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠙⠛⠛⠋⠉⠁⠀⢀⠤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠈⠀⡚⠠⠴⠋⠀⠈⠈⠀⠈⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣄⣠⣀⣀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣶⠄⢰⡆
⠀⠀⠀⠀⠀⠁⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣶⠀⠿⠐⢿⣽⣿⣿⣿⣇⠸⠗⠖⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠚⠛⠻⠋⣐⡆⠚⠉⠏⠁⠀⠀⠀⢀⣀⣠⣼⣿⣮⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠀⠀⠀⠀⠈⠉⢀⣠⣀⣀⠀⣀⣀⣀⣈⣥⣬⣿⣿⣿⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⡾⢿⣿⣿⣇⢿⣿⡏⣿⣿⣿⢻⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢽⠐⡟⣼⢛⠋⢸⣏⣽⣿⠈⣿⡇⢈⡀⢻⡇⢻⣿⣿⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢓⠇⠀⠂⠀⣏⠸⡿⠆⢻⣿⡞⠙⠿⣦⠨⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⢀⡀⠀⢹⠀⣤⠀⠀⠙⣷⣘⡀⣉⣢⣿⣿⣿⣭⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠈⠀⢻⣇⠀⠀⠹⣿⠁⢀⣿⣿⣿⣿⣯⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⣄⠈⢟⣤⣴⣆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠤⠀⠀⠀⣀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⢨⠀⢴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢁⣈⣤⣄⠙⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣞⣸⣿⣿⢳⣿⣿⣿⣿⡿⡟⣿⢛⣯⢀⣹⣷⣄⣛⣠⣤⣀⡉⣠⡎⢀⣷⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣛⠻⣿⣿⣿⣿⣇⣿⣿⠆⣿⣿⣿⠿⢛⣉⣛⣛⣛⣡⣎⣼⣶⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣄⣠⣜⢿⣯⣾⣿⣿⣿⣿⡽⠛⢁⠺⣿⡟⣿⣆⣹⡿⢛⡉⣍⡉⠙⠛⠷⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⢟⣿⣿⣿⣿⣟⠿⠃⠹⠟⠐⠚⠃⣿⣶⠤⠘⠿⠛⠛⠓⠒⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠐⠒⠒⠊⠒⠛⠉⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⠀⠀⣀⠀⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀⢀⣀⠤⢶⣶⣶⣤⣤⣄⣀⣠⣤⣤⣤⡀⢀⣀⣀⡀
⡀⣾⡆⣤⠀⣿⢆⣀⢀⣠⡤⠴⠾⣷⡶⣶⣄⡷⠶⠤⣤⣭⣬⣽⣿⣊⣩⢭⣭⣭⠛⢭⣟⡟⣻⣭⣆⡄⠠⢄⡀⠀⢀
⠀⢹⠁⠻⠁⢼⠀⢉⣿⣶⣤⣠⡖⠐⠚⠛⠉⠉⠳⢶⣿⣧⣈⠉⠉⠛⢿⣧⣤⠀⢣⡄⠨⣷⠅⣼⡇⡸⠇⠈
⠀⠀⢄⠀⠐⠒⠄⠈⠉⢉⣽⣿⣏⠀⠀⠀⠀⠀⣰⣤⣈⣿⣿⠛⠓⠓⠶⣾⣿⡄⠈⠙⠁⡝⠲⢒⣀⠷
⠀⢢⣿⣿⣆⠀⠀⣀⣴⣿⣿⣿⣯⣀⡤⠴⠚⠀⠀⠀⠀⠈⠓⢂⠀⠀⠀⠀⠙⢿⣦⣀⠴⠟⠀⠙⠁
⠥⣾⣿⣿⣿⠗⠃⠀⠀⠀⢀⠈⠙⠻⡀⠀⠀⠀⠀⠄⠀⠀⠀⢸⡇⠀⠀⠀⣀⣴⣏⠀⠀⠀⡚⠁
⠀⠀⠈⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣀⠀⠀⠀⠀⠀⣠⣿⣷⣴⣶⡿⠿⠟⠉⠀⢀⡎⠁
⠀⠀⠀⣿⣤⡀⠀⡀⠂⢀⠀⠀⠀⠀⠈⠙⠛⢿⣦⣴⣤⣾⣿⠿⠛⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠉⠙⠛⠿⠷⠦⠤⠤⠤⠴⠄⠀⠒⠐⠋⠀⠭⠉⠁⠀⠀⠀⠀⠀⡀⡀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡄⠊⠁
⠀⣀⠀⠀⠐⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡔⠉⠉⠉⠉⠁
⣀⣀⢈⣩⠤⡄⣀⣤⠀⠠⠖⠒⠒⠐⠋⠉⠉
⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⣿⣿⣿⣿⠿⠾⠿⡯⣿⣿⣿⠙⠋⣿⠿⠇⠀⠀⣀⢤⣤⣤⣴⣤⡶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⠙⠉⠀⡀⠀⠀⠰⣿⣷⣿⡏⠀⠀⠈⠙⡭⠆⠀⢸⠛⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣾⣿⡉⠛⠁⠀⠀⠀⢀⠂⢀⣠⣼⢇⣴⢀⡰⠛⠉⠁⢰⡄⠀⠀⠀⠀⠀⠚⠓⠿⣂⡄⠈
⠀⠀⠀⠀⠀⠀⠀⣤⢾⠟⠒⠛⠛⠓⢪⣿⣿⣿⡿⣡⣿⢷⡶⠶⣶⡶⠥⠴⢞⠛⠛⠿⠉⠁⣠⠤⠤⠞⠛⢓⣒⣒⢲⣄
⠀⠀⠀⠀⠀⣴⢽⣧⡜⠀⠀⠀⠀⠀⣾⣿⣿⣟⣡⢛⣫⣤⣭⣥⣼⣥⣤⠴⠿⠒⠒⠺⠯⠛⠉⠀⠀⠈⠉⠉⠉⠀⢸⣿⡄
⠀⠀⠀⠀⡾⠁⢸⣿⠀⠀⠀⠀⠀⢀⡟⢉⢘⣿⣿⡟⠛⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣿⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠃⠸⣶⠁⠈⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣸⣷⡄⢻⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡿⡇⠙⠢⢉⣢⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢷⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⢧

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡍⣠⣀⡀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⢀⣠⡤⠭⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣱⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣜⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠉⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠠⠀⡀⠖⠱⠄
⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣠⣄⠀⠠⠀⠀⠀⠐⠛⠃⠀⠄⡀⠀⠀⠀⠀⠁⠀⣤⣤⠄
⠀⠀⠀⢀⠁⠷⠀⣀⡀⠐⠀⠀⠀⡠⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡔⠀⠀⠀⡠⠊
⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠂⠀⣀⠤⠒⢢
⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠉⠠⢤⡞⠁
⠀⠀⠀⠀⢠⠄⠠⠤⠤⠤⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠗⠒⠒⢓⠳
⠀⠀⠀⠀⠀⠙⠶⠖⢂⣀⣀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢿⡐⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠊⠥⢤⣴⠤⢺⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠔⠛⠁⠀⠈⠐⠂⠀⠠⠄⠀⠐⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠻⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⠦⠶⣴⢀⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡄⢘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⣄⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣆⣿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⡸⠳⠛⢯⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡏⠀⠈⣿⡆⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠙⠿⠒⢀⡀⢀⠀⡀⢀⠠⠠⢤⢀⣀⣀⣀⠀⠀⠀⠏⠀⠀⢀⣿⡇⠀⠀⢀⣡⡇
⠀⠀⠀⠀⠀⢷⠃⠀⠀⣷⠀⠀⢀⡼⠄⢃⣈⠀⠀⠤⠤⠤⠶⠀⠴⠶⣿⣧⣭⣤⡳⠶⣷⣶⣤⣾⡿⠁⠀⠀⠈⢹
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⣤⣴⠛⠀⣰⠎⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠸⣿⡋⠻⣿⣷⣍⢻⣿⠟⠁⠀⠀⠀⢠⡞⠀⠀⠀⠀⡀
⠀⠸⡆⠀⠀⠈⣦⠶⠀⠀⠈⠙⠂⣴⠃⠀⠀⠀⠀⣿⠀⠀⠀⡆⠀⠀⢀⣿⡃⠀⠈⣿⣿⠛⡇⠀⠀⢀⣼⣷⡟⠀⠀⠀⢀⡾
⠀⠀⢳⡴⢄⠀⠈⢦⠈⢠⡤⠀⢠⡿⢆⣀⢀⡀⣀⣼⡷⣄⠀⡇⢀⣀⣼⣷⡁⠀⢀⣿⣿⡆⢿⣤⡎⣠⣿⠋⠀⠀⢀⣜⡽⠁
⠀⠀⠀⠑⢆⠐⠄⠀⠉⠃⠇⠀⠸⠁⠀⠈⠉⠀⡼⠉⠀⠀⠀⠇⠉⠉⠙⢿⡛⠛⠛⣿⣿⢇⣼⠿⠛⠋⠁⠀⢀⠔⣼⡏⠁
⠀⠀⠀⠀⠀⠳⢤⣹⡇⠐⢾⣐⡆⡆⠀⠀⠀⠀⠳⣄⠀⠀⠀⡆⠀⠀⠀⡸⠁⠀⢀⣿⣿⣾⣿⡄⣤⣴⠒⠒⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠂⡙⣄⠀⠀⠀⠀⠀⠛⠒⡄⡅⠀⠀⢾⣅⡀⢀⣾⣿⣿⠿⠿⠛⠋⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠊⠀⠉⣻⣮⡛⢶⣖⠀⠀⠀⢈⡅⠁⠀⠀⠀⠈⣹⣿⣿⣿⣧⠤⠤⣀
⠀⠀⠀⠀⠀⠀⣠⣅⡘⠉⠉⠉⠉⣀⡀⡬⣦⡙⢦⣄⣀⣺⡄⠆⠀⠀⣀⣤⣿⣿⡉⠛⠛⠻⠶⠾⣿⠶⡄
⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⢰⣁⠀⠀⠐⠋⠙⠿⠶⢒⣉⣁⣀⣰⣿⠿⠿⠟⠻⠿⣍⣐⣄⠀⠀⠘⣷⠛
⠀⠀⠀⠀⠀⠰⣞⡀⠀⠀⠀⣈⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠇⠀⠀⠘⣧⡇
⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠟⠀⠀⣠⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠉⠉⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡠⠖⠋⠉⠉⠙⠲⢄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡞⠀⡴⠊⠉⠉⠑⢄⠀⢣⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢰⠀⠀⠀⠀⠀⠈⡆⠈⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡆⠀⢆⡔⠊⠐⢲⠉⠁⠲⡁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠦⣀⠀⠀⠀⠈⡆⠀⠀⢣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡈⠀⠀⠀⠀⣏⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣸⣆⠀⡏⠀⠀⠰⠋⠉⠉⠉⠉⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢹⠉⠀⠀⢰⡀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠐⠂⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠓⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠇⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣠⠛⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎⠩⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠜⠀⢀⡀⠀⠧⠤⢄
⠀⠀⠀⠀⡠⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠰⣾⣷⠄⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢄
⠀⠀⢀⠞⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⢤⠀⢸⡇⠀⡤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠣⡀
⠀⢀⠎⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢪⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠱⡀
⠀⡜⠀⣰⡆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢸⠀⣼⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⣆⠀⢣
⠸⠀⢀⣿⣿⡄⠀⠣⡠⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢰⢠⣿⣿⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠜⠀⢠⣿⣿⡄
⡇⠀⣸⣿⣿⣿⡆⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡌⣾⣿⣿⣧⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⢰⣿⣿⣿⡇
⡇⠀⠈⠉⠻⣿⠀⠀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢰⣹⣿⣿⣿⣿⣆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠀⢀⣿⠟⠉⠁
⠰⣀⠔⠢⡀⠈⠁⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⣸⣿⣿⣿⣿⣿⣿⣆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠁⢀⠔⠢⣀
⠀⠀⠀⠀⠈⠦⡀⠀⠀⠈⠑⠢⠤⠄⢀⣀⠀⠤⠤⠒⠙⢄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠑⠒⠤⠤⠀⣀⡀⠠⠤⠔⠊⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣄⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⣀⠀⠉⠉⠛⠻⠿⣿⣿⣿⣿⣄⡀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⣀⠠⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠤⣀⡀⠈⠙⠻⢿⣧⣄⠀⠀⠀⠈⠹⢿⣿⠟⠋⠁⢀⡀⠤⠔⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠉⠻⢿⣆⡀⠀⠀⠀⠀⠋⡱⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠙⠋⠐⡢⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠒⢒⣿⡀⠤⠤⡇⠀⠀⠀⢀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣳⣤⣤⣷⣴⣶⣾⣿⣿⣿⠁
⠀⠀⠈⠀⢀⣀⣀⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣋⣯⣟⣿⣆⠀⠀⠀⠀⠐⠦⠄⠐
⠀⠀⠠⠴⠶⠶⣾⣟⣛⣛⣿⣿⣿⣿⣏⣽⣿⠟⠿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠉⢉⡉⠁
⠐⠚⠛⠿⠷⠶⠾⣟⠛⠛⠹⠛⠿⣷⣿⠟⢻⣿⣿⡿⠿⠽⠛⠛⢛⣖⣿⢿⣿⡟⠉⠉
⠀⠀⠉⠛⠛⠀⠰⠦⢭⣍⣉⣁⠀⠈⠛⠋⠬⠽⣿⠅⠀⠒⠒⠒⠒⣂⣉⡭⠉⠉⠀⠒⠒⠒⠢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠛⠁⠀⠈⠉⠑⠒⠀⠀⠀⠉⠉⠓⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠁⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠶⠀⠀⢨⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢡⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠁⠀⡀⠀⠀⠀⠈⠉⠀⠀⠈⠑⠀⠀⠀⠀⠀⠙⠒⠀⢻⠂⠀⠸⢠⡀
⠀⠀⠀⠀⠀⡒⠠⠀⠈⡍⠹⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⣹⡏⠀⠠⠖⢠⠇
⠀⠀⠀⠘⠀⠘⣆⡠⢀⠑⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⢀⡔⠀⢀
⠀⠀⠀⠐⠂⠀⠈⠢⢤⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠋⠠⢀⠉
⠀⠀⠀⠀⠀⣀⠀⢀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⣂⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠈⠀⠤⠵⠆⠀⠴⠶⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣵⣖⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠠⠔⠚⠋⢱⣄⣀⠀⠀⠀⠀⣀⣠⣤⢾⡑⠀⠀⠉⠉⠛⠓⠦⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠰⠊⠀⠀⠀⡶⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠙⠓⠞⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠋⢠⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⣉⠘⠀⡰⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⢡⠀⠀⠀⠀⠀⠀⡎⠰⠁⠐⠐⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡠⠀⢀⡜⣆⣰⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⡐⠁⢀⠊⢇⠳⡈⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠐⠀⠀⠀⠀⢀⡜⠀⢠⠃⡰⢈⠜⡸⠆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡰⠁⠀⡇⢸⠘⣀⠇⠀⢀⠔⠁⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢠⠁⠀⠀⠀⠀⠀⠀⡠⠃⠀⡔⠁⠀⢰⡉⡞⢀⠇⢀⠴⠃⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⡐⠁⢀⠜⠀⠀⠀⡎⢸⣠⡎⠠⠁⠀⠀⠈⠀⡌
⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⢀⠜⠀⢀⠎⠀⠀⠀⠀⢇⠸⠉⢃⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠸⡀⠈⡄⠀⠀⢀⠊⠀⠠⠃⠀⠀⠀⠀⠀⢨⢜⠇⢸⢀
⠀⠀⠀⠀⠀⠀⠣⠀⠈⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣀⡳⠟⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠒⠒⢈⣁⠴⢺⣍⢦⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠤⠤⠤⠤⠄⠒⠊⠁⠀⠀⠀⠛⠛⠲⣶⢝⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⠆⠀⠀⠀⠸⠁⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡼⠛⠰⣊⠉⢹⣄⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⢠⡇⡀⣿⣦⣤⣼⠿⠽⠤⠤⠴⠦⢤⣀⡀⠈⣆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢘⠉⠀⣿⣄⠀⣾⠀⠀⢀⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠐⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠆⠀⣷⡀⠘⣯⣤⣿⡖⠟⠤⠷⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣲⡤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡔⠈⠳⣤⣈⣿⡯⢉⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠉⠉⠄⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⣤⡀⢌⠙⣿⣇⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠺⣇⣠⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠨⡙⠶⣥⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢾⣻⣿⣧⣤⣴⣾⡿⠋⠀⠀⣀⡿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠀⠈⠳⢿⣶⣤⣽⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢐⣿⣟⣛⠛⠉⠁⠀⢮⠴⠛⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠀⠀⠀⠀⠀⠀⡈⠙⠛⣿⡽⠀⠰⠀⠀⠰⠂⡁⣷⣾⣿⡟⠙⠛⠶⠶⠶⠟⠛⠉⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠜⣶⠲⠶⠚⠛⠳⣅⠊⠀⣀⣢⣾⣿⣿⡟⠉⠙⠳⠶⠦⠤⠴⠴⠷⣦⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⡏⠀⠉⠙⠳⠶⢤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⠄⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠰⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡰⠟⠃⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠒⠒⠒⣿⡂⠤⠤⣯⠀⠀⠀⠀⣀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣦⣤⣼⣴⣶⣾⣿⣿⣿⠏
⠀⠀⠈⠁⢀⣀⣀⡀⠙⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣋⣯⣽⣻⣇⠀⠀⠀⠀⠀⠲⠄⠐⠒
⠀⠈⠡⠴⠶⠶⣾⣟⣛⣛⣿⣿⣿⣿⣟⣽⣿⠟⠿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠈⠍⣉⠁
⠒⠚⠛⠿⠷⠶⠾⣟⡛⠛⠫⠛⠻⣷⣾⠿⢛⣿⣿⣿⠿⠯⠟⠛⢛⣷⣛⡿⣿⣿⠉⠉⠁
⠀⠀⠉⠛⠛⠀⠐⠦⠭⣍⣉⣁⠀⠈⠛⠋⠥⠽⣿⡧⠀⠐⠒⠒⠒⣒⣀⡭⠉⠉⠀⠐⠒⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠛⠃⠀⠀⠉⠑⠒⠒⠀⠀⠈⠉⠙⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠘⠁⠀⠰⠖⠠⠤⠀⠖⠀⠀⠛⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⡂⠀⡞⠣⢀⠀⣺⠁⢠⠄⠀⠠⠒⠂⣠⡤⠀⠀⠀⠉⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠷⣼⠓⣒⡠⢋⣴⣦⣶⣧⣤⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠈⠞⣽⣿⣿⣾⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡤⢀⠃⠀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠛⢀⠀⠀⢲⠇⣠⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⢀⣡⣶⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠁⣶⡆⢼⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⡁⠀⠙⢻⡈⢧⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠻⣿⣦⣄⠀⣰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠧⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠛⠁⡙⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⢀⠀⣤⠺⣻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢈⣿⡒⠀⠈⠳⣤⠸⢦⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡋⠀⠀⠙⢿⣷⣄⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢬⡉⠃⠀⠀⠀⠳⣀⠀⠘⠿⣿⣿⣄⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣄⠀⠀⠉⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠘⠙⠚⠛⠛⢷⣤⢿⣿⣿⣆⡀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠐⢦⣄⡌⠉⠈⢿⢿⣿⣿⣷⣦⣄⣀⣠⣾⡟⠁⠀⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠁⠉⠛⠿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠷⣿⡿⠒⠲⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠉⠉⠉⠉⠙⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⡿⠟⠻⢿⣿⣮⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡏⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⡷⠔⠒⢖⢲⣞⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⣤⡗⠛⣯⢹⣿⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⠛⡇⢰⣿⣾⢻⡅⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⡗⠺⣴⡇⠀⣿⣿⣦⡇⢸⠁⢀⡤⠤⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣹⠶⠶⠶⢶⣒⣋⣹⣄⡹⢧⣀⣿⣿⣦⣏⢻⣷⣿⣿⣷⠐⣷⣏⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⢳⣮⣉⠛⣿⣟⡧⢽⡶⢳⣼⣿⣼⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢙⣿⣶⡏⠻⡧⠶⠞⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⡇⣿⡆⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⣿⡇⣿⣱⠞
⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠀⢘⣿⡇⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⣠⢋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣴⡟⠀⢸⣿⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣄
⠀⠀⢀⣼⠇⣼⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣐⣳⣇⡀⠸⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢳⣜⢧⡀
⠀⢀⣾⠏⢰⣿⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣄⡋⢙⠿⢿⣼⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣎⢷⡀
⠀⣼⡏⠀⣼⣿⣿⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠻⣷⣿⢦⣞⡉⣣⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⣿⣿⣿⣧
⢰⡟⠀⢀⣿⣿⣿⣷⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢨⣿⣿⠀⣏⣿⣹⠆⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣿⣿⣿⣿⣿
⡼⢀⣴⠿⡿⣿⣿⡛⠷⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠠⣿⣿⡆⣿⣡⠏⠀⠀⠀⠀⠀⠀⢠⡾⠷⢛⡿⣹⢻⠿⣟⠻
⡷⠋⠁⠀⢳⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡏⠸⠦⠐⣿⣿⣿⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣰⡿⡟⠀⠈⠙
⠀⠀⠀⠀⠘⣆⢈⡻⢿⣿⣶⣤⣄⣀⣀⣠⠞⣧⡞⠀⠀⠀⡼⢿⣿⣿⣧⡑⠦⢤⣀⣀⣀⣤⠤⠖⠉⣠⣾⡿⣱⠃
⠀⠀⠀⠀⠀⠘⣞⣿⣷⣍⡛⠿⠿⣿⣿⡟⢤⣿⣖⣀⠤⠚⠀⠀⠙⠿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣷⣿⣿⢟⣴⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣝⡛⠛⠁⠀⠀⠀⣷⢬⡏⠉⠙⠲⡄⠀⠀⢠⠴⠛⠛⣻⣿⣿⣿⣿⣿⡿⢟⣫⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⣹⡦⣽⡀⢀⣠⣾⣆⢰⠃⠀⠀⠐⠛⣛⣫⣭⣿⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠼⡗⠿⢿⣿⣿⡇⣀⡤⠶⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠢⡇⠀⠀⠈⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠛⠻⠿⢿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢹⣿⠛⠛⠛⠛⠂⠀⠀⠀⠉⠿⣆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⣀⣀⡀⡀⠀⢀⣀⣀⣠⣴⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⡀⠢⢿⡿⠟⠛⠁⠀⠀⠙⠛⠿⠭⠄⢀⣲⣤⣠⡀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⣤⣾⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠿⣿⣷⡟⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⢂⢲⣭⡖⠂⠂⢮⣶⠊⣄⠀⠀⠈⠉⠁⠀⣿⣾⠀⢀
⠀⠀⠀⢰⡀⠀⢸⣠⣧⣼⡇⠀⠠⢀⠰⠀⠁⢨⣌⠙⠀⠀⠀⠈⣯⣤⠀⠈⠰⣤⢤⡤⠄⣿⡏⠀⢸
⠀⠀⠀⠀⢇⠀⠀⠻⢿⡟⣤⣤⠣⡈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠏⢿⡿⠋⠀⣰⠟
⠀⠀⠀⢀⠈⠲⣴⡀⡀⠈⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠏⢀⣤⠾⠁⠀⡔
⠀⠀⠀⠈⠳⣄⡈⠛⠷⠷⠴⢶⢦⡀⣄⡀⠀⢤⣠⡀⠀⠀⠀⠀⣠⣤⣴⣦⣄⣀⡼⠿⡛⢡⣠⠾⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⠶⠶⠬⣿⡿⠿⣿⣿⡿⠟⠀⠀⠀⠀⠉⢉⠛⠫⠟⠛⠟⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⣤⡶⠶⠦⠾⠿⠷⡄⠀⠀⠀⠀⠀⠀⣰⠛⠛⠓⠀⠉⠲⠦⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠢⠷⠟⠒⠤⠤⣀⣀⢀⡈⠋⠙⠛⠺⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠂
⠀⠀⠀⠀⠀⠌⠀⠀⡀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠴⠓⠁⠀⠀⠀⠀⣀⡤⢴⠒⡖⣲⢦⣀
⠀⠀⠀⠀⠀⠀⢠⣞⡍⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣀⡀⠤⠊⠀⠀⠀⠀⠀⢠⠺⡅⠃⢰⠀⠀⠁⠈⡘⢹⢦
⠀⠀⠀⠀⠐⠀⠈⠙⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠂⠱⠀⠀⠀⠀⠀⠐⢀⠊⡼⡇
⠀⠀⠀⠀⠀⠘⠒⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡢⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⠠⢤⠄⠁⠀⠀⠀⠀⠀⠙⠦⣬⠁⢀⡠⣤⢄⡀⣤⠔⠛
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢰⡍⠀⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠿⣤⣶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢸⠂⢐⢧⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⢸⡌⢀⣤⡞⠐⠃⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠐⡂⢀⠔⣸⣿⡽⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⡉⣰⡿⣿⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣤⠾⢂⣾⣿⠀⡟⣧⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⢈⠘⠁⠀⣾⣏⡏⠀⡇⢿⡀⠀⠀⠐⠒⠀⠀⠀⠸⠆⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡤⠁⠀⠀⣾⣿⣿⡇⠀⡇⢸⡇⠀⠀⠀⠀⠑⡀⠀⢠⠅⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⢎⡀⠀⢀⡿⣿⣿⣿⠄⠠⡇⠀⣿⡂⠀⠀⠀⠀⠈⠆⢀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣤⣉⣀⠀⡀⠀⡇⠀⢻⣏⠀⠀⠀⠀⠀⠀⠸⡷⡶⢦⣤⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣤⣼⣶⣤⣤⣤⣃⣀⣀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⢠⣴⠶⠾⠿⡭⣤⣒⡚⠛⣛⠛⠛⠻⠿⠟⠛⠻⠿⠿⠿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⠁⠤⠤
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠃⠀⠤⠖⠊⠉⠩⠵⠖⠛⢳⡖⢀⣤⡴⠶⠶⢟⣩⠴⠤⣤⣴⣶⡆⠨⢤⡤⠉⠀⢀⢀⡤⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠴⠤⠈⠭⢉⣉⡥⠴⠶⠒⠒⠒⠲⠄⠈⠉⠉⠀⠀⠀⠉⠛⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⢡⠀⠀⠀⠀⠠⠂⠄⠈⠀⠀⠜⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⣰⢁⠀⡈⠡⠔⠩⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠘⠈⠁⠒⢠⠃⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠳⣄⠀⠀⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⣓⡔⠀⠀⠀⠀⠀⠀⠒⠤⠀⠒⠁
⠀⠀⠀⠀⠀⠒⠂⠀⠁⠀⠀⠀⡅⠀⠀⠈⠠⠆⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠋⠇⡀
⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠒⠀⠑⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠈⠀⠀⢀⢷⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠪⠥⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⡀⡀⠀⠀⠂⠀⠀⠄⠀⠀⠰⡙
⠀⠀⠀⠀⠀⠀⠀⠘⣠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⡆⠀⠀⠀⡀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡀⠈⠡⠤⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⡀⡇⠰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⠎⡄⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠙⠒⠂⠂⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⢀⢤⣤⣤⢤⣤⣀⢀⡀⠀⠀⠠⠌⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡀
⠀⠀⠀⠀⠀⠀⠸⠟⠁⠁⠀⠀⠈⠓⠇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⢀⠀⡰⠊⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸
⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠂⠠⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⢁⠜⠁⠀⠀⠁
⠀⠀⠐⡵⢋⠞⠡⠄⣀⢀⠀⣀⠀⡄⢢⡘⢮⡳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠼⠠⠇⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢡⠏⢀⠉⠐⠃⢈⠀⠸⠀⢠⠀⢱⣌⡇⠀⠀⠀⠀⠀⠀⠀⠈⠘⢸⠸⠀⠀⢀⠄⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠳⢾⡀⢼⠀⢸⠀⠠⡄⠀⣧⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⡇⠀⣀⠔⠁⠀⠀⠀⠀⠈⠀⢔⠢
⠀⠀⠀⠀⠀⠀⠑⠚⠐⠚⠀⠐⠃⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⣛⣛⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣡⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣿⣿⡏⠁⠀⠉⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣄⠀⢀⣤⢾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟⣡⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⣿⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⣾⣿⣷⣶⣖⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⢙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣲⣿⣿⣷⣄
⠀⠰⣿⣯⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⠆
⠀⠀⠈⠻⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⡇⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇⣿⣿⡄
⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⣿⡇⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⣿⣧⣿⣮⡻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣾⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⢻⣿⣿⣿⣿
⢹⡟⠉⢻⣿⣿⣦⡙⢷⣤⡀⠀⠀⠀⠀⠀⣿⠇⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣤⠾⢋⣴⣿⣿⡟⠉⠻
⠀⠀⠀⠀⠻⣿⣿⣿⣶⣌⡙⠳⠶⣤⣶⡾⠟⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣟⣋⣥⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣭⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣷⣬⣙⠻⠟⣡⣴⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⣴⡦⢀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠄⣀⠀⠀⠀⠈⠒⠠⡀⢱⠀⠤⡄⠀⠀⢀⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠂⠐⢚⣒⣒⡢⠤⠀⠀⠉⠓⠢⢄⣀⣠⠃⠀⠋⠀⢸⠀⠀⡎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣋⣀⡀⠀⠀⠀⠀⠸⠥⠀⣤⡤⠀⠀⣀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠦⠚⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⣠⠔⠊⣉⠤⠐⣪⠅⠀⠀⠀⡠⢊⣁⡤⠆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠇
⠀⠀⠀⠀⠠⠮⠔⡲⡎⣀⠔⢉⢎⣀⠤⠂⠀⠉⠩⣁⠔⠊⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠉
⠀⠀⠀⢀⢠⠔⠉⠀⠉⠀⠀⣠⡌⢀⠠⠲⠖⠒⠒⠦⣄⠀⠒⢄⢀⠀⠀⢄⡀⠀⢴⠄⢰⠂⣤⠀⡀
⠀⢀⠞⣥⡇⢀⣴⢀⡠⢀⣞⠅⠉⠃⠀⠁⠒⠢⠀⠉⠲⢝⠒⢄⠀⠢⡀⠀⠈⠀⠀⢀⠤⡀⠀⠀⠡⠀⠀⠀⠀⡁⢀
⠀⠤⠂⠀⣼⠇⢀⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠙⣄⠱⡄⠑⢢⠀⠰⡀⡎⠀⢂⣄⠀⠀⠀⠸⠀⠀⠇⢸
⠀⠀⠳⠊⡀⢀⠇⠉⠁⠀⠀⠀⠀⠀⣠⡴⣢⣷⠀⠀⠠⠀⠁⠈⡀⠙⡄⠈⢄⠀⠙⡇⠀⠃⡘⠴⠛⠃⢀⠀⠀⠀⠈⠀⠀⢠
⠀⣏⠆⠀⢨⢥⠀⠀⠀⠀⠀⠀⠀⣪⡥⢤⠈⠚⡇⠀⡠⠤⢀⡀⠁⡀⠀⢀⢮⡀⠀⠇⡘⠘⠀⣠⡀⠀⠊⠀⠀⡇⠀⢐⠀⠈
⣴⠆⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠱⠛⠉⠀⡕⢧⠜⠀⡤⠤⣌⠉⠸⡆⠈⢀⡇⢠⠠⠁⣠⢊⠜⢀⣠⡴⠀⠀⡇⠀⢸
⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢚⠄⠀⠀⠘⢄⠀⠨⠘⠜⠀⡄⢮⠀⠆⢀⠔⢁⡅⠀⠊⠉⠀⠀⠰⠁⠀⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠫⠀⠞⠑⠀⡰⠁⣀⠀⠀⠀⠀⢣⠀⠐⢰⠀⡇⡜⠀⠀⢊⢔⠕⣁⢤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣀⠠⢁⢈⣀⠜⠈⠑⣄⣀⣀⣠⣦⠀⢀⠀⢎⠀⢀⡔⢁⠄⢀⠀⣀⢀⡄⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠔⢲⡍⠀⣨⠿⠟⠀⠈⢀⠜⣠⠊⠐⡡⠊⠀⠉⠀⠁⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣄⠸⡽⠤⠷⠖⠄⠴⠾⣁⠔⠁⠀⣠⣴⡦⠛⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢊⠀⠀⢀⠜⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⢀⠄⠀⠉⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⢋⡩⠀⢀⡠⠊⠁⠀⠀⠒⠉
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⢲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠗⠠⠞⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠒⠷⠧⠤⣤⡶⠶⠷⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠷⢧⣤⣤⣶⠶⠷⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠈⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣇
⠀⠀⠀⠀⠀⡏⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣄⠀⠀⢀⠀⠉⡏⠙⢶⡀⢀⡔⡇
⠀⠀⠀⠀⠀⠳⡉⠀⠀⠀⠀⠀⠳⠀⠀⠀⢈⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡉⠀⠰⡏⠀⠀⠻⠀⢸⠇⢈⡽⠃
⠀⠀⠀⠀⠀⠀⠙⢦⣥⣀⣀⠀⢀⣁⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⢋⣀⠀⢀⣉⣯⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢴⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⣷⣟⡿
⠀⠀⠀⠀⠀⠀⠰⠾⠶⠶⢌⠉⣹⠦⠶⢦⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣌⠉⣹⠷⠶⢦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⡇
⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⣿⣟⠳⣤⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠛⣏⠀⠀⠀⠐⠈⢿⠀⠀⠀⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠀⢰⠞⠀⠘⠿⠀⢸⡇⠈⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢧⣤⣀⣤⠤⡦⣐⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣜⣃⣠⠤⣦⣐⣛⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢰⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢰⡿⠛⠋⠁

⠀⠀⠀⢀⢀
⠀⠀⠊⠉⠳⢄⠈⠙⠋⠉⠉⠀⠁⠀⣁⠨⠜⠋⠁⠀⠀⠀⠀⣠⠀⠤⠈⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⡀⠴⠒⠋⠁⠀⢀⠐⠊⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠈⢃⣀⣀⣀⡴⠏
⠀⠀⠀⠈⠐⠠⠖⠋⠀⠉⠀⢀⡀⣀⠀⠀⠀⠈⠁⠀⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠚⠁
⠀⠠⢀⣀⠠⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡠⠂⠛⠂⠃
⠀⠀⠀⠀⡠⠴⡄⠤⠠⢨⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠸⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⢅⠀⠀⡄⠀⠀⠀⡤⢀⡄⡄⡄⡀⠀⠀⠀⠀⠀⠘⠦⠀⠠⣔⣰
⠀⠀⠰⡀⠠⠆⠠⠤⠴⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠋⠁⠉⠀⠁⠉⠙⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠐⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⢀⡞
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢆⠀⠀⢠⠅
⠀⠀⠀⠀⢀⠐⠒⠂⠄⠀⠀⠀⠈⡄⠀⠄⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠐⠀⠀⠀⠈⡀⠀⠀⠀⠐⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠐⠄⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⢀⡴⠞⠁⢠⣀⣀⣠⠤⠖⠒⠲⠶⠶⠶⠄⠀⠉⡑
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠉⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡠⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠱⠄⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⢦⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠂⢾⡷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇⠙⠆⢆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠸⢻⡏⠀⡇⠸⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠁⡌⡇⠀⢧⠀⢩⣄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢸⠇⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢘⡀⠀⡈⢿⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⣇⠀⠦⠈⠟
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢀⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡑⢀
⠀⠀⠀⠀⣸⣿⢦⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠷⢀⠀⠀⠀⠀⢀⡹⢦⣶⡀
⠀⠀⠀⢀⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⠶⣽⣾⣿⣿⣦⡀
⠀⠀⠀⠈⠛⠛⠷⠾⠷⣤⣤⣤⣀⣀⣸⣻⣰⣀⣀⣠⠀⡀⠀⠀⠈⢂⠀⠀⠀⠀⠈⠉⠙⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣄⣼⣭⣭⣿⣿⣡⣶⠰⣧⣧⣤⠧⣀⣸⣦⣤⣴⠶⠒⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣴⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠙⠹⠟⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣤⣴⡴⠿⠿⠿⣛⣃⡛⠒⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠔
⠀⠀⠀⠀⢌⠋⠠⠑⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⢤⣉⣉⡒⠲⠤⢤⣀⣀⣀⣀⣀⣀⣀⣤⠶⠗⠊⠁
⠀⠀⠀⠀⠃⠀⢀⣤⣤⣄⣀⣀⣀⡀⠀⠀⢀⡀⣀⣀⣈⣉⠙⠛⠒⠚⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠧⣍⠙⠛⠫⢭⣽⣮⣅⣘⠲⠿⢿⣻⠖⠶⠤⠤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢌⣙⣿⣶⣄⡙⠋⠛⠛⣁⡀⠀⠀⠀⢀⡀⡀⡄⢀⠀⠀⠉⠉⠙⠛⢲⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠾⠿⠛⠻⠿⣿⣶⣄⢈⣁⣀⡀⡀⠀⠀⠁⠉⠉⠉⠈⠁⠉⠉⠻⣿⣿⣧⠚⠉⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⣷⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠈⢕⠌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡿⣻⠟⠙⣛⣻⣿⣟⡿⠟⢉⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⢐⣴⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣗⡻⠋⣵⢿⣿⢛⣛⣛⣿⣿⣶⣭⣚⠦⠦⣀⡆⠀⠤⠠⣦⣾⡃⠀⡀⠀⢀⡴⠛⠑⠻⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠉⣠⡾⣧⠟⣡⡟⢿⢟⣭⣭⠿⢿⣿⣿⡿⠿⠷⠒⠒⣻⢿⣽⣧⣭⣴⣶⠏⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⢋⢊⡣⣪⣿⣱⢿⣾⣽⣝⡛⢿⣯⡁⠀⠀⠀⢀⣼⡷⣫⣷⣾⣯⣤⣴⠟⠢⠦⠤⣦⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠙⢣⠛⢡⣹⡕⣹⣦⣀⣘⠛⠽⠿⠛⣦⣀⣠⣿⣿⢿⣟⣹⡿⣦⡀⠈⠈⠈⠉⡀⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠹⣷⠃⠳⣄⣄⡀⠈⠙⠷⡒⣿⡏⣿⣿⠟⠛⣻⣽⠆⠈⠙⢿⣳⣴⠈⢼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠳⣟⡓⠒⠶⠭⢿⣿⣿⠀⠀⠁⠈⡻⣼⣄⠈⢉⢻⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠿⠼⢾⣓⢔⠀⠤⠀⢼⣫⣽⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣐⠄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠤⠀⠀⠀⠀⠀⠄⢈⠀⠀⣀
⠀⠀⠀⠀⠀⠐⠊⢠⡔⠉⠀⠀⠀⠀⠀⠀⠈⢻⣽⣯⡉⠀⠎⠀⢀⠀⠀⠀⠀⠀⠶⠤⣀⡋⢀⡄⠀⠀⠀⠀⠀⠀⠤⣄⠒⠀⣱⠂
⠀⠀⠀⢈⠥⠒⠒⠦⠿⣶⠶⠀⣀⣀⡠⢔⠴⠯⡀⠑⠂⠴⠤⠔⠁⠀⠀⠀⠀⠀⠀⡠⠤⠨⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣦⡀
⠀⠀⡴⠉⠁⢠⠈⡀⠀⠇⠀⠁⡉⠉⠁⠨⢠⠂⠈⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠔⠁⠑⠒⠀⣄⠀⠀⠀⠀⠀⠀⢀⠰⠦⠈⠛⢯⠂
⠀⠀⠂⠀⠀⠁⠀⡏⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡀⠀⠀⠀⠀⠀⠁⡀⠐⠀⣤⡤⢈⡁⠀⠀⠓⠀⠐⠋⠁⣀⠀⠈⡻⠉
⠀⠀⠀⠠⠁⠀⠀⢣⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠸⠋⠓⠒⠀⢐⡒⠴⠛⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⢔⠉⡐⠀⢈⡦⠒⡂⠀⠀⠀⢰⢆⡴⠂⢶⡍⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠀⠘⠀⠈⠀⠠⠉⠠⠤⠐⠁⠀⠑⠂⠾⠤⢾⢈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢘⠄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠁
⠀⠀⠀⡀⠀⠀⣀⢠⠀⠱⣰⡦⠂⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠐⠠⠤⣀⣀
⠀⠀⠀⠄⠀⠈⢤⡈⢢⣐⠋⠀⠀⠀⠀⠀⠘⣆⠀⠀⠋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠚⠉
⠀⠈⠀⠀⠀⠀⡀⢁⣄⣻⠀⠀⠀⠀⠀⠀⠀⡷⠐⠐⠃⡇⡀⠀⠀⠀⠀⠀⠠⠜⢋⡁⠒⣳⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠳⠢⡀
⠀⠀⠀⠀⡀⠊⠀⠀⠈⠙⠄⠀⠀⠀⠀⢀⠼⠋⠁⡘⠈⠃⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠈⠀⠀⠀⠡⣤⠶⠤⠶⠶⣚⡂⠀⡄⡁⠙
⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠠⠯⠔⠂⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⡀⠁⠀⣀⣼⡐
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠦⠁⠀⣀⢎⣠⠟⠁⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣡⣤⣼⣿⣶⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢤⣤⡶⠯⠷⠆⠈⠁
⠀⠀⠀⠀⠀⣠⡴⢞⣫⡥⠀⠀⣀⣀⡈⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⢀⠏⣤⡤⠤⣵⠄
⠀⠀⠀⣠⢾⣿⠞⢉⡿⠗⠋⢉⣽⣋⣀⣀⣀⣀⠈⠲⣄⠀⠀⠀⠀⣎⣀⣬⣽⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠁⠀⢀⣴⠯⠭⣄⣀⠈⠳⡀⠀⢰⠀⢉⡻⠂⢀⣠⠴⠒⠉⠁⠀⠀⣀⡀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢯⣍⡀⠙⡄⠘⣠⠎⣠⠖⠉⣀⣀⡐⠿⣖⠢⢄⣙⣍⠓⢦⣽⡷⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢤⣌⡓⢌⣾⣣⠞⣁⣤⢭⣿⣬⠉⠉⠻⠆⠀⠈⠛⠀⠀⠈⠇⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⢀⡄⢀⣠⡶⢂⣀⣿⣶⣇⣨⣟⢿⠛⢒⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⣾⣠⠋⣧⠎⠶⠚⠉⢰⠉⢹⢨⠀⢿⠳⣄⢸⡇⢰⡆⢈⡓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣴⣼⠈⠟⠀⠋⠀⠀⠀⠀⠈⣹⠅⣾⠓⠋⠀⠘⠟⢻⡞⣿⢸⡇⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣿⠀⠀⠀⠀⠀⠈⠀⢽⡜⢷⣿⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⢿⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡿⣷⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠠⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠤⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣠⠴⠻⡄⠀⠀⣀⢀⡾⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡄⢹⡀⠀⠀⠀⣀⠙⣆⢰⣿⡎⣧⠊⠔⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢦⣣⠀⢷⠒⠚⠋⠁⠀⠈⢷⣇⠁⠹⠸⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⠀⣸⠦⠀⠀⠀⠀⢀⡤⠬⠿⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠁

⣏⠉⣹⣋⣛⣛⣛⣛⣏⣛⣛⣛⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠽⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣀⣀⣀⣤⡙⢛⣷⣶⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠒⠂⠋⠉⠑⢒⡒⠒⠓⣶⠀⢂⡒⠘⢽⣿⣿⣿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠻⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠞⠛⢓⣛⣫⣽⣿⣿⣿⣿⣽⣭⡿⠷⣦⢤⡀
⠀⠀⠀⠀⠀⢠⠖⠀⠄⠠⢄⠽⠠⠦⠤⠼⠬⣥⠬⣿⣿⣽⣿⠭⠿⠿⢿⣿⣟⣽⣿⣿⣷⣮⣄⣙⠋⠙⢷⣄⡀
⠀⠀⠀⠀⣰⠋⠁⠠⢤⣤⣀⣆⣀⣐⣀⣄⢤⣤⠠⠤⠤⢤⣴⣶⣶⣶⣿⠿⣾⣿⡷⣿⣿⣿⣿⣿⣷⣦⣄⠙⠻⣤⡀
⠀⠀⠀⢀⠎⠀⠂⠉⢁⣈⣈⣁⣀⣉⣁⣈⣉⣙⣏⣉⣉⣉⣉⣭⡤⠤⢶⣶⣿⡿⠗⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⢸⣷⣄
⠀⠀⠀⢠⠀⠀⢌⠐⠒⠂⠐⠳⠒⠒⠒⠒⠒⠲⠗⠒⠒⠒⠖⠛⢛⣋⣉⣯⣿⢷⣫⣿⣿⠛⠛⢿⣿⣻⠓⠂⠀⠐⠉⢻⡆
⠀⠀⠀⠘⢀⡠⢼⣒⣒⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣬⢭⣭⡭⢽⠿⠿⢿⣟⣯⡥⣶⣿⣷⣦⣛⣿⣿⣖⠲⠤⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢀⠀⠂⠹⢯⣀⡠⠤⠌⣃⠀⠀⠀⠠⠠⠟⠂⠀⠀⠒⢒⣚⣻⣿⠿⢷⣾⣿⣿⡿⣿⣿⣿⣿⣉⠄⣒⠀⠄⣀⠀⠘⡙
⠀⠀⠀⢨⠀⡄⠈⠻⡄⠀⣀⡠⠄⠀⠀⠀⠀⠀⣚⣉⣀⣉⣭⣥⣼⣗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠈⣀⠀⠀⠀⢈⡓⢱
⠀⠀⠀⠀⣅⠋⠘⠑⢨⡥⠤⠐⠂⣐⣀⣀⠈⠭⠭⣥⣀⣖⣾⣿⣯⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡛⠛⠷⢮⣍⡒⠀⡀⠉⠹
⠀⠀⠀⠀⢨⡀⠼⠂⣀⠴⠒⠈⣉⣠⡤⠤⠶⣒⣺⣻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣈⡁⠂⠤⡙⠳⢬⣑⠀⠂
⠀⠀⠀⠀⠀⠉⡼⠋⣠⡴⠚⣁⣭⠴⢒⡻⠭⣵⣶⣿⣿⣿⢿⣿⣿⣿⣿⣽⣭⣿⣿⣟⣿⣿⣿⡍⠉⠉⠛⠛⠶⣤⡑⠢⣙⠻⠃
⠀⠀⠀⠀⠀⠀⠐⢾⠁⡠⠚⣁⠴⣚⡽⣶⣿⣿⠿⠛⠉⣱⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠤⢈⡒⠤⣀⠉⠛⢷⣄⡑⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢅⡼⣡⠞⣡⣮⢞⠁⢀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⠀⠀⠈⠹⢷⣮⡛⠦⣄⡉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⣼⢵⠁⠈⠔⡁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉⠋⠙⣶⣤⣀⠠⢉⠛⣷⣬⡙⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⡀⠀⢐⢿⣿⣿⢿⣿⣿⣿⣿⠿⢿⣿⣿⣟⠋⠉⢷⠦⣄⠀⠙⠛⢿⣦⣁⠀⠈⠻⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣾⡄⠹⣿⢸⣿⣿⣿⣿⣻⣿⣶⠬⠉⠁⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠄⡀⢿⣿⡆⠟⣾⣿⣿⣿⣿⣿⡿⠗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡅⢸⣿⣿⢠⣿⣿⣿⣿⣿⣿⡗⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⠊⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⢸⣿⣿⣼⣿⣿⣿⣿⣯⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠛⡷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⠻⣿⣿⣿⣿⡿⣿⡧⡉⠢⡀⠀⢢⡀⠀⢠⡀⠀⠳⣄⠀⠸⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⣿⣿⣷⡙⣮⠳⡙⠄⠈⢆⠀⠹⣆⠀⢹⣄⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠏⠉⠛⢿⡜⣧⢹⡈⢆⠈⢧⠀⠸⣧⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠱⢸⣎⡇⠘⣇⢸⡇⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⢀⠤⠤⠤⠤⣀⣬⠄⡠⠊⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠼⠗⠐⠒⠫⠚⠓⢊⣂⠁
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⢀⡀⠀⠠⡂⠄⠀⣄⢠⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⡤⢁⡲⢟⣛⣛⠛⠉⠍⢉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢢⢠⡑⢃⡱⡐⢊⡭⢦⠆⡤⢄⠐⠀⠤⠀⡀⠀⡀⢀⣠⣤⡒⠩⢛⠖⠀⢠⡉⡈⠸⠛⣤⡀⠀⠈⠀⠀⠱⣀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠓⠈⡀⡆⠀⣃⢡⠐⠌⠡⠉⠐⢪⡁⡴⢊⡦⣨⡄⣓⠓⠚⠉⠉⠀⠀⠀⠐⠁⠠⢀⡀⠘⠚⠛⠀⠀⠀⠀⠀⠀⠐⢤
⠀⠀⠀⠀⠀⢠⠀⠀⠢⠀⢄⠐⢀⣆⡀⢲⣼⣄⠀⠀⣷⠀⢘⡷⠘⢆⠑⣦⠁⠨⣉⠓⠄⣀⠈⠉⠉⠉⠒⠢⠍⣛⡂⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠱
⠀⠀⠀⡀⣠⠁⠀⠀⢰⠀⠰⣷⠈⣏⣿⢰⣿⠟⠀⠚⠃⠀⠈⠀⠀⠀⠑⡌⠓⠄⡉⢧⡀⠀⠙⢢⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⢀⠐⠁
⠀⠀⠀⠈⠀⡆⠀⢂⠜⠀⠘⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠙⠄⠙⡄⠀⠀⠇⠀⠀⠀⠀⠀⢀⠠⠐⠊⠁
⠀⠀⢀⠀⠄⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠉⠀⠀⠀⠀⡘⠰⡘⡀⠀⠔
⠀⠀⢸⠀⡖⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠰⠁⡇
⠀⠀⠘⣼⣧⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡐⢦⡀⠀⣸⡇⢀⣤⣤⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠿⢿⣿⣿⣄⣿⣷⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⣾⣿⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⢻⣿⣿⣿⣿⣿⣍⠉⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⢠⣾⠃⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⣾⣿⠟⠿⠁⢻⣿⡄⠀⠀⠉
⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣿⣷⣿⣷⣿⣟⠻⢿⣷⡄⠀⠀⠀⠀⣸⠏⠀⢀⣼⣿⡗⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠁⠀⠀⢀⣠⣴⡿⢿⣿⣿⣟⠙⠛⢷⡄⠈⠁⠀⠀⠀⠀⠟⠀⢠⡾⢹⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⡿⠉⠀⢠⡿⣿⣿⣿⣆⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠈
⠀⠀⠀⠀⠀⠻⠀⠀⠀⣿⡇⠀⢻⡝⢿⡄⠀⠸⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠻⣎⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣿⣿⠀⣿⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣾⣿⣿⣾⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠻⠿⠿⠟⠛⠛⠃
⣿⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠑⢶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣠⡤⠼⠗⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠉⠁⠀⠀⠀⠀⠀⢠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣠⡤⠤⠶⠛⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠖⠛⠋⠉⠁⢸⠃⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣻⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡏⣄⣀⡠⠤⢲⠚⠀⠈⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣤⡤⠶⠒⢺⠏⠁⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢿⠁⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡆⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠈⠀⠈⠆⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⢀⣴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⢀⡠⣆⣴⡟⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣠⠴⡶⠛⠭⠠⢠⣏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⡾⡔⠚⠋⡀⠟⢀⣠⣴⣶⣖⣾⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠉⠀⢀⡧⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣗⣊⣥⣴⣾⣿⣿⣿⠿⠛⡿⣿⣿⣿⣋⢀⡀⣀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠈⣛⠛⡛⠋⣋⣯⣍⢓⣻⣷⡎⢿⣻⣯⠙⣿⣿⣿⡟⠛⠶⣶⣿⣷⣞⣉⣀
⠀⠀⠀⠀⠙⠉⠀⠀⡠⠙⣉⣠⣿⣭⣽⣿⣿⡞⢛⣯⣥⣶⣿⣟⡛⠟⢛⣩⣶⡈⠀⣍⣛⣛⠛⠻⣿⣿⡿⢶⣿⣯⣟⡛⠁⡤
⠀⠀⠀⠀⠀⠒⠒⠂⠉⠉⢭⣥⠄⢘⣿⣛⡉⠉⠥⠼⠛⠋⠉⠩⠔⠂⣸⡿⣟⠁⠀⢶⣭⣹⡧⣠⠤⢴⣶⣶⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⣦⠄⠀⠉⠁⠀⠀⠀⠉⠙⠉⠀⠀⠒⠬⠭⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⡉⡛⢿⣙⣠⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣠⠶⠛⠉⠀⠙⢀⣁⣒⣤⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣽⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣿⠀⢠⠾⠋⠨⠛⣤⠀⠀⠀⠁⣸⠛⢻⢿⢿⣎⢥⢴⣶⣶⣦⣤⣀⠈⠛⢿⣿⢛⡟⠂⠔⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢰⣿⣉⣴⣿⠀⠀⢀⣤⠛⣷⠀⢻⡀⠛⣶⣿⣫⣬⠘⢾⡌⠉⠀⠀⠈⠙⢻⠒⣤⣽⣿⣿⡆⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⣿⡇⢿⣿⣿⣤⠴⣫⣀⣶⣼⠻⢮⠇⢈⣽⣿⣿⣷⡀⣨⡷⡀⠀⠀⠀⠓⢀⢤⡟⠈⠻⣿⣿⡄⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡬⣿⣿⠇⣄⣽⣿⣿⣿⢆⣈⣰⣦⠙⣿⣿⣿⠗⠎⠀⢨⣿⣧⡤⠀⠐⠮⡿⣴⠃⠈⠿⡧⠀⢀⡾⢘⠇
⠀⡤⠀⠀⠀⠀⢹⣧⡘⠷⢶⣾⣿⣿⣿⣧⣶⣿⠛⢀⣴⠟⠛⢿⣿⣄⣴⣿⣿⣿⡿⠀⠦⠀⠈⢫⠙⡋⠀⡰⢂⣾⢣⠈
⠀⢳⣾⣦⣤⣄⠀⠈⠻⡖⢶⣿⣿⣿⣿⣟⠛⠧⠸⠟⠁⡄⢠⣿⣿⠛⠙⣿⢿⣟⠛⢷⡀⠀⠀⠀⠀⠀⠀⣠⠾⠡⠎⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠹⣿⣿⣿⡷⠤⢠⣦⣀⣿⣿⣿⣿⣿⣞⠀⠀⠀⠀⠀⢸⣿⠘⠀⣾⢻⣼⡿⠀⠈⠀⠀⠀⠀⠁⢠⣄⠘⢂⣠⡔⠒⢈⣩⠄⠀⠀⠀⠁
⠀⠀⠀⠈⠛⠾⠿⣶⣤⣼⣿⠻⣶⣿⣛⠙⠇⢗⡤⠀⠀⠀⠛⢠⡄⠀⢸⠂⠙⡿⢶⡀⠀⠀⠀⢈⣡⢶⢿⡟⡋⠉⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⡟⠻⠿⠿⣿⣿⣦⡋⠁⠁⠀⠀⡀⣸⡇⠀⠾⠀⠀⠀⠁⠈⣀⣴⡾⠯⠄⡍⣌
⠀⠀⠀⠀⢠⠴⢋⣩⣤⡞⢡⣴⣦⣤⣄⣹⣿⣿⣄⣀⣠⠤⠉⠀⠀⠀⠄⢀⣀⣀⣠⣞⣋⣀⣴⡖⠚⠓⠛⢶⣌⠒⣄⡀
⠀⠀⠀⣴⠟⠚⠋⠁⠀⠀⣠⠰⠙⠉⠉⠛⢛⣏⣯⣻⠽⣿⣿⢿⣭⣿⣽⢿⣿⣾⣿⡀⠉⣩⣉⣛⣷⢆⠀⠀⠈⠓⠪⣱⣄
⠀⠀⠀⡷⠀⠀⠀⢀⣤⠾⠋⠛⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠓⠌⠉⠉⠛⢧⠱⠄⠀⠀⠀⠀⢬⣃
⠀⠀⠀⠀⠀⢀⢤⡿⠉⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣠⠿⠟⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⢤⣤⣄⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⡀⢤⣤⠀⠤⢤⠤⢤⣤⣤⣤⣤⣀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤
⠩⢽⣛⣛⣻⣿⣽⣿⣿⣿⣽⣿⣿⣿⣭⡿⣿⣿⣷⠶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠿⠷⣒
⣬⡹⢟⣟⡛⣛⣛⣿⠿⢿⣿⡿⠛⢷⣾⣿⣿⣿⣿⡭⠽⠛⣙⡻⠿⠛⠛⠻⢿⣯⠉⠙⠛⠃⠰⠒⡋⠛⠒⠂⠀⠠⠤⠈⠉
⣉⠉⠁⠀⠁⠐⠒⠚⠒⢂⣀⡀⠙⠛⠛⠁⠄⠀⡀⠀⠤⠀⠀⠁⠒⠢⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⡀
⠉⠁⠀⠀⠀⠐⠂⠄⠈⠉⠉⠙⠀⠀⠀⠀⠤⣩⣁⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠒⠒⠚⠀⠀⠀⠀⠠⠄
⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣷⣶⣄⣠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⣀
⠿⠿⠛⠛⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠁⠀⠀⢰⣿⣿
⢠⣦⣄⠈⣹⡋⠁⠀⣠⣤⡃⠀⢼⣿⠇⠀⠀⠀⠉⠁⢀⡀⠀⠀⠀⠀⠀⠙⠛⢓⣆⠐⠐⠂⠀⠀⠀⠐⠻⠿⠒⡂⢈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠺⠶⠄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣤⣦⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣄⠀⠹⣿⣿⣇⠀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⣐⣺⣿⣿⢀⣿⡿⢂⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣅⣘⠿⣿⣷⣼⣿⣿⣾⣏⣴⣿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣫⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⣠⣶⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⡿⣿⣿⣗⠙⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣯⣾⡿⠛⣵⣿⡟⠀⣿⣿⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⢀⣀⡀⢀⢱⡇⣰⣴⣶⠦⠀⠀⢠⠟⢁⣾⡿⠛⣡⣾⣿⡿⠀⠀⣿⣿⠈⠘⢻⣿⣿⡄
⠀⠀⠠⢄⣛⣿⣿⣿⣆⣿⣿⣿⠁⠀⠀⢀⠀⠀⠸⠉⠠⣾⣿⣿⡿⠁⠀⠠⣿⣿⠀⠀⠀⣿⡿
⠀⠀⠀⠀⢈⣹⣾⣿⣿⣿⣿⡗⠀⣠⣿⣿⣿⣷⠄⠀⠀⠉⠉⣿⠃⠀⠀⠚⣿⣿⠀⠀⠀⠉
⠀⣠⣤⣴⣾⣿⣻⣿⣿⣿⣿⢇⣼⡿⠟⠉⠉⠉⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠁
⠀⠈⠉⣉⣭⣭⣿⣿⣿⣿⣿⣾⣿⣷⣾⣿⣿⡿⠇⠀⠀⢰⡟
⠀⠀⠺⠟⢋⣻⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣷⣧⡀⠀⢀⣿⠁
⠀⠀⢀⣴⠿⠻⢫⣿⣿⣿⣿⠈⣿⠿⣿⣦⡉⠻⠻⡀⣼⡇
⠀⢀⡾⠁⢀⣴⡿⣹⣿⢻⣿⣇⠘⣇⠘⣿⡏⠀⠀⢰⡿
⠀⠈⠀⠀⢸⡿⠱⣿⡿⠈⣿⣿⠄⠘⣆⠉⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⠀⢹⡄⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣿⣾⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣿⣿⣴⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣼⣿⣿⣿⣿⣿⣿⣿⣯⣉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠋⠛⣿⣿⣿⣿⣿⣿⡿⣛⣿⣯⢿⣟⣩⣿⣿⣿⣿
⠋⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠛⠁⠀⠀⠀⢀⣾⡞⢉⣠⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⠀⠋⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠋⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠰⢷⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣶⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣉⣉⣙⣛⣒⠃⠀⢀⣀⣀⣤⣤⡤⡤⠀⢈⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⢿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠟⠛⠿⠿⣿⡿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⡿⠭⠿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣤⣤⣤⣄⣠⣠⣶⣤⡄⠀⣀⡉⠘⣍⣺⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠤⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠺⢷⣶⣶⣤⡤⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠤⣀⣄⣀⣀⣩⣟⡓⠲⢶⣤⣤⣤⣿⣿⣿⣿⣿⣦⠹⣿⠙⣿⡋⢹⣿⡟⠀⢹⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⢩⣭⣭⠿⣦⣀⡉⠉⠉⠛⠛⠛⠛⠛⠛⠃⠀⠀⠉⢻⡀⠉⢱⡀⠀⠀⠀⠉⠉⣷⣆
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣤⣄⣤⣤⣤⣤⣠⠎⠀⣼⠇⠀⢀⣿⠀⢠⡞⠀⠀⡹⠛⣦
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⠟⡿⡿⡟⡿⠙⢛⡿⠉⢳⡖⠛⡷⠚⢿⢤⣀⣴⠇⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⢉⠔⢀⠇⡅⡇⠁⣼⠀⢀⡏⢀⠟⢀⡞⢁⡤⠛⠹⣤⢞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢠⠔⠁⢀⠎⠰⠁⢡⠈⡇⠀⠘⠀⠈⠀⠈⠀⠈⠀⠀⠀⢸⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢠⠇⠀⢀⡟⠀⢸⠀⠸⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠃⠀⢸⠀⠀⢇⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠀⠀⢸⠀⠀⠘⢆⠀⠀⠀⠀⠀⢀⣶⡟⠁⡀⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠛⠉⠁

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

⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⣤⡤⣀⡀
⠀⠀⠀⠀⡈⠘⢄⠠⢔⡲⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠓⢺⣅⠀⠏⣷⣰⠉⢲
⠘⠯⠭⣉⠁⡷⡆⣰⠋⠀⠀⠀⣰⢹⣭⡭⠷⠦⢙⣶⣶⣶⠆⠀⠀⠀⣀⣀⠠⢤⣀⠀⠘⢯⡿⣿⠇⢰⢻⡁⠠⣁⣿⣿⣆⠘⡇
⠀⠀⠀⡜⣠⠖⠢⡘⣆⠀⠀⠈⠻⣇⣛⣛⣋⠉⣈⠿⠋⠀⠀⢀⠴⠿⠤⣍⢢⡀⣿⠈⢢⡀⠉⢿⡇⠘⣗⠀⢘⢠⡅⠙⢿⣶⣿⣄
⠀⠀⠈⠚⠁⠀⠀⠙⠚⠃⠀⠀⠀⠘⠛⠒⠂⠉⠀⠀⠀⠀⠀⢮⣿⣿⣷⣮⣿⣧⣇⡠⠏⠁⠀⠈⠋⠙⣯⠀⣰⣰⣀⡴⠶⠻⠿⣿⡧
⠀⠀⠀⣀⢤⠴⠶⡶⡤⣤⡀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠤⢤⠠⠀⠀⠉⠙⢫⠞⠻⣟⠒⠒⡏⠀⠀⠀⠀⠈⠋⠉⠉⠁⢠⡴⠶⢤⡀
⠀⣴⠻⠹⠀⠀⡄⠀⢀⠃⠞⣳⣄⠀⠀⡠⠊⠣⡀⢣⡀⣹⠀⠇⠀⠀⠀⠘⠣⣧⠈⠹⠞⠀⠀⠀⠀⢀⡄⠀⠀⠀⡠⣿⢏⠢⠡⢹⡄
⠈⡎⢂⠁⠃⠀⢀⠀⠈⠈⡰⠡⢋⠆⡐⠙⠢⣤⠟⠉⡏⢻⢆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣸⣷⠀⠀⠀⡇⡈⠘⡄⡂⠆⣧
⠀⠘⢆⢢⢠⣠⢘⠀⠀⠐⠐⡡⠋⠀⡗⠲⠴⡟⢒⣶⡷⣯⠤⡇⠀⠀⣰⠇⡢⠀⠄⡁⠨⢳⡄⠀⠀⡯⠰⡄⠀⠀⢻⢁⠀⠀⠀⠀⢻
⠀⠀⠀⠳⣕⢷⣿⣾⠌⣠⠊⠀⠀⠀⠘⢦⠒⢛⠥⣮⢶⠛⣤⠇⠀⠀⣥⠊⣠⠎⡔⢠⠁⡅⡇⠀⣘⣋⠉⢵⠀⠀⠸⡼⡠⡀⠀⠀⡿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠙⠻⠤⠧⠜⠛⠁⠀⠀⠀⠋⣴⠃⣼⠁⣾⠀⣇⡿⠀⠫⣤⣺⣿⡷⠀⠀⢿⡵⣣⢇⢢⡇
⠀⠀⠀⠀⣠⡤⣀⢤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠒⠛⠴⠟⠒⠉⠁⠀⠀⠸⠿⠋⠁⠀⠀⠈⠻⣵⣯⠞
⠀⠀⡄⠊⠀⢳⡀⠈⢁⡆⠙⢦⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠻⣟⠛⠒⠚⠳⠲⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣼⣀⣀⣠⡜⢋⠓⠺⢀⠠⠈⢳⣄⠀⠀⡼⣛⣫⡈⠛⣷⣄⠀⠀⠉⢢⡀⢀⠀⠃⠈⠙⠦⣄⠀⠀⠀⠀⠀⢴⣾⣿⣿⢧⣄
⠀⢱⠀⠏⣇⡼⣟⡖⠈⢹⠶⠛⠉⣿⠀⠀⡏⣴⣾⣿⣮⡟⣿⣆⠀⠀⠀⠙⠒⠒⠓⠒⠒⠒⠚⠓⠀⠀⠀⣠⡾⣿⣛⣫⣤⠛⠙⡦⡀
⠀⠘⣦⠚⠙⠑⡎⠉⠺⡇⢠⠀⣆⢺⠀⠀⢣⡻⣿⣿⣿⣿⡿⠛⠀⣀⠤⢤⠞⠛⠙⣟⠓⠦⣄⣀⠀⢠⢞⡙⠀⠈⠀⠀⠀⠐⡜⡄⢽
⠀⠀⠈⠙⠲⠶⠧⠖⠲⠶⠚⠒⠛⠛⠁⠀⠀⠙⠚⠛⠋⠉⠀⠀⢰⡡⠊⠀⢂⠀⠀⠘⣄⠳⣈⡟⠀⠸⡕⢡⢀⠁⠃⡛⡈⡁⡐⡜⣼
⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣔⣚⠉⠓⢔⠄⡄⠈⣳⠋⠀⠀⠀⠘⠧⣼⢐⡃⡃⡇⢡⣱⡴⠋
⠀⠀⠀⣴⣿⣿⣛⡷⡄⠀⠀⢀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⠈⠳⣝⣛⢛⣿⣚⣷⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠛⠉⠁
⠀⢠⢚⢝⡉⠀⢉⣶⡿⡆⠀⡟⡩⠋⡥⠼⣧⡈⠉⢀⣴⣦⡌⢷⡀⠈⠈⠉⠀⠉⠀⠀⠀⠰⡒⡖⡲⠉⡟⠉⢹⠖⠘⠳⢦⣄
⠀⣇⢏⢮⠸⣼⣿⡿⣿⠁⢠⡋⢰⠺⢷⡄⠈⠻⣦⠈⢿⣿⣿⡎⣧⠀⠀⠀⢀⡖⠒⠦⣄⠀⠘⢶⠃⢀⠀⠀⠀⠀⢐⠈⠀⢘⢆
⠀⠈⠲⢽⣽⣿⣫⡞⠁⠀⢈⣶⡛⠳⣆⠙⠷⣄⠙⣦⢸⣿⣿⣧⣽⠀⠀⣀⡼⠅⠀⠀⠠⠛⣄⠈⠛⣇⠀⢠⠀⠀⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠘⢏⠙⠶⢌⣙⡓⢮⣿⣌⢣⣻⣿⣯⠏⠀⢸⠙⠌⠀⠀⠀⠀⠄⢹⠀⠀⠈⠛⣇⠀⠐⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣭⣭⣴⣍⣰⣇⠉⣿⡟⠀⠀⠈⠣⣀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠘⢶⣄⡀⠀⠀⠠⠀⢛⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⢠⣼⣦⣠⣾⣥⡄⠀⠀⠀⠀⢀⣭⣿⣦⣴⣶⣂⠀⠀⣈⣿⣧⣾⣯⡄⠀⢤⣶⣿⡆⣠⣴⣤⡀
⠀⠀⠀⢀⣼⣿⠛⣿⡿⠗⠂⢀⣦⣄⣾⣾⠉⡟⠉⣿⢹⠀⠈⢩⡿⢻⣿⠏⠋⠀⠿⣿⠁⡆⢿⡍⠻
⠀⠀⠀⠀⢰⠃⠀⠈⠛⠀⠀⠛⣟⠙⡏⠃⣼⠁⠀⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⡇⠀⠛
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⠀⠀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣀⠐⣦⣼⣀⡤⠀⠀⡼⣿⣷⣴⣿⣿⠂⠀⢐⣶⣿⣷⣴⣿⣏⠀⠀⢀⣠⣤⣾⣿⣯
⠀⠀⢀⣰⣦⡿⠿⢍⣿⡿⢿⣗⠀⣾⢹⡏⢩⠉⠙⠃⠀⠈⢹⡟⣹⠻⣿⠿⠁⠀⠽⢿⠛⢋⠙⠣
⠀⠀⠠⠟⣿⠙⢦⠏⠀⡟⠀⠈⠀⠈⠀⠁⠘⡆⠀⠀⠀⠀⠈⢁⡇⠀⠉⠀⠀⠀⠀⠈⠀⠘⡄
⠀⠀⠀⠀⠈⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢶⣄⣂⣂⢈⣑⡚⢲⣄⠈⠉⠉⠉⠛⠛⠋⠉⠁⠲⣤⡤⣤⠤⠤⠄⠰⠒⠲⠿⠛⠋⠙⠹⢻⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠛⠟⡛⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠀⣿⣿⠀⠀⣿⠀⠀⠠⣧⠀⠀⠀⢰⡟⠀⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⣉⣉⡅⡾⢧⡀⠀⠀⠀⡀⠀⠀⠀⠀⣼⢢⣿⣿⠀⠀⠸⡄⠀⠀⣿⠀⠀⠀⣾⡇⠀⠀⢀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠒⠉⣁⠠⠔⠋⠉⠛⠛⠢⣄⠠⣬⣩⣦⣤⣵⣴⣿⣽⣧⣀⣀⣠⣷⣄⣄⣿⣇⣀⣀⣿⣇⣄⣴⠟⠛⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠈⠀⠀⠀⠠⠐⠀⠁⠀⠀⠀⣀⣀⣀⣤⣶⣿⣿⣾⢿⣿⣿⢿⡟⢫⣹⠟⣟⣿⠟⡿⡿⠟⣿⣿⡿⣻⣿⣿⣿⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⢀⡴⢋⠽⠛⠉⣩⢟⡽⢋⣴⣟⡼⢫⡎⡼⠋⠀⡆⠋⢀⡾⠋⢀⠸⢋⠜⠉⠀⠀⣠⣜⣿⣷⣮⠟⠃⣧
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣰⣿⠖⠁⣠⠟⣿⠗⠁⡸⠫⣿⠟⢠⠟⠀⠁⠀⠀⠁⠀⠘⠀⠀⠋⠀⠈⠀⠀⠀⠞⠉⠉⠉⢹⣿⣤⡀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⢀⡔⣡⠞⠁⣰⠎⢠⡞⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢰⠋⣠⠞⠁⠀⣾⣧⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠿⠂⠁⠀⠀⠀⡏⢩⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⢰⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⡿⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣨⠏⠀⣧⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠄⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⡀⠀⠐⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⣀⠀⠀⣀⠤⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡘⠀⢀⠞⠳
⠀⠀⠀⠘⠀⠙⢦⡀⡇⠀⠀⠱⡀⠀⠀⠀⠀⡰⠉⢀⣴⣦⣄⠑⡎⢀⣴⣶⣤⠀⠈⢆⠀⠀⠀⠀⢠⠎⠀⠀⠀⣩⠔⠁⠀⠀⡇
⠀⠀⠀⢡⠀⠀⠀⠙⠃⠀⠀⠀⠘⡀⠀⠀⠀⡁⠀⣿⣿⠙⣿⡆⡇⢸⣿⡋⣻⡇⠀⢸⠀⠀⠀⠀⡈⠀⠀⠀⠀⠁⠀⠀⠀⡜
⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠱⣀⠘⢿⣿⠟⣡⠗⢌⡻⣿⠿⢁⡴⠃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠙⠢⠄⣀⡀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠁⣒⡲⠈⠧⠤⠤⠏⠉⠏⠠⠄⣀⠀⠀⠀⢀⠎⠀⢀⠤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠙⠢⣄⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠑⢤⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⢎⠀⠀⠀⢀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⣈⠑⠢⠤⣀⠇⠀⠀⠀⠀⠀⣀⣠⡤⠤⢴⠒⡎⢋⣹⠼⡟⠀⠀⠀⠀⠀⠈⠦⠔⠊⣡⠤⡄⠈⠢⡀
⠀⠀⠀⠀⠀⡴⠁⢀⠜⠀⠉⣒⣦⣄⠀⠀⠀⠀⠀⠀⠙⢗⠷⠺⠔⡖⢻⠙⣠⠞⠀⠀⠀⠀⠀⠀⢠⡴⠖⢧⠀⠀⠈⢆⠀⠱⡀
⠀⠀⠀⠀⡜⠀⢠⠎⠀⢀⠔⠁⡤⢉⣃⠀⠀⠀⠀⠀⠀⠀⠙⠲⠧⠵⠜⠊⠁⠀⠀⠀⠀⠀⠀⣀⠤⠐⠲⡀⠱⡀⠀⠈⢆⣀⠅
⠀⠀⠀⠀⠈⠒⠁⠀⢀⠎⠀⡜⠀⠀⡰⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠱⠀⠀⠀⢱⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠜⠀⠀⢠⠁⢠⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠀⠀⠀⢡⠀⢃⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡃⢸⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⣰⢋⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣛⠒⠶⣄⠀⠀⠀⠳⣿⠀⠀⠀⡴⣫⣭⣝⢷⠀⠀⠟⠋⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠙⠉⠛⠳⣬⡳⡄⠀⠀⠀⠀⠀⡾⣹⠋⠀⠁⢸⡇⠀⠀⡴⠿⠓⡿
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠻⠙⣆⠀⠀⠀⢰⡇⠟⠀⠀⠀⡼⠀⠀⣸⠁⢀⡼⠁⢀⡠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⡅⠀⠀⠀⡼⠁⠀⠀⣿⠖⠁⠀⠀⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠹⡄⠀⠘⡇⠀⢀⡞⠁⠀⠀⠀⠉⠀⠀⠀⢀⣠⡤⠴⣖⣒⡶⣤
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠈⠉⠛⢦⣄⣷⠀⠀⢷⠀⡾⠀⢠⣤⡀⠀⠀⠀⢀⡴⣫⠴⠚⠉⠉⠉⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠈⠳⠶⠤⠿⠷⠄⠀⠀⠀⠀⠙⠛⠀⠀⠘⣦⡇⠀⢸⣼⠃⠀⠀⣰⠏⠴⠃⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠘⠃⠀⠘⠃⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣛⣻⡏⠛⠛⠓⠶⢤⣀⠈⠻⠆⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⡴⢋⡴⠞⠋⠁⠀⠀⠀⠀⠀⣀⣤⣼⠷⠄⠀⠀⠀⠀⠀⠀⣸⠁⢀⣤⠖⠛⠉
⠀⠀⠀⠀⠀⠸⣇⠺⠇⠀⠀⠀⠀⣀⣤⠶⠛⠉⣠⣶⣶⡦⢄⠀⠀⠀⠀⠀⣿⠞⠋⠀⠀⢀⡠⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠛⠋⠉⠀⠀⠀⠀⠿⠭⠭⠶⠦⣷⠄⠀⠀⠈⠁⠀⠀⠀⠾⠯⠤⢤⠴⠋





⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⠠⢄
⠀⠀⠀⣄⣻⣿⣠⠀⠀⠀⠀⠀⢸⣷⠄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⡀⠣⡀
⠀⠀⠀⠀⣽⣿⣋⠀⠀⠀⠀⠠⣽⣯⡀⠀⠀⢰⣿⡆⢀⡤⣤⠦⠐⠚⠉⠓⠦⡄⠒⠁⠀⠀⠀⠀⠀⠀⠈⠀⠙⠑⠢⣀⣀⠀⠀⠀⠀⣸
⠀⠀⠐⣦⣿⣿⣿⠔⠙⢶⡋⢻⣿⣿⠟⡋⣷⣼⣿⣿⠁⠀⠀⠑⠢⡀⠀⠀⠀⠈⠲⠒⠤⠤⠀⠀⠴⠒⠉⠀⠐⠂⠀⠀⢠⠉⢂⡄⠀⣿⡆
⠀⠀⢸⣯⣿⣿⣿⡀⡀⣼⣧⣻⣿⣿⣿⣷⣿⣿⣛⠛⠒⠳⠦⠤⢤⣤⣠⣤⣶⡶⢦⠀⣷⢀⣤⠴⠒⠉⠈⠑⠒⢴⡦⠆⢾⡆⢺⣗⣪⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣤⡀⠀⠀⠀⠈⠻⣿⣿⡇⢾⣾⣿⣿⣧⣆⣠⡄⣀⣀⡄⣼⣷⣧⣾⣧⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠻⠆⠬⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢒⢺⣿⠒⡂⣿⣿⡿⣿⠭⢿⣿⠀⢸⣿⡟⣿⢻⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣼⣿⣤⣅⣿⣿⡇⡟⠥⠿⢿⠂⢸⡿⠒⠛⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠟
⠀⠀⠀⠹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣤⣼⢧⣤⢤⣘⣛⣻⣿⡿⠿⠿⣿⣿⣿⣿⣿⣭⣭⣭⣁⡀
⠀⠀⠀⠀⢹⣿⣿⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⣉⡙⣛⠛⣛⣟⠳⢷⣤⣍⣉⣭⣭⣽⠿⠿⠟⠛⠛⠁⠠⡤⠤⡦⣀⠛⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣿⣿⣿⣆⣽⠿⣛⣭⡛⠻⣿⣿⣿⣙⣛⣻⣿⣯⣍⣤⣁⣐⡒⠾⣉⡉⠓⠒⠲⠾⠤⣀⣤⣤⠤⠼⠛⠊⠉
⠀⠀⠀⠀⠀⠘⠇⠘⣿⣿⣿⣿⣿⣽⣿⣇⣾⣿⣿⣷⣄⢛⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣟⡛⢿⣟⣛⠛⢶⣤⡤⠶⣯⣤⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣶⣴⡶⢹⡇⢻⣿⣷⣶⣿⣿⣿⣤⣷⣤⡀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠔⠋⠵⠛⠉⠽⢿⠿⢿⣿⣿⣿⣿⣿⣿⢧⣿⣶⣶⣿⡿⠯⠍⠉⠈⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠚⠋⠉⠉⠙⠈⠛⠛⠒



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣠⣴⡶⠷⠶⠦⣄⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣿⠏⠀⠀⠀⠀⠈⣧⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣀⣠⠤⠤⢤⣀⠟⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⢰⡇⠀⠀⢸⣿⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢽⠿⡇⠀⠀⢸⣿⠟⠋⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⢲⣳⡟⠊⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠶⢰⣯⣀⡀⢀⣀⣀⣀⡀⢀⠀⠀⠀⡀⣄⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣴⣾⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢻⡜⣿⠛⢻⠻⢿⣿⣿⡛⠋⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣹⣦⣸⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣿⡟⠳⣟⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⣿⡇⠀⠈⢧⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⢯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⣲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⡀
⠀⢸⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⣠⢳⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⢹⣧
⠀⣾⣇⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⢻⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣽⢁⣎⣿
⠀⣿⣿⡈⢻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⣄⣷⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣾⠃⣸⣯⣿
⠀⣿⣿⣷⡌⢿⣧⡵⠶⠀⠀⠀⠀⠀⠀⠀⢀⡠⣣⠴⠊⣵⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠃⣰⣿⣹⡿
⠀⠸⡿⣿⣿⣆⠳⣄⠀⠀⠀⠀⠀⠀⢀⣔⣣⠿⣇⡤⠆⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠙⢿⠃
⠀⠀⠹⠏⠘⢿⣧⡀⠑⠦⣀⠀⢀⡔⣣⠟⢁⡜⠉⠀⠀⢸⣧⣿⣿⣦⡀⠀⠀⢀⡀⠄⠂⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠉⠉⠙⠓⠊⠁⠀⠀⠀⠀⢸⣿⡟⠝⠛⠉⠀⠉⠁⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⢀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣷⣶⣤⣀⠀⠀⠀⠀⢸⣿⡫⣴⣦⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⠀⠀⠉⠛⠿⣿⣶⣤⡀⣸⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠷⠀⠀⠀⠀⠀⠉⠻⣿⣷⣿⡿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣛⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⡈⠙⠻⠿⠿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⡿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣤⣀⢰⣄⠀⣿⡆⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⢰⠂⢀⡴
⠀⠀⠀⠀⠀⠛⠷⣿⣧⣿⣧⣾⣯⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣆⣼⣧⣿⢁⣠⠴⠒
⠀⠀⠙⠛⠶⢶⣤⣿⣿⣿⣿⣿⣥⡶⢃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣦⣽⣿⣿⣿⣾⣿⣵⠶⠖⠛⠁
⠀⢀⣒⣚⣒⣒⣶⣿⣿⣿⣿⣿⣿⣾⣭⣥⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⣾⣿⣿⣿⣿⣿⣧⣖⣒⠛⠒⠂
⠀⠈⠁⠀⢀⣠⠰⣛⣻⣿⣿⣟⠻⣎⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⢈⠺⢛⣿⣿⣿⣿⣿⣉⣙⠒
⠀⠀⠀⡴⠛⠁⠀⠔⣿⣿⣿⡧⠄⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠹⢿⣿⣿⠁⠉⠋⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣧⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣷
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠘⠛⠉⢉⣭⠍⣀⣀⣰⠀⡤⣤⡬⣬⡖⢊⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠹⣶⢷⠀⣠⠄⣀
⠀⡤⣴⣂⢀⡾⠫⡤⢤⣶⠶⢦⣴⣦⡾⠋⢳⣾⣿⣷⣮⣿⣿⣿⣯⣡⣿⣼⣽⣦⠦⠴⢾⣶⣶⣾⣬⣌⢀⣤⣴⣿⣷⡄⠛⢉⠂⢀⣒⣤⣄
⣤⣬⣭⣤⣤⣤⣭⣭⣭⣭⣭⣥⣭⣭⣭⣭⣭⣽⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣬⣭⣭⣭⣭⣭⣭⣥⣤⣬⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣭⣥
⣿⣿⣩⣝⣫⣍⣉⣏⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢚⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠟⠉⠉⠀⠀⠀⢁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠁⠐⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⢀⠠⠀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠐⠀⠀⠑⠈⢀⠀⠠⠀⠀⡤⢘⠢⠀⠀⠰
⠀⡘⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢟⠃⠃⡀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠙⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠂⠀⠀⠀⠀⠀⠀⠑⠀⠄⠀⠀⠄
⠀⠴⠦⣖⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠿⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠈⠣⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⡀⠀⠈⠀⠈⠀⠂⠁
⠀⡆⢐⠀⠈⠀⠤⠀⠐⠀⠋⠀⡄⠂⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⡀⠀⢠⡀⠀⡄⠐⢀⠀⠀⠀⡐⠌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠈⠏⢂⡈⠀⠀⡠⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠠⡀⠀⠀⡀
⠀⢀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⢀⠄⠁⠐⠄⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡄⠊⠁
⠀⠀⡈⠉⠀⠀⠀⠀⠑⣄⠀⠀⠰⠀⡋⠀⠀⠀⡀⠐⠐⠁⠀⣐⠾⠛⠁⠀⢀⡀⠄⠀⢦⠢⣄⠀⣠⢀⠈⠈⠀⡄⠈⠓
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠑⠚⠓⠒⠂⠛⠀⠀⠃

⠀⠀⠀⢢⡀⠀⠀⢠⠖
⠀⠀⠀⠀⢳⠀⠀⠸
⠀⠀⠀⠀⠘⡆⠀⡏
⠀⠀⠀⠀⠀⡇⢀⠁
⠀⠀⠀⠀⠈⡇⢨⠄⣀
⠀⠀⠀⠀⠀⣷⠘⣧⣨⣵⣄⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠈⠳⣌⠻⣍⠻⢿⣖⢦⠙⠷⢥⣄⡈⠀⠀⠁⠢⠤⠤⠤⠄⠀⢀⡦⠀⢠⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠭⣥⣚⣓⠦⣵⡢⣴⢟⣿⡔⠂⠀⣀⠀⠀⠀⣀⣀⢷⣄⣀⡀⢘⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠭⣽⣛⣿⣿⣷⣧⣼⢀⠉⠻⣿⢞⣻⣿⣾⣿⣿⡧⢸⣿⣶⣦⣊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⣁⢔⢛⠛⣿⣿⣟⣛⠓⢬⣭⣂⣀⣀⣈⣩⣼⣿⡛⢁⣼⡿⢻⡿⠻⠗⠆⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢠⠎⢄⡼⢛⣯⡿⣽⣟⣿⣿⡿⠿⠿⢿⣿⣿⡿⢛⣷⢟⣋⣥⣦⣼⠃⣠⡔⠝⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⢡⡴⠛⠁⡰⣿⠉⣤⣤⣶⡶⢛⡿⠕⠀⢠⠞⢁⠜⠉⣠⠤⠾⣿⣹⡗⠀⡴⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠝⣺⠇⠀⢀⡔⠀⡿⠀⣥⠜⢥⡔⠉⠀⠀⣠⠃⡴⢁⡔⠉⣠⠖⠊⠉⣙⠿⢾⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠞⠀⠁⠀⠀⠌⠀⠀⣁⢠⣯⡀⠀⡇⠀⠀⠀⠁⠀⠀⠈⠀⠈⡥⠚⣉⣭⣹⣻⠀⠐⠢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠹⡎⠀⠉⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⡀⠀⠘⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣻⠟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣄⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠤⠤⠴⠿⡿⠷⠤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠷⠶⠤⣶⣶⣶⣶⡶⠴⠶⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣶⣶⣶⣶⣴⣦⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⢦⠀⠀⠀⠀⠀⠀⠀⡇⣸⡇⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⣸⠀⢧⡀⠀⠀⠀⠀⠀⡇⢹⣷⠀⠀⠀⠀⠀⢠⠏⢸⡆⠀⠀⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿
⠀⣿⡄⠉⣳⡄⠀⠀⠀⢀⡇⢸⣿⠀⠀⠀⠀⣰⡋⠀⣼⡇⠀⠀⣿⣿⣿⡆⠀⣹⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡋⠀⣸⣿
⠀⠻⠻⣄⠈⠦⣀⠀⢀⣸⠃⣼⣿⣄⠀⢀⣠⠞⢁⣾⠻⠃⠀⠀⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿
⠀⠀⠀⠙⢷⣄⡈⠛⠛⠁⠀⣿⣿⣿⣭⣭⣤⣶⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⡈⠛⠛⠁⠀⣿⣿⣿⣭⣩⣤⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠷⣶⣶⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣾⣿⣿⣿⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣷⣿⣯⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⣿⣿⣿⣍⡁
⠀⠀⠀⠀⠀⠉⠉⣛⣿⣿⣿⣿⣿⣿⡛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⣾⠿⢛⡽⣿⣿⣿⣿⠻⣿⣦
⠀⠀⠀⠀⠀⢠⣾⡿⣿⣿⣿⣿⣿⡻⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣴⠋⣸⣿⠏⢿⣿⡄⠀⠙⠇
⠀⠀⠀⠀⠀⠟⠁⠀⣿⣿⠹⣿⣏⠻⣄⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡴⠟⠁⠀⢘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⠛⠦⠘⢦⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⣿⣴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣝⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢦⣀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣈⠁⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⡠⢀⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣧⡅⠢⡀⠐⣿⣦⡀⠀⠠⢊⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣻⣧⠬⠢⡙⣿⡰⠠⢁⣾⠛⢁⢄⣤⡶⠦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⠿⡷⣶⣤⣐⠺⣗⡑⣘⣧⡇⣽⣭⣯⣴⠟⠛⢉⣁⠠⣄⢤⡨⣤⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣈⠍⣉⠁⡛⠻⠿⣵⠢⡘⣯⣅⠟⡵⢋⣰⣦⡙⠧⠙⠳⠙⠃⠙⠈⠑⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣵⢟⡾⣻⠇⡾⢃⡾⣱⡾⡄⠨⣙⢺⣼⣿⠏⠔⢃⣨⣍⣭⣭⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⠀⠀⢠⣂⣥⡦⣶⡶⠖⠐⣾⣡⠛⡏⢙⡭⠥⠬⣍⢩⡉⠛⠫⢛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢛⠝⠃⢡⣠⢞⣩⠿⣀⣸⢈⡿⠫⡐⢦⣦⣼⣳⡗⢰⡄⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡄⠿⢈⣧⣾⢃⡈⠀⡇⡁⢨⡌⢦⡄⠩⠙⠋⠇⠘⢱⣷⡲⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣴⣆⠻⠀⠘⠇⡇⢬⡏⠀⠁⠁⠐⣯⡈⡼⡂⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⠉⠀⠀⠀⠀⠡⡸⠍⢀⡄⣀⠠⠞⠀⠘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠦⠘⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠒⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

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

⣿⣿⣿⣿⣿⣿⣷⣿⣦⡄⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣶⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⢀⣤⣿⣿⣶⣾⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⡓⠀⢈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⠛⠻⡟⠃⠀⠀⠀⠀⠀⠀⢡⡅⠀⠀⠀⠀⠀⠀⢪⣭⡅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣿⣯⠉⠀⠸⠷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⡦⠤⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣶⠓⠂⠀⢐
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠃⣾⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣙⠁⠀⠸⠃
⣿⣿⣿⣿⣿⣿⣿⣟⠛⠿⢿⣿⣭⡻⠿⣿⣿⣿⣶⣤⣿⠿⠛⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⠀⠀⢠⠄⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⠟⠉⠐⠈⠉⠉⠀⠀⠀⠉⠉⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠼⠃
⣿⣿⣿⣿⣿⡿⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠒⠀⠀⠺⠀⠀⢠⣶⣿
⣿⣿⣿⣿⣿⣷⣄⣠⣊⣄⡀⢀⣠⣴⣾⣦⣤⣄⣶⣦⣴⣿⣷⣴⣿⣿⡇⠀⠀⠀⠀⠀⠴⠾⣿⣷⠖⢘⣦⣦⠀⠀⣀⠀⠀⢀⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠛⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣷⣤⡤⠀⢀⡤⢶⣶⠈⢀⡀⠉⠛⠁⣠⡽⢟⣿⡷⠈⠉⠉⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠉⠉⢹⣿⠏⠉⠡⢾⣿⣿⣿⡇⠈⠀⠀⠛⠛⠛⠡⣴⠋⠁⣀⠀⠘⠛⠓⠾⢿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⣼⣿⣿⡿⠛⠀⠂⠰⠦⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⠀⠀⢠⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣆⠀⠈⠙⠛⠋⠉
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣿⡿⢛⠿⠿⠄⣀⣀
⠀⠈⠏⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠿⣄⠀⠀⠘⠂⠀⠀⠀⠈⠙⠓⠖
⠀⠀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢠⣶⣿⠛⠒⠈⠙⠃⢩⣠⡤⢤⡦⠦⠂⠀⠀⢤⣤⣤
⢧⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⡿⠈⠻⣿⣇⠀⠀⢻⡟⣁⣤⡀⠀⠀⢀⠎
⣦⢀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣶⡄⣿⣿⣇⠀⢠⣾⡿⠋⠁⠀⠀⢀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⣿⣦⠓⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠉⠀⠀⠀⣠⣄⣹⡟⠁⠀⠀⠀⠀⠀⠀⢀⣰⣿⠃
⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢶⣤⣀⣴⣿⣿⣿⣿⣿⣶⣄⡀⣀⡀⠚⠛⠛⠿
⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠿⠋⠉⠉⠀⠀⠹⢿⣿⡿⣍⠁⠀⢀⣴⣶⣤
⣿⣿⣿⣿⣧⣽⣯⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢷⣷⣤⣀⣀⣤⣾⣶⣿⠷⠆⢄⡀⠀⠀⢀⣾⣿⣿⣿
⠿⣿⣿⣿⣿⠿⠿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⡟⢿⣿⣷⣶⣿⡿⠀⠀⠀⠘⠿⢿⣿⣿
⡿⣿⣿⠟⣡⣾⣶⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣟⠁⠀⠀⠻⣿⡟⠁⠀⠀⠀⠀⠀⠀⣠⣿⣧
⣾⣿⣿⡿⣿⡿⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡆⠀⠀⠀⠈⠀⠀⠀⠀⢰⣿⠀⠀⣾⠈⠙
⣿⣿⣯⣾⡿⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠉⠀⣸⡟⠀⠆
⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣧⣀⣂
⣿⣿⣿⡟⢱⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠁⠀⠀⠀⠀⣀⣀⣀⠀⠀⠼⠦⣤⣀⣉
⣿⡿⣫⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⠤⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⠀⠀⠀⠀⠀⠀⣿⣿⠋⠁⠀⠀⠀⠀⠁⠉
⣿⣾⠉⠛⠛⠛⢻⣿⣿⣿⣿⠟⡚⠢⠤⠤⠧⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣤⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⠀⠀⠃⠀⡆
⣿⣟⣀⣀⣀⣀⣸⣿⡿⢡⠟⣶⣿⣆⣀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠐⠁⠀⠀⠀⢠⣿⣿⠛⠀⠐⠂⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⠿⢿⣿⣿⡆⠀⠀⣄⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠉⠀⠀⠀⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣇⡴⠉⢓⢐⣿⣿⣿⣦⣾⣿⣿⣦⣄⢸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢡⡀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢄⣠⡤⠶⠶⠒⠚⠛⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡡⠚⢉⠤⣄⣠⢖⣦⣹⣢⠤⠤⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠐⠫⠀⡰⠃⠒⠚⠁⠀⠉⠉⠀⠘⠓⠊⠠⠥⢤⣤⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⡴⢀⡖⠐⠞⠁⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠶⠶⢦⣌⡁⠒⠒⠌⢷⡄
⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⢀⠔⢋⢄⣠⠖⠀⠄⢒⣀⣈⠙⢶⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢋⣬⡀⠀⠀⠀⠀⠀⠈⠱⢶⡙⣦⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠻⠒⠊⠉⠉⠒⠲⢤⡀⠀⠈⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⣤⣾⣄⣀⣛⠆⠀⢸⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠉⠀⠈⣉⣿⣿⣿⣿⡇⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢸⣿⣿⡿⠟⠁⠀⠀⠀⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠁⠘⠻⣵⣾⣶⣤⣤⡆⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⣄⠀⢰⠇⠀⣸⢇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠈⠂⠙⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⡄⠀⠀⢠⡆⠀⠀⢷⣀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⠳⢄⣀⣀⣀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢧⠀⠀⠀⠓⠒⢠⡄⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⢠⣦⣤⣭⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠒⢶⣾⡿⣼⣿⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⣤⣽⣿⣿⠋⠀⠠⡇⠀⠘⠲⣄⡀⠉⠓⣆⠀⡀⣠⣿⡇⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠠⡌⣿⣿⣿⣿⠀⢤⠀⠑⣄⠀⠀⢸⡇⠀⢠⣿⣿⢡⣿⣿⡧⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢹⣮⣿⣿⣿⠀⢠⡆⠀⢸⡇⠀⢿⣷⡆⣿⣿⣿⣤⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣈⣻⣿⣿⠇⠀⣾⣇⠀⠈⢷⡄⠘⣿⣇⠻⣿⣿⣽⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⢼⣿⣿⡿⠋⢠⡀⣼⣿⠂⠀⢠⡇⠀⢸⡿⢠⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣶⣼⡿⠿⠟⠋⡴⢀⡜⠠⣿⣏⠀⢀⣾⠁⢠⡿⠁⣠⣾⣿⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡟⠋⠀⢠⠄⡗⠰⡃⠀⢸⠀⣿⠹⠀⢸⣧⡀⢸⣇⢠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠋⠀⠀⣇⢸⣆⣻⣢⠟⡤⠿⠶⠿⠧⣤⣾⡿⢃⣤⣿⠸⣧⣤⣤
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢠⢸⣧⠿⠊⠉⠉⠀⠀⠀⠀⠀⠒⠒⠒⠈⠽⢿⣿⣿⠶⠿⠛⠛⠁
⠀⠀⠀⠀⠀⣧⣤⣴⣶⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⣦⡀⢀⣴⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠻⠿⠿⣷⣬⣿⣾⣷⣶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠿⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠶⣤⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣟⣻⣿⣿⣿⣿⠏⣿⣿⣿⣿⡿⣿⣷⣌⠛⢿⣦⣀⣀⡀⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⣾⢩⡾⢋⣿⣉⣸⣿⠇⢰⣿⣿⣿⣿⣧⡈⠙⠻⣆⠀⠙⢿⣍⡙⠻⣶⣤⣼⣯⣶⠶⠺⠲⢤
⠀⠀⠀⠀⢲⣴⣿⣿⣿⣟⣁⣤⣼⣧⣄⣸⣿⡟⠉⣿⣿⣷⠀⠀⣠⠶⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⡛⠓⠦⡀
⠀⣠⠖⠟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⢿⡇⠀⠀⠙⣿⡆⠐⠋⠀⠀⢀⣤⣶⣿⣿⣿⣿⣹⣿⣽⣷⣤⡀
⠀⠀⣴⡾⣻⣿⣿⣿⣿⣿⣿⠿⣿⣦⡈⠀⠘⠃⠀⠠⠀⠈⠃⠀⠀⣠⣴⠿⣫⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣆
⠀⣼⠏⡼⢻⣿⣿⣿⣿⣿⣿⣷⣄⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣴⠟⠛⣿⣿⣿⣿⡇⠸⣿⠈⠻⠀⠉
⠀⠇⠀⠀⣸⡏⣼⡟⢸⣿⡇⠙⢿⣇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⣿⡇⠀⠸
⠀⠀⠀⠀⣿⠁⠹⠀⠀⢿⡇⠀⠀⠉⠣⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠿⠁
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠨⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣇⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⢀⣷⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠧⠿⠛⠾⠿⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣠⣀⠀⠐⠋⣀⣀⣀⣀⣀⣠⣤⣤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣾⣿⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⠣⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⢄⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⡀⠀⣿⠽⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢂⡀⠀⠀⠀⡜⠀⠀⠠⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣭⣿⠉⣷⣦⣅⣀⣀⣀⢀⠀⢘⡇⠀⠀⠈⣀⠀⠠⣀⣑⣤⡶⠂
⠀⠀⠀⠀⣀⣒⡒⢶⣶⣦⣤⣤⣦⣄⣸⣿⣿⣿⣿⡶⠶⠿⠤⠿⠤⠬⠥⠼⠷⣶⣾⣟⣛⠉⣛⠽⣁⡀
⠀⠠⠄⠀⢻⡛⠛⣿⣯⣽⠟⢛⣻⣏⣙⠛⠃⣿⣿⣿⣍⣻⣶⠿⠀⣲⣿⢗⠓⠿⠯⠀⠿⢿⡋⠤⠟⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠈⠀⠰⠴⠂⠨⠤⢈⣭⠉⠷⠾⢤⣀⠈⠑⡊⠁⢤⠘⠻⠿⠓⠂⠈⠁⠐⠂⠀⠠⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠄⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠁

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⡀⡠⠤⠄
⠀⠦⣄⣤⡀
⠀⠀⠀⠉⠛⠶⢴⣤⣤⣤⠤⡶⣶⡖⠶⠶⠦⣤⣤⣄⣀⠀⠀⠀⢀⣀⡤⠤⢄
⠀⠀⠀⠀⠀⣀⠒⠛⠿⠿⠾⢷⣏⡀⢀⣀⣀⠀⠀⠀⠉⠙⡞⢹⠀⠈⢦⡀⢰⡀⢠⡖⢦⣄⠀⠀⣀⣀
⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣀⣈⣉⠲⢤⣬⣉⣥⠀⣠⣶⡁⣸⣀⣠⣼⢷⡾⢷⡾⠻⡖⠉⠛⠛⠻⠿⣿⣷⣶
⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡠⠭⠹⠟⣩⠟⡿⠟⡛⠉⢹⡟⠉⣿⠀⢸⠇⢸⠃⠘⡇⠀⠀⠀⠀⠀⠀⠈⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⠀⡤⠒⠉⡠⢊⠞⡹⠁⠀⠨⠇⢀⡇⠀⢸⡆⢸⠆⠀⠁
⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢠⠃⣠⠊⡰⠋⡴⠁⠀⠀⠀⠀⠘⠃⠀⠀⠁
⠀⠑⠠⠀⡀⣀⡀⠀⠀⠀⠀⠀⠸⡸⣅⠘⣅⠀⠣⣄⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠺⢖⡬⢗⣒⠬⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠉⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠔⠈⠁⠢⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡐⠂⠒⠓⠀⠬⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠒⠵⠤⠄⠀⣀⡐⠁⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠒⢁⠶⢀⢀⡠⠁⠈⢀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠄⠂⠀⠐⠐⠀⢦⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢅⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡐⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠄⠊⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠋⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠁⠀⠀⡄⠂⠁
⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⡠⠐⠉⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⡔⠀⢷⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡘⠀⠀⠈⠕⠄⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⠂⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌
⠰⠁⠀⠀⠀⠈⢆⠂⡀⠀⡇⠀⠰⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠜⠀⢀⠌
⡀⠀⡀⠀⠀⢔⢦⠓⠁⠀⢁⠀⠀⠂⢀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠊⠤⡤⠀⠀⠀⠀⡃⠀⠀⠀⠀⡠⠂⠀⠠⠂
⠗⠈⠱⠀⠀⠀⠢⡑⢄⠀⠀⠄⠀⢈⡎⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⡠⢀⠔⠁⠀⠀⠉⠐⠂⠀⠀⡠⠊⠀⢀⠔⠁
⠀⠀⠀⠱⡀⠀⠀⠈⠐⠬⠀⠈⠗⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⠀⠥⠒⠁⠀⠀⠀⠌⠀⠀⢀⠠⠊⠀⢀⠀⠁
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠠⠄⠊⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⡀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠖⠀⠀⡤⢀⡄⢠⠀⠀⠴⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢠⠞⠄⠀⣸⠀⡀⠀⠀⠄⠒⣀⡄⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢇⣾⣑⡀⣊⣴⣦⣾⣦⣶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠋⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⡤⠀⠀⢑⡠⠀⣈⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢁⡔⠀⠀⣁⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣄⢙⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢠⣤⡴⢶⣶⣦⣤⣤⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠈⠁⠈⠿⡄⠯⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠙⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⢛⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⡟⢿⣧
⠀⠀⠀⠀⠀⠀⠴⠂⠉⠁⠀⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⣦⢀⠀⢢⡘⢨⣚⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⠿⠟⠁⠈⡟
⠀⠀⠀⠀⠀⢐⠻⠆⠀⠀⠩⠀⠈⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠿⠛⣦⡄⢿⣿⣆⣤⣶⣶⣶⣦⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠲⣤⡌⠁⠐⠹⣏⣻⣿⣿⣿⣿⣿⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠈⢹⠉⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⣤⣁⣰⢭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠙⠁⠀⠀⠙⠛⠿⠖⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣻⣉⣏⣻⣋⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⢶
⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠓⠒⠀⠀⠀⠀⠐⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠂
⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⣶⡾⠿⠿⠷⠆⠀⠀⢀⡀⢀⣠⣤⣴⣶⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⣿⣥⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡶⣦⣷⢛⣡⣶⣶⣶⡯⣮⣅⡀⣤⣢⣝⠿⢿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⣀⣹⣿⣧⠀⠀⠀⣀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣿⣛⠻⣿⣿⣼⣿⣿⣶⣿⣿⠛⠛⣻⣿⣿⡿⣽⡿⣿⣿⣿⣿⣷⣤⠀⡠⠄⣾⣀
⠀⠀⠀⠀⢺⣿⡋⡹⣿⣄⢀⣼⣧⡜⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⢃⣄⣼⣿⣿⣏⣴⣿⣿⣜⣿⣿⣿⣟⣫⣦⠉⣼⠿⢹⣷⡆
⠀⠀⠀⠀⠈⢿⣧⢳⡙⠿⣾⣿⣿⣿⣿⡿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣊⣿⣝⣿⣿⣿⣧⣿⠆⣼⡿⠁
⠀⠀⠀⠀⠀⣈⣻⣷⣝⣀⣿⣿⣿⡾⣿⣧⣿⣿⣾⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⠐⣨⢿⣿⣿⣟⣠⣼⢟⡁
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⢿⣿⣿⣿⡰⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣼⣿⣧⢹⣏⢿⣯⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣯⣯⣽⠋⣙⡻⢿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⡿⣟⣿⣌⣽⣿⣭⣿⣥⣿⣭⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣿⣶⣿⡿⢶⣶⣝⠷⣍⠙⢿⣿⣷⡈⢋⣙⣿⣟⣩⣋⣰⣿⣿⣿⡿⢋⡀⣻⣧⣘⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⣿⣿⣿⡽⢿⣿⡿⠟⠀⠀⠘⢿⣿⣷⣿⣴⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣯⣿⣿⡿⠃⠀⠀⠉⠁⠀⠀⠀⠀⠈⠙⠻⠿⡟⢿⠇
⠀⠀⠀⢀⣿⡿⣻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠛⠃⠈⠙⠛⠻⠿⠿⠟⠛⠋⠉⠁⠛⠛⠛⠋⠁
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣤⣶⡶⠶⣤⡀⠀⠀⢒⣛⣛⣛⡻⣿⣶⣄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣻⣿⣷⣶⣶⠶⠀⠀⠀⠀⠈⠙⠻⣿⣿⣳⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢿⢟⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⣾⣿⣿⡏⣿⣿⣿⠿⠋⠁⠀⢰⣤⣀⣄⣤⣤⣀⣀⣀⡶⠀⠀⠀⠉⠛⠿⢷⢺⣿⣿⣿⡄⠀⠀⠀⠀⣠⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠰⣿⣿⣿⣻⣭⣤⣄⠀⣠⣴⣿⣿⣽⣿⣿⣧⣿⣿⣿⣿⣿⣿⣦⣄⢀⣠⣤⣶⣮⣿⣿⡿⠁⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡈⢛⣿⣿⣿⣿⠯⣾⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⠈⢀⣤⣾⡿⠋
⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣀⡈⠻⣷⣬⣝⡛⠿⡟⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⠿⣛⣻⣿⡖⠿⠟⠉⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡾⠛⠛⠛⠛⠋⠙⠻⠿⠿⠷⣶⣮⣹⣿⣿⡧⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣛⣫⣭⣶⣿⢿⡿⠿⠟⠛⠻⠿⠟⠛⠛⣴⡄
⠀⠀⠛⠀⠀⠀⠀⠀⣀⣠⡄⣤⣤⣤⣤⣦⣬⣍⣛⣛⡿⠿⢿⣿⣟⣻⡿⣿⣹⣿⣿⠸⣿⣿⣿⣽⡿⣛⣵⣾⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⣰⠿⠿⠛⠉⠈⠉⠉⠀⠈⣉⣉⡙⣛⣻⡿⠋⠰⠋⣙⢿⣿⣿⣿⣿⣻⣷⠾⣻⣹⣿⣻⣻⣟⡛⣻⣟⠉⢉⡀⠀⠀⠉⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⣠⣶⣶⡆⢿⣿⠛⠿⢿⣿⡧⠿⠆⠙⠃⠩⠿⠟⠹⠿⠇⠘⠉⠻⠿⣿⡿⠿⠏⠛⣯⣾⣿⣿⣿⢶⣄⠀⠀⠈⠉⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡻⢿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢟⣯⣿⣿⡄⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠤⠈⠙⠄⠀⠈⢠⡄⠀⡀⢰⠺⣍⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀⣱⠀⠀⠑⢀⣀⠀⣀⠀⠀⣇⡙⠢⠘⡻⣦⣈⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣦⣤⣛⣁⡀⠉⢻⡞⢛⣛⣛⣛⣛⣓⣾⣛⣭⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠳⠶⠶⣒⣚⣻⣿⣿⣯⣭⣯⣥⣴⡖⢛⡻⠿⠥⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠫⣄⡃⣤⣀⠀⠀⣏⣀⠝⠲⢬⡀⡀⢑⠛⠀⣝⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡢⢍⣽⠟
⠀⠀⠀⠀⠀⠀⡀⠊⠑⠀⠐⣈⠝⠀⠀⠍⣶⣄⠠⠶⠍⠳⣀⠷⡒⢌⢦⠀⠀⠀⠀⠀⢰⠮⣈⣐⣤⡶⠖⠶⢤⣤⣀⣩⣤⡟⢁⠀⡀⠀⠠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠟⡟⠀⠛⠑⠀⠙⢆⡈⢛⡂⣄⠐⠀⣠⠞⠉⢛⣍⠠⢀⡉⠑⠀⠈⠀⣈⣛⣅⢐⠣⣚⠐⠓⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⡐⠣⠄⠀⠀⠀⠀⠈⠆⠈⡪⠿⠀⣰⡇⠀⢀⣯⡀⠑⠀⢀⢀⣠⡤⠂⢼⣿⣻⣧⡰⢇⡸⠿⡆
⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠶⠆⠀⠀⠀⣠⠈⡃⠹⠿⠷⣭⣻⣏⣭⣭⣽⣷⣶⢉⣻⣉⣒⣒⣚⢿⣷⣿⣷⣭⡙⣹⠇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠘⠠⣀⣤⡤⠖⠂⠑⠀⡛⠀⠀⠑⡤⠙⠾⠉⠓⢲⣤⣬⣧⣤⣩⣭⣭⡝⠷⢒⡛⠷⠶⠶⡿
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⢀⡉⠀⠀⠀⠀⠘⡖⠁⠀⠀⢀⠄⣠⡤⠆⢀⣤⡄⠠⠞⢀⣶⡃⣆⢠⡈⡄
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠈⠀⠀⡅⠀⠀⠀⠉⠀⠤⠄⠀⠀⠀⠀⠀⠈⠐⠁⣀⣈⠐⠨⣟⠽⢡⣾⣤⠟⠛⢋⠀⠚⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠔⠀⢠⠀⠀⠀⢠⢈⠣⢜⡂⠀⠀⢀⣹⠟⣡⠶⢛⣛⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣰⠆⠀⠀⠨⠀⠄⠈⠀⠀⠈⠘⠀⠀⠀⠀⠂⠀⢀⢤⡐⣼⠟⠁⠀⠠⠞⠋⠏⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠔⡄⠀⠀⠀⠀⡀⠄⠀⠀⢰⠀⠀⠀⠘⠷⠁⣯⡿⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠂⠀⠀⠈⠂⠀⠀⢸⡀⠀⠀⠠⡀⠒⠹⠷⠆⠐⠄⠀⠀⠀⠀⠠⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠘⡄⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⡆⢶⡀⠃⣰⡀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠄⠄⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠉⠉⠻⢿⣷⣶⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⣀⡀⠠⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡭⡳⡄⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⢀⣴⣾⠩⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢆⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢘⠀⢀⡀⠄⠐⣫⣵⡿⣟⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠈⠣⠀⠘⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣠⣴⣾⠟⡙⠂⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠛⠋⠉⠉⠉⣠⡖⠀⠂⠀⢀⠀⠈⠙⠛⠶⠶⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣀⣀⣠⣤⣶⡿⠿⠀⠈⠳⣬⣂⠀⠀⠠⠐⠤⠤⠀⠀⣀⣤⣤⣤⣤⣤⠤⠶⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⠋⠉⠀⠀⠀⠀⢸⠲⠤⠤⣤⣙⡻⠶⣤⣀⠀⠀⠈⠙⠚⠛⠛⠛⠓⠊⠉⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⣀⠀⠀⠀⠈⠉⠛⠛⠃⠀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⢠⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⠠⠀⠀⠐⠁⠄⠅⠐⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⣀⠄⣴⠣⠐⣀⠤⢀⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡅⠊⠀⠀⠀⠀⠀⠀⠀⠐⡲⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠀⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠣⠀⠔⠢⠤⠀⢀⠀⠀⠀⢐⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⡀⠀⠀⠀⠠⠣⠈⠢⢀⠠⠐⠁⠀⠀⠀⠤⡀⠐⠂⠀⠰⢿⠡⠂⠀⡄⠀⠠⠒⡂⠑⠃⠀⡐⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠊⢢⠢⡀⠀⠀⠀⠁⢄⢡⡀⠀⢀⡾⢞⡱⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢅
⠀⠀⠀⠀⠀⠀⠁⠼⢀⠀⠀⠀⠈⡠⠀⠄⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠂⠀⡈⣸⠓⢂⠉⠈⠂⡠⠤⠀⠀⡃⢀⢃⡟⢱⡒⡱⢌⠀⠐⠌⡉⠲
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠁⢄⠰⣀⡐⠃⠀⡠⢠⡟⡆⠁⠀⠀⡢⢈⡴⡁⠪⠅⠀⠠⠖⡐⠋⠀⠀⢔⣡⡮⠀⡑⠀⠂⠁⠤⠄⠠⡃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣀⡣⠄⠠⠀⠕⠐⣬⡂⠊⣤⠋⢈⡀⠈⡐⠪⡴⠛⣁⢤⣄⡈⢤⢶⠮⠤⠤⢤⣆⣁⣼⣠⢨⣥⠀⣉⠈⠠⠔⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢘⢯⡱⠞⠫⠒⠀⠒⢁⣭⡈⢀⡙⢡⠆⠀⠈⠓⢓⠈⡀⠀⠀⠠⠌⠀⢀⡴⠀⠈⣳⣰⢀⣾⡷⢎⡡⠁⡰⠀⢆
⠀⠀⠀⠀⢆⠆⠀⠀⠀⠀⠑⠉⠑⢥⣀⠄⢠⡞⠁⡭⠿⣉⠉⠀⠀⣠⡤⠗⠚⠺⣤⡤⠋⠷⠔⠉⢳⠀⠁⠈⠡⠈⡋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡄⠀⣊⠆⠀⠆⠐⠂⠀⠠⡙⠒⠋⠉⢁⣤⠴⠃⠂⠐⠓⢀⡤⠐⡒⠖⠠⠤⠒⠒⠀⠂⠉⣀⠀⠄⠁⠉⠠⡩⠂⠀⠓⠀⠀⠀⠀⠀⠀⣀
⡠⠀⠀⠀⣂⡄⢠⠀⢀⠀⠀⠀⠀⠀⠀⢸⠊⠁⠘⠚⠑⠶⠟⠉⠙⣻⡪⢔⣒⡭⠁⢠⠐⢖⠂⢀⠅⠂⠂⠤⠥⠬⠂⠐⠂
⠀⠁⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⣞⠠⠂⠀⠀⢁⡄⢰⢧⢭⡁⠉⠀⠀⢀⠀⢔⠀⡆⢀⢀⠀⠀⠀⠀⠰⠀⠘⠜
⠀⠀⠀⠁⡆⢠⣄⡨⡗⠀⠤⡬⡤⠔⠻⠄⠀⠀⠀⣄⡵⢀⡿⠞⠛⠀⠒⠒⠂⠁⡀⠀⢇⡱⠈⠈⠀⠤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⡠⢄⡁⠀⠀⠀⢂⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠘⣆⠈⠀⠀⠁⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⡀⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠁⠀⢌⣡⠟⡹⢛⢒⠀⡀⠀⠐⠀⠠⠤⠊⠬⠄⢒⠆
⠀⠀⠀⠄⠰⠾⡏⠁⠢⠁⠒⠂⠀⠂⠠⠄⠂⠠⠄⠂⠢⢣
⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠘⠄⠀⠐⠀⠀⠁⣀⣘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣧⣤⢿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣿⣿⣿⣿⣿⣿⣿⡞⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠛⠛⠛⠿⠛⠛⠿⡿⠿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠈⠁⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣀⣂⣠⣄⣤⣤⣶⣤⣤⣄⣀⣐⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⡟⠉⢙⡋⠉⠀⠈⣛⢉⡋⠀⢉⡉⣻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣛⣿⣿⣿⣉⣙⣿⣿⣿⣿⣿⣟⣉⣙⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢐⣿⣿⣿⠟⠉⠀⠀⠀⠉⠹⠿⣿⣿⣿⣯⣄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣴⣾⣿⣿⣤⣤⣤⣤⣄⢠⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣤⣤⣈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⡧⢹⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⢸⡟⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢺⣿⣿⣿⣿⡇⠀⣻⣿⣿⣿⣿⣿⣿⣿⣄⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⣿⣿⣿⣿⡏⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠰⠿⣿⣿⠛⠃⠀⠘⠛⠛⣿⣿⣿⠿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣿⠆⠀⠀⠀⠴⣾⣿⣿⣿⡆⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⣿⠆⠀⠀⠀⠀⠀⣤⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⣿⡆⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⣿⣿⣧⠀⠀⠀⢀⣀⣨⣿⣿⣿⣭⣄⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⡀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣻⣿⣿⣿⠟⠓⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⡷⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⣿⡗⠂⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣉⣿⣋⡉⠀⠀⠀⠈⠉⠹⣿⣿⠟⣋⡉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⣯⡄⠀⠀⠀⠀⠀⠠⣤⣤⣤⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⢠⣬⣭⡅⠀⠀⠀⠀⠀⠀⠤⢶⣶⣶⣶⠤⠄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣤⣭⠅⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡶⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣶⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣬⣭⣤⣤⣦⣤⣤⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣾⣿⣿⣿⣿⣿⠯⣭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣿⣿⣿⣿⣷⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⣙⠻⠛⠉⠁⠀⠀⠈⠉⠛⠛⢛⣛⣛⣻⣭⣭⣭⣍⡉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠚⣋⠿⠖⠂⠀⠀⠀⠀⠀⠀⣀⣬⣭⣽⣿⣿⡶⠦⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣭⣽⣿⣿⣷⠦⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠛⠷⠂⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣾⠇⣶⣶⣿⣶⣷⣤⣄⠀⠀⠀⠀⣀⣀⣀⣀⣒⣒⣲⣶⣦⣤⣴⣤⣤⣤⡄⢸⣷
⠀⠀⠀⠀⠀⠀⢀⠏⢸⣿⣿⣿⣿⣿⣿⣧⠠⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠙⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠈⣾⣿⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠰⢿⢀⣿⣿⣿⣿⣿⣿⡟⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⣿⣿⣧⢀⣀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢻⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣤⣀⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣾⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠒⠀⠉⣙⣻⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣴⣧⣾⣷⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡹⢿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⡩⠿⣿⣿⣄⠀⠀⣠⣾⣿⣟⠻⠿⠟⢛⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠑⣞⣿⣿⣦⣴⣿⡿⠯⠝⠋⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣟⣻⣷⡶⣶⣶⣾⣿⣟⢿⡷⡶⣖⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⡿⣿⢿⣟⡿⣿⣿⢒⣿⣿⣟⣽⣿⣿⣿⣿⣯⣿⡷⣿⡉⠛⠟⠚⠓⠁⠙⠈⠎⢿⠛⡏⢻⢶⣄
⠀⠀⠀⣀⣴⣿⢿⣿⡿⢫⢞⠞⡡⠋⠊⠐⠋⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠙⡷
⠀⢀⣼⣿⣿⢫⡾⠋⠐⠁⠈⠈⠀⠀⠀⢀⣾⣿⣿⣿⡿⢱⠏⢻⣿⠻⠟⢻⣿⣿⣿⣿⣆⡀
⢰⣿⡟⣹⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⠙⠀⠀⠀⣸⣿⠀⠀⠠⠘⢸⠘⢿⣿⣿⣦
⠨⡟⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠀⠘⣿⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡝⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠁⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢻⡿⣄⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⠈⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣚⣩⣵⣶⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠋⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣊⣥⣶⡿⠿⡿⠟⣛⣉⣭⣤⣴⣶⣶⣶⠾⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠉⠀⣰⣿⣿⣿⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠴⠛⠉⠀⠀⠴⠞⠛⠉⠁⠈
⠀⠀⠀⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣭⣕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⠉⠛⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⡢⠀⢎⠘⠿⢷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⢐⠂⠀⠀⠀⠀⠨⠳⡄⡀⡼⠁⠙⣀⣠⠀⠁
⠀⠀⣀⠠⢀⡲⠖⠚⠋⠉⣈⣡⠤⠖⠒⣒⣈⠭⠍⣉⠉⠁⠇⠘⡧⣈⣋⣒⣓⡲⠦⢬⣥⣌⢉⠉⠲⠤⢔⣂⠤⢤⣀⡀
⠐⠉⢀⡔⠁⠀⠀⠠⠂⣹⠋⠀⢰⡞⠉⠁⠀⢎⠀⠀⠐⠈⠀⣁⡒⠤⣀⡀⢭⠁⣡⣄⠈⠻⣧⣀⡄⠀⠀⠈⣻⣦⡀⠉⠓⢦⣀
⠀⠀⠸⡤⠀⠀⠀⠀⠀⣻⡀⠀⠐⠷⣄⡀⠀⠀⠁⠲⢤⡄⠠⠈⠉⠋⠒⠀⠚⣠⣦⣤⠄⠀⡠⡤⡄⠀⢀⡤⡻⡿⡇⠀⠀⠀⠈
⠀⠀⠀⠈⠲⠤⡀⡀⠀⠀⠙⢮⡢⣀⣀⠈⠑⢓⠰⠶⠶⣶⣶⠻⠿⣿⠾⠿⠥⠒⢊⡁⠤⡒⣠⠞⠀⡀⡠⣐⡼⠊
⠀⠀⠐⠄⡀⠀⠀⠀⠘⠒⠢⠦⢤⣀⣀⡀⠀⠚⠚⠯⠼⢿⣶⠶⠧⣤⢤⢤⠬⠝⠓⠒⢈⣁⣤⠤⠦⠖⠋⠁⢀⡠
⠀⠀⠀⠀⠀⠉⠁⠒⠂⠀⠀⠤⠀⠀⠀⠉⠉⣉⠙⠒⠛⠛⠛⠉⠉⠒⠚⠒⠒⠊⢉⠁⡉⠠⠄⠀⠐⢒⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠠⠤⠀⠀⠀⠀⠀⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠄⠃⠉⠈⠀⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⣾⠀⢴⡏⠉⢻⡇⠀⡟⠛⠛⠋⢀⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⢀⡼⠃⠀⡟⠛⢻⣦⠈⠁⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⠀⠀⠀⠀⠀⠀⣿⠀⢀⣔⡋⠀⣠⠠⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⡄⢀⠀⠄⠀⠐⠛⠂⠚⠉⠛⠛⠃⠀⠛⠓⠚⠁⠀⠀⡠⠀⡄⣨⠂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠊⠁⠈⠁⠈⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢀⡀⡉⢉⠁⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⢀⢇⠘⢪⠈⢢⠈⠘⡌⢸⠅⣫⡁⠪⠁⢧⠨⢜⠸⠀⠀⠀⠠⡟⠀⠀⠀⣤
⠀⠀⠀⠀⢷⣀⠀⠀⠘⣇⠤⢠⠤⠤⠬⠤⠌⠀⠈⠁⠁⠈⠁⠁⠉⠉⠈⠀⠤⠨⠀⠀⢀⡀⣨⡇⠀⠀⢀⡯
⠀⠀⠀⠃⢠⡗⠀⠀⠀⠀⠀⠴⠀⠀⡀⡀⠀⠀⠀⠀⡀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠰⠇⠘
⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢀⢀⠀⠀⠀⣀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠒⠀⢀⣠
⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠚⠋⠉⢉⣉⣉⡉⠭⠭⠭⠽⠿⠶⠶⠶⠤⠤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⢀⡤⠚⠉⣀⡤⠴⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠭⣑⡒⠢⠤⢤⣀⣀
⠀⠀⠀⣠⠋⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⡀⠀⠉⠁
⠀⠀⢠⠇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠒⠒⠒⠒⠒⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⡉⠉⠉⠉⠑⠒⠲⢤⣀
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡴⢊⠕⣋⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢳⠉⠢⡀
⠀⠀⠀⠀⠀⠈⠻⣏⠉⠉⠉⠁⠀⠸⡸⢧⠄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣸⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠉⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⠇⠀⠀⠀⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡭⠿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡀⠀⠀⣠⠎⠀⠀⠀⢠⠃⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣊⡥⠊⠉⠉⢉⣉⣶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠭⠤⢤⡭⠗⢲⣅⠀⠀⠀⣠⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡜⢰⢡⠤⠖⠚⣉⣁⣤⣽⡮⠽⠛⠓⢲⣦⣤⡤⠤⠤⢶⡒⣫⠝⠉⠈⣓⣒⣶⣶⣛⠯⠟⠋⢳⣄⠔⠁⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⡇⣶⠏⡳⠋⠉⢁⣞⣉⡤⠤⠒⢚⣟⢋⣉⣤⠤⠖⠒⠉⠉⠀⠀⠀⠈⠓⠤⠤⠤⠼⢒⣪⠟⠋⣹⡄⠀⠀⣠⠞⢸
⠀⠀⠀⠀⠀⠀⠀⠻⣽⠤⣇⠀⠀⡸⠀⡏⠀⠀⠀⡞⢈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣋⣉⣉⣩⠶⠒⣩⠞⣳⠒⠉⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⢆⠀⢧⡴⢧⠀⠀⢸⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢉⡷⠚⢹⣄⣀⡤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣓⢬⣦⠀⠳⡀⠘⡍⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠉⠁⠀⠀⣸⣄⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⢄⡈⣳⠾⣆⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠛⢧⣼⣋⣓⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡅⢠⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡋⣡⠎⣠⠃⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⢁⡴⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⣤⠴⠋

⣿⣧⡄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠈⠀⠐⠀⣀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠐⠝
⣤⣤⣤⣴⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠈⠉⠐⠂⠀⠌⠳⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⣀⣄
⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⢤⣄⡀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠚⠀⠠⣌⠹⣷⣧⡞⠁⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣄⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⡘⢋⣴⣬⣦⢹⡆⣹⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠈⢻⣿⣦⠄⠂⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠛⣿⢏⡾⠃⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣄⢢⡀⣹⣦⣀⣠⣿⣿⣾⣛⣁⣠⡴⠏
⠠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠉⠛⠿⠛⠛⠉
⠀⢇⠑⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀⠄
⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⣾⡧⢀⡀
⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠈⠀⠀⠀⠀⠸⠏⠀⢸⡿⢦⣄
⠀⠘⣦⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣧⠀⠀⠀⢰⡀⠀⠈⡄⢈⣄⡽⠶
⠀⠀⠘⢟⣀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢀⣀⣿⡗⠠⠖⠛⠋
⠀⠈⠉⠉⢿⣧⣤⣴⠞⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⠿⠃
⠀⠀⠀⠀⠈⠙⠛⣿⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡴⠀⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁⠀⠁⢀⡀
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣦⠤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠦⠴⠾⢿⣷⣦⡤⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡎⠀⣠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣍⡉⠝⠉⠁
⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⣿⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡿⠌⠑⠄⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢸⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⢀⡼⢃⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢀⠀⠀⣠⣾⠿⠋⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢠⡟⣀⡾⠋⠁⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⡟⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⡟⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⡜⡿⠁⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⡼⠁⠈⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⡼⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⡼⠁⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⡼⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⡾⠁⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠋⣠⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⠀⠀⠀⠉⠉⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⢀⣴⢟⣁⣤⠀⠀⠀⠀⢀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠸⣦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠂⠀⢀⣴⣿⣿⣿⣭⣤⠴⠶⠖⢻⣿⣯⣤⣤⣤⣀⣀⣀⣀⣤⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁
⠀⠀⠲⠿⠻⠭⠉⣡⣀⣀⣀⣀⣴⣁⣀⣀⣠⠤⠤⢤⠤⠤⡌⣿⡓⠒⠂⠀⠤⠠⠤⠀⢀⣠⣤⣤⣤⣤⣤⣶⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣛⡛⠻⠿⣯⣭⣄⣀⡈⠃⠸⠧⣬⣍⣀⣀⣀⣀⣀⣀⣀⣀⣐⠆⠀⢉⡿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢰⡿⠛⠛⠛⠻⢿⣿⣟⠶⠶⠶⠶⣬⣭⣭⣭⣤⣤⣤⡶⠶⠚⠋⣁⣠⡾⠀⠠⠔⢒⣋⣥⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⠁⠀⠀⠀⠙⢿⣷⣤⡀⠀⠚⠁⠀⠉⢻⣿⣿⣿⣿⠿⠛⠛⠉⣀⣤⡶⠾⠛⠉⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠀⠀⠀⠀⠉⠻⢿⣟⣓⠛⠀⢀⣠⡟⠈⠙⠛⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠾⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⠤⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠈⢷⡀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⣦⣈⣷⣶⡾⣃⢶
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠙⠷⠿⣥⡖⠀⠀⣠⠴⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢛⠩⠍⠩⠥⠀⠩⠞⡇
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠱⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠐⠀⠀⠀⣉⢹
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡆⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠀⠀⠄⠀⠀⢰⢸
⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢬⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⠴⠤⢾⡶⢦⣿⣷⡷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⢀⡀⠀⠸⣻
⠀⠀⠀⠁⢠⡇⡘⠀⠀⢀⣨⣔⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⠶⠋⠁⠀⠀⠀⠈⠀⠀⠀⠈⠳⠾⠢⡀⠀⠀⠀⠀⠀⠀⢀⡎⠉⢀⢀⠉⠳⢾⣿⠀⠀⠀⠀⡀
⠀⠀⡶⠀⢸⣿⣿⠞⠉⣅⣈⠙⡁⠀⠀⠀⠀⢀⠤⢤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣼⠇⠀⠀⠈⠀⠤⢼⡇⠀⠀⠀⠀⡇
⠀⠀⣿⣤⠘⣿⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⡄⠀⠀⢿⡀⠠⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡁
⠀⠀⢣⠀⢰⣿⠁⠀⠀⠀⠤⢼⡇⠀⢀⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡄⢸⣇⢀⡀⠀⠐⢀⡟⠀⠀⠀⠀⢁⡇
⡀⠀⠈⢧⠀⣿⣅⠠⠄⠀⢀⢸⡆⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⣿⠀⠀⠀⣛⡿⠁⠀⠀⠀⠘⢻
⡇⠀⠀⠘⣧⣾⣿⣄⡂⠀⡀⢠⣿⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣯⣿⣷⣭⣶⣿⠁⠀⠀⠀⠀⢠⠏
⡇⠀⠀⠀⠘⣆⠹⣿⣿⣷⣦⣿⡿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣟⠛⠛⠛⣫⠃⠀⣠⠀⣰⠟
⠸⣤⠄⠀⠀⠈⢧⡙⢿⣭⣭⣽⣾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣭⣿⡿⢁⡀⠀⣭⡟⠋
⠀⠹⡄⠀⠀⠀⠀⠳⣄⠹⢿⣟⠿⢿⣿⡦⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠟⢉⠔⠉⢠⡴⠏
⠀⠀⠹⣦⣈⣄⣀⠀⠈⠻⢽⣯⣅⣲⣤⣿⣦⢄⡦⣤⣵⢀⠀⠀⠀⠀⣄⠀⠆⣠⡀⠠⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⠄⣹⣯⡥⢤⣴⣁⣠⠶⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠻⣅⠀⢀⡈⢀⣈⣉⣽⡛⠛⡛⢷⣼⣿⣷⣯⣶⠗⠢⡠⣼⡾⢁⢸⣿⣿⠉⠙⣆⢀⠐⡠⣴⣤⣶⡯⢛⣫⣿⣏⣠⣬⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⡶⣧⡾⠁⡐⠂⠠⢹⡋⠀⠀⠈⢢⣷⣻⣼⠟⣩⣾⡟⣛⠋⠉⠁⠀⢠⡄⠀⠀⠀⠀⠀⣀⣼⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣁⢌⡁⠁⠀⢒⢢⣌⣳⡹⣿⣿⣅⠣⠅⠉⠄⠛⠅⠀⠀⢀⣴⣿⣿⢏⣾⣿⠟⠛⠛⠛⠋⠙⠻⡋⠛⠻⠶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢀⡐⠀⣠⡼⠛⠛⠛⠛⠛⠛⠿⠿⠿⢿⡿⣟⣿⣿⣾⣿⣾⣿⣿⣾⣧⣶⣿⣾⡿⢣⣾⣿⣷⣿⡶⠦⢦⣤⣶⣷⣟⠐⠠
⠀⠀⠀⠀⠀⠀⢾⣷⣿⠟⠀⠀⣀⣼⣯⣄⣄⢀⡀⣴⣦⣧⣿⣷⣮⣟⣻⣛⣿⣻⣛⣛⣛⣛⣭⣶⣿⣻⣇⣀⣀⣀⣠⣦⣼⡏⠢⡙⠷⣄⠀⠂
⠀⠀⠀⠀⠠⣃⣠⣿⠃⠀⠀⣰⠋⢸⡟⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠉⠉⠉⠙⢷⡆⠀⡀⠘⣷⠶⠆
⠀⠀⠀⠀⣵⣿⠟⠁⠀⠀⢠⣟⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣴⠀⠹⡆⠠
⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠈⣿⡯⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⡇⠀⠀⠹⣦⣥
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⢻⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⡇⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢻⣿⣅⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠋⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠛⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣭⡛⢶⣄⣠⠾⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⠶⠋⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠀⠀⡾⠃⣠⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⢷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⡿⠛⠋⠀⣠⣏⣀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣱⣾⣯⣭⣭⣤⣴⠟⠉⠀⣠⣤⣶⣿⣯⣭⣭⣭⣭⣛⣶⣿⣿⣌⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⢿⣿⣿⡿⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⠁⢠⡞⢹⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠰⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣿⣿⣿⣿⣿⡷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠙⢲⡄⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠙⣇⠀⢷⡀⠀⠀⠀⠀⢀⣤⠞⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⠈⢓⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠸⡆⠀⣷⠀⣀⣤⣾⡛⢁⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢿⣿⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⡧⠀⢸⠀⠈⠉⢉⣽⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⡷⠀⢸⠂⢀⣴⣾⣿⠟⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡙⢦⣀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⡇⠀⠀⠀⣰⠃⠀⣼⠖⣻⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣌⠳⢤⣠⠞⠋⠉⡉⠉⠙⠛⠿⠷⢤⠴⠞⠁⢀⣼⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣉⠓⠾⣏⣻⣿⣷⣶⣤⣤⣠⣤⣤⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣿⣿⣷⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠲⠛⠁⣠⡼⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠦⠶⠛⠉⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⡇⠀⠀⡈⠑⠦⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠤⠤⠤⠴⠖⠒⠒⠲⡟⣉⡁⠀⠀⠹⠤⢀⡍⠳⣔⢦⣀⣘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠋⠉⠉⠛⠛⠛⢉⠍⠁⠀⠀⠀⠙⠋⢀⠀⢸⡆⠈⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠱⢤⣤⣤⡴⠋⢳⣞⠱⠏⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⣆⠀⣀⠰⡄⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⠴⢾⡏⠁⠀⠹⡄
⠀⢀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡇⠀⠀⠀⣿
⠀⠘⣆⠀⠈⠉⠓⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡴⠃⠀⣸⠇⠳⡄⠉⢹
⠀⠀⠈⠳⣌⠢⢄⠀⠀⠀⠉⠛⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⡾⠁⠀⢸⠁⠀⡼⠃⠀⠀⢹⡆⠈⣇⠀⠀⣀⡠
⠀⠀⠀⠀⠈⠙⠦⣍⡲⢄⡀⠀⠀⠀⠈⠉⠓⠶⢤⣀⡉⠁⠀⠀⠀⡿⢀⡾⠁⠀⠀⠀⢸⠇⠀⣟⡤⠊⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣝⡲⢤⣄⣀⠀⠀⠀⠈⠉⠓⠦⢤⣸⢃⡞⠀⠀⠀⣀⣀⣿⡤⠖⠁⢀⡴⠶⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢭⣝⡒⠶⢤⣀⣀⠀⠀⠁⠘⣀⡴⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣭⣉⣩⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠒⠲⠶⠤⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠤⢤⣤⣤⣤⣤⣤⠤⠶⠖⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⣤⡀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠙⢿⣿⣦⠀⠀⣤⠀⠀⠄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣤⣄⠘⠃⠀⠀⠄⢠⣿⣿⣆⠀⢻⣇⠀⠀⣿⣿⠇⠀⠀⣴⣤⣔⡄
⠀⠀⠀⠀⠀⠀⠀⡉⠙⠃⠀⣿⣿⣶⣾⣿⣿⣿⣷⣿⣿⣦⣀⣽⡏⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⣿⠏⠀⣋⣤
⠀⢀⣀⠀⠀⠀⠀⠉⠉⠻⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣠⣿⡟⠀⢰⠟⠋⠀⠀⠀⢠⣤
⡔⣴⣿⣿⣿⣶⣤⣈⡁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣀⣤⣤⣤⡀
⠙⠻⠿⠛⠛⠛⣉⣉⠍⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠿⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠾⠟⠁⢀⣀⣀⣀⣬⣭⣭⣉⣛⣛⣛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣭⣉⣛⠛⠿⠿⠄
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣴⣶⣶⣶
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡏⢹⢀⡴⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⢾⡇⢸⣻⠇⠀⣇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⢤⣀⣸⣵⠃⢸⡇⣜⣻⢀⡾⠁⣿⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠚⣉⣿⣍⣙⣶⣄⠀⠀⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⣾⠁⣠⠿⡛⠲⣾⡙⣆
⠀⠀⠀⠀⠀⢴⣿⣒⣋⣉⣁⡀⠀⠉⢧⠀⠳⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⡴⠁⢰⠛⢦⣬⠿⣾
⠀⠀⣀⣶⠋⠉⢉⣹⣤⣄⣈⣭⡽⠶⣌⠳⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣷⠖⠓⠲⠶⣽⡄⠈
⢠⢞⡥⠞⠋⠋⠉⠀⠀⠀⠀⠀⠙⢦⡀⠙⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡇⠀⠀⠀⣰⠃⠙⣶
⠿⠋⠀⠀⠀⠀⣀⣠⢤⡴⠶⠶⠶⠦⠬⠷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠑⣾⠷⠤⠤⡾⠁⠀⠀⣸
⠀⠀⠀⠀⣰⠊⢁⣠⠞⠓⢒⣒⣶⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢐⣽⣿⣿⣄⠤⣈⡿⠃⠀⠀⣰⠛⠒⠒⠚⡏
⠀⢀⡴⠚⣉⡿⠋⠀⣰⣞⣉⣁⣀⡤⢴⡃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢞⣽⣿⣷⣼⡻⠿⠟⢨⡧⠋⠀⠀⠀⣠⡇⠀⠀⠀⢰⠃
⠀⣏⣤⠞⠋⠀⠀⡜⠁⢸⠁⠀⠀⠀⡴⠙⢾⣁⣰⠃⢠⠄⢀⡄⠀⡄⢸⡙⠿⠛⢈⡗⢢⣶⡋⠀⣀⠴⢚⣉⡞⠙⠢⣤⣴⠏
⠀⡇⢸⠀⠀⠀⢠⣧⢴⡇⠀⠀⠀⢰⠇⠀⠀⠉⣿⠚⠿⠦⠾⠧⠴⠧⠬⠿⠶⠒⢻⣿⠉⠙⣣⣾⠵⠚⣉⡿⠁⠀⢀⠞
⠀⢧⢸⠀⠀⠀⠸⡀⢰⡇⠀⠀⠀⠘⣦⠤⠖⠚⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⣀⡴⠏⢀⣠⠴⠋
⠀⠈⠻⡄⠀⠀⠀⠱⡞⢧⡀⠀⠀⠀⢷⠀⠀⠀⢀⡿⠒⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣙⣦⣀⠀⠈⠳⠦⡖⠋⠀⠀⠀⠀⠀⣤⣭⣭⣭⣽⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠘⠢⢤⣀⣀⣀⣀⣀⣀⡤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣤⣤⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⢹⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠲⠖⠋⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡶⢶⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀⡀⣀⣸⡇⢸⣇⣀⢀⣀⣴⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣀⣀⣠⣭⣭⣍⣹⡇⢸⡏⣉⣭⣭⣅⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⢹⡇⢸⡏⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡄
⠀⠀⠀⠀⠀⢀⡎⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠸⡄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢷
⠀⠀⠀⠀⢀⡟⠀⠀⢄⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣠⠀⠀⢸⡆
⠀⠀⠀⠀⢸⡇⢀⡄⠘⣧⡴⠾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⣴⠏⠀⠀⠘⡇
⠀⠀⠀⠀⢸⣧⠞⣧⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⣸⠓⢦⡇
⠀⠀⠀⠀⠈⠀⠀⠘⣧⡀⠀⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣠⡴⠟⠉⠀⠀⣰⠏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠈⠙⠛⠷⠶⢶⣤⣤⠁⠀⠀⠀⣧⣤⡴⠶⠾⠛⠋⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⡀⠀⠀⠀⡞⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣥⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣷⣶⣾⣿⣿⣶⣶⣶⣶⣾⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⢦⡴⠶⣤⡴⠲⠲⠶⠶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠉⠹⣿⠇⠀⠀⠀⣿⠇⠀⠀⠀⠀⢹⠍⠁⠈⠁⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢲⣿⡏⠁⠀⠀⠀⣿⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠿⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⣿⠁⠀⠀⠀⠀⢸⡆⢀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⣧⣄⡀
⠀⠀⠀⠀⠀⣠⠴⣾⡇⠀⠀⠘⡌⠀⠀⠀⠀⠀⢸⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠇⠂⠀⠀⠀⣸⠿⠿⠉⢷⡀
⠀⠀⠀⠀⢰⡇⠀⠹⣧⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠈⡧⠀⠀⠀⣸⡀⠈⠀⠀⡼⠃⠀⠀⠀⠀⡟
⠀⠀⠀⢀⣯⠀⠀⠀⢱⠀⠀⠀⡀⢱⠀⠀⠀⠀⠀⣧⠄⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠀⡿⠁⠀⠐⣾⠓⠀⠀⠀⠀⣼⡿⢦
⠀⠀⢠⣟⠉⢦⠀⠀⠀⢣⠀⠀⠀⠘⡂⠀⠀⠀⠀⢸⡄⠀⠀⠀⡅⠀⠀⠀⢸⠀⠀⠀⣰⠀⠀⠀⢰⠉⠀⠀⠀⠀⠼⠋⠀⢈⡇
⠀⠀⢸⡇⠀⠈⢲⣄⠀⠠⢦⠀⠀⠀⠱⡀⠀⠀⠀⠈⣡⠀⠀⠀⣧⠀⠀⠀⣾⠀⠀⢀⠏⠀⠀⢠⠇⠀⠀⠀⢀⡜⠀⠀⢀⣼⣷⡄
⠀⠀⢨⢧⡀⠄⠲⡿⣆⠀⠉⢧⠀⠀⠀⢷⣖⠀⠀⠀⡏⡆⠀⠀⡿⠀⠀⢀⢿⠀⠀⡜⠂⠀⢀⣞⠀⠀⠀⢀⡜⠀⠀⢠⠎⠀⢀⡇
⠀⠀⠸⡆⠘⢄⠄⠈⠻⡆⠀⠀⢳⡆⠀⠘⣿⠀⠀⠀⢇⠃⠀⠀⠁⠀⠀⢸⠂⠀⣠⡇⠀⠀⡟⠄⠀⠂⢀⡞⠀⠀⡴⠃⠀⡠⠎⡇
⠀⠀⠀⣿⠳⣾⢷⡀⠀⠙⢶⣄⠀⢿⠀⠀⢹⠀⠀⠀⠸⡀⠀⠀⡀⠀⠀⣿⠀⠀⡿⠁⠀⣼⠇⠀⠀⡠⡣⠄⢀⠞⠀⣰⣯⠕⣾⠁
⠀⠀⠀⠘⣷⣈⣷⡙⢦⠀⠀⢿⣆⠀⢣⡀⠀⢧⠀⠀⠀⡇⠀⠀⠁⠀⠀⡏⠀⣸⠁⠀⣰⠏⠀⢠⡞⠁⢀⡴⢅⡠⣪⣞⣵⢴⠋
⠀⠀⠀⠀⠙⢮⡻⣝⢮⡳⣤⠀⠹⣆⠀⠹⡄⠈⡆⠀⠀⢣⣠⢀⢠⡤⢴⣁⢀⡧⠀⢰⠹⠀⢀⡏⠀⣠⢞⡱⣫⣺⠖⣫⡷⠋
⠀⠀⠀⠀⠀⠀⢹⣚⢷⣝⣮⡣⣄⠘⢷⡄⢹⣤⢿⠁⠁⠸⡅⠸⣂⠀⡾⠀⡼⠛⢲⣿⣀⣠⠋⢠⢞⡕⣫⢞⣩⢶⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⡿⠿⣮⣢⣌⠿⣦⠙⡄⢣⠀⠀⡇⠰⠁⢠⠇⢠⡇⢀⡾⠉⡵⢷⣶⣗⣡⣎⣪⠭⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡶⣭⣻⢷⣬⠳⡘⣌⢇⠀⢿⢸⡇⢸⢀⣾⢀⣾⢀⣼⣵⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⠷⢽⠿⢿⣦⡹⣞⢞⣆⢹⠸⣇⡏⢈⡟⣾⣫⣾⣿⣿⣿⠇⣡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢲⣮⡟⣁⡀⠀⡿⣿⣯⣿⡌⡄⣹⡟⣾⡼⣱⣿⡟⠉⢡⣴⡏⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⠤⠽⠋⠉⠙⠳⠦⢌⣿⣿⣇⣿⣴⣿⣾⡟⢡⣷⠖⢚⣿⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠷⠖⠶⠾⠗⠒⠒⠋⠉⠻⠿⠿⠟⠓⠛⠛⠛⠒⠿⠿⠶⠚

⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡌⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡠⢂⠝⢛⠟⠛⠛⡻⢷⠟⣫⠿⣛⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠈⠀⠋⠸⠁⡞⢉⣬⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⡿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡷⠀⠀⠀⢀⣤⣾⣿⡿⠿⠿⠿⣿⣿⣿⡄⠀⠀⢀⣤⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠇⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠴⢿⣿⣯⠀⢀⣠⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋⠀⠀⠀⢸⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⣿⣿⣿⠇⠀⠀⠿⠧⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⠋⠉⢻⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣾⡿⠋⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⢠⣿⠟⠁⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⢀⣴⣾⣿⡿⠋⢀⣀⣀⣤⣤⣴⣶⣶⣦⣤⣀⠈⠉⠁⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⣠⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠿⠿⠃⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢾⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢄⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠷⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠀⠀⢀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⡟⠀⡀⠀⠠⠴⠀⠀⠀⢀⡀⢀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⢀⣀⣠⡄⠀⠠⢄⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⣀⣤⢾⡟⣿⠛⢛⡀⠀⠈⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣩⠿⠼⠓⠋⠻⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠁⠀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢀⡀⢴⡶⠚⡿⠇⠀⠀⠸⣗⠒⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡨⠀⢀⣤⣾⡯⣶⠚⣿⡾⣤⣼⡦⠴⠶⠾⣶⡿⠦⠴⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⣷⠾⢛⣉⡥⠷⠞⠒⠛⢟⠉⠉⠀⠀⠀⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠚⣟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠠⣄⠸⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣀⠀⢰⣶⣶⣿⣿⣿⣿⣛⠿⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣤⠶⢺⡿⢸⣶⣞⢫⣾⢿⣿⣿⣿⣙⡻⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠛⡉⠀⢸⣷⢞⣵⣞⣷⣾⣿⣿⣿⣿⣿⣿⡦⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣟⣷⡿⠺⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⠇⠀⠁⠀⠀⠈⠹⡿⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠴⠦⣤⡶⠶⠖⠒⠲⢶⣶⣤⡽⠿⠿⠉⣤⣦⡀⢀⣀⣰⣶⣇⠀⢠⡀⠃⠐⠶⠖⠀⠀⠀⠀⠒⠶⠖⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣿⡗⣰⡿⣶⣶⣶⣤⠀⠐⠒⣯⣷⡷⠈⣽⣾⣿⡏⣀⡘⠻⣿⣿⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠡⣶⣾⣍⠙⣃⣠⡔⢛⡿⠃⠀⠚⠿⣻⣥⠙⢿⣷⣄⠀⠀⠀⠀⠈⠁⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠙⠛⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣶⣦⣀⢀⠀⠀⡀⠀⢀⠀⠀⣠⣾⣿⡏⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⢿⣿⣿⡟⢁⡤⣄⣦⣁⢨⣷⢠⣾⣿⣿⠅⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠉⠀⠛⠛⠋⢠⣿⣵⣶⣯⢽⣷⣿⣾⣻⣿⠟⠀⠀⠀⢘⠂⢈⠃⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⣿⣿⠟⣻⢇⣠⣂⠀⠀⠈⠀⠐⠅⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢠⡴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣟⣿⠟⣡⣾⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣈⣀⣉⣉⣻⣿⣿⣿⣿⣾⣿⢟⣀⡀⡠⣄⣤⡤⠄⢀⣀⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⢿⣿⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠘⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⡿⡯⠊⣿⢶⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠃⠁⣀⡾⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⡨⢿⣯⡙⢿⣿⣯⢿⣿⣆⠀⠢⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⣠⣿⠉⠀⠀⢀⣼⡿⢈⣼⣿⢿⡷⠙⢰⣿⡇⢿⣻⣿⣷⣾⣧⣀⠻⣿⣿⡟⣿⣅
⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⢀⣾⠟⠀⣾⠏⠟⠏⠁⠀⢸⣿⠇⠀⠻⡟⣿⣿⣿⢯⡀⠉⠹⠻⠹⢿⣦
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣸⠃⠀⠈⣿⠊⠀⠀⠀⠀⢸⣿⠀⠀⠀⠁⠊⣷⣯⣟⠆⠀⠀⠀⠀⠸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠸⡽⠀⠀⠀⠀⠀⠉⢹⣾⡇⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣤⢶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣏⠉⢉⠁⠉⢀⠀⠀⠉⣲⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⡶⠚⠻⢿⣿⣿⣟⠻⠻⣿⠿⠲⢶⣶⣖⢒⡒⠲⣦⡠⠴⠖⠛⠉⠁⠘⣇⢘⣠⣀⡈⣳⡖⠂⡺⠙⢇
⠀⠀⠀⠀⠀⣿⡛⠛⠋⠉⠸⠿⠛⠛⠛⠛⡻⠋⠀⠀⢀⣐⣒⣂⠀⠭⢿⣷⣿⢩⠠⠤⠔⠒⢒⣋⣩⣤⣤⡭⢍⣁⢆⢀⡄⣩⡄⡇
⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣺⣿⣿⣿⣿⣿⣗⡦⠈⠻⣿⠃⠀⠀⠀⠀⠠⠬⢿⣿⣟⢉⣙⢶⡡⢄⣀⠒⡇
⠀⠀⠀⠀⢸⣿⡄⠀⠀⠂⠀⠀⠒⠒⢢⠀⢀⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘⣧⠀⠀⠀⠀⠂⠉⢙⣿⠙⣋⠀⢺⣉⣬⡓⣾⡇
⠀⠀⠀⠀⢸⡿⢡⣀⠀⠀⠀⡀⠀⠀⡎⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⡄⠀⠀⠀⠈⢉⠉⠿⠛⠿⠛⠉⠀⠀⣀⠜⠁
⠀⠀⠀⠀⢸⡇⠀⢻⠀⠀⠀⣁⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⡇⠀⢧⡶⠀⠀⠠⠦⠇⢃⣀⣀⣴⠤⠒⠉
⠀⠀⠀⠀⢸⣇⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠈⢀⣴⣶⣶⡞⣿⣿⠁⠻⠛
⠀⠀⠀⠀⢸⡿⣶⣾⡇⠀⠀⠀⠀⠀⠲⢄⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢠⣦⣴⠞⢋⢉⢁⣳⠈
⠀⠀⠀⢀⣾⣿⣛⣿⡇⠐⠀⢀⡤⣤⠤⠬⠓⠀⠙⢳⣶⠶⡶⢶⣶⣾⣫⣿⣁⣧⠘⣿⡻⣧⣿⣾⡿
⠀⠀⠀⠀⠉⠉⠉⠉⠙⠷⠾⠿⠿⠷⠶⣶⣤⣭⣤⣼⣭⡿⠿⠟⠓⠻⠷⠊⠉⠁⠀⠈⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

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

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠷⠾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢦⣄⠀⠀⠠⡇⠀⠀⢹⣦⣀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⣯⠑⢦⣀⣇⠀⠀⠸⣿⡷⡑⢢⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡃⠸⠀⠀⣹⣿⠄⠀⢸⠇⡇⠀⠀⠻⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢼⣿⠀⠀⠸⠁⠀⠀⠀⠀⡼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⣿⢇⠀⡚⠀⠀⠀⠀⠀⠀⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⣷⢸⠀⠇⠀⠀⠀⠀⠀⠀⢹⣜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣻⡋⡞⠀⠀⠀⠀⠀⠀⠀⢸⣻⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⡄⠀⠀⠀⢀⠀⣸⢾⣶⠃⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⣈⢂⠀⠀⢢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⣅⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡅⢣⠀⠀⠀⠀⠀⠈⣀⠚⠉⠃⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⢀⣠⣤⣽⣿⡧⠖⣂⣉⣀⠀⠀⠀⠀⠸⣿⣧⢸⠀⠀⣀⡀⠀⢈⣵⡖
⠀⠀⠀⠀⠀⠀⣰⣿⠁⣀⠔⠀⠀⠀⢀⡈⢻⣷⣶⣭⣿⠭⠍⣡⠄⠀⢀⣈⣿⣿⣬⠁⠀⣁⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠨⠷⠩⠿⣁⣈⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⣃⣤⣶⣿⣿⣿⣿⡟⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⡏⠉⣿⣷⠂⢀⣠⣤⣾⣿⣿⣿⣿⠿⠋⠉⠙⠿⢀⠴⠃⠀⠄
⣠⣄⣄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣴⡿⢓⢀⣷⠀⢸⣿⡇⢀⣆
⣿⣿⣿⠿⢿⠂⠉⠚⠦⡀⣶⣬⣀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣇⠉⢈⣉⣶⣾⣿⣿⣿⡀⣿⡿⠛⢸⣿⣦⣤⣀⣀⣠⠤⢢
⠉⠉⠉⠛⠢⠀⠀⠀⠻⣦⣤⣈⠉⠛⢿⣷⣾⣶⣷⣿⣿⣷⣾⣿⡿⠿⠿⠿⣾⣟⠿⠿⣿⡿⠿⠁⠹⠆⠀⢿⣿⣿⣏⠛⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⡤⠌⠉⠉⠉⠛⠶⡿⠍⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠘⢿⣿⡯⠀⠘⢷⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⠀⢁⠀⠀⠂⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠂⠀⢠⠁⠀⠀⠀⠀⠀⢄⠀⠀⢀⠠⠂⠀⢀⠀⠀⢄⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⢀⡀⠀⠤⠐⠈⠪⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠑⠀⡀⠀⠀⠘⠄⠠⡤⠚⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠐⡄⠐⢞⠛⠁⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠊⠢⠘⢤⣀⠀⣀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠛⠓⠒⠒⠤⠭⢦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠶⠶⠤⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠒⠒⠲⢖⠚⠉⠄
⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⡿⠋⠀⠀⠀⠈⠂⡀⠀⡀⡀⠀⢀⡀⠀⢀⠤⢄⠀⠀⠀⠀⡄
⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠉⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⢂⠐⡌⠢⣈⠀⢂⠀⡆⠀⠑⠀⠀⠊
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⣷⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣷⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣴⡖⠛⠛⠛⠛⢶⣶⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣶⡞⠉⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⢻⣶⠶⢦⣄
⠀⠀⠀⠀⢀⣠⣴⠁⠀⠀⠈⢿⡄⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⡇⠀⠀⠉⠳⣦
⠀⠀⠀⣰⠋⠀⠹⣆⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢹⡆⠀⠀⠀⠈⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡷⢦⣄
⠀⠀⣰⠛⣆⠀⠀⢹⣆⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠏⠀⠀⠈⠻⣄
⠀⠀⣿⠀⠸⣆⠀⠀⢿⡄⠀⠀⠀⠸⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⡿⠀⠀⠀⣿⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢿⠀⠀⢻⡄⠀⠘⣷⠀⠀⠀⠀⢻⡆⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⢰⡏⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢀⣴⣿⢷⡀
⠀⠀⠸⡆⠀⠀⢿⡀⠀⢻⡆⠀⠀⠀⢸⣇⠀⠀⠀⣿⠀⠀⠀⣾⠀⠀⠀⣿⠁⠀⢀⣼⠃⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠙⣆
⠀⠀⠀⢳⡀⠀⠈⣧⠀⠸⣇⠀⠀⠀⠀⣿⠀⠀⢠⡇⠀⠀⣸⠃⠀⠀⣼⠃⠀⢀⣾⠃⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⢧⠀⠀⠸⡆⠀⣿⠀⠀⠀⢀⡇⠀⠀⣼⠁⠀⢰⠏⠀⠀⣼⠃⠀⢀⡾⠁⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⣾⣇
⠀⠀⠀⠀⠘⣧⠀⠀⢻⡀⢿⠀⠀⠀⢸⠇⠀⢠⡇⠀⢠⠏⠀⠀⣼⠃⠀⢠⡿⠁⠀⠀⣠⠟⠁⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠘⣦
⠀⠀⠀⠀⠀⠸⣆⠀⠈⡇⢸⠀⠀⠀⣾⠀⠀⡾⠀⢠⠏⠀⢀⡼⠃⠀⢠⡿⠁⠀⢠⡾⠁⠀⠀⠀⢀⣠⣶⠟⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠹⡆⠀⢿⢸⠀⠀⢀⡏⠀⢰⠇⢀⡞⠀⢀⡼⠁⠀⣰⡟⠁⠀⣰⠏⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⣾⠀⠀⢸⠃⠀⡟⢀⡞⠀⢠⡞⠁⠀⣰⡟⠀⢀⡼⠁⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⣤⡴⠾⠛⠋⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣷⠘⣿⠀⠀⡿⠀⣸⢁⡞⠀⣠⠏⠀⠀⣼⠏⠀⢠⠟⠀⠀⢀⣴⠟⠁⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⣀⣠⡏
⠀⠀⠀⠀⠀⣠⠞⠋⠈⣇⣿⠀⢰⡇⢠⠇⡼⠀⡴⠃⠀⢀⣼⠃⢀⣴⠋⠀⢀⣴⠟⠁⠀⣠⠶⠋⠁⠀⠀⣀⣤⡴⠶⠞⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠻⡀⠀⠀⢸⣽⠀⢸⠇⡞⡼⢁⡾⠁⠀⣠⡾⠁⣠⡞⠁⠀⣠⠟⠁⢀⡴⠋⠁⢀⣠⡴⠞⠛⠉⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⣿⡇⢸⣼⣱⢣⡞⠀⢀⣴⠋⢀⣼⠋⠀⣠⠞⠁⣠⠖⠉⣀⡴⠞⠋⠁⠀⣀⣀⣤⣤⣶⡶⠶⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢻⣷⢸⣿⣧⡟⠀⣠⠞⢁⣴⠟⠁⣠⠞⢁⡤⠚⣡⠴⢛⣁⡤⠴⠖⠛⠋⠉⠉⠁⠀⠀⠀⢀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣾⣿⡿⢀⡜⣁⡴⠛⢁⡤⠞⣡⣴⣫⢔⡫⠔⣚⣉⣀⣀⣤⣤⣤⣤⡤⠶⠶⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⠿⠚⣉⡤⠖⣫⣴⣞⣻⡽⠾⣿⠟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⢿⣵⠶⠟⠛⠋⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠲⠤⣼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣾⡇⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢧⢿⠟⢬⠋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣸⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⠆⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢾⣥⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠘⠛⢻⡟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣨⣿⡿⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠳⣶⣎⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢯⠁⣾⡷⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢙⣢⣄⣽⢀⡀⣀⢴⠒⣎⠏⢙⣫⠛⠂⢏⢿⡖⡻⡗⡶⠢⣤⣤⠼⢦⣿⣴⠀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠸⣿⣄⠀⠻⣌⣽⣿⣿⣤⣮⡷⠛⠛⠙⠿⣿⣵⠠⣶⣿⠽⠧⣤⠀⠉⠟⣳⣦⠏⠁⣴⢿⠃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠹⣧⡴⠀⢈⣯⣉⡟⣿⣿⡛⠁⠀⠀⠀⢈⡆⠀⡟⠃⠀⠀⣐⣿⣿⣆⠙⣣⣤⢶⡧⠈⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡷⠄⠀⠀⠈⣧⣿⣿⣛⣿⣾⠻⠻⢻⣦⣤⠀⠀⠀⢡⠠⠁⠀⢀⠛⢁⡿⠇⠀⢏⣘⡻⠋⠀⢀⣀⣴⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢧⡠⡄⡀⠈⣷⠶⠉⠀⠀⠀⠀⠐⠟⠿⣷⣢⣤⣤⣴⠞⠛⠛⠋⠀⠀⠀⣾⢧⢘⣿⣾⠏⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠻⠿⠻⢇⡀⠀⠈⠁⠠⡄⢀⠀⠀⢀⣥⣾⡆⠀⠀⡀⠴⠐⣉⣁⣉⣤⢞⠉⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣶⣦⣼⠷⠾⠿⠛⠛⠛⠓⢶⣴⣷⠄⢹⡇⢈⣤⣢⠌⣀⣴⠋⠉⢉⣙⠐⠾⠒⠊⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⠛⠁⠀⡀⡀⠀⢤⣴⠾⢦⡬⠛⠋⠉⠛⠊⠓⠒⠉⠑⠚⠻⠷⠆⠆⠘⠏⠐⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⢀⣹⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢴⣆⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣇⠠⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠈⠀⠀⠀⠀⠀⡀⢀⣀⣧⠜⠖⠶⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠦⠶⠒⠚⡟⠉⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢬⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⡢⠐⠀⠀⠈⠉⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠚⠋⠉⠁⢸⠇⠀⠀⠀⠀⠰⡆⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢻⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡃⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡇⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠁⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣴⠟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠷⣾⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡀⢤⠖⠻⠟⠁⠂⡂⡾⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣀⠦⡜⡾⠉⠉⡠⠘⢁⣤⣴⣾⣿⣯⣷⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⡀⠿⠐⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣬⣥⣶⣿⣿⣿⣿⣿⣋⠻⢿⣿⣿⣿⣁⣀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⢀⡖⢮⣙⣛⡛⠉⢩⣽⣭⣙⣋⣿⣶⠳⣿⢿⣯⢙⣿⣿⣿⠝⣛⠷⣴⣶⣶⣾⣍⣀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠠⠌⠋⣁⣬⣿⣭⣿⣿⣿⡞⢛⣯⣥⣶⣿⣿⣛⠛⠛⣉⣵⣆⠁⢨⣉⣛⣛⠙⠻⣿⣿⠿⠶⣿⣯⣟⡛⠁⠠⠄
⠀⠀⠀⠀⠀⠐⠒⠂⠉⠉⠉⣭⣥⠀⢼⡿⠛⠋⠁⠤⠼⠛⠋⠉⠩⠕⠂⢀⣿⠿⡋⠀⠰⣮⣍⣻⠥⠤⠤⢴⣶⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠁⠁⠀⠀⠀⠘⠻⢿⣷⣶⡄⠀⠒⠉⠀⠀⠀⠈⠙⠛⠉⠁⠀⠢⠭⣭⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣾⢛⣥⣶⣵⡯⢒⣡⡤⠞⣁⣴⢞⣾⢿⣿⣿⢿⣾⡷⡟⠂⣬⠅⠐⠒⢂⠉⢩⣅⣒⢦⣤⡳⠾⣅⣀⡀⠛⠛⠛⠛⠷⢯⣷⣶⠦⣤⠆
⣿⣯⣾⣿⡯⠖⡩⣥⡶⢟⣻⠝⣫⠴⡿⢋⣾⣿⢯⣒⠨⠥⣄⣒⡒⠮⠭⢛⣓⠬⠙⣃⠤⠑⣀⠤⢉⣉⡉⢉⠛⡉⠰⠒⡬⢙⠳⢦⡋
⣿⣿⡿⣋⣄⣵⣿⡵⠞⣫⡴⢚⣥⢞⣵⣿⣿⣿⢿⣭⣍⠁⠠⠤⠶⠒⠋⢁⣤⠔⢋⡡⠂⣉⠴⢊⡥⠀⠀⠀⠀⠐⠒⠚⣤⣥⡔⠀⣉
⣯⣻⡞⣱⣿⣋⡥⣖⣯⠗⢚⣩⠶⢋⣵⣿⣋⠩⠤⠤⠐⠒⠒⠒⠒⠒⣉⠥⠔⣊⠥⣒⠭⠒⣉⠤⠒⣒⡀⠍⠁⡈⠉⠉⠄⠘⣻⣶⣄
⣿⡿⣰⣿⣿⡿⢛⣥⠶⢚⣉⣤⣶⣿⡿⠍⠉⠀⠐⠀⠀⠒⠒⠛⢋⣉⡠⠶⣛⠵⢚⣡⠖⢋⣡⣴⠿⠏⣉⣃⣤⣤⣤⣤⣀⡀⠤⢌⡙
⣿⢁⣿⣿⣯⡞⣉⣴⣾⣿⡿⠟⣛⣡⠖⠂⠬⣭⣉⣉⣉⣩⣭⣭⠴⠶⢚⣩⠴⢞⣩⣴⠾⢛⣩⣴⡾⠿⠿⠿⣿⣯⣁⠒⠭⠛⠷⠶⣤
⡟⣼⣿⣿⠏⣴⣿⣿⡷⠤⣀⡛⠩⢄⠛⠥⣒⠀⠤⠤⠤⠤⠤⠄⠒⣊⡬⢕⡶⠟⢋⣠⣾⣿⣿⣷⡶⣦⣖⣂⠤⢈⠉⠟⡶⠥⣦⡅⠂
⣳⢹⣿⣿⢠⣿⠟⠁⠀⠀⠀⠉⠲⣄⠙⠢⢤⡈⡉⠀⠀⣀⣉⣩⡽⠶⠛⣉⣤⣾⣿⣿⣿⣟⣻⣭⣭⣄⡐⢉⣙⠻⢾⣧⣤⡁⠀⠹⣿
⣟⢸⣿⣿⣾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⢈⣉⣉⣉⣉⣠⠤⠤⠚⠛⢿⡻⣿⢿⣿⣿⣉⠥⠼⠛⣙⡛⠷⣥⣛⠱⣊⢝⡻⡳⡢⠈
⡇⣼⡏⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣴⣫⣭⣙⠒⢦⢄⣀⠐⠦⣙⠷⣄⠰⣍⢦⢈⢶
⢱⣟⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣳⢖⡩⢟⡱⢎⣷⣌⠳⣄⠢⡑⢏
⣯⢿⡌⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣭⠔⡣⠄⡙⢷⣌⡣⣈
⣿⣍⢷⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡄⠀⠰⡿⠿⡿⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢧⣤⡐⠈⠻⣿⣷
⣿⠋⣼⣾⠀⠀⠀⠀⠀⠀⣄⠀⠀⢾⠁⡀⠁⠄⢰⣷⣶⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡖⠀⡈⢟
⠏⢼⣿⠃⠀⠀⠀⠀⠀⠀⠈⡀⠀⣻⠃⠐⠀⠶⣾⡟⠿⡿⢿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠄⠉
⢎⣾⠟⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠋⠁⠀⠀⢉⡄⠀⠀⠀⠸⠁⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀
⣾⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠈⡇⠀⠀⠀⢀⠀⠀⡀⣇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⡄⠀⠀⠀⠸⠆⠨⠃⡌⠀⡀⠆⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣿⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢀⡈⠀⠀⣶⠀⠀⠀⠀⠀⠂⠀⡇⢀⡀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡏⣼⠀⢀⣀⡟⢀⡀⠀⠀⠀⠹⡆⢱⡈⢀⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡇⣿⢻⣾⢿⡿⢿⣿⣧⡄⠀⣴⠁⢸⡇⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⣧⣿⣾⡿⠞⢛⠛⠋⠿⠿⣿⣿⣧⣼⣇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡌
⣿⢸⣧⢸⠀⠀⠀⠀⠀⠀⠀⣬⠿⡿⠿⠟⠋⠀⠀⠉⣤⠀⠀⠀⠈⢿⡟⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠽⠁
⢻⠇⣹⡌⡄⠀⠀⠀⣀⡐⠻⠟⠀⠀⠛⠋⠀⣤⡀⠀⠈⠁⠀⠀⠠⠀⠀⠉⠉⠉⣹⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⠁⠀⠀⢔⣤
⡘⣤⢙⣧⠁⡀⠀⠀⠹⠿⢿⣷⣶⣾⣶⣿⣶⣾⣷⣶⣶⣦⣤⣤⣄⢀⣤⣬⣥⣤⣯⣥⣼⣿⣿⡿⠀⠀⠀⢀⣤⠊⡱⣓⣤⠖⠞⠉⠑
⣷⢀⠜⣻⡆⢡⠀⠀⠀⠀⠀⠙⣿⣿⣯⡿⣿⢿⣿⣿⠿⠿⣏⠉⣿⣾⣿⣿⣇⣹⢻⣏⣿⡿⠿⠁⠀⠀⣠⠪⣡⣾⡽⢋⠥⠒⡠⠔⠊
⣟⠀⠐⢹⣿⡀⢣⠀⠀⠀⠀⠀⠙⠿⡛⣷⢿⣿⣷⢹⡷⠾⣿⣿⣿⠿⣿⣿⣟⣿⠞⠋⢹⠀⠀⠀⢀⡴⣡⡞⠉⠉⠀⠀⠀⠈⠐⣠⣴
⣼⡆⠠⢁⢻⣿⡄⠳⡄⠀⠀⠀⠀⠈⠻⢾⣿⣾⣟⣿⣿⣾⣿⣯⣫⣾⢻⣿⣿⠁⠀⠀⠈⠀⢀⡴⢋⣴⠯⠗⠀⠀⣄⠔⢀⡀⠾⢋⡼
⡠⠶⠀⢂⠆⡹⢟⢦⠘⢆⠀⠀⠀⠀⠀⢀⡈⣻⣿⣿⣿⡟⢻⣿⣿⣿⣿⠇⣿⠀⠀⠀⢀⠴⡉⣰⠟⢁⣤⠆⡀⠀⣠⠒⠁⢀⡴⠋⠰
⣧⠞⢋⠌⠄⢱⣏⢾⡷⡌⢆⠀⠀⠀⠀⡟⠉⣿⢻⡿⠈⣧⣻⠙⣿⡿⠃⠀⠉⠀⡠⢚⡵⠊⡴⢋⡴⢛⡱⢊⣤⠊⠔⢁⠴⠋⠈⠁⠈
⠶⢛⡁⠄⠔⡀⢫⣆⠱⣞⡎⢦⡀⠀⠀⠃⠀⣯⠘⣿⠀⠛⡟⠀⣿⠀⠀⠀⣴⡾⠶⡋⢀⢄⠰⣫⡜⣡⡞⠻⢁⣤⠂⠁⠀⠀⠄⡠⢊
⡔⠙⢷⣈⢆⠱⣤⠻⣦⠈⢾⣦⠑⡄⠀⠀⠀⠋⠀⢿⠀⠀⠁⠀⠁⣠⣶⠿⡁⠄⢊⢀⡷⢋⡴⢋⣾⠟⠁⠀⢊⠀⠀⠀⢠⣎⠚⣠⢞
⠴⡀⠌⢿⡈⠵⡹⣣⠙⣧⠈⢿⣡⠘⡄⠀⠀⠀⠿⢸⠀⢀⣠⡴⠞⣩⠧⠀⠀⠢⠁⢈⡴⠋⣠⠞⠁⠀⠒⠁⡀⠀⣠⡾⠁⠀⠚⣡⡛
⠃⡥⠈⢊⣷⠀⣴⡹⣧⡘⢷⡈⢧⢣⠘⠆⠀⠀⣀⠌⠚⡉⢉⡤⠚⠁⠀⠀⢀⣤⡶⠋⠁⡐⠁⢀⢆⠀⠀⠀⣠⡾⠋⡠⠀⣰⣺⠋⠵
⢦⡹⣏⢄⠱⣿⡌⢸⣥⠻⣎⣷⣈⢆⠠⠘⣄⣼⠃⣠⠂⠔⠡⠀⠁⠐⡡⠞⡩⢊⠔⣠⠟⢀⢤⡀⠃⢀⣠⡾⠋⢄⣡⣶⠿⠊⠁⡠⢂
⠨⢷⡜⢮⡱⣌⠻⣦⡙⢷⡜⢿⣯⢺⣆⠰⢹⠧⣄⠄⢂⠀⢀⡢⢉⣀⠄⡣⢀⣤⠞⠁⢀⣮⢛⠀⣠⡾⢋⣤⡾⡛⠍⠡⡐⠄⢊⣔⡡
⢸⣧⣟⣦⡻⣎⠳⣌⢷⣄⠹⣮⣹⣧⡹⣷⣇⣿⠚⠠⡑⠀⢊⡴⠋⠑⣠⡴⢋⡁⢀⣴⡿⠑⣡⣾⣿⣼⠿⢋⣡⠶⠋⣁⢠⣼⠾⣌⡴
⠀⠿⣿⣿⢷⣌⢳⣌⠳⣝⢦⣄⠻⡿⣿⣎⣿⣿⣄⢁⣴⣶⠏⢦⡵⠚⠁⠔⡨⢾⡿⠋⣠⣾⢿⣿⡟⠥⣒⣩⣠⠶⢙⣪⠭⠖⠃⠁
⣶⣷⣾⣿⣷⣿⣷⣿⣷⣾⣷⣿⣷⣾⣿⣷⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣿⣶⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠋⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⠁⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡬⠿⢿⣿⣶⣦⡀⠛⣸⡇⢀⠖⣁⠀⢈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠙⠿⣷⠀⣼⣣⠏⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣭⣭⣶⣦⣬⣵⣟⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣡⣤⣜⣿⣿⣾⣶⡭⢙⣷⣶⡶⠦⣝⣣⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠿⢿⣿⣿⣯⣾⣟⡟⢩⣿⣿⣦⡘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠸⣿⠋⣿⢛⣿⣿⢸⠈⠉⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢻⠋⡰⠈⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⣤⣏⠀⠀⠀⠀⢀⠀⣀⡀⡀
⠀⠀⠀⠀⣠⣤⡤⣤⣤⣤⣤⣤⡉⣭⣭⣽⡿⠿⠩⠈⣿⠤⢬⣕⣛⠚⢻⠉⠉⠩⠁
⠀⠀⢠⣤⡴⠦⣴⣿⣿⣿⣽⠻⠷⠛⠉⠁⠀⠀⠀⢀⣹⣧⠀⠉⠙⠻⢛⣛⣛⠓⢲⣖⣀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠠⢶⡶⢶⣶⣦⣭⣙⢂⣀⡀⡀⠀⠀⠀⠀⠈⠉⠳⣄⠐⠀⠀⠙⠻⢿⣷⠍⠉⡉⠈⠉⠉⠛⠛
⠀⠀⠀⠀⣤⣤⣶⣿⣶⣬⣭⣾⣶⣿⣿⣿⣧⣈⠓⠒⠓⠈⠀⠈⠳⠄⠀⠀⠀⠴⠾⠒⠿⠿⠶⠆⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠤⠒⠶⠶⣾⣯⣤⣀⡈⠛⠋⠀⠲⠾⠿⠿⠿⠻⠒⠒⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠈⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣋⣉⣏⣩⣭⣭⣭⣭⡤⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⢠⣿⡆⠀⠀⠀⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠛⠛⠛⠛⠛⡷
⣼⣿⣿⣷⣷⣶⣧⣶⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣴⣧⣬⣯⣯⣿⣟⣰⣧⣨⣿⣯⣏⢹⣿⣿⣧⣦⡆⣤⣼⣿⣶⣶⣶⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⠿⠙⠯⠹⠯⠻⢟⡻⢶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠏⠣⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠋⡝⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠟⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠃⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡽⢳⣄
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡀
⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢀⣠⣴⣶⡶⡾⣶⢶⣶⣤⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⢀⣿⠉⠁⠀⠀⣠⣾⡟⠿⠙⠀⠀⠀⠈⠉⠘⠻⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⢸⡏⠁⠀⢀⣼⠿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⣾⠒⠀⠀⣾⡯⠁⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⢿⡄⠀⢰⣿⠄⢠⣾⠟⠋⠉⠉⠘⠙⢻⣧⡄⠀⠀⠀⠀⠀⢈⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⡇
⠀⠀⠀⠀⢸⣟⠀⢸⣿⢠⣿⣡⡶⣶⠶⣶⡄⠄⠀⠙⣿⡄⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠈⢿⡆⠀⣿⣼⣿⣿⣿⣿⣦⣽⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠈⢷⣄⠘⣿⣿⣿⣿⣿⣟⣰⡿⠀⠀⢠⣿⠁⠀⠀⠀⠀⣘⣿⣿⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣌⡻⢿⣿⣿⡛⢉⣀⣤⣼⡿⠁⠀⠀⠀⠠⢠⣼⠟⠛⠻⠶⠶⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣍⠙⠛⠛⠛⠋⠁⠀⠀⢠⡀⣠⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣷⣦⣤⣦⣶⣦⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢳⠀⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡜⣧⠀⢻⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⡄⠈⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡀⢨⣿⡄⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠠⠐⠒⠁⠈⠉⠀⠀⠀⢸⣧⣼⠟⢻⡄⠀⠰⡀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣿⣄⠀⠀⢀⡄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⢿⢸⡆⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣀⣶⡾⠿⠯⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡀⠀⢠⡾⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⠛⠃⠒⠶⠶⠦⢤⣤⣴⡇⠀⣠⣾⣦⣴⣯⣀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠉⠙⠻⠿⢯⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⠸⢿⢰⣶⡤⣬⣤⣤⣀⣄⡄⠀⠀⢀⣀⣀⣤⠤⡶⠃
⠀⠀⠀⠀⠀⠰⣬⣵⣾⣿⣿⡏⠉⠉⠀⠀⠀⣀⡀⣀⣒⣒⣴⠺⣥⡿⣿⠶⠞⠛⠫⠉⠑⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠿⣿⣦⣦⣤⣤⡤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠬⠽⠷⠶⣿⣿⠠⠔⠒⠤⠄⣀⣀⣤⠤⠀⠀⠤⣤⣤⣤⡼⠛⠛⠁⢠⣤⡤⠤⠄
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⣉⡉⠀⢀⣀⣀⣀⣉⣉⣉⣀⠀⠄⠀⠀⠀⣉⡥⣤⣄⣓⡙⠉
⠀⠀⠀⠀⠀⠀⠠⠔⠛⠋⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠊⢀⣼⣿⣛⣻⣋⣭⣭⠍⠀⠀⠀⠀⠀⠿⢶⣭⣙⠻⣻⡋⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣞⣀⣴⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⣶⣌⣷⣤⡘⢆⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⢸⡆⠀⠀⣧⣄⠀⢠⡏⠉⣹⣿⠿⣋⣠⣴⡶⠖⠛⠟⠿⠟⠿⠿⠿⠶⠶⢦⣤⣀⠈⠻⣿⣿⡿⠚⢓⡄⠀⠀⢨⡜
⠀⠀⠀⠀⠀⠈⢧⢦⠀⣷⢸⡀⣿⣷⣿⡟⣁⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣌⠛⣦⣠⣼⡿⢀⡄⣸⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠸⣏⢤⡘⣧⡳⠈⣿⡿⣧⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠗⣿⣿⡿⣱⢛⣶⠇⠀⢠⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣎⢳⡝⠿⣶⣠⠿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣷⡞⣻⣿⣮⡷⠛⢉⡠⢎⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣽⣓⠦⣭⣿⣿⣿⣿⣿⣦⣄⡀⢀⢀⢀⣤⠀⣠⣄⣠⡀⣠⣴⣾⣿⣿⣿⣿⢋⣭⠥⣄⣐⣠⣽⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢙⡛⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠵⢿⣛⠛⣛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⣯⠼⠟⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠾⠭⣗⣦⣕⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠀⠀⣰⠚⣷⣷⣿⣭⣿⣿⡿⠿⠿⠿⣿⡿⡿⠿⣿⣷⣶⣶⣶⣷⣯⣀⡀⢠⣄⠈⠙⠻⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠁⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠙⠿⣿⣿⣷⡄⠀⠀⠀⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⢀⡏⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⢽⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁