Robots Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠃⠀⠐⠀⠠⠀⠂⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠍⠀⠀⠀⠒⠀⢸⠀⠡⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡀
⠀⠀⠀⠀⠀⠀⠔⠂⢉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠂⠀⠀⠀⡆⢈⠀⠒⢄
⠀⠀⠀⠀⠀⠁⠔⠉⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⡀⠀⠀⢀⠋⠁⠈⠑⠄⠁
⠀⠀⠀⡀⠑⠎⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠈⠖⠁
⠀⢠⠩⠩⠈⡆⠀⠀⠀⠀⠀⠘⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠊⠀⠀⠀⠀⠀⠀⡞⠈⠉⢢
⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠐⠀⠀⠀⠀⢸⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⡃⠀⠀⠀⢘⠂⢺⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣉⡇⠀⠀⠀⢨⢉⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠈⠑⠀⠀⠠⠊⠀⠈⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⣀⣀⣀⣀⡂⠀⣔⣀⣀⣀⡀⡀⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠐⠲⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣿⣧⣄⣀⣿⣆⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡅⡠⠀⠁⡤⡲⣦⣀⣠⢔⢢⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠞⠛⠇⡀⠀⣗⠦⣼⢿⣯⡳⢴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣒⡀⣀⣸⠁⠀⠙⠛⠁⠘⠈⠉⠉⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡇⠀⡀⢰⣗⣶⣼⣼⣭⣵⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡱⣃⣀⡅⠘⠓⢒⣒⣚⣒⣁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠠⢿⣿⣿⣿⣿⣟⣩⡗⠀⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⠄⠉⠣⡀⢹⠈⠉⡩⢤⠀⠀⠀⠀⠀⠀⠀⣆⡆⠀⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⡆⢉⠈⠀⠀⡇⢸⠀⠀⠘⠊⣀⣀⣠⣄⣀⣠⣤⡤⡄⠀⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣧⡈⢁⣴⣶⣿⢠⡞⠀⠀⡎⠛⣛⢿⡿⠻⠯⠭⣄⢹⠁⠀⡇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⢻⣿⣿⣿⡟⣾⢱⠀⠀⡇⠐⣅⣀⠵⠀⠠⠠⠎⢸⡀⠀⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⢤⡀⢀⣾⣿⣿⣿⢧⣿⡸⠀⠀⡇⡀⠨⠀⠀⠀⠀⠰⣆⣾⡅⠀⣧⠀⠓⣄
⠀⠀⠀⠀⠀⠀⠀⡘⢠⠒⡄⣷⣿⣯⣿⣿⣿⣼⣿⢀⠀⠘⡇⠓⠘⡒⢀⠉⠁⡄⠰⢸⡂⠀⣿⡟⠁⠀⠉⠑⠂⢤⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢡⡈⠒⢁⡯⠀⡼⣿⣿⣏⣿⡇⣼⠀⠀⣇⣈⣡⣥⣤⣴⣶⠶⠛⠛⠀⠀⣿⣷⠷⠖⠀⢠⢛⡙⢄⡀⠀⢨⢵
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠟⠇⠀⢸⣿⣿⣿⣿⢸⠇⠀⠀⠈⠉⢰⠋⠁⠀⠀⠈⠀⠀⠀⠀⡇⠈⠁⠒⠤⢼⡐⢂⣼⠿⢏⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡄⠀⠀⠈⣿⣿⣿⡇⣿⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠙⢅⣀⣁⣰⠮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⡄⠀⠀⢿⢿⣿⣾⡏⢸⡇⠀⠀⠀⠘⣒⣒⣢⣽⣥⣶⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢉⢳⣶⠓⢤⠀⣿⣿⣿⣾⡷⠾⠿⢻⣿⡿⠟⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣛⣸⡀⠃⠀⡇⢸⣿⣿⠃⠀⠀⠀⠘⡿⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠋⠘⣇⢸⣤⠇⢸⢹⡟⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠉⠉⠀⠀⠀⣼⣇⣄⠀⠀⠀⠀⣏⠹⡗⠚⠋⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡠⡈⠻⠒⠊⠉⠙⠀⣷⠾⠾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢆⡸⠀⣧⡶⠿⢻⡞⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣄⣀⣴⠇⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⢹⠇⠀⠀⠀⢸⠁⠀⠀⠈
⠀⣀⣤⣶⣶⣶⣦⣴⣶⣿⣿⣿⣿⣷⣶⣶⣶⣾⣇⡟⣾⠀⠀⠀⠀⢸⠀⣀⣠⡷⠒⠤⠄⣀
⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠣⡃⠀⡇⠀⠀⢸⣿⣭⣄⡀⢀⣀⣤⢴⡇
⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣁⡚⠲⢧⠶⠞⠛⠉⠙⠛⢿⡭⠓⣊⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣶⣦⣄⣠⡤⠶⢚⡳⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠛⠻⢷⣿⣽⣟⡻⢿⠇⣐⠠⠵⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠒⠋⠉⠉⠉⠉⠉⠙⠒⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠐⠀⠀⠀⠘⢢⠐⠀⠀⠀⠂⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡀⠀⠀⠀⠉⠁⠀⠴⡀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠉⠢⣄⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⣀⠴⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠉⠒⢢⠤⠤⠤⠤⡄⠒⠘⠁⠀⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠔⠁⠀⠀⢸⠁⠀⠀⠀⠀⢠⡄⠒⡄⠀⢰⡏⠉⢲⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠒⠃⠉⠁⠀⠀⠀⣀⡤⠞⠀⠀⠀⠀⠀⠀⢷⡀⣹⡤⢾⣀⣠⠎⠈⣳⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠈⣧⠴⠚⠁⣳⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢄⡤⠛⡿⠿⠛⠛⢿⡟⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⢀⡟⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⠏⠀⠀⢸⡁⠀⠀⠀⠀⠀⠈⠁⠀⡝⠉⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣑⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⡜⠀⠀⠀⠀⠙⠦⢄⣀⣀⣄⡠⠴⠚⠁⠀⠀⠀⠹⡋⠛⢤⠀⠀⣀⡤⠚⠅⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠉⢸⠁⠀⢠⠃⠀⣠⣴⣤⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣯⠉⠀⢁⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⢀⡷⠚⢧⢀⡏⠀⠀⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⣽⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠋⠀⠀⠀⢹⠁⠀⠀⠘⢿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠃
⠀⠀⠀⠀⠀⣀⡤⠯⣼⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡸⠁⠀⠀⠈⢳⠀⠀⠀⢠⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⠴⠚⣉⣧⡀⠀⠀⠀⣼⣄⠀⠀⠸⢇⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⢸⡇
⠹⣤⠞⠁⢠⠛⢲⠶⠻⣄⠘⢦⡀⠀⠀⠘⡋⠲⠦⢤⣄⣀⣀⣀⠘⠂⠀⠀⠀⣀⣀⣀⣠⣤⡴⠖⠊⠁
⠀⠈⠳⢴⡇⠀⢸⠀⠀⠈⠚⠋⠀⠀⠀⠀⣱⠖⠒⠶⢄⠈⠉⠉⠉⠉⠉⢉⡽⠿⠿⢤⡀⢀⠃
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⡶⠒⠠⡀⠳⡀⠀⠀⠀⣴⠃⡠⠄⠲⣾⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⣿⠴⠁⠀⠀⠀⢹⣴⣇⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠓⠛⠓⢾⡁⠀⠀⠀⠀⠀⠀⠀⣹⠶⠒⠛⠛⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣀⣀⣠⠗⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠛⠋⢽⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠛⠩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠚⠛⠒⠒⠤⢄⣧⠀⠀⠀⠀⠀⠀⢠⣇⠤⠄⠒⠚⢓⠶⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⢠⠈⠁⡠⠀⠀⠈⠳⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⣁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⢤⠤⠤⡤⠤⠤⡶⠴⠤⠤⠤⠤⢾⠀⠀⣾⢠⡤⠤⡤⠤⢤⡤⠤⢤⡤⠤⣤⠄⢾
⣀⣀⣀⣀⣀⣀⣀⣀⣽⣶⣾⣶⣶⣷⣴⣴⣷⣦⣶⣶⣴⣶⣾⣄⣀⣿⣴⣦⣤⣷⣤⣴⣦⣤⣴⣦⣤⣶⣤⣾⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠳⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⣾⠹⠏⣻⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⡽⠑⡟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⡟⠀⠀⡏⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠾⣅⣀⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⠖⠲⢦⣀⡇⠀⠀⡇⠀⢀⣀⡀⠙⢦⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠐⣇⠀⠀⢠⣿⡇⠀⢸⡇⡼⠉⠀⠙⣦⠘⣇⠀⠀⠀⠀⠀⠙⡦⠤⣄⣀⡀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣀⣈⣙⣚⣉⣸⣀⡀⣸⠀⠹⢤⣠⡴⠋⠀⣻⠀⠀⠀⠀⠀⣸⠃⠀⠀⢨⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠓⠒⠶⠶⠶⠤⢾⠀⠀⠀⠀⢀⡟⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠀⠐⠒⠒⠒⠀⠠⠤⠤⠤⢄⠀⡰⠀⠀⠀⠀⣸⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⣶⠰⢰⣶⠒⠒⠓⠒⠒⠒⠒⠲⠖⠒⠶⠲⠶⠶⠶⠿⠶⠶⣶⢠⡀⢀⡇⠀⠀⠀⡼
⠀⠀⠀⠀⠀⣠⠏⠀⠀⢠⠏⠃⢸⣿⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⣿⠰⠉⡽⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢠⠏⠀⠀⠀⡾⠤⠆⠈⣿⡀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢸⡟⠸⢶⠃⠀⣀⡴⠃
⠀⠀⠀⠀⡞⠀⠀⠀⢰⠇⠀⠀⠀⢹⣧⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⣿⠁⢰⠉⠛⠋⠉
⠀⠀⠀⢰⠇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠹⣧⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⡾⠀⢠⡾⠃⠀⢸
⠀⠀⠀⡾⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠙⠦⣄⣀⣀⣀⡤⠞⣁⡴⠛⠀⠀⠀⢸
⢀⡴⠚⠷⢤⣄⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣭⣭⡤⠶⠞⠋⠀⠀⠀⠀⠀⢸
⢏⠀⠀⠀⠀⠀⠈⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡄⢠⡦⢤⣄⡀⢀⡗⠀⠀⢠⠀⣀⢠⣄⣤⣤⣤⣀⣤⣀⣀⣀⣀⣀⣠⣄⣠⣤⣤⣄⣀⣠
⠈⣇⠘⡇⢀⡞⢁⡾⠀⠀⠀⠀⠀⠁⠀⠙⢦⡀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⡴⠋⠀⠁
⠀⠈⠙⠛⠚⠦⠾⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢦⡀⠀⣿⣄⣀⣀⣼⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⡤⠤⠤⢤⡉⠙⡇⠀⠈⠀⢸⠉⣽⠧⠤⠤⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠞⠁⠀⠀⠀⠛⢶⣇⠀⠀⠀⣸⠞⠃⠀⠀⠀⠘⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠶⠚⠋⠙⠒⠦⣄⣿⠀⠀⠀⣇⡤⠖⠛⠉⠓⠲⢤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣀⣀⣀⣀⣀⣀⣀⣨⡇⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠒⢂⠀⠀⠀⠠⡁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢸⣶⣵⡄⠀⠀⠀⠀⢉⠂⢠⢦⡤⣠⣴⡀⣀⣀⣆⣬⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠏⢿⠃⠀⠀⠎⢸⠶⠀⠁⣀⣴⠆⠲⣾⠩⡀⠡⠀⠀⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠂⢘⣂⡴⠖⣀⡈⣶⢀⡆⠚⣯⠷⠶⣿⠀⣧⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣠⣯⣉⡙⠾⣏⣼⣧⠎⠱⠊⢿⣤⣼⡿⠀⠘⣧⣷⡄⢶⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡍⡷⠤⡙⢲⣿⠿⠋⠀⠀⠀⠈⠻⢿⠃⠀⠀⠈⢻⡏⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠉⡇⢻⠀⢬⣽⣿⡄⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⢀⡿⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣠⣅⠈⠁⣾⢿⣿⣧⡀⣨⣧⣤⣤⣤⣁⣀⠀⢀⣾⡝⢿
⠀⠀⠀⠀⠀⠀⠀⢉⢁⣿⡿⠉⠹⠀⠋⠀⣈⣿⡙⠛⠩⠉⠭⠬⣽⣿⣿⣿⡿⢅
⠀⠀⠀⠀⠀⠀⠐⣯⢽⠈⡏⠀⠀⠀⢰⣞⣱⠉⡙⠁⠀⠀⠂⠛⠛⢟⠻⣧⡈⠢⣅⢠
⠀⠀⠀⠀⠀⠀⠀⣥⠞⠀⠈⠀⠈⠀⢸⠀⠉⠙⠀⢰⠀⠀⠀⢌⣀⡙⠉⣿⡅⠢⠈⢹⣖⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠁⠀⠀⢸⠀⠀⠀⢂⣸⣷⠀⢠⣾⣧⠁⠀⠘⣧⠀⠀⡼⣻⢋⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡇⠀⣴⣶⣾⠀⠀⠀⢸⣿⠟⠐⢾⣿⣧⠄⠀⠀⢿⣷⣾⣿⠹⢹⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣰⢏⣴⣿⣿⡿⣏⠀⠆⠀⢸⡟⠀⠀⠈⢿⣷⡀⠀⠀⠈⢻⡿⢧⠜⠟
⠀⠀⠀⠀⠀⠀⠈⠱⣯⣾⠛⣿⠛⠀⣿⠀⠀⠀⢺⣇⠀⠀⠀⢀⣇⣿⡈⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢀⣿⠀⠀⢸⠛⠲⣶⣿⣿⠀⠀⣠⢿⠟⡛⣿⠙⠛⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠀⢸⡏⣱⣿⣿⡟⠀⣶⣿⣊⡔⠁⣮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⣠⣴⣇⠙⢹⣾⣿⣶⣯⣝⣽⣅⣴⣶⡒⠒⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣴⣾⣿⣿⣿⡿⣤⡯⠼⣿⣿⣿⡆⠀⠈⠁⠹⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⢤⣖⣹⣿⣿⣿⣿⣿⣿⡿⠁⠋⠀⠀⠸⡟⣿⣧⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣤⣄⣀⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣠⣴⣾⣿⣿⣿⡿⠧⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⢿⣿⣿⣿⣿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣾⣭⣛⡋⠂⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠉⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡿⠿⣿⣿⠀⣴⡾⣶⣄⣸⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⣠⣿⣿⠈⣿⣤⣼⡿⣿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠆⠈⠉⠉⢠⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣻⢿⠿⣀⡀⠠⠤⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣾⡧⠠⣤⣖⣲⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⢰⣾⣿⣷⣿⣿⣿⣷⣦⣿⣿⣿⣾⡟⠑⠆⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣤⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠊⠀⢰⣠⣤⣿⡎⠇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⠇⠉⠉⠙⠛⠏⠉⠀⠀⠀⠀⢠⣼⣿⣿⣿⣷⢸⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣧⡶⣖⣶⣶⣶⣲⣖⢦⠀⠀⠀⠙⣿⣿⣿⣿⢸⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡿⡘⢰⣲⠀⠀⣿⣿⣿⡟⢸⡇⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠉⠀⢡⣿⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡿⣿⣿⠏⣰⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠐⠁⠀⠀⢂⣾⣿⣄⠹⣿⣿⢏⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠂⣹⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢷⡀⢀⣾⣿⣿⣏⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠉⠻⠟⠉⠀⠀⠀⠀⠀⠀⠈⠛⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠋⡴⡶⡄⢠⢶⢦⡙⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⣏⠀⠣⠷⠃⠘⠮⠞⠁⣿⣨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠭⠟⠯⠯⠭⠭⠭⠽⠿⠭⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡇⢠⠒⡖⢲⠒⠖⡖⢲⠒⡆⢸⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢎⢣⡱⡇⠈⠓⠓⠚⣚⣓⠛⠚⠚⠁⢸⢞⡜⡱⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⠤⢌⠙⣦⡷⠋⡇⠀⢀⠀⠀⠑⠊⠀⠀⡀⠀⢸⠙⢾⣴⠋⡡⠤⡭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⡍⠀⠀⡇⠰⡊⣳⠒⡎⢱⠖⣞⢑⠆⢸⠀⠀⢩⠐⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡇⠀⠉⠉⠀⠉⠉⠀⠉⠉⠀⢸⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⣒⣒⣲⠒⠒⣖⢒⣒⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⣺⠀⠀⣷⠒⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠖⢿⠀⠀⡿⠰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠥⢽⠀⠀⡯⠬⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠭⠭⠽⢤⡤⠯⠭⠭⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⢄⠼⠧⠤⠤⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⣀⣲⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢤⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣠⠖⠉⠄⠀⠀⠌⠉⠲⣄⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⣠⣒⣉⣥⠤⢤⣽⣶⣶⡀⡀
⠀⠀⠀⢰⡗⠋⠉⠉⠉⠉⠉⠉⠛⢮⡆⠀⠀⠀⢠⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢢⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢰⣻⡇⢪⣿⡆⠀⠀⣼⣿⠂⢸⣷⠄⠀⠀⢸⣸⣿⠀⢿⡽⠀⠀⠀⠸⣯⠇⢸⣿⢸⠀⠀⠀⡎⡇⢰⣿⣳⡄⠀⠀⣾⣟⣲⢸⠉⡂
⠀⠀⠀⠹⣧⣀⣀⣀⣀⣀⣀⣀⣠⣾⠋⠀⠀⠀⠀⠙⠛⣦⣀⡀⠠⢤⠤⠀⣀⣠⡾⠛⠁⠀⠀⠀⢿⣧⠈⠛⠛⠁⠈⢀⠙⠛⠋⢸⣶⠇
⠀⠀⠀⠀⠈⠢⡤⠤⠴⠶⠶⢦⠐⠁⠀⠀⠀⠀⠀⠀⠀⡹⣶⠬⠭⠭⠭⠭⢴⡾⡁⠀⠀⠀⠀⠀⠀⠈⠦⣀⡀⠀⠀⠀⢀⣀⡤⠋
⠀⠀⠀⢀⣴⣿⠄⠀⣶⣦⠀⠨⣷⢦⡀⠀⠀⠀⠀⢀⠞⣾⣿⢠⡖⠒⠒⣦⢸⣷⡞⢆⠀⠀⠀⠀⠀⢀⣴⣶⠛⠛⠛⠋⠉⢳⣶⢄
⠀⠀⠀⡮⣤⠟⡀⠀⠉⠉⠀⢸⠫⣤⢧⠀⠀⠀⠀⢜⣴⡟⢇⠀⣛⠛⠛⠃⢀⡟⣷⡜⡄⠀⠀⠀⠀⢯⣀⢿⠀⠀⠀⠀⠀⢸⢧⣤⡧⡀
⠀⠀⠸⣀⠾⠀⢹⣶⠦⢴⣶⠋⠀⠧⣠⠇⠀⠀⠀⠘⠟⠃⢸⡛⠛⡶⣖⠛⣻⠀⠛⠟⠀⠀⠀⠀⢻⣿⠏⢀⣿⣄⡀⣀⣤⢏⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⣸⠛⡇⢼⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣛⡇⣟⣛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣠⣤⠏⢣⣬⣨⡄
⠀⠀⠀⠀⠀⠮⠤⠵⠇⠲⠏⠬⠧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠤⠽⠟⠿⠥⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠥⠼⠀⠸⠷⠥⠬⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⢀⣓⣎⣘⣀⣀⣃⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣵⢹⡲⡖⡾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣿⣷⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡟⠀⠀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠂⠒⠢⣄⠀⠀⢻⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡟⠁⢸⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣇⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⢠⠊⢲⠀⠀⠀⠀⠀⠀⡖⠑⡄⠀⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠳⣤⣼⡄⠘⡄⠀⠀⠀⠀⠢⢄⣀⣀⡠⠔⠂⠀⠀⠀⢠⠇⢀⣧⣤⠞⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠉⠃⠂⠀⠀⠀⠀⠐⠚⠋⠁⣀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⢛⡛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⠉⠙⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢐⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠄⠀⠀⠀⠈⢠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⢄⠀⠀⠀⠀⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠡⠀⠋⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣠⣄⣀⣤⣀⣠⣄⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠕⠀⠀⡠⠤⠤⠄⠒⠐⠠⢤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢲⣶⣤⠒⣶⣢⡄⡆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⡏⠁⠀⠈⠓⠀⠀⠀⠃⢆⡞⠚⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⡇⢰⠐⠒⠒⠒⠂⢤⢀⢸⡇⠀⠀⣿
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⠈⠢⢄⣀⣀⡠⠏⢸⠀⠳⠤⡞⠁
⠀⡴⢢⢫⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⠟⠛⠒⠒⠲⠶⣶⠾⠤⢶⠒⠃
⡴⢜⢦⣱⡵⠛⢣⠤⣤⡠⡤⠤⠤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢀⢳⣀⣀⡀
⠈⢪⡗⠁⠀⠀⢸⠊⠁⡇⣇⠀⢠⡇⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡇⣏⠀⠀⠀⢸⠛
⠀⠀⠓⢄⡀⠀⣸⠀⠀⠇⡏⠉⢹⠀⠀⠀⠠⠀⠀⠁⢸⢘⠀⠀⢿⠈⠉⢹⠉⢹⢸
⠀⠀⠀⠀⠈⠉⠡⡳⡀⠀⠇⠀⡿⠀⠀⠀⠀⠀⠀⠓⠃⠊⠀⠀⣼⠀⠀⢸⠀⡼⢸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢾⢳⡀⠀⡇⠀⠀⠀⢀⠀⠤⠀⠨⣈⡄⠀⡇⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⢩⢱⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⡗⣾⠤⠤⠤⡔⠊⠁⠀⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⠀⠉⠉⠉⠉⠉⠙⠢⣄⣀⣿⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣱⡖⠒⠤⠤⠤⣤⣄⣀⣀⣀⣸⣃⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠏⡝⠉⠓⠚⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠦⣀⠀⠀⠀⠀⠀⡜⠀⢦⡤⠤⠤⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⡆⠀⠀⠀⠈⠉⠒⠒⠒⠛⠂⠉⠁⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⣧
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠺⣀⠓⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡏
⠀⠀⠀⠀⠀⠀⢸⠉⠁⠒⠤⡤⠒⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣇⡀
⠀⠀⠀⠀⠀⢴⡋⠑⠠⢄⡀⡇⡠⠔⠊⢉⠦⠀⠀⠀⠀⠀⢰⠚⠭⢄⣀⠲⠃⣈⠥⠶
⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣈⣻⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠈⢙⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⣄⡉⠒⠢⢼⡤⠒⠊⣁⡼⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣲⣤⣧⣖⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⡦⠀⠀⠀⣎⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠵⢏⠀⠀⠀⠀⢈⡿⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⢉⣩⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣭⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⠴⠚⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠤⣿⢀⣤⣴⣀⣀⠀⠀⢀⣀⣠⣤⡉⢹⡇⠀⠀⡴⠒⡺⢳⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠈⣹⣿⣷⣿⠀⠀⢠⣿⣿⣭⡟⢸⡇⠀⠀⡇⠀⡇⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠤⡟⠀⠨⣉⠁⠉⠀⠀⠘⠉⠉⠙⠃⠈⡇⠀⠀⠳⠤⠷⣼⠒⠁
⠀⣖⣒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠺⠤⣤⣤⣀⣀⣉⣽⡁⠀⠀⡇⠀⠀⢀⣠⠶⠋
⣠⣄⣈⣧⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡻⠓⠒⠶⠶⠦⠤⣤⣤⣤⣀⣀⣧⣤⠞⠉⠈⠳⣄
⠳⣌⣩⠼⣏⣹⢒⣆⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢳⣼⠈⠻⠶⢤⣤⣤⡤⠴⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣴⠛⠉⠉⠙⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⣄⣀⣠⣤⣼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢷⣤⠴⣤⣤⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠂⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠄⢹⠀⠀⠀⢀⣸⡏⢻⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠐⠒⠒⠒⠲⠶⠤⠤⠤⠭⣤⣌⣈⣐⣂⣃⣸⡤⠴⢊⡥⠒⣇⢀⣟⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠈⠉⠀⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⣤⣄⣼⠤⠚⠁⠀⠀⣿⢯⣉⣉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡿⢯⣬⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣀⡴⣾⢰⠁⠈⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡿⠛⠛⣿⠗⠒⠶⠶⠶⠶⢤⣤⣤⣤⣤⣿⡴⠞⠉⠀⠹⡼⠂⢰⣃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢿⡀⠀⠀⠀⠀⠀⣾⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠙⠓⣻⢷⡀⠀⠀⠀⢀⡽⣄⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣛⣁⣀⣀⣠⠞⠁⣘⡇⢀⡤⠖⠋⠀⠈⢉⡿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠭⠭⠷⠖⠋⠁⣾⣛⠲⠦⠤⢤⣤⡞⢀⡼⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣷⣶⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠛⠛⠛⠲⠶⠶⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣠⠖⠒⠒⠶⠶⢤⣤⣄⣀⡀⠀⠈⠉⠉⠛⠛⠶⢦⣄
⠀⠀⢀⣠⣤⠴⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠒⠒⠦⢤⣀⠀⠈⢿⢦⣄
⣴⠛⢯⠁⢇⢀⣷⠞⢹⡇⠀⠀⠀⠀⠀⢀⡟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢸⠀⠙⠳⣄
⠈⠳⡄⢧⡼⢾⣃⣴⠋⠀⠀⠀⠀⠀⠀⣸⠃⠀⡏⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⡾⠀⡇⠀⠀⠀⠘⢷
⠀⠀⢹⡞⣛⣉⣹⣧⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠁⠀⠀⠀⠀⡦⣤⣀⣀⣀⠀⠙⠁⠀⠀⠀⠀⠀⢠⡇⠸⠀⠀⠀⠀⠀⣸
⠀⠀⣸⠉⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⠇⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠀⠀⠀⠀⣼⢀⠇⠀⠀⠀⠀⠀⡟
⠀⠀⣿⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣾⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢸⠖⠛⠀⠀⠘⣇⠀⠀⠀⠀⢰⡏⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⣾
⠀⠀⢸⡇⠀⢀⡀⠀⢹⡄⠀⠀⠀⠸⡇⠀⠙⠲⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⠇⠀⠀⠀⠀⢰⡇
⠀⠀⠀⢿⠉⠁⠀⠀⠀⠻⡄⠀⠀⠀⠙⠶⣤⣀⣀⡀⠀⠈⠉⠉⠛⠒⠶⠦⣤⣄⣀⣀⠀⢀⡼⠁⣼⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠘⣧⠀⣠⠞⠀⠀⠙⢦⣀⠀⠀⠀⣼⠋⠉⠉⠛⠓⠶⠦⣤⣤⣀⣀⡀⠀⠉⠉⠉⠉⠀⣰⣃⣤⡴⠖⠛⠉
⠀⠀⠀⠀⠘⢯⠀⠀⠀⣀⠀⠀⠉⠓⠶⣿⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠲⠒⠚⠉⠁⠹⡆
⠀⠀⠀⠀⠀⠀⠳⣤⡴⠁⠀⠀⡰⠀⠀⡿⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⣀⣠⡿⠴⢶⠶⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣴⡁⠀⣰⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠉⢹⡀⠀⠀⢸⠀⠀⢀⡎⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡷⠊⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠈⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⢠⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⣶⠛⠋⢹⡋⠛⠛⠷⢦⣄⠀⠀⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢰⡏⠁⠀⠈⠉⠉⠙⠛⠒⠒⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⠄⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣠⣏⣙⣓⢚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠠⡶⠚⢋⣩⡼⣭⡽⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠛⠚⠋⣸⠁⡎⡃⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⣧⣴⡤⠶⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢼⣷⣄⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸
⢴⠦⣤⡀⠀⣀⣀⡴⠒⢾⠁⠀⠸⡀⠉⠛⠒⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠦⠶⠶⠶⡶⠚⠋
⠘⢧⠀⠉⠻⡉⠙⢆⠀⠀⢳⠀⠀⠑⠀⠀⠀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠹⠶⢶⠶⢦
⠀⠈⢷⠀⠀⠹⡀⠘⠆⠀⠀⡀⠀⣠⠶⠤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡤⠊⠁⠀⣠⠂⠀⡀⠀⢨⠷⣦
⠀⠀⠈⢧⡀⢀⣧⣀⣀⣠⠞⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡋⠀⠀⠀⠰⠁⠀⢠⠁⡰⠁⠀⣿
⠀⠀⠀⠈⠷⠞⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣀⣠⣇⣀⣀⣾⣤⡇⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠁⢸⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢀⡠⠤⠄⡀⠀⠀⠀⠀⢀⠠⠤⢄⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢠⠊⡰⢒⠢⠈⡆⠀⠀⢠⠁⠔⡐⢆⠑⡄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⠀⠀⠀⠐⢄⠑⠬⠔⢠⠃⠀⠀⠘⡀⠢⠥⠊⡠⠀⠀⠀⠀⡗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠑⠂⠒⠁⠀⠀⠀⠀⠈⠐⠐⠊⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⡤⠀⠀⢤⠀⠀⠀⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢸⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⡏⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡀⢰⠀⡀⢘⠒⠖⠒⠒⠒⠒⠒⡖⠒⠒⢲⠒⠒⠒⠒⠒⠒⠒⡃⢀⠀⡆⢀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⣀⡅⠀⠀⠸⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢡⠀⠀⡌⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢀⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣁⣀⣀⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣠⠶⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⢶⠄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⡤⠒⠢⣤⣄⠀⠀⠀⠀⣠⡔⠒⢠⣤⣄⠀⠀⠘⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠋⠃⠀⣀⠉⢀⣤⣤⠈⢻⣷⠀⠀⠸⠃⢠⣤⡄⠈⣿⣇⠀⠐⢰⡿⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⣦⠀⠙⣦⡀⠛⠋⢀⣾⡿⠀⠀⢸⣆⠈⠛⠃⣠⣿⠇⠀⠀⠙⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⡀⠀⠀⠈⠛⠀⠘⠛⠋⠀⠀⠀⠀⠙⠛⠀⠚⠛⠁⠀⠀⠘⢴⣇⠀⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠛⢶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣤⣤⣶⣦⣤⣴⣦⠄⣴⣦⠀⣴⡄⠀⠀⠀⢀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⢀⣿⣿⠀⣿⣿⡄⣿⡅⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠉⠀⠉⠉⠀⠉⠉⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⠀⠈⣿⣏⠀⢈⡁⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠛⣦⣤⠘⠛⠛⣿⣀⣀⣀⣿⡟⠋⢻⣇⣀⣀⣸⣿⡛⠛⠛⠛⠛⠛⣿⠟⠛⠛⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⣴⣾⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⣿⠁⢀⣴⣦⡄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢿⣿⣿⠀⣴⠂⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠉⠀⠐⣶⠀⣿⠀⠸⣿⣿⡿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣹⣦⣀⣈⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣻⠷⣀⣀⣀⣀⣴⡟⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢇⡸⠃⠀⢻⠋⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠘⡟⠻⡟⠀⠸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢇⡸⠃⠀⣴⠏⠀⣿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠇⠀⢹⡆⠀⠸⠇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠸⢇⡘⠃⠀⢰⡇⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⡆⠀⠀⠸⡆⠀⠸⢇⡸⠇
⠀⠀⠀⠀⠀⠀⡸⢇⡘⠃⠀⣸⠇⠀⠀⠀⣿⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⣿⠀⠀⡿⠀⠀⠀⠸⡆⠀⠘⢇⡸⠇
⠀⠀⠀⠀⠀⡘⢃⡜⠃⠀⢸⠇⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠸⡆⠀⠘⢇⡸⢇
⠀⠀⠀⠀⣸⢷⣌⠃⠀⠸⠇⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠸⡆⠀⠘⣧⣾⣆
⠀⠀⣠⠞⠋⠀⠉⠛⢿⡇⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡟⠛⠁⠀⠙⠳⣄
⠀⠀⠛⠀⣴⣿⣷⡄⠘⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠁⢠⣾⣿⣦⠀⠻
⠀⠀⣷⠀⠙⠿⠟⠁⢠⡎⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡄⠘⠿⠿⠋⠀⣶
⠀⠀⠘⣷⠦⠀⢠⣶⠊⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠱⣦⡄⠀⠴⣿⠁
⠀⠀⠀⣧⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣼
⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠛⠓⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠚⠟⠀⣿⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠒⢲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣟⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢹
⠀⠀⢀⣏⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣄⣀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣽⡀
⠀⠀⢟⣋⣉⣉⣉⣉⣉⣱⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠁⠉⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣉⣉⣉⣉⣉⣙⣿
⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⣶⣶⣄⠀⠙⣄⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣰⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣠⣶⠀⠘⣿⣿⣷⠦⣀⣀⠀⠀⠀⠀⣀⣼⡇⠀⢿⣿⣿⠟⠀⢀⣿⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠟⠿⢿⣿⣿⣿
⣿⣿⡟⠉⠀⠀⠀⠀⠀⠙⣿⣿⡇⠀⠀⠀⣴⣟⣩⣶⣄⠀⠉⠀⣀⡙⠛⠀⠐⠂⠘⠛⠉⠳⣆⡀⠈⠁⢀⣤⣾⣽⣿⡀⠀⠀⠀⢸⣿⣿⠋⠁⠀⠀⠀⠀⠉⢻⣿
⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⠀⣰⣿⠿⠋⢉⣹⣿⠿⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣿⡿⣏⠉⠛⢿⣿⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢃⣤⡾⠋⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣍⠻⣧⣄⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠟⠋⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡌⠛⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⣀⣴⠟⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡙⢷⣤⠈⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣛⢡⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣉⣻⣧⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠉⠙⠛⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠙⠉⠹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣶⣶⡄⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣠⣶⣦⡀⠈⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⢿⣿⣿⡟⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⣿⣿⠇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠉⠉⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⠈⠉⠁⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣱⠿⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⣿⣿⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣶⠶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⣶⡶⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠘⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠋⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢷⣶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣷⣀⣉⣉⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣉⣀⣠⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⡟⠛⠛⠉⠛⠛⠛⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠉⠛⠛⢛⡃
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣷⣿⣷⣶⣿⣷⣾⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣾⣿⣾⣿⣶⣶⣿⣷⣶
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⠿⠟⠙⠿⠟⠿⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣯⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣟⠛⣷⠀⠀⠄⠁⠀⠈⠀⠀⡀⠂⠀⠀⠀⠀⠀⠔⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣼⣻⣿⡋⠄⠀⠀⠠⢀⠠⢀⠀⠄⡁⠂⠈⠀⠀⢀⠀⠐⠊⠁⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠂⠐⠈⣉⣉⠛⠉⢁⠀⠀⠨⠕⠢⠀⠤⠔⠠⠄⠤⠆⠉⠒⠁⠀⠀⠀⣰⢶⠀⠀⠐⠀⠈⠑⠂⠤⢀⡀⣤⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⢞⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢷⡻⡧⣬⣤⣧⠌⣁⠂⢄⠨⢘⣿⡔⠟⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⡋⣰⣤⣀⣶⣿⣿⠴⠀⠉⠀⠀⣏⣻⣧⣤⣙⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠃⠐⣰⢃⣯⣠⣼⠛⠁⠀⠀⠀⠀⠀⢰⡯⣷⡄⠍⡁⠸⠅
⠀⠀⠀⠀⣠⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣴⠛⠓⣘⠛⣿⣡⣬⡿⠋⠀⠀⠀⠀⠀⠀⣠⠀⠀⠛⡳⢷⡆⠆
⠀⠀⠀⠀⠘⡀⠀⠀⠐⣚⠃⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⠿⡡⢠⣨⡁⣾⣷⣶⠏⠀⠀⠀⠠⠄⡐⠲⡌⢿⣧⠀⠀⣯⡈⡇
⠀⠀⠀⠀⠀⠈⣦⠀⠥⠏⠀⢀⡪⠁⡔⠊⠛⣛⠛⣻⢻⡓⠄⢲⣓⣾⡿⠷⠋⠉⠁⠀⠀⢀⠐⠁⠀⠀⠠⠗⢈⣾⠀⠀⠶⠁⠁
⠀⠀⠀⠀⠀⠀⠈⢔⣻⡁⠀⠰⠁⣾⠂⠀⠀⠿⢷⢶⡾⣡⣼⣾⣏⠒⡀⠀⠀⠐⠀⠀⠀⠀⠈⠐⠀⢨⠔⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠰⠀⣷⢀⡀⠀⠀⢀⠈⣿⣿⢟⠛⡉⣴⣧⣤⣀⡐⢀⠀⠀⠀⠀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠉⠁⠀⠀⣰⣿⠁⠴⠇⢙⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣰⡟⠀⡁⠀⣈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠓⠛⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⣿⠤⢚⠦⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠄⠐⢶⣖⣴⣌⣷⡀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣑⢂⠠⢈⣿⡀⠀⠀⠀⠀⠀⣀⣤⣾⣸⡽⣧⡀⠀⢨⡿⡿⢿⣿⡟⠀⠆⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡎⣴⢧⣺⢿⣦⣤⣤⡴⠿⠋⠁⣡⣿⣧⢸⡇⠀⠈⠀⠀⠀⢹⣿⠶⠀⢠⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠲⠶⣤⣀⠻⣿⣶⣌⡿⣶⣿⡿⣥⢀⣀⢰⣟⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡈⣬⡟⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣦⣴⣴⣶⣧⣡⠀⣉⡛⢮⡝⠻⢿⣿⣿⣷⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⢀⣠⠠⡴⣿⣤⡔⣷⣿⣥⣤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡼⣿⠟⠁⠀⠈⠙⠻⣿⣿⠆⠈⡓⢦⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡟⢻⣿⣿⡾⣿⠻⣿⣤⠶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠻⡀⠀⠀⠀⠀⠀⠈⠛⢧⣲⣍⡂⢈⠑⠦⢤⣀⣀⣀⣀⣄⣠⣤⡴⢘⣛⣤⣤⣰⢶⣋⣿⠟⠙⠷⣦⣤⣴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⢳⣄⡀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⣀⣤⣤⣄⣨⠿⣿⣡⣿⢔⡠⠿⣿⢟⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⢻⡶⢤⣀⡀⠀⠀⠀⠀⣀⠉⠛⠿⠿⣷⣿⣽⣶⣷⣾⡶⠀⠀⠈⠿⣟⡁⠀⠀⠀⠆⠀⠓⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⣾⣿⣿⣿⣿⣿⣷⣿⠿⠿⠿⢿⣶⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣍⣾⢽⣿⣿⣛⠿⡿⠿⢻⣯⣡⣾⣟⣼⣭⡶⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣿⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣹⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡹⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣩⣿⣿⡟⠛⣩⠉⣹⢫⡮⢹⣿⣿⣿⣿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣷⡝⢎⣹⣶⣿⣿⣿⣿⣟⣻⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣮⣿⣿⣿⣟⣭⡉⣶⠏⠉⣻⡿⣿⣽⣿⡶⣚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⣿⣿⣿⣹⣫⣿⣸⡕⢎⣶⣿⣴⣿⣿⣿⣿⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡿⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣮⣿⣯⣾⢿⣿⣿⣿⣷⣤⣴⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣻⣿⣿⣿⣿⣿⣥⣷⣾⣿⣿⣿⣿⣿⣿⡟⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡿⣿⣿⣿⣿⢹⣿⣿⣿⣏⣾⣿⣿⡿⠵⠁⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣴⣾⣗⣽⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⢿⢿⡿⠿⠛⠿⠿⠦⢿⢿⣿⡟⠭⠿⢿⠷⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⡿⠟⠛⠋⡉⠉⣙⢻⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠋⣲⣿⣿⣿⡄⣬⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣤⣤⣽⣿⣿⣿⣿⣿⣦⣴⣶⣤⣈⣍⣈⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡤⣿⣿⣽⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢺⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣬⣙⢿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⡾⠻⣿⠃⠀⠀⠀⢀⣤⣶⣦⣄⣀⢀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠮⡻⣄⠀⠀⠀⠀⠀⠚⠛⠛⢛⣹⣿⣿⠙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣿⠀⠀⠀⣀⣀⣦⣀⡀⠐⠋⣁⠈⠁⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣿⣿⡿⠀⠀⢸⠀⠀⡀⠘⣿⡿⠛⠁⠀⢞⣡⡀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⢿⣿⣟⢷⣄⠀⠈⣄⠀⠠⠀⣿⠃⠀⠀⠀⠀⠙⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣯⣿⣷⣤⣼⣷⣦⣤⠋⠀⢸⣔⣛⣨⣽⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⠋⠀⠁⠸⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣙⠛⢉⡈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠉⠀⢀⠀⣐⢄⠳⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠲⢦⣬⣙⢿⣷⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣈⢻⣷⣽⣿⡜⣿⣿⣿⣹⣿⡿⣳⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣝⣿⣿⣿⣹⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡹⣿⣿⣷⣝⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡹⣽⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣏⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣟⢾⢳⢻⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣾⣿⣿⣿⣯⡻⣿⣿⣿⣿⣷⣿⣾⣿⡷⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣼⣿⣆⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡀⢸⣿⣿⣿⡇⢠⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⣀⣀⠀⣧⠀⠀⣟⡉⣿⢈⡳⢠⣤⣿⠀⣀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠈⠛⡣⢵⡄⠀⡸⢧⣤⡼⢇⠈⢱⡤⣞⢛⠠⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢡⣾⡇⢀⡄⠸⠻⠇⣌⡀⣼⣷⡌⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⠰⡀⢺⣅⢐⠏⣶⡷⣉⣷⠦⣹⠀⣼⡇⢠⠆⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⠿⣿⠂⠀⠐⡢⠀⢉⣸⡛⠃⠛⠛⢰⣏⡉⠀⡔⠀⠀⠐⣿⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣈⣻⣛⣟⣃⣀⡀⠀⡢⠸⡿⠃⠀⠀⠀⠸⣿⠧⢂⠀⢀⣀⣘⣿⣛⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⠙⣿⠉⢏⢏⢦⠆⠀⠈⠁⠀⠐⠺⢧⣶⠹⠗⠀⢀⠈⠁⠀⢐⡴⡼⡹⠉⡿⢋⡄⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠱⣄⠀⢀⡀⣠⡄⡀⠀⠀⠱⡆⠀⢘⢛⡁⠀⢰⠏⠀⠀⢀⢠⣄⢀⡀⠀⣡⠎⠀⠀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⢧⡀⠣⠈⠺⠇⠀⠀⠂⢉⠠⠀⣿⠀⠄⡉⠀⠂⠀⠸⠇⠁⠌⢀⡾⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡆⠀⠀⡼⠁⢶⡟⡒⠓⠢⠤⠴⣏⣴⠏⠉⠹⣆⣹⠦⠤⠆⠓⢚⢻⡶⠈⢇⠀⠀⢰⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠄⣠⠃⠘⢧⡀⠀⠁⠀⢤⡙⠷⠦⣄⡀⠀⠀⠯⠀⠀⠀⠏⠀⠀⣀⣠⠴⠞⢋⡤⠀⠈⠀⢀⡼⠂⠘⣄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠙⠂⠀⠀⠀⠈⠀⠀⢠⡄⢸⠀⢠⣤⣤⣤⡄⢀⡇⣠⡄⠀⠀⠁⠀⠀⠀⠒⠋⠀⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣿⠀⠸⣿⣿⣿⠇⢸⣿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠖⠀⠀⠀⠲⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠠⢤⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⣰⢰⣶⠀⠀⠀⠀⠒⠂⠀⢤⠀⠀⠐⠂⠀⠀⠀⠐⠀⠈⠉⠡⠈⠉⢠⡴⠶⠿⡀
⠀⠀⠀⠀⠀⠀⣿⣉⠇⠠⠤⠀⠐⠀⠲⠀⠈⠉⠂⠀⡀⠀⠄⠀⠃⠀⠈⠀⠀⠀⢀⣈⣤⡴⠖⠃
⠀⠀⠀⠀⠀⠀⢎⣿⠗⠸⡆⢀⡆⠀⠰⠀⠄⠀⠀⠰⠶⠀⠄⢐⣀⣠⠔⠲⠚⠋⠉⠀⠘⠃⠀⡀
⠀⠀⠠⠀⠀⠀⠈⠙⣾⣷⡀⡈⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⢛⡉⠹⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⡄⣲⡀⠄⢐⣒⡉⠙⠛⣏⠸⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠀⣳⡋⠉
⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠

⠀⠀⠀⠀⢀⡀⠐⠀⠐⠀⣤⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣀⠀⠀⠘⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠛⣝⠷⢚⣳⡀
⠀⠀⠀⠖⠓⠀⠀⠀⠀⠀⠀⠃⠀⠀⠓⠡⢿⣧⡀⢀⠀⣰⠿⠿⠾⡟⠁⠀⠄⠀⠀⠀⠀⠋⠿⡿⠁
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢶⠲⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠙⠀⠀⠀⠀⢸⠍⢠⠀⠀⠄⠀⠀⠸⡟⠘⡛⣷⠖
⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠈⠐⠀⠆⠁⠰⠀⠀⠈⠁⠘⢦⣄⣿⠀⠀⠀⠶⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠐⠿⠋
⠀⠀⡀⠀⠀⠀⠚⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢀⢠⠆⠐⠀⠀⡀
⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣼⡛⡷⣦⣀⠴⠂
⠀⠈⢀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⣿⣷⣷⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣼⣥⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣄⠀⠀⠀⡀⠀⡠⠇
⠀⠂⠀⠀⠈⠳⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠾⠷⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⡄⠁⠀⠀⠀⢀⠀⠀⠐⢀⣹⣿⣿⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡀⢀⠀⡆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢬⡙⡆⠀⠀⡉⠻⠏⠁⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⢀⠀⢲⢃⠘
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠁⠀⠀⠤⠀⠀⠀⠀⠉⣿⣿⣿⠿⢀⠈⠀⠈⠆⠀⠘⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⣤⠐⠀⠀⢠⡶⠂⠀⠀⠀⠀⠀⠀⢀⣐⠂⠀⠀⠀⠐⠀⢀⠁⣰⡽
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⠔⢚⠟⠁⠀⠀⠀⢠⣀⣤⠂⠀⠀⠃⠀⠀⠀⠀⠆⠐⠂⠀⠈⠀⢠⠞⠃
⠀⠀⠀⠀⢀⡞⠀⠀⠀⣴⣋⠤⠚⠁⠀⠀⠀⠀⠀⠘⡿⠿⠄⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⡎⢀⣠⣴⡴⠿⠓⠒⠒⠲⢤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⢩⢿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣿⡟⢠⣿⣧⠸⣷⠘⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠋⣴⣿⡟⢠⣿⣿⣿⡆⠹⣧⠹⣿⣿
⠀⠘⠟⣻⣿⣿⣿⣿⠿⠋⢙⢩⣝⣿⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⣼⣿⡟⢀⣾⣿⣿⣿⣿⡀⢻⣧⣿⣿
⠀⢰⣿⣿⣿⣿⣿⠁⠀⠐⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢀⣾⣿⣛⣿⣿⣿⣷⠀⢻⣿⣿
⠀⢠⣶⣿⣿⣿⣿⣄⣈⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠲⣤⣾⣷⠀⠀⠀⠈⠋⢹⣿⣿⡟⢀⣾⣟⣻⣿⡿⢿⣿⣿⣇⠈⣿⣿
⠀⠸⣿⣿⣿⣿⣿⠿⣍⣷⢿⠫⢿⣿⣿⡅⠀⠀⠀⠀⢀⠴⠚⠻⠋⢹⣌⣧⠀⠀⠀⠀⢸⣿⣿⣦⡀⢻⣤⢹⣿⡁⢸⠍⣿⡿⠀⣿⣿
⠀⢼⣿⣿⣿⣿⢫⣀⢹⢟⢹⡐⣿⡿⣻⡁⠀⠀⠀⠀⠁⠀⠀⣘⢷⣿⡇⢹⠇⠀⠀⠀⠙⣿⡟⣿⣷⡾⠋⢸⣿⡏⣿⡆⠉⢠⣾⣿⣿
⠀⢸⣿⣿⠛⠒⣿⣿⡿⣸⡞⠃⠈⣼⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣖⣿⣿⠉⠃⠀⠀⠀⠀⠹⣧⠈⠻⣷⢰⣿⣿⠃⣿⡷⠀⠘⠛⠿⣿
⠀⠈⠻⢿⣤⣿⣿⡾⢧⣿⣿⣶⣷⣿⠇⠀⠀⠀⠀⠀⠙⡿⢿⠿⣼⢿⣿⠀⠀⠠⣶⣴⠆⠀⠻⢻⣦⡈⣿⠏⣼⠀⡿⣯⢀⣄⣠⣾⡿
⠀⠀⠀⠀⠻⣿⣿⣿⣛⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣾⣁⣀⠀⠀⠀⠀⡀⠀⠀⠻⠿⣿⣧⡀⢹⠀⣧⢹⣾⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠒⠉⠛⢻⡁⠛⠁⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⣼⣿⣿⣿⣿⣿⢠⡤⠀⠊⠀⠀⠀⠀⠀⠐⠿⠿⢿⣷⣿⣶⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢸⠟⠉⣀⣤⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣿⣿⢿⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⣿⣿⠿⣿⣷⡿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣿⣿⣿⣿⣟⡉⠙⢛⠭⣈⢻⣷⢈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⣿⣿⣿⡿⠉⠀⠀⠉⢹⣿⢯⠟⡽⠛⠉⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢺⣿⣀⠀⣤⣖⣂⠀⠀⣿⣼⣼⠁⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠘⠛⠋⠉⢠⣿⣿⣿⣷⣿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⣠⠶⠀⠀⠀⠸⠿⢻⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⢆⠀⠀⠀⠀⢀⣼⢿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠀⣀⣤⣶⣿⠃⢘⡿⠟⠋⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⡿⠒⢉⡤⢖⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠛⠛⠁⢀⢰⣃⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣎⢿⠀⣠⠔⢁⣾⣽⣿⣿⣿⣏⣩⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⡏⣿⠧⠤⣽⣿⣿⣿⡿⣿⠟⠛⠉⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⣻⣿⣿⣿⣴⣪⣾⣿⣿⣿⣿⢋⠀⠁⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢹⡿⠑⣿⣿⡿⢃⠞⣹⣿⠟⣿⠟⠁⠃⠀⠀⠀⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⢀⡿⠋⠀⠔⣰⠟⠃⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⠤⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣀⡀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⢉⣀⣤⡤⠶⠖⠒⠚⠉⠁⠀⠀⠈⢹⣿⣶⣤⣀
⠀⠀⢀⡴⠚⠉⠉⠉⢲⢮⡁⠀⠀⠀⠀⠙⢦⠀⣠⠴⠋⠁⠀⣠⡴⠞⣋⣭⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣦⡀
⠀⢠⠏⠀⠀⠀⠀⠠⣿⠆⠱⡀⠀⠀⠀⠀⡠⡟⠁⠀⢀⡴⠛⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠙⢦⡀
⠀⢸⣤⡀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣠⠎⠀⡇⢀⡴⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⢀⣞⣁⡀⠰⠣⠎⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠉⠍⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠃⠀⠀⠀⠀⣠⠶⣶⣦⡀⢳⡄⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣿⣤⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⣶⣿⣿⣿⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⡄⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠀⢹⡀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹
⠀⠀⢀⡟⠂⠄⡀⠀⢀⠀⠔⠁⠈⠢⡀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣧⠀⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢸⡄
⠀⠀⡼⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠸⣿⠟⢻⣿⣿⣿⣿⠀⠀⠀⡴⠓⣟⠻⣿⣿⣿⣿⣿⣿⠟⢳⡀⠀⢸⡇
⠀⢀⡇⠀⠀⠀⠈⠢⡁⠒⠀⠀⠐⠂⡉⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠋⣠⡀⢻⣿⠿⠃⠀⢀⡞⢁⣄⠘⣆⠀⠈⠉⠉⣠⠋⣠⡄⢳⠀⢸⡇
⠀⢸⣇⣀⣀⣀⣀⣀⣀⣈⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡷⠃⣴⣥⣷⡀⢳⣤⣤⣤⠞⢠⣾⣼⣆⠘⣦⢤⢤⡴⠃⣼⣥⣷⡀⢧⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢹⠿⠿⣿⣿⣿⣿⠟⠁⠀⢀⣼⣿⠏⡜⢁⣼⣿⡟⠀⢳⡈⢿⣿⠋⣠⠋⠀⣼⣿⣆⠹⡄⣼⢁⣾⣿⠿⠓⢳⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠀⠀⠉⠛⣳⠒⡶⠻⠿⠧⠤⠽⠿⢿⡿⣤⣤⣤⣧⠈⢯⠿⠥⠤⠾⠿⠛⠻⣆⠹⡍⠉⠀⠀⠀⠀⠀⢳⡈⢧
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠀⠀⠀⠀⣰⠃⡼⠁⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠈⢧⠈⢧⠀⠀⠀⠀⠀⠀⠹⡄⠹⡄⠀⠀⠀⠀⠀⠀⢧⡈⣇
⠀⠀⠀⠀⢠⡤⠤⠞⢠⠏⠀⠀⠀⠀⣀⣀⣰⠃⡸⠁⠀⠀⠀⢀⣀⣀⡞⢀⠞⠀⠀⠀⠀⠈⢧⠈⢦⣀⣄⡀⠀⠀⠀⠹⡄⠹⣄⣀⣀⠀⠀⠀⠈⢧⠘⠶⠶⢦
⠀⠀⠀⠀⠈⠓⠒⠒⠋⠀⠀⠀⠀⠸⢥⣤⣤⡴⠁⠀⠀⠀⠀⠯⣤⣤⣤⠎⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⠴⠃⠀⠀⠀⠀⠙⠦⠤⠤⠞⠀⠀⠀⠀⠈⠛⠛⠛⠋
⣶⣶⠶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣋⣯⣭⣭⣭⣭⣍⣬⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢦⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠴⠶⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢄⣀⡈⠙⠛⠛⠛⠋⠉⣁⣀⣀⡴⠶⠛⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⠗⠀⢸⠀⠀⠉⠁⠀⠛⠚⠛⠛⠫⣤⣍⢡⡀⠀⠀⢸⡇⠀⡀⢨⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢀⡀⠶⢾⡀⢶⣿⣟⡄⢱⠄⠀⡎⣿⣶⡞⢀⠙⠀⣰⠚⠓⠐⡇⢸⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣤⠛⢛⠀⢶⣽⣿⣧⠴⠂⠀⢷⣈⣿⣣⣼⠶⠀⠙⢳⡶⠺⣷⠸⠁⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⢸⠀⠀⠁⢀⡤⠀⠠⣄⠀⠉⠀⠉⠀⡀⠀⠀⢸⡇⠀⠘⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⣤⠶⣍⠀⠀⢀⣼⠂⠀⠀⣠⡾⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺⣄⠀⢀⣷⠴⠛⠁⣀⣠⠾⠋⠀⡃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠉⠛⠿⣥⠤⠚⠟⠋⠉⠀⠀⣿⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠿⢧⡀⠀⠀⠀⠈⠢⣄⠀⡀⠀⢂⣴⠏⠀⠀⣠⡾⠉⠉⠀⠀⠉⣩⡟⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠋⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠙⠛⠚⠛⠁⣴⣿⠶⠛⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⡴⠷⢦⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢀⡴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⣴⠶⠒⠶⢾⣧
⠀⠀⠀⠀⢰⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢠⡞⠁⠀⠀⠀⠀⠙⣧
⠀⠀⠀⢠⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠘⠷⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠂⠛⡈⢳⠀⠀⣿⠀⠀⣿⡀⠀⠀⠀⣀⣀⣤⣴⡶⠿
⠀⠀⠀⠀⠀⠀⡏⠀⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠦⣤⠀⣄⠀⣀⠁⢸⠀⠀⣿⠀⠀⠉⠛⠛⣿⡟⠛⠉⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⠛⢿⡷⠶⠶⢦⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠉⠀⠀⣿⡀⠀⠀⠀⢀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠳⠶⠶⠶⣶⣤⣤⣤⣀⣀⣽⣇⣠⡶⠞⠋⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⣿⠀⠀⠙⠛⠶⢦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠉⠁⢀⣠⣾⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣠⣾⣦⣤⣤⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠿⠿⠶⢶⣶⣦⣤⣤⣤⣄⣀⣀⠀⠀⣘⣠⣴⠿⠋⠀⣿⠀⠀⠀⣾⡇
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠄⣀⣀⣀⣀⣀⣉⣉⡿⠟⠛⣿⡟⠛⠋⠀⠀⠀⢀⣿⣄⣀⣀⣽⣇
⠀⠀⠀⠘⣧⣀⣀⣀⠀⢀⣾⠀⢀⣀⣠⣤⡤⣾⡇⠀⠀⠀⠤⠀⢉⡉⠉⠉⣉⣉⣤⡴⠛⣿⣇⠀⠀⠀⠀⠀⡏⠁⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠈⣿⠋⠉⣿⠋⠁⠀⣾⢿⡄⠀⠀⢹⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢁⣤⡴⠚⣿⠋⠉⠛⣻⣶⣶⢃⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⡟⠀⠻⣦⠀⠀⠻⢶⣤⣀⣀⠉⠁⠀⠀⠙⠋⣉⣀⣴⠾⢋⣠⡶⠟⠁⠀⡟⠘⣷⠶⠶⠶⣶⠿⠃
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣷⡀⠀⠘⢳⣄⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⣉⣡⡶⠛⠁⠀⠀⠀⣀⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡯⠀⠀⣟⠀⠀⠀⠙⣷⡄⠀⠀⠙⣷⠶⠶⠿⠟⠛⠛⠚⠛⠛⠛⢿⡏⠁⠀⠀⠀⢀⣠⣶⠟⠁⠀⣿⠀⠀⢨⣿
⠀⠀⠀⠀⣰⡷⠶⢶⣏⠀⠀⠀⠀⠈⠻⣦⡀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⢀⣠⣶⣿⡏⠀⠀⠀⠀⣿⠀⠀⢸⡟
⠀⠀⢀⡾⠁⠀⠀⠀⠙⡿⣆⡀⠀⠀⠀⣿⠻⣾⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣧⣾⠟⠋⠀⣿⡇⠀⠀⠀⠐⣿⣤⣤⡾⢷⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⣰⡟⠀⠀⠀⠀⠀⠀⣇⢸⣧⠀⠀⠀⣿⠀⠈⠉⢹⣟⠉⠉⠉⠉⠉⠉⠉⠉⣿⡏⠀⠀⠀⠀⣿⠁⠀⠀⠀⢰⠋⠳⢦⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⣩⡿⣦
⣰⠏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣻⡀⢀⣿⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⢦⣀⠀⠀⠙⠛⠶⣄⣀⡰⠶⠛⠋⠀⣠⣿⣀⣀
⡿⠒⠦⢤⣤⣤⣀⣸⡀⣿⠿⠾⣿⠇⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡽⠳⣤⣀⠀⠀⣼⠋⣀⣀⣤⠞⠻⣯⣤⣩⡽
⠘⢧⡄⠀⠀⠀⠀⠉⣻⣿⣤⠾⠁⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢰⣯⡀⢠⡞⠛⠳⢾⢿⡖⠉⠉⢳⣤⣤⢾⠇⣏
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⣀⣼⣏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢸⡇⢹⡇⠀⠀⠀⠀⠀⠙⣶⡶⢾⡏⠀⢸⠄⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠿⢷⣶⣿⣧⠀⠀⠀⠀⠀⠀⢰⡿⠿⣶⣶⣶⣶⡿⢿⣷⠀⠀⠀⠸⣧⣄⣳⠀⠀⠀⠀⠀⢰⠏⠀⢸⡧⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⠿⣤⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⢷⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠿⠿⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠛⠿⠷⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣿⡁
⠀⠀⠀⠀⠀⣀⣠⠶⠞⠛⠁⠀⠀⠀⠀⢀⣴⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣏⠉⠉⠉⠛⠶⢦⣀⡀
⠀⢀⣤⣾⣛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣤⡀⠀⠀⠀⠉⠙⠛⢦⣄⣀
⠀⣼⠋⠉⠉⠙⠛⠶⠶⠶⣤⣤⡶⠋⠁⢀⣴⡟⠉⡃⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣄⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠈⠉⣷⣦⡀
⠀⢿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠇⢀⣴⠟⠉⣥⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⡷⣦⡀⠀⠉⢻⡶⠶⠚⠛⠛⠛⠁⢹⡃
⠀⠀⣹⠉⠹⠛⡿⢳⡶⣤⣤⣴⡿⢻⠅⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣶⣾⠃⠈⠻⣦⣄⣸⡇⠀⣀⣀⣀⣀⣠⣼⠇
⠀⠀⠹⣄⣼⡄⠁⣼⡀⠠⡄⠀⢀⣼⠷⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠃⣴⠏⠛⠛⠿⠙⡟⢛⠙⡏⢹
⠀⠀⠀⠀⠀⠙⠶⠜⠛⠶⠼⠷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠹⣆⠀⢀⣸⣠⣷⣀⣶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⢰⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠜⠓⠒⠚⡙⣾⠻⡛⡛⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢗⠄⠪⠊⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⢸⠑⢕⠕⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢘⡢⡠⡢⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠑⠂⠀⠒⠁⠀⢨⠆⠔⣀⢨⡄
⠀⠀⢀⡀⢎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠤⣤⢤⣤⣼⠗⠒⠛⠋
⠀⠀⠈⠻⢬⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠈⠉⠀⠉⠉⠉⠉⠉⠩⢹⡛⠉⡭⠭⡍⢽
⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⢬⡇⠀⠠⡶⢈⡶⠮⠶⠀⠀⠀⢸⡦⢞⣣⣶⣘⢼
⠀⠀⠀⠀⠀⠀⠀⠑⢜⢄⠀⠀⠀⢀⠠⠴⢸⣦⠵⢊⡉⡃⠀⠀⣤⠶⠶⠶⠖⡺⠗⣢⢥⡖⠋⠝⡡⡉⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡕⣲⡬⠕⠋⠈⠁⠈⠉⠓⠢⢭⣤⣆⣿⢤⠔⣎⡥⠖⣿⠁⠀⡣⡢⡪⡢⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠠⠛⢁⡴⡁⣿⠀⠠⢆⢔⡺⡢⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠐⠀⠉⠀⠂⢻⠁⢸⢗⢕⢰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠀⠀⠁⠀⢸⢗⢕⠋⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣂⣀⡀⠀⠀⠀⠀⠀⠠⣸⣗⣇⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠏⣽⠉⠋⠉⡿⠉⠉⠉⢈⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠊⡛⠓⡞⢣⠀⠀⠑⠕⢑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⡀⣢⣨⡇⢸⡀⠀⠀⣢⣢⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠈⠉⠉⡉⡁⡽⠁⠉⠉⠉⠫⡉⠁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⣀⣀⡀⣀⣬⣮⣇⠄⢀⠀⠀⠈⡴⣋⣰⡇
⠀⠀⠀⠀⠐⠂⠐⠀⠐⠀⠂⠀⠒⠒⠂⠒⠀⠒⠒⠂⠄⠞⠲⠞⠒⠛⠖⠚⠈⠛⠫⠟⠫⠛⠭⠿⠥⠐⠄⠠⠀⠄⠠⠤⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠉⠓⢧⡄⣸⡄⠀⢀⡆⡀⠀⢠⣼⢀⣤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⡿⢹⣇⡅⣸⡾⣧⣀⣯⠙⣁⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⡜⣧⣼⠛⣿⣯⢀⣽⣿⡿⢾⢿⡄⠀⠀⠀⠹⡂⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⠀⠠⡔⢀⡼⢷⣛⡈⢣⠘⢿⣿⣿⢿⢇⠌⢰⣱⠾⣷⣀⡠⠼⣂⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢲⡟⠉⠘⢳⠀⣿⣅⣸⠟⣧⣄⠈⠿⢿⣶⣿⠟⣀⡾⠙⢶⡿⠃⡀⠀⠈⢧⢀⡈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⡰⠊⠀⡌⠀⠀⠀⠘⣶⣵⡿⣿⣄⠈⢉⠛⢿⣴⡿⣦⡼⠋⡁⣠⢾⠟⠌⠀⢀⣀⣀⣯⡁⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⡇⢠⠖⠀⢰⣋⡝⠻⣻⡛⢧⣚⠀⠰⡀⢹⠁⣼⣀⣉⣠⣛⢸⡃⠀⠈⠃⣹⡻⣷⣄⡀⠀⠀⠘⢧⣀⠀⢠⠂
⠀⣴⡿⣄⣾⢿⠀⠀⢀⣻⠟⠀⠀⠀⢻⠁⠀⠀⠙⢿⣿⣟⣿⣿⣁⣼⣧⣤⣷⣿⣿⠛⠚⠣⢄⡀⣤⣍⣹⢈⡙⢿⣦⡀⣠⡏⠹⡴⠃
⢌⠛⢡⣟⣿⣿⣿⠞⠉⢠⣿⣿⡝⣤⠃⠀⠀⠠⢠⡞⢿⣿⡿⣯⣭⣥⣼⠶⢧⣨⡜⢆⠀⠀⢸⣿⣯⡼⠻⣿⡇⠀⠙⢻⣿⣿⣾⠇⠀⠀⡈
⢸⡀⠀⠉⢝⣿⠃⠀⠀⣿⣧⣽⣻⣿⡆⠀⠀⢠⠟⠀⢪⣿⡵⣅⠀⠀⠀⢀⡞⣿⡄⠀⢂⠀⠀⠛⠃⠚⢛⡡⠶⢆⠀⠀⠉⠿⡏⠀⡰⠉⠱⠆
⢸⠈⠢⡀⠈⢻⠀⠀⣸⠎⣀⠀⠛⡋⠀⠀⢰⢋⠀⣠⢿⠉⣷⣈⠁⠀⠈⠀⢼⡻⡟⠢⣀⣇⠀⠀⢇⢰⡏⠀⠀⠈⠆⠀⠀⠐⣥⡼
⠘⡀⠀⠘⣆⣹⠀⠀⡟⠀⠈⢳⣰⠇⠀⠀⣏⠈⠛⠁⡟⠀⣿⣄⠙⠒⠂⠀⢸⣇⠿⣦⡀⠈⡀⠀⠈⣿⠀⠀⠠⢀⡼⣦⡀⠀⢛⡇⠀⡄⢺⡆
⠀⠿⠋⠲⡌⢿⠀⠀⣷⠶⡂⣀⣿⠀⠀⠀⠙⡀⡠⠃⢠⡘⠈⠛⠑⠀⠀⢠⠏⠘⢧⣣⠙⠢⡇⠀⠀⠸⢧⡘⣠⠎⠀⠈⠁⠀⠸⣧⠞⠀⢨⡇
⠀⢸⡄⠀⠈⣿⠀⢰⠏⠀⠹⣹⠇⠀⠀⢚⢧⡉⠁⢰⡟⠀⠀⠀⠐⠠⣤⠖⠀⠀⠀⠙⣦⣠⣷⠀⠀⠀⠈⣷⠃⠀⡠⢄⡀⠀⡠⢿⠀⠀⣸⠃
⠀⠘⠟⣄⠀⢸⣄⢸⠇⠲⣄⠹⣆⠀⠀⠀⠛⡓⠺⢻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣻⣯⡿⠀⠀⠀⠀⠀⢃⡼⠃⠘⡇⠀⢇⢸⡆⢠⣿⠁
⠀⠀⠆⠈⢃⢀⣴⣾⠀⠀⠘⢷⡿⠀⠀⠀⠀⠘⢿⣏⣀⠟⣦⠀⠀⠀⠀⠀⠀⠀⡔⢃⡾⠃⠀⠀⠀⠀⠤⠀⢸⣧⣤⣤⣿⡀⢾⢹⣇⠍⢙⡄
⠀⠀⠘⣄⣰⣿⠉⢙⡶⡛⢛⣻⣷⡄⠀⠀⠀⠀⠀⠈⠳⣄⣇⠑⡄⠀⠀⠀⢠⣾⣠⡟⠀⠀⠀⠀⠀⠀⢀⣀⡞⢃⣠⣈⠉⠛⣧⡾⠁⠤⣾⠁
⠀⠀⠀⠈⣿⠉⣀⣞⡠⠙⢈⠉⠙⢗⠠⣤⠀⠀⠀⣀⣴⢟⣳⣿⣟⣀⣀⣀⣴⣾⣿⣿⣇⡀⠀⠀⣠⠒⣮⡀⠻⠋⢀⠍⠐⠀⣌⠙⠛⠚⠋⠉⠒⠂
⠀⠀⠀⠀⠙⠛⠛⠿⠿⠛⠾⠿⣿⠞⠛⠣⠠⠤⠠⢻⣟⣛⣛⣟⣛⣉⣉⣉⣭⣭⣭⣭⣭⣁⣀⣀⣉⡉⢛⣓⣋⡒⠊⠒⠓⠋⠉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠀⣀⣽⣤⣭⣤⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣭⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣄⣽⣧⣈⠃⣿⣗⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⣿⣿⣿⣿⣿⣛⡿⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣶⣾⣿⣿⣿⣿⣿⣷⡶⣦⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢸⣿⣷⣿⠀⣨⣟⠛⠛⠛⠋⡀⢸⣼⡟⢹⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣿⠛⣿⣄⣧⣿⡀⢰⣿⡆⠀⣼⢻⣇⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⢿⣿⣿⣿⣃⣀⣉⣀⣠⡟⠀⢻⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⠆⠀⠸⣿⣯⣴⣶⡶⣾⣭⣽⠇⠀⢺⣿⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⣀⠛⣛⡷⣷⡛⢛⡀⠀⠀⠀⠹⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⢸⣿⡿⠿⠷⠾⠿⣿⣽⡄⠀⠀⢀⣜⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠛⠆⠀⣿⡿⣿⠀⠀⠀⢸⣏⠉⠁⠀⠠⠟⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠁⠀⢿⣿⣿⡀⠀⠀⢸⣿⣷⠏⠀⠈⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢿⡿⠟⠀⠀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡏⣸⠖⠀⠀⠻⣿⣩⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡴⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢾⠆
⠀⠀⠀⠀⠀⠀⣂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⣀⠀⠀⠙⠒⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⠙⠛⠳⢦⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣠⡾⠁⢀⡴⠞⠋⠉⠙⢻⡟⠺⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡤⢀⠀⠘⠀⠀⠀⢸⡿⠁⠀⠀⣼⠿⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⢿⣧⠤⠀⠁⠀⠀⢸⡇⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢡⣾⡇⠀⠀⠀⣿⣇⠀⢠⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⣾⡏⠀⠀⠀⢀⣿⣿⠷⠾⠏⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⡏⠀⠀⠀⢰⣿⠣⣈⠀⢤⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢈⢷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⣿⠇⠀⠀⠀⠀⢸⡏⢰⡷⣯⠘⠀⠀⠀⠀⠀⣀⢀⡀⣼⣤⣢⣤⣠⣦⡆⣴⣿⣦⣤⣴⣶⣶⣿⡿⣿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣣⣶⠀⠀⠀⣠⣴⡾⣿⣁⠀⠀⠀⠀⠀⢠⣵⠟⠛⠛⠻⢿⡇⢩⣿⣿⣿⠛⠟⠋⠉⠉⠁⠄⠀⠈⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⠀⣰⣿⡼⠂⠁⠀⠀⠀⠀⢀⣴⠿⣿⠿⠟⠛⠛⠛⠛⠙⠛⠉⠀⠀⠀⠀⠀⠀⣀⢀⣀⣨⣶⣿⣧⣾⣷
⠀⠀⠀⠀⠀⢠⠀⠀⣠⣶⣿⣿⣿⣿⣯⡻⢄⡿⢁⡸⠧⢴⠄⠀⠀⢠⣿⣿⣟⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠶⠄⢀⡀
⠀⠀⠀⠀⠀⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡾⢯⡘⠃⠀⠀⠀⠀⠀⢨⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⢿⠿⠻⣿⣿⣿⡿⣟⣉⣁⡐⣶⡶⣶⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣭⣿⣿⡙⠛⠁⢠⣗⠫⠽⠖⠀⠀⠀⢺⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠙⢛⣿⣿⣿⣯⠿⣯⠳⡄
⠀⠀⠤⠀⠀⣾⣿⣿⣿⣿⡟⣋⣭⠏⣡⣶⣾⠄⣩⠜⠂⠀⠀⠀⠀⠘⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⠀⠙⠛⠻⠷⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⡙⢻⣿⡆⠙⠉⢱⠎⣿⣼⡦⠄⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣠⠦⣄⡀⠀⠀⢹⡆⠀⠀⠀⠄
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣤⡙⠋⢷⣿⡷⣷⣿⢈⣀⢀⣼⣿⠋⠀⠄⢀⣦⠔⠂⠀⠀⠀⢀⣼⠁⢀⣤⣤⡾⠛⠀⠀⠘⣀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣯⣴⠿⠛⠉⠉⠙⣿⠳⠷⠀⣴⡟⠁⠀⠀⠀⠀⢠⣾⣿⣴⣿⣿⣍⠁⠀⠀⠀⠀⠘⢿⣼⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡟⢻⣧⣼⣶⡟⠁⠀⢀⡀⠀⠀⠈⠓⠓⢾⡟⠁⠀⠀⠀⠀⠀⢳⣿⣿⣿⡟⠠⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⡄
⠀⠀⠀⠀⢹⣿⣿⣿⣍⣽⣿⡿⣿⠫⠘⣷⠀⠀⢸⠇⠀⠀⠀⠀⠀⢈⡇⠰⣀⣠⡀⠀⠀⡈⢿⣿⣿⣇⠀⢠⡅⠀⠀⠀⠐⠀⠀⣿
⠀⠀⠀⠀⠈⣿⡟⣿⣿⢿⢏⡝⠀⣴⠾⡏⠁⠀⢻⠀⠀⠀⠀⠀⠀⠾⠁⠀⠈⠁⠈⠉⠀⠀⠙⣗⣿⣿⡄⠘⠷⢎⢀⡀⠀⣸⣶⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣴⣾⡏⣾⠁⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⣿⢷⣦⡹⢯⣿⣻⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⡇⣿⠀⠀⠧⠀⡆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠀⢷⠻⠀⠀⠙⠛⠟⠛⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⡏⢰⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣶⣼⡧⡴⠞⢠⣼⣾⣷⣄⠀⠀⣤⣤⣴⣶⣦⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣹⣿⣧⡟⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⠉⠁⠀⡀⠚⡟⠁⠉⠛⡇⠈⠉⠉⢿⡥⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠴⠀⠹⣿⣷⣿⣿⣷⣿⣾⡙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⢠⡶⠿⠀⢠⣿⠶⢡⣄⠀⢰⣄⠀⢶⣾⠟⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣟⣻⡿⠆⠀⠀⠀⠀⠀⢀⠤⠤⣾⠇⠃⠀⠀⠀⣤⣷⡀⠀⢀⣠⡀⢸⣧⡈⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠀⠀⢀⠟⠀⠀⠀⠀⢠⡿⠏⠀⠈⠀⠙⠁⠸⣇⠉⠉⠉⣹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⡻⠁⠢⠀⠀⠀⠀⠀⠀⠈⠂⠀⢠⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣯⣿⡤⣿⣦⣀⡀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⣞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⡄⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⣤⣤⡤⠲⡎⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣗⣻⣣⡀⠣⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⡾⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣏⣌⢀⡛⡉⠓⠀⢀⡄⠀⠀⠀⢰⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡐⢦⠀⡄⠀⢀⣠⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡏⣠⡝⣿⣿⣿⠟⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⡿⢿⣶⡆⠀⣿⡟⢿⣿⣿⣿⣿⣽⣡⣿⣿⣧⣿⣻⣿⠀⠀⠀⣀⡀⠰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣤⣿⡇⢸⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢾⣿⣁⡀⡈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣈⡛⠛⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣿⣛⣿⣿⠀⠀⢢⣽⣭⣾⣥⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻⣿⠿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠸⡿⠀⣿⡇⣤⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⣷⡐⡘⣿⣿⣿⣷⣿⡟⢿⣽⣿⡿⠛⠻⣾⣗⡄⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣶⣶⣦⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠀⣿⣿⣇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣶⣦⣤⣭⣿⣿⣿⢻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣼⣏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣷⣶⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣤⣾⡟⢿⣿⠟⣡⣶⣼⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣉⣙⠛⠉⠀⠈⠻⢸⣿⣿⣇⣽⣿⣇⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣿⠿⣿⡇⠀⡈⢺⣟⣲⣍⢹⣿⡏⠃⢸⣷⣮⠉⣿⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣾⠿⣶⣿⡇⠈⡏⠀⣿⣯⣾⣾⣿⡷⠃⢸⣿⣵⣷⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣈⠉⠀⠀⣍⠀⣿⣿⣿⣿⣿⢧⠀⢸⣿⣿⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠉⠉⠻⣷⡀⠀⠶⠀⣿⣿⣿⣿⣿⣿⡗⣾⣿⣿⣿⣿⣿⣟⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⠀⠀⣼⡇⠀⡷⠘⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠁⠘⠛⠀⠟⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠟⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣤⣠⡆⢀⣰⢞⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠐⣿⢿⠧⠤⠴⢛⠁⠀⠀⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⣔⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣄⠀⠀⠀⣀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠓⠠⡖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠃⠀⠀⠀⢝⣆⡁⠖⠲⢠⠀⢠⠀⠀⠀⠀⠰⠺⠂⢍⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢋⠁⠸⠀⠀⠀⠀⠠⠆⢰⡟⠀⣀⠀⠈⠀⡄⠀⠀⠀⠀⠁⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⢰⠀⠀⠘⢋⣁⠈⠀⠃⠀⢸⣀⣢⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠛⡶⠀⠄⠘⠋⠑⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣌⣠⠔⠃⠀⠀⠀⠀⠷⢤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⣠⡾
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠁⠂⠁
⠀⠀⠀⠀⠘⣦⠀⠀⣉⡀⠀⠀⠀⠀⣨⢀⡛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠠⠁⠉⠁⠀⠀⠀⠈⠛⠚⠋⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣶⠶⠒⠖⠒⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠐⢁⣄⢢⠄⠀⠀⠀⠀⠀⠈⠉⠙⠷⠶⠜⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⢄⣀⠉⣬⣀⣜⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁

⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠀⠉⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠢⠀⠀⠁⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠶⠀⢀⠀⠶⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠐⠂⠀⠀⠠⠀⠉⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠃⠀⠀⠀⣀⢠⠀⠀⠶⢤⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠁⠀⠂⠀⠀⠒⠐⠈⠀⠀⣠⠔⠀⠀⠀⠀⠱⣀⡧⠴⣁⠴⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⣤⣄⣀⣠⣤⡤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⣖⣋⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠒⠈⡋⠈⠉⠁⠈⠓⠈⡶⠠⢄⠀⠀⠀⠀⠀⠀⢀⡕⢤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠋⢠⠃⠀⠀⠙⠲⠤⠤⠤⠴⠊⠁⠀⠀⢻⠐⢄⣀⡠⡔⢋⡠⠃
⠀⠀⠀⠀⠀⠀⡠⠚⣀⣤⠀⡎⠀⣴⣿⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣇⡀⠴⠊⠁
⠀⠀⠀⠀⠀⣘⢁⠖⠁⠈⢳⠁⠀⠙⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠁
⠀⠀⠀⢀⠴⠣⢿⠀⠀⠀⠸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⡠⠤⡇⠀⠀⢀⡇⠀⠀⡇⠀⢂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⡆
⠫⡤⠊⡽⠲⠶⢮⠙⣆⠀⠈⢑⠒⠤⠤⢀⣀⣀⠂⠀⠀⣀⣀⡀⢬⠴⠚⠁
⠀⠉⠲⠆⢨⠇⠀⠉⠁⠀⠀⢀⢒⡁⠒⢄⠀⠀⠀⢀⠔⠋⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⣱⣤⠂⡀⠀⣇⣔⠁⠈⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠒⠒⢎⠀⠀⠀⠀⠀⢨⠗⠒⠚⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣶⡶⡎⠀⠀⠀⠀⠀⢘⡶⣶⡶⡏
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠰⠴⠣⠄⠤⢱⡀⠀⠀⠀⠀⣸⡡⠤⠤⠣⠤⠤⢄
⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠐⠈⠠⠀⠀⠙⢆⠀⢀⠞⠁⠀⠀⠈⠄⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⣏⣩⣉⣹⣉⣹⣉⣩⣉⣹⣹⠀⢸⣸⣁⣋⣈⣛⣀⣙⣈⣋⣘⡃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣾⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠤⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠀⠀⠰⠖⠀⣘⠀⠶⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⠀⠠⠀⠠⠐⠀⠀⠀⠠⠐⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⠢⡅⠉⢸⠀⠀⡒⠀⠀⡏⠁⠨⠐⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠓⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠂⠒⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡔⠒⢺⠀⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⢸⠒⠂⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡊⢀⡱⠒⠙⢤⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣇⠜⠐⠴⡁⠈⣢⡀
⠀⠀⠀⠀⠀⡐⠀⡸⠋⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⠀⠀⢀⣀⠀⡀⢀⢀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⢺⡀⠐⡀
⠀⠀⠀⠀⠰⠉⣰⠁⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢱⠈⠡
⠀⠀⡔⢋⢉⡐⢣⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠶⠶⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢴⠖⠁⠀⠀⠀⠀⠀⠀⠀⢨⠓⡈⠉⠲⡀
⠀⠐⠉⠉⢀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡀⢀⡄⠀⠀⠀⠀⠀⣆⠀⣰⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡄⠠⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡇⠀⠀⠀⠀⠀⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⡇⠀⠀⠀⠀⠀⡋⠉⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠛⠒⠀⠧⡀⠀⠀⠀⣠⠧⠐⠚⠃⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⠀⠀⠀⠀⠀⠀⠘⡆⠀⢜⠁⠀⠀⠀⠀⠀⠀⢳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⢶⡏⠉⠉⠲⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣾⢿⣇⣒⢐⢀⣻⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠘⡏⠉⠉⠛⠉⢉⣍⣈⡉⠉⠉⢉⣉⡉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣏⡴⠃⠀⠀⠋⡈⠉⣾⣧⡞⢰⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠁⠈⠘⣠⠆⠀⣶⣙⣂⡝⢋⠻⣬⣙⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⡀⢄⣠⡏⠀⠀⠈⠉⠉⠀⠈⠀⠈⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⢿⡃⠀⠀⢰⣟⣛⣚⣿⣘⣋⣹⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⡾⠀⠀⠀⠸⡯⠽⠥⠯⠼⠷⠻⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠒⢳⣶⣶⣷⣶⣶⣶⣶⠶⢶⣒⣒⡒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣬⣿⣿⣿⠛⠛⢫⡉⣿⠿⠿⠿⠿⠿⠶⠶⠣⠤⠤⠀⠘⠛⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⡀⠁⡀⠀⠡⠀⠀⡇⠀⠀⡐⠲⡄⠀⠀⠀⠀⠀⠀⠀⠰⣄⡿⠀⢸⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⡁⢹⠀⠠⠀⠀⡀⠀⡇⠀⠀⠙⠀⢀⣀⣀⣀⣄⣀⣀⣤⣤⡤⣄⠀⢸⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡄⠙⠉⢀⣤⣶⣾⢀⡄⡇⠀⠀⣎⠶⠿⠛⣯⣽⠟⡟⣫⣉⡁⢹⡇⠀⢸⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠻⢶⣶⣿⣿⣿⡟⣼⡇⡇⠀⠀⣿⠀⡞⠙⠁⣆⠀⡀⠀⠀⡟⢠⡇⠀⢸⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠀⠀⣼⣿⣿⣿⣿⢃⣿⢰⡇⠀⠀⣿⠀⠘⠆⠊⠁⠀⠀⠈⣉⣀⢸⡇⠀⢸⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠳⣄⠀⣼⣿⣳⣿⣿⣿⢸⣿⡆⡇⠀⢀⡏⡴⠂⠀⠈⠀⢠⣀⡀⣈⠦⢸⡇⠀⢸⣷⣈⡉⠳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠐⠛⠆⢸⣿⣿⣵⣿⣿⣿⡇⣿⣧⠀⡇⠀⠜⣧⠉⠂⢉⡀⣀⠀⡅⠠⠄⠶⠸⡇⠀⢸⣿⡏⠀⠀⠀⠀⡉⠓⠢⣤⣀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠀⠂⢸⡀⢠⡼⣿⣿⣿⢳⣿⠃⠀⡇⠀⢀⡏⠀⢉⣈⣁⣀⣤⣤⣤⡴⠟⠖⠁⠀⢸⡿⣷⡶⡶⠖⠀⢠⠛⠛⢣⡀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣴⣯⠀⠀⣷⣿⣿⣿⣼⡿⢠⡇⡁⠀⠀⠙⠛⠋⠹⠛⠉⠋⠁⠈⠀⠀⠀⠀⠀⢸⠛⠋⠡⠄⣀⡀⠃⠠⣈⠂⣿⣷⡖⠴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠁⠀⠀⠀⢹⣿⣿⣿⣿⡇⣾⠇⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠳⡶⠚⠛⠉⠉⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣧⠀⠀⠀⠈⣿⣿⣿⣿⢠⣿⠀⡇⠀⠀⠀⠀⠀⡅⠀⠀⠀⢀⣀⠄⠀⠀⢀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠄⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡸⣧⠀⠀⠀⠿⠿⢿⣿⣾⡏⠀⣷⠀⠀⠀⠀⠀⣉⣉⣉⣤⣴⣶⣶⣟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠩⡙⢶⣖⠓⢢⡀⠈⣿⣿⣿⣶⣿⣶⡾⠿⠿⣿⣿⣿⡟⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⠣⠇⣨⠈⠆⠀⠱⠀⣿⣿⣿⡏⠁⠀⠀⠀⠀⢿⡟⠁⡅⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢱⠊⢻⡄⠀⠀⣈⠀⠿⢿⣿⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⣀⣸⡧⠴⠾⠋⠀⢀⢸⡟⠀⠀⠀⠀⠀⠀⢸⣤⢀⡄⢀⣀⣠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠧⠦⣄⡀⠀⠀⠀⢈⣄⠘⡏⠉⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⢄⠈⢿⠒⠊⠉⠁⢹⠀⣷⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠈⡇⠈⣦⣤⣶⠶⢧⡴⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡈⠘⠁⣠⠋⠀⠀⠀⢼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⡶⣾⠟⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡂⣸⠇⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡃
⢀⣀⣤⣶⣾⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⡇⣿⢰⡇⠀⠀⠀⠀⠀⠋⢀⣀⣠⣼⠇⠒⠂⠤⠤⣀
⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⡈⠀⠀⠀⠀⠀⠀⢹⣿⣭⣁⠀⠀⠀⢀⣀⣤⣼⡂
⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⡛⠠⠭⢲⣤⣀⣀⣤⡴⠟⠛⠻⠿⣿⣶⢛⡫⠭⠒⣚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣉⡉⠀⠀⠀⠀⢀⣀⣤⣽⡩⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⢿⣿⣾⣭⣟⣻⠿⣿⣿⣿⣿⣿⣦⠶⠛⠉⠁⠐⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣬⣭⣛⡻⡟⣒⡨⠤⠒⠂⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⠿⣋⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣭⣭⣭⣵⣾⣿⣿⣿⣿⣟⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⠀⠈⣿⣶⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⡇⠀⠀⠀⠙⠛⢿⡟⠝⡔⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣠⣤⡄⠀⠀⠀⠘⠁⠀⡿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⢿⣿⣦⡆⠀⠀⠀⠀⢀⡤⠀⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡉⠉⠀⢴⡇⠀⠀⢸⠀⢀⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣍⣈⡳⣄⠈⠃⠀⣠⡞⣰⡿⣱⣿⣿⡿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡤⠼⠀⣠⣾⣿⣿⡟⢰⡿⣽⣟⣽⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⣴⣾⣿⣿⠟⡽⢡⣿⣽⢏⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠁⡼⢡⣿⣿⣏⣾⣿⡿⢟⣻⣿⠯⠻⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡼⢡⣿⣿⣿⣸⣿⣡⡶⠛⢁⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⢟⣿⣿⠟⢀⣠⡤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣿⣿⣿⢣⣿⢟⣯⠾⠻⡿⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠛⠛⠛⠛⠃⠙⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡅⠒⠶⣤⢠⠱⣈⠀⠀⢉⡜⡀⣤⠴⠒⢪⣥⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣮⡃⣾⢢⡤⢿⠔⣷⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠥⠈⣀⣄⡍⠨⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠈⠁⠀⢤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⠁⠈⢿⣶⠄⢀⠀⠀⡀⠠⣶⡿⠂⠀⣡⣤⣤
⠀⠀⠀⠀⠀⠀⣠⢤⢤⡆⣤⣀⣀⡈⠻⠿⠇⠀⠒⠸⣆⢈⡁⠈⣁⣠⡏⠐⠀⠐⠿⠿⠁⣀⣀⣤⢐⡤⣤⢄
⠀⠀⠀⠀⠀⠐⡿⠩⠼⠃⡿⠦⣍⡇⠐⠶⠦⢴⡄⠈⠙⠻⣷⣾⠿⠛⠁⢂⣦⠰⠶⠆⢸⣯⡴⢾⠀⠷⠍⢻⡇
⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⡷⢦⣄⣹⣎⠛⠛⠶⡛⢚⠁⠢⢈⣃⠄⠈⠑⢛⠶⠟⠛⡣⣏⣡⡴⢾⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠿⡶⣬⡏⢸⡇⣼⡎⠑⡿⢷⠤⠤⠤⠤⠴⣿⠞⠱⣯⢸⡇⢸⣥⠶⡿⠀⠀⢰⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣷⣷⠀⠀⠇⠈⠁⠀⠀⠈⠁⠸⠀⠀⣺⣿⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡖⠠⠆⠀⠀⠀⠀⠀⠀⠘⠤⠰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⣀⣀⣀⣡⠡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠠⠐⡖⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⢂⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢧⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠀⢄⠺⠿⢃⠀⠀⠀⠠⡾⢃⠀⠀⠀⢸⣼⣠⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⠉⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⢸⠹⣾⠟
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠐⠄⡀⠀⠀⠐⠂⠀⠀⠀⢀⠔⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⢘⡀⢆⡀⠀⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⠁⠒⠀⠀⠀⠀⠂⠁⣀⡀⣼⠤⠏
⠀⠀⠀⠀⠀⢈⠀⢸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣛⣟⣿⠅
⠀⠀⠀⠀⠀⠐⡒⠒⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠒⠃⠀⠛⠛⢳⠶⡖⠠⢀
⠀⠀⠀⠀⠀⠀⠈⢻⠛⢻⡀⠀⠀⠀⠀⢀⠠⠐⢨⡁⣦⡥⠊⠀⠐⣛⡛⠛⣿⡋⡗⡶⠉⠉⠁⠐⠂⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⢣⡀⠀⡠⠐⠁⢰⣠⡇⠀⡇⠀⠀⠀⡠⢊⠈⠀⣿⠉⢹⠙⢦⡤⠤⣄⣀⡀⠀⠠⣨⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠁⣷⠊⡀⣀⠔⠋⠀⠀⡀⣇⠀⠀⠘⠄⠁⠀⠁⣿⡀⢸⠁⢸⠆⠀⠀⠀⠉⠁⢳⠋⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢤⡱⠚⠁⠀⠀⠀⡆⠀⠂⠀⡘⠛⠿⠿⠿⠿⠿⠉⣸⠈⣘⠀⠀⠀⠀⠀⢀⠆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⢀⡟⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⡀⠀⠀⢀⣀⣀⣤⣸⣿⣿⡿⠁⠀⠀⠀⠀⡼⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣠⣤⡄⠀⣶⣤⣤⣍⠁⠀⠀⠀⠀⠀⣀⢀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⣻⠀⠀⠘⠙⠛⡏⠀⠀⠀⠀⠖⢉⠙⠟⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⠀⠀⠀⢄⡇⠀⠀⠀⠀⠑⠀⠁⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢸⠀⠀⠀⠀⢊⡅⠀⠀⠀⠀⠀⡐⣡⠂⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⣷⠆⠀⣸⠀⣥⣇⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠳⣄⣀⣀⡄⡰⢀⡀⢀⣁⣡
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⠀⠀⠀⠐⠂⠄⠰⡅⣤⣴⣠⣤⡿⣇⣀⣄⣉⣉⣳⠖⢀⠴⠀⠀⠠⠐⠀⠔⠰⠀⠀⠄⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⢴⠂⠉⠉⠉⠉⠉⠉⠀⠒⡤⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡿⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠑⠂⠸⣇⠈⢦⡀⠀⠀⠀⠀⠀⢀⡜⠀⣼⢇⠰⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠎⠉⠢⡀⠀⠀⠀⠻⣷⡀⢻⣶⣶⣷⣶⣾⠋⣠⡾⠋⠀⠀⠀⢀⠜⠓⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⢄⠀⠈⠀⠀⠀⠀⡈⢿⣄⣀⡽⠛⣧⣀⣸⡏⡀⠁⠀⠀⠀⠁⠀⡠⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠻⣶⣍⡒⠳⠦⠤⡄⢠⣾⠋⠁⡄⠀⣄⠈⠛⣦⠀⣀⡤⠴⠒⢂⣩⣴⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⠁⠒⠘⠛⡛⠻⡷⠶⣧⣼⠁⠑⠂⠀⠀⠀⠑⠂⢸⣆⣴⣶⡶⠿⣛⠟⠀⠄⢹⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⠀⠀⢀⣠⢀⡀⠀⠀⠋⢿⣀⢤⡀⠀⠀⠀⣰⢄⣸⡏⠗⠀⠁⠀⣀⡀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣤⡿⠉⠀⠀⠀⠈⠑⠀⣈⠻⣄⠸⠆⠀⠴⢁⣼⠋⠀⠐⠋⠁⠀⠀⡉⠓⣦⣸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠸⠇⣰⣮⣿⣶⣦⣄⡀⠈⠡⡈⢷⡀⠀⣠⠏⡠⠋⠀⢀⣤⣴⣶⣒⣧⡀⣽⠙⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢠⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⣷⠈⠀⢰⡏⣀⣤⣾⣿⣿⣿⣿⣿⣿⣧⠰⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⡦⢹⣿⣿⣿⣿⣿⣿⣿⡿⣯⡴⣩⣬⡳⣌⠿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⠃⢄⣉⢝⠻⠿⠿⢟⠯⣿⡫⣾⣿⣿⣿⡏⣷⡯⠻⠿⠿⠿⢿⡛⢁⡔⠿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⡿⢌⣃⠀⢉⣀⣹⣤⠺⠄⣿⣿⣿⣿⣿⣿⡅⡷⢤⣝⣀⡀⠁⢀⣋⡴⡿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢏⣉⣚⣭⣉⣡⡵⠆⢻⣿⣿⣿⣿⡿⠱⢶⡅⣉⣩⡗⣒⡘⣧⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣘⣮⣻⠫⡰⣷⠿⣆⣾⡶⣝⣛⣛⣋⡵⣿⣠⠿⢎⢯⡸⡿⣣⣟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠁⠀⣟⠻⣇⣿⠳⢮⣅⣠⡻⣙⢋⡻⣄⣠⣿⠾⢿⣜⡿⢛⠉⢨⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢰⣖⠐⣡⠀⣿⣰⣿⣏⢷⡈⠉⠉⠛⠛⡟⠻⠛⠋⢉⡴⣻⣿⣇⣼⠇⢘⠢⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡹⢤⡿⠓⣌⢈⡷⣿⣾⣷⣶⣴⣦⣶⣶⣷⣿⢵⡋⣡⠜⢿⣤⠎⣷⣸⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢄⠸⢹⣿⣾⠃⢠⣾⣾⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣿⡄⠸⣷⣾⡟⡏⢀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣱⠏⠀⠘⢄⣿⣿⠀⣸⣯⢩⡋⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⢱⠋⣹⣇⢀⣿⣿⠁⠆⠁⡸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡾⡉⠂⡀⢀⠳⣿⣿⣄⣈⣻⣦⣉⣾⣬⡛⡛⢻⡟⠛⢛⣩⣷⣈⣴⡟⠁⣠⣿⣾⠕⡡⢊⠔⢈⠦⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠢⠈⠢⡑⠂⡜⠻⡷⢄⡈⢋⠙⠃⣞⡉⠁⠈⢩⣣⠘⠛⡙⢉⡀⢼⠟⢏⠁⢎⠀⢀⠄⢁⠔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣢⣈⢆⣠⠎⠀⠀⣿⣾⣿⣿⡈⢠⡿⠀⢢⡄⠂⢿⡀⢂⣿⣿⣷⣾⠀⠀⠳⣌⠠⡀⡠⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⠁⢠⣉⡈⠻⢿⣿⣿⠻⠸⣇⠒⠐⢀⠒⣾⢃⠿⣿⣿⡿⠿⡉⣑⠀⠙⣷⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣿⡿⢷⣶⣶⣴⣶⣇⠹⡄⣤⠸⢠⠋⣸⣴⣤⣤⣶⣶⣿⣿⣤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠳⠬⢉⡉⠥⠞⠉⠁⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠂⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣁⣤⠔⣾⡷⠒⠒⣒⣲⡶⠶⢾⣶⣶⡒⢒⣲⡶⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢉⣙⣷⡶⠾⣟⣁⣠⠴⠚⢁⠨⢛⣉⠫⡀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢒⣿⠟⠋⣁⣤⣶⣛⣉⣐⣀⣿⣥⣬⣭⣤⣴⣮⣦⣄⡀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⣋⣶⣿⡾⠿⠛⣛⣩⣿⣿⣿⡿⠷⠟⠋⠙⡩⣭⣟⠉⣠⠞⠁⠠⠂
⠀⠀⠀⠀⠀⠈⠲⠀⠀⣀⣤⣾⣿⠿⠟⠛⢹⣧⡴⠶⢿⣿⡿⠞⣋⠄⠈⠋⠀⠀⠠⢀⠙⠯⠿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠐⣴⡿⠟⠋⣁⠤⣴⠞⠋⠉⠀⠀⠀⠈⠱⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⢈⡷⠖⠻⠯⠛⠛⠿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣉⡉⠉⠉⢉⣉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡚⡅⢜⢱⡛⡇⢰⠡⡏⣳⢠⢗⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣣⣇⣈⣓⣋⣁⣀⣑⣚⣁⣘⣜⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠚⠓⠒⠒⠒⠒⠒⠒⠚⠓⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡇⠀⡴⢶⠶⡶⠶⡶⢶⠶⡶⠶⡄⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢮⡑⣌⡇⠀⠧⠼⠤⠷⠴⠧⠼⠤⠷⠴⠂⢸⣥⠎⡵⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠲⢷⡌⢦⣱⠼⡇⠀⠀⠀⠀⠀⣎⣱⠀⠀⠀⠀⠀⠈⠧⣎⡴⢃⡾⠖⠒⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠉⢢⡀⢻⠞⠁⠀⡇⠀⢀⣄⠀⠀⢠⡅⠀⠀⣠⡄⠀⢠⡀⠈⠳⡟⠀⡔⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⡼⠀⠀⠀⠁⠘⣏⣨⡏⠙⣅⣨⠋⢹⣅⣩⠃⢸⠃⠀⠀⢧⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⠈⠁⠀⠁⠈⠀⠈⠁⠈⠀⢸⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⡤⠤⠤⡤⠤⠤⢤⠤⠤⠤⡤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⡇⠀⠀⢸⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⡇⠀⠀⢸⠒⠒⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠭⠭⡇⠀⠀⢸⠭⠭⢭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣒⣒⡇⠀⠀⢸⣒⣒⣚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠧⠤⠤⠷⣄⣠⠾⠤⠤⠼⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡟⣹⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠗⣠⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⠀⠀⠀⠀⠀⠀⠀⣠⡄⢠⡄⢠⣀⣀⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⢀⣸⡆⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⢧⠟⠸⠇⠈⣯⣉⣉⣉⣉⣩⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⠛⠉⠉⠁⠀⢀⡀⠀⠉⠉⠙⢻
⠀⠀⠀⣶⣸⣿⣿⢸⣿⣿⢠⣾⠂⠀⠀⠀⠀⡤⣄⡤⣄⠀⠘⢿⣭⡉⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠏⠿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⢸
⠀⠀⣰⣟⣿⣿⡏⢼⣿⣿⢸⣿⠀⠀⠀⠀⠀⠷⠟⠣⠟⠀⠀⣠⡍⠻⢶⣄⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⠿⠿⠿⢷⡆⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸
⠀⢀⣽⣿⣽⣿⣷⣾⣿⣿⣼⣿⣨⠀⠀⠀⠀⢰⡄⡴⣦⠀⡾⠛⠒⠒⠚⠻⣆⠀⠀⠀⠀⠈⢁⠀⠈⠸⠀⠀⠐⡈⠀⠀⠀⠀⠿⠛⠋⠉⠉⢩⣿⣯⠉⠉⠙⠛⠻
⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠘⠃⠳⠋⠘⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠣⠤⣀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠉⠛⠳⠆
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⠄⢿⣿⠇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢋⣍⠉⠳⢦
⠀⠀⠀⣞⡟⠉⣠⣿⣀⠉⢟⡦⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣌⢹⣄⠀⠈⢳⡀
⠀⠀⢴⠻⠀⣾⣟⢙⣿⡆⠠⠟⠄⠀⠀⠀⡴⠿⢷⣤⣀⣀⣀⣉⣧⣀⣀⡀⠀⠀⠀⠀⣀⣴⣤⣤⣤⣀⣤⣤⣾⠟⣛⣛⡢⠀⠀⠀⠘⣿⣿⣿⣿⣶⣝⣲⢶⢂⡇
⠀⠀⣴⣶⡾⠛⠻⠿⠛⠻⣶⣶⡆⠀⠀⢠⡾⣛⢷⡌⣿⣿⣿⣿⣿⣿⢛⣭⣓⠀⠀⠜⣩⠶⢮⡙⣛⣫⡙⠿⢡⣿⣫⣝⢿⡄⠀⠀⣀⢈⡻⢿⡿⢟⣫⣭⣭⣭⠅
⠀⠀⠘⠛⠡⡵⣴⢶⡴⡥⠙⠛⠁⠀⠀⠸⣷⣟⣽⠏⠿⠿⠿⠿⠿⠇⢿⣒⣽⠂⠀⠀⢷⣛⣽⠇⠛⠛⠛⠛⠘⢿⣼⣯⡿⠃⠀⠀⢇⡈⢛⠁⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠆⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣮⠛⣤⠀⠀⠀⣠⣶⣶⣦
⢀⡄⠀⠀⣤⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⣠⣾⡋⣹⠏⠀⠀⠀⠀⠀⠀⠀⣿⠓⢺⣿⣿⠀⠀⠀⠀⠀⠀⠉⢣⡝⢪⡲⢄⣤⡹⣿⣿⡿⠁
⠈⠁⠀⣼⣿⣿⣿⣿⣿⣿⡿⠇⣉⣽⠇⠀⠀⠀⠀⠀⠈⠛⣫⣾⣯⣨⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢺⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣵⣿⣿⣿⠆⠁
⢠⡖⠀⠻⠿⣟⣛⣻⣭⣵⠶⠟⠋⠁⠀⠀⠀⠀⠀⠴⣾⣿⣿⣿⡿⣫⣾⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⠛⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⢟⣵⡿⣦⡀
⠈⠙⠛⠛⢛⣻⣭⣭⣷⣶⡿⠃⢀⣴⣄⠀⠀⠀⢰⠁⠘⠿⣿⣿⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠂⠀⠂⠂⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⠟⠁⠙⢫⣿⢏⣳⡆
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠙⠁⠀⠀⠀⠘⠂⠤⠆⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠶⠿⠁⠛⠤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠙⠟⠉

⠀⠀⠀⠀⠀⢠⡞⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢴
⠀⠀⠀⠀⢀⣬⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣥⣄⡀⠀⢀⣹⣁⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⡶⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠙⢻⠛⢹⡇⢰⡟⢟⠟⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣛⣛⣛⣛⡃
⠀⠀⠀⠀⡯⢸⣿⣿⣿⠀⡟⠀⠀⠀⠀⠀⠀⡟⢰⡾⣷⠘⡇⢸⣅⡾⣦⣸⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⣿⣿⣿⣿⣿⣿⢰⡆
⠀⠀⠀⠀⠻⢰⣿⣻⣿⠐⠇⠀⠀⠀⠀⠀⠀⠉⢹⡇⣿⠋⠁⠈⢹⡇⣿⠉⠀⠀⠀⠀⠀⠀⣀⣀⣐⠶⢖⣂⣀⡀⠀⠀⠀⠀⠀⠀⢿⠸⣿⣿⣿⣿⣿⡟⣸⠇
⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆⣴⠀⠀⠀⠀⡅⣥⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣿⡇⠉
⠀⠀⠀⠀⠀⢸⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣻⠀⠀⠀⠀⠇⠏⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⣽⡉⣹⡍⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡀⠀⠀⠀⠀⠀⠀⢐⠈⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢇⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠒⠒⠃⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣦⡀⠀⠀⣉⠐⠄⠀⠀⡀⠀⠌⣠⠀⠀⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠠⠉⠀⠂⠀⠀⢄⠙⠋⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣄⣠
⠀⠀⠀⣀⡴⠶⣤⡀⣀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠵⠋⣠⠇
⠀⠀⠀⠈⢮⠂⠀⠈⠙⠻⢶⣠⣀⢀⠀⠀⠉⢁⣀⣲⣶⢶⣖⣒⣐⣒⠊⠀⠀⢀⣀⠔⠉⠀⠀⡘
⠀⠀⠀⠀⠀⠑⢕⠄⠀⠀⠀⠈⢹⠋⢤⡄⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠙⢺⡋⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠸⠀⣖⠄⢸⠀⠀⠀⠀⠠⠀⠀⢨⣍⣩⡅⠀⠁⢀⡠⡢⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⢐⡀⢸⠀⠠⠤⠦⡠⠅⡀⠐⢒⡔⠂⠀⡏⢣⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢻⠀⢔⡀⠀⢉⣉⣉⣈⡀⠈⠁⠨⡵⠮⡅⠀⡇⣠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡸⠀⠀⠀⠀⠈⠁⠛⢸⠁⠀⠀⠐⠪⠭⠂⠀⣌⠛⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢡⡀⠀⠀⠀⠀⠐⠒⠚⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⠛⠛⠯⠭⠭⠭⠿⣩⡟⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠸⢄⡀⠀⠀⠀⠀⠀⡀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⠀⠄⡄⢹⠄⠀⠀⢀⡈⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠈⠒⠲⠶⠖⠒⠉⠀⠀⠀⠀⠁⠲⠆⠤⡤⡤⠪⠀⠀⠀⠀⠂⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣿⣿⣿⣼⣯⡟⣻⠶⣄⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣶⣄⣀⣀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢾⣻⣿⣿⣿⣾⣾⡏⠁⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⢄⠠⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣀⣀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠸⣿⡿⣿⣿⣷⣦⣤⣴⣶⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠿⣇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⡿⣿⣿⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠙⠿⣶⣼⣿⣧⡀⠈⠟⠿⣿⣿⣿⣿⠿⢃⡾⠚⠟⠉⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠷⠶⠤⠶⠏⠁⠀⠈⠀⠠⠀⠈⠀⠀⠀⠀⠉⠀⣀⣿⣿⣿⣿⢏⣴⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣠⠀⣤⣄⣀⣤⣤⣤⣤⣴⣾⣿⣿⣿⣏⢸⣯⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣾⣿⣿⣿⣿⣏⢸⣿⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣦⡙⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠰⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢨⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣺⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⣿⣿⡿⠯⠁⠀⠉⠻⣿⣿⣷⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⣿⣿⣿⣿⡧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣱⣿⣿⠏⠀⠀⠊⠀⠀⠀⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣘⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⡙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣵⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⣠⣴⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣠⣴⣾⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⠀⠀⠀⢀⢠⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠑⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⢃⠀⠀⠀⠊⠻⠶⠦⠤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠰⣒⢏⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠢⠀⢤⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢀⠚⠑⠠⠀⠠⠖⡑⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢐⣡⡄⡀⠂⡐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡀⠀⠀⠀⠁⡀⢤⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣷⣶⡄⠀⢀⡶⡆⠀⠀⠀⠀⠀⠈⠛⠲⠄⡀⣀⠀⠀⠻⠗⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⡔⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⡛⣻⣽⠀⢩⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘
⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠘⠋⠈⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣀⣦⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠗⠐⠎⠒⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⢀⡀
⠀⠸⠡⠦⠀⢀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣠⡘⠿⠀⠀⠀⠀⠀⠀⠀⡠⠊⢃⡀⠈⠐⠁⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸
⠀⡸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠰⠤⡆⠀⠀⠀⠀⠀⢈⠀⠂⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣼
⠀⠀⠘⡆⠀⣶⠡⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠔⠁⠀⠀⠀⠀⠀⠀⠁⠢⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠛⠀⠀⣿⣿
⠀⠜⠉⠔⣂⠩⠒⡻⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣀⣠⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⡀⠀⡘⠀⠀⠠⠓⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿
⠀⣏⠀⠈⠈⠠⠐⠈⢠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣿⡏
⠀⢸⠱⠊⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇
⠀⠀⢢⣄⠀⠀⠀⢀⣿⡉⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠈⣿⠇
⠀⠀⠀⠛⠿⣷⣶⣾⣿⣿⣾⣿⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⠋⠒⢁⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠘⠉⢩⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⣀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣀⣀⣿⢸⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⡏⢀⣆⣸⣦⡀⢹⣿⣿⣿⣿⠁⣰⣈⣷⣆⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣇⠈⢿⣿⡿⠃⣸⣿⣿⣿⣿⡀⠹⢿⣿⠟⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⣿⢸⣿⣿⣧⣄⣀⣠⣴⣿⣿⣿⣿⣿⣷⣤⣄⣀⣤⣾⣿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⣀⣀⡀⢸⡇⣿⡟⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠙⣿⣿⣿⡇⢀⣀⣀
⠀⠀⠀⢠⣿⡿⢿⣿⣿⣇⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣷⣿⡿⢯⣻⡄
⠀⠀⠀⠘⣿⣿⣾⣿⣿⣿⣿⡇⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⣿⣿⣿⡿⣿⣿⣿⣿⠁
⠀⠀⠀⢠⣿⣿⣿⠃⢸⣿⣿⡇⠀⢀⣤⡀⢀⣤⡀⢀⠤⣄⢠⢤⡄⣠⢤⡄⣿⣿⣿⡇⠈⣿⣿⣿⡄
⠀⠀⠀⣾⣿⣿⠇⠀⢸⣿⣿⣇⠀⠘⠛⠃⠈⠛⠃⠈⠛⠋⠘⠛⠃⠙⠛⠁⣿⣿⣿⡇⠀⠘⣿⣿⣷
⠀⠀⢰⣿⣿⡏⠀⠀⢸⡏⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡇⠀⠀⢹⣿⣿⡆
⠀⣠⣿⣭⣿⡀⠀⠀⠘⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣿⣭⣿⣄
⣾⣿⣧⣨⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⠁⠀⠀⠀⠀⢰⣿⣿⣅⣽⣿
⠻⡿⠀⠀⠘⡿⠃⠀⠀⠀⠀⢰⡶⠶⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⣾⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣶⣄⡀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣾⣿⣿⣿⣿⣿⣿⣷⡄⢠⡾⣡⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⢀⣀⣤⣤⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⣼⠃⣠⣾⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⠀⠀⠀⠀⢸⣧⣾⣿⡟⢀⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢱⣟⣿⡆⠀⠂⢸⣿⣟⡁⠖⠻⢿⠛⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠛⠛⠁⢠⠆⢸⣿⣿⣿⣦⣠⡀⠸⢷⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣷⣶⣾⣿⣦⠾⣿⣿⣿⣿⣿⡿⢔⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢹⣿⣿⣿⣿⣷⣀⠈⠛⠿⣿⣿⣿⣦⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣴⣿⣿⣻⣿⣿⣿⣿⣿⣖⠠⠀⠀⣉⠉⢙⣉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⣭⣛⣿⡻⣟⢯⣻⣿⣿⣿⣶⣶⣶⣶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠉⠙⠓⠮⠛⢯⣿⢿⣗⡿⣿⣿⣿⢿⣶⣾⣿⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢫⣤⡀⠀⠀⠀⠀⢦⣻⣦⢿⣿⣿⣾⡿⣿⣿⣿⠗⢻⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃⠿⠿⠇⠀⠀⠀⠀⡈⢳⣻⣿⣥⠙⠛⠿⠆⠀⠀⠀⠘⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡀⠀⠀⠀⠀⠀⠀⠘⢼⣧⢻⣿⢧⣄⠠⠀⠀⠘⡇⠨⣼⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠐⡈⣿⠀⢻⣷⣷⠀⣀⣀⠀⠻⠒⠉⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠛⠀⠘⣿⢿⠉⠙⠀⠀⠀⠀⠀⠁

⣤⣤⣄
⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⣶⡀
⣤⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠃⣼⠶⠷⡀⢄⠀⠀⢢⣄
⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠠⠁⠤⠀⠖⠃⢰⡀⠀⢀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡔⠒⠒⠶⢢⣬⣑⣾⠁⠈⠁⠀⠀⢂⠀⠀⠀⠀⠄⠀⣳⠀⠀⠀⠀⠀⠄
⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠂⠀⠈⠂⠀⠀⠀⠀⠀⣈⣱⣶⠖⠉⣠⣵⣿⣧⠀⠀⠀⢠⠖⡁⠉⠀⠀⠀⠀⠘⠖
⣤⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣽⠋⠀⡰⢿⣿⡿⢿⣯⡄⠀⠀⠀⠀⠘⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⢨⢀⣰⠏⢙⡷⠏⠀⠀⠀⠛⠻⢣⣼⣿⠀⢻⡆⠀⠀⠀⡀⠀⠀⠴⠚⠃⠠
⣤⣶⣦⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⠆⡾⣀⠀⡰⠂⠘⢿⣿⠟⠁⠀⠐⠌⣹⠀⠀⠀⠙⠻⢶⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀⡀⠀⠀⠀⠐⠃
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⢰⡇⣀⡞⠁⠀⠀⠺⣿⠀⠀⡀⢀⡎⠁⠀⠀⣄⣀⣀⣸⡟⠟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠜⠂⠈⠁⣀⠠⡤⢶⣿
⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⡧⠞⢀⡜⠖⢿⡄⠀⣤⡙⠀⠀⠀⢰⠈⠉⠉⠉⠹⠁⠀⣄⡀⠤⠖⠂⠉⠉⠃⠀⠀⠀⠀⡛⢠⡗⠦⠉
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⣠⠃⢤⣈⣁⠆⠘⢇⡀⠉⠀⢘⣃⠀⠼⡆⠁⠀⣀⠠⠤⠚⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡾⠘⢀⣠⡀⠈
⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠈⠷⣩⠃⠶⠘⠷⠤⠶⠾⢉⣿⣷⡿⡙⠛⢠⠄⠰⠆⠃⠃⠀⠀⠀⠀⠀⠀⢀⠂⢀⠁⠀⠁⠀⠈⠁
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⠀⠀⠀⠀⣈⠁⠀⠄⠀⠰⠀⠀⠀⣾⢁⣈⣿⣾⠆⠈⠠⠀⠂⣀⣀⢀⡀⠠⡤⠔⠒⠈⠀⠈⠀⠀⠀⣤
⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⣝⢷⣤⣄⠴⠃⠀⠀⢸⡆⠀⢡⣀⣄⣹⣮⡽⣏⣽⡅⠴⣦⣜⠿⢻⣟⡿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡌
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⣶⠃⠀⠀⠀⠀⠀⠙⢡⣀⠀⣒⣤⡀⠈⠛⢃⠀⠀⣀⡈⠁⠘⠈⠳⣦⠀⠀⠀⢠⣀⠀⠀⠀⠀⢠
⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⠀⠀⠀⠀⠠⡾⡟⢉⣋⠀⠀⠀⠴⠌⢷⣁⢉⠁⢄⠠⠖⣸⠏⠀⠀⠀⠀⠘⠁
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠄⠒⢩⣉⡉⠉⠀⠀⣶⠲⠒⠀⠛⠓⢷⡶⠤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠚⠦⠀⠠⡄⠀⠀⠀⢦⣴⣾⣿⠿⠿⢷⣶⣤⡿⣼⡶⣶⣦⣴⣶⣦⣤⣶⣾⡀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠦⠴⣄⡀⠀⠀⠀⣠⣾⡿⠁⢡⣶⢣⣼⣟⠉⢲⡈⠃⠈⠉⠉⠙⣿⣥⣶⠤⣿⣧
⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⢦⡀⠘⢿⡀⠀⠀⣿⢬⣿⣠⡴⠛⠛⠁⣰⠆⢸⡇⠀⠰⠀⠀⠘⠞⣿⡈⢃⡈⠋
⠁⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⢀⡐⠎⣉⡄⠈⠁⠀⢀⠛⠋⠀⠀⠀⠀⣰⡶⠿⡿⠚⠃⠀⠘⠀⣨⠷⠟⠉⠹⣆⠀⠀⠀⠀⡄
⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠈⠸⣇⢀⠛⠉⠉⠉⠉⠉⠁⡀⠀⢀⣀⣿⠀⠀⠀⠀⢳
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢘⠟⠀⠀⠀⠀⠀⠀⠘⡏⠉⠛⠉⠃⡀⠀⠀⠀⠌⢻⡆
⣶⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⡹⠀⢘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⣤⡾⠛⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠠⣀⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢰⢤⠃⠀⠘⣷⠿
⣶⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣤⣌⠁⠄⢀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⡀⣨⣤⣤⠞⠀⠀⢀⣀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⡿⠃⡀⠀⠀⠈⢉⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠿⠵⣎⠉⡬⠆⠀⠀⠀⠀⠀⠄
⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠿⠂⠘⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣤⠆⡐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⠃⠀⠀⠀⠀⠙⣆⠀⢸⣶⡶
⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠖⠋⠀⣀⠀⡀⠀⠀⠀⠀⣠⠗⠀⠀⠀⠀⠀⠀⠀⢀⠛⠃⠀⠀⠀⠀⢠⡴⠉⠈⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡏⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠚⠁⠆⠛⢹⡟
⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣧⡀⠀⢀⣴⢠⣴⣾⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠠⠛⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⣳⡀⠘⣻⠟⠁⠀⠀⠀⡀
⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢈⠁⣴⣶⣶⣤⣄⣀⣾⠀⠀⠀⠀⠀⠀⢀⣤⣶⠀⠀⠀⢀⡟⠙⠻⢿⣆⠀⠀⠀⠀⠊⠉⠑⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣀⣼⡏⠀⠀⠀⢹⣏⢁⣠⣤⣤⣷⡒⠲⣞⣷⣿⣧⣤⣀⠘⢀⠀⠀⠀⡀⠐⠂⠀⡀⠀⠀⠀⢰⡇
⠿⣿⡿⠀⠀⠀⠀⠀⠀⢀⠀⠇⠀⠹⠿⠀⠀⠀⠀⢸⣿⣷⠿⣿⡿⡿⠛⠃⠉⠀⠀⠘⢉⠻⣿⣶⣿⣶⡄⠀⠈⠀⠀⣈⣉⣰⣌⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣆⠀⠀⠈⠙⠀⠀⠀⠉⠹⣿⢛⠻⢤⡶⣶⣴⣦⣽⣶⣤⣼⣿⣷⣦⣌⣈⢱⣤⣀⡀⢸⣷⡟⠋⠁
⠿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣠⠶⠀⠀⠐⠂⠠⡜⢻⠛⠿⠟⠀⠀⠀⠉⠙⠛⠛⠧⡤⠀⠉⠉⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣅⠀⠈⠉⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠿⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢔⠀⡀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⣠⢉⣠⣴⣤⣔⣷⣦⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠙⣿⣿⠟⠉⠻⣿⣽⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠀⠀⠀⠀⠴⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣴⣾⣿⣆⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⡶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⠀⠀⠀⠠⠄⠈⢻⣿⣿⣿⣷⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠖⠀⠀⠀⠐⠀⢀⣀⡀⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⢂⠀⠂⠡⠀⣰⣿⣿⣿⣶⣌⢻⣿⣿
⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⣾⡿⠋⠋⠀⠀⠉⠉⠉⠻⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠂⠀⡾⣿⣧⠄⠂⠀⡀⠄⡉⠤⠀⠈⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠶⣾⣽⡶⣆⣴⠿⠠⠀⠐⠀⠀⠀⠀⠀⠂⡄⠀⠀⠙⠻⣿⣿
⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣆⣀⡒⣚⡭⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻
⠀⠉⠺⢷⣀⠐⠂⠦⡤⢀⢀⠀⠀⣀⠀⢀⣠⣤⣾⡾⠛⠛⢛⠿⠉⠙⠁
⠀⠀⠀⠀⠉⠙⠛⠚⠅⠦⠖⠖⠀⣠⠤⠀⡀⠊⠊⠀⡠⠖⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⣋⡴⢚⣭⠀⡠⠀⠠⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⣀⣴⣞⣯⠟⠋⠉⢠⣎⣑⡸⠁⠀⡖⠊⣧
⡀⠀⠀⣠⣴⣾⠿⠛⠉⠀⠀⠀⢀⣾⣹⣿⠂⠀⠀⠙⣯⠜⣇
⣿⣶⣬⣍⠉⠀⠀⠀⠀⠀⠀⠀⢸⣷⡼⠃⠀⠀⠀⠀⠘⠛⠿⠿⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣾⠅⠘⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠄⣤⠴⣿⣿⣿⡆⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠉⢸⣿⡿⣇⣻⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⢶⣶⠿⠋⢠⣿⠉⣹⢶⣤⢲⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⡇⠀⢸⠁⠀⠀⣘⣿⣿⠉⠲⣇⠘⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠇⠀⠀⣀⣤⣿⣿⣶⣶⣶⢾⣯⡙⢿⣿⣷⡄⠙⣷⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡔⠁⠺⢿⣿⣿⣿⣿⣿⣿⠈⣿⣷⡠⢻⣯⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⢟⣱⣦⣼⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⣿⣿⣿⣤⣄⣀⣤⣴⣿⣿⡿⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣰⣶⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠦⠀⠀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡔⠓⢦⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠘⢧⣤⡞⠀⠐⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣧⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣎⡆⢾⡆⠀⠀⠀⠀⠀⠀⠀⢸⡁⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⣧⣀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⢿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢿⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⡇⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣸⡆⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⣼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣛⣛⣛⣻⣏⣉⣻⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢒⡾⠁⠀⠀⠀⡤⢤⡀⠀⠉⠉⣷⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡹⠊⢩⡇⠀⠀⠀⠀⡹⢯⠀⠀⠀⠀⢸⠓⢞⠙⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠯⣈⡿⠁⠀⠘⡇⠀⠀⠀⠀⢙⣚⠀⠀⠀⠀⣸⠀⠀⢳⣏⣈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠄⢀⠈⡇⠀⠀⠀⢻⣄⣀⣀⣀⣙⣴⣁⣀⣀⣠⠇⠀⠀⠀⡟⠀⡠⣄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢾⣠⡇⠀⠀⠀⠀⣼⣅⣽⠈⠉⠉⠉⢹⣅⣽⡄⠀⠀⠀⣄⣸⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣄⣼⡀⠀⠀⠀⣸⣄⡤⣇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠺⠧⣤⡼⠧⡄⢀⠴⠿⣤⡤⠿⠒⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⢐⣒⣒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣩⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣮⣗⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠉⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣷⣮⡳⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢀⠉⠀⠉⠉⠉⠉⡛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠛⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣤⣤⣤⣀⠀⠐⠺⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⢟⠝⠒⢄⠨⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠋⠁⠈⣝⢷⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣿⠀⣆⠀⠀⠀⢠⣤⣼⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡿⢟⠂⢄⠀⠀⠀⣸⡇⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢚⡷⠃⣑⣥⣤⡶⢻⣀⠘⠗⠷⠶⠶⠶⠶⠶⠲⠶⠶⠶⠶⠒⣶⠶⠶⢖⣶⠖⠲⠞⢓⡾⢂⡐⠂⠐⣈⠟⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⡤⡖⠒⠯⣠⠊⠁⠀⢠⠏⠃⠀⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⠁⠀⠈⣧⢔⠯⠟⣣⡴⠋⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢸⡏⡀⠀⡀⣙⠀⠀⠀⠸⣷⢀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⠰⢙⣿⣃⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⢳⣬⠸⣃⡕⠀⠀⠀⠀⠙⢮⡌⠱⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠂⠉⢮⣳⣻⠛⣣⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠹⣄⠀⠀⠀⠀⢸⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠃⢸⡇⢹⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢇⣈⢀⡇⠀⠀⠀⠀⠈⢷⠤⠰⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠘⣦⠼⠇⠀⢨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⢙⣒⣒⣒⡒⢦⣿⣀⣀⣮⣤⣦⣤⣌⡙⢦⠀⠀⠀⠀⠀⠀⢠⢷⣤⣴⣿⣿⣟⠀⠀⣙⣛⣒⣒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣯⣷⣶⣿⣿⣿⣿⣿⣦⣸⡿⠿⠿⠿⠿⠿⠿⠿⠾⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⣫⣶⣶⣿⣿⣿⣿⣿⣧⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⢰⡶⠀⠀⠀⠀⠀⡠⢶⡆⠀⠀⠀⣠⡶⣆
⠀⠀⠀⠀⢿⣯⣷⣄⠀⠀⠀⢁⡈⠒⠀⠴⠄⠰⣇⡈⠀⠀⠘⢧⠀⠰⣶⣶⡜⠉⡼⠃⠀⠀⡰⢀⢀⡻⠆
⠀⠀⠀⠀⢸⡛⡿⣿⣆⠀⢀⠈⠓⠂⠀⠀⠀⠀⠛⠁⠀⠀⠀⠰⣿⣴⡿⡍⠃⣩⠞⠀⡀⡘⢓⣄⣏⢻⣦
⠀⠀⠀⠀⠈⠳⠅⢈⣻⠏⣻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⣌⣛⣷⣾⣿⡿⢀⣿
⠀⠀⠀⠀⠀⠀⢠⣮⣿⠿⠛⢿⣷⣶⣦⣀⡀⠀⠀⠀⠈⢹⠋⠉⠀⠀⡒⠀⠙⣦⣴⣿⣿⣿⠻⣮⠁⠋⠁
⠀⠀⠀⠀⠀⠀⠟⠛⠁⠐⠦⢄⣩⡿⣿⣼⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠜⢿⣦⣉⣉⣭⠆
⠀⠀⠀⠀⠀⢀⠀⠀⠦⡘⠀⠀⠀⠀⢰⡤⠤⡀⠀⠐⠒⠚⠒⠒⠀⠀⣠⢤⣤⠀⣉⡉⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠲⡶⠂⠀⢀⣴⣿⡇⠈⢿⣄⠚⠶⢄⠀⠀⠠⢀⡠⠶⠒⣸⡏⠀⣿⣿⡄⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣼⣧⡀⠀⢿⣦⠁⠈⢃⢀⣶⣌⠀⢀⣼⡷⠀⠀⣿⢻⣿⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⡿⠿⠃⠀⢐⡦⠀⠀⠘⠙⡿⠀⠀⠀⢰⠆⠀⢘⡇⢻⣿⣇⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠿⣦⣄⠀⢌⢉⠀⠀⠀⢂⠀⠀⢻⣴⡣⠀⠈⠀⠀⣠⣜⠋⠀⠀⣽⡇⣾⣿⡏⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠙⠟⠃⠈⠸⣄⣀⠀⠊⠀⠀⡼⠇⠀⠀⠀⠀⠀⠐⠿⡀⠀⢀⣾⡥⠛⠉⠀⠀⠀⠙⠛⠂
⠀⠀⠀⠀⠀⢴⣷⣤⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⢠⣾⣦⣀⠀⠀⠈⠂⢀⣄⣀⠀⠀⠀⠀⠀⢀⣤⣶⡄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣆⠀⠀⣀⣤⣶⣾⣷⠀⠀⠀⠀⣈⣋⣙⠛⢲⣄⠀⢹⡞⣷⢿⣦⡀⢀⣴⣟⣾⣿⡆
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠋⠀⠘⠶⡌⣻⣿⣿⣿⣿⣿⣷⣿⡁
⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⡟⣿⣿⣿⣿⠉⠀⣀⠀⠀⠈⠂⠂⠀⠐⠀⠀⠀⠀⠸⣷⣽⣿⡃⠛⠘⣿⣷⣿⠅
⠀⠀⠀⠀⠀⠀⠙⠋⠁⠈⠀⠻⣿⡿⠋⠀⠀⡿⢴⣦⣀⠀⠀⠀⢀⣀⢠⣤⠀⠈⠙⠛⠛⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣄⠀⠀⢠⣄⠀⢸⡟⣻⣷⡶⣶⣬⣿⣿⡭⠃⠀⣾⣤⣄⠈⠃⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡸⣷⠛⠋⠉⠀⢰⣿⣿⣆⠀⠉⠙⠃⠹⠃⠹⠿⠿⠃⣠⣴⣿⢟⣻⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⠰⠋⠀⢀⣀⠀⠀⠀⠸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣸⡿⢛⣵⣸⡏⠀⢈⠀⠀⠐⠩⣿⡆
⠀⠀⠀⠀⠀⢀⣀⢶⣿⡽⠃⠀⠀⠠⣮⡙⣿⣿⣿⣶⠀⠀⠀⣴⣿⢛⡱⣾⢻⡯⡿⠁⠰⠁⠀⠀⠐⣀⣹⡇
⠀⠀⠀⠀⠀⠸⣿⠟⠋⠀⠀⠀⢀⣴⣿⣿⣿⣼⢿⣿⠀⠀⢸⡟⣡⣾⣷⡟⠿⠟⢃⡀⠀⠀⠀⠀⠈⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣿⣿⡎⠝⢆⠈⠈⠁⠀⢀⠀⣉⣉⣁⠀⠐⠒⠹⣿⣅⡒⣦⢤⣄⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢲⣶⣿⣿⣿⣿⢿⣿⣿⣆⠀⢀⡤⢐⣲⣤⡞⠛⠛⠉⣠⣶⣿⡉⢻⠟⠟⣛⣢⣩⣵⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣽⠿⠛⢠⣾⡿⠟⠁⠈⠁⠀⠀⠀⠳⣶⣶⣶⡶⠏⠙⠛⠛⠛⠙⠛⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⡄⠀⠀⢠⠏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⡘⡇⠀⠀⢸⢃⠀⠀⠀⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠈⠛⠀⠀⠀⠧⣵⠀⠀⣮⠼⠀⠀⠀⠛⠁⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣦⠰⣶⣤⣀⡀⠁⠈⢀⣀⣤⣶⠆⣴⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠉⠀⠈⠉⠉⠁⣴⣦⠈⠉⠉⠁⠀⠉⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠰⠀⢀⡴⣆⠀⠿⠿⠀⢰⢦⡀⠀⠆⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠢⠰⡁⢰⠀⠀⠀⠀⠆⢈⠆⠐⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⡁⠠⠄⠈⠀⠈⠀⠀⠀⠀⠁⠀⠁⠠⠄⢈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣧⢤⠀⠀⠀⠹⠷⠾⠏⠀⠀⠀⡤⣼⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠻⣽⡤⣷⠁⠀⠘⠃⠀⠘⣾⢤⣯⠟⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢙⠲⡤⠄⠀⠀⠠⢦⣖⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠁⠀⠰⠆⠀⠈⠉⣠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠄⢀⡀⠠⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣷⣶⣶⣶⣶⣶⣦⡀
⠀⠀⣠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢃⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣥⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢏⠁⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠋⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠁⠀⠘⠋⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⢘⠓⠄⠀⣠⣿⣿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢠⣄⡐⠁⠙⡀⠈⠉⠉⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠁⡠⠐⢲⣿⡟⠀⠀⠀⠐⢤⣀⣠⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⣉⣽⡶⣶⣤⢴⣤⣎⠀⢠⡿⠃⠃⠀⠀⠀⣠⣾⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡝⠀⠀⣰⠇⠀⢰⠞⣉⠆⠀⠈⠻⣿⣽⠈⠠⠑⢢⣴⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⡟⢷⡀⠀⢿⠀⠀⠀⠀⠉⢀⠵⠀⠀⢹⣿⣧⣠⣴⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣉⡀⢨⣿⣦⣼⣷⣆⠀⠀⠀⠀⠰⣄⠀⣤⣙⣿⣿⣿⣿⣿⠋⠛⠓⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⡩⣩⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣿⣾⠛⠙⠉⠻⣿⣿⣿⣧⣄⡤⡁⢼⣿⠿⠿⠿⠛⠛⠛⢛⠉
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⣼⣧⡍⢸⠇⢾⣿⣿⣿⣇⠀⠀⢀⣾⡛⠁⠀⢀⣤⣴⣿⣿⡿⠛⠿⣷⣶⣯⣿⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣛⣻⠿⠳⣆⢸⣿⡇⢀⣼⠋⠉⠉⠉⢿⡇⠀⠀⡀⠙⠿⣿⣾⠂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠳⣮⣻⣿⣿⣾⡏⠀⠀⠀⠀⢸⣇⢀⣀⣀⣰⡄⠙⣿⡔⠀⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣄⣀⣨⣿⣿⣿⣿⣿⣦⣄⣠⣤⣿⣿⣿⡿⢿⣿⣿⣿⣿⣟⣀⣿⣷⣶⡂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⣈⣛⣛⣛⠻⢿⡿⠉⣀⢛⣿⣿⣿⣿⣿⣿⣾⣿⠟⠻⣿⣿⣿⣿⣟⠿⣧⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣀⠴⠾⣿⠿⠛⣭⡗⠈⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣾⣿⣿⣿⣟⠛⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣠⡴⠛⠁⠀⢢⣝⣻⣿⠿⢻⣿⣿⡿⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠢⠴⠛⠁⠀⠀⠀⢾⣆⣈⣭⣴⠾⠛⣉⣤⣶⣿⣿⡿⠟⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠡⠏⠔⠇⠀⠀⠀⠀⠹⠿⣿⣿⣛⣩⣴⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⣿⣯
⠀⠀⢸⣿⣿⣻⣿⣟⣿⣾⣅⣀⢀⠀⣂⣀⣀⣠⣼⣾⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠉⠈⠙⠻⣿⡟⢿⢿
⠀⠀⠘⣿⣿⡿⢿⡿⢿⢯⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠋⠁⠀⠀⢀⠀⠈⠀⠈⠏⠑
⠀⠀⠀⠈⠙⠁⠈⢙⣛⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠠⠌⡽⠛⣿⡳⠾⠿⠛⠻⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⢀⠀⢡⡇⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⠌⠁⠢⣀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⢁⠉⠀⠀⠐⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⣠⠂⢀⠀⠀⠀⡨⠓⠒⢛⣞⠛⠩⠰⠁⠀⠀⠀⠀⢀⠜⠀⠀⡠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠀⣀⡠⣞⠇⣭⠁⡵⠂⠈⢳⣈⢃⡸⠀⠀⢸⠁⠀⠀⠀⠀⡠⠁⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢥⠀⠈⢳⠒⠦⠂⠀⣀⠄⠀⠁⢎⡀⢀⠂⠀⠀⠀⠔⠈⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡀⠯⠗⠀⠀⠀⠀⠀⠀⠀⢀⡔⠳⡒⢆⠉⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠜⠒⠠⢄⠀⠀⡄⠄⠐⠊⢠⠂⠀⣨⠀⠂⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⠄⡀⠀⠑⠄⠐⠖⠘⣠⠁⡆⠊⠀⠀⠸⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠀⠈⠀⠁⠠⡘⠢⠀⠁⠔⠁⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠂⠀⠀⠀⠀⠀⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠂⡇⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡞⠀⠀⡈⠀⠀⠀⠀⠀⠀⢐⣃⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠄⢰⠒⠀⠀⠈⠉⠀⠀⠀⣽⣀⠀⠤⠤⠀⠲⠗⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⠤⡄⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠌⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⠤⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢫⢞⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡀⢠⡆⠀⠀⠀⠀⠀⠀⠙⠦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠞
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣴⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⠤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡔⠀⢸⠛⠓⠒⠒⠒⠒⠒⣶⠋⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⡇⡜⠉⡧⠀⡠⠒⢦⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠓⣾⠀⠀⣞⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⠚⠁⠀⠑⠠⠊⢸⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⠀⠹⣤⠜⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⠖⡤⣀⣀⣠⡤⡄⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣸⡛⢦⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠓⠧⠿⣼⠤⠦⠊⢸⡖⣃⣀⡷⠀⠀⠀⠀⠀⡰⢧⣈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠲⠶⣶⠶⢦⠤⠤⠼⠷⠛⠁⠀⠀⠀⠀⢀⣼⡙⢲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⢉⣽⣤⣤⣄⣀⣀⡀⠀⠀⠀⣠⠞⢤⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⢋⣉⣀⣀⣈⣓⣉⠀⠀⠀⣀⣩⡭⢿⣧⠾⡅⣱⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠁⢰⡏⢫⣘⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢦⣸⠀⠀⠋⡉⡉⠉⠙⠛⢲⠀⢻⠀⢸⣙⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣀⣸⡆⠀⡟⢤⠞⠀⠀⠀⢸⣿⢸⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⡟⡇⠀⣇⡠⢤⡀⠀⠀⢸⣧⢸⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢻⠁⣧⠀⢸⡓⢚⣀⣀⣀⣸⠟⢸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠒⣾⠀⣿⠀⠀⠉⠉⠉⠉⠉⠉⠀⢸⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⡿⠀⢸⡆⠤⠤⠤⠤⠤⠤⠤⠤⢼⠖⣋⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⣠⡟⢦⠘⠓⡖⠲⡞⠒⠒⠒⠒⠲⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡎⠀⢃⠈⡆⢠⡉⢙⡇⠀⠀⠀⠀⠀⣧⡤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠇⠀⠸⠿⠇⢸⣉⣹⠀⠀⠀⠀⠀⠀⢿⠤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⣼⠀⠀⠀⠀⠀⠀⢸⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⣾⠀⠀⠀⠀⠀⠀⢸⡓⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠦⣯⡀⠀⠀⠀⠀⢠⠾⡋⢹⠆⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠙⠶⠃⣹⠀⠀⠀⠀⢸⣄⠉⠉⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣤⠤⠴⢒⡽⠀⠀⠀⠀⠘⠮⣍⣓⡒⠒⣊⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠐⣿⣷⣄⢰⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⡆⣠⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠳⣽⣟⣉⣻⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣯⠞
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⢻⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⡟⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠘⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠃⠸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⡰⢛⣿⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣿⡛⣆
⠀⠀⠀⠀⢀⡾⢡⣾⣿⣿⣆⡀⠙⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⢀⠉⢀⣠⣿⣿⣷⣌⢷⡀
⠀⠀⠀⢰⣿⣴⣿⣿⣿⣿⣿⣿⡆⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⣿⣿⣦⣿⡆
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⢀⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠻⣿⠿⠋⠻⠿⣻⣿⣿⣦⣄⠏⢸⣿⣿⣿⣿⣿⣿⣷⣄⣠⡤⢤⣄⣀⣾⣿⣿⣿⣿⣿⣿⣏⠹⣰⣤⣿⣿⣿⡿⠟⠙⠿⣿
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡿⢦
⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⣽⠻⠿⠿⠛⠻⠋⠁⣠⣿⣿⡄⠈⠛⠛⠛⠿⠿⠟⣯⠀⠀⠘⢿⣿⣿⣿⣷⡎
⠀⠀⠀⠀⠀⠙⢿⣿⡿⣧⢠⣄⠀⠀⠀⠀⠀⠈⠛⠃⢠⣿⣿⣿⣿⡄⠘⠛⠁⠀⠀⠀⠀⠀⣠⡄⣼⢿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣤⣤⣶⣶⣆⢤⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⡤⣰⣶⣶⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠛⡏⢹⠛⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢿⣦⡀⠈⠀⠃⠀⠂⠘⠀⠁⢀⣴⡿⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⣦⠈⢇⠉⠛⠐⠤⠠⠄⠄⠒⠛⠉⡸⠁⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⠀⠘⡓⠡⠔⠄⠠⠢⠌⢊⠃⠀⠀⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⣀⣠⣶⣶⣆⣀⠁
⣿⣿⢟⠻⣛⡛⢻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⠀⠀⢠⡴⠏⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠁⠀⠀⣿⣿⣿⡄⠈⣹⠾⠃⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠋⠉⠉⠉⠠⠔⠀⠀⠀⠀⣀⢤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢳⡀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣍⡳⠂⠀⠀⠠⣄⠠⠀⣀⣊⠁⠀⠀⠀⠀⠀⠁⢣⢈⣄⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⢠⠀⠁⠀⠀⠀⢀⠺⢦⡴⠀⠉⠸⠠⠒⠂⠀⠀⠀⠈⠩⡁⠈⠂⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠠⠀⢸⠀⠀⠀⣠⣀⠋⠀⢻⠐⠄⠆⠀⠀⢐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⡀⠂⠸⠀⠀⠀⢣⣤⣄⡀⠈⠀⠀⣣⣴⡴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠚⡛⠉⢙⠗⠾⠶⠛⣿⠈⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢤⣠⠶⠋⠀⠀⠀⠀⠘⠛⢤⣇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠁⠀⣠⡾⡁
⠀⠀⠀⠀⠀⠐⠃⠀⢀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠉⠂⡁
⠀⠀⠀⠀⠀⢷⡀⠀⠈⣁⡐⠁⠀⠀⠀⢠⢀⡘⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢍⡜⠈⢋⠀⠀⠀⠀⠘⠻⠞⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⡰⣤⣤⠤⡀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠠⠃⠀⠤⡀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣧⣄⣠⡄⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠄⠀⠀⢻⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠚⠁⠛⠲⠗⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠⠂⠀⠀⢀⣠⠿⠛⠛⠙⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣾⣁⠀⠒⢻⠋⠁⠀⠀⠀⠀⠀⠀⢄⠸⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢶⡜⢿⡟⢿⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⣗⠊⣿⢸⡇⣫⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣰⣿⣧⣟⣼⠁⠴⣇⣁⠀⠀⢀⣈⣉⣓⣦⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢄⠀⣯⡿⠀⠀⠘⠻⠯⠿⠇⠃⢠⣾⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣛⣳⣺⣾⣿⠗⠌⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡏⠹⡦⠀⠀⠀⠀⠀⠀⠀⠠⠤⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠀⠂⠀⠀⠀⣀⣤⣌⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡋⢹⡿⣿⣷⣄⡴⠀⠀⠀⠀⠉⣭⣭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⡇⢈⣿⣿⣿⣶⣦⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣝⢿⣿⣿⡿⠋⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠉⠻⣿⠿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠿⠿⣷⣦⣀⣤⢙⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠦⣄⡀⠈⠻⠁⠙⢭⡙⢿⣷⣽⣿⣿⣿⣿⣿⣏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣠⡔⠈⠄⠀⠙⠆⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⡦⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠠⣄⠈⢿⣿⣿⣿⣿⣿⡿⠿⠾⢿⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⣼⣷⠦⠀⠐⠃⡜⠓⠖⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠄⢨⡀⠟⠆⠈⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠠⡀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢬⡅
⠀⠀⠀⠀⠀⠀⡃⠂⠁⠐⠟⠀⠀⠀⠐⢦⠀⠈⠂⠀⢀⠀⠀⠀⡄⠀⢄⡰⣄⠉⠲⡘⠀⣠⠁
⠀⠀⠀⠀⠀⠀⢓⣠⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣄⠀⢳⣄⠀⠰⡀⢀⢳⣮⢠⠈⠄⢀⠔
⠀⠀⠀⠀⠀⠈⠀⠀⠈⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⡁⣄⠩⠃⠀⡘⢶⢦⡽⠇⠀⠅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠃⠀⠀⠀⠐⢦⡄⠀⠀⠀⠀⠈⠀⡍⠣⠀⠠⡀⠓⢂⣐⠒⣀⣠⠶⠂
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠂⠀⠄⠀⠀⠀⠱⠋⠤⠀⠀⠀⠢⡂⡂⠀⠰⢀⣈⠈⡉
⠀⠀⠀⠀⠀⠀⢁⠐⠠⠄⢀⠀⠀⠠⠄⠀⢀⠀⠀⠁⠄⠀⠀⠀⠈⢠⡁⠁⠉⠀⠉⠈⠋⠙⠓⠂
⠀⠀⠀⠀⠀⠠⠀⠁⡀⢀⠀⠀⠁⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⢀⠌⠀⠀⠀⡈⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⢠⠈⠁⠈⡄⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⢀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠃⠈
⠀⠀⠀⠀⣀⠁⠈⠀⠄⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀
⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⢅⠁⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠡⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢱⡀⠀⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣋⣷⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠶⠟⠻⠬⣝⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠄⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣏⣀⠀⠀⠀⠀⠀⠀⣴⠚⣒⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠶⢒⠒⡛⢛⣉⡙⠷⣌⣾⣿⢛⡛⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠘⣿⣿⡇⢠⣿⡋⢻⠀⣿⣽⣿⣼⢰⣋⢺⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠐⣿⣿⡅⡟⢿⣿⡿⠀⠟⣿⣿⡿⣟⢟⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣒⣒⣿⣦⣠⣴⡿⠚⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠛⣷⡈⣿⡏⠈⣽⠛⣙⢳⣠⣤⣤⣴⣾⡾⣿⣽⠿⠿⠿⣻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣏⣽⣧⣿⣿⢸⣿⣿⣯⣾⣿⡿⣿⣷⠿⢋⣤⡶⠖⠒⠦⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣾⡿⣡⡶⠋⣠⣶⣾⣿⣿⣦⣬⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣻⣿⢯⣾⠿⢿⡆⣷⣾⣿⣿⢟⣾⠋⣠⣾⣿⣿⡟⠁⠀⠉⠉⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢱⡿⢡⣿⠃⢠⣿⠓⠛⠻⡿⣣⡿⠃⣼⣿⠿⣿⣿⡇⠀⡶⣡⡠⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣶⣿⣿⣂⣾⣃⣀⣀⣼⢳⡟⢁⣾⡿⠛⠠⣌⣿⣧⠘⣧⣫⣃⣸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠹⣿⣿⣿⣿⣟⣛⣿⣧⣿⢀⣾⣿⣁⠀⠀⢢⣿⣿⣄⠈⠙⠚⠋⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⠈⣿⣿⣿⣿⣶⣴⣿⣿⣷⣾⣿⡿⠉⠨⣷⣦⣿⣿⣿⣷⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⢸⣿⣾⣍⣩⣿⣿⣿⣿⣿⣿⡏⠠⣄⠙⢻⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠛⠿⡿⠟⠋⠙⠛⠉⠱⠄⠈⡗⠼⡛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣼⣿⣿⣾⣿⣟⣠⡤⠶⡏⠕⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠏⠁⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠟⠁⠀⠀⠀⢠⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⣿⠟⠁⠀⠀⠀⠀⣴⠞⢁⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣛⣷⣾⠫⠀⠀⣀⣠⣴⡿⠗⠚⢣⣠⣶⣿⡿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣶⣾⡏⠉⠉⠀⠀⣠⣿⣿⣿⣿⡿⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢳⡄⠀⣠⣾⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠿⠿⣷⢷⡾⠿⠿⠛⡽⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡇⠀⠈⠀⣀⣴⠮⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡚⢢⡀⠀⣜⠡⠂
⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⢰⢃⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠁⠀⠁⠀⠀⣀⡴⠆
⠀⠀⠀⠰⠀⠀⢰⠁⣠⢴⠶⢛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⢀⡀⡰⢃⣴⡷⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡴⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠟
⠀⠀⠀⠀⠛⠻⠞⠛⠋⢃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠏⠀⠀⠐⠈⠀⠀⠠⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣎⠀⠀⠀⢀⡠⠔⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢘⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⡀⣈⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠛⠂⠀⠀⠀⠀⠘⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠐⣄⠘⠗⣀⠖⢀⠐⠗⢀⡼⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠋⠉⣁⠀⠀⢉⡉⠉⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⠯⣥⢤⣭⠟⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⣤⣤⣤⣤⡴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠒⠚⠒⠚⠚⠛⠒⠛⠛⠚⠒⠒⠒⠓⠚⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⠠⡆⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡾⠀⢨⠀⣷⠀⢠⡞⠁⢀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢀⣽⣄⠈⠋⣿⠀⠀⢧⡀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣟⣁⣴⡋⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡴⠋⠀⠉⠛⠶⠾⡆⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠙⢦⡀⠙⢦⣀⣀
⠀⠀⠀⠀⠀⠀⢀⠶⠉⠉⠃⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⠀⢸⡇⠀⠀⠀⠀⠙⢷⡎⠁⡀⠙⠆
⠀⠀⠀⠀⠀⠀⠘⣀⠐⠗⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣾⠀⠀⠀⠀⠀⠀⠐⣆⡈⢉⣠⡆
⠀⠀⠀⠀⠀⠀⢀⡟⢳⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠁⡄⠀⠀⠀⠀⠀⠀⡰⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⠙⣏⠹⣇
⠀⠀⠀⠀⠀⢀⡞⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘⠢⠄⠀⠀⠤⠚⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸
⠀⠀⠀⠀⠀⣾⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣧
⠀⠀⠀⠀⡸⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢠⡿⠶⠶⠶⠶⢶⡟⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡄
⠀⢀⡴⠞⢳⠔⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡾⠁⠀⠀⠀⠀⠀⢻⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⣟⠙⠓⢄
⢰⠏⢀⡴⠛⠶⢄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢧⡀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡎⠉⠙⢦⠀⢇
⣇⠀⡞⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⠀⠀⣘⠀⡈
⡟⠀⢻⡄⠀⢀⡼⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠿⢤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⣹⣤⣀⠀⠀⠀⠀⠀⠈⢳⡀⠙⡇⠀⠙⠶⠁
⠹⣆⢀⡿⠀⠀⠳⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠀⠙⣧⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠛⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⢾⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣤⣄⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣉⣉⠁⡄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡀⣼⡀⣤⣠⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢠⡄⣸⣠⡇⢀⠀⣀⣀⣀⡀
⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠉⠳⣄
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠿⠀⠤⢤⠤⡤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣦⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢻⢟⣿⠟⠻⠿⠿⢿⣿⣿⠟⠛⠛⠛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⡟⠁⢸⠟⠀⠀⠀⠀⠀⠀⠙⠲⢷⡀⠀⢿⣄⣙⣿⣄⡿⠉⠙⢿⡟⣿⣿⣿
⣿⣿⣿⢿⣿⣿⣽⢏⣿⠏⠸⣆⠞⠀⡰⠋⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣿⣶⣾⣿⣿⣿⣿⠁⠀⠀⠀⠹⣮⡟⢻
⠉⠉⣈⠓⠮⣿⣯⣼⣯⠀⠀⢻⢀⡼⠁⠤⠖⠒⠂⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠈⣷⠋
⠁⠀⠀⠀⠀⠘⣿⡏⠈⠙⠲⢼⡟⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠐⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢻⣿⣿⡆⠀⠀⠘⣧
⠀⠀⠀⣀⣀⣀⣸⣿⡄⠠⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣷⠀⠀⠀⢿
⠀⠐⠋⠉⠉⠉⠀⢿⡇⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⢿⣿⣿⣿⣿⡟⠋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡟⠀⠀⠘⣿⣿⣿⣿⠇⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⢹⣇⠀⠀⣰⣾⣿⣿⣿⠿⠛⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣸⣦⡀
⠀⠀⠀⢘⣶⣦⣤⡄⠘⣿⡆⠀⠀⠀⠸⣿⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢰⣤⣸⣿⣿⣿⣦
⠀⠀⠀⠈⢿⣿⣧⠀⠀⢻⣷⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣿⣿⣿⣿⣶⣾⣿⣿⣏⢿⣿⣿
⠀⢤⠀⠀⢻⠁⣙⣳⣄⣸⣿⣤⣤⣶⡀⠘⢿⠀⠀⢸⡏⠀⣿⣿⣿⣿⣿⣿⠿⢿⡿⡿⠿⠛⠋⠹⣿⣿⣿⣿⣿
⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠀⠀⠀⠀⠘⢩⡿⠯⠁⠀⠀⠀⠚⠋⠁⠀⠀⠐⣶⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡋⣀⠀⠉⠛⢒⣲⣷⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⠤⠤⠉⣻⣅⢹⣿⣿⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⡐⠂⠈⠻⠿⠟⣫⣽⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠂⣿⣿⣆⠀⠀⢀⡤⠀⡰⢊⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡅⠡⢷⣾⡟⠋⠀⠀⠳⠌⢹⣂⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠓⠒⠒⠚⠿⠇⢀⣴⣧⡀⠀⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣤⣤⢤⣄⡀⠀⠀⠙⡿⠋⣀⣴⣾⣿⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⢉⣿⣿⡿⠁⣐⣶⠋⣴⣾⣿⣿⣿⣇⣀⣠⣶⠏⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡻⠖⠛⠁⠁⠘⠿⠿⣦⡙⢿⣿⣿⣿⣿⠿⠿⠋⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⢀⣀⠀⣀⣀⡀⢻⣿⣶⣝⡻⢿⣤⣴⢦⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⣿⣿⣿⣁⢿⣟⢻⣿⣿⣿⣿⣿⠃⠨⢯⣤⡅⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋⠽⠛⠁⠈⠙⡿⢿⣿⣻⣽⣶⣿⡿⠁⡸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣠⡼⢀⣘⣿⡿⠿⠿⣿⢧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢻⡽⠿⠇⢸⣿⡋⠀⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⢀⣴⡾⠃⣼⣿⠃⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠙⡿⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⣺⡁⠀⠀⢠⠏⠀⠀⠀⣠⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⡞⠀⠀⠀⣼⣿⡿⠛⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⣠⣅⠀⠀⠀⠹⣿⣿⣿⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣤⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⠛⠛⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⣀⡎⠀⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠀⣄⠀⣠⡀⠸⡇⡀⠀⠀⢀⢸⠇⢀⣄⠀⣠⠀⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠻⢰⣝⣻⣄⠇⠇⠀⠀⢘⠸⣠⡟⣫⡆⠟⠁⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⠀⢌⠻⡇⡘⠲⣇⡄⢠⣸⠖⢃⠸⠟⡡⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠐⢙⠁⠁⢠⠏⢸⡇⠹⡄⠈⠈⠃⠂⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠉⢀⡭⡁⠀⠘⠀⡜⢣⠀⠃⠀⠈⣭⡠⠉⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠁⠈⠐⢡⣄⠀⠀⠀⠀⠀⠀⣠⡌⠂⠃⠉⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠻⠷⠶⠶⠾⠟⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠈⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠁⠰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⢿⡷⠈⡌⠀⠀⠀⠀⡠⠀⠀⡀⢠⠀⠀⢆⠀⠀⠀⠀⢡⠁⢾⡿⠖⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣖⠄⠀⠠⠁⡄⠀⠀⠀⠀⢠⠈⠄⠀⠠⣢⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⡔⠂⠀⠀⣻⣦⣀⣀⣴⣟⠀⠀⠐⢢⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⢠⡄⠰⡾⢡⣾⠏⠹⣷⡌⢷⠆⢠⡄⠙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠻⣷⣾⡻⢇⡨⢛⣷⣾⠟⠚⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣤⣤⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠰⠒⠀⢠⢦⠀⠀⠒⠂⠀⢀⡴⢤⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣖⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠂⠀⠐⠂
⠀⢰⡆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⠈⠀⠀⠁⠀⠀⢰⡂⠐⣴⠆⠈⡁⠀⠀⢹⠀⠀⠐⠀⠀⠈⠱⠂⠀⠁⠀⠀⠙⠋⠀⢾⠇⠀⢸⠃
⠀⠘⠃⠀⢀⣀⠀⢠⡥⣥⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠁⠀⠀⠀⢤⣤⠄⢀⣀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠛⠃
⠀⣀⢀⠀⠈⠋⠁⠘⠃⠛⠀⢀⠀⠁⠀⠀⠸⠹⠀⠀⢀⡀⡀⠀⠀⠀⠀⢸⠈⡇⠀⢀⠀⠀⠤⡿⡇
⢰⠖⠀⠀⠀⠖⠂⠀⠀⢄⠀⠀⢠⠲⠀⠀⠀⠀⠀⠒⠀⠀⢴⢓⠆⠀⢠⠖⣆⠀⠔⢦⠀⣠⠄⠀⠠⠶⠀⠀⠀⠀⠀⠰⠀⠀⣤⢶⠄
⠀⢀⠀⢰⣿⣾⣦⠀⠀⠀⠀⠀⠾⠂⠀⠀⠒⠀⢀⡄⠀⠀⠀⠁⠀⠀⠉⠀⠈⠀⢤⣬⠀⢶⣶⠀⠀⣷⠀⣴⣦⠀⠀⢀
⠀⣨⠇⠸⡟⠛⠛⠃⢰⣶⣶⠀⠀⠀⠀⠸⣿⡇⠀⠁⢠⠀⢰⣿⡆⠀⣤⣤⣤⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠉⠉⠀⠀⠉⠁⠀⠰⠂
⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢀⡀⡈⠀⠘⠉⠃⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⣶⣆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡄⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⢹⣦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣤⣭⣦⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠋⠉⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣡⣄⣤⣤⣤⡀⠉⠁⠀⢳⠀⠀⢀⣶⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡧⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠏⠁⠈⠛⢻⣂⡀⠀⠲⣷⠀⢼⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡆⠀⠀⢀⣀⡈⠻⣗⠀⠀⠜⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⢶⠶⠞⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⣀⢀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠿⣿⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠟⠉⠀⠑⣄⠀⢀⢀⣠⣠⣄⢤⡈⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠺⡟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⣽⣿⠟⢻⠉⠛⠯⠟⠳⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⣴⡾⠤⠋

⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢀⣤⣤⡀⡀⣀⠄⣐⣛⡯⠁⠹⣟⣓⡀⢄⡀⡀⣠⣤⣀⡀⠀⠀⢠⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⠆⡀⠀⢧⡺⣿⣿⣿⡟⠻⠿⠋⠁⢀⠆⢀⡀⢀⠀⢆⠀⠉⠿⠿⠛⣿⣿⣿⣿⣢⠅⢀⡀⢤⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠾⣼⣷⣦⢍⡛⠿⣿⣷⡈⡤⠀⠠⠔⠶⣼⣿⣿⡶⠇⠤⠀⠠⡌⣰⣿⡿⠟⣛⢥⣶⢿⡼⢆⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢿⡆⠁⡀⣶⠋⠉⣤⣤⢹⡏⠀⢠⠂⢀⠀⠀⠀⠀⠀⢀⠀⢢⠀⠈⣿⢡⢄⣌⠉⢻⡆⡀⠁⣾⠇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡜⢧⣹⡆⢴⣽⠋⣾⡉⠓⢭⣄⡈⠃⠀⠀⠀⠄⠋⢀⣬⠕⠋⣹⡄⢻⣵⠤⣾⣩⠟⣼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢌⠈⠉⣁⣋⡱⡆⢿⣄⠰⠀⠈⠙⠶⠶⠤⠴⠶⠞⠋⠀⠰⢀⣼⠇⡶⣉⣃⡉⠉⢈⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠈⠐⢤⡉⠓⢈⣬⠿⠷⠄⠀⢀⣴⡀⠀⢀⣨⣄⠀⠀⠼⠿⢿⣜⠘⠋⣁⠔⠈⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠠⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⢹⠟⡟⠚⠚⡟⢿⠁⠀⠀⡀⢸⣿⡿⠋⠀⠀⠀⠐⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠁⠁⠀⢼⣶⢿⣿⡇⣀⠀⠀⢈⠃⠀⠀⢋⠀⠀⢀⡈⣿⣿⢷⣾⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⠀⠀⠀⠀⠀⡿⢠⣿⣿⣿⣇⡋⣿⣧⡈⢳⡄⣴⠋⣠⣿⡟⣣⣿⣿⠻⣧⠸⡇⠀⠀⠀⠀⠀⠠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⣾⣿⣿⣿⣇⢙⠙⢁⣿⣿⣿⣿⣿⣇⠹⢛⢁⣿⣿⣿⣿⡆⠀⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠐⢾⡿⠛⡈⣿⡿⣡⣴⡿⠛⠛⠛⠛⠛⠿⣷⣤⡹⣿⡏⠙⠻⣿⠖⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⣿⠂⠀⠀⣡⠍⠘⠛⣟⡷⢡⡞⠉⠛⢦⠱⣟⡟⠛⠈⢥⡁⠀⠀⢺⡇⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡟⢠⣄⣦⣿⣦⠐⠦⢻⢷⡈⠦⠀⠠⠌⣴⢿⠳⠆⢠⣾⣧⣆⣤⠘⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠰⠀⠈⠉⠁⠀⠀⠀⠆⡾⣼⠟⠐⠶⠖⠘⢿⡸⡆⠄⠀⠀⠀⠉⠉⠀⠡⠀⠀⠢⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠄⠈⠀⠋⠀⠀⠀⠀⠀⠈⠃⠈⠀⠄⠀⠄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠖⠚⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣈⠉⠉⠉⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠰⠞⠋⠁⠀⠸⠿⢦⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⣾⠀⠀⠀⢰⣹⡆⠀⠰⣽⡷⠀⠀⣾⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣦⣀⠀⠀⠉⠀⠀⠀⠀⠀⣠⣼⣿⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⡿⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠦⣤⣉⣙⡋⠀⠀⠀⠀⠀⠙⠛⠉⢁⡰⠏⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠠⠄⠀⣀⠉⠉⠉⠛⠛⠓⠂⠉⠉⠉⡀⠀⠤⠐⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠈⡂⠈⠀⠃⡄⠀⠀⠀⠀⠀⡇⠀⠈⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣏⠀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡊⠀⣠⡄⠠⠀⠀⠀⠀⣀⠀⡇⠀⢀⣇⠀⣹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠻⣄⡼⠋⠙⢦⡀⡸⠀⠀⠀⠀⠀⠀⠀⠐⢇⠘⠋⠀⠐⠀⠀⠀⠀⠁⠀⣇⣠⠞⠉⠳⣇⢨⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡿⣀⡟⠀⠀⠀⠀⠉⡛⠀⠀⠒⡀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⣇⢈⣇
⠀⠀⠀⠀⢀⠴⠚⠓⠾⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠙⠀⠀⠂⠀⠐⠀⠈⠁⠈⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⣹⠤⠿⢤⡀
⠀⠀⣀⣤⡏⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⡄
⣴⠋⢁⡴⠛⣄⣀⣀⣠⠛⢦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡤⠻⣄⠀⠀⢀⡼⣍⠓⢦
⠈⢳⣏⠀⡼⠀⢸⠇⠘⢦⣠⠷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⢾⣀⡴⠋⢹⡏⠙⣆⠈⢳⣠
⠀⠀⠈⠙⠧⠤⠼⠆⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⢶⠀⠀⠀⠀⠀⠈⠀⠀⢸⣇⣀⣽⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠉⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣈⣀⡀⡸⣄⠀⠀⠀⠀⠀⠀⡧⣐⠀⢀⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⠙⢳⠾⠿⠗⠂⠉⠑⢤⢀⡴⠊⣙⣳⠶⠤⠬⠒⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠎⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⢹⡔⠉⠀⠀⠑⢄⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⡀⠀⠈⢢⣀⣀⣀⠠⢤⠂⠀⠀⠀⠀⠀⢣⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢰⠁⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⢠⠃⠈⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡀⡀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠸⡀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⢀⡌⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢀⠜⠁⠀⡀⠄⠊⠘⢄⠀⠀⠀⢀⡴⠁⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⡟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣶⣿⣿⣿⣿⣿⡿⠻⠿⢿⣮⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠿⠿⣿⡙⠛⠟⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣦⡈⢻⣦⠀⠀⠀⡀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣸⡇⠀⣿⡆⠀⣤⡬⠙⠂⢠⣖⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣾⡏⣧⠀⠀⠀⠀⠀⠉⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢿⣿⣿⣿⡿⠃⢻⠀⠀⠀⠀⠠⠄⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⢻⣿⣷⣄⢸⡀⠀⠀⠀⡠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣿⡏⠻⢿⣷⣤⣀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣄⡀⢸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⣘⣿⣿⣿⣿⣇⣩⣙⢾⣿⣟⡻⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠛⠭⣙⠻⣿⣿⣿⡏⢸⣿⣿⣧⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣼⣿⣷⣿⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣭⣿⣿⠿⢿⣿⣿⡐⠳⢸⣿⣿⣯⡿⣿⣏⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠐⠋⠉⢿⡍⢺⣿⣷⡹⣿⣽⣽⣿⡜⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⠛⠙⠋⠁⠉⠁⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡄⠀⠢⠠⠂⡱⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢠⣶⣶⣶⡾⠇⠉⠁⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡊⣯⢉⠝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⣤⡐⠄⠀⠀⠀⠀⠄⡠⠀⠀⠀⠀⢸⡀⣅⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠘⡀⢾⣯⠇⡀⠀⠀⠀⢸⡼⠟⠀⠂⠀⠀⢸⣷⡏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣷⠀⠀⠈⠒⠐⠊⠀⠠⠤⠄⠀⠀⠈⠀⠀⠀⠀⢸⡟⣯⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠰⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⢄⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠠⠀⢸⣇⠌⡋
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠑⠠⢄⣀⠀⠀⠀⠀⠀⢀⡀⠔⠁⠀⢸⡕⢩⡏
⠀⠀⠀⠀⠀⠀⠀⣀⡆⡀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⠤⠀⠀⠀⣀⣉⣉⣉⣉⣁⠀⠠⠤⠤⠼⠷⠒⠁
⠀⠀⠀⠀⠀⠀⢩⠀⠤⣧⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣙⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠤⠤⠾⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠒⠚⠒⠒⠛⠛⠓⡶⢤⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⢶⡾⠧⣮⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠁⣄⣓⡶⠒⠂⠢⠶⠦⠶⢿⣶⠗⣦⣤⠜⠒⠒⠂⠠⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⣆⠀⠀⠀⠀⣀⠔⠈⡀⢠⠙⠂⣿⠘⠀⠀⠀⡐⡑⠡⢀⢺⣿⠒⢫⢷⣄⡀⠀⠀⠠⠀⠀⠀⠁⠁⠂⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠐⠈⣆⠀⡠⠊⠀⠄⣈⠖⢻⠀⠀⣿⠀⠀⠀⠀⡜⡔⠀⢢⢻⣿⠀⢸⡢⠉⢽⠑⠓⠳⠦⠦⣤⣁⣈⣶⣧⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡼⢏⢀⢀⡼⠊⠁⠀⢸⠠⠄⣿⡀⠀⣀⣚⣀⣈⢀⡡⣸⣿⣀⢸⡪⠂⣽⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⡤⠖⠋⠀⠀⠀⠀⠈⡀⠀⠓⠒⢀⡛⠛⠛⠛⠛⠻⠻⠏⠀⣸⡈⣂⡍⠀⠀⠀⠀⠀⠀⣰⠃⢀⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠘⠃⠀⠀⠀⣿⡿⣿⠇⠀⠀⠀⠀⠀⢰⡟⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣂⢁⠆⠀⠀⠀⠀⠀⢀⡟⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⣂⠤⠄⠐⠛⢚⠛⡋⡿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠞⠁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢾⣶⣶⠀⠀⢸⣶⣷⣷⣶⠀⠀⠀⠀⠀⢀⣠⣴⣄⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⡯⠀⠀⠈⠀⠈⡉⡇⠀⠀⠀⠀⢠⠋⡠⣄⠙⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢁⣿⠀⠀⠈⠀⠨⢪⡇⠀⠀⠀⠀⠀⠉⠀⢀⢠⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃⣿⠀⠀⢠⠀⠨⢪⡇⠀⠀⠀⠀⠀⠀⡠⢊⡼⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣴⣽⠀⠀⢸⡀⠸⡘⡇⠀⠀⠀⠀⠀⠀⠑⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠀⠈⠉⡆⢀⡎⠉⠉⠓⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠃⠈⠓⠶⠒⠒⣧⡜⠓⠠⠤⠤⡴⡾⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠠⠐⠠⠐⠦⠂⠔⠈⠐⠉⠐⠡⠛⠖⠶⢶⠧⠶⢶⠏⡧⢤⣼⣬⣤⣥⣽⡫⠐⠬⠊⠅⡊⠑⠪⠐⠁⠊⠕⠊⠒⠘⠂⠑⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠾⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣾⣿⣿⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡇⠀⠀⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⢸⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠙⠛⠷⢶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠉⠀⠀⣠⡴⠾⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡛⠛⠛⠿⠶⠶⠦⡦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⢀⣰⡆⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⢀⠀⠀⣤⣤⣤⢀⣠⣒⣢⣼⠶⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⢳⡀⠀⠀⣴⣆⠀⠀⠀⠀⠀⢠⡴⠊⠻⠋⠛⢰⠋⠘⢀⠁⠈⠉⠉⠉⠛⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠐⠀⠀⠀⠀⣠⠆⠀⠐⢲⡟⣧⠀⠀⠈⢶⣌⡀⡀⣀⡤⠏⠁⠀⠀⠀⣀⠘⠀⠀⡈⠀⠀⢀⣠⣰⣾⡿⠋
⠀⠀⠀⠄⠀⠀⣤⣰⣶⣶⣶⣶⣿⠙⠛⢿⣷⡶⠿⠦⠞⢇⣀⡀⠴⠿⠻⠋⠀⠀⠀⠀⠉⠙⠙⠛⠀⠀⣀⣄⣠⣤⣼⣷⣷⣾⡿⠾⠿⠿⠿⠟⠉
⠀⠀⠀⣀⣿⣶⠿⠛⣫⣾⣿⣿⣀⡀⠀⠀⣿⠶⠦⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⠆⠴⠶⣦⣌⡁⠀⠀⠙⠉⠹⡏
⠀⠀⠸⣾⣿⣁⠀⢠⣿⠟⡉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠯⠽⢜⣦⣅⣀⠁⢈⣷⣄⠀⠀⠁
⠀⠀⠀⠙⠉⠛⠛⠛⠁⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣶⣿⣿⣿⣿⣤⣔⣛⣙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣐⡆⠀⠀⠈⠙⠇⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠽⠿⠿⠿⠿⠛⠁
⠀⢀⠀⣠⣤⠀⠀⢀⠀⡄⠈⠈⠁⠀⠀⠀⢀⡄⠀⠀⢠⣔⠻⣿
⠀⠀⠁⠋⠉⠁⠉⠛⢋⠀⠀⠀⠀⠠⠀⠉⠘⠁⠀⡹⠊⣿⡒⠻⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠄⠀⣠⠼⢃⡀⠀⢲⣤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠒⠚⠋⠀⠀⠁⠀⢭⣿⣴⡆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣄⣦⡖⣷⡀⣠⠇⢀
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠆⣀⣤⢴⣦⠾⢷⠷⠾⢿⢛⡗⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⣀⠀⠘⣺⠯⣿⣽⠟⣈⡿⠿⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠐⠋⠟⠢⠜⠆⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⠀⠀⣟⡧⣄⣾⠃⠀⠀⠀⡀⠀⠀⠀⢸⣠⠆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣰⠄⣄⡀⢠⡀⣆⣾⣇⢠⢾⠀⣰⣟⣟⣴⣿⢇⢤⠀⠈⡿⠀⠀⠰⢸⠯⠶⣼⣷⡀⢸⣿
⠀⠀⠀⠀⠀⣖⠀⢸⣿⣿⡆⢰⣦⣀⣿⣷⣾⣧⣼⣸⡿⣿⣧⣿⣼⢠⢿⣧⢳⣿⣿⡞⣿⣦⠀⣿⣠⣤⣾⣿⣠⠼⢿⣿⢧⣼⣿⠀⣤⣼⡄
⠀⠀⠀⠀⠀⣿⣤⢼⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⡿⢿⣾⠀⣼⣧⢸⣿⣿⣿⣿⣿⡌⣿⣿⣿⡟⢛⣹⣾⣽⣿⣿⣿⣿⣇⡝⣿
⠀⠀⠀⣄⣐⣿⣿⢸⣿⡿⣿⢿⣿⣿⣾⣿⣿⣿⣿⣸⡇⢠⣿⠻⢿⠖⣿⠓⢸⣿⣿⣿⣿⣿⣧⣟⡿⠎⢯⣿⣿⣿⣷⣿⣏⣿⣿⡿⠀⢸⣿⣦
⠀⠀⠾⣿⣿⣿⣿⢸⣿⣽⡿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣟⠋⠓⠚⣡⡄⠻⡆⠀⠋⣿⣿⣿⡿⠛⠉⠙⠇⠶⣛⣮⣿⣽⡍⠁⠻⣿⢷⣶⡾⠿⠿⡀
⢠⣶⣎⣿⣿⣿⣿⣿⣿⣭⣭⣯⣿⣭⣽⣿⣿⣿⣿⣤⣬⣤⣤⣤⣤⣥⣼⣍⢠⡈⠉⠉⠁⠁⢠⡴⠾⠴⢯⢿⠿⠉⢡⣤⢧⣆⣸⠭⣉⣽⡛⡉⠁
⠀⠁⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣤⡀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⢀⣐⠂⠐⠂⢺⡌⠀⠈⢧⡀⣶⣶⡉⡞⠀⢀⡦⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠛⣷⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠘⠁⢠⣮⣷⣷⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⣿⠾⢷⣦⣤⣀⡀⠀⠀⠐⠖⠂⢀⡄⠐⣮⣽⣿⣿⢿⡉⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡘⠂⠞⠛⠾⠛⠀⠀⣀⣀⡀⠀⠉⠋⠛⢾⣭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠈⠀⢠⣤⠈⢷⡕⣦⡀⠀⢀⠀⣠⠚⣿⠃⣶⣄⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠘⠷⠄⠈⣆⣴⡎⠈⢼⠇⢀⣿⣿⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡿⠟⠛⠁⠀⠹⣄⡤⠈⠟⠠⡐⡼⠀⢸⡏⣿⡷⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠳⣆⡀⠰⠀⢀⡟⠁⠀⠀⠈⠻⡀⠀⣾⡿⠟⠁⠀⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣀⠀⠀⠁⠀⣀⡀⠀⠀⠸⣷⣄⡀⠈⠀⣤⣄⡀⠀⠀⣀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣧⣤⣶⣿⣿⠃⠀⠀⠀⠒⠢⠉⠇⠀⢯⠻⣿⣦⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⢿⣿⣿⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡏⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠟⠋⠀⠘⣽⣷⣄⣀⢀⣤⢴⠆⢀⠉⠉⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠶⠛⠀⣼⣿⡄⠛⠻⠹⠟⢿⡿⠇⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⢀⣄⠀⠀⢻⣿⣿⣄⠀⠀⠀⢀⣀⡸⢟⣽⣽⠁⠈⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠉⠀⠀⣸⣮⣿⣿⣷⠀⢠⡿⢫⣼⣿⣿⡇⠀⠁⠀⠠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣤⣄⣤⣾⣿⠻⡍⠙⠋⠀⢈⣙⣉⠉⠀⢳⣿⣢⣤⣄⣀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠿⠿⣿⣿⣿⣧⠀⣴⣬⠶⠞⠉⠉⡴⠾⠏⣟⣛⣭⣮⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠘⠋⠁⠀⠀⠀⠀⠈⠛⠛⠋⠀⠉⠀⠀⠈