Smallish Emojis & Symbols

⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡄⠀⠀⢀⣀⡟⠛⢻⣄⡀⠀⠀⠀⣴⡿⠿⣆⠀⠀⠀⣰⣾⠿⢷⣄⠀⠀⠀⠠⢂⢦
⠀⠀⠀⠀⠀⢻⠺⠟⠿⠇⠀⠀⠈⠙⠀⣀⡀⠛⠉⠀⠀⠀⠛⠀⠀⠻⠀⠀⠀⣿⡇⠀⠒⣿⠀⠀⠀⢼⠋⠉⠹⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⣤⣤⣶⡄⠀⠀⠹⢷⠀⢰⠟⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠀⡑⠀⠀⠀⠐⠀⠀⢘⠠⠀⠰⠀⠀⠀⠋⠙⡝⡏⠀⠀⠀⡠⡄⠀⢀⠄⠀⠀⡀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠠⠿⡝⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⢀⣴⠶⣦⠀⠀⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⣰⠿⠿⣿⡄⠀⠀⠀⢠⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠈⠛⠋⠉⠃⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⢜⠀⠀⢹⠀⠀⠀⠀⠨⠀⠀⠨⠁⠀⠀⠀⠸⠉⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠈⠾⠷⠏⠀⠀⠀⠀⠀⠐⠀⢂⠀⠀⠀⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠀⢀⣠⣤⣠⣤⣦⡀⠀⠀⠄⣆⠀⢘⠻⡤⠀⠀⢴⡀⠀⢰⠆⠀⠀⠀⠈⡁⠒⢨⠁⠀⠀⠀⠘⠀⠀⠈⠙
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⣀⡀⠀⢀⣀⠀⠀⠀⠄⠁⠀⠁⡆⠀⠀⠀⢀⠁⠀⠈
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⡿⡿⢿⡆⠀⠀⠀⣰⣿⡿⢿⣆⠀⠀⠀⠠⠾⠛⠱⠄⠀⠀⠀⢠⣶⣿⣶⡄⠀⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠈⠐⠒⠒⠃⠀⠀⠀⣟⠙⠀⠊⣿⠀⠀⠀⠰⠏⠉⠙⠏⠀⠀⠀⢸⡿⠿⢿⡇⠀⠀⠀⣿⡿⠉⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⣧⠀⢰⣿⠀⠀⠀⠀⡠⠀⣠⡀⠀⠀⠀⠈⢷⣾⣾⠁⠀⠀⠀⢿⣇⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠊⠰⠈⠂⠀⠀⠀⠛⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠨⠉⠙⢀⡄⠀⠀⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣾⣿⣷⣄⣶⣿⣿⠀⠂⠀⠐⢲⠓⠀⠐
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠛⠛⠀⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣞⠫⡹⣿⣿⣷⢤⡶⣖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣂⠔⠒⠚⠛⠉⠉⠁⠀⠁⠀⠀⠘⢧⡙⠮⠭⠶⠒⢷⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢲⡖⣛⢻⡟⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⢸⣗⣲⢾⡥⠐⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡽⢁⣤⢶⠺⣦⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⠇⠬⢹⡇⠘⠋⠉⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣾⠁⣸⡀⣼⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠉⠀⠒⢉⠥⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠠⣾⠟⠁⠀⠀⣿⡈⢉⠀⣿⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠊⡁⠨⡄
⠀⠀⠀⠀⠀⢔⡿⠋⠀⠀⠀⠀⠈⠛⠿⠟⠁⠀⠀⠸⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⠹⣆
⠀⠀⠀⢠⣾⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⣀⠠⠄⠂⡹⣆
⠀⠀⡴⠟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣂⡥⠶⠚⠁
⠀⠐⠀⠀⢸⡇⠠⣦⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣂⣩⡤⠶⠚⠉⠉⠹⡇
⠀⠀⠀⠀⢸⡇⠀⣿⢉⠛⠷⠤⣠⡄⠀⡀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣀⣠⠤⠶⠚⠛⠉⠁⠀⠀⣀⣀⣤⣴⣬⡇
⠀⠀⠀⠀⠸⡇⢸⣿⢺⠤⣇⠀⣸⡄⠀⡿⠻⠶⣤⣄⣀⠀⠀⡷⠀⠀⠈⠁⣀⣤⠀⢠⡴⠖⠲⠯⢉⢹⢩⠹⠀⣿⡇
⠀⠀⠀⠀⠀⡇⢸⣯⣌⣀⡇⠈⣿⠃⠀⡷⢸⣄⣰⠁⢹⠃⠀⡇⠀⢰⡟⢿⠉⣿⠀⢸⣿⠀⠀⠀⢸⢸⢸⠛⠉⡿⠁
⠀⠀⠀⠀⠀⡇⢀⣃⣀⡀⠅⠘⣿⠀⠀⣹⢸⡀⢸⠁⢸⠀⠀⡇⠀⢸⡗⢛⠀⣿⡇⠸⣿⠀⠀⣀⣸⢊⢺⡖⡂⣿⡆
⠀⠀⠀⠀⢀⡧⣄⡉⠛⠳⢧⣖⣿⠀⠀⡿⢈⠉⠙⠂⢼⠀⠀⡇⠀⢸⡧⠰⠒⣿⡇⠀⢹⠀⠀⠉⢹⠄⣿⣳⣺⣿⡇
⠀⠀⠀⠀⢠⣿⡇⠨⡑⠒⠀⢀⠉⠀⠀⠛⠶⣤⣠⣀⣺⡆⠀⡇⠀⠘⣗⢠⣤⡿⠇⢠⣿⠀⣀⣠⠼⠶⠮⣥⠶⢺⡇
⠀⠀⠀⠀⠀⠀⠉⠑⠃⢦⣄⠀⠈⡙⠲⠦⣄⡀⠉⠉⠙⠃⠀⣇⠀⠀⠛⠉⠈⠀⣀⣸⠛⢉⡁⠤⠄⠲⢘⣦⠤⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⡤⣀⠀⠉⠓⠦⣤⣀⠀⡇⠀⣀⣤⠴⠖⡻⠟⣻⠀⠰⡠⠰⠚⠊⢉⣈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠠⣄⠀⠈⠙⡟⠈⠁⠐⢀⡠⠄⠛⠉⠁⠲⢬⣉⠐⣟⣹⡬⠴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣤⣧⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣴⡄
⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠘⢿⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠈⠓⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠻⣦⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢺⣷⣄⠀⠀⠀⠙⠻⣷⡀⠀⠈⠑⠀⠀⠈⠻⣷⣆
⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠈⠑⡶⠀⠀⠀⠀⠀⢀⡀⠉⠙
⠀⠀⠀⠀⠀⠀⠙⠿⣧⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡤⠀⠙⢿⣦⡀⠀⠻⣿⣷⣦⡀⠀⠉⠉⠉⠃⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⠀⠀⠀⠈⠙⠢⠀⠈⠛⠻⠿⢷⣤⣀⡀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠛⢋⣉⣀⣀⣀⠤⠀⠀⣠⣴⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠴⢶⡦⠤⠠⠀⠀⠐⠛⠛⠛⠉⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⡀⠉⠻⠿⣦⣀⠀⠀⠀⠠⢶⣤⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⡤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠈⠛⠻⠿⠿⠿⠶⠦⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂

⠀⠀⡄⠀⠀⠀⠠⡤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢈⣩⡣⠄
⠀⢄⡄⡦⣀⢀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁⠀⠀⠀⠐⠒⠄⠀⡀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠠⠤⠀⢀⣀
⠀⠉⣉⢉⡉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠸⠥⠤⠀⠀⠀⠀⠀⢀⣈⣀⣈⣀⡀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠆⠀⠠
⠀⠀⢘⣟⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠄⠉⠂⠘⠀⠤⠄⠄⠁⠁⠉⠋⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⢦⠠⠀⠄⠀⠐⠀⠆⠘⠂⢠⡄⠀⠀⠦⠀⠁⠀⠀⠀⣀⣀⢠⢠⠀⡚⠀⢈⡀⡀
⠀⠀⠍⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠸⠇⠈⠁⠀⠃⡄⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⡀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠋⠢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠤⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠄⢀⠀⡐⠀⠁⢤⠁⠄

⠀⢀
⠀⢸
⠀⣿⣇
⠀⢻⣿⡄
⠀⠀⠛⠃⠀⢀⡀
⠀⠀⣀⠀⠀⠀⠑⣦⡀
⠀⠀⠀⢳⣄⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠻⣷⣆⠀⠈⠻⢿⣷⠀⠀⠈⠑⢶⣤⣀⡀⠀⠀⠐⠶⣤⣄
⠀⠀⠀⠺⣄⠈⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⡄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠻⣶⣤⡀⠀⠀⢄⡀⠀⠀⠀⠠⣄⡀⠈⠉⠁⢀⠀⠀⠀⠀⠀⠠⢄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⡀⠙⢿⣦⡀⠀⠈⠛⠃⠀⠀⠈⠙⠳⣶⣶⣤⡀⠀⠈⠛⠿⠿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⡤⢤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠓⢶⣤⣤⣄⣀⡀⠀⠐⠢⢤⣤⣤⣤⣀⡀⠀⠈⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣦⡀⠀⠈⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣴⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣶⣤⣄⡀⠈⠉⠙⠋⠁⠀⠀⠀⢠⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⠄⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠈⠉⠙⠉

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢮⣥⣀⠈⣉⠉⡖⢤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠀⡠⠐⠓⠂⠠⢀⡐⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠰⣤⢀⠻⢕⠄⠀⠀⡠⢂⡁⢀⣀⣤⣤⡑⡄
⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠃⢴⠘⣗⡤⡫⠆⡇⢀⡤⣁⠁⠀⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⢄⡄⠀⣀⢐⠙⠒⠄⣣⣝⢰⣼⡶⠯⠡⠤⠿⠶⢄⣀⡀
⠀⠀⠀⢀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⢀⡀⠶⠘⠁⠂⣀⣽⣼⠤⡾⣤⣽⠗⢊⣡⣀⡠⡤⣠⡠⣤⡀⠀⠀⠁⠒⠄⡀
⠀⠀⠀⠀⠠⠀⠈⠦⠤⠄⢀⠠⠔⠊⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢚⣽⣻⢫⡶⣻⡾⣿⣿⢛⡏⠋⠁⠘⠦⣸⠭⠣⠝⡔⠧⣰⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠄⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢠⠟⢐⠁⠻⠟⡻⢊⡵⢯⢸⡇⡡⣧⢃⡄⠢⡀⠈⠢⠀⠀⠈⠁⠉⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⡾⠙⢀⢌⡴⠃⢸⡅⣼⠂⣝⠋⠚⠃⠀⡰⡀⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠁⡠⢣⠎⠀⠀⠐⣀⡿⢸⠞⠀⠀⠢⢾⡔⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢰⠏⠀⠀⠀⠀⢹⡿⡟⠀⠀⠀⠀⠈⢄⡜⢧⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⠁
⠀⠀⠀⠀⣀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣏⠀⣀⡀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⣀⡄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠐⡎⣒⠈⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠤⠂⠉⠉⢉⠽⡇⢀⣿⠒⢒⡂⠤⠤⢄⡀⠀⠀⠀⣀⣀⡀⠤⠠⠔⠒⠂⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁⢀⠁⠀⠀⠄⠀⠉⠓⠦⡀⠀⠀⠀⠈⠉⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⡴⡞⠁⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠂⠀⠀⠈⠀⠄⠀⠂⡠⠂⠀⠹⡶⠠⢒⠠
⠀⠀⠀⠀⠀⠨⡀⢇⠈⠄⣑⠨⢍⡢⠀⠠⠀⢐⡤⢚⠀⡄⠡⢀⢄⡁⠀⠀⡊⠁⠠⠀⢄⡀⣢⣠⠆⡃⠁⢀⡹⠈⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠀⠀⠀⠁⠀⠐⣂⠠⠄⠉⠀⠀⠀⠒⡂⠁⠈⠈⠉⠈⠁⡉⠉⠀⠀⠡⠔⢊⠀⠀⠀⠀⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠤⢀⡂⠈⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⣶⡄⠀⠀⢀⣀⣸⣿⠠⠄⢈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣠⠈⢛⣿⣷⣾⣿⣿⣿⣿⣦⣴⣿⣿⣷⣶⣤⡀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣼⣿⠇⠀⠉⢹⡿⣿⣿⣿⣿⣿⡄⠉⢟⣉⠙⠛⠻⣿⣼⣿⣿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⡿⠿⠶⣦⠀⠙⠁⠘⢿⣿⣿⣿⣷⠀⣘⢛⡑⢿⠇⡋⢹⡟⢻⣘⣀⠁
⠀⠀⠀⠀⠀⢠⣶⣾⠿⣿⣷⣫⣤⠀⠶⠾⣷⣿⣿⠟⠈⠀⠀⠉⠉⠀⠻⣿⣽⣶⣾⣿⣿⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠸⣏⣠⣴⣶⣿⢉⣡⣶⣦⡄⠋⠛⠛⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⣘⠟⣅⠈⡙⣿⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣛⣿⡦⣸⣤⣿⣿⡟⠀⠀⠀⠀⠀⠈⠁⠈⢡⣾⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⡇⠀⠈⢹⠇
⠀⠀⠀⠀⠀⠀⣀⣬⣷⣾⡿⠟⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠁⢹⣿⣈⣥⣤⣀⣿⡇⠈⠀⠈⠁
⠀⠀⠀⠀⠀⣾⡿⠿⣿⣿⣶⠀⠀⠀⢿⣷⠀⠀⢀⣤⣤⣀⠀⠈⠄⠀⠃⠐⢿⠟⠻⠿⠿⣯⡁
⠀⠀⠀⠀⠀⠈⠀⠠⠙⢿⡇⢀⣴⣶⣿⠿⢇⣀⣾⠛⠛⠛⢶⣦⣠⣤⣤⣤⣸⠐⣀⠐⢸⠉⠳⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠁⣼⣿⣿⡇⠀⠐⠭⠇⠄⠀⠤⠸⣿⠛⠛⣿⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⣘⣛⣛⣾⣿⣶⢾⣿⡄⠀⠻⣿⠟⠃⠀⠘⠃⠀⠀⠀⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⣭⡭⠭⠍⣉⡉⠿⠿⢿⣶⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⣹⡀⢀⣴⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣖⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⢾⣿⣿⣟⣻⣿⣿⣿⣿⣿⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣭⣧⠿⡼⣿⣾⣯⣿⣽⣟⣻⣟⡧⡭⢀⡂⠲⣄
⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⢿⣿⣾⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⣧⣭⣀⢥⡀⠀⠀⠀⠀⠀⢀⠄
⠐⠒⠀⠈⢠⣤⣤⢐⠀⠀⠀⠀⠉⠓⢷⣾⣯⣿⠿⢻⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡄⠀⠀⢀⡜
⠀⠀⠀⠨⢩⠿⡉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣯⠀⠸⠟⠿⠿⢿⣯⣛⣿⢽⠿⠿⠋⠀⠀⢀⠊
⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠲⠿⣳⠤⢄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⢦⣎⣩⢉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠄⠀⠀⠀⠀⢸⣿⠿⡞⠙⣠⡤⣴⢶⠶⡶⢶⡤⣀⣀⡀
⠠⠂⢐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⠁⠈⡐⠃⢹⣤⣼⣾⡞⢻⣶⣿⣵⣯⣫⠳⣄⢀⣀⣀⣀⡀
⠀⠀⢀⡠⠤⠤⠤⡀⣀⣰⡀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠀⣷⣶⣿⣟⠿⣄⠄⢀⣀⣈⣽⣟⣛⣿⣿⣿⣷⣿⣿⣟⢷⡀
⠀⠀⢘⠀⢤⣀⡠⠭⣭⣥⠏⢳⠀⠀⠀⠐⠀⠀⣿⢿⡀⢀⣿⣿⣿⠿⠿⠛⢿⣾⣯⣼⣷⣿⣽⣏⣿⣷⣶⣾⣿⣿⣖⣿
⠀⠀⠈⢳⣤⣤⣄⠀⣁⣤⢤⡾⠀⠀⠁⠈⢲⢳⣍⢎⡇⢸⣿⣯⣳⣾⣵⣜⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣿⣯⣾⠟
⠀⠀⠀⠀⠁⢿⡏⣷⣿⣠⠂⠀⠀⣄⡀⣰⢌⣿⣿⢹⠁⠀⠈⠙⠛⠿⢿⣿⣻⣿⣿⣿⣿⣿⣽⡿⠟⠋⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣤⣙⣌⠓⠿⣀⣿⣇⢸⡙⣄⣿⣿⣿⢸⠂⠀⣟⢲⡀⠀⠀⠈⠻⣿⣧⠁⢿⡿⠋
⠀⠀⣠⠖⠋⣉⣁⣉⠙⡆⠨⣿⣿⣺⡇⣿⠧⣼⣿⣿⠄⠀⣷⣾⣧⣠⠒⣀⣀⣿⡿⠀⠘⣶⣖⣘⣂⣠⣄⣀⡀
⠀⢰⠃⣴⣿⡟⠋⡍⢡⣷⡀⠱⣿⣿⠁⣿⡆⣿⠿⡿⠀⠀⣿⣿⢾⣟⣃⣿⣿⣿⡧⠀⣴⣯⣭⣩⣭⣽⣿⣍⣿⣷⣦⡀
⠀⠈⠣⠻⠛⢷⠀⢷⢿⢻⢳⡀⠈⠻⡿⡉⠙⣻⣷⢲⠁⢀⣻⣾⣼⣘⣻⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⣷
⠀⠀⠀⠀⢰⢮⡇⠸⣮⠁⠀⢟⡢⢤⣸⣅⢠⣿⣤⠿⣰⣿⣿⣻⠃⠙⣿⣿⣟⡟⠀⠀⢸⣿⣿⣿⡿⠏⠫⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠂⠈⠈⡇⠀⢻⡀⢀⣾⡟⠓⣬⢡⣾⡟⠙⠠⡝⢿⣿⡿⢿⣿⣿⣽⣿⣷⣀⢀⣾⣥⣷⣿⣿⠂⠀⣟⠛⣋⢉⠉⠐
⠀⠀⠀⠀⣀⣶⢿⡀⢀⢃⢈⣿⣧⡀⠰⣿⠟⠗⣚⣃⣓⣞⡏⠀⠘⢿⣿⣿⣿⣾⣏⣽⣿⣿⣟⠿⣇⡠⣼⠟⠿⠠⠮
⠀⠀⠀⠀⣈⠁⠐⠉⠉⣤⣤⣼⣿⡿⠋⠙⠻⠛⠁⣮⣙⣿⣇⡀⢀⣸⡿⣿⣧⣉⣭⣍⠉⠀⠀⠀⠐⠀⠀⣀⢀⡈⠒⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠾⣾⣿⡇⠀⠀⠀⣠⣶⣿⣿⣿⣿⣯⣿⣿⡿⠟⠶⠬⠄⠤⠤⠄⠀⠀⠀⠈⣀⢀
⠀⠀⠀⠀⠐⠀⠀⠚⠁⠀⢱⣿⣧⡣⣄⣠⢶⣟⣿⣿⣿⠟⠛⠛⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⡻⠯⠿⠛⠑⠏⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠛⠛⠛⢈⣉⣉⠙⠻⢻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⣋⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣌⣛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⡏⢭⡘⠇⡵⢘⡟⢿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⢿⣿⣢⣿⣧⣦⢠⣶⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠿⠿⠿⠿⣿⣿⣿⣿⢱⢈⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⣠⣪⣥⣬⣿⣷⣶⣦⣄⡀⠀⠀⠈⠙⠛⢻⠀⠸⡏⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢀
⠀⠀⣀⣤⣾⡿⠛⢛⣉⣙⣛⣉⣉⣛⡻⠶⡄⠀⠀⠀⠸⡃⠀⡇⠀⠀⠀⠀⡀⠀⣠⣤⣴⣶⣿⣿⣿⣿⣶⣌⣦⡀
⠀⣬⡾⣻⣭⣴⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⠆⠀⠀⠀⢠⠁⢰⣿⠀⠀⠠⣿⣥⣶⣶⣶⣶⣽⣦⣒⣶⣲⣶⠮⢥⣙⣄
⠀⢋⣴⣿⣿⣿⣿⣶⣷⣿⣿⠏⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇
⠀⠈⠛⠛⠋⠉⠀⠀⠈⢛⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡄⠀⠀⠸⠿⠿⠿⠛⠛⠿⣝⡿⢙⣭⣉⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡇⢸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⡟⡁⠸⠛⠛⠻⠿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣮⣿⣝⣛⣿⣿⣶⣄⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠸⡄⠀⠀⠀⠀⣸⠄⢀⣿⣿⣯⣽⣯⡩⣿⣿⣶⣦⠀⠀⣿⠀⠀⠀⠀⠀⠀⡾⠅
⠀⠀⡆⠄⡀⠀⠀⠀⠅⠂⠀⢷⣦⢸⡆⠀⠀⠀⢺⣋⢸⡟⣿⣿⡇⢸⡇⠉⠉⠉⠉⠀⢀⣿⠐⡆⠀⠀⠘⡷⠇
⠀⠀⠴⠂⠁⣤⠀⠀⠉⠀⣀⠈⠻⣦⠣⡀⠀⠈⣉⣇⣼⡇⠉⠉⡇⢸⢁⠀⢼⡀⠀⠀⣾⡇⢰⠀⠀⠸⡷⠷
⠀⠀⠀⠶⡶⠇⠀⡀⠑⢒⠀⡄⠀⠹⡇⠑⣥⠀⠤⡇⢸⡇⠀⠀⡇⣿⡘⢈⣻⠂⠀⣾⠏⢀⡆⠀⠲⣦⣧⠀⡀⠐
⠀⠀⠀⠀⠠⠤⠉⠃⡀⢈⡀⣴⠀⠀⣹⣤⣌⠳⣄⣏⣹⡇⣸⣠⣧⡟⡇⠐⠷⢦⣾⠏⠈⠈⠀⡄⣠⡅⠀⠶⠀⠵
⠀⠀⠀⠆⠀⢾⡅⢀⣞⡀⠓⢀⣤⣈⢿⡶⢾⡦⡿⣿⣿⡥⢸⣿⣻⢃⡍⢃⣤⡾⢡⠆⣴⢠⠰⢿⡏⠑⡇⡠⡀⡀
⠀⠀⠛⢀⡖⣀⢩⣽⡞⠗⠈⢴⡌⠀⣾⠁⣸⡧⠿⣿⣿⣿⣼⡿⢇⡾⣻⣿⡿⣱⡇⢱⣏⢨⣀⡈⠯⢧⠨⡤⠈⠉
⠀⠀⠀⠀⠈⢉⠆⢊⠸⠘⣤⠘⣋⣷⣿⠀⣿⡀⠀⣻⠟⢻⣿⣴⣾⣿⣿⣿⠁⠉⢀⣿⣿⣿⣾⣵⣄⣼⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠉⣀⣾⣦⣥⣾⣿⣿⣿⣛⣿⡋⠻⣿⣠⡾⠁⠈⣹⣿⣿⡇⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⠌⠐
⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢧⣿⣧⠶⠷⡿⣿⣿⠷⣾⣿⠻⢿⣿⣿⠟⠛⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠋⠛⠛⠋⠑⠒⠿⠃⠙⠲⠦⠂⠈⠁⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡄⡤⣘⣁⡀⠀⢄⡤⡞⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⡈⢀⠋⠐⠋⢣⡭⣤⡾⠁⢀⣀⠨⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠲⠀⢶⡰⣤⠅⠃⣦⡗⢀⢂⡤⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⢉⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠙⠄⠃⠸⣧⠘⠢⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣤⢻⡶⠓⠆⠀⠋⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢀⠺⠃⠁⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⡄⠀⡀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠈⠂⠀⠀⠀⠀⣼⣅⠂⠀⠼⣄⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠁⠁⠰⠀⠀⠀⠀⣀⠄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠴⠚⡤⠀⠄⠀⠀⡀⣠⠄⢸⡧⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡇⢀⠳⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠠⠞⠠⢶⣦⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⣈⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠒⠀⢠⠀⠀⠀⡀⠊⠀⠀⠐⠂⠈⠀⠀⠀⠀⠑⠀⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢒⡩⠝⠛⠛⠛⠲⣄
⠀⠀⠀⠀⠀⣀⣠⣤⣄⣠⣴⣴⣟⣿⣿⣿⣿⡟⣿⢿⣿⣆⠀⠀⠀⠀⠀⣀⠤⠄⠂⣀⡄⡒⠙⡡⠒⠁⠀⠀⠄⠤⠤⡀⠈⠃
⠀⠀⢠⣶⣿⣿⣾⣯⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣹⡟⠀⠀⠀⢠⠎⠁⠀⠈⠁⠛⠁⡠⠊⠀⠶⠀⠀⠀⠀⠀⡀⠘⠂
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡶⠀⣀⡀⡀⠀⠀⠂
⠀⠀⠸⣿⣿⢿⣿⣿⣿⣿⣻⣿⣷⣬⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠈⢅⠀⢤⠀⠀⠀⠀⠔⢆⠀⠙⠏⠀⠄⠀⠀⠂
⠀⠀⠀⠈⠻⠿⣿⣿⢿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣤⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠉⠉⠙⢳⣦⣤⠀⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣽⣟⣻⢿⣿⣾⣿⣿⣷⣶⠖⢫⣵⣤⡀⠀⠀⠀⠀⠀⠀⠼⡚⠑⠃⡖⢤⡀⠉⠀⠀⠀⠝⣭⠿⠟⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⠠⠀⡀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣷⣿⣽⣭⡿⠟⠋⠀⠀⠀⠀⠀⠀⣦⠀⠀⠁⠢⢀⠀⠀⠀⠀⠀⠁⢉⡙⠉⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⡄⠅⠈⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠆⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⡀⠀⠠⢀⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣿⢿⣿⣿⣿⣿⣷⡧⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣸⡇⢰⡀⢠⠀⠀⠁⣝⣄⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠃⠿⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠨⣛⣾⣷⣿⣶⡀⣰⠟⠉⠀⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣠⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⡿⠿⡿⠛⢿⠛⢿⢻⠻⠿⠿⢿⠿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⢛⡁⠀⠤⠠⣬⣔⣵⣄⣦⣄⣨⢞⠁⠂⡀⠉⠙⠛⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣤⣀⣈⣅⢀⣙⣿⢉⢬⠠⢉⣿⣿⣶⣟⣻⣶⣶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⣛⠂⢀⠀⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡴⣶⡴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⡀⠀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠐⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡞⠀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⡼⠁⠠⢀⣾⠀⠀⢸⡄⣠⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣷⠃⣼⠀⡞⡏⠀⢰⠿⡷⢁⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡈⢿⣿⣄⡿⣾⢡⣵⣧⣷⣾⠗⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠈⠟⠃⠛⠸⠏⠟⠁⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⢐⣘⡆⠐⢀⠀⠀⡀⢠⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⢊⠒⠆⡩⣉⠇⠀⠁⠔⠒⠙⠁
⠀⠀⠀⠀⠀⠠⠀⡠⠀⠀⡒⠀⠥⠜⠛⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠢⢠⠀⣤⡄⠰⠚⠂⠀⠀⢀⣤⠤⠔⠊
⠀⠀⠀⠀⠁⢀⡄⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⢀⣤⣈⡦⠄⢀⠁⠄⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢦⡀⠁⠀⢤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠒⠠⠤⡷⠂⠠⠌⣠⠂⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣋⣝⢡⠒⣡⠬⠦⣀⡀⡽⠦⢀⠨⡁⠠⠴⠂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠃⠈⠀⠀⠀⠿⠒⠓⠒⠾⠄⠽⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠊⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⣎⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠛⠿⠁⢙⠆⢠⣴⣥⠶⢀⣠⣤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⠀⠁⠀⠀⠀⠘⠋⡫⣁⡄⠘⠃⠀⠀⠀⠀⠘⠃⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⡀⢉⣅⣈⣉⢀⡴⠈⡿⢿⡆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡀⠿⣥⠬⣉⣀⣠⡟⠛⢏⡫⠋⠀⠀⡀⠀⢀⡀⠀⠀⠠⠈⠆⠲⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠲⢢⠀⡉⢈⠡⠌⠀⠀⡰⠈⢀⣄⡀⠀⠀⠀⣀⣀⡤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⠀⣄⡀⠀⠀⠀⠀⠋⠀⠀⢀⠐⠛⣦⡒⢚⡛⠈⠛⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠠⠾⠀⠀⣼⡿⠲⣤⣄⣠⣴⢛⡀⠹⠂⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢇⡻⠩⠀⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⢠
⠀⠀⠀⣰⠉⢳⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠁⠤⠉⠠⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐
⠀⠀⠀⠺⠰⠉⠆⠀⠀⠀⠀⠀⣶⣏⣳⡖⠀⠀⠀⠀⠀⢀⣘⣁⠀⠀⠀⠀⠀⠀⠹⠺⠈⠁⠀⠀⠀⠀⠀⢨⡀⠗⣈⠂
⠀⠀⠀⠐⠿⠗⠃⠀⠀⠀⠀⠀⠁⠙⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠺⠏⠁⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢘⣏⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡉⣯⠩⠀⠀⠀⠀⠀⠀⡑⣇⣿⡋⠀⠀⠀⠀⠀⣾⠃⢷⡄⠀⠀⠀⠀⠀⢸⠦⠿⠀⠀⠀⠀⠀⠀⠠⠛⡈⠓⠆
⠀⠀⠀⢓⣿⡀⠂⠀⠀⠀⠀⠨⠃⣇⡿⢡⠀⠀⠀⠀⠀⢻⠀⣾⠃⠀⠀⠀⠀⠀⡅⠀⠀⣏⠀⠀⠀⠀⠀⢐⠀⠁⠘⡂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⢸⣁⡿⠀⠀⠀⠀⠀⠀⠹⣀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠴⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠰⣤⣤⡔⠢⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢐⡀⠀⠀⠀⠀⠐⢀⡊⣒⠀⠀⠀⠀⠀⠀⠀⡤⢢⠄
⠀⠀⠀⢐⢠⢰⠂⠀⠀⠀⠀⠀⡜⠀⠂⠀⠀⠀⠀⢀⢀⡻⠻⡟⡀⠀⠀⠀⠀⠀⠈⠀⠃⠔⠀⠀⠀⠀⠀⢀⣄⣀⡀
⠀⠀⠈⢹⡸⢀⠁⠀⠀⠀⠀⠐⠢⢝⢡⠃⠀⠀⠀⠀⠀⢿⡘⣿⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠘⢖⣲⠗
⠀⠀⠀⢻⣒⣚⡗⠂⠀⠀⠀⠀⢶⣌⣰⠦⠀⠀⠀⢀⠁⠪⡄⠼⠉⠀⠀⠀⠀⠀⠀⢇⡼⢀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡀⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⠀⠀⠀⠀⠀⠀⢨⣡
⠀⠀⠐⡘⠁⠋⠪⠀⠀⠀⠀⠀⢈⠇⣺⠀⠀⠀⠀⠀⠀⠒⠐⠀⠠⠀⠀⠀⠀⠀⠼⠃⠷⠀⠀⠀⠀⠀⠀⠀⠚⠳
⠀⠘⠒⠄⠀⠀⠔⠀⠀⠀⠀⠀⣰⠐⡰⠀⠀⠀⠀⠀⠬⠀⠀⠈⠤⠀⠀⠀⠀⠈⢦⣛⣠⠆⠀⠀⠀⠀⠀⠀⠂⠘⠂
⠀⠈⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⢤⠔⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠚⠘⠆⠀⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⣶⣤⡴⡇⠀⠀⠀⠀⠀⠘⢡⠛⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡆⣖⠀⠀⠀⠀⠀⠠⠌⠀⠀⠠⠄
⠀⠀⠠⠄⣿⠅⠃⠀⠀⠀⠀⠀⢂⠸⠀⡂⠀⠀⠀⠄⠀⡌⠈⣄⠀⠀⠀⠀⠀⠀⣠⡞⠤⠀⠀⠀⠀⠀⠀⢴⡄⠀⣼
⠀⠀⠀⠶⠊⠛⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠑⠃⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡀⠀⠀⠒⡻⢿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣤⣄⣀⣠⣈⠀⣺⣿⣿⣿⣷⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣨⣭⣿⣿⣿⣻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠈⠉⠉⠉⠉⠁⠁⠀⠀⠘⣿⣿⣷⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣦⣈⢻⠊⢳⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠀⢴⡶⠶⠶⠤⠀⠀⠀⢸⣿⡟⢉⣉⠒⢈⡇⠄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠑⠾⣿⠃⠀⠀⠠⠟⠻⠛⠓⠀⠀⠀⠈⠉⢰⠿⡸⠠⢠⣇⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠖⠂⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⡟⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⣼⠁⠀⠛⠷⣦⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⢠⠃⠀⠀⠀⠀⠢⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⣤⣴⠾⠛⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⣀⣤⠖⠋⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠉⢠⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⢸⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁

⠀⠈⠻⡇⠀⠘⠇⠀⠀⠆⡰⠈⠉⢈⣧⣾⣿⣿⣿⠟⠉⠉⠙⣾⣿⣿⡿⠆⠀⠀⠀⠀⣁⣠⣤⣤⡻⢿⡆⠀⠀⠸⠟⠃⠀⢠⣿⣿⣿⣧⠈⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣷⠀⠀⢸⡟⠀⠈⠙⣿⠑⠂⠈⠀⡋⠈⠛⠃⠀⣴⣶⣶⣦⠹⣿⡿⠿⠿⡆⠀⠀⠀⠀⠀⠁⠉⢻⡏⠀⠉⡿⠀⠈⠉⠁
⠾⠻⠷⡆⠀⢀⣿⣿⣿⣿⣏⢰⣾⣿⣷⣤⠀⠀⠀⢦⣶⣩⣾⡇⢀⣤⣄⠀⠇⠀⠀⢹⡇⣡⠀⠀⢠⡷⢖⠶⠲⣄⣦⣤⣦⡆⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠿⡿⠛⠻⣿⣿⣟⠋⣭⡟⠀⠀⠀⠈⠙⠻⠋⠀⣿⣿⣿⣿⠀⣀⣀⡾⠃⠁⠀⠀⢰⣷⠀⠀⣠⣾⣿⣿⡟⣷⣿⣿⣿⣦⠀⠀⣤⢶⣶
⢀⣴⣦⣤⣄⢠⣦⠀⠀⠀⢠⣤⣀⣀⣉⣁⣀⣾⣶⡦⢴⡿⠆⠀⢉⣉⣉⠉⠀⠉⠀⠹⠂⠀⠀⠀⠘⢣⡧⠸⣿⡿⠿⣿⣾⣿⠿⢿⣿⡟⢀⠀⠁
⣿⡟⠛⠻⢿⡿⢿⠦⡀⢠⣾⣿⣿⣿⡿⠿⠿⣿⣷⣄⢀⣰⣿⡶⠋⠉⠻⣷⣀⣴⣶⣤⠀⣴⣦⡄⣤⣴⡆⠸⠚⠋⠈⢻⣉⠁⠀⠀⠁⣄
⣿⡄⠀⠀⣸⣿⣿⣾⣷⡞⢻⣿⣿⣿⠀⠀⠀⢻⣿⡿⠀⢸⣿⠀⣠⣤⣀⣰⣾⣿⠿⣿⢠⠤⠉⠀⢩⣽⣿⣿⣿⣿⣷⣄⢹⡆⠀⢀⣴⡟⢀⣶⣶⣶⡄
⡿⠛⠁⠺⡿⢻⡿⡿⢿⡟⠀⠉⣉⢀⠀⢀⣤⣿⣿⣴⣿⣭⣝⢻⣿⣿⣿⠩⠉⠀⠐⠘⡇⢀⠀⠀⣿⣿⡏⠀⠀⠘⣿⣿⠀⠔⣠⣷⣾⣆⢈⣿⣿⣿⠁
⠁⠀⠀⠀⠀⠈⠃⠀⠀⠗⣘⠀⠀⢀⣷⡿⢿⣿⣿⠏⠀⠀⠈⣿⣿⣿⡿⠃⠀⠀⠀⠀⣧⣰⣶⣦⣝⠿⠇⠀⠀⠸⠛⢁⣀⢠⣿⡿⣿⣿⣄⠙⣿⣿⣆
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡟⠀⠀⠈⡿⠁⠃⣈⠁⡃⠀⠉⠁⠀⣾⠿⢿⣷⡝⢿⠿⠟⠛⠇⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⡽⠋⣈⢷⡱⢯
⠛⠛⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢳⣾⣤⣿⠇⢠⣴⣤⣀⠃⠀⠀⢸⡇⢰⠄⠀⠠⡷⠊⠛⠛⢦⣷⣮⣷⡆⠀⠀⠀⠁⠀⢻⡖⢙⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣿⣿⣿⣿⠀⠀⠀⠚⠁⠀⠀⠀⢰⡟⡂⠀⠀⢸⠿⠿⠟⠀⠀⠀⠀⠀⠘⢣⡾⠿⠿
⢠⣶⡶⠶⣦⡀⣒⡆⠀⢀⣤⣤⣤⣤⣤⡀⢀⣴⣾⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣷⠀⣠⣆⣀⡀⠀⠀⠀⠀⠀⠀⠘⢃
⣿⡏⠀⠀⢸⣿⢻⠃⢴⣿⣿⡿⠋⠉⠉⢙⣾⣿⣿⠟⠻⣿⣿⠇⠀⠀⠀⣴⣿⣿⣶⣄⠀⠀⠀⠀⠈⠷⣿⡟⠛⠋⢿⢿⣦⠀⠀⠀⣠⣤⢈⣧⢦⣤⣄
⣿⣅⡀⣀⣼⣿⠃⠀⠈⠛⠿⣷⠀⠀⠀⠀⠉⢿⠇⠀⠀⢈⣿⡀⠀⠀⣼⣿⣿⠿⠿⣿⣦⣀⣠⣤⣀⡀⢸⣷⠀⠀⠈⣾⣿⡇⢀⡀⢿⢿⣿⠁⠀⠙⢿
⣿⡟⠀⠻⣿⡿⠋⠉⠈⠳⠀⠀⠶⢀⣤⡀⠀⠀⠀⠀⠀⠈⠉⠁⠉⢰⣿⣿⠃⠀⠀⢻⡟⠉⠉⠉⢉⣿⡄⠀⣠⢤⡄⠉⠁⠰⠏⠀⠀⢰⣿⡀⠀⠀⣸
⢹⣷⣾⣶⠏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⣀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⢻⡿⡿⣖⣺⠿⢻⡇⡞⠁⠀⠹⡆⠀⠀⠀⠀⠀⢸⣿⣿⠇⠸⣿
⣿⣅⠈⢻⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣦⠎⠈⢳⣖⡄⠀⠀⠀⠀⢠⣤⣀⣠⣾⠧⠀⣏⣙⡆⠘⠃⣿⣄⠀⠼⡇⠀⠞⠀⠀⠀⠁⠀⣯⣤⣤⣼
⠋⠉⠀⠉⠁⢀⣴⠀⠘⠻⣷⣿⡿⠿⠿⠿⢿⡟⢃⠒⢋⠙⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣉⣭⣥⣴⣶⠀⡥⠀⢀⣄⠀⠐⠿⠿⠿⠄⠀⠻⠟⠉⠛
⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠋⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣡⣤⣶⣶⣶⣾⣯⣿⣿⣿⡛⢻⣿⠿⠿⠿⡇⠈⠉⡥⣌⠉⢀⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣿⠿⢻⠟⠀⠈⠙⢧⠘⠃⠀⠀⠀⠁⣀⢀⠀⠀⠀⠘⠛⠁⠈⠛⠛
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣰⠛⠋⠙⢳⡀⠀⢀⣤⣶⣶⣶⣌⠁⠀⠀⠀⣀⠀⠀⠀⠀⢸⠀⠰⠶⠆⢠⡿⠟⠻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡷⣶⣶⠿
⠀⠀⠀⠀⠘⠀⠀⠀⠐⠀⣿⠀⠀⠀⢸⡇⠀⠘⠉⠉⠉⠁⠉⠃⠀⠀⠀⢸⠆⡀⢀⡔⠈⠀⠀⠀⠀⣸⡇⠀⠀⠀⡧⠀⠀⠀⠀⡀⢀⢀⡀⠓⠒⠒⠛
⠀⣀⣀⣀⡄⠀⠀⠀⠀⠀⢿⠷⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣴⣿⣿⣿⣿⣾⡻⠀⠀⠀⠿⡿⠀⠀⢘⡟⠋⠙⢳⡗⠚⠉⠈⠀⡀
⣾⠋⠉⠻⠀⠀⠀⠒⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠃⠀⠉⠀⠀⡀
⠇⠀⠀⠘⠀⠰⠂⢠⡆⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣶⣶⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⡀⠀⠀⢀⢀⣾⡄⠈⠛⣳⣶⣶⣶⣆⠀⠀⠀⢀⠀⢀⣀⡉⣉⠉⠉⠉⣿⠀⠀⠀⣀⠀⡀⠀⠀⢀⣀⡀⣀⠀⠀⠀⢀⡀⣀⡁⠀⣀⠀⢀⡀⠀⢀
⣿⣿⠿⠻⢿⠛⠿⢿⠿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⡿⡇
⠀⠀⠀⢰⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣴⣇
⠀⠀⠀⠀⠙⣿⣿⣿⣽⣿⣶⣤⣀⢀⠰⠃⠘⠆⠀⢀⣤⣶⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢻⣙⣻⣿⣿⣿⣿⣿⣷⡇⠡⠄⠸⣾⣿⣿⣿⣿⣿⣟⣋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠉⣵⣿⣿⣿⡽⣛⢵⡂⠙⠋⢐⡪⣝⢯⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠈⠷⠀⠀⡀⢀⣰⡄⠀⣸⢿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢰⢹⣿⣾⣿⣿⡏⣆⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰⠂⠁⣠⣴⣿⠿⠋⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠶⠭⠭⠷⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⣵⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠸⢿⠿⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢢⢂
⠀⠀⠀⠈⠖⡼⢀⣃⡤⢠⡄⣀⠀⢄⡘⠃⠘⠃⣀⠀⢀⠠⡄⢰⣔⣇⢧⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢳⡧⠀⡄
⠀⠀⠀⠀⠘⡍⣑⡒⠽⡶⠳⢺⢧⣦⡍⠁⠈⢉⣴⡤⣟⠺⢳⡮⠔⡚⡉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠮⢷⠢⠌
⠀⠀⠀⠀⠀⠀⠨⠦⠎⠑⠘⠋⢋⣹⡄⠀⠀⢠⣏⠙⠋⠋⠚⠹⠴⠏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣮⡯⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⣴⣤⣤⣤⣶⣦⣉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⡀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⣛⡻⠿⠿⠿⢛⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠀⢠⣀⣼⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢩⣙⣛⣛⣋⡭⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢻⣛⣩⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣤⣄
⠀⠀⠀⠀⠀⢸⣿⠛⠉⠛⠿⢿⡟⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢰⣿⠻⠿⠿⣿⢿⣿
⠀⠀⠀⠀⠀⠸⡏⠒⠀⠀⠐⢈⡇⠀⠀⠀⠀⠀⠀⣿⠋⠀⠉⠁⠈⣿⡇⠀⠀⠀⠀⠀⢸⣯⠐⠀⠀⠐⠀⣿
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢻⡀⠤⣤⠤⣰⠇
⠀⠀⠀⠀⠀⠀⠈⠳⠶⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⡾⠟
⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠆⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠘⠛⠛⣓⠚⠛⠛⠓⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⢰⣿⣿⢿⠟⢿⣿⣇⠀⠀⠀⠀⠀⠀⣾⡏⠈⠛⠛⠛⣿⠆⠀⠀⠀⠀⠀⢀⣿⢿⢿⣿⢿⣿⣷
⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⢹⠊⠁⠀⠀⠁⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠀⠈⣿⣷
⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠄⠀⢀⣼⣿⡄
⠀⠀⠀⠀⠘⣿⣿⣷⡄⠠⣾⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢘⠛⠋⠀⠀⠉⠛⡃⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣤⣤⣼⣯⡍
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⣠⣶⣄⣤⣤⣤⣤⣤⣦⡄⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⣿⣿⠛⠛⠉⠙⣿⣧⠀⠀⠀⠀⠀⠙⣷⣿⠿⢿⡿⢻⣿⣾⠃⠀⠀⠀⠀⣾⣿⠟⠛⢿⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠻⡿⠀⠀⠀⢀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣄⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠷⠄⠠⠾⠟⠛⠁
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣿⣿⣿⡷⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢾⣿⣿⣿⣯⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⣿⣿⣿⡀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣉⡀⠀⠀⣿⠿⠿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃⠀⠀⣀⣠⣾⣿⣿⣿⢷⣤⡀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣦⣀⠀⠀⠀⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⣀⣰⣿⣿⡿⣋⠹⡏⣼⢹⣿⡄⠀⠀⠀⠀⣰⡿⣿⣮⡿⠩⣿⣿⣿⣧⣀⣀⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣻⣿⣿⣟⠁⡁⢙⡀⠄⠠⠎⠞⠛⢦⣾⣆⣼⢋⣇⡱⠀⠀⢰⡋⢈⠙⣿⣿⣿⣯⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣯⣿⣿⠄⠓⠸⠧⠀⠀⠂⣆⡄⠈⣿⣿⣰⣄⡏⡃⡀⠀⠞⠁⠄⠰⠿⣿⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣶⣶⠲⣶⣾⣶⡷⡖⠒⢦⣿⣟⡻⢿⣶⣶⣶⣻⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣟⠻⢿⣷⣿⢿⡇⢑⠍⣉⠀⣿⡏⠐⣻⣿⠉⣹⠿⣷⣾⣿⢿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡉⢶⣊⠁⠉⠒⠀⡍⠀⠀⠀⢽⡇⠀⠀⢷⣧⠐⠊⠉⠙⣻⣾⣸⣳⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⢷⣟⠲⠍⣀⠖⠀⣤⢀⠀⠀⠀⠸⠃⠀⠀⢀⡆⣤⠑⢟⣈⡹⢟⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⢿⣿⣶⢾⣥⢴⣆⠤⠀⠀⠀⠀⠀⠀⠀⠈⢧⣾⢿⣬⣶⣶⣯⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠙⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⠏⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠹⡀
⠀⠀⠀⢾⠓⠦⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢷⠀⠀⡠⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⢆⠀⠈⠳⡀⠀⢀⡎⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡼⠀⡜⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢳⠀⠀⢀⡴⠒⠉⣹⠃
⠀⠀⠀⠀⠈⠳⣄⡀⢳⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⢹⣸⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⡰⠋⠀⢀⡴⠁
⠀⠀⢠⡤⠖⠒⠲⠭⢟⣦⣿⠀⣸⠁⡠⢦⠀⠀⠀⠀⠀⢇⠀⠑⢆⢸⡇⣠⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢳⠀⢸⣰⠧⠴⠚⠁
⠀⠀⠀⠉⠓⠦⠤⠤⠤⠬⢿⡰⠃⡼⠁⢸⠀⠀⠀⠀⠀⠈⢦⡀⠈⢿⡛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⢱⡀⠘⣆⣾⠓⠚⠉⠉⠙⠒⣢⠄
⠀⠀⠀⠀⠀⢞⡉⠉⠙⠒⢬⣇⢸⠁⢀⡞⠀⠀⠀⠀⠀⠀⣀⡈⠓⠺⡇⣠⠖⢻⡆⠀⠀⠀⠀⢸⡀⠀⣇⢀⠞⣀⣉⣓⣒⡒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠶⠺⣟⡠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠙⢆⠀⡷⠁⠀⡼⠀⠀⠀⠀⠀⠀⢣⠀⢸⣾⣉⣀⣀⣀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⣦⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢱⡏⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢠⣄⠴⣶⢲⡆⢤⣴⣤⠄⣰⢲⡶⢂⣤⠄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠻⣧⣄⠀⠹⠀⣽⡇⠰⠉⠀⣠⣾⠋⠀⠀⠀⠀⢠⣾⠏⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⢻⣿⡄⠁⠏⢀⠁⣰⡿⠉⠀⠀⠀⠀⠀⠀⢊⣽⣿⣯⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠈⡿⠞⠀⠈⠻⣏⠀⠀⠀⠀⠀⣴⣶⣶⡽⢻⡿⡿⢻⣹⣾⣶⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⣤⣄⡘⢿⣿⠇⠀⢇⠃⠀⣿⣿⠋⣀⣴⡄⠀⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⡠⠀⠀⢠⠤⠾⠟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⢧⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡶⢶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢶⣦⠆⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡾⣏⡙⣃⢈⡄⠀⠀⠤⠄⢀⠀⠀⡀⢠⣶⡄⢀⡀⠀⠀⡠⠤⠀⠀⠀⢈⣠⣼⣧⣄⡃
⠀⠀⠀⠀⠀⠀⣨⣭⣿⣿⣟⣯⡭⡄⠀⠀⠀⠀⠀⠈⣓⢉⣿⣿⣿⡉⢉⠀⠀⠀⠀⠀⠀⠀⣈⣉⣿⣿⣋⣀⡀
⠀⠀⠀⠀⠀⢀⠃⣾⣿⣿⣿⡇⢇⠰⠀⠀⠈⠙⠱⠋⠁⠞⠈⣿⠉⠣⠀⠙⠟⠛⠁⠀⠀⠎⠀⣿⣿⣿⣿⠁⠰⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣇⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣿⣿⣿⣿⡷⠦
⠀⠀⠀⠀⠀⢸⠁⣿⣿⣿⣿⡟⢸⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⠇⠹⣷⣿⣍⣠⠊⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠉⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⡛⡛⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣴⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠿⠿⠻⣿⣿⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠻⠻⠿⠡⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠀⠀⠀⠀⠀⡴⠿⠟⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠇⡀⠀⠘⡏⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠⠀⠀⠀⠠⠃⠃⠀⢰⠇⢀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⡀⣰⣦⠀⠀⢀⡏⠀⣼⠀⠀⢀⠤⠨⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⢿⠇⣽⠋⠀⠀⠈⠀⠀⡗⢀⡴⠁⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠊⢘⣼⣿⠀⠠⠀⢠⠖⠚⢱⡟⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⣿⡇⠈⠀⠀⡈⠃⠀⣸⣇⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣠⡄⣼⣿⠃⠀⡤⠊⠀⡀⠸⢿⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣙⣟⣿⣹⡇⢠⣞⡁⠀⢷⣸⣄⣰⣷⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⢀⠀⣴⣿⣿⣿⣿⠁⠈⠀⠈⢻⣶⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠂⠀⠈⠻⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠁⠈⠙⠛⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠀⠀⠀⠀⠀⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢫⠳⠦⠀⠀⠀⢰⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣰⠄⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣄⣀⡀⠀⠀⢆⠳⠤⠀⠀⠀⡼⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣬⣑⠂⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠏⠀⠀⠑⠈⠉⠻⢛⡛⠛⠛⠛⠲⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣄⣀⡀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋⠋⠁⠌⠛⢷⣤⡄⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡿⠁⠀⠀⠀⠀⠀⣀⡠⠄⠋⢉⠀⠀⠀⠀⠀⠀⠐⠛⠛⠻⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢀⣀⣰⡄⠀⠢⡈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⢀⡤⢖⣋⣵⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠿⣆⡀⠀⣰⣿⣾⡿⢋⢉⡉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠠⠴⠲⠶⠿⠿⢿⣶⣿⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⢀⣠⠬⠿⠿⣿⠏⡇⠀⠀⣿⣿⣩⠥⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠒⠠⠃⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠁⠀⠹⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠆⠑⢧⠘⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡇⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠃⠀⠛⠿⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡏⠁⣀⠀⠀⠘⠢⢀⠀⠀⠁⠀⠀⠀⠀⡄⠀⣰⡟⠁⠀⠀⠀⠀⠈⠂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠻⠓⣿⠙⢣⣀⣱⣶⣄⣲⣀⠀⠀⢠⢸⡿⢀⠰⠄⠂⠀⠀⠄⣤⣷⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠀⠈⡟⠛⠁⠈⠉⠁⠀⠀⠸⡿⢟⠁⡀⣀⣤⡦⢄⡀⡀⡽⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠝⣿⣿⠅⠀⢸⣿⣫⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡇⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣧⠀⠀⠀⠀⢰⡿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢸⡄⠀⠀⠀⢸⡷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠐⠒⠚

⣿⣿⡿⠿⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢃⣤⡀
⠀⠀⠀⠀⠀⢸⢰⠞⠁⠁⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠈⣣⣼⡉⣠⠄⠄⠀⣀⠀⠠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⢸⡤
⠀⠀⠀⠘⠦⠀⠿⠂⠘⠃⠀⠀⠀⢆⠰⠀⠀⢀⢰⡆⠀⠀⣩⣷⠀⠀⠀⠀⣤⡀⣄⢃⣨⡶⠗⠀⠉⠁⠀⢀⠈⠳⠄⢀⠀⣀⠀⠀⠠⢀⠀⠘⠟⠁
⠀⠀⠄⡀⠠⡿⠤⠤⠔⠂⠀⢶⠄⠈⢶⡀⠐⠁⡘⠀⠀⠀⠉⢹⠀⠀⠀⠀⠈⠛⣤⣏⡏⠒⣧⠔⠀⠀⠀⠀⠁⠤⢮⢠⣤⡄⠀⠈⠦⠐⢂⡴⠆
⣀⠀⠑⠼⣶⠆⠀⣀⡰⠀⠀⠀⠀⡀⠘⣧⡔⠀⠀⠀⠀⠀⠘⠝⡇⠴⠀⠀⠀⠀⠀⢸⣽⠻⠃⣤⠀⠀⠀⠙⢦⡀⠈⣥⡅⠀⠀⠐⠶⠂⠾⠆⠀⢀⡀
⠀⠀⢠⣤⠅⠀⢀⡴⠀⠀⠀⠀⠠⡈⣤⡼⠀⠐⠒⢰⠇⠀⢀⢠⣧⠤⡄⠀⠀⢰⡟⠂⢸⠀⣀⡰⠆⠀⠀⠀⠀⠐⠄⠘⡁⠀⠀⡀⢠⡀⢀⠀⠀⠘⠊
⠀⠀⢰⠉⢁⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⣸⣀⣁⠠⠂⠀⠀⠘⠈⡁⠀⠀⠀⠀⠀⠘⠟⣛⣯⡌⠁⠀⠀⡶⠀⠀⠀⠈⣿⠀⠀⠀⠙⠛⣁⢨⠦⡤⢀⠬
⠀⠀⠀⠀⠻⠀⣀⠀⠀⠀⠁⠀⢢⠀⠀⠉⠀⠀⠤⢺⠃⠀⠀⠻⠟⠀⠃⠀⢀⠀⠀⣀⠈⢃⠀⡠⢪⡞⠁⠀⠀⠀⠀⠘⠀⠀⠠⣠⠀⠻⠂⣤⠀⠾⠆
⠀⠀⠃⣀⠀⠀⠀⠑⢄⠀⠈⣦⠀⠁⢀⡄⣃⠠⢀⠂⠀⠀⠸⠽⣷⡰⠀⠀⠈⠙⠦⠬⠽⢶⣔⡤⠊⠀⠀⢠⠎⡆⠀⠰⠀⠀⢠⠼⠦⠀⠀⠉⣀⡀
⠀⠀⠀⡌⠀⠂⠀⠀⠀⢢⠀⠈⠳⣇⣸⠁⠉⢀⠁⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠁⢃⡅⣀⠀⠀⠀⠈⠚⠁⡤⣄⢠⠉⠋
⠀⠀⠀⠆⠀⠩⡀⠀⠀⠀⡆⠀⠀⠈⠏⠁⠀⠁⠀⠀⠀⠀⠑⠊⠋⠀⠀⠀⠀⡤⠄⠀⠀⠘⠀⠀⠈⢇⠀⡆⢸⠃⡋⢀⠀⠀⠀⠐⣀⣿⠁⠀⡤⢌⡆
⠀⠀⠀⠘⡀⠀⢧⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢐⠀⠀⠐⠀⡀⠀⠀⢰⠀⠀⠀⡀⠀⠲⣆⢠⠀⠉⠂⠀⠀⠀⠠⠈⠀⢹⠁⡤⡑
⠀⠀⠀⠀⠀⠦⠘⠦⠀⢀⠃⠀⠀⠀⡆⠀⠀⠀⢈⣋⠀⠀⠀⠀⠀⠈⠄⢁⢰⠀⠀⠀⠀⠈⠀⠀⠈⠙⠁⠀⡹⠊⢀⠀⠁⢠⠞⡇⠀⠀⠀⠌⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠁⠀⢀⠀⠀⠐⠀⢀⡀⠉⠉⠰⢄⠀⠰⡄⣄⠈⢠⡌⠀⢸⠇⠀⠠⠀⠀⠀⠑⢵⣀⢻⠀⢂⠀⠠⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢑⠀⠀⠀⠈⠳⠀⠀⠑⢬⡃⠀⠀⠫⡁⡀⢣⡀⠂⠃⠀⠀⠀⠀⠰⣄⠀⠙⢢⡦⠂⠀⡡⠀⢠⠊⠇
⠀⠀⠀⠀⠀⡀⠰⣆⠀⢀⠀⠀⠘⢆⣴⡄⠀⠳⠇⠀⠀⠀⠀⠀⢀⡀⠀⠳⣘⢁⣅⡀⠐⡦⠀⠀⠰⡈⢆⢂⠀⠡⡀⣸⠃⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠐⡀⢈⣆⠈⠀⠀⠰⠻⢀⡀⢠⠶⢀⠀⠀⠀⠀⡀⠀⠆⠐⠦⢜⡆⣑⣀⠄⠀⠀⠀⠀⠈⠒⠶⣄⠀⠡⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠋⠀⠈⢆⠱⣅⠀⠀⡸⠁⠀⠀⠄⠠⠘⡗⠁⠀⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠢⣇⠀⠋⢀⡶
⠀⠀⠀⠀⠀⠀⠃⠈⡒⢷⠀⠀⠾⠀⠀⠀⠦⠀⠑⢾⠇⠈⢀⡀⠀⠀⢆⢀⠁⠀⠰⣠⠁⠠⠀⠉⠀⠀⠈⠫⣆⡃⠀⠘⡀⢀⣨⠌
⠀⠀⠀⠀⠀⠀⣤⠀⡀⠹⡇⣀⠆⠀⠀⠀⠈⢌⠄⢸⡆⠀⠘⠁⠀⠀⢀⠑⣤⡔⡐⢡⡎⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠀⡟⠉⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡟⠡⢂⠏⠀⠀⠀⠀⠙⢾⡇⡴⠰⡄⠀⠇⠈⠃⢸⢪⡴⠊⠁⢀⡄⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⡧
⠀⠀⠀⠀⠀⠀⠀⣤⠈⡘⢿⠔⣃⠀⠀⠀⢂⠀⠀⢸⡟⢁⡠⠁⠀⠛⠷⣷⠀⡼⠀⠀⡄⠘⠀⠀⠀⠀⢊⡁⠀⠀⠀⠀⠀⠀⠐⠠⠀⢱⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⡇⠀⠁⠀⠀⠀⠈⠢⡀⢄⣧⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢰⠀⡠⠁⠀⠠⠀⡤⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠘⢢⡀
⠀⠀⠀⠀⠀⠈⠣⡀⠀⣀⡅⢐⠂⠀⠀⠀⠀⠄⢈⡀⡟⠠⠐⠹⠁⠀⠀⢰⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⢈⠁⠀⠀⠐⠃⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢈⠓⠻⠃⠍⣀⠀⠀⠀⠀⠀⠀⠈⢟⠐⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠃⠀⠀⠀⡤⡄⡠⠀⠨⠀⠀⠀⠤⠀⠀⠀⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢇⢰⣿⣠⠍⠀⠀⠠⢀⠤⠀⠀⡏⠀⠀⠀⠴⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠐⠼⠧⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⡠⢀⢍⠜⠀⠀⠀⠀⠀⠓⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⢶⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠶⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄

⠄⠠⠄⣀⠀⢄⠈⠀⠀⢀⣀⣤⣀⣀⡈⠉⠉⠀⣀⣀⠀⠀⠀⠀⠄⢀⢤⣀⠀⠡⠀⠈⠀⠁⢀⠀⠀⠀⠈⢇⠀⢀
⠀⠀⢠⠛⠐⠀⡄⠀⢰⠋⠀⡛⠻⠿⠹⠂⠀⢠⠉⠉⠉⡆⠀⠀⠀⡎⠉⠉⡆⠀⠀⠀⢀⣀⢈⣁⡀⠀⠠⡆⠀⠀⢡
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠠⠄⠀⠈⠀⠀⠈⠀⠤⠀⠃⠀⠀⠀⠁⠠⠀⠁⠀⠀⠀⠸⠀⠄⠀⣃⣀⣀⣃⠀⠄
⠀⠀⠀⢠⣤⣤⠀⠀⢠⢀⠸⠀⠈⡆⠀⡠⠀⠀⢀⡀⣀
⠀⢰⠉⠀⠀⠀⠀⠠⡄⣯⡀⠀⠀⠀⣉⠀⠐⠀⠀⠀⠀⠀⠴⠄⣄⠀⠀⠀⠀⡼⠛⢢⠀⠃⠲⠋⣶⢶⣂⡄
⠀⠀⢂⠀⠀⠀⠠⠤⠤⠤⡇⠀⠀⠰⡄⠀⠀⡀⠀⠀⠀⡌⠀⠀⠈⡄⠀⠀⢰⠃⠀⠘⡆⠀⠀⠐⠀⠀⠀⠃
⠀⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠁⠐⠀⠁⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠘⢄⠀⣠⠁⠀⠀⠀⠠⠀⠠
⠀⠀⣀⣀⡀⣀⡀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠥⠀⠇⢀⠀⢀⠐⠃⠀⠘⠂
⠙⠋⠉⣋⠁⣀⡔⠀⠀⠀⠀⠫⠀⠀⢀⠀⠀⠀⠙⠉⢳⠀⠀⠀⠀⠟⠤⠀⣡⠀⠀⡀⠀⠠⠉⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠁⠠⠀⠁⠀⠀⠠⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠁⠀⠊⠀⡀⠀⠀⢢⠁⠀⠀⠃⢀⡀⠰
⠀⠀⠀⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠌⠁⠄
⢤⣄⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⡤⠀⠀⠈⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⡀⡀⠀⢀⠀⢀
⠉⠐⡇⠀⠀⢐⡈⠃⠀⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⣶⠀⣩⡁⠀⠀⠀⣾⡁⢙⣷⠀⣀⣀⢐⠄⠀⠛⡄⠁⠀⠿
⠂⠀⠁⠀⠀⠈⡇⠠⠀⡐⠀⠀⠀⠠⡀⠀⡰⠀⠀⠀⠸⠃⠁⢈⠇⠀⠀⠀⡟⠓⠒⠙⠀⠀⠀⠘⠀⢠⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢸⣤⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠸⠂⠀⠀⠀⣟⠈⠁⢈⠀⠀⠀⠀⣀⠀⣀
⡀⠈⠀⠀⠀⠂⡀⠀⠀⠀⡠⣄⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠆⠖⢄⠀⠀⠀⠀⡠⠀⣀⠀⠉⠀⢠⠖⢒⠈
⠁⠀⠀⡀⠀⠀⢠⠀⠀⢶⡁⠀⠀⠉⡀⠀⠀⠀⡄⠀⠀⡰⠀⠀⠀⠸⠀⠀⠸⠈⠇⠀⢠⣦⠀⣫⡀⠀⠸⣧⠁⠈⠉
⠀⠀⠀⠘⡀⠀⣸⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁⡘⢀⠀⠀⠀⠀⠀⠇⠃⠆⠀⠀⠀⡘⠉⠀⢨⠁⠀⠀⠈⠣⠀⠐
⠀⠀⠀⠂⠀⠈⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠌⠀⢀⠀⢀⠀⠀⣀⠀⠀⣀⠀⠐⠆⠀⠻⠂⠀⠀⠐⠂
⠀⢀⠑⠄⣐⣤⡈⠀⠈⠀⡄⠀⠀⢀⠀⠀⠶⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠈⠈⠀⠠⡆
⠀⠀⢀⠒⠀⠀⠸⡄⣀⠠⠆⠀⠀⠠⡀⠀⠀⡹⠀⠀⠀⠈⠀⡀⠀⡃⠀⠀⠊⠀⠀⠀⠏⠀⠀⠀⡆⠀⠀⡶⠁⠀⠰
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⠀⠀⢀⠀⢐⠀⠄⠀⠀⢢⠐⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠢⡤⠤⡤⠤⣠⣀⣀⡀⣸⣴⣿⢹⣷⣼⠄⣀⡀⣠⡠⠤⡠⢦⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠀⠘⣦⣽⣿⣿⣻⣭⡍⡏⠈⡏⢩⢙⡹⠻⠙⣣⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⢀⣠⠄⠀⠰⠁
⠀⢂⡛⣯⠛⠘⠫⠝⣿⡛⠅⠣⠀⠓⠘⣛⣚⠠⠛⠈⠋⡏⣻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⢠⠀⢀⠀⠁
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠉⠈⠂⠀⠈⠁⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⣀⠠⣤⠀⠀⠀⣤⢄⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢰⠁⠀⠀⠀⠉⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠴⠀⠂⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠢⠀⠀⠠⠊⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠄⢀⠀⠀⠀⠐⠂⠀⠉⠀⠁⠂⠺⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡤⠀⢄⡀⠸
⠀⠀⠂⢬⠓⠉⠀⠄⠐⢒⡏⠈⠈⠀⢟⠒⠀⠀⠀⠑⢫⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠻⠈⠀⠠⠟⢃⡀
⠀⠀⠀⠀⢣⡖⠢⡍⠙⡻⢃⠀⠀⠀⠚⢛⠁⡥⠒⣦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠠⠃⣀⠘⠄⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠈⣰⡇⠀⠀⠀⢳⡈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡀⠀⠀⠀⣸⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢣⡀⠀⢀⠃⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠠⠀⠀⠀⠀⠀⢄⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠁⠇⠀⠀⠀⠀⠀⠰⠈⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢠⢠⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠤⠀⠀⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠆⠉⡞⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣀⠓⠀⢃⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠄⠀⠣⡀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⢈⠀⠀⠀⠉⠀⠉⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠁⠀⠎⠁⢱⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠒⠀⠐⢤⠊⠁⠀⠉⠓⠴⡀⠀⠒⣁⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁⢴⡃⠀⠀⠀⢘⡆⠌⢀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠃⠆⠀⠀⠀⠀⠀⢆⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⢈⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠜⠁⢂⠘⡀⠀⠀⠀⠘⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⠀⠀⠀⠀⠀⢲⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡤⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠼⠀⠀⠀⠀⠀⠀⠀⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⢀⠠⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡶⠙⢻⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠒⠻⠟⠀⠀⠀⠀⠀⣠⣤⣤⣴⡆⠀⠀⠀⠀⠐⠊⠁⠉⠁⣀⠐⠂⢀⠀⠀⠁⠋⠙⠀⠀⠀⠁⠀⣀⣀⣀⣀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⢠⡎⢴⡉⢹⡿⠓⠐⠢⡀⠐⠖⠾⠿⠟⢻⠿⠯⠛⠛⠃⠀⠀⠀⠀⠊⠃⠀⠐⠉⠉⠉⠛⢹⠋⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠀⠙⠋⠂⠀⠀⠀⠀⠀⠂⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣄⢀
⠀⠀⠀⠀⣤⡀⣠⡄⡀⠀⠀⠀⠀⠀⢷⣴⠖⣀⣈⠀⠂⠀⠀⠀⠀⡀⣸⡄⠀⠀⠀⠀⠀⠀⠀⣤⢠⡆⠀⠀⠀⣄⡀⠀⣰⠨⠜⡇
⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠉⠉⠀⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠈⡝⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⢠⣤⣄⡀
⠀⠀⠀⠀⣠⠴⢒⣶⣦⣄⠀⠀⣤⣤⣤⡤⣶⠶⠶⠶⣶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠋⠉⠉⠀⢀⠀⠀⠐⠶⠄⠂⠀⠀⢀⡆
⠀⠀⠀⠀⣉⠒⢚⡿⠝⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠙⠲⢶⣤⣶⣤⣤⠜⠉
⠀⣤⡚⠽⠝⠋⢉⡃⣠⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠟⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠈⢉⣻⠃
⠀⠈⠁⠀⢣⠀⢰⡆⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⠇⢀⣀
⠀⠀⠀⠀⠈⣦⣾⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠁⠸⠁⠻⠿
⠀⠀⠀⠀⠈⠛⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣤⣤⣿⠄⠀⠀⠀⢀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠑⢶⠁⢐⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⣤⣾⡟⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣎⡇⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠒⠒⠒⠒⠒⠂⠒⠆⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠄⠀⢀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⡔⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡚⠀⣀⣀⢘⢲⡀⠐⠀⠀⠀⢰⠒⢢⠤⠄⠀⠀⠠⠄⠀⠀⠀⠀⢃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡔⠃⠊⠙⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠘⢀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠂⠘⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠖⠲⠖⠒⠒⠒⠂⠒⠂⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⡠⠒⠂⠐⠒⢤⠀⠀⡴⠂⠑⢢⠀⠖⠉⠙⢆⢸⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠇
⠀⠀⠀⡇⡘⠀⠀⠀⠀⠀⠀⢱⡀⢣⡀⢀⡼⠀⢦⣀⣀⠎⢸⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⡃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⣠⠤⢤⠄
⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⠆⠠⢄⠀⡴⠒⠢⡄⢸⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠧⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⡇⠱⡀⠀⠀⠀⠀⠀⣰⠁⣇⠀⠀⣸⠀⣄⠀⢀⡄⢸⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠰⠄⠀⠀⠠⠄⠄⠤⠀⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡇⠀⠈⠢⠤⠀⠤⠞⠁⠀⠈⠉⠈⠁⠀⠈⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⡀⠀⢀⣄
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠳⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠉⠒⢄⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⣀⣀⣀⡀⠠⠊⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣅⠼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠴⠊⢠⠘⠅⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠀⡀⠤⢀
⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠄⡀⠀⠀⠀⠈⠀⠄⠀⠀⠤⠂⠀⠘⡆⠀⢀⡔⠚⠒⠒⠢⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⠀⠀⠀⢨⣀⠠⠀⠠⠐⠨⠀⠐⡇⠀⡎⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⢀⡠⠀⠀⠀⠀⠑⢤⡐⣀⢶⠡⡈⠎⠀⠀⠡⡀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠔⠉⠑⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠈⠛⠂⠑⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⢐⠀⠀⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡦⠀⠀⠠⠄⠀⢠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠢⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠊⠀⠁⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⣀⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢛⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡠⠊⠀⠀⢈⣹⣿⣿⠓⠊⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠒⢿⣿⡟⣀⡰⠁⠀⢠⠞⠛⠛⠻⠿⠦⡀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣯⣤⣄⠀⢀⣾⠯⠾⣿⡇⠀⠀⠌⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣞⠁⠀⠀⡟⣣⠄⠀⠜⠀⠀⠀⠀⠀⠀⠸⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⠜⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣏⠙⡓⢸⣚⢽⠁⠀⠈⠫⡀⠀⠸⡀⠀⠀⡌⢀⠄⠐⡀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠬⠤⡆⠺⡾⢻⡀⠀⠀⠀⠈⠢⣀⠑⠄⣪⠀⠄⠀⠀⠰⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⣄⡐⠂⣂⠛⣾⠊⠢⡀⠀⡀⣄⢄⣵⣴⣿⣶⣶⣶⣴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⠙⠓⠀⠁⠖⠐⣢⡌⠑⠝⠮⠗⣻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣊⡵⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠀⣀⣀⣀⣰⣖⡮⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⠀⠀⢀⠀⡀⠀⣠⣿⣄⠀⢀⠀⠀⠀⠀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠻⣄⠈⢲⣷⣶⠿⠿⠿⣿⣿⡶⠁⣠⠶⠁⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠂⠉⠉⠁⠉⢉⣛⣿⣿⣶⣤⣄⣀⡈⢓⣿⣿⣿⣧⣰⣤⣿⣿⣿⣛⣁⣤⡴⠶⠟⠻⠿⠍⣉⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠁⣀⣤⣎⣁⣈⣹⣿⠟⠙⢻⣿⣿⡿⣿⣿⡿⠋⠻⠍⠲⣤⣤⣤⠷⠦⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠔⠉⠀⠀⡤⢻⠛⠀⠀⠀⠀⢸⣿⣟⣛⣿⣿⣷⡀⠀⠀⠀⠈⢦⠉⠲⣄⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⣟⠀⠀⠀⡠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠢⣀⠀⠀⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡤⠖⠊⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄⠈⠛⠶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠻⣿⣿⣿⡿⠛⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⢀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠘⣿⡆
⠀⠀⢰⢻⠋⠀⠀⠀⠀⠀⠀⣔⠔⡀⡀⣐⡎⢠⠤⠀⠀⠀⣁⣈⠀⠀⢀⣀⣀⡉⠁⠁⢈⣀⣁⣀⠈⠀⠀⢿⠇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠛⠋⠀⠀⠀⠀⠀⠈⠙⠋⠉⢡⠖⠀⢀⠄
⠀⠀⠰⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠆⢰⠀⡠⠄⡰⢃⠀⣀⣀⢤⡴⠋⢀⡄⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡎⠀⠀⠀⡰⠃⠐⠁⠘⠃⠘⠀⠛⠀⠘⠁⡜⠁⠰⠉⢠⠏⠐
⠀⠀⠀⠈⠀⠀⠂⠂⠀⠠⠓⠂⠁⠂⠚⠁⠀⢰⠇⠤⠀⠰⠁⢠⡾⠹⡼⠢⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣯⢿⢜⡜⣾⢽⠀⠉⠉⠐⠂⠐⠐⠢⠤⠴⣶⡀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠘⠫⢷⡮⠵⠋⠀⠠⠠⠄⠴⠆⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠂⠀⠀⠀⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⠞⣠⠀⠀⡰⠂⢀⡠⣄⡀⠀⣰⢋⡆⠀⢀⡤⠄⠀⣤⢀⡀⢀⡴⢁⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⢀⠜⠁⡼⠁⣠⠞⠁⠐⠎⠠⠏⢀⡞⣹⠉⠀⠰⠉⠀⠀⠀⠃⣏⠔⡼⠱⠏⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡄⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀⡀⠀⠀⠀⠀⠀⠀⣧⣀⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠂⠀⣀⡀⠀⠀⡠⡄⠀⠀⠀⢀⣃⡀⠀⣀⡀⢀⠄⠀⠀⢀⣀⠀⠘⠂⠛
⠀⠀⠀⠘⡥⡄⠀⠀⡁⠀⠀⠀⡰⠉⠀⠴⠯⠍⠙⡜⠀⠀⠀⠀⠀⠒⠒⠒⠂⠔⠠⠆⠆⠴⠶⠲⠰⠶⠴⠆
⠀⠀⠀⢠⠀⡂⠀⠀⡁⠀⠀⡰⠣⡇⠀⠀⠀⡼⢰⢇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠂
⠀⠀⠀⢸⠀⡅⠀⠀⠀⠀⢸⠇⡰⠁⠟⠺⠂⡗⢛⡏⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⠁⠀⠀⠀⠀⠘⣷⡶⠶⠶⠆
⠀⠀⠀⠈⠂⡅⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠁⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠅
⠀⠀⠀⠀⠀⡇⠀⣘⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣄⣀⠀⡀
⠀⠀⠀⠠⠀⠇⠀⠿⠦⠤⠶⠶⠶⠶⠶⠾⠛⠁⠋⠉⠀⠄⠀⠈⠀⠀⠀⠠⠠⠊⠇⠂⠀⠀⢰⡤⠀⠉⠉⠐⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡿⠛⠛⠻⡟⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠏⢿⠇⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠡⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠲⠀⠎
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⢠⣤⣤⡀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⢀⣰⣿⠀⠀⠀⠀⢸⣿⣿⣿⣧⢠⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠉⢹⠃⠀⠀⠀⠀⠀⠀⠀⠟⠙⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⣿⣿⣿⣿⣦⠀⠀⣠⣶⣶⣾⣷⣴⣿⣶⣶⣦⡀⠀⠀⢠⣴⣶⣿⣦⣾⣷⣶⣄
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇⠿⠿⠿⠿⠿⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣋⠀⢙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠠⣿⣇⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣄⡀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠈⠰⣿⡇
⠀⠀⠀⠀⢠⣶⣿⣿⣄⣀⣀⣿⣿⣷⣦⠀⠀⣾⣿⣿⣿⣿⢻⣿⣿⣿⣿⡄⠀⠀⠀⠘⣧⡀⠀⠀⣰⡇
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠛⠛⠒⠚⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄⠀⠀⠀⠀⠀⣠⠊⠉⠒⠉⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡁⢘⣆⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⢀⣠⡤⣤⡀
⠀⠀⠀⢀⡴⠊⠉⢠⡰⢏⠃⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣘⣁⠙⣉⣏⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⣾⣿⢶⣤⠁
⠀⠀⠀⢺⢀⡷⠚⠃⠀⡜⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣍⢤⣩⣀⣯⠀⠀⠀⠀⠀⠀⢸⠟⠈⢲⡷⠊⠀⠘⠃
⠀⠀⠀⢸⠻⢥⣤⠤⠊⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⢲⠓⠂⠚⠀⠀⠀⠀⠀⠀⠼⠥⣌⣡⣨⢽⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚

⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⠀⠀⡄⡀⠀⢰⣀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⢀⣠⡴
⠀⠀⢀⢤⠀⡊⠀⢈⡀⡏⣷⠀⠀⠀⠀⠀⠀⢠⡀⠀⡰⠟⢠⠖⠁⠀⠀⢈⠙⣶⠂⣿⡃⢀⣞⣥⠾⠤⠀⠀⠀⠀⢹⣧⠀⢀⢔⡿⠉
⠀⠀⢸⣸⣦⣰⠀⢸⣾⢟⠟⠀⠀⠀⠀⣀⣈⢳⣿⣠⢶⣟⣿⣀⠀⠀⠀⠋⠉⠙⠫⡹⡴⠟⠛⠻⢯⠀⠀⠀⠀⠀⠀⣯⣇⢎⡞
⠀⠀⠈⠳⠥⡟⠀⢰⠏⠁⠀⠀⠀⠀⠀⠈⢎⠳⡽⢡⢻⡟⣥⠴⠀⠀⠀⠀⠀⠀⠀⢧⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡏⣿⣼⡀
⠀⠀⠀⠀⠀⠱⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣇⣬⣬⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣇⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⣸⣿⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⠤⡄
⠀⠀⠠⠶⢤⡄⡾⢁⣤⡆⠠⠀⠀⠀⠀⠀⣴⣸⣿⣷⢟⣾⣶⡁⠀⠀⠀⠀⠀⢀⠮⠭⠥⠬⠭⢷⠀⠀⠀⢀⡴⠅⣤⢤⠒⠶⠻⣄⣄
⠀⠀⠀⠑⠂⠉⣧⠟⠭⠔⠀⠀⠀⠀⠀⠀⣿⣿⣾⡏⣼⣿⣹⣃⠀⠀⠀⠠⣦⠞⠀⠀⠀⢀⠀⢈⢣⣄⠀⢨⠧⡶⠿⢀⡀⢐⠶⠀⠘⢶⡄
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⡟⣾⡛⣿⢗⡿⡏⠁⠀⠀⠘⠓⠚⠙⢦⡜⠉⠑⠊⠀⠀⠀⠀⢱⠃⣠⠋⠉⠙⠋⠙⣦⠀⢣
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠟⠚⠃⠛⠂⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⡤⠢⡔⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢟⣻⠒⢤⠀⡴⢫⠝⠋⠉⠀⠀⠀⠀⠀⠀⣀⠤⢤⡀
⠀⠀⣀⣀⣃⢀⣀⡸⠊⠉⢿⠀⠀⠀⠀⠀⠠⡀⠀⡆⢀⡠⠔⠂⠀⠀⠀⠀⣀⠤⡤⡭⣿⣿⣧⡀⣲⢦⡀⠀⠀⠀⢀⠞⠁⠀⠀⠙⡄
⠀⠘⠢⡤⠈⣿⢉⡷⠠⢤⠚⠀⠀⠀⢀⣀⣀⡈⢦⣷⢞⣡⡤⠤⠀⠀⢠⣮⢴⠎⣰⠇⠈⠘⡆⢷⠈⠉⠁⠀⠀⠀⠸⠀⣄⠀⠀⢀⢱
⠀⠀⠘⠤⠚⢈⡉⡧⣸⠞⠀⠀⠀⠀⠉⠀⠀⠉⠻⣷⠋⠁⠀⠀⠀⠀⠈⠀⠈⡼⠀⠀⡇⡆⠙⣾⠀⠀⠀⠀⠀⠀⠘⠚⢻⠒⢲⠛⠋
⠀⠀⠀⠀⠑⠊⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢻⣿⣿⣿⣏⣀⣀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢯⠈⠋⠉⣿⣿⣿⣿⣿⣿⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⣤⣄⣈⣉⠉⠉⠁⠀⠀⢹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣫⡯⣿⣾⣻⣿⣯⣿⣶⣄⠀⡀⠞⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢷⠟⣿⣿⡉⠊⢳⣀⠱⡙⣿⣀⣀⡀⠈⢀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣿⠂⣸⡇⣻⣿⣇⡋⢎⣧⣰⣳⣹⣇⢨⣿⣠⣬⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⡁⡏⣷⣿⣿⣿⡷⣶⣻⣏⣩⣼⣾⡿⡋⠄⠸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠷⣿⣻⣿⣽⣿⣶⣿⣿⢿⢯⣿⣿⢿⢻⣇⢡⢡⢠⢷⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠹⣿⣤⡛⣯⣿⣿⣿⣾⣿⢿⡋⠘⡎⣿⣀⣄⠆⠾⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡇⠈⡿⠿⠿⡟⠻⣿⣇⣠⣤⣧⣤⣷⡾⣿⣿⠓⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣀⣱⣤⣶⣷⠿⣿⡟⣿⣯⣭⣿⡿⢻⣿⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⣿⣶⠾⡖⡋⢙⣹⠉⠉⠁⠀⠀⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⡆⠀⠉⠀⠀⣹⣿⣿⣿⣶⣦⣄⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣆⠀⠀⣾⣿⣿⣿⣟⡿⠉⠉⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠉⠛⠛⠁⠀⠀⢀⣾⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣄⠀⠀⠀⢀⣀⣴⣿⣿⡇⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠿⡻⢿⢻⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⣻⡿⠞⢛⠉⠉⠉⠉⠛⠻⢿⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣿⠟⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠓⢌⠳⣽⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠣⠙⣟⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠇⡇⢹⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠰⠘⠀⠘⢰⠀⡎⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢏⡎⠁⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠺⣷⡀⢀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠚⣡⠞⢀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⢿⣦⡈⠒⠆⠀⠀⠀⠀⠀⡴⠏⢁⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⠟⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡛⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡊⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠈⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣇⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣻⠦⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⡍⠇⠨⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣽⣬⣿⣫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⢖⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢿⠸⣠⠇⠀⠀⠸⣆⠈⣩⡆⠀⠀⠀⠀⢀⠄⠀⠀⠀⣿⠀⢛⡛⠁⠀⠀⠀⠀⠀⠀⢀⡤⠘⢁⡀
⠀⠀⢈⠐⢋⡶⠀⠀⠀⠈⢈⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡈⠚⡏⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠴⠛⠁
⠀⠀⠸⣦⡈⢠⢴⡆⠛⣩⠃⣞⣴⠃⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠇⠀⠀⠀⠀⠁⠀⠀⠸⣷⡄⣰⡦
⠀⠀⠰⣄⠙⣠⠶⠀⢤⣘⠧⢁⣴⠇⠀⠀⠐⠀⠀⠀⠀⣄⡻⠄⣀⡶⠀⠀⠁⠀⠀⠀⠀⣤⣽⣡⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠚⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡶⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠀⡀⢠⠆⠀⠀⠀⡈⠗⠾⠇
⠀⠀⠰⡆⢚⡋⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⠰⠈⠛⠋⠀⠀⠀⠀⠀⠊⠸⣀⠁⢋⣠⠄⠀⠀⣻⢇⢴⣦
⠀⠀⢀⠐⢋⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⣠⡄⠀⠀⠀⠀⠀⢀⠀⠈⠁⠋⢡⠆⠀⢀⡹⠀⢰⠇
⠀⠀⠘⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⡾⠍⡉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣗⡏⢠⡶⠘⣛⠁⢉⡙⠁
⠀⠀⠰⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠋⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⣈⣹⣏⣤⠀⠙⠳⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠄⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⢀⣀⠀⠀⠀⠠⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⠏⠁⠀⠘⠛⠀⠀⠀⠀⠀⣠⡌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠁⠀⠀⠀⠙⠒⠦⠤⣄⡀⣀
⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⠀⠀⠀⠚⠚⠀⠀⠀⠀⠚⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠐⠦⣄
⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⡦⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⢀⣸⡧
⠀⠀⠀⠀⠙⠻⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠞⣉⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣛⣿⠶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⢤⡴⠶⠒⠒⡨⠛⣉⡠⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣼⣿⣮⣬⣉⣙⣭⡉⠉⠉⠉⣹⡙⠄⣀⣴⣤⣶⡮⠶⠶⠛⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⢻⣿⠟⠛⠃⠐⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠄⠀⢸⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣼⠻⠳⠦⠆⢶⡊⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⣿⣷⣾⣤⠶⠚⠛⢧⠀⠀⠀⢠⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⡀⡇⠀⣿⡿⠃⠀⠀⠀⠀⠘⡆⠀⠀⡟⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠹⢄⢳⡇⠀⣿⡏⠀⠀⠀⠀⠀⠀⠹⠀⢸⡇⣠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡎⣾⡙⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⢠⡇⢰⠁⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣧⡿⣷⣸⣿⣦⠀⠀⠀⠀⡇⠀⠀⠀⣿⢸⣀⡾⠊⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⠑⣯⣿⡏⣿⠀⣀⠀⢠⠀⠀⠐⠀⢻⡞⢠⠇⢰⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣸⢧⣿⡇⣿⠂⡇⠀⡆⠀⠀⠃⠀⣿⡇⡾⢠⡟⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣾⣿⡗⣿⣰⠀⢸⠀⡄⠀⢸⠀⣿⣿⣵⡾⢋⣉⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣿⣿⡄⢸⡀⣷⢰⣸⣾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠷⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠂⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⠀⠀⠀⣿⣿⠋⣿⣆⡹⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⠀⢸⡄⠀⠀⠟⠁⠀⣽⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢸⣿⣿⣿⡆⢸⡇⢀⡆⠀⣠⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⠀⠀⢻⣿⣿⣟⢀⣬⣱⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⣀⡀⠈⣿⣿⡿⠸⠏⠁⣤⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣿⣇⡀⢿⣿⣿⣄⣠⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣾⣿⣿⣹⡟⣋⣠⣶⡿⠻⠋⣹⣿⣿⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣋⡻⠿⣹⡧⠀⠶⠖⠉⠀⢀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣇⠙⣿⣷⣬⣇⠀⡀⢢⡀⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠈⣿⠟⠉⠉⠙⣄⠰⢶⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠿⠶⢷⣆⣨⣧⣼⣿⣿⡿⠿⠿⠛⠛
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡀⠀⣸⠇⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠘⠋⠉⣤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⠀⢸⡅⢰⣿⠄
⠀⠓⣤⡄⣿⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠸⣿⡟⠁⣀⡴⠀⣠⡞
⠀⠀⣿⠁⠙⣯⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣴⣾⣶⠚⠉⠠⢞⠛
⠀⠀⠀⠰⠃⠀⠀⠀⢀⡀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡌⠻⣿⣿⣿⣁⠤⣠⣾⡵⠊
⠀⠀⠀⠀⠀⠀⢤⣴⣀⡷⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠙⣿⡄⠘⣿⣿⣿⡿⠟⠋⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡟⠁⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣧⡸⣇⠀⣾⠉⠁⠀⡀⢀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⣠⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣶⠏⣠⡄⠟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠀⠛⠀⠀⠀⢀⡀⠀⠘⣿⣿⡟⣿⣿⣟⠰⣿⣀⣠⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣴⡎⡹⣶⣄⢿⡟⠀⠈⢻⠋⢁⣴⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣷⣠⡆⠉⠘⣧⠘⣿⣿⠁⢠⠀⠀⣠⣟⣱⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣣⣄⠀⢠⣄⠸⣿⡄⠸⡆⠀⣹⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢿⣏⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⡋⠉⠙⠁⠈⢿⣧⢿⣷⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⢰⣦⣴⣄⠀⠉⠠⣿⣧⣿⠇⠀⠀⢀⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢀⣾⣀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⡌⢻⣿⣿⣷⣤⣀⠙⣿⠇⢀⡄⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣼⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡃⣼⠏⠻⣿⣿⠱⠀⢾⣠⡎⢡⡾⠃⠀⠀⠀⠀⠀⢦⣄⠀⢀⡁⠀⠙⡏⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠀⠀⠀⠙⠟⠀⠗⠀⢿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄⠀⠀⢶⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣦⡀⠀⣾⠀⠀⢴⣧⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⣄⣸⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣴⡟⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠿⣿⣿⡅⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠙⠟⠀⠀⠀⡀⢻⡷⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡟⢻⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠐⡇⠀⣾⡇⢸⣿⡀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠘⣟⣧⡴⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣄⡀⠀⠀⠉⠻⢼⡆⠻⡇⠈⣿⠁⠀⠀⠀⣴⠀⠀⠈⠁⠀⢠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣈⠳⣎⣁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠄⠃⢰⣧⠀⠆⣠⣴⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣮⣿⣦⡄⠀⣦⠀⠀⠀⢀⣴⣷⡆⠘⣿⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠙⢿⣿⣿⡇⠀⡷⢀⡄⢠⡾⠋⢹⣿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠙⢿⣧⣤⣥⢀⣴⠟⢀⣶⣾⣿⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠉⢿⡇⠈⢡⡄⢸⣿⠀⣈⢠⣶⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠈⡇⢀⣾⡇⢸⣿⣦⣈⡵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡀⠀⠀⣿⠈⢿⡇⠀⡻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣷⣄⠀⣿⡆⠘⠃⠀⢃⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⢿⣿⣿⠿⢇⣿⠀⢀⠀⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⡆⠈⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠛⣿⡄⢰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣶⣦⣀
⠀⠀⠀⣇⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠘⣞⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⢃⠀⠀⠀⢐⠉⢹⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠘⣾⣟⣹⣿⣿⣦⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢦⡙⢿⣿⣿⣷⡄⠀⠀⠀⣶⠶⠶⠶⣶⣶⣶⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠻⣿⡿⣿⣦⠀⠀⣿⣇⠀⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡃⠀⢿⣿⠷⢀⣩⣭⣶⣶⣶⣶⣶⣶⣤⣄⡸⢽⣿⠀⠀⠀⢀⣾⠛⠻⣿⣿⣿⣿⣿⣿⠟⠁
⣀⡤⠴⣶⣿⣷⣄⠀⠀⠀⠹⣄⢈⣥⣾⠿⠛⠉⠁⠀⠀⠀⠀⠈⠙⠻⢿⣶⣝⠀⠀⢀⣾⠃⣰⠃⠀⣽⣿⣿⡟⠁⠀⢰⠚⠿⣶⣄⡀
⣻⣷⡞⢹⣿⣿⡽⡆⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣀⣾⢃⣼⠏⢀⣾⣿⣿⣿⠃⠀⠀⡄⠀⠀⠈⠙⢷⡀
⠹⣿⣿⡰⣿⣿⣿⡹⡄⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢣⣾⡟⢀⣾⣿⣿⣿⠃⠀⠀⣼⣴⡷⢀⠀⢀⠜⠁
⠀⢻⣿⣷⡹⣿⣿⣧⢻⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠉⣾⣿⣿⣿⠃⠀⢀⣼⣿⠟⣰⣯⡴⠁
⠀⠀⢿⣿⣧⠹⣿⣿⣧⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣅⣸⣿⣿⣿⠃⢀⣴⢿⠟⢁⣼⣿⡟⠀⠀⠀⠀⢀
⠀⠀⠈⣿⣿⣦⡜⣿⣿⣾⣿⡄⠀⠀⠀⠀⣀⣤⣶⣶⣿⡿⠿⠿⠿⠿⢿⣿⣶⣶⣿⡇⠀⠉⣿⣿⢃⣾⣟⠁⠈⢀⡞⣽⡟⠀⠀⣀⣰⣾⡏
⠀⠀⠀⠘⣿⣿⣷⠸⣿⣿⣿⣷⡀⣠⣶⣿⢿⡻⠝⠒⠋⠉⠉⠉⠉⠉⠉⠛⠛⠽⣿⣿⣦⣴⡿⢁⣾⣿⢿⣷⣦⡈⢠⣿⠁⠀⣴⠅⣽⣿⠁
⠀⠀⠀⠀⠹⣿⣟⢣⡹⣿⣿⣿⣿⣿⡫⠊⠁⢀⣠⠤⠖⠒⠒⠒⠒⠒⠒⠲⠤⢄⡀⠙⣿⣿⣧⣿⣿⡃⢸⣿⣿⣿⣿⡿⢀⡼⢋⣾⣿⠋
⠀⠀⠀⠀⠀⢻⣿⣠⠿⣮⣿⣿⣿⠋⢀⠴⠊⠁⠀⠀⠀⢠⣆⠀⢀⡀⠀⠀⠀⠀⠉⠂⢸⣿⣿⣿⡉⢉⣾⣿⣿⣿⡿⢡⡞⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢻⣷⣀⠈⣿⣿⡏⠰⠁⠀⠀⠀⠀⢙⣳⠶⢿⣞⣩⠐⣒⠶⠀⠀⠀⠀⣼⣿⣿⣿⡇⣾⣿⣿⣿⡟⠁⠀⠈⠉⣿⡅
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣿⣿⡹⡄⠀⠀⠀⢴⣞⣛⢤⣠⢼⣟⡵⠞⣋⡤⠂⠀⣠⣾⣿⣿⣿⣿⢗⢿⣿⣿⡟⣅⠀⠀⢀⢞⠞
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠾⢿⣿⣿⣌⠓⠤⣀⠀⠈⠉⠉⠛⠛⠛⠒⢋⣁⣤⣶⣿⣿⣿⣿⣿⣿⢏⣿⣿⡿⢟⡞⣿⠀⢠⢮⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡻⣿⣿⣿⣾⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⠋⠉⠉⠀⢬⡙⠿⣶⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠫⣉⣻⣿⠟⠀⠀⠀⠀⠀⠀⡄⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣮⣭⣿⣿⣿⣿⣿⣧⣤⣤⣶⣶⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⢣⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⡠⣫⣿⣿⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠘⢦⣄⣠⣦⣿⣿⣿⠟⣡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣷⠎⠋

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣤⣼⣿⣶⣿⣶⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠛⠉⠉⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⠿⠿⠛⠛⠧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⢠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⣴⡿⡶⣤⡄⡀⠀⠀⢀⢀⠀⢿⡖⠀⠀⠥⢾⣿⡒⢿⠛⠷⣷⡆⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢻⡀⠀⠈⠀⣀⡈⠁⠈⠁⠀⠀⢈⠀⢸⡇⠀⠀⠀⠀⠃⠀⡀⠀⠀⠛⠱⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⣼⣿⣿⣈⣤⠶⠟⡀⠀⠀⠀⠸⣃⣇⠈⣿⣤⣄⣀⢠⡀⠀⠒⢻⣷⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡻⣿⣿⣿⣧⣻⡂⠀⠀⠀⠀⠠⣆⢻⣤⣿⣿⣿⣿⣸⣠⣤⡾⠿⡏⢃
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⢷⣝⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠿⡆⢻⣿⣿⣿⣿⣿⣿⠙⠷⣶⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠳⡆⠀⠿⣎⣿⠁⡇⠀⠀⠀⠀⠀⠀⠀⢿⣄⠻⣿⣿⣿⣿⣿⣿⣶⣿⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣧⢳⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡘⢿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⡆⠀⠀⠀⠀⢀⡀⠀⠀⢻⣿⡌⢿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣦⠀⠀⠀⠀⠘⠻⠓⠀⠀⠀⠀⠘⢿⡄⠀⠈⢻⣿⣆⢻⣿⠀⢻

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣄⣀
⢀⣾⣷⣀⢠⣶⠖⡀⠀⠀⠀⠀⠀⣬⣣⣒⣻⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⢀⠰⠞⠁⢀⠀⠀⠀⠀⠀⠀⣿⣭⣧⠼⠿⢟⠃
⢹⢿⡿⣿⣵⣥⡩⠒⠀⠀⠀⠀⢓⣑⣢⣶⠗⠆⠀⠀⠀⠀⠶⠀⣀⠈⡲⢤⡆⠒⡀⠀⡤⠆⠀⠀⢠⣶⣿⡿⣿⣶⣷⣿⣿⣄
⠠⠉⢲⣿⣿⢬⡟⠛⠃⠀⠀⠀⣷⣾⣶⣶⣾⠆⠀⠀⠀⠀⠀⠀⠖⣠⣄⣁⣴⠖⠒⢤⣀⠀⠀⠀⠸⠟⢠⣾⣿⣿⡿⠉⠛⠁
⠀⢂⡾⠜⣿⣷⡽⠲⠢⠀⠀⠀⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿
⠀⠸⠁⢈⡿⠹⡗⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟
⠀⠀⢀⣶⠅⢺⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⣀⣤⣴⣴⣦⣤
⠀⠀⠚⣯⠀⣿⠇⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⣾⠟⠘⠁⠀⠀⠀⠀⠀⠀⠀⣦⡀⣦⢀⣴⠟⠀⠀⠀⠀⠀⣠⢿⣿⣿⣷⣽⣖
⠀⠀⠀⢀⣸⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣇⣿⣿⣿⣥⣾⠃⠀⠀⠀⠀⠞⠋⣿⢻⡉⠙
⠀⠀⠀⠀⠱⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣼⣿⣿⡧⢾⣒⠀⠀⠀⠀⠀⠀⠀⣿⡄⠁
⠀⠀⠀⠀⠀⠃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⣽⣿⣿⣱⣆⠈⠁⠀⠀⠀⠀⠠⠶⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠈⠈

⡶⠉⠀⣀⠀⠀⠙⣿⣿⣿⠉⠐⠠⣤⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦
⡇⠀⠀⠙⠀⣤⣶⣿⣿⡿⠀⠄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⠉⣿⠋⠉⠉⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⢹⣶
⠀⠀⠀⠀⣰⣿⣿⣿⣏⣀⣠⡀⠀⠀⠀⢸⡿⠋⢿⣿⣿⣿⣿⣿⣿⣷⠂⠀⠃⣾⣿⣷⠀⣤⣤⡀⢸⡿⢹⣿⣿⡿⣿⠐⠖⠀⠐⠇
⠀⢀⠀⢀⣿⣿⠿⠿⠿⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣇⠀⣦⡉⢋⣁⡈⠻⣿⠟⢘⣁⣼⣿⣿⣶⣾⣷⣶⣖⡶
⣿⣿⣷⣸⠏⢀⡀⠀⢠⣤⣶⣿⣧⣤⣴⣾⣿⣿⠿⠒⠈⠀⠈⠉⠉⠉⠻⠀⠹⣿⣯⣉⣻⣦⣤⣶⣿⣿⣿⠿⣿⣿⣿⣿⣿⣋⣤⣴⣶
⠙⡻⣿⣿⢾⣿⠿⠂⠈⠛⠋⠙⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠉⠉⣿⣿⡿⠀⠀⠉⠙⠛⠛⣿⣿⢿⡛
⣶⣿⣿⢿⠀⠀⣠⡾⠿⣦⣤⠔⠉⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠀⠀⠛⠀⠀⣿⢫⣀⣬
⠏⠩⠑⡼⠈⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠀⠀⠶⠄⣠⣤⣼⣿⣷⣶⣶⣾⣿⣶⣶⣤⡀⠀⠀⠠⠀⠀⠀⠀⠘⣿⣿⡿
⡿⠇⣢⣾⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠆⠀⣀⣼⣿⣶⣦⣀⣑⣅⠌
⢆⣾⣿⣿⡿⠛⠛⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡿⠋⠙⠛⠉⠈⢹⣿⣿⡏⠀⢀⡀⠀⠀⣿⡿⠿⣿⣿⠿⣿⣿⡇⣑
⣿⣿⠛⠁⠀⠀⠀⠘⠛⣿⣿⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⣇⠀⠀⠀⣀⡀⠀⣿⣿⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣸⣿⣿⣿
⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡆⡀⠁⠶⠂⠀⠈⠁⠀⢻⢿⡟⠉⠻⢿⠿⠿⢿⡿⠀⠀⠛⣿⣿⣿⠿⢿
⢿⣿⡿⠃⠀⠰⠀⠀⣀⣽⣿⣄⡈⠀⠀⠸⣿⠋⠁⠀⠀⠈⠙⣿⣷⡄⠀⠀⠀⠀⠀⢠⣸⢁⣄⠀⠀⢀⠀⢼⠃⠀⠀⠀⠘⠋
⠘⢿⡇⠀⢀⣴⣾⣿⣿⣿⣿⡿⢧⣤⣄⣠⡏⢀⠀⠀⠀⡀⠀⣿⣿⣁⣀⣀⡀⠀⠀⣠⣼⡄⢀⣀⣀⠈⢁⡏⠂⠀⠀⠀⠀⣄⡀⠀⠛
⠀⠀⠁⠀⣸⣿⣿⣿⣿⡏⠀⠀⢀⣤⣉⣉⣅⠉⢁⣀⠈⠉⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣷⣿⠁⠀⠀⠀⠀⠀⢹⡀⠀⡀
⠀⣀⣤⣉⣿⣿⣿⣿⡟⠃⠒⠀⠉⣅⣉⣉⣼⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⣿⣿⠿⠿⠿⢿⣿⣿⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠂⠀⡀
⠀⠛⠁⠀⠀⠀⠀⠙⣿⣿⠀⢸⣿⠿⢟⠿⠿⣿⣿⣀⣠⠄⠚⠛⠛⠿⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠒⠀⠀⢤⠄⠀⣿⣋⣀⣸⣯⣾⣯⣶⣷⣿⣿⣼⣯⣤⣀⠀⠀⠀⠀⠀⠀⠸⢫⣿⣿⡟⢀⣴⣶⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠄
⣤⡰⣤⣤⢦⡀⠀⢈⣹⣿⣿⣿⣿⡛⣻⣯⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⡟⠁⢸⣿⣿⠿⠿⠿⠿⣿⡿⠀⠀⣠⣶⣶⣷
⠿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⡿⠛⠻⠿⠿⠿⣿⠀⣀⣤⣤⣄⡁⠀⠀⠐⠀⠀⡏⠀⠀⠀⠀⢀⠈⠃⢠⣾⣿⡿⠻⠿
⠀⠻⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⢀⡀⠀⢀⠀⣻⣾⣿⣿⠿⠿⠿⢶⣶⣿⣿⣦⣄⠙⠁⢀⠀⠉⣠⣘⣿⣿⣿
⠛⠀⠀⠀⠀⠀⣀⠀⠀⠉⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣸⣿⣿⠏⠀⠀⠀⢠⠌⣿⣿⣿⣿⣿⣷⣿⣯⣽⣿⣿⣩⣿⣿⠟⠰⠆
⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⣾⠿⣿⣿⣿⣿⣇⣀⣀⣨⣶⣾⣿⣿⣿⡀⠴⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⡟⠛⠛⠛⠁⠈⣿⣷⡄
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣙⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢀⣾⣋⣽⣿⣿⣿⣟⡛⠋⠁⢤⡤⣀⣴⣶⣾⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⡰⠃
⠀⠀⠹⣿⣿⣦⣀⠀⠀⡰⢆⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠙⠻⣁⡐⠙⠆⠓⠈⣇⣀⣀⣼⣇⣀⣈⠧⠀⠠⠐⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⠀⢀⠀⠤⠀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠙⢛⢠⣴⠖⠋⠉⠙⠄⠈⠀⡄⠀⠁⠒⠂⠀⠀⠀⠀⠀⢸⠄⠀⠀⠈⠉⠁⠀⠀⠀⡀⣠⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⠀⠀⠀⣀⢤⡆⠀⠅⠂⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠽⠍⠙⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒⠒⠂⠀⠀⠐⠂⢄⡀
⠀⠀⠀⣠⣴⣶⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠉⢓⡐⠒⠂⠀⠀⠈⠉⠻⣦
⠀⡴⣾⠿⠿⡶⡤⣄⠈⠀⣖⢲⡖⢲⣴⣤⣄⠀⠀⠀⢀⠀⠴⢶⠋⢻⡀⠀⠀⢨⣿⠁⠀⠀⠀⠀⠀⢀⣻⠄
⠀⠀⣙⠻⡱⣓⡗⠀⠘⢼⠃⣜⠣⡞⠥⠗⢽⣳⠄⠀⠀⠀⠀⠘⢷⠮⢶⣦⣤⣸⣷⣤⣶⣷⣶⣶⡶⠿⠛
⠀⠀⠈⠋⠀⠈⠈⠱⠤⢥⠔⢤⠞⣥⠖⠀⠲⠟⠁⠀⠀⠀⠀⠀⠘⠀⣀⡿⠙⢇⠈⠉⠉⠉⠉⠙⠶⠶⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣴⠠⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠡⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⢀⠀⣰⡓⣐⣄⣶⣷⣰⣄⠀⠙⠻⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠘⠳⣄⡀⠀⠀⠀⠀⠈
⠀⢀⡏⣴⣆⡀⣿⣿⣍⣽⣿⠏⡷⢧⣀⡄⢋⠁⠦⠀⠀⠀⠀⠀⣀⣤⣠⣴⣤⠴⠊⢉⣿⢟⡶⣺⢿⣷⣾⠉⠈
⠀⠘⢦⡟⠁⠀⠉⡿⠛⠁⠉⠙⠃⠀⠺⢣⣄⠱⠀⡁⠀⠀⣠⣾⣿⣿⡇⣼⡿⢿⢸⢧⡿⣻⠟⢯⡞⠙⡾
⠀⠀⠀⠑⢰⣶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢠⠄⠀⠈⠙⠛⠉⠉⠁⠁⠀⠀⠉⠉⠉⠑⡄⠀⠀⠠⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂

⠀⠀⠀⠉⠻⣟⠓⢛⡟⠉⠀⠀⠀⠀⠀⢸⠿⣶⢒⡛⢻⡛⢙⣧⢰⡘⣇⠀⠀⠀⠄⠤⠀⢠⠶⢚⡋⡉⠀⠀⣠⠀⠛⠀⢃⢈⠆⢠⡈⠻⣆
⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⣻⢸⣷⡞⡇⡎⠘⠜⣧⠸⣆⠀⠐⣦⣴⠀⠈⠒⢬⡙⢡⣄⡆⢨⡄⠱⡀⠘⠋⠀⡈⢁⡄⠈⠛⢲⡄
⠀⠀⠀⠀⠀⢰⠟⢻⠀⠀⠀⠀⠀⠀⢀⡟⢀⠸⡎⠟⣷⢀⡇⢶⢠⡙⣧⠹⣆⠀⠈⠁⠀⠀⠀⠀⠛⠶⢦⣄⡀⠹⠶⠟⠀⠆⠈⠃⢀⠙⠦⠴⠀⠻⢄
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⣼⡷⣼⠀⢿⡀⣿⠘⣿⠸⣧⢷⣘⢧⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣶⣶⣾⠋⠛⠓⠒⠒⠉⠉⠑⠋⠁
⠀⠀⠀⠀⠀⢸⠀⠸⡆⠀⠀⠀⠀⠠⣿⣧⡏⢸⢸⣇⡟⣠⢟⣼⣧⣠⣽⡷⠟⠁⡠⠶⠒⠶⠒⠦⣄⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⢸⡀⠀⢷⠀⠀⠀⠀⠀⠈⠿⠷⠷⢾⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⣿⡇⠀⠀⠐⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠘⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⢻⠀⠀⠀⠀⢸⡇⠂⠠⣤⡄⠀⠀⠀⣷⠀⠀⠀⠀⣼⣿⡇⢠⣀⠀⡀⠀⣱⠀⣇⡀⡄
⠀⠀⠀⠀⠀⠀⠹⣆⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡸⣿⡇⠀⠀⠀⠈⠳⠶⢤⣿⣀⠀⢠⡞⠁⠀⠀⠀⢠⣿⣿⠁⠘⠛⠛⠛⠛⢉⠋⠉⠉
⠀⠀⠄⠠⠀⠀⠀⠈⠲⠾⠃⠀⣰⣿⣷⣆⠀⠀⠀⠀⠀⣿⡧⠿⢿⡀⠀⠀⠀⠀⣰⡿⠁⠈⠉⠉⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠉⣁
⠀⢔⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⢧⡉⢉⡼⣣⠀⠀⠀⠀⡏⢠⣦⣤⣷⡀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠠⡀⢠⠤⠔⠀⠀⠀⠀⠀⣿⠀⠋⠍⠀⠸⡆⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣰⡟⢴⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠈⠁⢔⡀⠀⠀⠀⠀⢠⣇⣩⠖⠦⠦⠴⢹⡀⠀⠀⣿⣿⠿⢉⣿⠙⡇⠀⣿⡇⠀⠀⠀⢀⣠⣶⣿⠏⢀⣾⠃⠀⠀⠀⠀⡀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠦⣤⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⠀⢱⡀⠀⢹⡇⡄⣿⣿⣦⣷⠀⠙⠃⠀⢀⣴⣿⣿⠟⠁⢀⡾⠃⠀⠀⠀⣤⣴⣷⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠦⢴⣦⣤⠖⠛⠈⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⡿⣁⢂⣴⠟⠁⠀⠀⡠⢿⣿⠟⠿⠿⢛⣷⡆
⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⢸⠉⠈⡀⠀⠀⠀⠀⠀⠘⣿⣇⣉⣉⡿⠀⠀⠀⠸⣿⣿⣷⡴⠟⠁⠀⠀⠀⠈⠢⠤⡤⣤⢤⢄⡈⠐⠛⡄
⠀⠀⠀⠀⠘⣾⠱⢧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢀⣰⠉⠁⠁⠁
⠀⠀⠀⠀⡄⢊⠈⠀⢄⠀⠀⠀⠀⠀⡇⠀⠰⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⠶⣶⠲⣤⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿
⠀⠀⠀⣘⠊⠉⠉⢈⣠⠇⠀⠀⠀⠀⢡⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⠷⢏⣦⠸⠟⢡⡍⠛⠓⢄⠀⠀⠀⠀⠀⠀⣸⡧⡟⠀⠀⠀⢀⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣽⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣤⠀⡿⠀⠀⠀⠀⠀⠀⠀⢺⣿⡿⠿⠟⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡅⢀⣀⠀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⢠⣴⣶⣾⣿⣦⣤⣤⣾⣷⣶⣶⣄⠀⠀⠀⢰⣾⣿⡇⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠺⠿⠿⠷⠶⠴⠶⠿⠿⠿⠃⠀⠀⠀⠸⠿⠿⠿⠷⠶⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⢿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⢀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠤⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣤⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠻⠿⠟⢁⡀
⠀⠀⠀⠀⠀⢀⣶⣿⡀⠀⠀⠀⢀⣿⣶⡄⠀⠀⠀⢰⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣷⡄⠀⣴⣿⣿⣿⣷⣄⣀⣠⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠦⠤⠴⠿⠿⠿⠇⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠋⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠒⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠻⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⣤⡀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠂⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠐⣿⣿⡧⠀⠀⠀⠀⠀⢀⣀⣀⣀⠉⠉⠁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠂⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣯⣿⠀⠀⠀⠀⣀⡇⢸⣇⡀⠀⠀⠀⠴⡶⢆⣀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠼⣏⣽⡀⠀⠀⢸⠯⣯⣰⡿⠓⢦⡀⠈⢉⣉⣉⣁⢹⠁⠀⠀⠀⠀⢋⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠤⣄⠀⠀⠀⣸⣇⠀⠀⠀⣧⠀⠀⢸⡄⢀⠈⠀⡄⢀⢳⡀⠀⢀⣀⣀⠀⠆⣾⣷
⠀⠀⠀⠀⠀⠀⠈⠇⢀⡿⠀⠀⢰⡏⠉⡾⠀⠀⢹⠀⠀⣾⣷⠈⢲⠀⢻⠀⠸⡇⠈⢩⣭⡥⠤⡔⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⢸⠁⢰⡇⢆⠀⢸⡆⠀⡇⣿⠀⠀⠀⢸⠀⠀⡇⢀⣐⣂⣀⣒⠃⢘⣃⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⡇⠀⢸⡄⢸⠃⠀⠀⢸⡄⢀⣇⢹⣦⣤⣰⡜⣡⡖⠃⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠘⣃⠚⠃⠑⠂⢸⡇⠀⠉⢸⣿⣿⣿⣷⢸⠉⠐⠶⠶⠶⠒⠒⠺⢻⠛⠛⠛⡛⡏
⠀⠀⠀⠀⠀⠀⠈⣉⣉⣉⡀⠀⠀⠉⣤⡤⣤⢤⡞⠃⠀⠀⢸⣿⣿⣿⣿⢸⡇⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⢿⣷⣿⢸⡇⠀⠀⠀⢸⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠈⢉⣉⠉⠁⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠀⠀⠀⠀⣸⡗⣿⢸⡇⠀⠀⠀⠸⣿⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠬⠉⠁⠀⠸⠇
⠀⠀⠀⠀⠠⠤⠤⠤⠬⠭⠀⠀⠀⠀⢹⡴⠏⠰⠷⠄⠀⠀⣰⣿⣿⣠⡶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠈⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠂⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢁⣀⣤⣤⡴⢴⣶⠶⢤⣤⣄⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⣞⣋⣯⢦⢺⡇⠀⣿⡴⣌⢿⣿⣿⠄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⠻⢿⣷⣯⣼⡇⠀⢸⣿⠽⠿⣛⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⡀⠀⠀⠀⠈⠙⠒⠲⠶⢾⠃⠀⢸⠗⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠏⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣾⠀⠀⠀⠀⣠⣴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠉⠉⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⢰⠇⠀⢀⡏⠀⣤⡶⢛⠉⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢾⣿⣯⣿⠉⣿⣻⣳⣦⠀⣷⠀⠀⠀⢀⡟⠀⠀⣼⠁⢸⡇⠁⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠬⡏⠀⣯⣭⡭⠥⠞⠁⠀⠀⢀⡞⠀⠀⢠⠇⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡟⠀⠀⠀⠀⠀⢨⡟⢳⡶⠶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡄⠀⠀⠀⠀⠀⢰⠇⠀⠀⣼⠁⠀⠀⠀⠀⣠⠏⠀⡼⠀⠀⢀⣀⡴⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⡀⠀⠀⠀⠀⡾⠀⠀⢀⡇⠀⠀⠀⠀⣴⠃⠀⣼⠁⣴⠚⠋⠉⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⡀⠀⢳⡀⠀⢳⡄⠀⠀⢰⡇⠀⠀⢸⠁⠀⠀⠀⡼⠁⠀⣼⠃⠀⠻⣜⢯⣹⡗⢢⣄⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠛⠶⣄⢳⡀⠀⠙⣆⠀⢸⡇⠀⠀⣼⠀⠀⠀⣸⠃⠀⣸⠃⠀⠀⠀⠈⢱⠟⢀⣟⣺⢟⣠⠟
⠀⠀⠀⠀⠀⠀⣼⠟⠅⠀⠀⠀⠀⠀⣺⠇⢻⡄⠀⠈⢧⣾⢀⠀⡄⣿⠀⠀⢠⡟⠀⢠⡏⠀⠀⠀⣀⡴⠋⢠⠞⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⡤⢴⡖⠋⠁⠀⠀⢻⡄⠀⠘⣿⡸⢠⢲⣻⡆⠀⣸⢷⢰⢸⡇⢀⣤⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠻⣄⠀⠀⠀⠀⢷⢁⠄⢻⣏⡿⡘⣿⣷⠀⣿⣾⣾⢸⣷⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠻⣶⣶⣶⡜⣟⢄⡠⢿⡹⡱⡣⠹⡇⣿⠿⢡⣾⣿⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⣶⣶⣶⠏⢻⣃⡆⠊⣿⣴⣥⡼⠃⢹⡇⡴⠰⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⢧⣥⡼⠏⠀⠀⠀⠀⠀⠙⠶⠞⠁

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣤⣥⣾⣧⣬⣤⡄⠠⣤⠀⡀⠀⢰⠀⠀⠀⡡⠤⠁⠀⠀⠐
⠀⠀⠳⡀⠀⠀⠀⢀⣴⠋⢳⣆⠀⠀⠀⠀⡠⠊⠀⠀⠀⠊⢱⡄⠀⠀⠀⠀⠀⠀⣀⠘⢺⠿⢣⡜⠿⡖⠃⣀⠀⠀⠀⠀⠀⠢⡠⣆⠀⠀⢀⣠
⠀⠀⠀⠀⡘⠂⠂⣈⢻⡄⣼⢋⡀⠀⠐⡂⠀⠀⠀⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠛⡿⠋⠉⡄⡀
⠀⠀⠀⠀⠀⠈⠁⠉⢁⠸⢀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠑⣄⡳⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡈⠀⠀⠀⠀⢀⢁⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⢄⣀⠴⢊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⠆⡀⠑⢦⢠⠖⠁⠰⠄⠀⠀⠀⠀⠀⡀⢡⠌⠉⠁⡎⠀⠀⠀⠀⠂⠉⠉⠀⠀⠀⠙⣗⢸⡏⠁⠀⠀⠀⠉⠐⠂⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⢀⡡⢬⡇⠀⣿⠤⣅⠀⠀⠀⠀⠀⠀⠈⠴⠈⠹⠁⠶⠒⠁⠀⠀⠈⠂⠀⠀⠀⡀⠀⣯⢸⡃⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⢇⠈⠳⢤⠛⠈⢃⠴⠋⣀⠆⠀⠀⠀⠀⢠⠀⠛⠂⠛⠀⢤⠀⠀⠀⠀⠀⠀⢹⣏⢇⣀⡘⣏⣀⣠⢯⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠙⠂⠂⠀⡉⠀⠒⠒⠁⠀⠀⠀⠀⠀⠄⠈⠓⠐⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⢹⡛⢟⠻⠌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡠⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠂⠒⠲⠤⠐⣄⣧⣥⡔⠡⠤⠒⠂⠀⠀⠀⠀⠀⡆⠀⠀⡥⠀⢀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠈⠀⠀⠀⠀⣶⡐⠙⡆
⠀⠀⠀⠀⠀⠀⠐⢻⠂⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⠔⠛⠢⣈⠆⠀⠀⠀⠀⢀⠘⣇⠀⠀⢰⡤⣴⠀⠀⢀⡟⢀⠀⠀⠀⠀⠀⣹⣇⢀⡇
⠀⠀⠀⠀⠀⠀⡠⡖⠋⠙⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠦⠤⠏⠀⠀⠀⠀⠀⠀⠀⢼⡄⢀⣻⣿⣿⢒⡀⣼⠁⠀⠀⠀⠀⠀⠀⣿⣷⣾
⠀⠀⠀⠀⠀⠌⠐⢇⠀⢀⠟⠀⠣⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠸⠐⣄⠀⠀⠀⠀⡌⠸⣇⠀⠙⢿⠿⠋⢀⣿⠞⡄⠀⠀⠀⠀⠀⢸⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢀⢴⡄⠈⠀⠀⠀⠀⠐⢀⠘⡦⠀⠈⠀⠠⡞⢀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠑⠂⡀⠀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠰⠆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡽⠤⡧⠬⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠗⢏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠤⠐⠂⢆⡇⠀⠀⠀⠀⢤⡀⠀⠒⠒⠀⢀⡤
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠹⣦⡾⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠴⠖⠢⠔⢣⠤⠀⠀⠀⢰⠒⠇⢹⠋⠉⠉⠸⠐⡆
⠀⠀⠀⢀⠄⠁⠀⠀⠁⠀⠀⠊⢃⠀⠀⠀⠀⠀⠀⡆⡠⠊⠤⠆⠤⠄⠢⡀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠆⠀⠀⠀⠀⣄⠀⢠⡕⠀⣠
⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⢰⠂⠀⠀⠀⢠⠀⡀⠀⠹⠁⠨⠖⠀⠐⠲⠀⠹⠅⠀⣀⢠⠀⠀⠀⠀⠃⡀⠀⢈⠀⠀⠀⠀⠀⠢⡀⢦⠂⠐⡄⢀⠔
⠀⠀⠀⠀⠀⠀⢙⠒⠶⠚⡘⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠏⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⠉⠁⠀⠀⠉⠁⠂⠓⢤⡁⡘⠆⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢅⠀⠄⣀⠒⠾⠰⠀⣠⡀⠀⡀⡀⠀⢠⢆⠠⢠⠀⠀⣀⣤⡜⢯⣳⣷
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⠐⡶⡤⣤⣄⣀⣠⣁⡤⠀⠤⠤⠀⠠⠄⠁⠉⠠⠤⠀⠋
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⡐⠶⠐⠆⠂⠀⠀⡐⠀⠂⠀⠀⠀⠀⠀⢱⠀⠀⢠⣅⠀⠈⠉⠻⢇⣿
⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠠⠿⣦⠗⠛⠉⠛⢾⣿⣤⡄⠈⠓⠀⠀⠀⠀⢃⠀⢸⠇⠉⠐⣄⢀⣠⣽⡇
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⣸⡃⠀⠀⠀⠀⠘⣿⣆⠐⠂⠀⢯⠀⠀⠀⢸⡄⢈⠀⠀⠀⠀⠈⠀⣿
⠀⠀⡞⠀⠀⠀⠀⠀⠀⣾⡥⣿⡇⠀⠀⠀⠀⠀⠈⢻⣄⡈⠁⠭⠄⠀⠀⢸⠇⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠛⣷⢹⡇⠀⠐⠤⠠⢄⠀⣸⣷⡥⢌⠍⠁⠀⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢊⡚⢻⡆⠀⠀⠀⠀⠛⢻⡇⠝⠃⠀⠀⠀⡀⠃⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠘⡅⠀⣀⡀⠠⠊⠀⠀⠀⢸⠄⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠈⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠢⣤⠀⠸⣼⠀⠀⠀⠀⠀⣼⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠲⢷⣦⣿⠇⣰⠀⠀⡀⢹⢧⣮⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡜⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⡤⣀⣡⣾⡜⢷⢣⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠰⣀⠏⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⠁⠯⡟⡁⢣⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠠⡀⠀⠀⠀⠀⠀⠀⡘⠱⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⢦⡆⠟⡧⠹
⠀⠀⠀⠀⠀⠀⡀⠸⢸⢋⠀⢘⡆⠘⡂⣨⠀⣶⢟⣸⡀⢀⠀⠘⠀⠀⡀⠀⡀⣠⢏⡛⣆⠇⢀⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠾⠖⠿⣧⡇⠿⡈⡇⡟⠈⡏⠃⠎⠐⡀⢠⣰⡾⢶⡟⠉⠀⢳⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠰⠈⠃⠘⠠⠇⢾⠚⠷⠔⠋⠀⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⣿⣼⣾⣇⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠆⢠⣷⢠⣾⡇⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡠⢿⠟⠉⣿⣿⣧⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⢸⣻⣾⣿⡏⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣶⣦⣴⣶⡶⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣁⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⡞⠙⣿⣶⡄⣿⡀⢰⢸⡏⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣿⣿⡿⢄
⠀⠀⠀⠀⢀⠴⠛⢀⣴⣿⣿⣿⣿⣧⣼⣦⣷⡄⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⠀⣿⣿⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠹⣿⣿⣿⣿⣿⡟⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠰⢷⣤⣿⣿⡿⡆⠁
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠈⠛⠋⠉⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⡈⠻⠿⠟⢠⠇
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡆⢰⠀⡄⠀⣿⠀⠀⠀⢣⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⣦⣷⣶⣿⣶⣷⣴⠇⣸⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⢤⣤⣄⣀⣀⠀⠈⠦⠠⣄⡀⠙⢦⡟⣋⣙⣿⣿⣿⣿⠶⠃⣀⡴⠰⠀⠉⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⡠⠐⠋⠁⠁⠀⠈⠉⠛⠛⠲⠤⠌⣹⣶⢤⣀⣀⣉⣗⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣀⣠⣤⣖⣋⡩⠭⠛⠛⠒⠀⠐⠀⠂⠈⠑⠢⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠀⠐⠲⠖⠚⠛⢲⠟⠽⠿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠿⠶⠤⣤⠄⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⣠⢴⠏⠀⠄⠀⣼⡇⢠⣿⣿⣿⡌⢿⠁⣙⢿⡉⢹⣦⣀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⡼⢀⡄⠀⠘⠁⣭⣿⣿⣿⣿⣿⣦⡀⠘⠆⢁⢹⡇⠈⠑⠦⡀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠊⠀⠀⠀⣀⣿⡀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⢜⣶⣇⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠘⣿⡾⢻⡏⢿⣿⣷⣿⣿⣿⣿⣿⠟⢹⣦⣿⣿⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠙⢿⣿⣿⣿⠟⠋⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⣶⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣻⠿⢿⡁⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠿⢛⠿⠦⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⠶⣿⣡⣚⡷⠊⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠿⠗⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⣤⡶⣾⣿⣿⠶⣦⣄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠎⠹⠃⣤⡄⠀⢀⣤⡀⠟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⣘⠀⡀⠀⢼⠆⠀⡀⢸⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡿⣴⢿⣦⢸⢠⣾⠷⡜⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡊⠺⣦⠀⠀⠀⢾⡷⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⠀⣆⠘⡟⢰⢈⡿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠈⠀⠐⠀⠁⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢹⠀⠹⠾⠞⠁⢸⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⡟⠋⠀⠐⠀⠀⠀⠒⠀⠀⢀⠒⠀⠀⠚⠂⠀⠆⠀⠂⠘⠀⠈⠑⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠄⠀⡠⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠰⠁⣺⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠘⣤⠄
⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⢰⢾⠃⠀⠀⠀⠀⠀⠀⢇⠀⣀⣀⢀⠀⠀⠁⠲⠤⣄⣀⣀⣀⣀⣠⠤⠒⠛⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡄⠀⡀⡄⢸⠈⢠⠀⠀⢠⠀⠀⢀⢠⢤⡂⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠀⠠⠄⠀⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠃⠲⠷⢷⠂⠀⢸⢀⠀⡀⠀⠠⠥⠜⡇⢸⠀⡃⠀⠀⠐⠂⠀⠀⣤⢶⣄⢀⡀⠀⠀⠀⠀⢸⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⣧⠀⢸⠀⠀⠃⠀⠀⠛⢟⠿⢍⠀⠐⠀⠀⢀⠀⠀⠀⠿⠤⡿⠈⡇⠀⠀⢀⡿⠜⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣿⠀⠺⠄⠀⢠⡆⡔⠀⠸⠀⢰⠀⡇⠀⣸⠸⠀⠀⢰⠒⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠈⠀⠀⠀⠀⠣⠀⢰⠏⡸⠇⠀⠀⠀⠁⡇⠀⠀⠀⣰⠀⠁⠀⠧⠺⡄⢀⡀⠧⠄⠰⠶⠀⠀⠀⠀⡇⠀⡇⠀⠘
⠀⠀⠀⠀⣀⠀⠀⠀⢐⣋⣀⠇⠀⢀⠀⠀⠀⠇⠄⡇⠀⡇⠀⢀⡴⠀⠀⠀⠈⠃⢀⠀⠐⡌⠁⠀⠀⠀⡇⠀⠁⠀⢰
⠀⠈⠉⠋⠁⠀⠀⠀⠘⡎⡘⢰⠀⠈⠃⠀⣴⠀⠀⠀⠖⠂⠐⠂⡇⠈⡲⠀⠀⠀⠈⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠲⠶⠶⠀⡇⠀⠀⠀⢰⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠤
⠈⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⢱⡶⢶⡶⠀⡄⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⢨⠀⡄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⢀⠀⠀⠀⢲⣤⡄⠀⠀⢀⠀⠀⡸⣆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⣿⣶⠘⣧⣿⡟⠀⠀⠈⠿⣿⣄⢐⣾⡷⢳⣽⡍⠀⠀⠀⠀⠀⢻⣖⠀⠀⣤⣶⠧⠀⠀⠀⢀⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⡤⣤⣠⡿⣌⢡⡄⢶⣷⣤⢨⣉⡱⣎⣠⡔⠊⠀⠀⠀⠀⠻⣤⣀⠘⣿⡦⠴⠛⣧⣴⠀⠀⠙⣾⣇⠈⣯⣵⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣴⡓⢿⡀⠘⡯⣞⠛⢿⠃⠠⣍⣡⣔⠛⣵⠋⠸⣿⡍⠀⠀⣰⡦⡿⠶⣏⢠⡟⢀⠀⠀⠸⠻⠄⢠⣄⡄⢉⣌⣰⡞⠟⠃⢰⣠⣆⡤
⠀⠀⠀⠀⠀⠀⡼⠟⠋⠀⢉⣭⡿⡿⢟⢧⣈⣯⣟⣯⠃⠀⠙⢻⡄⢀⠙⠠⣄⣤⠀⣀⣼⣄⡎⣻⠹⡿⠶⠂⠀⢀⡰⠀⠽⠻⡟⢣⠶⣀⢀⣤⣊⠃⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢠⣄⣰⠀⠀⠀⠀⠀⡿⡿⠙⣷⢼⡟⠀⠀⠀⠀⢻⣜⣿⣤⡽⢿⠆⡁⢀⠈⠁⢿⡘⣶⡂⣀⡴⣮⡛⢃⣀⣺⣡⡴⠞⣉⠁⠀⠨⠗⠢⣴⣧⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣽⠿⡃⢦⣾⣧⠄⠀⢀⠙⣲⣆⢺⣴⣧⢀⡄⠹⣿⢿⣟⢪⡴⣷⠿⣄⠀⡀⢹⡝⠁⣼⣟⣋⡉⢚⡿⠋⠀⢙⡘⠿⢾⡀⠀⢸⣷⠃⠉⠟⠷
⠀⠀⠀⠀⠀⠀⡼⠟⠋⢉⣿⢶⣿⡁⠀⠒⠻⠟⡿⠛⢿⡁⠀⢠⣿⣶⣿⠉⣽⢦⡶⣤⣤⠀⠀⠽⣾⢀⣔⠉⣀⠿⢳⢎⣴⠃⠀⢿⡿⠁⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠻⠃⠈⠙⣦⡷⣏⠀⢠⠄⠈⣷⠀⠀⠀⠀⠹⣦⡆⠈⠞⢳⢾⡇⠀⢀⡟⠋⠈⠽⣋⣠⢸⠋⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣁⣾⢷⣄⣀⣬⣷⡀⣹⣿⣶⠀⢿⡄⠀⠀⡽⡏⢻⡇⢀⠴⢶⡍⢉⡝⢺⡇⠀⢠⣴⣍⣇⢸⠀⠀⠀⠀⢀⢠⣤⠄
⠀⣀⣶⣼⣦⢤⣤⣄⡤⢿⣿⡿⠟⠋⠀⠈⠻⢶⡶⠿⠶⢾⣷⣦⣤⣅⣄⣼⡇⣸⠻⠎⢀⣻⡛⠀⣷⢢⣴⠚⠟⠙⣿⠀⠀⣳⢆⢸⠟⠁⢀⣦⡞⠀⢠⡀⣴⣀⡀
⠘⠉⠋⣸⠗⠋⢀⣹⠳⠈⠸⠆⠀⠀⠀⠀⢻⡻⠀⢀⢀⣤⣤⡋⠉⠉⠉⠛⢿⣷⡄⢤⡬⠉⣥⣠⣟⠁⠉⣉⣢⠀⡏⠰⣾⣈⣋⡇⠀⣀⣼⢷⠶⢖⣉⡛⠉⠈⠁
⢀⣀⡞⣋⣓⠀⠈⠏⠃⠰⣆⣀⡀⢲⡤⣶⣫⡝⢳⣿⠀⡴⣿⠏⠀⠀⣄⣲⠺⣉⣿⣎⣹⠏⣙⠈⣧⢼⡔⣊⢉⡾⢡⣸⡏⣙⣫⣷⡋⠀⠛⠻⠦⠈⢿⡦⠄
⣘⣽⠻⠁⣯⠀⠀⠀⢀⣏⣹⣻⣥⣿⠾⠧⣿⣛⣹⣏⣀⡀⠘⠂⠀⠀⠚⣫⡾⢹⠇⢻⣇⠰⣿⢷⡇⣰⣽⡛⢻⣇⣹⡿⠛⣉⣡⣸⣷⢀⣠⣦⣶⡀⠀⠛
⠉⡎⠃⠀⠀⠀⠀⠀⢶⣾⣣⡝⠁⣞⡆⠀⠀⣉⢹⡭⠛⠛⠻⠶⢦⣴⣿⣤⠈⣏⣻⣾⢻⣧⢀⣾⣥⡿⢟⡏⢹⡟⠻⢞⣰⡜⢻⣽⠖⢺⣿⢩⢧⣇⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠘⠁⠾⡿⠀⠙⠇⠀⠈⡽⠟⠃⠀⠀⢠⣠⡄⠈⢛⣿⣶⠋⠓⠀⠀⢻⣿⠋⣷⡤⠘⣿⣿⠘⢶⣸⣻⣶⠻⡄⢐⣤⠖⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣝⢫⣽⣍⠛⣦⡀⠀⠀⢰⣿⣆⠘⠿⠀⢠⡿⠤⠚⣹⡦⠈⢿⡉⠉⢩⢾⠃⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣯⠐⢻⡏⠀⠈⠻⣦⣀⡿⠀⣿⣄⣀⣀⣿⣁⣰⡧⠄⢹⢢⣼⣧⣗⢀⣼⣤⠤⠴⣚⣟⠻⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⣿⣉⣿⣿⣽⣫⡴⠶⠂⡉⢸⣯⢀⡟⠀⠀⡀⠀⠋⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡷⣰⠄⠻⣷⣄⣼⣿⣿⡇⢡⡈⢯⣥⡄⠙⠛⠛⢻⣾⠖⣰⡮⠷⠍⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣼⣿⣱⣦⣿⣿⣿⣿⣿⡇⠺⣟⡄⡿⠀⠲⣿⠋⣸⣿⣉⠀⣻⣆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣤⣸⣿⡎⢸⣿⣿⣿⡇⠈⠉⢙⠇⠀⢸⣠⡾⣅⠀⠞⣗⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⣹⣾⣿⣿⣿⣿⣶⣶⣾⠶⣤⣾⠻⣟⠟⠃⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡝⠀⢹⣿⣿⣿⣿⡏⢠⢷⡂⠸⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⣿⣿⣿⡇⠀⢫⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢄⣰⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣴⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠙⠛⠛⠛⠛⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠠⠀⠀⠄⠀⠈⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠰⣄⠀⠀⡠⠂⡠⠃⠀⠀⠀⠀⠀⣀⡀⠀⠀⠨⣿⣿⠷⠀⠠⠀⠀⠌⠉⠹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⡆⢨⡅⠲⠪⣀⠀⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠉⢆⠀⢄⠀⠀⣀⠄⣀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠁⠈⠁⠀⠀⠑⣦⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡲⠬⠉⠰⢪⡁⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠪⣤⣠⢟⠀⠀⠀⠀⢘⠢⡴⠉⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠃⠀⣈⠀⡉⠢⡀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠉⠀⠀⠀⢡⠀⠈⠢⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠇⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢸⢩⠬⠭⢭⠽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠀⢸⠜⠀⠀⠈⠒⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠶⡒⠃⠀⠑⡶⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠕⠁⠀⠀⠀⠑⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⢀⠜⣽⠋⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠜⠁⠠⠒⠉⠉⠁⠒⠤⠀⢀⣀⣤⡶⠏⠀⠀⠀⠒⡲⠖⠐⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⢀⢀⡴⠅⠀⠀⠠⢤⣤⣶⣶⢿⢿⠭⠭⠓⠒⣚⠡⠄⠊⠁⠀⠀⠁⠀⠄⠀⠀⠀⢠⣤⣦⡾
⠀⠀⠀⢀⣠⣴⡏⠀⠀⠀⣀⣤⠶⠚⠉⠀⠀⠀⠀⢀⠔⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢝⣿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⡴⠒⢒⣲⡖⠉⢉⣈⠉⠉⠋⠀⠀⡄⠐⠀⠆⠀⠀⡍
⠀⠀⠀⠀⠀⡸⣋⠄⠀⠀⠁⠀⠀⠀⠀⠀⠁⢐⠀⠀⠠⢀⡂⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠄⠀⠀⠀⠘⠀⠀⢀⠄⡉⢆⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠁⠔⠁⠠⠁⠀⢀⡀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢾⣦⡀⣀⡜⠊⠅⢰⠂⠰⠀⠀⠀⠀⠚⠋⠁⠀⡀⠠⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣦⣿⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⢀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠐⢂⡀⢬⣿⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡞⠀⠀⢠⢃⠄⣀
⠀⠀⠀⠀⠀⠀⠀⡁⢠⠊⠁⠀⠀⢈⠡⠴⢒⠂⠒⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡾⠉⡿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⢛⣶⣧⡀⠂⢀⠀⢀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠓⣄⢸⣶⢸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢉⡿⠿⡇⠄⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠞⠀⠐⢠⠄⠀⠀⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢠⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⣄⠀⢀⡀
⠀⠀⠀⣿⣉⣉⣉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⢸⣿⣿⠀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡄⠀⣠⣤⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⢨⠁⠀⠀⠀⢰⣶⡆⡆
⠀⠀⠀⢸⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠇⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⢸⠀⠀⠀⠀⢸⣿⡇⡇
⠀⠀⠀⢸⣿⣿⣇⡇⠀⠀⠀⠀⣴⡆⢸⣿⠀⠀⠀⠠⣿⡀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣾⢰⣾⣶⠀⠀⠀⣘⣿⣧⣃
⠀⠀⠀⢸⣿⣿⢸⡇⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣿⢸⠀⣿⢿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠸⣿⣿⠀⠀⠸⣿⣿⣿⡿
⠀⠀⠀⠸⣿⣿⣸⠇⠀⠀⠀⠀⠿⠗⠸⠟⠀⠀⠀⠸⠿⠺⠀⠿⠸⠀⠀⠀⠿⠃⠀⠀⠘⠿⠿⠀⠿⠿⠀⠀⠀⠿⠿⠿⠧
⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⠷⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣭⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⣀⣄⣀
⠀⠀⢠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⣴⡗⠀⠀⠀⠀⠀⣠⣾⣿⠟⢹⣶⣿⣿⣿⣿⣿⣶⣆
⠀⠀⢸⣿⣿⣿⣿⠀⠀⢀⣤⠶⠶⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣾⣿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢸⣿⣿⣿⣿⠀⢀⡞⠁⠀⠀⠈⢳⡀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠂⠀⠀⠀⠙⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⢸⣿⣟⣛⣛⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠆
⠀⠀⢸⣿⣿⣿⣿⠀⠘⣇⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡶⠶⠤⠦⠦⠶
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠙⢦⣤⣶⡿⠋⠀⢀⣤⣤⣤
⠀⠀⢸⣿⡿⣿⠿⠀⠀⠀⠀⣸⣇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠚⢛⡿⠿⢿⡛⠓⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⢸⣿⣯⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⡔⠁⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢰⡀⠀⠀⠀⢀⣤⣶⡇⠀⠀⠀⠀⣏⠀⠀⠛⠟⠃⠀⠀⡇
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠣⡀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠻⣄⡀⠀⠀⠀⢀⣼⠃
⠀⠀⢸⡿⠿⠿⠿⠀⠀⠀⠀⢻⡟⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⡿⠟⠁⠀⠀⠀⠀⠉⠛⠓⠓⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠙⠒⠖⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⢀⡀⠀⠀⠀⠸⡤⡀⢀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⣿⢿⣛⣿⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⣂⡤⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⠀⠀⠛⠎⢁⢀
⠀⠊⠍⠉⢁⠀⠀⠀⠀⠀⠀⠈⡁⠿⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⣁
⠀⢁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢸⡇
⠀⠒⠡⠘⠽⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⡀⠌⠀⠐⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿
⠀⠦⠄⠀⠀⢠⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡏
⠀⠠⣄⣀⠀⣸⣍⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⡇
⠀⠀⠈⠙⠛⠻⢿⡿⠯⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢂⢐⠞⠂⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⡀⢈⡄⡀⠀⢀⠀⠀⠀⠀⠀⣊⣀⢀⠀⠠
⠀⠘⠓⠀⠀⠈⢐⣁⠀⠀⠀⠀⠀⠀⣲⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⣦⡾⡉⡹⡓⠒⠁⠀⠀⢠⠆⠈⣹⠟⠷⡄⠊
⠀⠰⢹⡄⠀⠀⠀⠉⡧⠀⠀⠀⠐⠀⢣⢺⠃⡏⠀⠀⠀⠀⠀⠀⠚⠁⢠⣾⢋⡔⠘⠐⠊⠀⢠⡀⡘⠁⣠⡀⣠⠡
⠀⠀⠹⢄⠀⣀⠀⠄⠁⠀⠀⠀⠀⠠⠊⢹⠡⠁⠀⠀⠀⠀⠀⠊⠁⠀⠚⠕⠂⣸⠀⠀⠀⠠⠚⠀⠁⢀⡈⠀⡤
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠠⠀⡤⡼⠀⠈⠈⠄⠀⠒⠁⠀⠀⠀⠐⠀⠀⠐⠀⠁⠀⠁
⠀⠀⠀⢈⣉⠑⢨⡚⠠⠄⢐⠀⠀⠀⠀⢸⠀⠠⡀⠀⠀⣡⢜⠞⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⡀⢀⣈
⠀⠀⡔⠉⠇⠀⡡⠝⡠⡍⠁⠑⠀⠀⠀⢸⠀⠀⠁⠀⠀⢦⠈⠆⠑⠌⠀⠀⠀⠀⠠⠀⠀⠀⠤⡚⠈⢀⣤⡀⢀⢠⠄
⠀⠘⠀⠀⠀⠤⠔⣾⠸⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠀⠠⠶⡦⠤⠀⠀⡀⠀⠉⠰⢸⠀⣈⣭⣀⢀
⠀⠀⠀⠀⠀⠠⠔⠁⢀⠀⠀⠀⣰⠀⢰⡴⢻⡆⠪⡁⢠⣞⡹⢠⡎⠀⢸⠇⠰⣉⠂⠀⠁⠄⢰⣾⣛⣭⠾⠯⡲⡀
⠀⠀⠀⠀⠀⠂⠀⠰⠁⠀⠀⡄⢿⡠⢺⠀⢸⡠⢄⣸⠾⢧⡤⠚⢧⡤⠾⠤⢄⣨⡇⠀⠀⠀⠁⡇⠡⢶⣺⠁⠇
⠀⠀⠀⠀⠀⠀⢀⠰⣀⠀⠰⠀⠀⠀⢀⠀⡇⡀⢀⢀⢀⠀⡀⡇⢸⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡉⠁
⠀⠀⠈⠂⢀⡲⠊⠌⢶⠈⠀⠈⠀⠀⠸⠄⠇⠃⠘⠹⠰⠛⠆⠧⠸⠘⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⢶⣀⠊
⠀⠁⠀⠉⠀⠁⠂⠀⠼⡁⠂⠒⠀⠈⢠⢤⠤⡤⢀⠀⢀⠀⠀⠀⢀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠐⠀⢽⣿⡅
⠀⡀⠄⠂⠁⠀⠀⢠⠀⠈⠄⠀⠀⠀⠀⠀⠒⢩⣥⣄⣩⠇⠀⠁⠞⢦⠼⠧⣄⣤⠤⢄⡀⠀⠀⠀⢀⢚⣘⡶⡀
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⡀⠠⣎⣠⠀⠑⡌⠉⠆⠀⠀⠀⠁⠀⠀⠀⠀⠐⠄⠀⠀⠁⡠⠃⢰⣶⡺⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⢁⣆⠀⠀⠀⠀⠀⠀⠀⠠⡄⢠⣴⣆⣠⠂⠀⠀⠀⠀⢠⢼⡖⣄⣈⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠠⡤⠀⡏⠊⠀⠀⠀⣠⡄⠀⠀⠀⠈⣋⠉⠉⡇⢀⡀⠀⠄⣀⣀⣧⠙⢦⣌⣀⡐⣁⠀⠠
⠀⠸⠁⠀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠁⠀⠘⢢⣀⡯⠂⠀⠀⠈⠰⢼⠀⠠⡎⠁⠀⠀⠀⠀⠀⠸⠇⠴⠉⠁⠙⠈⠂
⠀⠀⠐⠦⠄⡀⠀⡀⠑⢢⡀⠀⠀⠀⠀⡐⡻⣿⡅⠐⠀⠀⠀⠂⢂⠀⠀⠙⡤⠀⡀⠀⠀⠀⠠⠤⠤⢠⣧
⠀⠀⠀⠀⠀⠈⠁⠈⠉⠢⣉⠄⠀⠤⠀⠸⡴⠋⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠂⡀⠸⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣸⡿⢿⡟⡿⠿⣋⣛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⡴⠖⠚⠛⢋⠙⢻⡟⣉⣑⡒⢂⣶⠀⣠⡀⠀⠂⠀⡸⠏⠃⠉⠟⢛⡒⠻⠿⠧⠃
⠀⠀⢀⣠⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣾⠗⠛⠛⠛⠙⠟⠻⢿⡿⣿⡿⠃⢯⡖⠭⣀⣂⣉⣥⣤⣷⣬⣶⣿⠯⠋
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⣭⣿⣿⣿⣟⢾⣢⠼⠫⠛⠉⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⠿⢿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠶⠾⠧⢤⣀⡀⠈⣙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢷⣾⣿⣿⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠢⠠⠄⠘⠂⠤⠄⠘⠿⠿⠿⠿⠷⢶⠖⠀⠂

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⢀⣔⣵⣿⠟⠋⠉⠀⠀⠈⠉⠛⢿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⣼⣿⡏⠀⣀⡴⢚⣿⠟⠛⢿⣟⠲⣄⡀⠈⣿⣿⡆
⠀⠀⠀⠀⣿⣿⡇⣾⡉⠀⢾⡇⣄⣽⡆⣿⠆⠈⣹⡆⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣇⠀⠙⠶⣬⣷⣮⣭⣴⣏⡤⠞⠉⢀⣿⣿⠇
⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⡀⠀⠀⠀⣀⣠⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣮⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠏

⣿⡯⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢦⡀⠢⣄⡀⠐⣦⡀⠀⣤⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠙⠻⣷⣮⣿⣦⣌⢻⣦⡘⣿⣆⣿⡆⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠈⠁⢀⠀⠀⢀⣀⣀⣹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣡⡇⠀⠀⠉⠀⠀⠉⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣏⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢟⣛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣶⣿⣿⣿⠇⠀⠀⠀⠠⠿⠒⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣷⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠓⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢸⡟⢿⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠁⠀⢀⣀⡀⠀⣠⣴⣾⣿⣿⣿⡿⠳⠀⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⡶⠶⠳⠏⠉⠁⣸⣿⣿⡛⠿⠻⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠐⠂⠀⠀⠀⠀⣶⠒⠐⠗⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⡼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣃⡾⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠈⠻⣧⣄⣀⣀⡀⠀⠠⢤⣴⡟⠀⣽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣾⡇⠈⠉⡍⠉⢉⢙⡻⠷⣶⣽⣇⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠗⠀⠀⠀⠀⠀⠀⠀⢨⡭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⣍⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⢂⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢶⣶⣶⣦⠀⠀⠀⠀⠉⠁⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣦⡄⠀⠀⠀⠀⠐⠷⠐⠛⢣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠈⣿⣿⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⣸⣿⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⡉⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿⡁⢀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠴⠶⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠎⠁⡈⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠂⣠⠀⠀⠀⠀⡁⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡈⢠⡿⠀⠁⠀⢀⣐⣐⣀⣀⠀⠃⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢁⣩⣬⣶⣾⡿⣛⢿⣿⠿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢚⢻⣿⠛⣿⣿⡇⢱⡯⣿⡄⣸⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠻⡨⢿⡾⠛⠁⠈⠠⡴⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⣠⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢀⠇⢸⡀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⡰⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⠁⠀⠀⢄⠀⠀⠀⢣⡀⠘⣆⣾⡷⠶⠶⠦⠼⠇⠠⡄⢠⣄⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠲⣌⠣⠀⠀⠀⢷⡀⠘⣿⣶⠀⠀⠀⠀⠀⠘⣀⣞⠁⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣧⠀⠐⡈⣷⡄⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡭⠻⣿⠃⠀⠀⠸⡟⢺⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡴⣿⡐⡶⠶⡄⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡇⣿⣷⠀⡀⢐⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢂⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣿⣿⣰⢻⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢈⣀⡀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢿⣿⣿⣴⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠠⢠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⡷⣺⣿⣿⣿⡿⠟⠉⠀⠀⠱⣄⠀⠀⠀⠀⠐⣰⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠿⠛⠁⠀⠄⢉⡉⢤⣠⡘⣾⠷⠀⠐⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣻⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⣾⠿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢦⣀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡀⠀⢀⡤
⠀⠀⠀⠀⠰⡀⠈⢳⡆⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢃⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢡⡎⠀⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣾⣿⠷⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⢰⠏⢀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠈⢻⣧⡀⠀⠀⠀⢀⣼⠀⠛⠀⠆⠛⠀⢸⣗⠀⠀⠀⣠⣴⡿⠋⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠙⢿⣦⡀⠀⠐⣼⡀⢠⣤⢤⣄⠀⣾⡭⠁⢀⣼⡿⠋⠀⠀⣠⡟⠁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠉⠻⣦⡀⠘⣧⡈⠛⠛⢁⣴⠏⠀⣠⣾⡟⠀⠀⣠⣶⢏⡴⠛⠛⠛⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠈⢿⡄⢸⡟⠳⠾⠛⣿⣴⣿⠿⠋⠀⢠⣾⡿⠉⡟⢠⢀⠀⠴⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠻⡄⢠⣷⣴⣷⠙⠋⠀⠀⠀⣰⡿⠋⠀⢸⡇⢀⣨⣤⡀⠀⢈⡽⠀⠀⠦⡀⢀⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢷⡈⠿⠟⠁⣠⡏⠀⠀⠀⡇⢿⠋⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣤⣀⠀⠀⠀⠀⠙⢶⡶⠋⠸⡇⠀⠀⠀⢣⡈⠀⢠⡟
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣄⢀⣠⡴⠿⣇⣀⣠⠟⠷⣤⣠⡿⠁⢠⡟
⠀⠀⡦⣀⠀⢰⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣶⣿⡿⠛⠛⠛⠛⢿⣿⣿⠙⠋⠉⠁⣈⣉⣉⣀⣈⣼⠋⠀⢠⡟
⠀⠀⢱⣌⠛⠶⡗⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⣠⣿⡟⠠⠄⠀⠀⠊⠂⢹⣿⣤⡉⣉⣉⣁⣁⣀⣴⡟⠁⠀⣠⡟⠁
⠀⠀⠀⠙⢦⣀⠀⠀⠳⣄⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢰⣿⣿⡇⢰⡆⢲⠀⠺⠇⠘⣿⣿⣿⣀⣤⡤⢤⣼⡟⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠙⠳⢶⣄⠈⠛⠷⣶⣤⣀⠀⠀⠀⣿⣇⠀⠀⠀⢿⣿⣿⡇⠀⠀⢀⣀⣀⠀⠀⢹⣿⡿⠀⣤⣤⣾⠟⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠉⠛⠿⢷⣄⢻⣿⠲⣤⣀⣈⡻⣿⣷⠀⢻⣧⣬⡿⠀⣰⣿⠟⠁⢀⣴⡿⠁⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠉⠻⣿⣆⣀⡈⠉⠁⠈⠉⠳⣀⠉⠋⢁⣴⡟⠁⢀⣴⡿⠋⠳⣄⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠉⢻⠿⢿⣷⣶⣤⣄⣹⡟⠛⠋⢸⡇⣰⣿⣿⣿⣦⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⡟⠀⢰⣿⣿⣿⣿⡿⢧⣄⣠⡼⠛⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⢶⣿⣿⣿⣿⣿⣇⠀⣸⣯⡀⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⣇⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⡻⢶⠀⡃⠀⠀⣤⡤⠀⣀⠀⠀⡀⠠⢰⣶⡆⠄⢀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠀⠠⣀⡤⣤⣀⡤⠀⠀⠀⠀⠀⠀⠀⢸⠀⡾
⡿⣿⣉⣁⠀⠀⠀⠊⠌⣉⠙⠯⠙⢛⣿⣿⣿⡒⠫⠖⠒⠀⠀⠀⠀⠀⠀⣀⣀⠴⣿⡀⢁⣿⡦⣀⡀⡀⠀⠀⠀⠀⣈⣉⣿
⣿⣿⢡⠸⡀⠀⠀⠶⣶⣤⡴⠖⢂⡾⠻⣿⡟⢷⡐⠒⢦⡤⣶⡦⠀⠀⠀⠈⠁⣼⢩⢿⡟⡉⢷⡈⠁⠀⠀⠀⠀⢠⠃⣾⣿
⣿⣿⢸⡄⠁⠀⠀⠀⠀⠉⠀⠀⠘⠁⠀⣿⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⣸⠟⠄⣸⣃⠠⠺⣧⠀⠀⠀⠀⠔⠀⢀⣿⣿
⣿⣿⢸⡏⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠃⠈⢫⡄⢂⠑⢿⡇⠀⠀⠀⠀⢼⠛⣿⣿
⡟⠃⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡀⠁⣀⣼⣆⡀⠀⢈⢿⠀⠀⠀⠀⠘⡄⢿⣯
⠿⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⡴⠊⠁⠈⠁⠈⠂⢠⣀⡄⠀⠀⠀⠊⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠁⠀⠁⠐⠀⢀⢀⣤⣶⣄⣠⠄⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⡷⢶⠊⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⠦⠀⠀⢀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡾⣟⡻⣦⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟
⣿⣄⣀⣄⣀⣀⠠⠴⠖⠂⠀⠀⣘⣛⣷⣿⡷⣯⣙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠁⠀⠀⠰⠒⠂⢀⡤⣄⣠⣄⣃
⣿⣿⠉⠈⡁⠀⠀⠀⠀⠀⠀⢰⠁⣿⣿⣿⣿⣏⢧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠙⢒⡈⣽
⡟⠘⢇⠀⠙⣻⡽⠟⠓⠀⠔⠀⢰⣿⣿⣿⣿⡿⢸⡀⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⢤⣠⣄⣤⠀⡠⠀⠘⠻⠿⡶⠋⡉⣼⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⣿⣿⣷⢸⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣿⣿⠁⢴⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣿⣽⣿⡟⠃⡸⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⣿⣿⣷⡼
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿⣧⣤⣤
⡇⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢂⠄⠀⢀⡎⠀⢸⣿⣿⣿⣿⡇⠀⢱⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⣤⠖⠈⠀⠀⠀⠀⠀⠀⠁⢀⣤⣿⣿⣿⣿⣿⣿⣦⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣻⡟⠂⠀⠠⠄⠀⠀⠀⠀⠀⠈⡄⢿⣿⣿⣿⣿⡿⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⠻⠿⡿⠟⠡⡆
⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠠⣄
⢾⠊⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⠀⢀⣤⠤⠴⠶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡘⠛⠫⣉⠀⠀⠈⠉⠙⠛⠷⢶⣤⣄⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⠁⠀⠉⠙⠻⢦⣄⡑⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⢁⠤⡀⠀⠀⠀⠨⢿⠃⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠠⠴⠒⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⠛⠛⠛⠛⠛⠶⢦⣜⡆⠀⠀⠀⠐⠋⠁⠠⠤⠀⠞⠐⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⠶⢦⡀⠀⠀⠀⠀⢀⡠⠭⣿⠀⠀⣴⠿⠛⠛⠻⠿⠿⠷⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠽⡄⠀⠀⢰⣏⣤⡴⠶⠶⠶⠶⢧⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢷⣦⡀⠐⢤⠀⠀⢸⣿⣅⡀⠀⠀⠀⠀⠐⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡛⢷⣾⣷⠤⠬⠷⠭⠿⢄⡀⢀⡏⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢟⣫⠭⠖⠒⠊⠉⠉⢙⡿⣿⣿⣧⣤⡤⣤⣤⣀⣀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⡊⠁⠀⠀⠀⠀⠀⠀⢀⣜⣾⠟⠉⠀⠀⠀⠀⠀⠉⠛⢿⣵⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠧⠠⡀⠀⠀⠀⠀⠀⠀⣠⠤⠺⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⢸⡆⠀⠀⠀⢠⡿⢧⣀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡀⢀⣀⣀⠎⠀⠀⠀⠀⡜⠀⢀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⣀⠀⠔⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⠀⠀⠋⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠂⠀⠀⠀⡀⠧⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣇⠀⠀⠀⠀⡇⠈⠀⠇⠀⠀⠀⠀⠀⠀⠈⡀⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠊
⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⢐
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠚⠅⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢀⡞⠑⠊⠀⠀⠰⠉
⠀⢡⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠦⠃
⠀⠰⣅⠁⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠘⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡄⠀⢀⢀⠀⠀⠀⠀⠀⠁⡀⠀⠃⠀⠄
⠀⠀⠐⠀⠀⠀⠀⠡⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⡄⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡤⡄⠁⠀⠀⡄
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠴⠓⠀⠀⠀⠑⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡱⠔⠠⠀⠀⠀⠀⠠⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠓⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠊⠀⠀⠀⠀⠐⢄⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠑⢠⡴⠤⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠰⣀⡢⠀⠀⢈⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠀⠁⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠻⠅⠀⠀⠀⠀⡀⠐⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⣂⠠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢢⠼⠁⠀⠀⠂⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⠒⠀⠀⠀⠠
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠦⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣤⠧⠄⠀⠀⠀⠀⠀⠀⠀⢜⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠌⢀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⣀⠀⠂⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢀⡀
⠀⠀⠁⠀⠀⠠⡶⢢⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡎⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠀⠀⠒⠂⠠⠌⠋⠙⠫⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠋⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢶⣶⣤⣴⣄⣄
⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⠀⠀⠐⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣿⣿⣿⣇⠀⠤⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⣿⣿⣿⣿⡀⠀⠀⠿⡿⠇⢠⣤⣶⠀⠀⠀⠀⠀⠀⢀⣀⡺⣿⣿⣿⠯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⠉⠙⠙⠿⠃⢀⣀⢤⣤⣤⣾⣿⠿⠁⠀⠀⠀⣠⣰⠋⢙⣿⣽⣿⠟⣀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣾⠛⠉⣽⣿⣿⠁⠀⠀⠀⠀⣠⣌⣻⡟⠲⠆⠉⠈⠻⠟⢯⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⣹⣿⡇⠀⠀⠀⠀⣉⣙⠋⠳⠀⠀⢠⠀⠀⠀⡉⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠀⠀⠀⠀⠙⠛⡠⠦⠀⠀⠈⢳⡄⢾⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣤⣄⢁⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡆⠀⠀⠀⠀⠀⠈⠐⣶⡄⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣿⣷⠀⠀⢀⡀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡧⣿⣿⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⢻⣿⣧⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢹⣿⣿⡆⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠶⡰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡟⡑⢿⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢧⣼⠟⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣆⢹⠆⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡡⢚⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⡠⢊⡤⠒⢛⠀⠀⠀⡀⠀⠈⢡⢈⠀⠀⠐⠐⠂⡀⠀⢀⠐⢂⡀⢀⠂⠀⠀⣤⠀⠀⢠⣄⠀⡄⢠⣀⠀⢠⠀⠀⠴⠒⠒
⠀⠀⠀⣠⡤⡈⠊⠀⠃⠐⠈⡀⠀⠠⠄⠈⠶⣚⠩⠉⠁⠀⢸⣀⢀⣁⣀⣚⢀⣘⣃⣀⡃⠀⠀⢿⠀⢀⣸⠿⠀⠅⠀⡀⠀⠀⠀⠀⢰⠈⣆⠀⢰
⠀⠀⠘⢁⠀⡤⠀⢂⠀⠐⢉⠠⢨⡀⠀⠀⠀⠀⠀⠀⡁⣀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠈⠁⣀⣤⠤⠀⠀⠀⠐⠀⠀⠀⡆⠀⠡⠰
⠀⠀⡀⠈⠁⠤⠀⠁⢁⠐⣍⡐⠌⠓⡤⠀⠀⠀⠀⠎⢭⠀⠀⠤⣤⢅⡥⡥⢤⠄⠀⠰⢱⠋⠄⠀⠘⣁⢠⠰⡇⠀⠀⢐⠚⡀⠃⡇⠀⢈⠀⠀⡀
⠀⠀⠘⠊⠀⡠⠀⠂⠁⢀⠞⠀⠀⠀⠀⢈⡀⠀⠐⠄⡀⠀⠈⡀⠁⡀⢀⠀⠠⠀⠀⠀⠩⠀⠀⠀⠀⢫⠰⠚⠢⠀⠀⡰⡀⠀⠀⠇⠰⢀⣠⡖⣊⠅
⠀⠀⢀⡈⡁⣠⠴⠂⠀⢀⠀⠀⠐⢀⣁⣠⠀⠀⠀⠂⠀⢂⠀⠇⠀⠀⠀⢈⠁⠀⠀⠐⢀⠀⠀⠀⠀⠈⢄⣤⢤⠀⠀⠀⣀⠀⠀⠁⠘⡉⠦⠉⢀⠂
⠀⠀⠀⠈⠒⣀⣤⠴⠶⠈⠄⠈⡀⣘⣥⠐⠀⢀⠀⠠⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠈⠃⠉⡀⢀⡈⠻⡄⢤⡄⢀⠀⠁⢉⣀
⠀⠸⡇⢰⠚⠉⠀⠀⠀⢀⠐⠀⠀⠀⣫⠐⡕⠀⠄⠠⡀⠄⠀⠀⢀⠀⠀⢄⠂⠀⢀⠀⢀⠀⠄⠀⠇⣠⡄⢀⠾⣽⣿⠇⡄⠹⡄⠀⠁⠋⢀⡐⠛
⠀⠀⣇⣈⠆⠰⣶⡆⡱⡄⠂⠀⠀⢂⠙⠋⠁⠀⠉⠀⠡⠀⠀⠀⢤⠀⠠⠠⠀⠐⠩⠄⠈⠆⠀⠊⠘⠛⠁⠉⠘⠃⠸⡼⠁⠀⠙⠀⠰⠙⠋⠹⢳⡄
⠀⠀⢸⣿⠀⠀⠁⠑⠐⡛⠀⠈⠀⠀⠀⠀⠀⡀⠤⠁⠀⠀⠀⠀⠀⠴⠖⠒⠒⠂⠀⠀⠀⠐⠒⠈⠁⠀⡸⠖⢶⡇⠀⠀⠀⡂⢂⠉⣧⠆⠀⢁⢀⣿⠁
⠀⠀⠸⣿⡂⠈⠂⠊⡀⣤⠤⢁⠐⠀⠐⠈⢁⣠⠀⠀⠀⠀⠁⡆⠐⢄⡄⣴⢀⡄⠀⢀⡀⣤⠄⠀⠀⠠⠛⠛⠛⠃⠀⣠⠞⠉⠊⠀⠙⠯⣚⣊⠕⠁
⠀⠀⠀⠀⠀⠠⠀⢁⠂⠀⠀⣉⡆⠀⢰⠊⠀⢘⠇⠀⠀⠀⠀⠁⢀⠀⣁⣉⣉⠀⠀⠁⠁⠀⠀⠀⠀⠘⠒⠒⠒⠂⠀⠛⢀⡚⠱⡄⠀⢀⡸⢄
⠀⠀⡄⠁⢀⠌⠀⢀⡡⠖⠉⢉⠀⠀⠀⠓⠈⠁⠀⠶⢦⠀⠠⠴⠀⢀⣤⣈⢉⣀⣀⡄⣠⡤⠁⢀⡤⣄⠀⠀⣤⠩⡏⠀⠀⣄⡀⠆⠀⠀⠀⠀⠀⠒
⠀⠀⢉⠉⠀⠀⡐⢀⠆⠀⡐⠁⠠⠀⠀⠀⡌⠉⡄⠘⠈⢁⠀⠀⠘⠀⠈⠑⠈⠁⠀⠁⠇⠀⠀⢿⣲⡭⠀⡠⠿⠟⠇⠀⠀⠐⠓⠀⠀⠊⢀⣀⣘⠠
⠀⠀⠸⢀⡣⠀⡁⠀⠀⠁⠈⠐⠀⠀⠀⠀⡀⠀⠠⠀⠕⠈⠁⠀⢰⠞⠓⠀⠀⠠⠂⢐⡄⠀⣀⣀⡀⠀⣸⣀⠶⠶⢐⣇⠀⠘⠚⠂⠠⠀⠉⠉⠉⠠⠃
⠀⠀⡀⠶⠄⠀⠈⢀⡴⠀⠀⠀⣰⠀⠃⠀⢷⠇⠀⠀⠀⢀⣀⠆⠀⠄⠀⠀⠀⡠⠀⡠⠁⢸⡅⣈⡇⠀⠏⣅⡔⠲⠏⢸⠀⠀⠀⠀⠀⠈⠉⢉⠉⠁
⠀⠀⢏⡩⠁⠀⠀⢀⣀⠀⢀⣀⡀⠠⢂⡀⠈⠀⣀⠀⠀⠈⠀⠀⠀⠈⠢⠜⢀⣁⠁⡘⠀⠀⠈⠉⠀⠀⣦⣟⣳⣶⡦⠶⠀⢠⣉⡄⠀⠀⠀⡸⠭⣄
⠀⣀⣤⣦⠹⠆⠀⠁⠀⡀⠁⠀⠀⠘⠉⠀⠀⢈⣉⠀⠀⠐⠀⠒⠤⢄⣀⣀⠠⠈⠉⠁⠀⢴⠀⠀⡀⡀⢀⣠⡤⣤⢶⠲⣶⣾⠟⢉⣿⠀⠸⡷⠿⢿
⠀⠸⡟⠫⠀⠐⡆⠘⠁⠀⠚⠃⠀⠀⠤⢀⠀⣺⠈⡄⠀⠀⠀⠀⠀⢘⡇⠤⠤⠀⠀⠀⠀⠘⠒⠂⠀⣀⢀⡀⠀⠀⠂⠂⠒⠲⠶⠶⠆⠀⠀⡇⠀⠸
⠀⠀⠀⠠⠀⠀⠈⠄⠀⠀⢀⠁⠁⠀⡀⣀⣠⢾⡄⢳⠀⢀⠀⡀⠔⠁⠀⢀⣠⠠⠄⢲⣆⠀⠀⡐⠁⡀⠲⠴⡁⡄⠀⢀⡀⠀⠀⠤
⠀⠀⠀⠐⠀⠀⠀⠐⢅⠀⠸⠰⠚⠈⠁⠁⠠⠀⠳⣾⡀⠀⡡⠠⠀⠔⠊⠹⢋⣁⠀⠘⢹⠀⠀⠠⠅⠀⢀⠀⣄⠆⠀⠑⠄⠀⠀⠒⢂⣶⠖⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⡀⠱⣎⡁⠀⠀⠀⠈⠀⢀⠀⠿⠂⠀⠐⠁⢀⣠⠁⠈⠀⠀⠀⠀⢂⠀⠀⠀⡄⠀⠈⢩⠁
⠀⠀⠀⠠⠢⣄⠐⣯⡔⠄⠀⠀⢀⠀⡐⠀⠀⠁⡄⠀⡇⢻⠀⠧⠡⠘⠀⠂⠈⠁⠀⢀⣀⠀⠀⠀⠠⡀⠀⣄⢀⣉⢀⣄⠀⠸⠄⠀⠀⠦⠀⠀⢼
⠀⠀⠀⢠⠃⡜⡤⡧⡶⢸⠀⠀⠐⡀⠀⣀⢀⡀⠂⠀⠁⠈⠀⣠⠔⠒⠒⢦⢀⠀⠈⠀⠛⠀⠄⠈⢀⡰⠂⠀⠘⡇⣀⢸⠀⡀⠦⠀⠀⡴⠀⠀⠰⡇
⠀⠀⠘⠢⣁⠰⠦⠀⡡⢸⠀⠀⠀⠀⠉⢀⣠⣤⠐⠢⠀⠀⢞⠁⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠅⠀⠀⠠⠀⠀⢧⠈⠸⡇⠀⠀⠠⡤⠷⠐⠒⡂⠁
⠀⠀⠀⠂⠄⠈⠈⠁⠄⠊⠀⠐⠀⠀⠀⠈⠙⠛⠋⠜⠀⠀⠈⠙⠒⠒⠋⠀⠀⢰⡀⠤⠤⠤⠤⠶⠀⠀⠂⠀⠀⠸⠒⠊⠁⠘⠄⠐⠡⠥⠤⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣠⣤⣶⣶⣤⣄⡀
⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⠢⢄⡀⠀⠖⠃⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⢄⢠⣯⣀⠀⠙⣿⡟⠁⣀⣬⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⢸⣿⣿⣿⣦⣾⣷⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠻⣿⣿⡟⠋⠁⠀⣸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⠀⠀⠀⠀⠈⣛⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⡆⣠⡤⡖⠲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⢿⣿⣷⣿⣿⠿⠷⣮⣥⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢰⣾⣿⣿⣿⡿⣥⣿⣶⣶⣍⡻⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢠⣿⣿⣿⢏⣾⠡⠜⠛⠻⢿⣿⣿⣮⣻⡝⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣿⣿⡿⢋⣾⠟⠀⠀⠀⠀⠀⠙⢿⣯⣻⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣻⣿⣿⡿⣡⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⣾⣿⣿⠏⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⣿⣿⢏⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢶⣦⡀⠀⠀⠀⣀⣤⡾⢿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⡟⢛⣼⣿⣿⣿⣿⣿⣿⣏⠰⠆⣹⣷⡄⠠⠀⣽⣿⣡⣩⣿⣿⣿⣿⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣇⠀⠀⡄⢹⢿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠀⠀⠁⣼⢀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠘⠡⠄⠁⠨⢸⠉⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢶⣿⣿⡶⣶⠸⢶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠙⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠛⢻⡁⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⡟⠀⠹⢁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢽⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⠛⢿⠏⣿⠀⢹⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣼⣿⣿⣿⣿⡿⣿⡟⠀⣿⢿⡇⠀⠀⠀⣿⠁⣼⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣇⣿⡇⣿⠃⠀⡏⢸⡇⠀⠀⠀⠸⢀⡿⠛⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⢸⠃⣿⠀⠀⠁⠈⡇⠀⠀⠀⠀⠘⠁⠀⢀⠘⢛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠉⠙⠻⠿⢿⠀⢸⠀⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠈⢀⡞⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡣⠈⠃⠀⠀⠀⠀⠀⠀⠀⢸⠃⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿