Sausage Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⠁⣠⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠋⠉⠀⠀⢀⣡⣶⡶⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠄⢰⡴⠂⠒⠠⡀
⠀⠀⠀⠀⠀⡠⠄⠀⠠⠤⣀⣡⡀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⣠⡴⠿⠛⠒⢠⣔⠂⠐⠲⡖⠉⠀⠀⢸⠃⠀⠀⢠⠞⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠎⠀⠀⠀⠴⠚⠉⠀⠉⣠⣶⡶⠦⠠⢄⣀⡀⠀⠀⠁⠀⠀⡴⠾⠛⠋⠁⣲⣟⡀⠀⠀⠀⠏⠀⠀⠰⠋⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠠⠞⠋⠁⠀⢀⡴⠟⠋⢉⡡⠶⣤⠔⠒⠒⠒⠒⢘⡾⠛⠛⠛⠒⣦⠔⠒⠒⠂⣤⠄⠀⠀⠀⣰⡖⠡⢄⡀
⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣰⡋⠀⠀⠀⠀⠀⠼⠁⠀⠀⢀⡼⠃⠀⠀⠀⣰⠏⠀⠀⢀⡴⠋⠀⠀⠀⠘⡄
⠀⢀⣠⡴⠞⠋⠉⠉⠀⠒⠤⢀⣀⢀⡠⠒⢰⡖⠉⠉⠉⠉⠁⣼⠋⠁⠀⠀⣰⠔⠉⠉⠉⢩⡗⠉⠉⠉⣩⡟⠀⠐⠲⡋⠀⠀⠀⠀⠀⢀⠟⠶⣤⣀
⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⢠⠏⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⡴⠋⠀⠀⠀⢠⠏⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⣀⡀⠀⠤⠄⠊⠀⠀⠈⠹⣷
⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠐⢍⠻⢷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠀⠶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠄⠐⠒⠒⠉⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⢛⠝
⠀⠀⠉⠒⠬⢉⡛⠻⠶⠶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡴⠶⠾⠛⡋⠭⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠂⠤⠍⢉⡉⢙⣛⡛⠛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⠛⢛⠉⠉⠉⠉⠥⠀⠒⠀⠈⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣿⣷⣶⣶⣶⣶⣾⣷⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣶⣿⣿⡿⠿⠻⢿⣿⢿⣭⣷⣤⠀⠀⠀⠀⠀⢀⣀⣤⣖⣲⣶⣶⣿⣿⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡴⢞⣿⡟⠋⠉⠹⣿⠀⠀⠀⠸⠇⠀⠀⠸⣿⠁⠀⠀⣠⣺⡽⣿⡏⠁⠀⣿⠃⠀⠀⠀⣿⠋⠙⠶⣽⣦
⠀⠀⠀⠀⢀⣰⣏⡴⠋⢿⡇⠀⠀⣀⣭⠴⠶⠶⠶⠶⠶⠶⠿⠛⢀⢔⣿⣿⠙⠀⢿⡃⠀⢀⣿⣀⣠⣀⣀⣟⠀⠀⠀⣰⣟⠂
⠀⠀⠀⢠⣿⣿⠏⠀⠀⠘⣣⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠘⣿⠀⠀⣸⡶⠞⠛⠉⠉⠉⠉⠉⠉⠛⠳⠿⠟⠁
⠀⠀⢀⡿⣼⣿⡇⠀⢀⡼⠋⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⣰⣽⣿⠀⠀⢿⣴⠞⠋⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⣼⣿⠃⢹⡇⢠⡞⠀⠀⣴⣾⡽⠟⠋⠻⣷⠀⠀⠀⢸⣿⡇⢻⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢰⡿⠚⠻⢿⣆
⠀⠀⣿⡏⠀⠀⢠⡟⠀⢠⣾⢿⣅⠀⠀⠀⠀⡿⠀⠀⠀⣿⠇⠀⠸⢃⡾⠁⠀⢀⠀⠀⠀⠀⣠⣾⡷⠶⠄⠸⣿⠀⢀⣤⣤⣤⣀
⠀⠀⠹⣧⣀⣠⠞⠁⢠⡿⠇⠀⠙⠂⠀⢠⡞⠁⠀⠀⠀⢿⡀⠀⣀⡼⠁⠀⠀⠈⢤⡀⢡⣸⣿⣏⣁⠀⢀⣼⠏⢀⣾⡓⠋⠁⠹⡆
⠀⠀⠀⠀⠉⠁⠀⢠⣿⡿⣦⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⠛⠚⠋⠁⠀⠀⠀⠀⢈⣿⣿⣧⣾⣯⣽⣦⣾⠋⠀⣼⣧⣀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠻⠀⠈⠁⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⢻⣭⣿⡿⠛⠛⠓⣺⣿⠿⢿⣿⣿⣿⣯⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣧⡄⠀⠀⠀⢸⠁⠀⠀⠀⢀⡴⠶⣾⣿⣿⢿⡿⣿⣿⣿⣧⡈⠿⠋⠀⣀⡐⢟⠋⠀⠀⢤⠿⠿⠛⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⢸⠀⠀⠀⢰⣿⠊⠉⢁⣹⣿⣦⣤⣬⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⢠⣬⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣼⠀⠀⠀⠈⠷⣤⣤⣿⣿⡿⣍⣻⣿⣿⡉⣛⣻⣿⣿⣧⣶⣾⣿⣿⠏⢻⣿⡿⠿⣦⡀⠀⠈⢯⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠹⣏⠙⠒⠚⠛⠻⠿⠿⠿⠿⠿⠿⠏⠉⢁⣠⠴⠚⠉⠀⠀⠈⢻⡄⠛⠛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣤⣤⣤⣤⡤⠤⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⡳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠚⢳⠶⣄⡀
⠀⠀⠀⠀⠀⢀⣠⠶⠒⠒⠺⢤⡺⣻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠥⠧⢼⣏⡟⠃
⠀⠀⠀⠀⣠⠟⢁⣀⣀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⣰⠏⣰⠏⠀⡿⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡞⠙⡆⠀⠀⠀⠀⠘⣆
⠀⠀⢰⠏⢀⡏⠀⡜⢀⣀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠁⣼⠁⢰⠁⠀⠀⠀⠀⠀⢻
⠀⠀⡾⠀⣸⣠⣼⠿⠛⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣾⡶⠟⠋⠁⠀⠀⢸⡀
⠀⢰⠇⠀⡏⢹⠃⠀⠀⠀⢀⠔⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡾⡏⠀⠀⠀⢀⠞⠀⢸⡇
⠀⣿⠀⢸⠃⡎⠀⠀⠀⠀⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠀⡇⠀⠀⠀⠂⠀⠀⢸⡇
⢸⡇⠀⡼⢰⠁⠀⠀⢀⡠⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡇⠀⠀⠀⣠⠀⠀⢸⡇
⣸⠃⠀⡇⢀⣠⣤⡶⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠇⠀⣠⡼⠁⠀⠀⠈⡇
⣿⠀⢸⠛⠛⠉⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⣴⠾⠋⠁⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠸⡆
⢿⠀⠀⠀⠀⣀⣠⣴⡿⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⢳⡀
⢸⡄⠀⠾⠛⠛⠉⠁⠀⠈⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⣀⣴⡾⠛⠁⠀⠀⢀⠀⠳⣄
⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠘⣆⠀⢀⣤⡴⠾⠟⠋⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀⣤⠾⠛⠁⠀⠀⠁⠂⢀⠀⠈⠙⠶⣤⡀
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⢠⣶⠆⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠉⠀⡇⠀⡔⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⡠⠀⢀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠀⠁⠀⣴⣠⡆⢀⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠉⠁⠀⠀⠀⠁⢠⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⢸⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⡴⠂⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠀⠈⠀⠀⠀⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⣠⡿⢿⠁⣰⠃⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⠛⠉⠀⠀⠀⠁⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣤⡤⠶⠖⠈⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠂⠒⠛⠻⣌⣨⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣀⡀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠗⠒⠛⠋⠉⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠭⠿⠷⠶⠄⠀⠀⠀⠀⣰⣃⡤⠴⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠈⠻⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢥⡀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠉⠲⣤⣄⣀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣶⠶⠦⠄⠀⠀⠀⠀⠀⠀⣠⠎⠀⠉⠑⠻⢷⡀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠙⣓⣶⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⢀⣨⡷⠤⢤⣤⢤⣤⣤⣤⣤⣤⣤⠶⠒⠛⠛⠛⠛⠛⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠈⠉⠛⠂⠀⠀⠉⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠒⠋⣁⣀⣀⣀⣀⣤⠤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠈⠳⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠖⠒⠋⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠘
⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠙⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣟⠛
⣤⣤⡬⠙⠓⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣦
⠭⣷⣤⣄⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠉⠉⠀⠀⠀⢠⠖⠒⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⡀⠙⠢⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠐⠛⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣋⠀⢀⡀⠀⠀⠈⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠠⠾⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢿⣤⣀⣤⣀⣀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣨⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⣄
⣤⣤⣶⣶⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢻⡏⠀⠀⠿⠉⠉⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⣧⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣾⠟
⡬⠶⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡋⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠕⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠴⡞⠉⠀⣿⡠⢊⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⠀⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⢠⠀⠀⢀⣷⣤⣴⡿⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⢿⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠷⠾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡶⠿⣦⡀⠀⠀⠛⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠉⡈⠄⠀⣀⡿⠃
⠀⠀⠀⠀⢀⣠⢤⠶⢶⡶⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠰⠄⠀⣠⠴⠁
⠀⠀⠀⠀⣾⠀⠀⠀⠈⠀⠀⠼⠋⠙⢱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠄⠀⠀⡴⠁
⠀⠀⠀⠀⠘⠲⠤⠤⠤⠤⢄⣀⠀⠐⠉⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠛⣿⠿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡽⣿⠁⠀⠸⣃⣀⣀⣉⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠉⠀⠻⣠⠖⠋⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⠜⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠤⡀⢀⣀⠥⢤⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠞⠀⠀⠀⠙⢶⡗⠋⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡀⢈⣀⠉⠄⢹⠀⠀⠀⠀⠀⠀⠀⡠⠾⣄⠀⠑⠆⠀⢀⠏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀⠧⣀⣀⠀⠀⡸⠀⠀⠀⠀⠀⢀⠼⣅⠀⠈⠑⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠉⠑⠒⡄⠑⡀⡇⠀⠀⠀⠀⣀⣇⡀⠀⠙⠂⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠂⠀⠈⠉⠄⠔⢹⠁⠀⠀⠀⣜⠀⠈⠛⠦⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠐⡀⠀⠀⠀⠀⡼⠀⠀⠀⡔⠉⠳⢤⡀⠀⠀⠀⠀⢀⠞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠐⢠⠀⠁⢆⡇⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠦⡄⠀⠁⠤⠚⣸⠁⠀⢨⠀⠈⠑⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣰⠇⠀⢐⠙⠲⢤⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡶⢙⡆⢀⡴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡼⠉⠁⠀⠀⠀⢀⡀⠳⢤⡀⠀⠠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡸⢠⠇⠀⠀⠀⠀⠀⠈⠙⠤⠜⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠢⣀⠀⢙⣋⡘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡀⠀⣇⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢠⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡂⠀⠀⠀⠀⣀⡤⠠⠤⠤⠤⠤⠤⠤⠖⠒⠊⠁⢠⠀⠀⠈⠢⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⠞⠁⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠂⠀⡠⣡⠏⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⡄⠀⣿⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⢧⠀⠀⠀⠑⠄⠀⠀⢀⡨⠞⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠰⠀⠈⠦⡀⠀⠀⠆⠀⠀⠘⡄⠀⠀⠈⠃⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢶⠖⠁⠀⠀⠀⠈⠑⠢⠤⢄⣀⣀⣀⣀⣠⡤⡤⠖⢛⠩⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢁⠀⠈⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⠄⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠄⠀⣀⡀⠤⣀⣀⣀⣀⠀⢀⣀⠀⠉⠢⢄⡀⠀⠀⠃⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠴⠛⠺⠁⠀⣀⣀⠀⠀⠀⢤⡀⠀⠱⡄⠀⠀⠑⡀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠐⠀⡇⠀⠀⠀⢡⠀⢹⡀⠀⠲⠿⠲⠦⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⣀⢀⣀⠀⣀⡠⠼⠥⠖⠊⠀⠀⠀⠀⠀⢇⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣷⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⢇⣸⣿⣿⣿⢻⣿⣿⣿⣶⣦⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⠿⣿⣟⣷⣧⣧⣿⣿⣿⡿⣻⠿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡉⠛⢯⣿⣿⣿⣿⣧⣿⣾⣿⣿⣿⡿⡷⣯⣿⣿⢿⣽⣿⣿⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⣬⣉⠛⠿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣧⣾⣿⣿⣷⣾⣿⣿⣿⣿
⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣛⣻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣷⣶⣶⣶⣶⣬⣭⣿⣿⣿⠿⠋
⠀⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠹⣿⣷⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠙⢿⣿⣿⣧⣿⣿⣿⡛⡿⣿⣾⣿⡿⣿⣻⣿⣿⣿⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠙⢿⣿⣿⣿⢷⡟⢰⣿⣿⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣾⣧⣿⣿⠏⣤⢼⣿⣿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⡿⠛⠟⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣾⣟⣾⣽⣟⠁⣾⣿⣿⡟⣾⣿⣿⣿⠟⢻⣿⣿⣿⢋⢿⡯⢿⢿⣛⣿⣿⣿⣿⣄⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣶⣝⣿⣿⡧⠙⣙⣿⣿⠅⣯⢿⣿⣿⡘⠎⡿⣚⣦⣌⠊⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣦⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⣥⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡙⢿⣿⣿⣿⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡆⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡙⢻⣿⣿⣯⣿⣹⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣯⡟⣻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣫⣾⣿⣿⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣏⣵⣽⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣽⠻⣿⣿⣿⣿⣧⣿⣿⣿⠛⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣝⣿⡏⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡏⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣾⣝⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣭⣾⣿⡟⣥⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣟⣻⣿⣿⡟⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣅⣡⣿⣿⡏⢳⢿⣿⣿⣿⠉⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣮⣿⣿⣿⣿⣶⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠙⠦⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣀⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⢶⡛⣶⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⠛⠹⡽⠈⠳⠟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢤⣈⠙⠒⠚⢙⠄⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⠳⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠙⠢⢤⣈⣉⡹⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠑⢄⡉⠓⠶⢤⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢯⠀⠀⠀⠀⠀⠉⠓⠒⠊⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠉⠈⢣⡈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⢻⠀⢧⠀⠀⠀⠀⠙⠲⠤⠜⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠳⡀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁
⡔⠶⢤⣀⡀⠀⢸⢷⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠈⠲⣄⠀⠉⠓⢿⣻⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉
⠀⠀⢸⠀⠀⣠⠞⣟⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠸⡄⣰⠃⠀⠈⠳⢄⣀⡀⣀⣀⡤⠴⠒⠉⠁
⣶⣶⠶⢾⡷⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⠤⠴⠒⠋⠁⠀⠀⠀⡎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⣾⢰⡇⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⠀⡆⠀⠀⠘⡇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢸⡀⢇⠀⠀⢸⡇⡎⠀⠀⠀⠀⡎⢠⠀⠀⠀⠀⢰⠃⡆⠀⠀⠀⢻⠘⡇⠀⠀⠀⢳⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠱⢄⠀⠀⠘⣇⠹⠀⠀⠀⠘⡇⢸⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠘⣆⠑⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠘⢆⠀⠀⠀⠀⢻⡈⠂⠀⠀⠀⠀⢳⡈⠀⠀⠀⠀⠈⠃⠀⠀⢀⡀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣄⣀⣀⡙⠂⠀⠀⠀⠀⠀⠈⣁⣀⣀⡤⠤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠢⠞⠣⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠵⡄⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠉⠛⠲⠤⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⢲⡄⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠒⠒⠒⠐⠊⠉⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣠⠞
⠀⠀⠀⢀⠤⣀⠀⠎⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⡣⠀⠙⢄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠘⢦⠀⢀⣜⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠐⠄⠊⠀⠓⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠤⠤⠤⠤⠄⠀⠒⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠉⠉⠒⢦⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣞⠕⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠊⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣱⣾⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢔⠭⢀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⣋⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠁⣠⣀⣀⡀⠀⠀⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⢆⠀⠀⠀⠀⡠⣾⢟⡥⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡤⠞⠃⠀⠉⠉⠉⠙⠓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡲⢻⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢛⡩⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠁
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠳⣟⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢰⡄⠀⠀⠀⢀⡏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠏⠀⢀⡜⠀⠀⠀⢀⡼⠁⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠙⠳⣦⣀⣠⡤⠤⠄⠘⠋⠀⣀⠐⠉⠀⠀⠀⠀⠚⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣄⠀⠁⠀⠀⣀⡀⠀⠀⠋⠁⠀⠀⣀⡴⠚⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢀⡀⠀⠚⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠋⠁⠀⠀⠀⠀⠀⠋
⠀⠀⢀⣇⠀⠀⠀⠀⠀⢀⣠⠤⠄⠒⠒⠒⠀⠐⠒⠲⠤⣄⡀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠰⡖⢋⢀⣀⣠⡤⠖⠚⠉
⠀⢀⢼⡘⠲⠴⢔⣒⠿⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠨⣓⡂⠴⠖⣡⣶⣀⠀⠀⠀⠃⠈⠉⠁
⠀⠀⠙⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠓⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠸⣟⢹⣇⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠈⢉⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⣾⠁⢀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢿⠀⣿⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠀⣿⠀⢸⡆⢈⡛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⠀⢸⡇⠸⡇⠀⡈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⢹⣦⠤⠴⠶⣤⣤⣤⣀⡀
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠙⠳⣬⣃⠀⠁⢔⣵⡷⠶⠦⣄⠀⠀⠀⠀⠀⠈⢻⠏⢤⠀⡀⠰⢤⡀⢉⡙⠻⢷⣦⣀
⠀⠀⠺⣞⣋⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⠏⠀⣠⠀⠘⢧⡀⠀⠀⠀⠀⠸⣆⣀⠈⠪⣤⠈⢋⡎⢳⢄⢲⣌⠻⣧
⠀⠀⠀⡿⠁⢀⡉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣼⠀⢀⡈⢳⡄⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠦⣥⣀⠢⠆⢮⠀⣹
⠀⠀⠀⢧⠀⣮⠇⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠿⠀⣽⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣤⠾⠿⣤⡀
⠀⠀⠀⠘⡆⢿⡀⢰⢢⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠿⠀⢸⡇⠀⡈⠓⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣩⡄⠀⢀⠈⣷
⠀⠀⠀⠀⠘⢾⡇⢸⢸⠀⢀⡉⠲⢄⡀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠸⡇⠈⣇⠀⠀⢹⣆⣀⣤⣴⠾⠟⠉⢤⡤⠤⠀⣁⠈⣽⠏
⠀⠀⠀⠀⠀⠈⠳⣌⢮⡄⢺⣃⠀⢤⡉⠲⣄⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠻⠀⢀⣰⡏⠁⢤⢄⡀⠻⠖⢤⣄⠀⢀⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡘⣿⡀⠸⣧⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠻⣶⣤⣁⣊⠒⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⣄⡙⠷⠀⢠⣿⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⠻⣤⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⡲⢮⡛⠿⠟⠳⠙⠛⠛⠻⠶⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣚⠷⠟⠋⢀⣀⣤⠴⠖⠒⠒⠒⠒⠲⢤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣾⡷⠂⠀⠀⠀⣠⣴⡶⢮⡟⠛⠋⣙⣉⣉⣁⣩⣤⣭⣭⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠟⠉⠀⠀⣠⣶⡾⠿⠉⢁⣠⣴⣶⣭⣅⡈⠍⠿⠿⠟⠚⠪⠷⠎⠻⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠧⠀⣀⣴⢯⣬⡙⠟⠚⠋⣁⣠⣤⠴⠶⠶⠶⠶⠦⠀⠚⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⢀⣾⣿⣿⡶⠉⠉⣠⣴⠟⡉⠅⠒⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⣴⣿⡿⠾⠿⠆⣠⡾⣫⠔⠁⠀⠉⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠃⠀⣴⢯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⢿⡿⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⣀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⠀⠀⠀⠀⠉⠌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⢀⠋⠒⠀⠀⠐⠒⠈⠉⠀⠀⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠃⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⢸⢀⣒⠤⠭⠀⠒⠠⢄⠀⠀⣀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠉⠢⡀
⠀⠘⠦⠀⠀⠀⠀⣀⠠⠤⠐⠊⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⡀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⡀⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⠒⢄⡀⠀⣀⡀⠤⠤⠔⠒⠊⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⢘
⠀⠀⠰⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠈
⠀⠀⠀⢇⠤⢀⡀⠤⠤⠒⠂⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠃⠐⢢
⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠘⠄⠀⢀⣀⣀⡀⡀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠈⠀⠀⢄⢠⠁⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⡰⠂⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⠂⠀⠀⠈⠉⠉⠉⢉⡉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣖⣲⡿⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠟⢷⠲⣺⡆⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢤⠘⣝⡦⢖⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠒⠻⢿⣣⣶⠐⠐⠠⢠⡤⠀⣄⠀⣀⡴⠤⢸⣟⣻⣧⣿⡿⣿⠾⡷⢊⠠⠄⠀⢀⠀⠀⠀⠀⣈⣴⡾⣷⠁⠀⠀⠙⠋⣓⣙⣦⡀
⢤⣀⣀⡀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯⠽⠿⠛⠋⠀⠀⠉⠛⠿⠶⠶⠟⣳⣾⣶⣾⣯⣽⠿⠟⢻⠀⡆⠀⠀⠀⠀⡗⢀⢻
⣼⡿⡆⠌⠻⣃⠉⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢳⡀⠀⠀⠀⡇⢸⠶
⢻⣏⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⢗⣤⡄⠀⠀⡟⡨
⠀⢻⠃⣅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡉⠉⠍⢛⣿⣿⠏
⠀⠀⠻⣿⡼⢃⠀⠀⠀⠀⠀⠀⠀⢀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢮⢷⣿
⠀⠀⠀⠙⠿⣎⡔⡂⠀⠀⡀⠀⠀⠀⠱⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡁⠄⠀⠀⠀⠀⠀⠀⠤⢼⠿⣿⠃
⠀⠀⠀⠀⠀⠈⠻⣧⣝⣦⡠⠧⣤⣤⣴⣸⣇⣀⣀⣐⠀⠀⠀⠒⠀⠀⠀⠒⠂⠉⠉⢉⠉⠉⠁⢀⡺⣷⣈⣆⠀⣀⡀⢀⠄⠮⠝⣁⣯⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⢿⣯⣟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣏⣿⣻⣿⣯⣼⣯⣶⡖⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣄⣀⣀⣀⣀⣀⣀⠀⡀⢀⣠⠤⠤⢀⣧⣾⡞⡻⠁⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠷⠿⠬⠭⠭⠭⣭⣙⣁⣉⣱⣈⠹⠏⠬⠿⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠷⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⢰⠚⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠑⡄
⠀⠀⠀⠈⠳⠤⠤⠤⠶⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠤⠞⠁
⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⢿⣿⠿⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⢶⣤⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠤⡀
⠀⠘⠶⠖⠛⢀⣀⣤⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⡾⠀⠐⠓⠒⠤⠃
⠀⠀⠀⣠⠾⢿⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠘⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠉⠙⠛⠲⠶⠦⠤⠤⣤⣤⣤⣤⣤⠤⠤⠶⣶⡶⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠓⠒⠲⠦⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡤⠴⠶⠶⠦⣤⡴⠚⠉⠙⠳⣦⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⣰⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣤⠾⠋⢠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⡤⠴⠶⠚⠛⠉⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠄⠄⠀⠀⠄⠀⠀⠀⠄⠂⠤⠤⠂⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠷⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆
⠀⠀⠀⢰⠋⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠉⢲
⠀⠀⠀⠈⠳⠤⠤⠤⠦⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⠤⠴⠋
⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⣻⡿⠿⠿⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡷⢶⣤⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠤⢄
⠀⠘⠲⠒⠊⢀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⢰⠇⠀⠈⠒⠢⠚
⠀⠀⠀⣠⠞⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠘⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⣶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠙⠒⠒⠲⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⠀⠠⢤⠤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣀⠴⠖⠒⠒⠶⢤⡖⠋⠉⠉⠳⣦⣴⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡼⠁⠀⠀⠀⠀⠀⠀⠙⠀⢀⣠⡴⠟⠁⣸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⠤⠤⠴⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢖⣶⣟⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠉⠉⠉⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠈⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣤⣀⣀⣀⣀⣤⠴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠐⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠋⠉⠑⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢉⠙⢦⡀⣿⠀⠀⠀⠀⠀⢀⣠⣤⣝⡲⣄⠀⠀⠀⠀⠀⢀⡴⣂⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⢆⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠟⠋⠉⠑⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣠⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠈⠁⠈⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣄⣈⣉⠉⠛⠲⢦⡄⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⠂⠈⢧⠀⠀⠀⠀⠀⠀⠸⣍⠙⠻⣿⠉⠉⠉⠉⡿⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣀⣠⠞⢳⡀⠀⠀⠀⠀⠀⠘⠦⣄⣘⡆⢀⣠⠞⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢤⣤⡤⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢀⡀⢀⠱⢈⣂⠀⠉⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡄⡍⢤⣑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠣⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠙⠒⠀⣡⠬⢁⡃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠹⣁⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡀⠍⠀⡔⢀⠒⠀⢀⠄⠀⠀⠀⠀⠀⠀⠘⢂⡖⠽⠞⠛⣶⡀⠀⠤⣤⠄⠠⢤⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠂⠀⠀⠐⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⡑⠶⢦⣲⠟⣂⠄⡂⠡⢆⠋⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠭⠌⠍⠀⠌⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠰⠂⡋⠰⠄⣀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢮⡏⠉⠀⠀⠐⢦⡀⠈⠑⠠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢼⠶⣄⡒⠤⠖⢾⠶⡀⠀⠀⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠘⠃⠲⣤⡀⣠⡤⢪⠀⠀⠀⠀⡃⡼⡍⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢬⠻⣉⣉⡸⠃⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠗⠃⠒⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠳⠀⠁⢉⢲⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠤⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢎⣀⣀⣀⣠⢠⣤⣴⢭⣧⣠⣤⡤⠶⢶⢞⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠶
⠀⠀⠀⠀⠀⠀⠀⢂⢉⣽⠛⠑⠂⢠⣼⣧⣤⠂⠀⠀⢨⢓⠆⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠃⠄⠀⠤⠄⠋⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢂⠄⠂⠐⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠨⠋⠁⠀⢀⡀⠤⠀⠀⡠⠊⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⠤⢄⠀⠀⠀⠈⠀⠤⠒⠁⠀⠀⣀⡀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠋⠉⠉⠂⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡠⠔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⢄⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⢰⠁⠀⡀⣠⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢡
⠀⠀⠀⠀⡠⠀⠃⠀⠀⠀⢸⣀⠀⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠨⣾⡇⠀⠘⠀⠀⠀⠀⠠⠔⠒⠅
⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠘⡆⠀⠀⠠⢖⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⠄⠀⠀⢠⡟⠀⠘
⠀⠀⠀⠀⢂⠀⢀⠀⠀⠀⠸⣷⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣶⣿⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⠀⢻⣷⣶⣦⣭⣩⣍⣉⣉⣩⣍⣁⣠⣦⣶⣿⠇⠀⠀⠀⡐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠻⣿⡿⠿⠛⠛⢿⠿⠿⠟⠿⢿⣿⡿⠋⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣀⠤⠋⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠠⣀⠀⠀⠈⠉⠀⠂⠈⠉⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠒⠂⠤⠤⠤⠤⠤⠀⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡄⢐⣦⣤⡀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟
⠀⠀⢠⣾⣤⠀⠸⠿⢿⣿⣆⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠈⣿⠀⠀⠀⠀⢸⣿⡇⠀⣾⠁
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣆⠰⠀⠀⠀⠀⣠⠊⣡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠁⠀⠀⠀⣿⡄⠀⠀⠀⢸⣿⡇⣶⡏⠀⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⡄⠀⠀⠀⢰⢁⡞⠁⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣿⡇⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⡇⣿⡇⢸⡿
⠀⠀⠀⠀⠙⠛⠓⠒⠈⠁⠀⠀⠀⡸⠙⣿⡀⠀⠀⠘⢿⡝⢤⣀⣀⣇⡀⠠⠄⠒⠒⠀⠀⠀⠀⠀⠈⢛⡿⠦⣀⠀⠀⠘⣿⡆⠀⠀⠛⣿⣥⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃⠀⢹⣿⠀⠀⠀⠀⠙⠿⣍⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡄⠉⣧⠀⠀⢻⣎⠀⠀⠀⣿⣸⣿⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡔⠀⠀⢼⡿⡀⠈⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣾⡆⠀⠈⢿⣆⠀⠀⢹⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣆⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠀⢠⣴⣇⣿⣿⡿⠃⠀⠀⠈⢿⣧⠀⢸⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⣶⣇⡀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⠀⠞⠓⢻⡿⠋⠀⠀⠀⠀⠀⠈⡿⣦⠀⣿⣿⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⢻⣧⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⢿⣶⣶⣿⣏⠀⠐⠀⠀⠀⠀⠀⠀⢰⡈⢲⣿⡟⢠⠆
⠀⠀⢀⠢⢈⡠⣤⣄⣀⡤⠀⠀⠀⠀⠸⣿⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡏⠀⠀⠀⠀⠉⠉⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣿⡇⡟
⠀⠀⠀⡐⣶⣿⣿⣧⣹⣶⣶⣀⣀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣠⠂⢳⣀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟
⠀⠀⠈⠈⠁⠈⠻⠋⠿⠿⠟⠻⠊⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⢀⣴⣮⡀⠀⠀⠏⠁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠛⠇
⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⡟⠀⠀⠶⡞⠠⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠘⡿⣍⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠀⠀⠀⠀⠀⠠⢠⡇
⠀⠀⠀⠀⠀⠁⠈⠂⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠃⠀⠄⠀⡀⠀⠀⢀⠎⠀⠀⠈⠀⠀⠀⢀⡄⣰⠀⠗⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠠⠄⣤⡴⠟⠁⠀⠀⢀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠴⠤⠓⠃⠤⠂⢾⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡝⣰⠁⢿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⡀⠐⠋⠈⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡿⠀⠀⠀⠀⠀⠀⠁⠀⠈⢰⣀⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⠿⠓⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡀⠀⠉⠛⠂⠀⠀⠘⠉⢋⠀⠀⠀⠀⢠⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠃⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⣀⠤⠄⠠⠄⣢⡀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⣰⡧⠀⡆⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠈⢠⠖⠂⠾⣤⣷⣀⣄⠀⣀⣤⣆⣰⣰⣦⢀⣤⡎⠁⠀⠀⠀⠀⠀⠹⣶⠤⠄⠀⠀⠀⣀⣠⠀⠀⣠⣾⠟⠁⢀⠟⠂
⠀⠀⠀⠀⠄⣀⠀⠀⠈⠀⠀⠈⣫⣽⣿⣿⣿⣿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⣾⡷⠞⠛⠋⠉⠉⠁⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣨⣿⠋⠉⠉⠛⢿⣿⣿⡇⠀⠀⢄⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⡄
⠀⠀⢀⣀⡀⠀⠀⢀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠘⠟⠛⠁⠀⠐⠂⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣇⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠘⠿⠃⠀⠀⢿⣷⡄⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⠄⢤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠤⠠⠄⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠴⠒⠒⠋⠉⠉⠉⠙⠊⠓⠊⠒⠚⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⠀⢀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠛⠀⠀⠁⠁⠀⠀⠰⠆⠀⠀⠀⠋⠀⠀⠀⠀⠀⡀⠀⠐⠂⠀⠀⠉⢢
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠉⠀⠀⠘⠃⠀⠀⠀⢣⡀
⠀⡴⠋⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣄⠀⠀⠀⠈⢦
⢠⠃⠀⠀⠀⠀⠀⠀⠠⠆⠈⠀⠀⠀⠀⠀⠑⠚⠛⠉⢉⠉⢉⡱⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢎
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠃⠀⠰⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈
⠘⡄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡴⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⣠⠀⠀⠀⠘⠃⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⠟
⠳⢄⡉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⢀
⠀⢸⢁⠀⠈⠐⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠋⠀⠀⢹
⠀⢸⠈⠒⠀⠀⠠⣀⠀⠀⠉⠀⠒⣒⡠⠤⠤⠄⣀⣀⡀⠀⠀⠀⣀⣀⣀⡀⠠⠤⠒⠉⠁⠀⠀⠀⣀⣠⠞
⠀⠈⢢⡀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⡀⢀⡀⠤⠒⠁⠀⢠⠃
⠀⠀⠀⠙⠢⢄⣀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠚⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⣿⡀⢜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡸⠁⢸⡇⠈⠢⣹⡷⢤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⣧⠀⠀⠙⡿⣦⠙⠢⡈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡾⠁⠀⠀⢸⣿⠀⠀⠀⠈⢾⣇⠀⠈⢷⡄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⢀⣾⠃⠀⠀⠀⢈⣿⠀⠀⠀⠀⠘⣿⡄⠀⠈⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⢠⢰⡿⠀⠀⢺⣿⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⣿⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣧⠀⠀⠻⣿⠀⠀⠀⠀⠀⢾⣇⠀⠀⠀⠀⢿⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⡄⠀⢰⣿⡆⠀⡀⣄⣦⣿⣿⠀⠀⠀⢀⣾⢇⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣦⣺⣿⣿⣇⣿⣼⣿⣿⣿⡠⠐⡆⣼⣿⣦⣾⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⣾⣿⣿⣿⣿⣿⡟⢻⡟⣽⣼⡡⣼⣰⣏⡐⠶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢙⢻⢿⣿⡿⣿⣿⣿⡷⣞⡞⣇⢻⠿⠇⢡⡝⣉⣡⣴⠛⠊⠉⠉⠉⠉⠉⠙⠒⠂⠶⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠿⠿⣻⣼⣢⣿⠧⠤⠄⠐⢿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⡀⠈⠁⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⣄⣀⡀⣀⣀⣠⣴⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠿⠻⠿⠟⣿⣷⣾⡿⢟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠤⠤⠤⠤⠤⣄⣤⣔⣲⠖⠂⠘⠀⠀⠈⠃⠋⠁⠛⢻⢿⣦⣄⡀
⠀⠀⠀⠀⡊⠀⠀⠀⠲⡲⠁⠀⠁⢶⠈⠁⠀⢳⠏⠀⠀⠀⠠⠂⢆⡠⢀⡀⠔⠊⠀⠉⣿⠿⣶⣄⡀⠀⠀⢠⡄⠀⠀⠰⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠓⠤⠤⠤⠤⠄⢠⣤⣤⣬⣀⣀⣠⠞⡰⠀⠀⠀⡹⣿⣾⣿⡗⠂⠀⠀⠀⠀⠀⠔⠿⢿⢿⡷⣶⡾⣿⢫⠋⠀⠹⡽⡽⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣀⢀⡀⠄⠈⠀⠍⢙⣽⣿⣀⡀⠀⠀⠀⠀⠀⠀⠚⠀⠇⠃⢿⠘⡆⠀⠀⠀⢷⠙⠉⠘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⢀⡠⡖⠋⠀⠙⠿⣦⣤⡀⡀⠀⠀⢠⣁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⣠⢤⡔⠚⠁⠀⠀⠀⠀⢀⣴⣟⢧⡀⢀⡀⣀⢸⡿⠀⢤⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⣠⠖⠉⠉⠑⢸⠃⠀⠀⠀⠸⡥⠁⠀⠀⠀⢠⠁⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡤⠔⠊⠀⠐⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡇⡆⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⡀⠀⠀⠿⠁⠀⠀⠀⠀⠀⣀⠤⠄⠂⠀⢹⡏⠀⠀⠀⠀⢸⡇⡁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠐⠒⠛⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⣇⣿⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤

⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⣀⣤⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⣿⠟⠉⠀⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠾⠛⠛⢿⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣾⣷⣶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣧⠀⠀⠀⠈⣇⠀⠀⢀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠘⠀⣠⣿⣷⡀⠀⠀⠘⡄⠀⢸⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠉⠛⠲⢄⠀⠙⠀⢸⡇⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡀⠀⠘⠀⣠⠏⠀⠀⠀⠀⠀⣀⡤⠶⠻⢇
⠀⠀⠀⠀⣴⠋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⢻⢿⣦⣤⡴⠶⠛⠋⠀⠀⠀⠀⠘⡆
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢈⡿⠁⠀⢀⡀⣠⠀⠀⠀⠀⢀⠇
⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣹⢋⣀⡀⠠⠬⠝⣹⣶⣶⣶⠖⠁
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⡀⠄⠤⠒⠛⠉⠁⢀⣀⡀⠤⠐⠀⣰⠃⠘⣿⡇
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣴⠶⠶⣄⠀⠐⠂⠀⠀⠀⠀⣀⠀⢂⣴⠇⠀⠀⣸⠁
⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣠⡟⠀⠀⠀⠈⠉⠉⠉⠀⣢⣿⢿⣤⣤⠶⠋
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢌⠡⡉⠳⠶⣤⣤⣤⣤⠶⣺⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠄⠀⠄⠀⡀⠀⠀⠀⠽⢮⣕⣀⣀⣀⣀⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠓⠢⠤⣄⣀⡀
⠀⠀⠀⠀⠀⣠⠒⠋⠭⠭⠅⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠈⠁⠂⣀⣀⠀⠉⠙⠒⠢⣤⣀⡀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⡠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⡐⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠒⠒⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠡⡀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠆⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⡀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚
⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠂⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⣀⠤⠔⠒⠊⢳⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠒⠒⢤
⡎⠁⠀⠀⠀⠄⠀⢀⣉⠉⠉⠉⠉⠉⠓⠋⠀⠀⠘⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠀⣤⣤⡤⢄⣀⠗⠂⠀⢀⡠
⠈⠓⠢⠤⢄⣀⠀⠀⠀⠈⠉⠉⠁⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠒⠂⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣒⡤⠦⠤⠭⠖⠒⠚⠉⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⢤⠄⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣞⠛⣳⠄⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⢁⣠⣎⣉⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢯⣁⠡⠝⠃⠀⠀⠀⠀⣐⠲⣤⠤⠀⠒⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣩⣤⡤⣀⡀⠀⠀⣀⣀⡀⠤⠤⠐⠒⠂⠉⠉⠈⢁⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣡⡼⣇⠈⢊⡾⠋⠉⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⢠⠞⠻⠮⡺⣝⢷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⣯⠀⠀⢘⡩⣲⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢠⣞⣳⠄⠀⡀⠀⠀⠀⠀⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣴⡟⢫⡄⠐⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠉⠁⠀⠀⠀⠀⠀⢀⡞⠛⠥⣀⡝⢵⡄⠀⠀⠘⠀⠀⠀⠈⠈⠑⠈⠁⠀⠀⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢆⠦⠀⠘⣧⡀⠀⠀⢊⡼⠁⠀⠀⠀⢰⠒⣦⠀⠀⠀⠀⠀⠀⠀⠓⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠠⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠋⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠴⠒⠒⠂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠒⠒⠃⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣧⠸⡇⠀⣀⠀⠀⠀⠀⣰⠃⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡹⡋⠉⠒⠉⠙⢦⠀⠀⣸⡆⢻⡏⣨⠇⠀⢀⡴⠃⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⣀⠾⣤⡀⠀⡰⠁⡠⢀⣠⡤⠤⣿⡿⢹⡇⠘⡿⢧⣠⠚⣡⠂⠀⠀⠀⠘⡄⠀⢄⠱⣄
⠀⠀⠀⠀⠀⠀⠰⡁⠄⠀⡹⠒⡳⠎⠈⠁⠂⢉⡽⠋⠀⠈⡇⠀⣯⡀⠈⡛⢧⡀⣼⠀⠀⠀⢱⡀⠈⢆⠈⣦
⠀⠀⠀⠀⠀⠀⠸⡔⠨⠐⢣⣒⣃⣤⡀⠀⣠⡟⠀⠀⠀⡇⣷⠀⢸⡴⠀⠀⠀⠘⢧⠀⠀⠀⡎⣧⠀⠘⡎⣾⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠸⡆⠀⢠⠪⠉⢇⢊⠋⠳⡏⡅⠀⠀⠀⠇⢹⡧⠀⡷⠀⠀⡇⠀⢸⠂⠀⢰⣰⣿⠴⠞⠛⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠦⢿⠀⠀⡘⢦⡑⠢⣇⣵⣄⠀⠀⠘⠈⣿⠀⣧⢃⢀⠇⢀⣾⣄⢀⣹⢧⣀⡀⠀⠀⣀⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣬⠶⠟⠓⠚⢮⣻⣷⣤⣄⣀⣼⡄⠸⣆⣈⡤⢞⣈⣉⡋⠻⣆⠄⣈⣉⡉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠓⠒⠈⠛⡽⢚⡩⠭⠲⠭⣷⣮⡑⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣋⣀⠠⠤⠀⠀⢉⣿⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠥⠶⠤⠤⠔⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣟⡉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣨⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣿⣿⣿⣿⠟⠛⣿⣿⣿⣀⣿⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣦⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⠟⠛⠋⠛⠛⠿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠙⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠋⣿⣿⣿⣿⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⠁⣼⣿⡟⠙⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣯⡀⢻⣿⣿⣿⣦⣾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⣤⣾⣿⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣃⣨⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣤⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣇⣰⣿⣿⣿⣿⡋⢉⣽⣿⣿⣿⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠛⣿⣿⣿⣿⠛⣿⣿⣿⣿⡟⣻⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡛⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣥⣾⣿⣿⣛⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣿⣿⣿⣿⣿⡿⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⠿⣿⣿⣿⣿⣏⣻⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⡉⢻⣿⣷⣦⣿⣿⣿⣿⠟⢿⣿⣿⣿⡿⠿⣿⣿⣿⣷⣶⣿⣿⣿⠛⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣾⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠈⠑⠂⠂⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠖⠢⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣀⣤⣄⢀⣀⡀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠻⠿⠿⣿⠿⠃⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠶⣶⣶⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣄⠀⠀⠉⣩⠟⠛⠻⢅⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⢡⣶⡿⠀⠹⣾⣿⣂⣀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⢀⣀⣄⡀⠀⢸⣿⡿⣶⡄⠙⣿⣿⡋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⡶⠷⠒⠀⠀⠀⠀⢽⣿⣿⣿⣿⡇⠀⠙⠛⠛⠁⠀⠈⠻⠿⠷⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⡇⠀⠀⠀⠈⠿⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢴⠂⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣮⠀⠀⠀⣸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠠⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡀⠀⡀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣀⠀⠀⠙⢟⠛⢋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⡳⠀⠀⠀⢤⣄⠀⠈⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⢀⣀⣿⣿⣧⠀⢿⣿⠏⠙⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠁⢻⣿⣿⣿⠏⠀⠈⠙⠀⠀⠖⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣠⠴⠋⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠶⡤⠀⠀⣀⣠⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⡀⠤⠐⠂⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡟⢦⠶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡠⠳⣄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢃⠞⢁⡴⠃⣰⣦⢾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠋⡠⠔⠁⡰⠋⠀⡴⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠀⢠⠊⠀⢠⠞⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋⠀⠀⠀⠀⡰⠃⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠃⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⠎⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢸⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⠎⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠋⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⠞⠁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢀⣀⡴⢋⠔⠉⢀⠔⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠤⠤⠴⢿⣿⠛⣁⡠⠖⠁⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢥⣄⣴⠀⣀⡠⠴⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠉⡉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣠⠴⠖⠖⠉⠁⠀⠀⠀⠀⠈⠐⠒⠂⠀⣀
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣯⣿⣇⢻⡅⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠹⢴⣀⣠⣄⢠⣄⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠀⠀⠀⠀⠀⡀⢀⣀⣴⣿⣿⢉⠈⠛⠦⠄⠀⠀⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⢬⣠⡿⠾⠿⠟⠩⡟⠉⠀⣅⡀⠀⠀⠀⢀⢀⠄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡠⠛⠃⣄⠀⠉⠉⠛⠒⠒⢲⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡄⠀⢄⣀⡀⠀⠀⠘⣌⠀⠀⠙⠲⠶⣤⡤⠤⠂
⠀⠀⠻⠿⠿⠿⠿⠿⠟⠛⠛⠉⠀⠀⠛⠿⠿⠦⠴⠴⠛⠓⠒⠢⠤⠀⠁


⣴⣶⣶⣶⣶⣴⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⡤⠶⢤⡀⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠍⠀⠀⠀⠈⠳⠤⠔⠁⠀⠀⠀⠀⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠈⢁⡴⠒⢲⣤⠤⠤⠴⠖⠂⠀⠃⠀⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠷⠖⠒⠒⠋⠉⠉⠳⠤⠼⠁⠀⠀⠀⢀⠤⢤⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡼⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⢠⠖⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⡼⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠐⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡋⠁⠉⢻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⠿⠟⢛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠿⠿⠟⠁⠀⠀⢩⣭⣭⣴⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⠋⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠑⠂⢄⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠔⠂⠐⠢⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢢⠀⠀⠀⠘⠀⡌⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠃⠀⠀⠀⡐⠀⠡⠀⠀⠈⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂
⠣⣀⠄⠀⠀⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⠀⠀⡀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠱⡆⠀⠀⠉⠀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠓⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢄⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡀⠉⠲⢄⠑⠀⠐⠚⠛⠛⠛⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⣏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣀⣤⣼⣧⠤⠤
⠀⠀⠀⠀⠁⠒⠢⢤⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠔⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⣰⠏⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠿⣶⣤⠀⠀⠀⠀⠀⠈⢷⡀
⠀⣰⠏⠀⠀⠀⠀⠀⠰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠈⢷⡀
⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣽⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣍⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⢀⣤⣶⠀⢠⠾⠛⠋⠉⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠉⠛⠻⢦⠀⢀⣀⠀⠀⠀⠀⢸
⣿⠀⢀⡴⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠳⣤⡀⠀⢸
⣿⠀⣿⠀⠀⠀⢘⡇⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⣿⠀⠀⠈⢻⡆⢸
⢹⡆⠙⢦⣤⣤⠾⠃⠀⣿⠈⠉⠛⠳⠶⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠛⠋⠉⣸⠇⠀⢿⣀⠀⠀⣰⠇⣾
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠙⠛⠛⠁⢰⡏
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⣿⠲⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⣾⠋⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢀⣤⣤⣄⣀⣤⣤⣤⡀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡟⠀⠀⠈⠉⠁⠀⠈⣷⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣓⣶⣶⣶⣶⣶⣶⣶⣖⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⣦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠚⠁⠀⠀⢄⣄⣀⠀⠀⣀⣀⠀⠀⠀⠀⢻⡀⠀⠉⣍⠁⠀⠀⠀⠀⠀⢘⡄⠀⣀⣠⣧⣀⣀⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢻⣿⣿⣧⡀⠀⣦⠘⣷⣄⣶⣿⣶⣗⣷⣾⣷⠿⠾⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠠⠦⠤⠶⠖⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣜⣦⠼⠟⠛⠉⠉⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⣿⣿⡿⠛⠁⣤⠀⢀⣶⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠿⠷⣤⣀⣛⣴⡼⠿⠶⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣧⣄⡀⡸⣼⣿⠀⠀⠀⢀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡐⠲⠿⠿⠛⣛⠿⣟⠿⣿⡟⠉⠉⡉⠙⢛⠛⠛⠓⠶⣷⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⣛⠤⠇⠛⠇⢠⣤⣈⠉⠁⠀⠀⠠⠛⣿⣷⣾⣟⣰⡏⠀⡀⠀⠀⠈⡂
⠀⠀⠀⠀⠀⠀⠢⣈⣛⣋⣬⣤⣤⣤⡶⠶⠾⣾⣼⡿⢷⢾⣶⣶⣴⣾⣷⣿⣿⣟⣿⣿⣿⣿⡿⠶⠘
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠿⠾⠿⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠿⠿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠁⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠻⣷⣄⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⠿⠷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠘⢿⣆⠀⠹⣷⡀
⠀⠀⠀⠀⢀⣤⣾⠿⠛⠉⠁⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠈⢿⡆⠀⠹⣷⡀
⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⡀⠀⠀⠀⠀⢈⣹⣿⠆⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣸⣿⠀⠀⢻⣧
⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠈⣿⡆
⠀⠀⠀⠘⣿⡿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠹⣿⡄⠉⠛⠿⠿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⠿⠟⠛⠉⣻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⣾⠟⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠁⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢷⣶⣶⣶⣶⣶⣶⠾⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡊⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡃⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢉⠄⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣼⣗⠀⠀⠀⠀⠀⠱⣄
⣠⠴⠒⠒⠋⠉⢉⠉⠉⠉⠀⣨⢀⠀⠀⠉⢉⡇⠀⠀⢠⡄⠼⢛⡇⢸⠀⠀⠀⠀⠀⠀⠈⢧⡀
⢿⣿⣾⣾⣴⣴⣿⣿⣷⣾⣶⣿⣿⣦⣴⣦⣾⣄⣀⡰⣬⣿⣀⢸⡏⠝⢀⣦⠂⠀⠀⠀⠀⠀⠻⣄
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠀⠉⠛⠛⠛⠛⠛⠋⠉⠽⢿⣖⢻⣾⠇⢂⠆⠀⠀⠀⠀⠈⠷⠶⢴⣶⣶⣤⣤⣄⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣀⠀⣤⣶⠀⠀⠀⣶⣿⣿⣞⣽⣾⣿⢡⡄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿⠿⣿⣿⠿⠟⠋⠉⠀⣈⣹⣿⣆⠙⣿⣿⡕⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣀⠀⠀⣠⣦⣤⣬⣿⣿⣮⠛⠶⢬⡉⠛⠃⠠⠤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠟⠛⠋⠉⣠⣼⣿⡟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⢶⣷⣤⣀⣿⣿⣯⡦⡪⣀⣀⣀⡀⠀⠀⠀⠀⡀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣿⣦⡥⢾⣼⣿⣸⣿⣿⣿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⣟⣿⣿⣿⣿⣿⣷⡀
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⠒⠒⠒⠒⠒⠒⠒⢶⣶⣒⣒⡲⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠀⠀⠀⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠂⢤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡏⠀⠀⠀⠀⠈⠉⠒⡤⣄⡀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠁⠂⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡂⠀⠀⠀⠁⠒⢤⡜⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⢀⣀⠀⠀⠀⠘⠓⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⡇⣿⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠰⣇⣵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⡇⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⡇⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⣷⠀⠀⠀⠀⡀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⢷⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⠰⠀⠀⠀⡀⠀⠀⠉⠀⡤⠀⠀⠀⠀⠀⠀⢧⣾⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⢈⠂⢄⠀⠀⠾⠃⠀⠀⠀⠀⠐⠀⡄⠀⠠⡿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣌⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⠇⡇⠀⠀⢠⠃
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣤⣤⣀⣀⠀⠈⠙⠓⠤⣐⡠⠐⢶⡄⠀⠀⠀⣣⠀⡠⠃
⠀⠠⣾⣿⣿⡟⣿⢻⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠉⠑⠒⠤⢤⣠⣼⣿⠔⠁
⠀⠀⣿⣿⣿⣏⣿⣣⣃⣕⣩⣼⣿⣿⣿⠇
⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣿⣆⠀⠉⡒⠤⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣾⣿⣿⡄⠀⢸⣿⣶⣶⣾⠀⠀⠀⣠⠴⠂⠐⢦⡀
⠀⠀⠀⠀⠀⢠⠊⠒⣒⣒⣒⣒⣒⣤⡀⠀⠀⠀⡜⢰⣿⣿⣿⣧⠀⠸⣿⣿⣿⡏⠀⣠⠞⠁⣠⣀⣀⠀⣷⡀
⠀⠀⠀⠀⠀⢁⣠⡀⠀⠙⠿⣿⣿⣿⣿⣦⡀⢠⠃⠈⢿⣿⣿⣿⣦⠀⢻⣿⣿⢃⡜⢁⣴⣿⣿⣿⣿⣷⣌⢿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣦⠀⠀⢿⣿⣿⣿⣿⣿⡞⠀⣠⣾⣿⣿⣿⣿⣷⡌⢿⣿⣯⣦⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢷⡀
⠀⠀⠀⠀⣨⣿⡿⠋⠉⡆⠀⠈⢻⣿⣿⣿⣿⠁⠀⠀⠠⣾⣿⣿⣿⣿⡷⠠⠿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⡘⣷
⠀⠀⠀⠀⣿⠏⠀⠀⠀⠑⣄⠀⠀⠸⢿⡿⠋⠀⠀⠀⠐⠺⠿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢹⡇⡟
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣶⣦⣬⣷⣤⣤⣴⣿⡿⢟⣿⡟⠀⠀⠀⠀⠀⠀⠘⠛⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⡤⢒⣭⠷⠒⠒⠀⠀⠈⠙⠛⠛⠛⠛⠛⠿⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⣁⣤⠒⠋⠉⠈⠀⣈⣉⠙⣻⣿⣿⣶⣄⡙⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣴⣿⣿⡿⠦⠄⠀⠀⠀⠩⠴⠛⣻⣿⣿⣿⣿⣿⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢿⡏⢉⣶⠷⣶⣤⡀⠀⢸⣟⣿⣿⣿⣿⣿⣿⡿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣤⠁⠘⠣⠌⠉⣿⣿⣿⠈⢿⣿⣥⡾⣿⡿⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠈⠉⠙⠀⢸⣿⡄⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣹⡆⠀⠀⠀⠀⣀⣤⠀⠀⢸⣿⣿⣄⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡀⢀⠀⡾⠉⠙⠶⣤⣼⠿⢟⣻⣧⣿⣿⣧⣶⣶⣶⣾⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣇⠸⡄⢿⣿⣟⢓⠒⠒⣛⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠰⡌⠋⣿⠈⣻⡟⠉⢹⣿⣿⣿⣿⣿⣿⠯⣉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠆⠈⢢⠸⠃⠀⢸⣿⣿⣿⣿⣿⡯⣉⠐⠊⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⡄⠀⢀⣿⣿⣿⡿⢟⠪⡉⠂⠁⠀⠀⠐⠄⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠙⡒⢛⣡⣿⠟⠀⠀⠑⠈⠀⠀⠀⠀⠢⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⡀⢰⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡵⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢣⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣤⠐⡃⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣛⣉⣀⣀⣀⣀⡀⠈⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠋⠁⢀⣀⣤⠶⠞⠒⠈⠉⠉⠉⠉⣉⡉⠉⠉⠉⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⡙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠛⠳⣆⠀⠀⠀⠀⠸⠀⠀⠀⣠⣴⡿⠟⠛⠛⠛⠛⠿⠿⣦⣍⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⣴⣿⣋⡴⠞⠋⠙⠛⠒⠀⠀⠀⠉⠓⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠜⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣠⣴⡿⠟⠛⣉⣉⣙⣛⣿⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⠹⣿⣆
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢠⣾⡿⠋⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠁⠀⠀⠘⢿⣧
⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣧
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣷⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⠐⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠉⠉⠀⠀⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⢠⣠⣀⡄⠀⠀⢆
⠀⠀⠀⡍⡉⠉⠉⣙⣛⣛⣛⣋⢙⡟⠓⠤⡤⠊⠁⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠻⣷⣤⡀⠸⣍⠀⠀⢠⣀⣀⠈⢀⠀⠐⠀⠠
⠀⠀⣀⣴⣦⣠⣴⣤⡈⠉⠏⠀⣾⠀⠰⠀⡇⠀⠀⡀⠀⠀⠀⠀⢠⣽⣿⠿⠷⠶⠾⠿⠿⠶⢦⡟⠿⠾⠿⠭⠧⠽⠿⢶⣶⠾
⠀⢠⣿⣿⣿⣿⣿⣿⣷⣶⣦⣾⣿⠀⢸⠀⡁⠀⠀⠀⠀⠀⢀⣰⣾⣧⣬⡶⣀⣤⠀⡀⢀⠀⠀⠈⡃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠠⠄⠀⠀⡀⠈⠁⣴⠯⣾⣭⣯⣉⡉⠛⠻⠶⠾⢿⣿⠷⠖⠁
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣧⠀⠀⠚⠃⠉⠉⠁⠀⣻⡟⠓⠛⠿⣷⣦⣤⣀⣀⢨
⠀⣿⣿⣿⣿⡿⢿⣿⠿⢿⣿⣿⡍⢅⡿⣀⠀⣠⢀⠀⠀⠀⠠⣿⣿⣶⣦⣄⣀⡉⠙⡋⠋⠁
⠀⢿⣿⣿⣇⠀⣰⣏⠐⢀⣽⣿⣶⠋⠀⠉⠑⠂⠤⠔⣂⡀⢠⣀⣙⠉⠹⠿⠟⠛⠛
⠀⠘⠻⠿⠽⠿⠭⠽⢿⣯⣭⣯⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠘⣆⡠⠔⠚⠋⠉⠁⠀⠀⠉⠉⠙⠒⠦⣄⣰⠁⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠈⠓⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⣀⣀⣀⡿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣧⣀⣀⣀⣀⣤⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠉⠁⢸⣏⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣩⣿⡆⠈⠉⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⢀⣀⠠⢄⣀⡀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠸⡷⠀⠀⣿⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣶⣶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠲⠆⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⣀⢀⠔⠁⠀⠆⡀⠰⠆⠀⣃⠈⠹⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠱⠀⠰⠃⠘⢀⠃⠀⠃⠰⠈⠁⢠⠈⠁⠈⠘⢃⡐⠈⡳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡋⠐⠃⠔⠀⠘⠈⠃⡞⠀⠀⠰⠀⠈⠄⠀⠀⠀⠀⠀⠐⠇⣉⠑⢈⣆
⠀⠀⠀⠀⠀⠀⣰⢉⡁⠇⠀⠀⠀⠀⠐⠃⠀⠀⠠⠖⠘⠂⠈⠉⠠⡄⠀⠀⠀⠀⣈⢀⡋⡌⢇
⠀⠀⠀⠀⠀⢀⠿⠈⢴⠀⠀⠀⢀⡆⠀⠃⠰⠀⠀⠈⠁⣀⠃⠀⠄⠈⠆⠀⠀⠀⠈⠀⣁⠀⠘⡆
⠀⠀⠀⠀⠀⢸⡐⢃⠀⠀⠀⠀⡤⠀⠀⠘⠀⢃⠀⠀⠀⢁⠐⢁⠀⡄⠙⠀⠀⠀⠀⢈⡉⢨⠃⠇
⠀⠀⠀⠀⠀⢹⠴⠈⠀⠀⠀⠔⠁⡤⠘⠆⠇⠀⢂⠀⠀⠁⠀⢩⠄⡆⠠⡄⢠⡄⠐⠂⠁⡄⠀⡆
⠀⠀⠀⠀⠀⠘⡄⠲⠀⢀⠀⠀⡄⢰⠆⠲⠐⠃⢀⠗⣄⠀⠀⠀⠀⠀⠋⠠⡄⣤⠘⠀⠃⠀⢠⠇
⠀⠀⠀⠒⠀⠤⠱⡠⠆⣄⠀⡄⠀⠦⠀⠴⠀⡠⠃⠀⠈⠓⢄⡀⠀⠘⠀⠀⠀⠀⠀⠉⠀⡠⠃⠀⠒⠀⠀⠀⢀
⠀⠀⠀⢠⣦⠀⠀⠘⠢⣈⠀⠀⠂⠀⣀⠤⠊⠀⠀⣤⠀⠀⢠⡌⠓⠢⠤⢀⣀⡀⠤⠔⠊⠀⢠⡄⠀⠀⠲⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠰⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⣴⡷⠀⡔⠉⠀⢻⠀⠀⠀⠀⠀⠠⠉⢀⠀⠀⠀⠀⠀⣔⢉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⣼⣿⡇⠀⠀⠀⠀⠀⣰⡿⠃⢰⠂⠀⢀⠄⢀⣾⡇⠀⣾⣿⣶⡄⣴⣿⠀⠀⡸⢿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⣼⣿⡟⣠⣶⣶⣄⠀⢰⡿⠁⠀⡸⠀⣠⠊⢀⣾⡟⠀⠀⣿⣿⠟⣸⣿⠃⠀⡰⠀⡸⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣸⣿⡟⣰⣿⣿⣿⠏⢠⡿⠁⠀⠀⣷⡞⠁⠀⣼⡟⠀⠀⣰⠏⠀⣰⣿⠃⠀⡰⠁⢠⠁⢀⣾⣨⠇
⠀⠀⠀⠀⠀⠀⣼⠏⠀⣼⣿⡿⠁⣿⣿⣿⠏⢀⡾⠁⠀⠀⣼⡏⠀⠀⣸⡟⠀⠀⣰⠏⠀⢠⡿⠃⠀⠠⠁⢀⠂⠀⣼⣿⡟
⠀⠀⠀⠀⠀⣼⠃⠀⣼⣿⡿⠁⢠⡿⠋⢀⣠⣾⡇⠀⠀⣼⡿⠀⠀⣰⡟⠀⠀⣰⡏⠀⢠⡿⠁⠀⢠⠃⠀⡜⠀⣰⣿⡿
⠀⠀⣴⣶⣿⣇⠀⠘⣿⠿⠁⢠⡿⢁⣴⣿⣿⣿⡆⠀⣸⣿⠁⠀⢠⡟⠀⠀⣼⡟⠀⢀⡾⠁⠀⢠⠃⠀⠰⠀⣠⣿⣿⠃
⠀⣼⣿⣿⣿⡿⠀⣼⠁⠀⣠⡿⢁⣾⢯⣟⣞⣿⠇⣰⣿⠃⣀⣴⡿⠀⢀⣾⡿⢡⣴⣿⡇⠀⢀⠂⠀⢀⠃⠀⣸⣿⠛
⠰⣿⣿⣿⠟⠀⠈⠁⠀⣰⣿⠃⠸⣿⣿⣾⣾⡏⠀⠿⢋⣾⣿⣿⡇⠀⣼⡿⠡⣿⣿⣿⠇⢀⠂⠀⠀⠌⠀⢰⢣⠇
⠀⠀⠉⠁⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠈⠙⠋⠀⠀⠀⠘⣿⣿⡿⠃⠀⠈⠀⠸⠄⠨⠉⠀⠈⠀⠀⠘⠀⠀⣯⡞
⠀⠀⠀⠀⠀⠀⠀⠰⠟⢡⣾⠆⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠁⠀⠀⠀⣀⡀⠀⠈⠁⠀⠀⠀⢀⡀⠀⣀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡄⠀⠀⢠⣾⣿⣿⣧⠀⠀⠀⠀⣴⡿⠀⣸⡿⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⢡⣶⠀⣿⣿⣿⣿⡏⢰⡿⠀⣼⣿⠃⢠⣿⠃⢀⣾⣿⣿⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⣾⠂⠀⣼⠋⠁⠀⢠⣿⠏⠀⣿⣿⠿⠋⢠⡿⠁⣸⣿⠇⠀⣼⡟⠀⣼⣿⣿⣿⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⢀⣾⠋⠀⣼⠃⠀⠀⢀⣿⠏⠀⢀⡟⠁⠀⢀⣿⠃⢰⣿⡟⠀⢰⣿⠁⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣾⠏⠀⣼⠃⠀⠀⠀⣾⠃⠀⢀⡞⠀⠀⠀⣾⠃⠀⢠⡿⠀⠀⣿⡏⣸⣿⣿⣿⣿⣿⠀⢠⢡
⠀⠀⠀⢀⣴⡟⠀⠀⠀⢀⣾⠏⠀⣼⠃⠀⠀⢀⣾⠃⠀⢀⡾⠁⠀⠀⣼⠃⠀⢀⣾⠁⠀⣸⣿⠀⠙⠿⣿⣿⣿⡿⢀⠆⡆
⠀⢠⣾⠿⢿⣧⠀⠀⠀⡜⠉⠀⣰⠏⠀⠀⣰⣿⣿⠀⢀⣾⠃⠀⠀⣸⠃⠀⠀⣼⠇⠀⢀⣿⠇⠀⠀⠀⡿⠁⠀⠀⡜⡸
⠀⣿⣇⣤⣾⡟⢀⠔⠊⠀⠀⣰⠏⠀⠀⣰⣿⣿⠃⢠⣿⠇⠀⣠⣴⡏⠀⠀⣸⡟⠀⢰⣾⣿⡀⠀⠀⢰⡇⠀⠀⡼⢠⠇
⠀⠙⠛⠋⠉⢰⣃⡀⠀⠀⢰⠏⠀⠀⣰⣿⣿⠃⢠⣿⡟⠀⢰⣿⣿⡟⠀⢰⡿⠁⣠⡟⡟⣿⠄⠀⠀⣾⠁⠀⣰⠃⡜
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⠀⠀⠀⣰⣿⣿⠃⠀⠺⠟⠀⢀⠳⣙⡟⠀⢠⣿⠁⠀⢺⣷⣷⡟⠀⠀⣸⣿⠀⢠⠇⢰⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠃⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⢠⣿⡇⠀⠎⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠋⠉⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠋⠙⠛⠛⠻⠿⢶⣤⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠋⠀⠀⣠⡴⠛⠉⠁⠀⠀⠈⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠈⠉⠛⢦⣄⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠈⡄⠀⠀⢠⠃⠀⠀⠀⠀⣠⣶⣶⣶⣤⡀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣾⠁⠀⢹⣿⣿⣧⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⣸⡏⠀⢘⣿⣿⣷⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣷⣴⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⢿⣷⣶⣾⣿⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⡤⠤⠖⠋⠁⠀⠀⠀⡴⠋⠳⠄⠀⠀⠀⠈⠙⠲⠤⢤⣤⠤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⡶⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢖⡿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣔⠋⠍⣙⠓⠒⠒⠲⠒⠒⢒⡚⠋⠉⠙⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣄⣀⣀⣉⣀⣀⣀⣤⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠒⠒⠲⢤⡀
⠀⠀⠀⢰⢊⢥⣀⣀⠀⠀⠈⠑⠢⢄⠀⣀⡠⠤⠒⠒⠒⠒⠒⠒⠢⠤⣀⣠⠖⠉⠀⠀⣀⠤⠖⠒⢳⢸
⠀⠀⠀⠸⡘⡆⠀⠀⠉⠒⢤⡀⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡴⠊⠀⠀⠀⢀⡇⡼
⠀⠀⠀⠀⢧⠹⡀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⡜⣰⠃
⠀⠀⠀⠀⠈⢣⡙⣄⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢞⡴⠁
⠀⠀⠀⠀⠀⠀⠙⠪⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠰⠛⠳⠀⠀⢀⡤⠴⠶⠶⠶⠦⢄⠀⠀⠋⠙⠂⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠉⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⢀⡤⡀⠀⠀⠀⡠⢄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠸⡀⢱⠀⠀⢰⠁⡸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠑⠊⠀⠀⠈⠓⠁⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠜⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠠⠂⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⣄⠠⠀⣀⣨⢶⠤⣄⡀
⠀⠀⠀⢀⠥⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠋⠄⠄⠘⠊⠀⠀⠈⠙⠀⡒⠒
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠐⢊⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠊⠀⢀⢼
⠀⠀⠈⣀⠑⠄⠐⠀⠂⠉⠀⠀⠀⠜⠀⠀⣠⠎⠜⠀⠀⠀⢀⠀⢈⡆
⠀⠀⠀⠉⠐⠠⠙⠀⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠻⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠉⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⠁⠀⠀⠀⠀⠄
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠆
⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣰⣜⠃⠀⠀⠀⢀⠃
⠀⠀⠠⠌⠙⠣⣄⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡕⠁⠉⠀⠀⠒⠂⠈
⠀⠀⠀⠠⠀⠀⡎⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠆⠀⠀⠀⠀⠀⣠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣴⣞⣠⢶⡆⠶⡶⣰⢋⡴⢋⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠟⠁⠈⠉⠉⠈⠉⠈⠁⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⣿⣿⣶⣶⣶⣶⣤⣀⠀⠀⠀⢀⣠⣤⠶⠛⠉⠀⠀⠀⠀⠉⠉⠛⠳⢦⣤⣀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⡀⣀⡀
⠀⣿⣿⣿⣿⣿⠟⢻⣿⣧⣴⡾⠏⠉⠀⠀⠀⠀⡀⣀⣀⣀⣀⣠⣄⡀⠀⠉⡻⠿⠷⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⣯⠽⣛⡿⠁⣰⣿⠏⠀⠀⠀⠀⡀⠀⡀⣀⣤⣜⣿⣿⣿⣿⡿⢿⣿⣦⣤⣓⣶⡶⣄
⠀⢰⣺⣽⠃⢠⣿⣿⣄⠠⡀⠰⣤⢸⣺⣿⣽⣽⡻⣯⣿⣿⣿⣤⣤⣷⠉⠉⠉⠁⠀⠀⠀⢀⡰
⠀⢾⣿⣿⠀⢸⣿⣿⣾⣴⣴⣷⣧⣯⣿⣿⣿⣯⣿⣿⣿⣿⣯⣰⣿⣿⣿⣶⣦⣤⣤⣄⡶⠞⠁
⠀⣺⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣠⣿⠻⢿⣿⣷⣀⠀⠀⠀⠀⢰
⠀⣿⣿⡇⠂⠀⣿⣿⣿⣿⣯⣋⢻⣮⣿⣿⣾⣿⣿⣿⣿⣯⡻⣤⠀⠙⠛⠿⢷⣶⠶⠖⠁
⠨⣿⣿⣇⠀⠀⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣰⠂
⠀⣿⣿⣿⠀⣠⡈⡿⢧⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠙⠛⠿⠿⠟⠃
⠀⣿⣿⣿⣆⡺⠿⢇⡴⠃⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠶⠒⠚⠋⠉⠉⣡⠞⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠒⠚⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⡿⠥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠉⠛⠲⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠿⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⢦⣤⣄⣀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡇⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣞⣱⠆⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣅⣠⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠺⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⢤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠦⣤⣤⣀⣀⣀⣿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀⡀
⠀⠀⠘⢧⠀⠀⠲⠦⣤⡉⠉⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠶⠶⠶⠶⠶⠦⣄
⠀⠀⠀⠈⣧⠀⠀⠀⠀⠙⢶⡄⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⣀⡤⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⢳⡖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠲⠤⣤⣤⣤⣴⡞⠁⠀⢀⡜⠁⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⢀⡼⠋⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠂⠀⢀⣤⠖⠛⠓⠲⢄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢄⡙⢧⡀⠀⠀⢀⣠⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⠀⠀⣀⡄⠀⠀⢠⡟⢁⣤⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⡞⢁⣀⠀⠀⠙⢆⢻⡟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⠟⠉⠉⠀⠀⠀⣿⠀⣿⣿⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⢸⣿⣶⠀⠀⢸⡄⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠈⣿⠈⠻⠿⠟⠀⠀⠀⢠⠇⠀⠐⠒⢛⣛⣛⣻⣿⣯⣄⠀⣼⣁⣘⡃⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠶⠿⠶⢶⣄⣀⣀⣠⠔⠃⠀⣠⠶⠛⠉⠉⠀⠀⠀⠉⠙⠛⠧⣀⠀⠉⠂⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⠞⠁⠀⢀⡤⣄⠀⠀⢠⡟⢻⡄⠀⠈⢷⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡏⠀⣿⠀⠀⠘⡇⠀⢻⠀⠀⢈⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⡇⢀⡏⠀⠀⠀⠘⠀⠂⠀⠀⣼⠁⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡴⠁⠀⠀⠀⠈⢿⣄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⡿⣦⣀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⣀⣀⣠⣴⡾⠋⠁⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣙⠻⠶⢦⣤⣤⣤⣤⣬⣽⣿⡿⠛⠉⠉⠁⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠀⠠⠤⠤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⢦⣤⣤⣄⣀⣀⣀⣤⣤⡴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠆⠐⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠂⠀⣠⠀⠀⠀⠀⠀⠀⢵⡆⠀⠀⠀⠀⠀⢀⠀⠤⠤⢀⣀⠤⢤⣤⡄⠐⠁
⠀⠀⠀⠀⠀⠈⢀⣤⡤⠔⠋⠁⠐⠋⢁⠀⠀⠀⣰⣤⣤⠟⣫⠩⠉⠀⠀⠀⢯⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠐⣿⣥⣀⣠⠀⢀⣢⡤⢽⣖⣻⠋⣩⡿⠀⣼⠉⡠⠀⠀⠀⠀⠀⠙⠒⠶⠤⠤⠦⠤⠤⠶⠶⠚⠋
⠀⠀⠀⠀⠀⢀⣿⡿⣍⣛⠾⡭⠵⢶⠾⢉⣉⠉⠔⣀⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠤
⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣾⣭⣤⣥⣤⠤⠤⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⡛⠛⠛⠶⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡤⠤⣤⢤⣄⣤⣤⣤⣶⠟
⠀⠀⠀⠠⠶⠏⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠒⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣾⣶⠀⠀⠀⠙⠷⣭⠶
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣾⣻⣦⢄⣀⠀⠀⠀⠀⠀⢀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠻⠟⣿⣿⣿⣿⣿⣳⠲⣾⡿⣏⡉⠩⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤
⠀⢀⠄⠀⠀⠀⣶⡒⣲⡷⠀⠀⠀⢠⣄⠀⠀⠀⠉⠙⠛⠛⠈⠻⠙⠋⠙⠉⠀⠀⠀⠀⠀⡀⢁⡔⠒⠓⢄⠀⠀⣄
⠀⠘⣆⠀⠀⠀⠉⠉⠉⣀⣀⣀⣤⡟⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡿⠀⠀⠀⠀⠀⢲⡇
⠀⠀⢉⡷⠶⢶⢶⣦⣷⡴⢞⡹⠿⣛⣹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠐⢀⠀⠠⢸⡧
⠀⠀⠈⠁⠲⠿⣿⡚⠻⣿⠚⣽⠷⠟⠉⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⡄⡀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠊⢀⣀⡐⣦⠀⠀⠂⠀⠀⠈⠁⠈⠙⠉⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠘⠿⣦⣄⣀⡀⠀⠈⠁⠀⠀⠀⢠
⠀⠀⡔⠀⠀⠀⣠⠶⠀⠠⢴⣆⠀⠀⢢⠀⠀⠫⣥⠠⡄⠀⡀⠀⠊⢉⡹⣖⣶⠖⠛⠁⠀⢀⡤⠄⠒⠒⠂⠤⢤⣄
⠀⢰⡇⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⣼⡇⠀⠀⠘⠳⠤⠭⠥⠤⠾⠿⠿⠛⠁⠀⠀⠀⠀⠘⠃⣠⠴⠶⠦⠐⢲⡟
⠀⠈⢿⣦⣀⠀⠀⠀⠀⠀⠀⡀⠀⣨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠐⠉⠉⠻⣤⣤⣤⡴⠟
⠀⠀⠀⠉⠛⠻⢶⣤⢭⣭⣤⠄⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⠖⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⡀

⠲⠶⠶⣶⠶⣶⠚⣿⢛⡟⠒⢦⣀⣀⣀⣠⡤⠤⠶⠖⠒⠒⠚⠛⠛⠛⠉⠉⠉⠉⠙⠛⠓⠲⠦⣄⣀
⠀⠀⠀⣿⠀⡏⢠⠇⡼⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⡀
⠀⠀⠀⠃⠘⠃⠾⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣨⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠓⢦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠙⠒⠒⠒⠲⢦⣀⠀⠀⠀⢉⣡
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⡀⢸⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣀⣠⠞
⠀⠀⡄⢸⠀⡇⢸⠛⠛⠓⠺⣷⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⣸⠋
⠀⠸⡇⢸⡄⣧⢸⡆⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⢿⡓⠦⣄⠀⠀⢻⡦⠶⠚⠁
⠶⠶⠷⠾⠧⠽⣤⣷⣄⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡈⢃⣠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡲⣔⡇
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⢸⣷⡴⣿⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠠⣦⠶⠦⠀⠀⡀
⠀⠀⠀⠀⠀⢐⡏⣠⢀⡀⠆⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⣠⠀⠛⡗⢆⣤⠀⠀⠀⢠
⠀⠀⠀⠈⠀⠚⣅⢀⠘⠙⣦⣤⡤⣾⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠆⠀⠐⠀⠈⠌⠙
⠀⠀⠈⢂⠀⠀⠀⢸⡀⢈⡸⠋⢠⡾⠀⠂⠾⠦⠀⠀⠀⠀⡀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠀⣠⠂
⠀⠀⠀⠈⠰⢄⣀⣸⣃⣾⡁⠀⠀⠘⠀⠂⠀⢄⠃⠀⠀⠀⠀⠈⠈⠉⠀⣀⣤⣀⠀⣀⣄⣜⡠⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢡⠀⠀⠀⠀⣼⠀⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⠤⠿⠰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠙⢇⠀⠀⠠⠀⢿
⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⡄⠀⠀⠊⠈⢧
⠀⠀⠀⠀⠀⠸⡄⡀⠀⠀⠀⠀⠀⡀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢳⣄⡀⡀⠄⠸⣷
⠀⠀⢀⠀⠰⠊⠁⠘⠲⢤⣀⡀⠀⠀⠀⣼⠋⠠⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠉⠻⢷⣄⣰⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠈⠆⢹⡇⠀⠀⠀⠀⡄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠩⣿⡿⠃
⠀⠀⡀⣾⠀⠀⠀⠀⠀⠃⢸⣷⣦⡀⡈⢷⡇⢀⣿⠄⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠈⠻⠀⢀⣠⣤⣮⡴⠟⠀⠀⠈⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠈⢣⣀⣀⣀⡠⣦⣀⢀⣶⠃
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣸⠅⠄⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢠⡀⠊⠀⠀⣠⠉⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠰⠂⠌⢷⣆⠰⡍⢧⡴⢚⠁⣠⣾⠀⢠⡀
⠀⠀⠍⠀⠀⠗⠀⡀⢣⡀⠀⡀⢰⣆⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠁⠁⠁⠘⠁⠛⠄⢻⠁⠌
⠀⠀⠀⠀⠀⠊⢉⠀⠀⠉⠉⠈⠁⠀⠀⡔⠈⠁⠆⠀⠀⠀⠀⠱⠤⠤⢧⣄⣠⡴⠀⢀⣀⠀⠀⣄⣀⢁⡄
⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠉

⠀⠀⠀⠀⠀⠀⢸⣶⣄⣀⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⡍⠳⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣶⣶⣿⣷⡿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡀⠄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣳⣿⣿⣿⣿⣿⣿⣷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣾⣶⣿⣿⣿⣿⣿⡿⣦⡀⣠⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⢜⣻⣿⣿⣿⣿⣿⣯⣀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣁⠄⠻⣿⣿⣿⣿⣿⣾⡷⠹⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⡄⣾⣿⣿⣷⢹⣿⣿⣿⣿⣿⣛⡁⢀⣾
⠀⠀⠀⠀⠀⠀⢀⡿⠻⣶⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠢⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⡟⣿⣿⣿⣿⣿⠿⠿⠟⠋⢙
⠀⠀⠀⠀⠀⠀⢸⢯⣄⣈⢁⡺⣿⣿⣿⣿⣷⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣿⣿⣿⣟⣛⣁⣀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⢘⣶⣄⣭⣉⣿⣿⣿⣿⣿⣿⣋⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠭⠛⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢻⡟⠉⢽⣻⣿⣿⣿⣿⣶⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣶⡄⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣂⠙⢩⣿⠿⠟⢿⣷⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠆⠀⣬⣮⡭⣿⡛⢧⣄⡝⡲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠘⠃⣼⡶⠈⠙⠓⠿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣵⣿⣿⣿⠂⡀⣶⣤⡴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⡸⣿⡄⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⠃⢺⡇⣞⡀⢀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠸⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡿⠋⣥⠏⠙⠁⠀⠁⠀⢉⡀⠀⠀⠈⠀⠀⠉⠓⠀⠁⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡟⣡⡯⠋⠀⣴⣶⣿⣶⣤⡀⠉⢢⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⠟⢁⣼⣷⡟⠙⠛⠛⢿⣿⣦⡀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢯⡤⠚⢿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣦⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⡀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠋⠁⠀⢀⣄⠀⠀⠈⢅⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡻⣿⣶⣤⣠⣥⣦⣦⣤⣤⡀⠀⠀⢠⡾⠃⠀⡀⡔⠁⠁⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣇⣹⣿⣿⣿⣿⡿⠿⠿⣿⡿⠛⠷⠉⠀⠀⡠⠙⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⣠⣼⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠛⠋⠀⠀⠀⠀⣐⡤⣴⣶⣆⣴⣌⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣷⣿⣿⣿⣿⣿⢿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠏⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠛⠛⠉⠉⠉⠉⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠓⠢⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣠⣤⣄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣼⣿⣿⣿⣧⢸⠀⠀⠀⠀⠀⢀⡠⠤⠶⠶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠹⣿⣿⣿⣟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠛⣩⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢰⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⡄⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠘⣿⣿⣦⣤⣈⣉⠉⠙⠛⠓⠒⠒⠒⠛⠛⣉⣉⣩⣥⣼⣿⡿⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠤⣤⣤⣀⣀⣀⣀⣤⣤⠤⠶⠚⠋⠁

⠀⠀⠠⠋⠛⠂⠸⡟⢿⣏⡇⢰⣿⣿⡆⢀⣾⣿⣿⣦⠀⣿⣿⣿⣷⠀⣿⣿⣿⡆⢀⣾⢿⣆⠀⣴⣿⣿⣦⠀⣴⣿⣿⡄⠀⣰⠀⠀⣾⡇
⠀⠀⠘⡄⣄⡄⠀⣇⠘⢻⡇⠘⣿⣿⠇⠘⢿⣿⣿⡿⠀⢹⣿⣿⡇⠀⢿⣿⣿⠇⠈⢷⣼⠏⠀⠛⢿⡿⠛⠀⠹⣿⣿⠃⠀⣿⠀⢰⣿⡇
⠀⠀⠀⢻⡿⠀⠀⠙⣿⠛⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⢹⡏⠀⠀⠀⣿⠃⠀⠀⠈⡏⠀⠀⠀⢸⡇⠀⠀⠀⢹⠁⠀⢸⣿⠀⢸⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠈⣿⠀⠈⢻⡇
⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠈⡋⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⠸⡇⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⢸⠇⠀⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⠀⢨⣯⠀⠀⠀⢸⡅⠀⠀⠀⢿⡇⠀⠀⣿⡇⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢦⡴⠚⠛⠛⠒⠲⠶⠶⠶⠒⠒⠒⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢰⣽⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⣨⠟⠶⠦⠶⠶⠦⣤⣤⣀⣀⣀⣀⣀⣠⠼⠧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⡟⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⣧⠀⠀⠀⠀⣠⠖⠋⠛⢦⡀⠀⢀⡞⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⡾⠋⠀⣀⡴⠞⠁⠀⠀⠀⢨⡇⣰⠋⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⢀⣴⠏⠀⡇⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢁⣠⣤⣤⢤⡴⠶⠶⠚⠛⠛⠛⢶⡛⣷⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⣀⣀⣀⣠⣤⣤⡴⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠄⠤⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠉⠉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠈⠓⠤⠟⠀⠀⠀⠀⠀⢀⠤⠄⡀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢠⠋⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢒⡄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠘⠒⠁⠀⠀⠀⠀⢠⠄⡄⠀⠀⠀⣇⡁⠀⠀⠀⠀⢠⠔⢄⠀⠀⠀⠀⠀⠑⠬⠃⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠴⠃⠀⢀⣀⣈⣀⣀⠀⠀⠀⠈⠲⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠉⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⠀⠀⠀⠀⢠⠈⡇⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⢀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡎⢰⠋⡆⠀⠀⠀⠀⠉⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠰⡋⡰⠂⠀⠀⠋⢣⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠰⠒⢄⠀⠀⠀⠀⢦⡼⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⢧⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠈⠂⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⢠⠄⢠⡄⠀⠀⠀⠀⠈⠑⠂⢤⢄⠀⠠⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠚⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠘⠂⠋⠀⠀⠀⠀⢠⡆⠤⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠤⢺⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⢰⣃⡽⠀⠀⠀⠀⠀⠀⠢⠂⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠘⢥⢀⡵⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠤⢄⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⢰⠋⡇⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠙⠄⠤⠛⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠘⠤⠖⠁⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠤⠄⠀⠀⠀⠤⠤⠄⠒⠊⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⢲⠶⠒⠂⠉⠉⢉⣍⠉⠉⠉⠉⠉⣠⢤⠒⠖⠂⠤⢤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢉⡁⠀⠀⠀⠱⠔⠋⠀⠀⠀⠑⠴⠗⠀⠀⠀⠘⠧⠞⠀⠀⢀⡔⣢⠀⠉⠉⣢⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠁⠀⠀⠘⣌⡳⠀⠀⠀⡔⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⢢⡈⠉⠁⠀⠐⠯⠤⠟⢀⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡎⢱⠀⠀⠀⠀⠀⢠⠤⣌⠓⠠⣤⠄⣀⣀⣀⣀⣀⣀⡀⠀⣤⣤⠤⠚⠁⣠⠔⢦⠀⠀⠀⠀⠪⡌⢣⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠑⠃⠀⢞⠉⢆⠀⠈⠢⠼⠂⠀⠀⠀⢠⠤⢄⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠙⠒⠋⠀⠀⣤⠄⠤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⣠⠒⡆⠀⠠⠧⠴⠋⠀⠀⠀⠈⠓⠤⠜⠀⠀⠐⡖⢤⡀⠀⠀⠈⠑⠒⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠶⠀⠤⠤⠤⣤⣤⣀⣀⣀⣤⣤⣤⡤⠤⠤⠤⠀⠰⠶⠒⠒⠈⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡽⣶⢮⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢦⠀⣞⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠸⠀⠉⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⢳⠈⢧⢸⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢮⣇⠈⢿⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢻⡆⠈⣿⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣆⠀⢻⡄⠘⠧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡤⠦⣄⡤⢤⠈⡇⠀⠀⠀⠀⢴⡾⠻⣿⡆⠀⠀⠀⠘⣆⣀⣁⡴⠛⠳⡜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⡇⣤⣿⡄⣸⠀⣇⠀⠀⠀⠀⠀⣹⠚⠁⠀⠀⠀⠀⠀⢾⡅⠈⣷⣆⣠⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠉⠈⠉⢁⠀⣿⠀⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡞⢁⣀⣀⡆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⠤⠤⠔⠋⠀⢸⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠀⠁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣾⡚⢻⡀⠀⠀⣀⣠⠴⠚⢹⡄⣿⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠻⠓⠋⠉⠉⠉⠉⠀⠀⠀⠀⡇⢻⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡤⠿⢤⡀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⣀⡼⠧⣄
⠀⠀⠀⠀⠀⠸⢿⣶⠏⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠛⢳⣾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁⠀⠙⣆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠰⣄⣇⡼⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠶⡮⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⣤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣀⣤⣯⣤⣦⣥⣵⣬⣷⣤⣵⣮⣼⣥⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣿⣿⣿⣿⣿⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡞⠁⠉⡦⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⢦⡀
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⠋⠈⣹⠀⠀⠀⠈⠉⣠⠴⠦⡀
⠀⠀⠀⠀⠀⠈⠣⣀⣠⠟⠒⠉⠀⠀⣏⣹⠆⠀⠀⠀⠀⠈⢧⣀⡗⠀⠀⠀⡔⠉⠃⠀⠀⢸⡄
⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠰⡖⠋
⠓⠞⠁⠀⠀⠀⠀⡴⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣨⡇⠀⠀⠀⠗⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠚⠛⠉⠉⠉⠉⠁⠉⠓⠓⠒⠲⢤⣄
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠋⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⠛⠉⠁⠀⠀⠄⠀⢀⣀⠀⠀⠁⢀⢄⠀⠀⠀⠀⠀⠶⠀⠀⠀⠠⠄⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣸⠛⠒⠚⠁⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⣠⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠙⠁⠠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠋⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠀⢠⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢐⡳⠀⠀⠈⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⢀⣀⣀⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⠴⠖⠛⠉⠉⢀⡼⠁⠀⠀⠀⡶⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡀⠀⠀⠘⠃⠀⠀⠠⠀⠀⠀⠀⠈⠋⠀⢀⡴⠋⠀⠀⢀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⣾⠀⠀⠀⠹⠃⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠁⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠈⠉⠓⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠃⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠘⠤⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠚⡆
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠖⠒⠒⠒⠉⠉⡩⠓⠦⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠚⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⡴⠊⠲⡄⠀⠀⠸⡄⠀⢀⡗⠀⠀⠀⠀⢠⠊⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠳⢄⡤⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠈⠉⠢⡀⢰⠁⠀⠀⢳⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡇⠈⠢⠤⠤⠃⠀⠀⡔⠉⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡴⠒⠀⠒⢦⡀⠀⣉⣒⣋⣀⠤⠤⠤⠃
⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⢀⣀⣀⣀⠠⠤⠤⠤⠒⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠘⠒⠒⠂⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡡⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡁⠀⠰⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠊⠰⢠⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⣶⣖⠚⠁⠀⠀⠀⠘⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⢀⡠⠖⠋⢡⠞⠟⠋⠉⠉⠳⣤⣐⠙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠐⠠⢊⣴⢋⠀⠐⣰⠇⠀⠀⠀⠀⠀⠀⣹⠉⠉⠳⡈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⡼⠁⠀⠐⢠⡞⠁⠁⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⡄⠀⠹⣟⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡏⢰⡇⠀⠀⠀⣿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢻⠀⠀⢈⢦⠻⡄
⠀⠀⠀⠀⠀⠀⣾⠁⢸⡇⠀⡀⢸⡿⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠸⣇⠀⠀⠈⡆⠹⣦
⠀⠀⠀⠀⠀⠀⢻⣴⡈⢧⠀⢇⠈⡇⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠹⡅⠈⠳⡌⠀⢿⣆⠀⠐⡀⢸⠀⠀⠂⠀⢀⠀⢰⠇⠀⠀⠀⠀⠀⢀⡾⠀⠀⢰⠇⠀⡿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣕⠘⢦⠈⠻⣦⡀⠀⠈⣇⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⠏⠂⢠⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣑⢤⣈⠻⣦⠀⠘⣆⢀⠀⠘⣸⡇⠀⠂⠀⠀⠀⡰⠃⠀⡠⠋⠀⢠⡿⠘⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠿⣾⣷⣶⣎⣷⣁⠤⣿⡇⢈⠀⢀⣠⠞⠀⣨⠞⡡⢀⡴⠋⠄⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠻⢿⣿⣿⢿⣿⣷⣶⣿⣿⡵⠞⠉⣀⣠⡴⠋⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⢙⠕⠋⠛⠻⠻⠻⠟⠯⠥⠤⠦⠒⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⣦⣭⣝⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⠶⢶⣶⣶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿⣿⣿⡌⢷⣤⡤⠤⠤⠤⣄⣀⡀⣴⠋⠀⢈⣴⣿⣿⡿⠋⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠨⣿⣿⣿⣿⡈⣷⠒⠀⠀⠀⠀⠀⠉⠛⢦⣇⣾⣿⣿⣿⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢿⣇⢘⣿⣿⣿⣿⢁⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠆⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠻⡗⢿⣿⡿⠿⣿⣷⡄⠀⢄⠁⠀⠀⠀⠀⠀⠀⠘⣿⣟⣡⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠙⠦⣤⣠⣤⠶⣿⣿⣿⣶⣷⣦⠁⠀⠀⠀⠀⠀⠀⠀⢸⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡅⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣟⠿⠟⠀⠀⠀⠀⠀⠀⢀⣀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⡞⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⠿⠀⠀⠀⠀⠀⢀⣶⠈⣉⣁⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⡇⠀⠀⠀⠀⡆⠀⠀⠀⣿⢀⠀⠀⠀⢀⡄⠀⠀⠐⡉⣠⠞⣭⡀⠀⣉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣷⡄⠀⢠⡀⣷⠀⠀⠀⣿⣿⣆⢄⠰⠃⠀⠀⠀⠘⣰⠃⢰⣿⠁⠀⢿⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⣿⣦⣻⣷⠸⣧⣄⠘⢾⣿⣿⣿⣶⣴⣴⣿⣧⠄⠻⣄⣈⣡⣤⣤⣤⣁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡹⡿⣿⣷⣿⣧⣻⣿⣷⣦⣝⣿⣿⣿⣿⣿⣿⣥⣤⣤⡬⣭⣍⣉⣭⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣡⣴⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠁⣀⣤⡶⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣟⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢴⣾⣟⡛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⣻⣆
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠘⢷⡀⠀⠀⠸⣿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⢉⣼⠏⠁⠀⠀⠀⣿⣇⠀⠀⣼⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⢷⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⢸⡇⠙⣧⣀⣿⠀⠻⣄⠀⠸⣷⣄⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠙⣷⣄⢸⡇⠀⢰⣦⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣴⠶⣶⠀⢲⣼⡇⠀⠈⠙⠛⠀⠀⠙⢷⣄⣻⡝⢦⣄⠹⣿⣦⣄⡀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣿⠀⠈⣟⠃⠀⠀⣀⣀⣀⣀⠀⠀⠈⠛⠛⠀⠙⣷⣿⣶⣿⣿⣿⡶⠶⣶⣤⣼⡷⢠⣶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⣿⠀⢴⡿⠿⠿⠿⣿⡷⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⣿⣿⣦⢸⡇⠀⣀⣬⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⣻⣿⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢸⣧⡾⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢛⣷⣿⠀⠀⠾⠋⢿⣿⡟⠙⠂⠀⣶⠀⠀⠐⠛⠹⣯⣿⠙⠿⠀⠀⢸⣯⠶⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡄⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠷⢸⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⣸⡇⠀⠀⢰⣶⣶⣶⣶⣶⣶⡏⢉⣿⣶⣶⣶⡶⢶⡶⠀⠀⢸⡇⠉⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠀⠀⣿⣷⣦⣤⣬⣉⣉⣀⣀⣉⣩⣤⣤⠴⢾⣿⠆⢀⣿⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢹⣷⣤⡀⠉⠉⠉⠉⠁⠀⠀⢤⡴⣿⠇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠙⠲⠦⢤⣤⣤⣤⣤⣤⠤⠶⠚⠉⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⣴⠤⣤⡀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣼⢶⡇⠀⠸⣷⣶⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠋⢠⡀⠀⠀⢁⠘⠟⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠿⣆⣰⠟⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣠⣤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣄⣀⠀⠉⠛⠶⢤⣀
⠀⠀⡇⠈⠉⠓⠲⢤⣄⣀⡀⠀⠀⠀⠀⠀⠙⠲⠦⢤⣽⠗⠀⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠉⠙⣳⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠉⠛⠶⣄⡀
⠀⠀⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠴⠛⠒⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀
⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⠖⠒⠛⠛⠒⠦⡀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣽⠒⠒⠒⠲⠶⠦⠤⠤⢽⡇
⠀⠀⡇⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⠁
⠀⠀⡇⠀⠀⠀⢀⡴⠛⠉⠳⡄⠀⠀⠀⠀⠸⣄⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡏
⠀⠀⡇⠀⠀⢰⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⣿⠀⠀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡏
⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠓⠶⠤⣤⣄⣀⡀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠐⠃⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠿⣅⣀⡀⠈⠉⠳⡄⠀⠀⠀⡀⠀⠀⠙⢦⣄⣀⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⣿⣀⡀⠈⠛⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠉⠙⠛⠲⠦⣤⣄⣀⢸⠀⠀⠀⠯⠭⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃

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

⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⡏⡇⠀⠀⠀⠀⠀⡏⣇⠀⠀⠀⠀⠀⢸⢹⡄⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠙⠓⢦⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡇⡇⠀⠀⠀⠀⠀⡇⣯⠀⠀⠀⠀⠀⢸⢸⡄⠀⠀⢠⡶⠶⠶⠞⠀⠘⠛⠛⠛⠃⠀⠹⠶⠶⠶
⠀⠀⠀⢸⠁⠀⠀⠀⡠⢤⡀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢳⠃⠀⠀⠀⠀⠀⢳⠏⠀⠀⠀⠀⠀⠘⡞⠀⠀⠀⢸⡇
⠀⣀⠀⢸⡄⠀⣀⠤⠇⠰⠧⣄⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⣞⠉⠳⢦⡷⠚⠷⠶⠶⠦⠀⠐⠛⢶⡃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠈⠓⢦⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⣿⢻⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⢸⠧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣷⠀⠀⠀⢸⠀⠀⠀⠀⠠⠱⠘⠘⠐⠀⠀⠀⣠⠞⠙⢦⡀⠀⢸⡇
⠀⠸⣄⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠁⣠⡟⠀⡤⠄⠚⠲⠤⡄⠀⠂⡄⠀⠃⠃⠀⠀⣴⠁⠀⠀⠀⢱⡀⢸⡇
⣀⡀⠈⠙⠓⠶⠤⠤⠤⠤⠤⠤⠴⠖⠛⠁⠀⢰⢳⣶⣶⣶⣶⣇⠀⠀⠁⡄⠀⢰⢠⠀⡇⠀⠀⠀⠀⠘⡇⢸⡇
⠙⠯⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢾⢼⢻⢿⢻⣸⠈⠀⠀⠇⡇⠸⠸⢀⡇⢀⣀⣀⣀⠀⣇⢸⡇
⠀⠀⠀⠉⠛⢷⣦⡤⠤⠤⣤⣤⣤⠀⠀⠀⠀⠸⣍⣘⣘⣙⣘⡽⠀⠁⢥⣤⣣⠄⠀⠘⠿⣽⣽⣿⣿⡿⠟⢸⡇
⠀⠀⠀⠀⠀⠈⢻⡒⠒⠒⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠘⠓⠒⠒⠒⠚⠒⠛⣓⣚⡒⠓⠛⠚⠒⠚
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠘⣆⠀⠀⠀⠀⠀⢸⠹⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠙⢦⠳⣤⡀⠀⠀⠀⠀⠀⢀⠴⠊⠉⢁⣀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣾⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⢿⣄⠀⠀⠀⡴⠁⢠⡶⠛⠉⠉⠙⠳⢦⡀⠙⣆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠸⢧⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢣⡀⠀⣼⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⢻⡀⠘
⠀⠀⠀⠀⠈⠳⣤⣄⣀⣀⣠⡤⠞⠀⠀⠀⠀⠀⠘⢤⣷⣶⣶⣶⣶⡶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠴⠃⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠹⣆⠀⠀⠀⠀⠀⠀⢀⡼⠁⢰
⠀⠀⠀⣠⠤⠖⠒⠛⡟⠛⠛⢻⠗⠲⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠛⠒⠚⠒⠶⠤⠤⣀⡀⠀⠀⠀⠳⡀⠈⠳⣤⣄⣀⣤⡴⠟⠁⡠⠋
⠀⠀⡞⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⣠⣖⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⣤⣤⠀⠈⠳⢤⣀⣀⣀⣀⣠⠴⠊⠁
⣠⡤⣿⠲⠭⣭⣖⣒⣒⣒⣒⣒⣒⣮⡭⠴⢺⡧⣤⡀⠀⠙⠛⣿⠻⠿⢿⣿⣶⣶⣶⣶⣶⣶⣶⣿⡿⠿⠿⢻⠟⠛⠀⠀⠀⢠⣤⢤⣤⣀⠀⣠⡤⢤⡄
⠉⠙⣻⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢻⡟⠉⠁⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢸⠂⠈⠙⠋⢿⠀⢀⡿
⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢳⡏
⠀⠀⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣠⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣶⣤⣴⣶⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠒⠒⠒⠒⠒⠶⠤⣤⣀⢀⣀⣀⣤⣴⣚⡭⠟⠀⠀⠀⢰⣍⣉⣉⣀⣀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣱⡆
⠀⠀⠀⠀⠀⠀⡾⠉⠀⢀⣀⠤⠤⠤⠤⠤⢤⣀⡀⠀⠈⢻⣭⠟⠋⠉⠁⠀⠀⠀⠀⠀⢸⡆⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢿⣦⣞⡁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣢⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣉⡉⠛⠓⠒⠒⠚⠛⠋⣉⡥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡤⠖⠒⠒⠒⠲⣄⣠⠞⠁

⠀⠀⠀⠀⠀⠈⣿⣷⡲⠦⣄⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⡤⣶⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡷⣦⣝⠳⣶⣿⠿⠋⠉⠉⠉⠋⢍⠛⠶⣤⡶⣻⣿⢋⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣷⡙⢷⡌⢮⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣣⢞⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡈⢿⣼⣷⣿⡌⠂⠀⠀⠀⠀⠀⠀⠀⠹⣿⢏⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⢿⣿⣧⢾⣿⣿⡿⠃⢀⣵⣦⠃⠀⠀⠀⠀⠀⢻⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡄⠉⣸⣿⣿⡅⠀⣛⡛⠟⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⣰⣿⣿⣿⣿⣦⣄⡠⠀⣰⠇⢀⡤⠂⢀⣀⣨⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣿⣿⡾⢋⣭⡀⢉⣍⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⠁⠈⢿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢹⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣞⠁⠙⢿⣦⣬⣤⣶⣶⣶⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣿⣦⡘⠻⣿⣿⣿⣿⣿⣿⣿⣶⣭⣿⡛⠋⣩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣛⣛⣛⣛⣉⣥⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠒⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠅⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡁⠒
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⠀⠀⡆
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣤⠀⠂⠰
⠀⠀⠈⠀⠀⠀⠈⠬⠑⠂⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠤⠐⠒⠒⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢠⡀⠀⠀⠀⠈⡄⠀⠈⠙⡄⠀⠀⠀⢀⠠⡎⠁⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢒⣛⠒⠠⠌⠀⠀⠀⠀⠱⡀⠀⠈⠀⠁⠆⠈⠱⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠁⠀⠛⠎⠂⠀⠀⠀⠀⠈⠢⢔⡂⠠⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠁⣤⠀⣁⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢰⠁⠃⠀⠈⠀⠁⠠⠉⠀⠀⠘⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠆⠀⠈⠀⠀⠀⠀⠀⠀⡆⢀⡄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠙⠀⠄⠁⡜
⠀⠀⠀⠀⠀⡞⠀⠀⠰⣂⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⡀⢀⠀⠀⢀⠁⠀⠀⠀⢀⠠⠨⢁⠀⠀⠀⠰⠿⠏⠠⠟⠁⣠⣚⣀⣤⢊⣀⣴⠋⠈⣵⢋⣭⠷
⠀⠀⢀⢜⠐⠦⠐⠛⠂⡀⠀⠐⠀⠀⠸⠄⠀⠠⢊⠡⠎⠀⠆⠀⠀⠀⠈⢇⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠗⡸
⠀⣠⡡⠆⠐⣊⡀⠀⠀⡂⠀⠀⠀⠀⠊⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡴⠁
⠀⠁⠀⠈⠉⠁⠀⠒⠊⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠥⠊
⠠⠔⠀⣄⡠⠆⠀⠀⠀⠀⠀⠀⠀⠊⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠭⡲⠒⠖⠖⠲⣒⠽⠁
⠀⠀⠀⠉⠉⢻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠀⠀⠀⡀⠀⠈⡌
⠀⠀⠐⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡄⣇⢀⠀⢐⠐
⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⡀⠔⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡸⠒⠠⠤⡧⠦⠀⠀⠀⠃⢃
⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠈⠀⣀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠏⠅⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠄⠠⠠⠄⠐⠀⠀⠠

⠀⠀⠀⡀⠀⠐⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠂⡀⠀⠀⠀⠀⠀⡀⣀⠀⠄
⠀⠀⠀⡒⠠⠆⡁⠀⢠⠀⠀⠐⠒⠒⠒⠀⠒⠒⠒⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠂⠐⠀⠀⠀⠄⠀⠀⠉⠀⠀⠀⠀⣸⣄⡙⢥⠄⢤⣢⡀⡀
⠀⠀⠀⣤⢄⡛⠁⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠁⠐⠆⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣠⠅⠘⢇⠈⢫⣤⠃⢠
⠀⠀⠀⢠⡀⠵⠺⠄⠂⠇⠀⠀⠀⠀⠀⢘⣄⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠹⠃⠀⠀⠠⢠⡻⣦⡀
⠀⡀⠐⠀⣡⡤⠄⢶⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡈⢀⠀⠀⠀⢀⡀⡀⠀⠀⠀⡎⠚⠀⠀⢀⡦⠊⠁⠈
⠀⢇⠤⣮⠁⠠⠀⣦⡌⠱⡈⠀⠀⠀⠀⢀⡀⠈⢡⠌⢀⡤⣈⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠚⡱⠗
⠀⠈⠀⢁⠰⠀⢼⡄⠄⠀⢡⠃⠀⠀⠠⡿⠤⠴⡴⢠⠄⠜⠻⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠈⠆⠚⣃⠈⠀⠄⠀⠀⠀⠀⠀⢢⡙⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠔⠻⠷⠒⠁⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⢛⣐⡂⠊⠀⠀⠀⠀⣠⣆⠼⣦⡤⠾⣶⠦⠂⠀⠀⠀⠰⠃⠰⠀⠀⠀⠈⠁⠪⠁⠇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⡀⠀⠀⠀⣿⡤
⠀⠀⠀⠄⠠⠔⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠂⠄⠒⠖⠒⠴⠆⢀⠀⠀⠀⠐⠛⠀⠂⠐⠉
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠀⠀⣀⣆⠀⠀⣀⠀⠀⡀⠀⢀⠀⠀⠀⡤⠁⠀⠀⠁
⠈⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠘⠂⠈⠙⣏⠁⠈⡙⠉⠉⡉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠉⠉⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠂⠒⠛⠀⠒⠛⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠾
⠀⠀⣀⢤⠤⠠⠤⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⡄
⠀⣄⡈⠂⡀⠂⡀⠘⢀⠘⢀⠀⠀⣤⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⡀⠀⠈⠁⠒⠂⠀⠀⠀⠀⠀⠈⠀⠀⢀⠤⠀⢀⡀⠀⠀⠀⠀⠀⣠⣍⣤
⠀⠈⠄⣆⣀⣶⣀⣰⣀⡐⣀⣰⠖⠀⢀⠌⠀⠀⠠⡌⠁⢀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣀⣀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢐⠀⠀⣇⠀⠈⠄⠀⣦⠀⠈⡇⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣮⠀⠀⣇⠀⠐⣧⠄⢶⣠⠀⣠⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣇⢸⠿⠀⡰⢿⡧⡐⢫⡬⡿⣻⠓⡉⠀⠀⠀⠤⡀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢄⠉⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠠⠀⠰⠷⠟⠃⠀⢄⠀⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⢐⣦⢀⣀⢠⡄⣴⠑⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢢⠛⠀⠀⡘⠁⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠁⠀⠀⠀⠠⣮⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⡜⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠀⠀⢀⠔⠀⠔⠁⠠⠀⠀⢠
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⢀⡤⣸⣧⡀⣀⣤⣶⡯⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣸⣵⠋⠀⠀⠚⠧⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢠⣀⠻⣇⠀⠄⠀⠀⠀⠀⠀⠀⢸⢦⠀⠀⠀⠀⠀⠜⠋⡁⠀⡀⠀⠀⠀⢈⠣⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠁⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠾⢬⣀⠀⠄⠀⠁⠀⠀⠀⠈⢹⠀⠀⠀⡀⠆⠁⠄⠀⠀⡉⠂⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⢛⣋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣠⢴⡿⢃⡴⢞⣿⠇⡾⢻⠟⣸⠟⢁⡴⠇⣠⠞⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣡⡾⠛⠉⠓⠋⠿⠊⠀⠛⠊⠹⠖⠉⠷⠊⠙⠖⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⣀⣠⣤⣤⣤⠤⠤⠤⢤⣤⣤⣤⣀⣀⣴⠟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢱⣤⠴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠦⣴⡁⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠘⢧⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣰⠏
⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⣴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠄⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠰⣿⡋⡇⠀⠀⠀⠀⣠⡴⠶⠚⠛⠛⠛⠛⠲⠶⣤⡀⠀⠀⠀⠀⡏⣻⡷⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣼⠥⢄⣀⣀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢠⠞⠁⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠈⠛⠁⠀⠀⠀⠀⢀⣀⡀⠤⢽⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠸⣿⣿⠇⠀⠀⠰⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⢀⣨⡷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠖⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣠⣴⠖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢷⠃⡆⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠘⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⡀⠈⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⣸⣿⡿
⠀⣇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣿⣿⢀⡀⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⢰⠃⠰⣶⡆⠀⠀⠀⢹⡟⠁
⠀⢹⡄⠀⠀⢥⣀⠀⠀⢰⠆⠐⠰⣿⡇⠀⣿⣿⠸⡧⠀⠀⠘⠛⢿⠀⠀⠙⠛⡟⠾⠆⣺⠂⠀⠛⠇⠀⠀⢀⣿
⠀⠈⣇⠀⠀⠘⣇⠀⠀⠘⡄⠀⠀⢸⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣆⠀⠀⣼⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠷⠤⣄⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢴⣎⣀⣤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀⣀⣠⠤⠴⠚⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡎⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠚⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠚⠉⠉⠀⠀⠙⢤⣀⡤⠋⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣀⣀⡤⠤⠶⠒⠊⡟⠉⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡶⠓⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠳⠤⠖⠃⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡴⠋⠙⢦⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠁⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠹⢤⣠⣞⣠⠇
⠀⠀⠀⠀⠀⠉⢲⡀⠀⠀⠣⣀⣀⡰⠃⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢹⠀⣀⣀⡤⠤⠒⠚⠉⠉
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠿⠔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⡇⠀⣀⣀⠤⠴⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠤⠄⡀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠀⠙⠃⠰⢤⠀⠀⠀⠀⡀⠀⠈⠉⠉⠉⠉⠉⣲⡀
⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠂⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⡨⠐⢾⣿⣿⣷⣬⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⣿⣿⡗⠒⠒⠒⠤⣌⡂⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⡿⠓⠤⠀⠤⣤⣀⣨⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠌⣿⠿⠿⢷⡀⠀⠀⠀⠀⣬⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠈⠀⠀⢨⠀⠀⠒⠦⢬⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣀⣀⣀⣠⣼⣤⣀⣀⣾⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠸⣃
⠀⠀⠀⠀⠀⣀⣴⣶⠾⠟⠛⠉⠉⠙⠛⠛⠛⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⢉⡄⠉⠥⠉⠉⠉⠑⣦
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⢼⣰⣆⠀⠀⠀⠀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠁⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⠂⠶⠶⠶⠒⠿⠃⠀⠀⠀⠀⠀⠀⣾⡟⠉⠁⠉⠉⠀⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⣄⢀⠄⠀⢠⣴⡅⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⣿⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⢸⣿⡿⠦⠀⠀⠀⢤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⢀⠃⠐⠂⠈⠈⢄⢰⡿⣿⣷⢶⣄⠈⠁⠂⢄⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⠛⢧⠀⠀⠀⠀⠀⣶⣿⠂
⠀⠀⠀⠀⠀⢠⠂⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢉⣙⠁⠀⠶⣾⠀⠀⠀⣿⣿⣿⣿⡀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠐⠲⠾⣿⠋
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠑⠠⠀⡀⣠⣧⢼⠀⠀⠰⠋⠀⠀⠀⣿⣿⣿⣿⣧⠀⢹⣦⣤⣤⣿⣧⣄⠀⠀⠀⠀⠀⣠⣧⣄⣤⣼⣿⡿
⠀⠀⠀⠀⠀⢠⠁⠀⢀⡀⠀⠀⠀⠀⠀⣌⠉⠙⠳⠃⠀⠂⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡭⠟⠁⠀⠈⠉⠛⠛⠛⠻⠷⠿⠿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠙⠒⠒⠂⢾⣿⡿⠖⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣾⣦⣤⣤⡮⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⣀⢀⡀⢀⣀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⣄
⠀⠀⠀⡐⠉⣧⠀⠀⢰⢻⡆⠀⠀⢻⡄⠈⠳⣄⠀⠀⠀⠀⠀⡴⠚⠓⢄⠳⡀⠀⠀⠘⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠘⠃⢈⣹⣦⣀⣿⣾⡿⠂
⠀⠀⠀⡇⠀⣿⠀⠀⢸⢸⡇⠀⠀⠀⠙⢷⣤⡟⠃⠀⠀⠀⠀⢳⡀⠀⢸⠄⡇⠀⠀⠙⠀⠀⢀⡍⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸⣿⣿⡟⡻⠿⠁
⠀⠀⠀⡇⠀⣿⠀⠀⠈⢸⡇⠀⠀⢀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠀⠙⢲⣤⡴⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠉⠁
⠀⠀⠀⡅⠀⣿⠀⠀⢰⢸⡇⠀⠀⠀⠻⡊⢀⠀⠀⢀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⣿⠀⠀⢸⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠜⠁⠀⢸⡆
⠀⠀⠀⡇⠀⣾⠀⠀⢸⢼⡇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⡔⠀⢙⣽⣿⣶⡧⠀⢰⠄⠀⠀⠠⣀⣄⣤⡤⢂⠐⠀⠀⠀⠀⠠⡄⢸⠀⠀⠀⠈⡇
⠀⠀⠀⣧⠀⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠠⠄⢰⡃⠀⠀⠚⢦⣐⣾⣿⣿⡯⠴⠶⠋⠀⠀⠀⠀⠠⠬⠼⠚⠉⠀⠀⠀⠀⠀⠀⠐⡀⠣⣤⠠⠞⡹
⠀⠀⠀⠈⠿⠋⠀⠀⠀⠿⠁⠀⠀⠐⠘⠉⠉⠙⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠰⣄⣀⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠀⠀⠀⠰⢦⣀⣀⣤⣀⣀⣤⣶⠀⠀⠒⠀⠀⢠⡶⠡
⠀⠀⠀⠀⠀⠀⠒⣠⡷⢄⠀⠀⠀⠀⠀⣀⡠⣖⣚⣯⡴⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠀⣀⢐⡶⣦⢤⢤⣤⣾⡶⠂
⠀⠀⠀⠀⣠⠶⠋⠁⡗⠸⠀⠀⠀⢠⠔⣉⠤⢋⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⡀⢤⡀⠀⠀⠀⠀⠀⠀⠁⢉⠁⠉⠁⢈⠀⠀⢀⠏
⠀⠀⠀⠀⢇⣀⡠⠊⣰⠃⠀⣰⠖⣡⡄⢁⣴⠗⠁⠀⠀⢱⣂⠀⠀⠀⠀⠀⢈⡀⡀⠀⠚⠒⠤⢬⣁⠦⡤⠀⠀⠀⠀⠄⠁⠀⢀⠃⣴⡾⠃
⠀⠀⠈⠳⠤⠤⠷⠊⠁⠀⠀⠀⠐⠪⠖⠛⠁⠀⠀⠀⠀⠀⠳⢅⡀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠾⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⡄⢼⣿⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⢿⢸⣿⣿⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠀⠀⠀⠀⠀⠚⠁⠋⠀⠋⠈⠀⠀⠀⠀⠀⠁⠂⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠡⣅⠴⠀⡤⠟⠁⠀⠀⠀⠀⢀⡔⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⣤⣤⠆⢀⣼⣾⣿⣗⠶⢤⣀⠀⠀⢸⠃⠀⠀⢠⣀⡀⠀⠀⠀⠀⢀⠀⢀⡀⣼⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠉⢀⣴⣻⣿⣿⡇⢾⡁⠄⠀⠑⠀⠘⢦⡃⢀⣾⣿⣿⣦⣄⣀⣀⡈⡿⣏⠉⠃⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⣱⡿⣿⠟⠻⡄⠑⠈⣳⡀⢰⡄⠈⣷⣿⣿⠀⢿⣿⣿⣿⠏⠅⢙⡄⠀⠄⣺⠏
⠀⠀⠀⠀⢠⡾⢁⠀⠀⠀⠀⢀⡼⠿⢀⡿⠀⣘⣿⣦⠰⣿⣿⡟⡧⠀⠘⣿⣷⣆⣈⣿⡍⢈⢀⣀⣼⠖⠛⠋⠁
⠀⠀⠀⠀⠀⣷⢺⣶⣄⣄⣴⣯⠏⠀⢸⡃⠀⣿⣿⣿⣆⣿⣿⡿⠃⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠉⠘⢿⣿⣿⣳⣿⣂⣠⡿⣇⢰⣿⣿⣿⣿⡯⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠩⣿⠹⣿⠟⢁⡟⠬⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⡒⠒⠋⠩⠉⠉⠉⠉⠉⠉⠙⠒⠒⠦⣄⡀
⠀⠀⠀⢀⡠⠔⠚⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣄⡀
⠀⢀⠔⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣤⢤⣤⠄⡤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣢⣥⠦⠤⠾⣮⣤⠞⠀⠄⠀⠀⠀⠁⠀⠀⠀⠈⢣
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸
⢻⣄⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⣤⣀⣀⣴⣿
⠈⢿⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠠⠀⠀⢀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⢤⣀
⠀⠈⠻⡌⠑⢦⣄⣀⠀⠀⠀⠀⠁⠀⠀⠒⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠘⢦⣀⠀⠉⠏⠛⠲⠲⠲⠤⢤⣤⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠳⡄
⠀⠀⠀⠀⠀⠉⠳⢦⣐⣄⡀⠀⠠⠠⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠒⢶⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⢉⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠰⡇⢀⠉⠀⠀⠙⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡀⠀⠀⠀⠀⠈⢣⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⡤⠤⠚⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠠⠔⠀⠀⠀⠰⠖⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣠⢶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⣒⣒⡶⠦⡤⠤⣴⣖⣲⡫⠝⠊⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢛⡩⠛⠹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⣿⠋⢁⡠⠒⠉⠈⢱⠀⣿
⡏⠉⡲⠤⢄⡉⠉⠛⢿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⡀⠀⠀⠀⡆⣿
⡇⠘⠀⠀⠀⠈⡡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠃⣿
⣿⢰⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣿
⣿⡀⢆⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⣿⣧⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⠉⠉⠁⠒⠢⢄
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠱⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠊⠑⡄⠀⠀⠀⡇⠀⠱⠀⠀⡇
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠢⠴⠃⠀⠀⠀⠈⠉⠁⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣀⡠⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⡀
⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠀⠀⠀⠚⠛⠃
⠀⠀⠀⠸⠀⠈⠓⣶⣤⣤⠄⠀⠈⠁⠀⠃⠄⠀⠀⠀⢀⣀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠣⢀⠀⠀⠉⢙⡀⠔⠀⠀⠀⠘⠀⠒⠀⠁⠀⠄⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⣉⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⠒⠚⠁⠐⠄
⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⠀⠈⠑⠄⠀⠀⠀⡀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡆⠁⢐⠆⠀⠊
⠀⠀⠀⠀⢀⠆⠀⢀⣖⣲⣦⠀⠀⠈⠀⠀⠈⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠁⠈⠰⠈⠤⢀
⠀⠀⠀⠀⠰⠀⠀⠘⠋⠉⠹⠃⠀⠀⠂⠀⠀⠀⠀⠀⠹⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠡⠤⠖⠁
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠄⠊⠅⠀⠀⠀⠀⠀⠑⠂⠠⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⠀⠀⠀⠀⠂⡀⠁⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠦⠄⠘⠏⣽⠀⠀⠐⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣄⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⡀⠀⠙⠀⠙⠀⠀⢰⠀⠀⠀⢀⣀⢀⡀⣀⣈⡁⣠⣤⣤⡅
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠳⣄⣀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠘⠈⠉⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠛⠉⠀⠀⠀⠀⠀⠈⠉⠛⠚⠋⠁