Bun Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠖⠈⠉⣹⡏⠁⠀⡀⠉⠛⡏⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⢻⡄⠀⠀⠀⠷⣀⠀⠐⠀⣀⡴⠃⠀⠀⠈⢱⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⡤⠈⠀⠀⠈⢧⡀⠀⠀⠀⠈⠛⠛⠉⠉⠀⠀⠀⠀⠀⣾⠃⠀⠈⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⠀⠀⠻⣦⡀⠀⠀⠀⠈⠓⠶⢤⣄⣀⣤⣄⣀⣀⣀⣤⠄⠀⠁⠀⠀⠀⢀⡏⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠹⠀⡀⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣼⠏⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠁⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠰⠶⠦⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⢀⣀⠄⠐⠉⠀⠀⠀⠰⠋⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢆⠀⠀⠀⠈⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⣰⠯⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠢⡀⠀⠀⠀⠀⠀⠉⠐⠢⢀⣠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢳⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠀⢀⡔⠲⡄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠘⠛⠛⢦⠀⠀⠓⠒⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠤⠐⠟⠋⠂⠀⠀⡾⠀⠀⠇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⣦⣀⣀⣀⠠⠤⠀⡀⠀⠒⠶⠶⠖⠘⠃⠈⠉⠉⣉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠐⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠙⣦⠀⠀⠄⠀⢀⣤⣀⡠⠚⠉⢳⣄⢀⣤⠤⣄⠀⠀⠀⢠⣸⠁⠀⠠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠊⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠈⢳⡄⠆⠀⡏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠜⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣠⠴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠀⢀⣀⣀⣠⡤⠴⠶⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠋⠉⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⢲⠴⠖⠒⠒⠒⠒⠒⠒⠒⣒⡲⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠖⠛⠋⠀⠁⠀⠀⠈⠛⠿⣶⣦⣤⣀⠀⠀⠀⠙⢿⣷⢦⣄⡉⠙⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠖⠚⣤⣴⠶⠶⠶⢤⣤⣄⣀⡀⠀⠀⠀⠙⠳⣯⣛⢶⣄⡀⠀⠉⠳⣮⡛⢦⣄⠀⠀⠈⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠁⠀⠀⠀⠈⠉⠛⠶⣤⣀⠀⠀⠉⠙⠳⢦⣄⡀⠀⠀⠙⠳⣬⡛⢦⣄⠀⠈⠙⢦⣝⠷⣄⠀⠀⣤⣈⠑⢄
⠀⠀⠀⠀⠀⣠⠚⠁⠶⣶⣶⣤⣤⣤⣀⡀⠀⠀⠀⠉⠛⢦⣄⠀⠀⠀⠈⠙⢶⣄⠀⠀⠈⠻⣦⡙⠳⣄⠀⠀⠙⢷⡈⢳⣄⠈⢻⣷⣄⠑⢦⡀
⠀⠀⠀⢠⣎⣁⣀⠀⠀⠀⠈⠉⠻⢦⣍⠙⠛⠶⣤⣀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⠻⣦⡀⠀⠈⢷⡄⠈⠳⣄⠀⠀⠹⣦⠘⢷⡄⠙⣿⢷⣄⠙⢆
⠀⠀⠀⡼⠋⠈⠉⠛⠳⠶⣤⣀⠀⠀⠉⠳⣤⡀⠀⠉⠳⢦⣄⠀⠈⠻⣆⠀⠀⠀⠀⠈⠻⣆⠀⠀⢻⣆⠀⠘⢷⡄⠀⠘⢷⡀⠙⣦⡘⢷⡙⢷⡈⢳⡀
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠈⠻⣦⡀⠀⠀⠉⠳⣦⡀⠘⢷⡀⠀⠀⠀⠀⠘⢷⡀⠀⢻⡄⠀⠀⠙⢦⡀⠈⣷⠀⠈⠻⣌⢻⡄⠻⣦⠱⣄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡈⠻⣄⠀⠀⠀⠈⠻⣦⡈⢷⡀⠀⠀⠀⠀⠀⢻⣄⠀⣿⠀⠀⠀⠀⠙⠶⣼⠀⠀⠀⠈⠻⣷⠀⠈⠁⠘⡆
⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣜⣧⠀⠀⠀⠀⠈⠻⣾⡇⠀⠀⠀⠀⠀⠀⠙⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⢹⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⡻⠇⠀⠀⠀⠀⠀⠈⣀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣤⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀
⠀⠀⠀⣿⠀⠀⠈⠫⡳⡄⠀⠀⣠⠶⢤⡀⠀⡴⠋⠉⢂⠙⣆⠀⠀⠀⠀⢰⠟⠌⣱⠖⠉⠺⡇⠀⢀⡾⢅⢳⠀⣠⠎⠀⠈⠈⢧⠀⠀⡤⠶⢤⡀⢀⡏⠢⡼⠁
⠀⠀⠀⢻⡀⠀⠀⠀⠡⡇⠀⢀⡇⠈⢄⣇⣠⠃⠀⠀⠀⠃⢸⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⢆⠳⣤⠞⠀⠘⠐⠛⠁⠀⠀⠀⠑⠨⠷⠚⠁⠈⠐⠊⠉⡀⢸⠁
⠀⠀⠀⠘⢧⠀⠀⡀⠀⠙⠶⠊⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠤⢳⣄⣀⡤⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡠⣔⣴⠆⠞⠀⢸
⠀⠀⠀⠀⠈⠻⢥⣂⠭⢂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠄⣠⠶⠖⣭⡄⡂⣔⣖⡠⣀⢄⠤⠦⠒⠓⠡⠚⠁⠈⠈⠈⠀⠀⢀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢬⡔⠪⠂⢀⠔⡠⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢢⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠮⠮⣤⣊⡄⠪⠐⡠⢐⠠⢀⣄⠀⡀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⢀⡀⢀⢀⢀⠄⠰⠖⣔⣠⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠙⠚⠿⠷⠾⠶⢵⣬⣦⣵⣬⣴⣥⣦⣾⣏⣴⣯⡴⠡⠦⠖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠒⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠈⠛⢶⡤⣤⣀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠶⢿⡁
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣀⣀⣽
⠀⠀⠀⠀⠘⠷⠶⠶⠶⢶⡄⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⠶⠦⣤⡤⠶⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣷⣼⠃⠉⠛⠛⣳⠀⣟⣻⠶⠟⠛⠛⠿⣦⠸⡇⠀⣠⡴⠶⠛⠛⠓⠶⢦⣄
⠀⠀⠀⠀⠀⣰⡏⠁⠀⠈⠉⠙⠳⣤⡀⠀⢿⣠⡿⠁⠀⠀⠀⠀⢰⡏⠀⣿⢸⡁⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⢿⡷⣄⠀⠀⠀⠀⠀⠀⢻⡤⢾⡁⠀⠀⠀⣀⣠⢾⣿⠷⠶⠛⠛⢿⣶⠤⣤⣤⣤⡤⣶⡿⠇
⠀⠀⠀⠀⢀⣤⠟⠾⣽⣲⡶⢤⣤⣴⡿⠀⠘⠿⣿⣿⣿⠽⠞⠋⠁⠀⠀⠀⠀⠀⠉⠙⠛⠚⠛⠛⢿⡄
⠀⠀⢀⣴⣯⣤⣀⣀⣀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⠿⡆
⠀⠀⠈⠛⣲⣶⠶⠭⣭⣽⣛⣲⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣞⣻⡭⠷⠶⢶⡛⠛⠛⠁
⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠈⠉⠙⠓⠮⣽⣳⢦⣄⡀⠀⢀⣠⡴⣞⣯⠵⠞⠋⠉⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢹⡀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠉⠓⠢⢍⣉⡩⠵⠛⠉⠀⠀⠀⠀⠈⠳⡀⠀⠀⢠⡏
⠀⠀⠀⠈⢷⣆⠀⠈⠀⢀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⡀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠈⠙⠓⠲⠾⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⠤⠴⠶⠷⣾⣋⣉
⠀⠀⠀⠀⣠⡴⠶⠖⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄
⠀⠀⣴⣟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣬⣷⡀
⠀⠘⠷⣭⣭⣭⣙⣛⣒⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⣚⣻⣭⠭⣷⡶⠶⣾⣇
⢀⡴⠋⠉⠙⠛⠛⠛⠛⠿⠿⠶⣮⣝⡳⢦⣄⡀⠀⠀⠀⢀⣤⠴⣚⣭⠷⠞⠛⠻⠗⠒⠋⠉⠀⠀⠹⡎⠻
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⣽⣓⢒⣺⡽⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠸⡇
⠹⣮⣛⣁⡤⠶⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⢀⡴⠶⢦⣭⣤⠞⠃
⠀⠀⠉⠁⠀⠀⠹⣆⠈⠛⠛⢁⣤⣤⣤⣄⡀⠀⠀⠀⠀⣀⠀⣀⣤⣤⣄⡈⠉⠉⣡⡞⠁
⠀⠀⠀⢀⡤⠶⠖⠛⠛⠓⠛⠋⠉⠉⠉⠉⠙⢦⣀⣀⣈⡥⠞⠉⠉⠉⠉⠉⠛⠛⠛⠓⠶⠦⣄
⠀⠀⠀⢸⡟⠒⠶⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠤⠴⠶⠚⣿
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⠶⠞⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣠⣶⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠿⠻⠟⢛⣛⠛⠿⢻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣛⠛⢿⣿⣉⣩⣶⣞⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠿⠿⢍⣉⣃⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣫⣿⣿⣿⣿⣿⠛⠿⢯⣴⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣾⣿⣿⣿⣿⡿⠍⠐⣲⣿⣿⣿⣿⣿⣿⣿⣟⡋⠛⢛⠛⡛⠛⠻⢿⣿⣿⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⠟⣋⣉⣩⣽⣿⣿⣿⡿⠿⠿⣉⠁⣠⣑⣤⣉⠻⣿⣿⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⢧⣵⣦⣤⣤⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠲⣶⣾⣿⣿⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⡿⢿⣿⣿⣷⣶⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢟⣫⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡻⢝⡻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣿⠟⣿⣟⣷⣿⣿⣿⣏⣿⣫⡛⠿⣿⣿⢿⠿⣿⢿⡿⢿⣮⣿⡆⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⣴⠛⠁⠊⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣼⣷⣿⣾⣿⣏⢿⣿⣷⠄⠢⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⢪⠟⠁⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣯⢣⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣽⣿⠋⢀⠔⠀⣴⢫⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣮⠣⢣⡀⢆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡳⠀⠀⢠⡾⣽⠂⣹⣿⡟⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡃⠀⠡⢹⣯⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡳⣱⣣⡿⣽⢃⣴⠋⠋⣠⢏⣿⢻⣿⢿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠳⡀⡀⢋⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣽⣷⣿⣿⢣⣾⠁⣠⠾⣉⠟⢀⡴⣡⠏⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⠠⢹⠎⣰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡿⢣⣞⣡⡞⠁⠠⢊⠞⡁⠀⢟⡿⣻⣿⣿⢿⣿⣿⣿⣿⣿⡟⠏⠹⡿⣇⢫⣿⣥⡿⣿⣇
⠀⠀⠀⠀⠀⠰⢧⡿⣿⣿⢿⣿⣿⡿⣫⣴⣿⣟⣋⠄⣠⣔⠅⠂⠀⠠⢋⠼⣫⡞⣻⣿⣿⣿⣿⡿⠁⠈⠀⠀⠸⣿⣾⣽⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⢛⣿⡏⣾⣿⣫⣾⣿⣿⣿⣫⣵⢞⣩⡵⠊⠀⠀⠀⢡⡼⣫⡾⡿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⢸⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣁⣤⢀⣤⠞⡁⠚⠁⠈⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢸⡏⢸⣿⣿⣛⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣉⢀⡀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⣾⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡵⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣏⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⡇⡇⢸⣿⣿⣿⡿⠋⠀⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠹⡏⣿⡇
⠀⠀⠀⠀⠀⠈⣷⣸⣘⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⣶⠖⠀⠲⢦⣤⣀⣀⣀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⠄⠼⢿⣿⣽⣰⣦⣿⡿⠿⡟⢿⣿⣿⣁⠃⠹⢷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣏⡣⠶⣿⣦⣌⠙⠻⢻⣄⠁⡀⠀⢡⡈⠻⣗⡀⠀⣀⣽⣿⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠃⠀⠀⠀⠀⠙⠛⡿⣿⣤⡙⢿⣿⣷⣿⣿⣿⣄⣀⣿⠿⠉⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⠂⠀⠀⠀⠀⠀⠀⡰⡴⠻⠻⢿⣷⡄⠛⣿⣿⠛⠉⠉⠛⠀⠀⠀⠀⠀⠂⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⢶⣤⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣗⢠⢠⠀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠊⠘⠚⠂⠃⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⠛⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠳⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣥⣽⢶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⢙⣿⣿⣶⣶⣶⣦⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠿⢿⣿⣿⣯⣴⣅⡈⣄⣠⣤⣬⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣠⣾⣿⣿⡟⡉⠋⠻⠊⠉⠁⠀⠀⠙⠏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⢦⣾⡏⠁⣼⣿⣿⣿⡿⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⣞⣿⣿⣾⣿⣿⠏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠲⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇
⠀⠀⠀⠀⠀⠀⠘⠦⢬⠳⡏⠉⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢷⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠶⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢀⠜⠋⡩⠀⣀⡴⢔⡂⢤⣌⠧⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠊⠀⠏⠀⠈⠀⡔⠁⢀⠀⠀⠀⠀⠑⠢⡀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢴⠊⠑⢦⡐⢀⠁⠀⠁⠀⠀⠀⠀⠠⠀⠘⡄⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡀⢠⠈⢦⡀⠀⢃⠸⢀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⠢⡀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠇⠈⠀⠈⠆⠀⠙⢦⣸⡇⠀⠤⠀⠄⠐⠂⠁⠀⠀⠀⠘⠀⢡⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠊⡀⠤⠀⠀⠀⠀⢠⠁⠠⠀⡈⢄⠀⠀⠈⠓⠦⣀⠀⠀⠀⢀⡠⣀⠔⢠⠀⠀⢨⠁⠢⡘⡀
⠀⠀⠀⠀⢀⡤⠚⠁⢀⡠⠄⠒⠂⠉⠁⢫⠀⠡⠐⠈⢆⠁⠀⠀⠀⠀⠉⠑⠛⠤⢤⣤⠒⠁⠐⢦⠘⠀⠀⢉⡇
⠀⠀⢀⠔⠁⡀⠀⠈⠁⡠⠊⠀⢀⡀⠄⠰⢆⢄⠁⠀⠀⠱⢄⠀⠀⠁⠐⠀⠀⡀⠀⠈⢣⢰⠀⠀⠳⡄⢀⡸⠇
⠀⠀⠄⠀⢀⠀⠀⠐⢀⠤⠒⠉⠀⠀⢀⡠⠚⠢⡑⢄⠀⠀⠈⠑⠀⠀⠀⠐⠀⡈⢂⠑⠀⡌⠀⠀⠀⢋⠁⡸⠁
⠀⠀⣄⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠢⣁⠀⠀⠀⠀⠀⣀⡤⠆⠀⣀⡠⡞⠀⢀⠂⡀⢸⠊
⢀⠀⠁⠔⠀⠀⠀⠀⠀⢐⡠⢚⠏⠀⠀⡠⠀⠀⠀⠀⠈⠉⠙⠫⡙⠒⠁⠒⠒⠋⢉⠴⠥⠐⢁⡴⢀⡆
⡆⠔⠀⠀⠀⠀⣀⠤⠊⠁⢠⠋⠀⠀⠌⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠁⠒⠲⡖⠞⠥⣀⣀⣈⡡⠴⡉⠃
⠈⠄⢀⡠⠰⢎⠁⠀⢀⡴⠁⠀⢀⠊⠀⢰⠃⠀⠀⠀⠀⡄⠀⠀⢀⡆⠀⠀⡇⠀⠀⠆⠀⠀⠀⠇
⠀⠀⠈⠡⡀⠀⠉⠉⠁⠀⠀⡠⠃⠀⡠⠃⠀⠀⢀⠀⠊⠀⠀⠀⡰⠀⠀⢠⠁⠀⠀⠀⠀⠔⠀⠃⠀⠿
⠀⠀⠀⡟⢄⠢⠀⠀⢠⠄⠈⠀⠠⢎⡀⢠⠴⠊⠁⠀⠀⡀⡠⠊⠀⠀⡠⠃⠀⠏⠀⠐⠁⠀⡘⢀⠜⠀⡄
⠀⠀⠀⢀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠱⡀⢣⠀⠀⠀⠀⠀⠀⣀⠤⢊⠄⠊⠰⠀⠀⠀⠠⢊⠔⠁⠀⢀
⠀⠀⠀⠈⡄⠀⠀⠀⠢⢄⠀⠀⠀⠀⠀⠘⡄⢣⠀⠉⠉⠉⠩⠄⠂⠁⢊⠀⠁⠀⠀⠈⠔⠀⠀⠀⠀⡌
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⢠⠀⢣⠀⠀⠀⠀⠨⢄⠐⠁⠀⠀⠀⠀⠌⠀⠀⠀⣀⠞
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢸⠀⠀⢣⠀⠀⠘⠤⠀⠁⠀⠀⠀⠀⠂⠠⠤⠐⡺⠃
⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡈⠀⠀⠀⠣⠀⠶⠲⠠⡰⠁⠀⠀⠐⠒⠂⠂⠉
⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠈⡗⠤⠤⠂⠁⠀⠀⠀⠀⠀⠀⠄⠈⠢⡁
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠲⠒⠒⠈⠙⢄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣠⢀⠀⢀⣠⣾⣹⣿⢿⣿⣿⡿⠫⠋⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡑⠤⣈⠻⣿⢦⡽⣿⡎⣿⣿⣀⣴⣶⡶⢿⣶⢶⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣙⢷⣮⣑⣌⠻⣿⣿⣿⣿⣿⡿⣿⣅⢧⡈⠸⡈⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠑⠄⠙⠌⢦⡈⡻⣿⠿⣿⣳⣬⡛⡳⢬⡱⣌⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣼⡷⠎⣠⣽⣶⢈⣷⣏⣛⢻⣷⡬⢢⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢃⠔⠋⣼⣿⣻⣿⣯⣿⣿⣿⣴⣏⣿⣍⣽⣝⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡾⢨⣴⣾⣫⣾⢡⡞⣽⢋⣿⣹⣽⢿⡟⣿⣶⡟⡴⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡖⣿⣿⣇⣿⣏⣼⣱⣿⣼⣿⣿⡏⢸⣇⢻⣿⡟⢦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⠉⠁⠉⠉⠉⠉⠛⠙⠛⠛⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢋⡿⣂⣠⢾⡿⣦⢤⡾⢉⡿⠶⠡⡿⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⢉⡴⠙⠙⠿⣿⠃⠀⢷⢏⠞⢄⡞⣡⢏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⡰⢃⣆⡼⠃⠀⠀⠈⠻⣴⣟⣼⢣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⢸⠀⣿⣸⡌⠀⢸⣠⣡⠀⠀⡇⣇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠃⠘⠑⠀⠀⠀⠚⠀⠀⠃⠋⠘

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣷⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⢿⡿⠻⣿⣿⣿⣿⣿⣿⣝⡄
⠀⠀⠀⠀⢀⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⠀⠀⣀⣤⣤⣤⣧⣿⣇⣼⣿⣿⢛⡿⢿⣿⣿⣹
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣶⣾⣇⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠻⣟⢯⣿⣿⠿⣛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢻⣿⣯⣿⣿⣛⡽⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣇⢸⣿⣿⣿⣿⣿⣿⣟⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⣿⣿⣿⢟⣽⣿⣿⠟⠋⠙⢿⣿⣿⣿⣟⢿⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⢟⢿⣿⣦⣳⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢎⢿⣿⣿⣷⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣼⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⣿⣿⣟⡟⠿⢿⣷⣤⠀⠀⠀⢀⣴⣾⠿⢿⣿⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⠀⠀⠀⠈⠙⠀⠀⠀⠘⠋⠀⠀⠀⠻⣷⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣀⣤⡄⠀⠀⠀⠀⣤⣤⣀⣀⣀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠿⠻⠟⠋⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠛⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠀⢂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⢸⣥⣴⡿⠉⣿⣷⣿⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡖⣆⣾⣿⣧⡀⢸⣿⣿⡃⣰⢻⣿⣿⣿⣞⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣧⣿⣿⣿⣿⣧⠈⣿⣿⢳⣿⣿⣿⡸⢻⣿⣦⣿⣷⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢻⢸⣿⡿⣿⣧⡹⣆⣼⣷⡿⢿⡻⠛⠋⠛⠋⣀⢁⣥⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⣸⣆⠘⣎⡙⣮⡉⢿⣿⣿⣿⡴⣯⠚⠑⠀⣀⣐⣴⢎⣿⣋⣤⠂⢰
⠀⠀⠀⠀⠀⠀⠀⢰⣟⣼⡿⠻⣦⡞⣛⣻⣿⣿⡿⣿⣏⡙⢄⣂⠀⣨⣟⡛⢿⣿⠛⠩⢄⣴⣷
⠀⠀⠀⠀⠀⣠⣄⠺⠿⠃⡀⠀⠀⠸⠛⣻⣿⣿⡀⠈⠁⣠⡼⠗⣾⣻⣿⡏⠰⣿⣦⣶⣾⣾⠋
⠀⠀⠀⠠⠶⠟⠁⡈⣟⢩⣀⠤⠀⠐⠺⢿⣾⠿⣅⣰⣷⣿⣟⣧⡛⢖⠲⢤⡛⠹⣿⣙⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⡆⠀⣴⢿⣦⡤⠀⣼⣷⣌⡙⢻⣯⣿⣶⣞⠋⠠⡘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⡞⡿⣇⠀⣾⢁⣜⣀⠀⣹⣍⣛⠛⠻⢿⣿⣄⣉⣷⣍⠻⠷⣴⣬⡾⢿⢷⡀
⠀⠀⠀⠀⠀⠀⠸⠀⢳⡅⠀⠸⡈⣿⡿⠿⠿⢛⠻⠷⠤⢀⡐⢮⣁⣿⣯⣧⡀⣿⡿⣯⣼⠘⡷
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣆⣳⡝⣌⡿⠶⠀⠈⠛⠷⠂⠤⡽⠿⠿⠿⡿⣿⣿⣿⡗⣽⠃⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⣭⡐⣶⡿⣷⣬⣆⠀⠀⠀⠀⢤⣀⠓⠀⣀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⣿⣿⠗⣀⠄⠀⠀⠀⠀⠀⠀⠙⠲⢼⡿⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⣿⣿⣷⣶⣤⣴⣾⡿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣿⠋⣾⡿⣿⣿⣿⢫⣾⢋⡶⢟⣯⠭⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡿⢡⠐⣯⣾⢁⡟⣯⣿⣷⡟⠱⠋⠀⣀⠁⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⡞⣄⢻⣿⣸⡇⢸⣿⣿⡗⢀⣾⣟⣻⣿⣷⣼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣿⣳⣿⣷⣼⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡙⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣤⣤⣄⡼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡧⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⢦⣍⠙⢯⣍⡉⠱⢦⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻⠏⠉⢿⣿⣿⣿⣷⣤⣤⣀⣚⣻⣿⣿⣿⡞⠙⠂⡈⣛⠷⣌⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⡿⢷⠒⠛⢻⣿⣿⣷⣦⣉⡉⠉⠛⠛⣿⣿⣶⣄⠘⣯⡆⢈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣻⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣷⣿⣷⡸⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣿⣿⡿⡿⢋⡞⡟⣿⡿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢊⣥⣻⣿⠏⣼⣴⡟⢠⡜⢹⢧⡿⢁⣿⣇⢹⡇⣿⣿⢹⡟⣿⣿⣿⢯⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⣾⣿⣿⢿⣸⣿⣿⠁⢸⡇⡆⣼⡇⢸⣿⣿⣸⢷⢹⢸⣼⡇⢹⣿⣿⡌⢢⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾⣿⣿⣿⢸⣿⣿⣿⣴⣾⡟⣇⣿⢹⣼⣿⣿⣿⢸⡘⢈⢸⣷⠀⣿⣿⣿⡇⠈⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⢿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣇⣿⣾⣿⣿⣿⣿⣿⣿⢸⣻⣿⣦⣿⣿⣿⣧⠀⣿⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⡁⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣷⣮⣻⡟⠉⢛⠋⠈⠈⠉⠛⠁⠉⠋⠺⠿⠋⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡘⣿⣿⣿⡿⣿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡍⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠉⠉⠙⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠀⠀⠀⠀⠹⢦⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⢀⣀⡤⠞⢋⣁⡤⢤⣘⡳⠦⣄⡀⠀⠠⠌⠙⢓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⢿⣏⣙⠓⠲⠤⠵⠞⢉⡥⠖⠚⣉⠀⠐⠓⢀⡍⠓⠦⣍⠓⠶⠖⢋⣩⠴⠛⠉⠛⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠐⠂⣘⡵⠛⣀⡤⣄⡘⠳⢤⡀⠈⠋⢀⡀⠀⢀⣀⡤⠖⢋⣀⣤⡀⠻⣄⠀⠀⠀⢀⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⣠⢞⣡⠖⠋⠁⠐⠂⠉⠓⠦⣉⠓⠦⠬⠷⠚⢉⡡⠔⠊⠉⠀⣄⠈⠓⢮⣳⡄⠀⠛⠃⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⣿⣍⣙⣳⣶⣤⠀⠘⠛⠩⠤⠀⠀⠋⠀⢀⠀⠀⣀⠼⠓⣀⣤⣀⠻⣅⠀⠀⠛⠃⠀⠉⠀⠀⣀⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⢞⡡⠖⠋⣁⠀⠈⠙⠺⠷⣄⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠶⠖⠒⠚⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⡛⣉⣝⣛⡟⣫⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⢲⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠴⠆⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢄⠀⠘⡣⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠂⢴⡈⠐⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠉⠀⣍⠉⠀⠀⠀⠀⠀⠀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⢈⣦⠁⠀⠀⠀⠀⠀⣀⠄⠲⠼⣠⠋⡐⠠⠄⡀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡖⠈⠀⠀⡀⠀⠀⠀⡀⠀⠀⡈⠁⡓⢴⠉⠑⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠁⠀⠿⠀⠀⠐⠀⠰⠔⡌⢡⡄⠠⠅⠀⡀⢅⠀⡈⠁⠀⠀⠀⠀⠀⠦
⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⢔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠁⠐⠒⡆⠀⠀⠁⠀⠀⠑
⠀⠈⢠⢈⠤⡀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠀⡰⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠒⡃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠤⠦⠄⡈⠀⠀⢀⠙⠻⡀⠀⠀⠀⡰⠁⡁⢴⢔⡄⠀⠀⠀⠀⠘⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠹⣄⢀⡔⠁⠀⠀⢐⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡀⠄⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⢴⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠴⠄⠠⠤⠴⠴⠦⠤⠴⠤⠤⠄

⢀⣴⠟⠻⣆
⣼⠇⢀⡾⠃
⣿⠀⣿⠁
⢻⣆⣿⣀⣀⣀
⢀⣿⠋⠉⠉⠙⠻⣦⣀
⢸⡇⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⣀⣀⣀⣠⡴⣷⡀
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣠⡿⠁
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡖⠒⠒⠒⠛⠻⣦⣀
⠀⠘⣷⡀⠀⠀⣾⠲⠶⠤⢤⣤⡽⠆⠀⠀⠀⠀⠀⠙⠷⣄⡀
⠀⠀⠈⢷⣄⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠙⢷⣄⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠉⠻⣮⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠶⢶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣖⢀⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⠶⢦⣄⠀⠀⢀⡀⠀⠀⠀⢰⣿⣦⣴⡇⣥⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢱⠁⠀⠀⠀⠀⠀⠀⢰⣾⣶⡆⣾⢫⣶⡿⠛⢿⣷⣝⣷⡀⣿⡇⠀⠀⠀⢸⣿⣿⢡⢳⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⣾⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢸⡏⣿⣿⣧⠀⣾⣿⣿⢸⡇⢻⡇⠀⠀⠀⠈⣿⣿⢸⠸⣿⣇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⡹⣿⣿⣆⢻⣿⢟⣾⠃⢸⡇⠀⠀⠀⠀⠘⣿⣼⡀⠈⠿⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠻⣮⣭⣟⣯⣵⠿⠃⠀⠈⠁⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⢠⡰⡄⣎⡦⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢰⠼⣰⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣶⠆⡦⡖⠂⢀
⠀⠀⠀⠀⠀⢀⡾⠃⠁⢀⠠⡄⠈⠈⢳⡄⠀⠀⠀⠀⠀⠀⣴⠜⠚⢁⡠⢆⠉⠒⠡⣆⠀⠀⠀⠀⠀⠀⢠⠙⠟⠁⣀⠤⡉⠀⠋⢥⡀
⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⣿⠟⠀⠀⠀⢻⡄⠀⠀⠀⠀⣸⠃⠀⠀⢸⣿⡟⠃⠀⠀⠸⡇⠀⠀⠀⠀⢠⡏⠀⠀⠀⣿⡿⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢻⠀⠀⠀⠘⣿⠠⠀⠀⠀⢀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⣷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠹⣧⠀⠀⢠⡟⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠘⢿⡄⠀⢀⡼⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⢻⣆⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣨⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⣥⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⣡⡴⠞⠁



⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣶⣶⣄⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢟⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣾⣿⡿⢟⢿⡿⣛⣿⣿⣧⡀⠀⠀⠀⠀⠀⣴⡿⢟⢿⡿⣛⣿⠿⡻⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⣼⣿⣿⡇⠀⣼⡀⠠⣾⣿⣿⣇⠀⠀⠀⠀⢰⣿⡇⠀⣼⡄⠠⣾⠀⣠⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⣷⡀⢿⣧⠘⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣷⡀⢿⣧⠐⣿⣆⠻⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣬⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣬⣿⣷⣼⣿⡟⠀⠀⠀⠀⠀⠀⠹⣿⣿⣬⣿⣷⣼⣿⣦⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠐⠀⠀⠀⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⡰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢿⠧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⣀⣀⣀⣀⡀⢁⠶⠭⣓⠦⡀⠀⠙⠔⢅⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠖⠀⠠⠖⠛⠉⢉⡔⠣⢌⢙⠣⣍⡒⢌⠳⡄⣄⠀⠀⡀⡀⡢⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⠀⠴⠐⠁⠀⠀⠀⠀⠀⠑⠰⢯⡈⠑⠠⣁⠡⡄⢉⠀⢣⠙⣌⠳⣄⠀⠘⠄⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠑⠈⠢⠑⡇⠘⡄⠠⡀⠑⢬⣑⠲⣄⠀⠂⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⡀⠀⠘⡀⠡⡘⢤⡘⠂⡀⡀⠂⢤⡑⣌⢣⠁⡀⠈⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠂⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡁⠑⣄⠳⡜⢆⠱⢌⠢⢤⡂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⡙⢅⠈⢢⠙⠀⡑⠦⡽⠊⠀⢦⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠣⡀⢡⠈⠢⢌⡀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⡐⠀⠌⠀⠀⠀⠈⠃⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢆⡈⠑⢢⠍⠀⠀⠀⠀⠐⠀⠑⡀
⠀⠀⠀⠀⠀⠌⠀⠘⠄⠀⠀⠀⠀⠠⠀⢳⡀⢀⠀⠤⢤⠤⠂⠀⠲⠂⠰⠂⠐⠢⠄⢀⣀⠀⠀⠀⢠⠏⠢⠌⠛⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠄
⠀⠀⠀⠀⠌⠀⡐⠀⠘⡀⠀⠀⠀⠀⡡⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠰⣏⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⢂⠈⠂
⠀⠀⠀⠐⠀⡐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠈⢄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣄⢰⡄⣿⡗⠃⡀⠀⠀⠀⠀⣀⣤⠢⣤⣀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⡰⠼⠷⢾⣶⣿⣿⣷⡄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⢀⠀⠀⠀⢠⣀⣄⡀⠀⣀⣠⡖⠀⠘⡟⠢⢀⣀⠀⣀⡀⠙⡇⠀⠀⠀⣄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⣠⣹⣤⡾⢆⠀⠀⠀⠀⠈⠁⠁⠉⠀⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠈⠀⠀⠺⣧
⠀⠀⠀⠀⠀⠀⠀⡴⢦⣀⣀⣀⣼⠟⣦⡀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠠⢤⠔⠋⠀⠀⠀⢠⣄⣀⣠⣿
⠀⠀⠀⠀⠀⠀⢀⠙⠒⠚⠈⠡⠔⠚⢋⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠉⠉⠉⠭⣶⡄
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠠⢾⣿⡯⠀⠀⠀⠀⠠⠄⠀⠘⢲⠶⣆⡀⠐⠒⠄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠚⠛⠉⠀⡀⠀⠀⠄⡀⠀⠀⠀⠀⢠⡴⠿⠓⠠⠶⢾⡦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠢⠅⠀⠈⠁⠀⠂⠁⠀⠀⠠⠊⠀⠀⠀⠀⠀⠈⠿⠟⠃⠀⡀⠉⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⣠⣀⡀⠀⢀⡀⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⣶⣶⢤⣀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠥⠌⠁⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠐⣤⡺⣿⣿⠗⠿⡆⠀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⡀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠓⠖⡠⠀⠉⠀⠀⠀⠀⠘⠏⠈⠙⠉⠙⠾⢿⡿⠉⠀⠀⠀⠀⠀⠰⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢬⡠⠈⠁⢀⢈⣁⡠⠁⠰⠄⡀⠀⠀⠀⣼⡇⠉⠁⢂⣤⣶⣾⣿⡇⠀⠀⠀⣠⣤⡘⠢⣶⣶⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠺⠁⠀⠉⠃⢳⡾⡟⠀⠀⠀⠠⠤⠝⠻⠋⣯⡝⠚⠛⣊⡵⠀⠀⠀⠈⠷⣸⣦⣄⡈⠉⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢠⡀⢰⢀⣴⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠐


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠠⠒⠒⠢⢄⠀⠖⡙
⠀⠀⠙⠖⠢⣄⠀⠀⢻⠸⡠⠀⢀⣤⣀
⠀⠀⠀⢨⠀⠈⠳⡀⠀⣯⣁⠻⣄⣀⠀⠀⢀⣀⣠⣤⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀
⠀⠇⠀⠈⠉⠐⠲⠽⠦⡏⠙⡿⣛⠭⣛⡛⠋⠉⠀⠀⠤⠄⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⢘⢀⣀⡠⠒⠒⠢⠔⢻⠹⡈⠢⡉⠙⠛⠫⠍⡑⠒⠢⢄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠈⠁⠀⠑⠉⢉⡇⠀⠈⡄⠁⠀⠈⠒⢄⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⣠⡞⠀⢰⠀⡇⢀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⣷⡀
⠀⠀⠀⢠⡞⠁⠀⠀⠃⠀⡅⠈⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⢷⡀
⠀⠀⣠⠏⠀⠀⠀⡀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⣧
⠀⢰⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠰⡀⢻
⢀⡟⠀⠀⠀⠀⠠⠈⠀⠀⡇⠀⠀⠐⡀⠀⠀⠀⡀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡆⠀⢇⢸
⢸⠃⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⢰⠀⢸⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢧⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠸⡄⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⣆⠀⠀⠀⠱⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡇⠀⡿
⣿⠀⢀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠘⣆⠀⠀⠀⠱⡀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣇⠀⡇⣸⠇
⢻⡀⠀⡀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠘⣆⠀⠀⠀⠈⢆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⣷⡟
⠘⣇⠀⠱⠀⠀⠄⠀⠀⠀⠀⠘⢄⠀⠈⢦⠀⠀⠀⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣷⡟
⠀⢹⡄⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⣇⣼⠏
⠀⠀⢻⣄⠀⠱⡀⠀⠀⡀⠀⠑⡀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠘⣄⡴⢿⣟⡢⠄
⠀⠀⠀⠙⢦⡀⠙⣄⠀⠈⠢⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣆⣠⠾⠋⠅⠂⠑⠀⢄⢀
⠀⠀⠀⠀⠀⠙⢦⣌⠳⢄⠀⠙⢤⡀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢳⣠⡴⠟⠁⡄⡄⠲⢤⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣭⣒⣀⡉⠢⢄⡈⠑⠲⠤⣄⣉⣐⣒⣤⣤⡴⠾⠛⠉⠉⠃⠠⠘⢀⡆⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣿⣿⣧⢀⣼⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣏⠾⠁⡉⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢛⣵⣿⣿⣿⡟⠻⠧⠶⠾⠟⠉⠿⠛⢋⣉⠀⠀⠀⠉⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠘⣟⡻⢿⡅⠸⣿⡿⠿⠁⠀⠐⠒⠀⠠⠄⠀⠀⢰⣿⣿⣷⣾⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣿⣽⣿⡆⠻⣦⡈⠡⣤⣠⣤⣤⣤⣤⣤⣀⡀⠀⠈⠙⠿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢿⡿⠛⠁⠰⠿⣁⡜⡛⢹⣿⣿⣿⣷⣿⣿⡟⠀⠀⠐⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣼⢀⣤⢀⣴⣿⡿⠄⠇⢸⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡟⠙⢿⣿⣶⣶⠶⠿⠿⣭⢯⣥⣽⣿⣀⠀⠀⠀⠀⠹⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠋⣀⠈⣿⠏⠄⢂⠀⠁⠊⠀⠀⡉⠛⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣷⠀⣧⡾⠋⠠⡀⠈⠉⠁⠀⠀⠀⠐⠁⠷⢬⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣙⢥⠏⠅⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠠⣦⡄⠀⠈⠂⢀⡀⠀⠁⠀⠀⠀⢀⠀⠠⠄⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡇⣾⡇⠀⣿⡟⠀⠀⠀⠀⠙⠳⠦⠴⠆⠒⣿⣷⡄⠀⠈⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠃⠀⣿⣷⠹⢤⣄⣥⣀⠀⠀⠀⠰⣶⣿⣧⠉⢀⠂⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣶⣿⣿⣶⣾⣿⣇⢐⡷⠶⣶⣿⣧⣾⣿⡆⡀⢺⣜⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⢧⣷⣧⣹⡿⢿⣿⣿⣶⡷⠲⠶⣯⣽⣿⣿⣯⣇⠸⠊⠩⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣉⣀⡀⢸⠓⢼⡛⡂⠀⠐⠛⠿⠿⢿⠏⣿⠠⡀⠨⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠘⠗⠛⢀⠿⠆⠒⠃⠀⠒⠶⣤⠀⠀⠀⠀⠈⢲⠙⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣀⣁⡀⠀⠀⠀⠀⢀⠀⣿⣤⣠⣤⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠃⠀⠀⠀⠚⠳⠿⠟⠻⠟⢅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡠⠀⡀⠀⢀⠢⢤⡀⠀⠀⡏⠀⢻⢀⡤⣠⠊⠉⠀⣀⠤⠄⠀⢞⢡⠀⠑⢆⠀⠐⠤⡀⠀⠀⡞⢳⡄⠀⢠⠟⠓⠦⣔⠒⠓⠦⣄⣀
⠀⠀⢀⠰⢀⠑⣄⢻⠀⡠⠓⠀⠀⠈⠓⠁⠸⡀⣀⠇⠀⢸⣇⠀⠀⠀⠈⠉⠁⠉⠁⠤⠤⠒⠳⡄⠀⣧⣸⠃⢀⡟⠀⠀⠀⠈⠙⠒⠶⢤⡈⠉⠉⢱⡀
⠀⠀⠀⠂⢢⣽⣿⡻⠋⡠⠔⠒⠉⠉⠓⠦⣀⠀⠀⠀⠀⢸⣧⠤⢄⣀⡀⠀⠀⠀⠈⢉⣀⡈⢽⡃⠀⠼⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢀⡟⡄
⠀⠀⠀⢊⢀⠔⠁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡅⢀⡀⠀⠈⣈⡉⢉⣉⣀⠀⠀⣰⠇⠀⠀⣠⠞⠀⠀⠀⠀⠁⠀⠀⣠⢎⡵⠁⠀⡼⡸
⠀⠀⠀⠀⠁⢀⡎⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⣇⠀⠙⠒⠦⠬⠤⠤⠄⠀⢀⣀⠠⠤⠘⠁⠀⠀⣧⣀⠀⠀⠐⠀⠀⠀⣰⢣⠏⠀⠀⢠⢇⡇
⠀⠀⠀⠀⠀⠸⣇⠠⡀⠀⠀⢀⡞⠁⠀⠀⠙⢦⠀⠀⡿⠀⡴⠾⠶⡆⠀⠀⠀⠀⠀⠠⢤⣄⣀⠀⠀⠀⠘⠫⢽⣲⡦⣄⠀⢠⠇⣾⣀⣀⣠⡼⢸⠁
⠀⠀⢀⣤⣄⠀⠙⢦⡀⠀⠀⠘⣇⠀⡀⢀⡄⣸⠀⢠⠇⢸⠁⠀⣰⠏⠀⢠⠒⠒⢦⡀⠀⠀⠈⣧⣀⣀⣀⣀⣀⡈⠙⠲⣝⣫⠞⠓⠶⠶⠶⠖⠋
⠀⠀⣾⠁⠈⠳⣄⠀⠉⢦⡀⠀⠈⠳⠬⠵⠞⠁⢠⠟⠀⠘⠒⠚⠁⠀⠀⠘⢮⣀⣠⡇⠀⢰⣯⣽⣏⣀⣠⣤⠤⠽⠇⠀⠀⠀⠀⠀⢠⠖⠒⠲⡄
⠀⠀⣿⡄⠀⠀⠈⠑⢤⣀⠑⢄⡀⠀⠀⢀⣠⡴⢋⣠⠴⠖⠒⢛⡓⠒⠲⢦⡀⠀⠀⠁⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⠀⠀⠠⢹⡀
⠀⠀⠸⣷⡀⠀⢀⣤⣤⡈⠑⢦⡈⠉⠉⠉⠁⡴⢯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠲⢄⡀⠀⠸⣄⠤⠤⢷⡤⠤⠤⣤⡇⠀⠀⠀⢠⣾⠀⠀⠀⠀⠃⢦⡀
⠀⠀⠀⡟⣧⠀⢻⠀⠀⡝⣦⠀⢹⡄⠀⠀⡞⠀⠈⣧⠈⠀⠀⠀⠀⠀⠀⡎⠀⠀⢠⢙⡆⠀⣟⠒⠒⠚⣷⠚⠛⢉⡇⠀⠀⠀⠸⡍⢙⠋⠙⠉⠉⣩⠁
⠀⠀⠀⢿⠘⣇⠀⠳⢤⣤⠟⠀⢰⠇⠀⢰⠳⣕⠀⠘⣆⢀⣒⣒⣀⣀⣸⣇⣀⠐⣡⠞⢳⠀⢻⠀⠀⠀⠸⣆⠀⢸⠃⠀⣀⡀⠀⠙⠾⣷⣤⡤⠞⠁
⠀⠀⠀⠈⢧⣈⠳⢤⣀⣀⣠⣴⠟⠀⠀⢿⣐⣨⠷⠞⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⢧⣀⣸⠇⠈⢧⣀⣀⣀⣉⣠⠜⠀⢯⠉⢧⠀⢀⣠⡴⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠋⣁⣀⣀⡀⠀⢉⣁⡀⠀⠀⠀⠸⡳⣶⣄⡇⠀⠀⠀⠀⠈⠁⢀⣴⣦⣍⡉⠉⠀⠀⠀⠀⠈⠓⠻⣄⣈⡓⠶⠴⠶⠞⠃
⠀⠀⠀⢰⡒⠦⣴⡒⠋⢉⣉⣀⠠⠀⠉⢶⡁⠀⡝⠀⠀⠀⣠⠽⠾⢿⣧⣠⠤⠤⣄⡀⠀⢸⣝⡻⠷⣯⣭⣍⣉⣉⣉⣛⠲⢤⣀⠈⠉⠓⠦⣄
⠀⠀⠀⠘⣧⠀⠱⢭⣒⣤⣀⠀⠠⠄⠀⠀⠱⡀⢨⠆⠀⣼⢡⠊⠈⠀⠈⠁⠀⠀⢠⠹⡄⠈⠻⢿⣷⣶⣶⣯⣭⣭⣽⣛⡻⠶⣌⢳⡄⠀⠀⠘⣆
⠀⠀⠀⢰⡿⡦⢤⡘⣿⣭⢻⢷⣦⣄⡀⠀⠀⠑⣌⠆⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠶⠤⢤⣉⣉⣉⣉⡉⠛⠻⢷⣼⠋⠀⠀⠀⠀⣿
⠀⠀⢀⣀⠓⢬⣒⢵⡀⠛⠉⣟⠀⠛⢿⢗⠦⣄⡈⣧⠀⠸⡆⠀⠀⠀⠏⠘⠆⠀⠀⢰⠁⠀⢸⠭⠶⠶⠶⠶⠶⠦⣭⣍⡓⠤⡉⠀⠀⢰⠋⠉⠉⣹⠂
⠀⠀⠀⠈⣳⡀⠈⠑⠮⣓⢦⣈⠉⠳⡀⢀⡙⠲⠽⠇⠀⠀⠹⣔⠤⡀⠀⠀⠀⠄⣠⠏⠀⠀⠘⠿⢿⣒⣒⣒⣒⣒⠢⠌⡙⣾⠀⠀⢠⡟⡶⠒⢲⣿
⠀⠀⠀⠑⢿⣧⣤⡠⠴⡌⠑⢬⡓⢦⣉⣁⡸⠂⠁⢀⣀⠀⠀⠈⠓⠒⠛⠓⠒⠛⠁⠀⣀⣤⠴⠶⠲⠶⠦⣄⡀⠈⠙⢦⣄⣷⠀⠀⠸⠶⢷⣬⣿⣸
⠀⠀⣠⢾⡿⠋⠛⠒⠋⠀⠀⠀⠈⠓⠖⠋⢀⡠⠞⠉⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⢉⢳⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⠁
⠀⠀⠉⢁⡤⠤⠤⠄⢤⣄⣤⠤⠤⡀⠀⠀⡿⠤⠤⠤⠤⢤⡴⢦⣄⣈⣓⠲⢄⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⡰⡠⠃⢀⡴⠞⣻⠋⠉⠉⠉⠓⢦
⠀⠀⡎⢹⠉⣿⠛⠛⠛⠛⠛⠛⠛⡏⠀⠀⣛⣷⠀⠀⣀⠀⠙⠋⠀⠀⠉⠉⢻⠀⠈⢧⡳⢤⡀⠀⠀⠀⠀⠀⢸⠁⢀⡞⠀⢎⠉⣠⠶⢶⣄⠀⠈⢧
⠀⠀⠹⣼⣧⠘⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣯⡀⠀⠸⠽⠂⠀⣠⣤⣄⠀⠀⢳⠀⠀⠀⠙⣆⢷⠀⠀⠀⠀⠀⠘⡇⠸⡇⣠⠊⠘⣇⠀⣸⣽⠀⠀⣸
⠀⠀⠀⢠⡛⣦⣀⡀⠀⠀⠀⣠⠗⡄⠀⠀⣾⠁⠀⠀⠀⠀⠀⠛⠶⠟⠀⠀⢸⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⢹⠀⢷⡸⢄⣠⣌⢉⣩⣁⡖⣲⠏
⠀⠀⠀⠀⠉⠛⠿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠙⠒⠒⠒⠒⠒⠚⠙⠒⠒⠒⠒⠚⠀⠀⠀⠀⠈⠳⣧⣄⣀⣠⣤⡤⠞⠀⠀⠙⠶⣤⣈⣉⣠⡭⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣄⠀⠀⠀⣦⠀⠀⠈⢻⣿⣶⣦⣤⣀⣀⣀⣀⣤⣴⣶⣿⠟⠓⠦⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡇⠀⠀⢸⣷⠀⠀⠈⣿⣷⣦⣬⣭⣩⣿⣿⣿⣿⣯⣄⣈⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⠇⠀⠀⣸⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⣼⠟⠠⠀⣸⡿⠀⠀⢠⣿⠁⠀⠀⢀⣿⠋⠉⠉⠉⠉⠉⠉⠁⢀⣤⣾⣿⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣴⡿⠁⠀⣠⣿⠋⠀⠀⠀⣼⣿⣄⣀⠀⣀⣀⣤⣴⣾⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠃⠀⣠⣾⠏⠀⣀⣴⠟⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠱⠶⠶⠿⠯⣤⣴⠾⠟⠉⠀⠄⠠⠴⠿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠄⠀⠂⢀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣁⡀⠀⠘⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⡀⢀⡀⠤⠤⠄⣔⠋⠁⡄⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣾⢟⣿⣿⣿⣷⣯⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⠋⣼⡏⣾⡿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠂⠀⠀⠀⡀⠀⠐⠊⠁⠀⠈⢳⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡾⠁⢰⡿⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡀⠤⠥⠴⠤⢀⣀⠠⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣸⠃⠀⣼⠇⢰⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠐⠊⣀⠀⠒⠀⠀⠀⠀⠈⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡏⢸⠀⣿⠀⢸⣿⠃⢸⣿⣿⣿⣿⣿⣿⡎⢿⡄⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠂⠀⠀⠈⡀⠀⠀⠤⠀⠀⠐⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠁⠀⢸⡏⠀⠸⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⡘⣿⡀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠁⡐⠀⠀⠀⠀⠀⢹⡃
⠀⠀⠀⠀⠀⠀⠈⠀⢠⠈⠀⡄⢸⡇⠀⠀⣿⡅⠀⠸⣿⣿⣿⣿⣿⣿⡇⣿⣧⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠃⢸⡇⠀⠀⢻⣇⠀⠀⠹⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⣁⡠⠀⠀⡀⠀⠀⠲⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⢸⣧⠀⠂⠘⣿⣆⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⠀⠀⠀⠠⠀⠈⠈⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⡀⠀⠀⠹⣿⣆⠀⠀⠈⠻⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠒⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠃⢹⣧⠀⠸⠄⠙⣿⣷⣄⠀⠀⠀⠀⠉⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣤⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘⡄⠀⠀⢿⣧⡀⠘⠀⠈⢿⣿⣿⣶⣤⣤⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠱⡄⠀⠀⢻⣷⣄⠀⠀⠀⠉⠛⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠹⣦⡀⠀⠙⢿⣷⣤⣀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠊⠉⠉⠉⠉

⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠐⣾⣦⡀
⠀⢀⣤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣢⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀
⠀⠻⠁⡜⢁⠐⠉⠐⢄⠀⠀⠀⠀⠸⡌⠀⠀⡞⠐⢢⣄⡀⠀⠀⠀⠀⢀⠘⠻⣿⡿⠀⡀⠀⠀⠀⠀⠀⢠⢻⡿⠻⠿⢀⡀
⠀⠀⠑⠣⣀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠈⠢⠀⠇⠀⠁⠀⣱⠀⠀⠀⠀⠸⣿⣶⣶⣶⣾⡇⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣿⠃
⠀⠀⠀⠀⠈⠑⠘⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠖⠁⠀⠀⠀⠀⠀⠛⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠈⠛⠻⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⣤⣤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠰⣿⡿⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠈⠀⠀⣹⠃⠀⠀⠀⠀⠑⠢⣄⢀⡄⢂⡀⠀⠀⠀⠀⠈⡕⠪⠉⢉⡭⠷
⠀⠀⠉⠁⠈⢱⣿⣿⣷⠀⠀⠀⠀⠀⠁⢀⣐⡢⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢁⠄⢉⠃⠀⠀⠀⠀⠀⠣⠀⠀⣿⡿⠅⠂
⠀⠀⠀⠀⠈⠈⠻⠿⠋⠒⠄⠀⠀⠀⠀⠀⡀⠀⠔⠉⠐⠀⠀⠀⠀⠀⠀⠀⠘⠃⠰⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠂

⠀⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠓⢌⠂⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣤⣤⢄⡀⠀⠀⠀⠀⠀⣤⡴⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡌⢆
⠀⢀⢔⣦⠁⠐⠡⠀⠠⠠⠀⠀⠀⠀⡿⣜⠧⠿⠾⢵⡿⡆⠀⠀⠀⠙⠤⣀⠀⠀⠿⠶⣤⡀⠀⠀⠀⢠⠀⠀⠀⠀⣿⣆⡆
⠀⠸⣈⠛⠿⠶⠶⠶⠚⣠⠃⠀⠀⠀⠳⢤⣉⣉⣉⣉⡴⠃⠀⠀⠀⠀⠀⠈⠑⠀⣃⠛⣋⡁⠀⠀⠀⠈⠓⢦⣀⠀⣿⣿⡇
⠀⠀⠀⠉⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣶⣦⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣀
⠀⢎⠌⠲⣁⠶⢤⡀⠀⠀⠀⠀⠀⠀⡠⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⠟⡠⢐⢿⣿⣿⣧⡄⠀⠀⠀⣤⣾⣿⣿⣿⣿⠊⠙
⠀⠀⠑⢥⠌⢲⣤⢎⡩⢲⠀⠀⠀⠀⣃⣀⠀⠉⠉⢀⣠⡀⠀⠀⠰⣿⣜⣧⡛⣹⣿⣿⣿⡗⠀⠀⢸⡿⣾⣯⣿⠟⠋⣀⣠
⠀⠀⠀⠀⠙⢤⢘⡦⠉⠉⠀⠀⠀⠀⠋⠙⠛⠻⠛⠛⠉⠀⠀⠀⠀⠠⢈⡁⠀⠁⠀⡁⠀⠀⠀⠀⠀⠂⢈⠀⠀⠀⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⢄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢟⢦⡀⠀⠦⠀⠀⠑⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡙⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠢⠥⠀⠀⢀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠠⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠂⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⡄⢀⡀⠀⠀⠀⠑⠢⠀⣀⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢋⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⡰⠆⠐⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡅⢉⠷⠦⡀⠀⠀⠀⠀⠁⢸⣿⡀⠉⠉⣿⣿⣿⣿⡿⠀⠀⠀⠀⡀⢄⠀⠠⡚⢾⢌⠑⠦⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠐⢇⠸⠁⣤⠀⣄⠀⠀⠀⠀⠆⠀⠿⢷⠀⠘⣿⡿⠻⠛⠃⠀⠀⠀⣠⣮⠎⡡⡀⠘⠂⠋⢵⢆⠂⢵⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠘⣱⠂⣡⡈⠳⠢⢀⠀⡠⠒⢢⠊⠉⢣⠀⣿⡠⢐⡒⢄⠀⢀⠔⠀⠰⣾⠀⠀⠀⠀⠀⠂⠉⠘⠙⠉⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠡⣴⡇⠀⠀⠙⠀⠀⢨⠀⠀⢀⠀⠙⠀⢁⣠⡾⠶⠊⢀⡠⠂⠔⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⡈⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⣠⣀⣤⣤⡤⠔⠒⠠⢀⡴⠂⠈⠉⠈⠳⠤⠌⠀⠀⠈⣿⠦⠀⢠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠸⢿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⣦⣤⣀⡀⠀⢩⡐⠘⠁⢛⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡤⠖⠠
⠀⠀⠠⠄⠀⠀⠀⢀⣞⣠⣄⠀⠙⢄⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢿⣿⣿⣿⣿⣶⣼⣇⡇⠡⡀⠙⣧⠀⠀⠀⠀⠀⠰⡁⠈⡎⡆⠀⠀⡀
⠀⠀⠠⢤⡄⠐⠂⠈⠃⠀⠨⠁⠀⡌⠁⠀⠚⠂⠤⠶⢆⣠⢀⣀⢉⢀⡠⠊⠙⠛⠿⣿⣿⣿⣿⣿⣄⠘⠄⠈⠻⠗⢄⡀⠀⠀⠈⣵⢠⠠⠀⠀⡄
⠠⠞⣀⠄⠀⠰⠦⣀⠀⠀⣘⡆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠛⠈⠑⢠⠋⢹⠿⢿⣿⣷⣌⠽⠃⠀⠑⡿⠆⠀⠀⠣⠄⠣⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠁⢀⠈⠑⢢⡴⡄⢀⣀⣀⡤⣴⠀⠀⠀⠀⢠⣄⡀⠠⠎⠀⡸⡀⠶⣷⠀⠉⠈⠢⠠⠀⠸⠿⠷⠄⣶⠄⠀⢀⠄⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⠀⠀⣸⠀⠀⢸⠀⢠⣠⠤⢷⠀⠀⣠⠲⠤⡄⣠⠊⢀⠀⠀⢠⠇⡄⣶⣼⣄⣀⡀⠀⠀⠀⠀⠀⠀⠮⠤⣄⠀⠈⠧⠢⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠⠀⠀⣸⠀⠀⡤⠂⡛⠀⢀⠃⠈⠤⢁⡌⠔⠁⠀⣴⣿⠾⠠⠆⠤⣇⠀⠀⠀⠂⠀⢄⠀⠰⠀⢠⡄⠑⠊⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣴⢏⠠⠊⠀⠀⠝⠁⠀⠳⣀⡀⠀⠀⢀⣤⠊⣡⣴⣾⠋⠈⡐⢾⠀⠀⠀⠀⠄⠀⠹⡘⢂⣟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠙⠿⣦⣆⠘⢡⢂⡀⡄⡲⢶⣼⠂⠁⠀⠀⠀⠀⠀⠈⠁⣏⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⣣⢸⠷⣮⣝⢰⣘⣂⣂⡅⠷⢶⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂⠒⠒⠒⠒⠒⠒⠒⠶⢤⣦⣶⠶⠠⢒⡆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⢀⣤⣀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣴⣿⣿⣿⡆⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠿⠿⠛⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡼⠀⠀⠀⣀⣤⡤⠤⠶⠤⣄⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⢺⡆⠀⠀⠀⠀⠀⢨⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠐⠒⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⢀⡤⠼⠧⠤⠤⠤⠤⠤⠼⠼⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠃
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⢀⣀⡠⠤⠔⣊⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠐⢬⠭⠉⠉⠭⠭⠁⠤⠐⠂⠉⠀⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠘⡄⢀⣀⣀⡀⠤⠤⠀⠐⠒⠈⢹⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⢀⣀⣀⣀⠀⠻⣃⣟⠘⠃⠀⣠⠒⠢⠤⠭⠍⠉⠐⢲⠋⡇
⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⣤⣦⣄⠀⠀⠀⢸⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠐⠀⠐⠃⠀⠀⡇⠀⠀⠢⠠⠐⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⢱⠀⠀⠰⣿⢿⣽⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠁⢀⡀⠀⠠⠤⠤⠄⠚⠃⢰
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⠻⠟⠋⢀⣀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣔⡉⣑⢏⡠⢶⠀⠀⠀⠀⠘⠁⢠⢰⡴⠄⠶⠦⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠐⢖⠒⠒⠂⠈⠉⠀⠀⠀⡄⠀⠀⠀⡠⠠⠤⢊⠭⠠⠕⠀⣈⠔⠨⠥⠄⠒⠤⠀⠀⠀⠀⢰⣈⡡⡀⠤⠤⠤⠤⠒⠃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⠀⠀⣀⡠⠄⠊⠀⠀⠀⠀⡗⢒⡒⢁⠔⠒⠊⠀⠀⠀⠀⣀⠀⠀⡀⡀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢰⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⠂⠑⢊⣐⢂⡮⠭⠱⢞⢄⡨⠮⡀⢴⠀⠀⠀⠀⠆⠀⠐⠒⠂⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠐⠠⣀⠀⠀⠀⠀⠀⢸⣁⢉⡈⡡⠤⠤⠊⠑⠒⠉⠑⠀⠒⠢⠼⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⠤⡐⠜⠋⠁
⠀⠀⠀⠀⢀⠴⠈⠠⠤⠤⠄⠤⠤⠤⠤⠄⠤⠤⠤⠬⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠧⠤⠶⡈⢁⡀
⠀⠀⠀⠀⠘⣷⣮⡴⡤⠰⢶⣤⣤⣄⣄⣠⣡⣤⣐⣲⣶⠃⠀⠀⠀⠀⠀⠀⠀⣔⠈⡡⠄⠰⡀⠠⡤⢀⣐⠢⠤⣄⡉⢲⡄
⠀⠀⠀⠀⠀⢏⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⢋⡁⠃⠀⠀⠀⠀⠀⠀⠀⢙⠛⠠⠬⢵⣏⣘⣏⣁⣬⣡⠦⠶⠚⢛⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣼⠁⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⣸⠃⠠⡏⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⢰⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡄⠀⠀⠀⠀⢸⠇⠀⢸⡇⠀⢰⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⢇⠀⠀⠀⠀⣾⠀⠀⢸⡇⠀⣾⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⡟⠀⠀⢸⡇⣰⠏⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⣤⠀⠈⣧⠀⠀⠀⡇⠀⠀⢸⣇⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⢹⠀⠀⠻⡄⠀⠀⡇⠀⠀⢸⣿⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⢳⣄⡦⠀⠀⣿⠀⠀⡇⠀⠀⢸⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣿⡀⠀⠸⣇⠀⣷⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⢷⡄⠀⢸⣆⣿⠀⠀⣸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠈⢷⡀⠀⢿⣿⠀⠀⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢿⡀⠈⣿⠀⠀⣿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠘⢧⡀⢹⡄⠀⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⣧⠈⢃⠀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣄⠘⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢻⡄⠸⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢃⠀⢺⡅⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣱⡄⠀⢸⡇⠸⣇⣀⡼⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣧⣤⡜⠳⣤⠟⠉⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠋⠁⣠⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠶⠛⠋⠁⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⣠⠤⠚⠋⠁⠀⠀⠀⢠⠀⠀⠀⠘⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠖⠈⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⢀⡾⠃⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠘⠃⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢦
⠀⠀⣠⠟⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠃⠀⠀⠈⢧
⠀⢰⡏⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠈⣵⠀⠀⠀⠀⠀⠈⣷
⢠⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⢸⡇
⡜⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⢹
⣷⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀
⠈⢷⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⣸
⠀⠘⢷⡀⠀⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠶⠶⠶⠶⢾⣦⡴⢶⣤⡶⣤⡾⠛⠛⠛⠛⠛⠛⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠁⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⠶⠛⠛⠛⠻⠶⠶⠶⠞⠛⠛⠛⠛⠛⠷⠶⠶⢶
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠠⢄⡀⠀⠀⠀⠀⢰⠀⢸
⣿⠀⣀⠀⠀⢀⣀⠠⠤⡤⣒⣽⣻⣇⠘⡆⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀⠄⠭⣷⠀⠀⠀⢼⠀⢸
⣿⢴⠉⡏⠻⠋⠀⠈⡡⠊⢁⠈⡽⣟⠾⣰⠞⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠁⠊⠀⠀⠀⡸⠂⡇⠀⣿⠀⢸
⣿⠸⠀⠗⠀⠀⡠⠊⠁⠀⠠⣸⠐⣽⠁⢹⢐⣙⠶⢮⠀⣤⣄⠀⠀⠀⣔⠈⡀⠀⠁⠀⠈⠂⠀⠠⠀⠘⠁⠀⢉⢐⠟⡖⢾
⣿⠤⠶⠖⠛⢻⠁⠒⠲⠦⠤⣌⣙⡊⠀⠈⡾⠍⠃⣉⣁⣠⣨⠄⠀⠁⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣰⣱⣼
⣿⠀⢸⣿⠀⢸⠀⠀⢀⡤⠰⢶⣶⢿⣻⡞⠿⠛⣿⠷⠋⠁⠘⢆⣠⣄⡔⡆⠀⣀⣸⣿⠠⣤⣀⣀⣿⡄⠀⠀⠀⣼⣹⣹⣿
⣿⠀⢸⣿⠀⠘⡴⣚⣗⡤⠉⣭⣤⠞⠩⣩⣍⣻⠃⠀⠀⠀⠀⡀⠱⡈⢿⡏⠭⣴⡒⣿⣳⣾⣿⣿⣿⣷⣄⠀⢀⣯⣜⡹⣿
⣿⠀⣈⣭⢀⡜⢱⠿⣪⠇⣾⣿⡝⣷⣿⡽⣹⠃⠀⠸⣿⠁⢸⡿⠀⢻⣜⣇⠀⠁⠁⡇⠈⠉⠙⠘⠃⠃⢎⡶⠫⢤⡀⢳⣿
⣿⠀⢸⣿⣟⠐⠚⠀⠀⢀⣉⣅⣀⣁⣾⣠⣧⡄⢰⣻⡇⠀⠀⠀⠀⠸⣄⣎⠳⣶⣶⡷⡆⢦⣶⣶⣆⣄⢸⣟⡒⢚⠳⡘⣿
⣿⠀⢸⣟⡏⢩⢹⣫⡽⣻⣿⡏⣿⣿⣿⣿⡏⡇⠀⢸⠀⠀⠀⠀⠀⢠⣦⠉⣿⢲⣡⡲⡴⢾⢸⣿⢸⡶⢺⣻⣡⠻⠿⠿⣿
⢿⡄⣀⣀⣀⣈⠸⠟⣋⣿⡿⡧⢭⠉⠉⣙⢁⣇⣀⣸⡀⣀⠠⠄⠠⢞⢛⡆⠘⠚⠛⢉⣉⣉⣈⠼⠘⠛⠓⠒⠒⠒⠉⢉⣿
⣿⡒⠲⠶⠶⠶⠦⠤⠤⠤⠤⠷⠸⢤⣄⡀⢤⡤⢄⣜⠦⠤⠤⠶⠖⠚⠛⠉⠉⠉⡀⠠⠤⠤⠀⠀⠀⠀⠀⠄⠀⠒⠀⠉⣿
⣿⠬⠤⠴⣦⣤⣤⣭⣤⣬⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣯⣤⠶⠦⣤⣤⣤⣤⣤⣴⡶⠶⠶⠶⣤⣤⣤⣤⣤⣤⣿
⣿⣿⣿⣿⣛⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣻⣿⣿⣿

⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠘⠀⢐⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⡀⢻⣿⠾⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠸⠂⠀⠀⠀⡀⠤⣦⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⠾⠃⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛
⠀⠀⠀⠀⠀⡏⣠⠔⡄⢀⣏⣄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠛⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡷⢤
⠀⠐⡆⠀⠀⢆⡈⣥⣤⣾⡆⠉⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢀⠤⢂⢩⠇⠀⠤⠠⡀⠀⠀⠉⠈⠁⠉⠉⠉⠉⠁⠀⠀⡀⠀⠀⡀
⠀⠀⠡⠀⠀⣎⡥⠙⠉⠀⠙⡀⠀⠀⠀⠀⠀⠀⠐⠝⠀⠀⠀⣨⡶⠛⠇⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠤⠂
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠀⠀⠀⠀⣴⢾⡯⠉⠀⠀⢤⠀⢤⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠐⠀⠀⠀⠀⠨⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⣠⣤⡶⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠚⠂⠀⠁⠀⠀⠀⠀⠀⠉⠀⣴⡀⠖⠂⢋⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⣶⣲⡏⠀⠀⠀⠀⡀⠀⠀⠦⠀⠀⠀⣄⠀⠂⠀⠀⠀⠀⡛⠋⣀⠀⠀⠈⠀⠀⣴⠟⠁⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⣀⠀⣠⣾⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣄⠀⠀⠀⡀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠃⠈⠉⠀⠀⠀⠀⠠⠄⡀
⠀⠀⠀⣀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⡄
⠀⠀⠀⠠⢤⠄⣤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠀⠀⠈⠀⠐⠆⠀⠀⠠⠀⠀⠐
⠀⠀⠀⠀⢰⠀⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠆⠀⠀⠐⢲⡄⠀⠀⠀⠀⠘⢳⡄⠀⠀⠀⠀⠘⢲⡀⠀⠀⠀⠀⠀⠶⠀⢠⠀⢀⠀⠀⠀⠀⠀⠰⢂
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠆⠘⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⣄
⠀⠀⠀⠀⠸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⠀⠉⠟⣘⣷⣨⡅⠀⠀⠀⠐
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢰⠀⣀⠀⢓⡚⠦⣤⢾⢿⣷⠾⠇
⠀⠀⠀⠀⠠⠈⠆⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⡼⠀⠀⠉⠒⠠⢀⡤⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠘⠧⣄⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣶⡆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡀⠀⠀⠀⣠⠞⠀⠀⠈⠓⣦⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠏⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠓⣄⠀⠀⠀⣖⠋⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡄⠀⠀⠀⠤⠞⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⣀⣀⠀⠀⠀⠀⠀⠋⠁⣀⣀⣀⠀⠀⠀⡼⠀⠀⡼
⠀⠀⠀⠀⣤⣤⠤⠤⠶⠶⡶⠋⠁⠈⠙⣖⠒⠒⠒⢲⠎⠁⠀⠈⠳⡖⢲⠇⠀⣸⠷⠶⠤⢤⣤⣄
⠀⠀⠀⠀⡏⠉⠉⠉⠉⠙⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⡇⠀⡴⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⡀⢸
⠀⠀⠀⠀⣧⠘⣇⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⣠⠇⡟
⠀⠀⠀⠀⢹⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⠋⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣸⠃
⠀⠀⠀⠀⠀⢳⡀⠀⡤⠶⣄⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡤⠶⣄⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠳⣄⠳⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠣⡤⠞⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠠⣄⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⠀⠀⠀⠀⡔⢲⡄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣏⣉⣉⣉⣉⣉⣉⣉⣉⣹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠓⠚⠚⠻⠴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⢆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠄⠐⠀⠈⠻⣝⠒⠛⣚⡚⢧⡄⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡏⢀⡤⢤⣉⠈⢷⡀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣔⣦⡀⠀⠀⠀⠀⠀⠀⢸⠹⠏⢱⠁⣾⣷⠈⣧⠈⣱⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢣⠁⠙⣄⣄⠀⠀⠀⠀⢨⠟⠚⠿⢄⡻⠏⣀⠟⠀⡿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⠛⠧⠉⠀⢷⢄⠈⢦⣦⠀⢀⣀⣱⠦⠀⠀⠈⠉⠀⠤⡚⠓⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⣆⠀⠐⢦⡤⠌⠁⠑⠿⠝⠒⠒⠦⣀⠀⠀⠀⠈⢉⡠⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⢲⠛⡖⡦⢤⡨⡙⠦⠤⣉⣀⡀⠻⠀⣀⢳⡢⠄⠈⢳⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣳⠤⢤⣘⣖⣢⠣⣼⡠⢷⡀⠀⠀⠉⠉⡍⣖⣈⠂⠀⠀⢀⡜⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣑⠦⠤⠲⣤⣜⡉⢉⡟⠀⠀⠀⠀⠀⠳⣄⠈⠉⠉⠉⢉⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣉⣉⣁⣐⡢⠖⢋⠀⣀⣒⠀⠀⠀⢀⠜⠉⠉⢹⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⠤⢤⠤⠤⠤⠔⠲⡃⢀⡠⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⣎⣠⠀⠀⠀⠀⠈⠁⠀⠀⣇⠀⠲⠤⠤⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣟⣒⠒⠢⣌⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⢀⠤⣄⣰⠟⠉⠈⡇⠀⡟⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣤⣀⣠⠏⠀⠀⢠⡞⣛⣚⠁⠞⠙⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠱⡀⠀⢾⣸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡠⠞⢅⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣤⣤⣄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡔⢲⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠓⠲⢄⠀⡀⠤⠒⠒⡲⠦⣀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠋⠀⠑⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠁⠀⠀⠀⠀⠀⠘⣦⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢠⠀⠀⠀⢻⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠠⠀⠛⠺⡄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⣈⠃⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⡀⣀⡀⠻⠗⠀⣣⠈⠀⠀⠀⠑⠄⣀⢀⠀⠀⠀⠀⠀⣴⠋⠊⢉⠉⠣⣀⠀⠀⡀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠐⠁⣠⢄⠹⢶⠤⠾⠃⠦⡀⠀⠀⠀⠀⠀⠈⠑⠂⠄⣰⣾⡀⠀⠀⠈⠀⠀⡤⠑⠛⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⣀⡩⠞⣼⠁⠀⣠⠀⠘⢂⡤⠐⠒⠀⠐⠒⠠⢄⣹⡞⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠄⠠⣄⠸⡀⣶⢌⢁⣠⡞⠉⣠⠀⠐⠀⡒⠀⠀⠀⠉⢧⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢸⡆⢧⣅⣀⢀⡀⠀⠀⠈⠣⠈⠀⠁⠔⠁⠀⠀⠈⣳⣦⠤⢀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠥⠎⠀⡟⠁⠀⠈⡇⠀⣀⡈⠁⠀⠀⠀⠐⢂⠨⠓⠝⠋⠀⠀⠀⠀⢘⠀⡠⠂⠀⠀⠉⠒⠤⠤⣀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣇⠸⠖⠀⣣⠀⠀⠤⠂⠀⡐⠠⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠾⠀⠓⠒⠒⠙⢄⠀⢂⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣡⠀⠀⠀⠤⠐⢞⠈⠑⠤⣀⣀⠀⣀⣳⡀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢿⠁⠁⠀⠠⣄⠀⠀⠀⠀⠀⢨⠀⠀⠈⠢⠵⠀⣠⣄⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⡀⠀⠉⠉⠳⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⠐⠂⢤⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⢀⣀⠤⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠉⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠐⢄⣀⣀⣀⠠⠜⢗⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣁⠠⠊⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⣠⠴⠖⠒⠉⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠉⠁



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

⠀⠀⠀⡠⠖⠒⠒⠤⡀⢠⢊⠝
⠀⠀⠚⢲⠤⣄⠀⠀⠘⡇⣆⠤⠀⢀⣀⡀
⠀⠀⠀⢸⠀⠈⠳⡄⠀⢱⣞⠒⢮⡁⠀⠁⠀⠀⢀⣀⣀⣤⣤⡴⠶⠶⠶⠶⠶⠶⢦⣤⣤⣀⡀
⠀⡎⠉⠉⠑⠒⠤⠬⢦⣸⠙⠛⣿⠭⣛⠛⠛⠛⠉⠉⠁⠌⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣄
⠀⢰⠂⠀⢀⠤⠠⠤⡀⠼⡖⢎⠢⣍⠒⠛⠶⠖⡒⠒⠠⢄⡀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠳⣄⡀
⠀⠀⠂⠉⠻⠔⠒⢲⠀⠀⢡⠘⡀⠀⠑⢄⠀⠀⠀⠈⠂⠄⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⣠⠏⠀⡆⢸⠀⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠙⣧⡀
⠀⠀⠀⠀⣠⠞⠉⠀⢀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢷⡀
⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠘⣷
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸
⠀⣸⠁⠀⠀⠀⠀⡀⡄⠀⢸⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠈⡀⠀⢣
⢠⡇⠀⠀⠀⠀⠀⠁⡇⠀⠈⡄⠀⠀⠈⡄⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢰⠀⠸⡄
⣸⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⢣⠀⠀⠀⠘⡄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡆⠀⣧
⣿⠀⠈⠀⠀⠀⠀⠀⠘⡀⠀⠈⢧⠀⠀⠀⠘⡄⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢇⠀⠀⡇⠀⣿
⣿⠀⢀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣆⠀⠀⠀⠘⡄⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡆⠀⣿⠀⣿
⣿⠀⠘⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣆⠀⠀⠀⠈⢆⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⢹⠀⢹⢰⡏
⢹⡄⠀⠄⠀⠀⠀⠀⠁⠀⠀⠑⡀⠀⠘⢆⠀⠀⠀⠀⠡⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡇⣼⡿
⠈⣷⠀⠈⠄⠀⠐⡀⠀⠁⠀⠀⠘⢄⠀⠈⢧⡀⠀⠀⠀⠈⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⡆⠀⣷⡿⠁
⠀⠸⣇⠀⠘⡄⠀⠐⠀⠀⠀⢄⠀⠀⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣇⣴⠏
⠀⠀⠹⣦⠀⠘⢆⠀⠈⠂⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢀⠀⠀⠘⣦⡼⠻⡧⣑⠤
⠀⠀⠀⠘⢷⣄⠈⢣⡀⠀⠑⢄⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣆⣤⠞⠋⠨⠂⠙⠂⠡⣀
⠀⠀⠀⠀⠀⠙⠳⣤⡙⠦⡀⠀⠑⢄⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⣧⣤⡶⠛⠡⡀⡀⠒⢦⡀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣕⣂⠠⠑⠦⣄⠈⠓⠦⢤⣄⣈⠐⠲⢦⣤⣤⠶⠿⠛⠋⠉⠆⠀⡖⢂⢆⠀⠀⢀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣛⣚⣿⣿⣶⣶⣾⣛⣛⣛⣋⣉⣀⣀⣀⣀⣊⣈⣂⣙⣋⣕⣀⣁⣀⣁⣀⣀⣀⣀
⣿⣿⠋⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣯⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⡄⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⠉⡻⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⣤⣌⠻⠿⠛⣉⣤⣶⣦⠾⢿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡷⣶⣿⣯⣿⣿⢿⡷⠺⢢⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡮⠻⠿⠿⣿⢿⣿⢻⣿⠖⠀⢀⠈⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠁⠙⠀⠾⣯⡇⣾⣧⣄⠈⣿⣿⡄⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢀⠀⠀⠈⣾⠁⠛⢻⣿⣧⠙⠻⠿⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢶⣺⣿⡿⢿⣶⣿⡝⠁⡌⢹⠏⠀⠀⢰⣟⠿⢿⣿⣼⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣍⣛⣫⣤⣿⠟⠘⡇⠀⠀⣿⠀⠀⠀⣼⠉⠀⡼⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣯⣿⡟⠁⠀⠀⢻⡀⠐⡇⠀⠀⠀⣿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡇⠀⠀⠀⠈⣧⠀⣿⠀⠀⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⡇⠀⠀⠀⠀⢹⡄⢸⠀⠀⠀⠿⠀⡇⠀⠀⣠⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠁⠀⠀⠀⠀⠈⣧⣨⠂⠀⠀⠘⠀⠃⠀⣼⣯⣿⣼⡿⣿⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡿⠿⠿⣿⠿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠘⠛⠋⢙⡓⢦⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⡀⣀⡛⢰⣉⣡⠄⡙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢿⣛⣛⡯⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠂⠢⠴⡀⠀⠀⠀⠀⠀⠈⣘⣥⡀⠀⠀⠀⠀⠂⠀⠀⠂⠐⠂⠀⠀⠘⢲⣶⣦⣤⣁⡀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠈⠂⠀⠀⠉⠣⣥⠂⠣⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⡋⢙⡖⢲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢿⡖⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦⠄⠀⣨⣜⠻⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⣽⠻⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣣⠐⠚⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢖⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠂⠀⠒⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠀⠘⠀⠀⠀⠰⠄⠠⠀⠰⠃⠀⠀⠀⠀⠀⠀⢀⢁⠈⠀⠀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⠀⠀⢠⣷⠈⢀⠀⡰⢖⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⣰⣷⣤⣸⣷⣠⣿⠿⠶⠀⠀⠈⠋⣤⠶⣶⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠙⠋⢠⠈⠛⠃⠀⠀⠀⠰⠀⠀⠉⠀⠘⣧⣀⢀⣠⡴⠾⠛⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⠀⠀⠈⠉⠉⠁⢀⡄⠀⠀⠂⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠠⡄⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠘⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠈⠇⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡐⢈⣠⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⣿⣿⠀⣶⣤⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣄⡀⠒⠀⣃⠈⣙⠁⡀⣠⠤⡌⠉⠂⠀⣴⡄⠀⠀⠀⠀⢀⠀⠀⠀⠸⣿⠀⢻⣿⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠀⢁⠀⠉⠁⠐⠄⠀⠀⠨⠀⠀⠰⠐⠀⠀⠀⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⡇⠸⣿⠁⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠬⠒⠀⠀⠀⠀⠀⢦⡀⠀⠀⠄⠄⠀⢀⠀⠀⠀⠀⠄⠀⠠⣀⣻⣿⡿⣿⣷⣾⣿⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠑⠀⠹⠛⠶⣄⡀⠁⠀⠴⠀⠀⠀⢠⡀⠀⠀⠚⠿⠛⠉⠉⠁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠏⣄⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠲⠲⠶⠆⠀⠀⠀⠀⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⠋⠉⠉⠉⢩⠏⣉⡉⠉⠉⡏⠉⠉⠉⠉⠙⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢧⣀⣰⡏⠉⠛⠋⠁⠉⠓⠶⠛⠓⠲⠒⠒⠶⣍⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢠⡇⣠⠶⠞⠁⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠸⣇⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⠷⣽⠆⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⢹⣽⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡗⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠠⣎⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡟⣇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠃⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠉⠉⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣄⣀⣠⠤⣤⡤⢤⣀⣠⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣎⢩⣭⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣾⣃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠤⠤⠤⠤⠈⢉⡙⠓⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠄⠒⠀⣐⠒⢤⣤⣤⣤⣤⣤⣤⣈⣑⠪⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠊⠀⠀⠀⢐⣚⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⣤⣶⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣒⣒⣚⣛⣛⣛⣛⣛⣛⣻⣻⣿⣿⣿⡿⢿⣿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⣖⣒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣛⣭⣭⣭⣭⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣄⡉⠻⣿⣛⢻⠿⠿⣿⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠋⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡈⠙⢿⣿⣆⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠙⠿⣷⠀⠹⣿⣶⣤⡀⠀⠀⢀⣀⣤⡾⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⠘⣇⠀⢹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡻⡆⠀⠀⠙⠻⣿⣿⣿⣟⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣆⠀⠀⠀⠀⠈⠉⠛⢦⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⣀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡟⠛⠋⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⠀⠀⠁⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠄⠂⠀⠀⠀⢀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣿⡣⠀⠀⠀⣠⣶⣟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⢦⣴⣾⣿⣿⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡴⠿⠟⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁



⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣥⣤⣦⣤⣶⣥⣶⣭⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢰⠸⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⣄⠀⣠⣆⠠⢙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠐⡦⡕⠒⠒⢣⡤⠀⠋⠖⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠉⣇⠈⣉⠃⣀⠤⢊⠛⢀⠄⡄⡀⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠓⠭⠀⠰⣁⠻⢈⢥⠪⠚⠉⢹⠣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠧⣤⠤⠴⠊⠑⠢⣤⣩⠴⡁⠀⠀⠀⢘⡵⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢅⠀⠀⠀⠀⠀⠀⣀⡀⠱⠴⠴⢒⠊⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⡤⢀⣀⣀⣁⣀⠀⠠⠤⠠⠀⠠⡄⠑⡀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠁⠈⠉⠉⠉⠁⠈⠉⠀⠐⠒⠐⠃⣠⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠀⠉⠁⠂⠠⠀⠺⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠷⣤⠄⡀⡀⠀⠀⠀⢀⠀⢂⠠⣤⠾⣫⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠳⢦⢩⢐⡋⠀⠀⠀⠀⣉⠀⠀⡴⡞⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠉⠀⠒⠀⠐⠒⠀⠉⠁⠀⡀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠋⠋⠉⠙⠓⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⢏⣤⣀⣠⣀⣀⡀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃⠈⠀⠀⢽⠎⠁⠐⠦⠙⣇⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⡠⣶⠾⡤⢪⣭⠒⡄⠸⢆⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡛⠓⡄⡀⠀⠀⠀⠀⢳⠭⢴⣇⢻⣿⠁⡸⠀⡟⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡴⠷⡖⠃⢰⣌⠙⢆⣄⠀⠀⠘⣤⡀⠈⠑⠒⠊⢁⠴⠣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⣤⡀⠈⢤⣅⡈⠋⠲⠿⡛⠦⠤⣀⠀⠀⠀⠐⠛⡠⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⢴⢳⠦⡤⣄⣙⠲⢄⣉⣀⡀⠲⠀⡀⢶⠤⠀⠑⡄⠀⡼⠉
⠀⠀⠀⠀⠀⠀⠀⢱⠦⠤⢼⣞⣶⣣⣎⠼⡆⠀⠀⠉⠉⡭⢶⣙⠈⠀⢀⡰⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⡸⢍⣒⡒⠳⠤⠯⠉⡹⠃⠀⠀⠀⠀⠱⣄⣀⠉⠉⢉⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠬⠭⠭⡝⠋⡆⠠⠭⠄⠀⠀⢠⡊⠀⠈⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠑⢲⠗⠒⠒⠋⠙⠤⠤⠚⢱⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣚⡴⠂⠀⠀⠀⠀⠀⠀⠘⠦⢬⣍⣉⡚⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠒⢦⠈⡁⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⠀⠀⢀⠎⠉⠛⢩⠤⠤⠇⡠⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠸⠞⠑⠲⡀⠀⣻⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠿⡁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣄⢠⡄⣠⢦⢠⡄⡄⢤⠄⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⢰⡿⡾⡇⣿⢸⠀⣷⡇⢸⠀⡷⠀⡾⠇⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⡠⠁⠸⠀⠀⢀⠀⠀⢀⠐⠁⠉⠐⠐⠄⡀
⠀⢀⠠⠀⢀⠀⠠⣀⠘⠃⠀⠀⠸⠇⠀⠇⠹⠾⠀⠻⠀⠸⠄⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⣀⢀⠐⠀⠀⠀⠀⡱⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⡐⠾⠿⠿⠿⠛⠈⠁⠀⠑⡄⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠒⠒⠀⠒⠒⠒⡄⠀⡀⠀⠀⡡⡀⠈⠀⠀⠀⠎⠀⠀⠀⠀⠈⠁⠈⠁⠈⠁⠈⠁
⠀⠁⠀⢠⣤⣤⡄⠀⠄⠀⣀⠂⠀⠀⠈⡀⢀⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠠⠀⠀⠀⠠⠰⠁⣀⠀⠋⠁⠀⠀⠀⠄⠐⠀⡠⠠⠀⠀⠀⠀⡠⠔⠂⠂⠀⠒⢄
⠀⠐⡀⠘⠿⠟⠀⡐⠉⠡⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⢠⠔⠁⠈⠐⡴⠤⠀⠌⠁⡘⠀⠀⠉⡄⠀⠊⠀⠄⠀⠀⠐⠂⠠⠆
⠀⠀⠀⠁⠐⠂⠈⠀⠀⡄⣾⣿⣿⢿⡿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠀⠀⠀⣄⢠⢀⢀⠀⠀⠀⠀⠀⠀⡒⠁⢲⠀⠀⣄⠀⣁⠀⠀⠀⣈⠀⠲⠀⠀⠀⠐⠀⡈⠔⠁
⠀⠀⠐⠀⠊⠢⡀⠀⠀⡇⣿⣿⣿⣽⣿⣽⣿⣿⣿⠀⠀⠀⠀⠀⠇⠀⠀⠉⠘⠘⠀⠀⠂⠀⠁⠀⠀⣁⠈⠉⠈⢁⡨⠀⢡⠈⠉⠀⠤⠀⠀⠀⠉⢀⡈⠀⠀⡀
⠠⠈⠀⠀⠀⠀⠈⡀⠀⡇⢿⣷⢟⣹⣳⣭⣛⣿⡏⠀⠀⠀⠀⠀⢠⡜⠀⠐⠂⢒⠒⣒⣀⣒⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⡄⠀⠀⡄⠀⠀⢀⠶⢿⣆
⠀⢧⡀⣀⠀⣀⢸⠁⠀⠇⠛⠛⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠀⠀⠀⠠⢄⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠐⠐⠘⠀⠀⠀⠄⠀⠀⠑⣮⡀⠀⠆
⠀⢸⡇⠟⠠⣿⠐⠀⠀⠸⠛⠉⠉⠉⠉⠉⠉⢁⠈⠉⠻⠅⢀⠄⢂⠤⣠⠀⡠⢄⠀⠀⢀⠤⠘⣃⠂⢀⠀⠀⠀⡠⠒⠐⠀⣄⠀⠀⠐⠄⠀⠐⠀⣀⢀⣿⠟⠋
⠀⢸⡇⣀⢠⡆⠆⠀⠀⠀⠰⠀⠀⠈⠁⣀⡀⠀⡆⠀⠀⢀⠂⠀⠄⠀⠀⠈⠀⠀⠀⠰⠁⠀⠀⠓⠄⢂⠱⡀⠀⠀⠀⠀⢤⢡⡄
⠀⠸⠇⣿⣸⡧⠀⠀⠀⠀⠀⠀⠀⠀⣈⡡⠖⠀⠠⠀⠪⡁⠀⢠⠚⢠⠀⢃⠀⠜⢀⠀⠇⡠⠉⡠⠄⡴⠀⠌⠂⢄⡀⠀⠈⠈⠁⠀⢀⠘
⠀⠀⠀⠀⠀⣀⣀⣄⡀⠀⠀⠃⠀⠀⣉⡀⢄⠀⠀⠀⠀⠈⠢⢀⠉⠀⠀⠘⠒⠒⠁⠀⠁⠀⠃⠒⢉⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⡶⣶⡶⢶⣶⠲⠒⣶
⠀⠀⠀⣴⡾⠟⣛⡛⠿⣷⡀⠸⣀⣀⠀⠤⠤⠤⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⣿⣶⣧⡴⠃⠘⢿⣴⣤⣿
⠀⠀⣾⢫⡾⠛⠙⠿⣷⡜⣷⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⢦⠚⢰⠀⠀⠀⠀⠀⢀⡤⠀⠓⠉⠉⣩⢀⡑⠀⠀⠀⡀⠀⠀⠀⢰⠀⠀⠀⢈⠀⠠⠀⠀⠄
⠀⠀⠋⠋⠀⠀⠀⠀⠀⠙⠙⠀⠀⣾⢿⠿⣿⢿⡿⠛⢿⣇⠀⠀⠒⠁⠀⢠⠊⠆⠀⠈⢀⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⢄⣤⠀⠀⠀⠀⢀⣆⠀⠀⠀⣿⠀⠀⠉⠘⡇⠐⢸⣿⠠⣦⠞⠷⠀⣠⣤⡀⠀⢀⠀⠈⢁⠔⠈⠉⠢⠄⠔⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣿⣧⣄⠀⣠⣾⣿⠀⠀⠀⣿⠀⢸⠀⡄⠇⠸⢹⣿⠀⠀⠀⠀⠶⢏⣩⠀⠀⠀⠉⠒⠁⠀⡀⠀⠀⠀⠀⠀⡀⠈⠐⢒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠐⠀⠈⠛⢙⠇⠀⠘⢋⡉⠀⠀⠀⣿⣷⣿⣾⣷⣾⣦⢾⣿⠀⠀⠀⢀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠜⠗⠈⠀⠀⠀⠐⠨⠆⠀⠘⠄⠀⠀⠰⢆⣀⡉⣀⢀⢈⣉⡁⠈
⠀⠀⢀⡈⠁⢄⠀⠀⠀⣈⡀⠀⢀⣿⡇⣺⡆⡇⠁⡿⢸⣿⠀⠀⡞⠲⠶⠶⠶⢶⡄⢄⠀⠀⠀⠸⠀⠀⠔⠔⠀⠀⠀⢰
⠀⠀⠾⠽⠃⠀⢰⣶⣖⡯⠷⠀⠉⢿⣷⣾⣧⣶⣿⣶⣼⡿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⡄⢠⠌⠀⡀⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢸⡀
⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⢉⣁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⠀⢀⡄⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠏⠀⠸⡆⠠⠀⠀⠎⠀⠀⠀⠑⠀⠀⠑⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠃⠀⢀⣀⠀⠜⠁⠀⢀⠀⠉⠄⡰⠀⠀⠀⠀⡄⠀⡴⠁⠄⠀⠀⣰⣿⣿⣶⣆⠀⠀⠀⠀⢀
⠀⡄⠂⠉⠉⠉⠉⠐⠄⠀⠀⢀⡼⠀⠀⠀⠀⠠⢄⠀⢰⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⢻⠿⠛⠉⠀⡔⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⡅⠀⠠⢤⣦⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⠀⢀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠈⠇⠀⠀⠀⡇
⠀⠀⠀⠀⢀⢀⠀⠀⢄⠀⠀⠁⠀⠀⠘⠉⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡠⢛⡡⠊⠀⠀⠀⠀⢀⢄⣠⡐⠤⠀⠀⠡⡀⢀⠔⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠙⠟⠀⠀⠡⠀⠀⠀⠀⠠⢀⣀⠠⢠⡀⠀⠀⠁⠀⠀⠀⡐⠡⠔⢋⠀⠀⠀⠀⠀⠀⢡⡿⢏⣻⠆⠅⢀⠔⠀⠈⠢⡀⠐⠅⠙⠉⠆⠀⠀⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠉⠉⠐⠘⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢊⠀⠀⠙⠟⠁⠀⠀⠸⡀⠀⠀⢀⠁⠀⠀⠀⢶⡆⠀⠀⢰
⠀⠘⠛⠛⠉⠉⠉⠉⠁⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠐⠒⠂⠄⠂⠀⠀⠀⠀⠑⠠⠔⠁⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠻⠲⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠖⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠈⡆⠀⢄⠠⡀⠀⠀⠀⠀⠀⠀⣠⠖⡩⠐⢀⠠⠐⠂⡉⠄⢂⠀⠀⠀⠀⠀⠰⠘⡄
⠀⠀⠀⠀⠀⢠⠇⡀⠠⡑⠨⢑⡒⠤⢤⣀⡔⠁⠀⠔⠈⣀⠄⠂⠁⡀⠔⠁⠀⠀⠀⠀⠘⡄⠀⠘⡄
⠀⠀⠀⠀⢀⠎⠃⠃⠀⠈⠒⠠⢀⣉⠐⠂⠨⢙⠲⢤⡈⠀⢀⠔⠈⠀⠀⠀⠀⠀⠁⠀⠀⠰⡀⠀⠸⡄
⠀⠀⠀⢀⠎⠌⡘⠠⠀⠀⠀⠀⠐⠠⢈⠁⠒⠄⡉⠢⡙⢎⢀⠔⠀⡠⠀⠀⠀⠀⠀⡀⠀⠀⢃⠀⠀⢣
⠀⠀⠀⡎⡜⠠⠁⠂⠀⡄⠀⠀⠀⠐⢄⠉⠂⠀⠘⡄⠱⡈⣇⠀⠊⢀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠇⢸
⠀⠀⡸⠰⢀⠃⡈⢀⠀⠁⠀⠀⠀⠑⠄⠑⠀⠀⠀⠰⠀⠃⢹⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡸
⠀⠀⡇⠇⡄⠀⠁⢸⠀⡀⢀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢲⠃
⠀⠀⡇⠀⠃⠀⠀⠀⠆⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⢃⠟⡇⠀⠒⠀⠀⠀⠊⠀⢀⠔⢁⠀⡰⠃
⠀⠀⢇⢇⠘⠄⠀⠀⠈⠢⡑⠀⠀⠀⠀⠀⠀⠂⡶⡡⢃⠞⣰⠃⠀⠤⠀⠀⠀⠀⠀⠀⢐⡡⠚
⠀⠀⠘⢎⠢⡈⠐⠤⣀⠀⠠⠄⠀⠀⠀⠀⠀⢀⠔⡡⠋⣰⣏⠀⠀⠀⠀⠀⣀⣀⠤⠒⠉
⠀⠀⠀⠈⠳⢌⡐⠢⠄⢉⣀⠐⠂⠀⠀⠀⣀⡤⢺⣿⡴⠋⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠶⢶⣖⣒⣲⣒⣚⣻⣿⡿⢿⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠉⠉⠉⠉⠛⠷⠶⣶⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⣨⣝⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⡴⠣⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⡿⠀⠹⣤⠶⠙⣆
⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣄⠀⠀⡼⠁⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⣴⡳⠋⠀⠀⠀⠸⡄⢠⠘⡇
⠀⠀⡠⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠙⢟⣮⠳⣼⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⢠⡼⠋⠀⠀⠀⠀⠀⠀⢳⠀⠀⢳
⠀⡐⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠐⡻⠋⠀⠀⢈⢆⣱⡟⠀⠀⠀⠀⠀⠀⠈⢁⣴⣒⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠇⣹
⢠⠇⠀⠀⠀⠙⢿⡆⠀⠀⠠⡴⠊⠀⠀⠀⠀⠀⣼⡀⣧⠀⠀⠀⠀⠀⠀⣠⠿⣶⠜⣩⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⣿
⢸⠀⠀⠀⠀⠀⠀⢠⡶⣟⣶⡀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢠⡾⢋⣀⠈⣼⠇⠀⢠⣄⠀⠀⠀⠀⠀⠐⡞⠐⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡽⠏⠙⢷⣄⠀⠀⠀⠀⢼⢃⠸⡆⠀⢀⣼⠟⢠⣿⡟⣰⡏⠀⠀⠈⠣⡽⢄⠀⠀⢠⠎⠄⠆⢠⠃
⠸⠀⠀⠀⠀⠀⣴⡷⠈⢻⣄⠐⢟⢿⣷⣄⠀⠐⣎⡈⢧⠉⣴⠟⠁⠠⣹⠏⣰⡏⠀⠀⠀⠀⠀⠈⠪⠵⠠⢊⢄⡐⢁⠎
⠀⢆⠀⠀⠀⣴⡟⠀⠀⠀⢻⣆⠀⠑⢿⡿⣝⣦⡀⠾⣾⣿⠋⠳⢄⠛⠁⣰⡟⠀⠀⠀⠀⠀⠀⠀⣠⣔⣕⣥⣎⣤⢏⣠⠤⠤⣀
⠀⠈⠣⡀⠚⠋⠀⠀⠀⠀⠀⠹⣆⠀⣀⣠⢌⣿⣷⣿⣟⠀⠤⠤⠄⠙⠒⠛⣷⠂⡌⠁⠙⡏⠉⠉⣩⠟⠁⠀⠀⠀⠈⠀⠀⢀⠤⠑⢠⣤⣀⣀⣀⣀⡀⡠⠐⠲⠂
⠀⠀⢀⠈⠳⠦⠤⠀⠀⠐⠒⠂⠉⠉⠉⠁⠀⠀⢀⢉⡻⠖⢠⣤⣴⢶⡖⡂⣿⠀⠁⠀⢰⠆⢀⢤⠋⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠆⠀⠀⠀⠀⠀⢰⠇⠀⠇⡉
⠀⠴⢡⠀⠀⢀⠀⠀⠄⠀⠤⠀⠀⠒⠒⠒⠒⠢⡌⠺⢃⠐⢸⠀⣠⣼⣿⡇⣿⠀⠀⠀⡀⠀⢈⡏⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⢰⣆⣇
⡆⢀⢺⠀⢠⡇⢤⠆⠀⠈⠡⠇⠀⠀⠀⠀⠀⠀⡀⠈⠉⠀⢸⣰⣭⣿⣿⡇⣿⠀⠀⠀⡇⠀⢸⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠈⠀⠂⠘⠷⡚⠛⢛⢛⡚⠳⠤⠯⠤
⡇⣸⢸⠀⢸⡇⢸⡉⡉⠁⠈⠂⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠤⠤⡀⠀⢻⠀⠀⠀⡇⠀⡇⠀⠀⠀⢸⠀⠀⣰⠈⠈⢥⣄⠀⠄⡴⠒⠳⠎⠀⣠⠥
⡇⣼⢸⠀⠈⡂⠘⠂⠒⠂⠚⠀⠀⠀⠀⠀⡆⠠⠂⠀⡄⢠⣠⠀⠌⡣⠆⠀⢸⡆⡂⠀⡇⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⡄⢲⢾⠠⢸⠀⠀⠈⠀⠀⡟⣶
⡇⢼⠨⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠧⣀⠀⠰⠤⠃⠀⣀⠀⠀⠀⢸⡆⠀⠀⡇⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⣇⠈⣿⠘⠸⡀⠀⠀⣇⠀⣱⢇⠂
⡇⠾⢈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⡆⠀⠀⢀⠞⢩⠄⠉⠲⡀⢸⡇⠀⠀⣧⠀⡇⠀⠀⠀⢸⠀⠀⠙⠲⠼⠷⠋⠀⠑⠌⠟⢻⠛⠂⠘⠂
⡇⠨⡆⠀⠨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⠃⣠⠔⠚⡃⣛⡄⡆⠀⡇⢸⡇⠀⠀⣿⠀⢿⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢠⢖⡔⠚⢣⠾⢰⠂⠀⠘⠛⡆
⡇⡗⡇⠀⠀⢠⣰⣖⣲⠦⠦⠤⠤⠽⠥⠤⠤⠊⡜⣡⠋⠀⠡⣈⠉⢁⡴⠁⢸⡇⠀⠀⢸⠀⢸⡆⠀⠀⠀⢳⠀⠀⠀⠀⠀⠂⡌⠀⠀⡇⠀⡇⠀⠀⡄⣆⢸
⡁⠷⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣿⡇⡼⣸⠃⠀⠀⠀⠀⠉⠁⠀⠀⠘⡇⢀⠀⢸⡆⠈⠿⠃⠀⠐⠻⣧⡆⠀⠀⠀⣧⣿⡆⠘⣧⠐⣷⢂⠐⣿⣿⣿
⠈⠻⡟⠈⠉⠉⠍⠩⠥⠤⠤⠤⠄⣀⣐⣛⣳⢣⣏⣀⣀⣤⣤⣤⣤⣤⣤⣤⣠⣧⠘⠀⠀⣿⠀⠰⠶⠀⠀⠀⠻⠷⠂⠀⠀⠈⠿⠿⠶⠿⠿⠿⠶⣦⣿⠿⠟
⠀⠀⠉⠒⠚⠻⠿⠿⣿⣿⡿⠟⠛⣉⡭⠥⡏⣼⠁⠤⠄⠠⡀⠒⠒⠒⠒⠐⠒⠘⣛⡒⠚⠛⠓⠒⠚⠷⠶⠶⣤⢲⡿⠶⠶⠶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣶⡈⠁⣤⠀⢠⢡⡏⠀⠀⠀⣐⠟⣀⣀⣀⢀⣀⣀⣀⣛⠚⠀⠀⠀⠀⠀⠀⠐⣶⡨⠏
⠀⠀⠀⠀⠘⢇⠙⠃⢊⡙⢻⣿⣄⠐⢐⣡⣾⡿⠟⢻⢿⠿⢿⠛⠛⠙⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣶⣶⣦⣤⡴
⠀⠀⠀⠀⢇⣸⢀⣀⣀⣂⡾⣟⠌⠉⠀⠀⡈⠀⢠⠟⠎⠀⠀⠄⠀⠀⠸⡸⣿⣭⠩⣯⠘⡏⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⠃⠀⡞⡼⠀⠀⠀⠘⡄⠀⠀⢻⢻⡿⣀⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣸⢱⠃⠀⠀⠀⠀⢱⠀⠀⠀⢏⣷⢉⣿⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣀⡀⠙⠊⠁⠀⠀⠉⠉⠉⠀⢠⣄⠘⡘⣯⢿⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠴⢶⣄⠀⢠⠃⠹⠋⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠚⢛⡟⠉⣧⠠⠨⠷⠄⠀⣀⣀⡀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⣠⡗⠀⠀⠀⢸⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⡇⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡔⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⡏⡆⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⢀⠎⡜⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⢶⡶⣄⡀⠀⠀⠀⠹⣦⠀⠀⠀⡇⠸⡆⠀⠀⠀⠀⢠⡞⢸⠀⣰⠃⡜⠀⠀⣴⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠲⢄⡀⠀⠹⣷⡀⠀⢠⠀⢻⡀⠀⠀⢠⡏⠀⡇⢰⠃⡾⠁⢀⡞⠁⣺
⣀⣀⣀⣀⠀⠀⠀⠀⠀⠘⢦⡀⠙⢦⣄⠹⣷⡄⢸⠀⠀⢳⡀⢀⡞⠀⢸⢠⠃⣼⠃⢠⠎⠀⠀⡇
⠉⠉⠉⠉⠛⠛⠷⢦⣤⣀⠀⠙⠢⡀⠈⠳⣭⣿⡄⡆⠀⠀⠱⡜⠀⠀⣸⠆⢰⡏⢠⠏⠀⠀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣤⣈⠠⠀⠈⢻⣟⣷⡄⠀⢀⠃⠀⢠⡿⠀⡏⢠⠏⠀⠀⢠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣓⢷⣄⠀⠙⡌⢳⡀⠘⠀⠀⢸⡇⢸⢠⡟⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡝⢷⡄⠘⠀⠹⡆⠀⠀⠈⠑⣼⡏⠀⠀⠀⡸⠀⠀⠀⣀⠤⠴⠒⠉⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⢦⠹⠄⠁⠀⠀⡀⠀⠸⠁⠀⠀⡰⠁⣠⠶⠋⠁⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢷⡘⡄⠀⠀⠉⢠⡄⠀⠀⣸⡵⠋⢀⣀⡤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⡡⠀⠀⠀⢨⠀⢀⡾⠋⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢱⢃⠀⣠⡌⢀⡟⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⡏⠻⠃⠹⠇⢹⡀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠑⢄⠀⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠳⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⢇⣤⣼⣶⣄⢶⣀⡴⠖⠚⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣛⣉⣉⣀⣿⣁⣽⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⢠⠞⠁⠀⠈⠠
⠀⠀⠀⠤⠤⢀⣀⣀⣀⣀⡰⡄⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⡀⠀⠀⢀⠁⠀⠀⠀⡤⠈⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⣠⠌⠉⠁⠄⠰⣏⠜⣓⠲⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠈⠢⠐⢇⠀⠀⠀⠀⠀⠀⢠⠀⢀⠔⠁
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠠⠔⠲⢄⠙⢿⣦⣤⠤⠤⢤⣄⡀⢀⡴⠒⢻⡟⠛⢲⣀⠀⠀⢀⣀⠑⠄⠀⠀⢤⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⡞⠀⠀⠀⠆⠀⠀⠀⠀⠀⢹⠞⢹⠇⠀⠀⠀⠀⣹⠎⠀⠀⠉⠀⠀⠀⡈⢲⣄⡜⠉⠓⠀⠀⠀⠀⠢⠀⠀⠀⠠⠀⠀⡴⠃⠀⠈⠓⢦
⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠎⠀⠀⠀⠀⣰⠃⢈⠔⠀⠗⠀⠀⠀⢂⠀⠈⡙⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⣔⠋
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠙⢆⠙⢀⠠⠀⠀⠀⠀⠀⠀⢀⠐⠉⠀⠀⠀⠀⢀⡞⠀⠐⠓⠉⠁⠀⠐⠂⠀⠀⠈⠁⢠
⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡦⣀⠀⠀⠀⠀⠈⠳⡀⠰⣄⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⣠⠄⠃⠐⠒⠒⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠈⠂⠄⣀⣀⣀⠤⠴⠚⠉⠀⠈⠑⠢⠤⣀⣀⣀⡽⠀⠀⠁⠀⠐⠁⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣛⡍⠉⠓⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢻⣷⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠘⣿⡄⠀⠀⠀⢀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣦⡀⠀⠀⠸⣿⡄⣿⡇⠀⠀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⠀⣿⡇⢸⣿⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡝⢿⣦⣿⡇⢸⣿⣴⣿⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⢻⣿⡇⢸⣿⡿⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡈⣿⡇⣾⡟⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢸⣷⡿⢡⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢻⡇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢸⣿⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢸⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣏⢠⣶⣸⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⢹⣿⣿⣿⣿⣟⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⣩⠋⣠⣾⣿⣿⢹⣏⣟⣿⣿⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⢋⡵⠊⢁⡜⡱⢻⡏⢻⠈⡟⡜⣿⢯⠳⡈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢉⡴⠊⣴⣠⢏⡜⢁⠇⡿⢸⡆⡇⢹⡘⣇⢳⠙⢦⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣠⠞⠁⣠⠞⢡⠞⠀⡎⢰⡇⠀⡇⢹⠈⣧⢹⡎⢳⡾⢻⡄⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⣰⠋⢀⡞⠁⢠⠏⠀⡸⠀⢸⠇⠀⡟⠋⡆⢹⡍⠘⡄⢧⠀⠹⡄⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⠃⢠⡞⠀⢀⡏⠀⢠⠇⠀⣼⠀⠘⡇⠀⡇⠸⡇⠀⢱⠘⣇⠀⢹⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⢸⠀⢸⠁⠀⢸⠀⠀⣾⠀⠀⣧⠀⠀⡇⠀⢷⠀⡇⠀⢸⠀⢸⡄⢸⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢸⠀⢹⠀⠀⢸⠀⠀⢿⠀⠀⣿⠀⠀⣧⠀⣸⠀⡇⠀⢸⠀⢸⡇⢸⠀⢀⣿⡇
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠚⣆⠘⣇⠀⠸⡄⠀⢻⡀⠀⢹⠀⠀⡟⠀⡇⠀⡟⠀⢸⠀⣼⢡⡞⣂⣼⡿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠘⣧⠙⢶⠲⠻⡄⢀⣣⠀⢸⡇⠀⡇⢀⡇⢸⣇⢀⠷⣴⢃⡾⢀⣾⡟⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣈⠳⣌⠓⢤⡙⢦⠉⣆⠈⣇⢰⡇⡸⢡⡟⢀⡟⡰⣣⠞⣡⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣮⣷⣤⣙⢮⣳⣌⣆⡟⣼⢠⢃⠎⣠⣿⣾⣟⣥⣾⠟⠁
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⡟⠋⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠡⠾⠋⠟⠋⠉⠙⠛⠉⠛⠷⠶⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⢁⣀⣤⠴⠶⠒⠒⠒⠒⠶⠦⣤⣀⡈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣤⠞⠋⠁⠀⠀⢀⣀⡴⠶⢤⡀⠀⣀⠈⠙⠳⣤⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⢋⣠⡴⠚⠓⠒⠛⠉⠉⢡⡤⠤⠙⠉⣉⡙⠳⢦⣀⠙⢦⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠟⢁⡿⠀⠀⣟⣉⠀⠀⣀⡤⠦⢽⣥⡀⠞⢉⠉⠀⠀⠙⣦⠀⢻⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢰⠏⢠⡟⢡⡗⠀⠈⠉⣠⡾⠉⠀⠀⠀⠀⠀⣠⠟⢠⡿⣦⠀⢸⣶⠏⣽⡆⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣿⣤⠟⢀⡾⠁⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⢰⡏⣠⠟⠀⠈⢳⡄⠀⠀⣿⢻⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⡀⢠⠟⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⢿⠀⡀⣿⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡿⠃⢿⠀⠀⣸⠃⠀⠀⢀⡴⠶⣄⣠⠞⠁⠀⠀⠀⠀⠀⠀⣠⠟⠀⡇⣸⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣆⠀⠘⣇⠀⠹⡄⠠⠴⠟⠁⠀⠈⠡⣤⣀⡀⠀⠀⠀⠀⠸⣇⢀⡼⠃⡿⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣿⢷⣀⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⡄⠀⡿⠾⠁⣸⠇⣼⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠘⣆⠉⠙⣦⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠃⣸⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⢧⡀⠈⠛⠛⡷⠘⠷⣤⣄⣠⣤⣤⣤⣤⠤⠶⠃⣠⡖⢸⡇⢀⡼⠃⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⠻⣤⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⣋⡴⢟⣤⠟⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠙⠾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡴⣚⣩⠶⠋⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠉⠛⠓⠶⠶⠦⠴⠶⠶⠛⠛⠉⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠶⢤⣤⣤⣤⣤⡤⠶⠶⠛⠉⠁






⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠓⠀⢀⢰⠂
⠀⡇⠀⠁⠁⠀⡈
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠘⠙⠯⠃⢀⠀⠀⠀⠤⠅⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⡆⠀⠀⠀⠀⠀⠀⠀⣠⣩⣉⣙⢓⢀⢀⣤⣾⡷⠀⠀⠀⡄⣶⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠑⠒⠇⠀⠀⠀⣀⣀⠳⠶⠶⠮⢭⠽⠄⢠⠀⠀⠀⠀⢀⠀⠀⠁⠈⠁⠃
⠀⠀⠀⡀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⡉⠀⠀⢀⠖⠀⠀⠀⠨⠁⠀⠮
⠀⠉⠀⢀⡀⠀⠠⠀⢀⠀⠀⠀⣐⡨⠋⠈⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠀⠐⠁⠀⠀⠐⠒⠒⢲
⠀⠀⠀⠀⡀⠀⠆⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠤⠀⠘⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠤⠀⠀⠀⡀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡢⣲⠸⠿⠑⠀⠀⠀⠀⢀⡄⢶⠙⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣎⡙⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠠⡖
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡃⢠⡁⠀⠀⠐⠒⠧⠤⠴⠖⢛⠀⠀⠏⠀⠐⠂⠰⠖⠂⣸⠀⠀⠘⠐⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⡈⠐⠒⠛⢫⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠲⡄⠀⠀⢀⣀⡰⠟⠀⠀⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠘⠭⣓⣖⣒⣋⣥⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠠⠄⠛⠶⠾⢟⣠
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢀⣤⣾⡇⠀⠀⠀⠀⢀⠀⠀⠦⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠤⠞⠛⠁⠀⠀⢀⡀⠀⠀⠀⠈⠁⣩⠷⠦⢤⢤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠔⡀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠺⠷⠆⠀⠈⠉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢀⡀⡀⠀⠀⣀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⣷⣶⣤⣀⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠬⠁⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠲⣜⢿⣿⢟⡟⠆⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⣀⠀⠀⠀⠰⣴⠄⣄⣀⡀⣠⣤⣜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠒⠀⡉⠣⠄⠀⠀⠀⠀⠀⠀⣑⠀⣀⡀⠈⠋⣻⣋⠀⠀⠀⠀⠀⠠⡁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠈⠢⡄⢀⠠⠀⠶⣌⣀⣙⡲⠀⠀⠀⠘⢷⣤⣠⣤⢽⣿⣿⣻⠀⠀⠀⠐⡿⠢⣈⠛⠿⢿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡙⠁⠀⠀⠀⠉⠁⠈⠁⠚⠋⠀⠀⠲⡇⠀⠀⠀⠚⠑⠛⠓⠂⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠘⠖⠘⠐⠛⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠠⠤⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⢰⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣊⡀⠘⠄⠱⠀⠀⡀⠀⠀⠀⣠⠊⠁⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠴⠴⣌⠀⠑⢄⡌⠉⠉⢲⢃⡆⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠀⠀⠀⠀⢀⠤⠤⣽⡇⠀⠀⢡⠀⠀⠾⡆⡁⠤⠊⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠠⢀⣂⠑⢤⠀⠀⠐⠌⣩⠈⠁⠀⠀⠀⠑⠀⠘⠓⢒⡒⠈⠉⠉⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠈⢢⡰⠁⠀⢀⠔⠁⠀⠀⣀⠤⣀⡖⡰⠂⡤⣈⡁⠂⠀⠀⡤⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⣗⡤⠐⠁⠀⢀⡤⠊⢀⠀⠀⠈⠑⠚⠤⢆⡨⠓⢤⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠹⣇⠀⠀⢠⠿⢦⣀⠄⠺⣀⢀⡐⠀⠘⠀⠟⠀⠤⠠⡀⢄⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠋⠁⠀⠀⠀⢃⠢⡀⢗⠈⠦⠔⢨⣠⣋⠀⠈⠠⠒⠉⠑⠄⠣⠀⠀⠉⠁⠀⠈⠁
⠉⠤⢀⠄⢒⡤⠛⢉⡹⠔⠚⢆⠀⠀⠀⠀⠀⠀⠀⢂⠙⠦⡑⠛⠫⠧⠇⠚⡤
⠀⠀⠒⠐⠙⡀⡴⠋⠀⠀⠀⠀⠑⢄⠀⠀⢠⢄⠀⠈⠑⠒⣬⠒⠀⠀⣠⢎⠑⠒⡰⣆⠀⠀⠀⢰⡡⣬⡂⣤
⠀⠀⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠈⠢⣀⣮⠁⠀⠀⠀⠀⠸⢅⠼⠳⡀⠀⠀⠀⠘⠐⢮⠥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠘⠓⠂⠄⣀⡀⠠⠌⣈⣄⠀⠀⠀⡠⠁⠰⠀⣕⡹⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⡀⠈⢦⣀⠤⠤⠤⣤⠀⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡄⢀⡀⠔⢹⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⣀⡴⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠤⠤⠦⠀⠤⢤⠦⠴⠦⠤⠤⡤⣤⠤

⠀⠀⠠⢴⡶⠶⠦⢀⣶⣤⣤⣤⣠⡆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠄⡀⢄⠀⠀⠀⠀⠀⠄⠴⠦⠀⠀⠀⠀⢀⠠
⠀⠀⢀⡾⡵⣶⢰⢦⣥⢶⡆⣰⠆⠀⠀⠀⠀⠁⠥⠈⠔⠀⠀⠀⠙⠶⠛⠁⠈⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠉
⠀⠀⡾⠹⠴⢧⠯⠞⠯⠾⢴⡾⠓⠲
⠀⠀⠀⢠⠀⠏⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢖⣶⢦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⡱⠔⠠⠀⠀⠀⣦⣶⣶⣴⠀⠀⠀⠀⣺⣿⢿⣖
⠀⠀⠀⠀⢣⣀⡤⠔⠂⠀⣤⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠈⠀⠀⠀⠀⠉⠈⠁⠉⠀⠀⠀⠀⠈⠀⠈⠉
⠀⠀⡄⡰⣯⢯⣟⣦⣄⣂⡥⣬⡖⢄⠀⠀⠀⣠⣶⣠⣠⡄⠀⠠⢄⣤⣴⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⣤⣤⣼⣠⣄⣤
⠀⠘⡈⠄⣣⠠⡯⣶⣉⠿⣶⡜⠕⡸⠀⠀⠀⠁⠈⠙⠉⠀⠀⠈⠈⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠚⠛⠛⠻⠛⠃⠀⠰
⠀⠀⠈⠑⠂⠩⠭⠁⠈⠬⠅⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠀⠐⠀⠀⠀⡐⣶⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⠀⠀⠀⠀⢐⢳⠄
⠀⠀⠀⢠⣖⣴⢦⣶⣦⣶⢶⠶⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⢀⠀⠀⠀⢈⣀⡓⠀⠀⠦⢴⠧⠶⠀⠠⠀⠀⡓
⠀⠀⠀⠉⠉⠉⠁⠉⠉⠈⠉

⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣏⣹⢦⢀⡴⢿⣉⡷⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⠠⠄⠆⠆
⠀⢀⣴⢿⣿⡁⠈⣿⣷⡄⠀⠀⠀⠀⠀⢻⠞⢧⠷⣴⠛⡷⢻⡼⢦⡇⠀⠀⠀⠀⠀⢸⣬⡇⠀⠀⠀⠀⣧⣼⠀⠀⠀⠀⠀⢀⣘⣈⡅⣁⢀⠀⡀
⠀⣸⣇⣀⣽⠟⢿⡏⠉⢻⠀⠀⠀⠀⠀⠀⠙⣿⣶⣥⣤⣬⣿⡗⠋⠀⠀⠀⠀⠀⠀⢸⣤⡇⠀⣷⡦⠀⣧⣼⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⠄
⠀⢸⣿⠋⠛⣿⠟⠻⣶⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣾⡇⠀⠀⠀⠀⠀⢸⣴⡇⠀⠋⠀⠀⣧⣼⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⢦⣶⣿⣦⣤⠟⠁⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⡏⠙⠃⠀⠀⠀⠀⠀⠸⠴⠧⠤⠤⠤⠤⠧⠼⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⣀⡀⣀⣀⢀⡀⣀⣀⠀⠀⠀⠀⢰⣷⣿⣿⣿⣾⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣠⣄
⠀⠀⢀⢼⠂⠀⠀⠐⢦⠀⠀⠀⠀⠀⢸⠒⠓⠛⠚⠚⢳⠛⠚⠚⠓⠓⢺⠀⠀⠀⠀⣸⣟⣛⠛⢛⣛⣻⣄⠀⠀⠀⠀⠀⣠⣴⣾⣏⣀⣛⣃⣙⣿⣦⣄
⠀⢀⣞⡌⠀⠀⠀⠈⢏⣧⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠙⢯⡉⣉⠟⠉⠁⠀⠀⠀⠀⢸⢵⣶⣶⣶⣶
⢀⡞⣰⠁⠀⠀⠀⠀⢘⡌⢷⡀⠀⠀⢸⣒⣖⣶⣲⣲⣞⣶⣲⣲⣖⣖⣻⠀⠀⠀⠀⠀⠀⣠⠟⠣⣄⠀⠀⠀⠀⠀⠀⢸⠸⠿⠿⠿⠿⣀
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠠⠞⠁⠀⠀⠈⠳⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡠⢚⠃⢒⠒⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⣠⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⡀⠀⠀⠀⠀⠀⣀⣠⠤⠖⢚⣩⣿⡇⠀⠀⠀⠀⢺⡗⢲⣶⠛⣶⡖⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀
⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⠀⠀⢸⣿⣋⣥⣴⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⢸⣿⠀⣿⠃⢸⡇⠀⠀⠀⠘⠻⣿⣷⡾⠋⡝⢻⣾⣿⠿⠃
⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣇⠀⣿⠀⣿⠀⣼⠃⠀⠀⠀⠀⠀⠀⢹⣧⣶⣷⣼⡏⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢿⡈⠉⣻⠻⠿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠛⢿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢢⢠⢤⠀⠀⠾⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⡠⠃⠤⢀⣦
⠀⠀⠀⠀⠀⡸⠁⠋⠰⢆⠀⣉⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢈⣶⡆⢰⠾⡏
⠀⠀⠀⠀⠀⢁⣯⠁⠀⡀⠀⠁⠀⠒⡨⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⢀⣠⣈⠈⠉⠀⡅⢈⠘⠁⠁
⠀⠀⠀⠀⠙⠴⠄⠀⠂⠀⠡⠘⠒⠞⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⠀⠡⠄⠘⠚⣀⡄⣡⠋
⠀⠀⠀⠀⢠⣐⢄⠨⠄⡤⠂⠢⢉⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡕⠀⠂⢀⣤⣤⣤⣈⣁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⢀⠈⠀⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⣹⠀⡤⠰⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣻⡤⡀⠱⡅⢄⣨⠼
⠀⠀⡀⡀⠀⠴⠀⠁⡀⠊⢀⡀⢀⠄⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠠⡈⠙⠖⡂⠮⢑⠀⢠
⠀⠘⠀⠑⠈⠀⠀⠅⠙⠂⠠⠝⠁⠀⠉⠠⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠁⠁⠐⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⠀⠀⠀⢀⠠⠄⢀⠀⠀⠀⠀⢠⠻⠞⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠁⠀⠀⢀⠁⠀⠀⠠⢾⠔⠲⠻⠠⣂⢁⠀⠀⠂⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠀⠉⣠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⢁⠀⠀⠠⠀⠈⠀⡟⠶⢤⢖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣭⣝⣦⣶⣻⣧⠤⠴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣴⣀⣳⣆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠉⠂⠀⠒⠈⠀⠀⠀⠀⢨⠞⠍⢁⠖⡦⠄⠀⠀⠀⠀⠀⣳⠉⠉⣁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡵⠋⠀⠔⣥⠟⠀⠀⠈⠢⠀⠀⠈⠅⣙⢋⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⡀⠀⠮⣾⡑⢠⢊⠞⠈⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠰⠀⠀⠈⠀⠀⠉⠒⠈⠀⢀⠄⠦⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⠬⠶⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⠀⠙⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⠄⣀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡖⠲⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⠖⢲⠆⠀⠀⠀⠀⠠⠐⢊⠉⠃⠀⢀⠀⠒⠀⠀⢀⢸⠁⡀⠉⣏⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⠒⠟⠀⠀⢰⣋⣀⣀⡀⠂⢀⣦⠈⠀⡂⢨⠀⠀⢈⠙⠒⠒⠒⠛⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠒⠂⠀⠀⣖⠨⠐⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢴⣄⡀⠀⠈⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⢄⡠⠀⢘⣷⠶⠄⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠀⠀⣻⠉⠉⢘⠀⠄⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣿⠗⠵⠶⢲⡒⡖⢚⠩⠆
⠀⠀⠀⠄⡀⠀⠀⠀⣀⡴⠧⠷⠞⠛⠾⢦⠤⣘⣄⣿⣼⣤⣇⡀
⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⣤⠀⠀⠀⠀⣄⣀⠈⠋⠉⠉⠉⠉⠉⠙⠓⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠓⠲⠤⠐⠞⠉⠀⠰⠈⢂⠀⠀⠀⠀⠀⠀⠀⣀⣀⡁⠂⠠
⠀⠀⠀⣠⠾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠷⠒⠈⠁⠀⢀⠀⠀⠀⠈⠉⠰⠀⠠⡀
⠀⠀⡾⠋⠀⢘⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡡⠀⠀⠀⣶⠆⠀⡀⠀⠀⢦⡼⠀⠀⠀⠀⠐
⠀⡼⠀⢠⡀⠉⠀⠉⡈⠓⠢⢤⣤⣄⣀⣀⣀⣐⣸⠀⠁⠀⢀⠀⠈⠓⠐⠗⠚⠛⠋⠀⠀⠐⠆⠀⠀⡈
⠀⡷⠀⠈⠁⠀⠀⠀⠉⢁⣀⡒⡄⠀⠀⠩⢩⠿⣇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄
⢀⠁⠀⠀⠀⠀⠀⡤⠐⠛⢻⡧⠅⠀⠀⠀⢸⡄⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⡴⠂⢀
⠘⣇⠤⣤⣤⠠⠈⠀⢠⠈⠈⠀⠀⠀⢓⠲⠋⠄⠀⠘⠶⣕⡂⢤⣄⣀⣀⢄⣀⣄⠠⠠⠖⡊⠁⠀⢠⠞
⠀⠘⢷⡠⣤⣦⣄⣀⠀⣂⣤⣤⣴⡶⠿⣊⠕⠀⠀⠀⠀⠈⠙⠛⠶⠶⠧⣦⠤⠖⠒⠂⠀⠀⡠⠖⠁
⠀⠀⠀⠙⠻⣶⣬⣭⣉⣉⡩⠭⠴⠒⠛⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠉⠋⠝⢒⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁

⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣳
⢰⣩⡤⣌⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧
⠀⠀⠀⠈⢳⡀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠘⣆
⠀⠀⠀⠀⠀⢳⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠤⣤⣀⣤⠤⢧
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡴⠋⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢸⠦⣄⣠⣤⠴⠫⢀⠔⠂⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣉⣳⠀⢠⠓⠐⣶⠁⡂⠐⡄⠀⢸
⠀⠀⠀⠀⠘⡄⠀⠀⡁⠰⠀⠈⡆⠃⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠎⠀⠀⠀⠰⡀⠀⣸⠤⠀⠔⢁⡄⢸
⠀⠀⠀⠀⠀⠱⡀⠠⣔⢂⠀⠊⠘⠀⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢸⠀⠀⠀⠀⠀⢸⠛⠒⠒⠂⠉⣀⡆⢸
⠀⠀⠀⠀⠀⠀⠹⡀⢡⣀⠉⠉⠉⠁⣁⣀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠘⢆⠀⠀⠀⠀⠀⢻⠃⠀⠈⠀⠐⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡜⠳⠤⣄⣀⣀⣸⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢸⠈⢯⠀⠀⠀⠇⠀⢩⠀⢠⠊⡉⠉⢣⡤⠒⠠⢄⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⢸⠀⠰⡀⠀⠀⣈⠀⠀⠁⢈⠀⣰⠁⠢⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠣⡀⠀⠀⡠⠂⠁⠀⠈⠉⡏⠉⠒⠄⠔⣊⢰⠃⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡼⠉⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣤⣀⣈⣀⣤⠄⢹⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡾
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠟⠙⠋⠛⡿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣰⠃
⠀⠀⠀⠀⠀⣾⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠚⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⣈⠽⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠊⣠⠊⠀⠀⠀⠆⠀⠀⠀⠐⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡰⠁⠀⠀⠀⡘⠀⡆⠀⠀⠁⢰⠈⢆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠃⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠴⡷⠆
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡠⠥⠐⠒⠒⠒⠒⢂⠐⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⣠⠤⠴⠒⣈⢔⠴⠀⡈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠔⠉⢀⣀⣠⠴⢊⡴⡱⡡⢡⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠤⠄⠀⠀⠒⢸⢠⠃⡔⠊⢉⣀⠠⠤⠊⡜⢱⠁⠘⠀⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⡘⠀⠚⠉⢀⣀⠀⢠⠃⡟⠀⡆⠀⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢋⠏⠀⡼⢀⠇⢀⠇⢀⠃⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠢⡄⠀⠂⠀⠚⠠⠊⡀⠎⣠⠎⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠠⣤⣀⡐⠀⠨⠔⢚⣡⣾⠿⠃
⠀⠀⠀⠀⠀⠀⢠⡶⠐⣆⣀⣀⡀⡆⢀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠘⢧⣠⠏⢸⢸⠇⡇⢻⡸⠇⡇⡿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠘⠛⠛⠛⠿⢧⣤⠀⢀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣿⣆
⠀⠀⠀⠀⠀⣦⣄⣠⣾⡿⠿⠿⠿⠿⣿⠙⠃⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢈⣻⣿⣿⣿⣿⣿⠃⠀⢀⣠⣴⣶⡇
⠀⠀⠀⠀⠀⠿⠛⠛⣿⣧⣤⣤⣤⣤⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠈⠉⢽⣿⣿⣿⣿⠟⡿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣟⠀⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠠⢶⣿⣿⠶
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⡀⢀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠘⠋⠙⠃⠰⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⠁⠸⣧
⠀⠀⠀⠀⠙⠿⢿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⡀⠶⠄
⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⠈⢛⠁⠈⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣿⣥⣾⣧⣾⡇⠀⠀⢀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⡠⡿⢄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡞⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠊⠀⠃⠀⠑⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠐⠁⠀⠉⠉⠉⠛⠛⠻⠦⠄
⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣆⠀⠀⠀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⡿⡿⣿⣿⣿⣿⣹⣻⢻⣿⣿⣿⣿⣿⣟⡟⡿⡇
⠀⠀⠀⠀⢹⡁⠉⠁⢙⡇⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣧⣿⣧⣿⠁⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⡤⣤⣤⢤⢤⢤⢤⢦⢤⢤⣤⣤⡤⡤⡤⡤⡤⡤
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢦⡔⢀⠎⠀⣤⢀⠄⠀⡱⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⠀⠁⠊⠀⢀⠁⠈⠀⢀⠅⡀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⢤⣡⠎⠀⢀⠀⠄⠀⢀⢎⠜⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⡄⠀⢀⡖⠀⠀⠀⡠⠂⠀⢀⠔⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡃⢐⠎⠀⢀⠊⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠐⠀⠀⠀⠀⠁⠀⠠⠔⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⡀⠀⠀⠀⡁⠀⢀⢂⠅⢄⠈⠒⠀⢠⠄⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⡀⢖⠁⠄⠀⠀⠁⠀⢨⡌⠓⠃⠀⠀⠀⠈⠡⠂⠀⠀⠇⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠙⠸⠒⢁⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⡄⠤⢉⢀⡔⠘⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠈⡀⠀⠀⣰⠆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⡀⢉⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⢀⠄⠉⠁⠻⣚⠠⠧⠄⣄⠀⠘⢆⠀⠀⠀⠀⣀⠀⠁⠈
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⡠⠇⠀⠀⠀⡁⠀⠀⢄⡀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠴⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠍
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠎
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢄⡀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣿⣿⣻⣿⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⠄⢀
⠀⠀⠀⠀⣠⣤⢄⣤⣀⡀⠀⠘⡿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣻⣿⣿⡛⠃
⠀⠀⠀⠘⠿⣽⣿⣯⣾⠃⠀⠀⠛⠻⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣴⣄⡈⠀⠘⠳⡄⢠⡀
⠀⠀⠀⠀⠱⢸⠉⠙⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠿⢯⠿⣩⠘⠀⠀⡘⢪⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠐⠃⠂⠷⠀⢲⠙⢦⠤⠶⠓⠃
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢀⣀⡀⠄⠔⠚⠃

⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⡀⢈⡀
⠀⠀⠀⠀⠀⠠⢾⣟⣟⡦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠄⠈⠑⣺⢁⠀⢀⣀⠀⠀⢀
⠀⠀⠀⠀⠁⠐⢀⣀⣉⣁⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⣠⣤⣶⣦⡄⠃⠀⠀⣿⢻⣿⡿⠧⠀⠄
⠀⠀⠀⠀⠒⠈⠒⢀⡘⠈⣵⠀⣀⣤⣤⡐⠉⠀⠀⠀⠀⠀⠀⠈⠙⡟⠀⠛⣄⣤⠿⠁⠀⡝⠀⡄⡀
⠀⠀⢀⣤⡶⣶⣆⣁⣙⠁⢠⠘⠹⣷⠿⢷⡏⡉⠀⠀⠀⠀⢀⠘⠀⠁⡀⠀⠉⣯⠀⠀⠐⠀⠀⠀⡆
⠀⠰⣎⠈⡿⠟⠻⠏⣤⠀⠀⠛⠲⣒⣒⡄⠆⠇⠀⠀⠀⠀⠈⢀⠀⠀⠁⠀⠀⠁⢤⠀⠀⡀⢸⠂⠁
⠀⠀⠈⠒⡭⠍⠆⠋⠏⠀⠀⠑⠠⠅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠑⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠀⢀⡀
⠀⠀⠀⠄⠖⠠⠀⠀⠀⠀⠀⠀⢀⡢⠶⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢒⢇⠈⠀⣄⡀⠀⠀⣤
⠀⢠⢒⡄⠂⠀⠀⠷⢀⠀⠀⠊⠐⡀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⢶⣾⣷⢏⡻⡛⠋⢈⡉⠀⠀⠲
⠀⠀⠀⠀⠀⠒⣌⠀⢀⢃⠀⠀⠠⡇⠀⠀⠀⠰⠄⡀⠀⠀⠀⣰⣷⣾⠻⣿⣆⠁⠛⢷⣾⣽⡔⢃
⠀⠀⠀⠀⢠⣿⠛⢆⠛⢦⡀⠐⠂⠠⠤⢦⡀⢀⠾⠀⠀⠠⠊⠀⡸⠉⡶⣔⡾⢢⡘⠂⣷⡿⠟⠈⢦
⠀⠀⠐⠜⠋⣭⡀⠀⢣⠀⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⠆⣼⠁⢊⠁⠀⠀⠀⣠⡅
⠀⠀⠀⠀⠀⠘⠹⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠘⠀⠀⠀⠈⠀⢹⠀⠸⣝⠀⠀⢀⡿⠁
⠀⠀⠈⠆⡀⠀⠠⠉⣄⠀⣧⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣆⠀⠀⠀⣼⠁⠘⠈⢄⠀⠈⣢⠞⠁
⠀⠀⠀⠀⠈⠁⢄⣀⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠀⠸⠃⣀⣀⣸⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠹⣿⠿⠿⠿⣶⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠛⠈⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⣿⡇⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⡿⠀⠀⠀⣼⠃⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠛⠛⠛⢛⣿⣿⣿⡏⠉⠉⠙⠋⠀⠉⠉⠉⠙⠁⠀⠀⠀⠉⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣼⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠀⠈⠀⠐⠚

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠙⣀⠀⠀⠀⢀⡀⠀⣠⠖⠒⠲⠤⣤⣀⠀⠀⠀⠀⠀⠀⢠⡓⣀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⠑⠀⠀⠀⠀⠀⠛⠛⠀⣙⢋⣛⣷⠶⡶⢄⢀⡀⠀⠀⠀⠀⠠⠀⠁⠀⣠⠀⠸⠿⠃⠀⠀⢠⣔⣒⡄⠀⠀⠀⠀⠀⠀⠘⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⣦⠀⠀⠀⠀⢠⣼⣍⣭⣉⢈⠙⠛⠿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢴⣿⣷⡄⠀⢠⣾⠛⢛⡃⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣄⠿⠯⠭⡝⠻⠷⠶⢶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠈⠉⠀⠻⠟⠀⠀⠀⠀⠘⠓⢻⣿⣿
⠀⠀⠀⢛⣻⣶⣦⣤⣄⣀⠻⢿⡿⠇⠀⠉⣠⣶⣂⣀⣐⡢⠄⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠉⠋⠙⠛⠛⠛⠿⠿⠷⠶⠦⠀⠈⠳⣄⣉⣹⣿⣵⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠁⡆⡆⠛⠀⡇⢸⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠃⠇⠘⠀⡇⠀⢣⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢦⡀⠠⠀⢠⠀⢇⢀⡸⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⠶⢦⣬⡙⠻⢿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⡴⠛⠁⠀⠔⠂⠂⠀⠀⠈⠉⠛⣷⣄⠙⢷⣾⣤
⠀⠀⠀⠀⠀⣠⡟⠁⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⢀⡉⠈⡉
⠀⠀⠀⠀⠀⣿⠃⡴⠄⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠜⢿⣇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠖⠀⠀⢀⠀⠂⠑⠀⠀⣤⠄⠁⠀⠀⠀⠄
⠀⠀⣆⠀⢸⣿⠀⡇⢰⠃⡼⢂⠔⠀⡞⠁⢻⠀⠃⠀⠀⠦⠀⠀⠀⠁⠀⠀⠀⠀⠘⠗⠀⢠⣆⣀⡂⠐⠀⠘⠿⠀⠂⠀⠀⡀⠀⢀⡀⠀⠀⢰⡶⠶⠒⡆
⠀⠀⣿⣦⠀⣿⡀⢳⠀⠒⠀⠀⣂⡀⠀⢰⣿⡀⠄⠀⠀⠀⠛⠀⢋⡀⠠⠀⠀⠀⠀⠀⠀⡸⡯⣿⢷⠄⠀⣘⠣⣔⢀⡀⠔⠁⠀⠛⠁⠀⢰⣤⠟⠀⠈⡇
⠀⠀⠹⣿⣿⢿⣷⡈⠣⡀⠀⠀⠉⢁⣴⣿⡿⣷⣖⠴⠀⠀⠛⠀⠁⠀⠀⠊⢀⡀⠀⣷⡿⢣⠀⠁⣸⠆⠸⡇⣰⡿⠿⠆⠈⠀⠀⠀⠀⠀⠸⠣⠖⠀⠘⠁
⠀⠀⠀⠙⣿⣟⢿⣿⣦⡈⠑⢂⠀⠊⠙⠡⢮⣸⣿⣷⡀⠀⠀⠀⠀⣀⣤⣶⡿⠀⠀⠸⠿⠟⠀⠄⠴⣶⣾⣿⡧⠀⠀⠰⠄⠉⠁⡘⠀⡆⠁⠀⣀⠀⠈
⠀⠀⠀⠀⠈⠻⣿⣝⢿⣿⡦⠊⢴⡢⢀⡀⠈⡟⡽⢯⣿⣠⣤⣶⣾⣿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠮⡅⠠⡀⠘⠣⢧⡀⠚⣲⠏⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠵⠿⠛⠋


⣿⣿⣿⣿⣿⡷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⢿⣶⠿⠿⠷⠿⠷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢉⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⠣⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⣦⣽⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣿⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣧⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢨⣿⣿⡶⠹⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣿⡇⢹⣿⣷⣤⠀⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣷⣶⠜⠃⠸⣿⣿⣿⣦⠀⠈⠁⠖⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⢋⣵⠟⠉⠉⠀⣠⣼⣿⣿⣿⣿⣷⣦⣀⠈⢀⠀⢉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠟⠋⠁⣠⠟⣁⣀⡈⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠛⢦⡛⠱⣤⡀
⠀⠀⠀⠀⠀⣰⡾⣽⠋⢀⣴⣿⢋⡤⢁⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⢻⠻⠻⢿⡎⠻⣿⣿⣶⡄⠀⠙⢢⠈⠽⣦
⠀⠀⠀⠀⣾⣿⣿⣷⣴⡿⣫⣿⣟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣯⡟⢿⠘⠃⠁⠀⠑⠀⠹⠉⠻⣧⠀⠀⠀⠠⠀⢎⣷⡀
⠀⠀⠀⣼⣿⣿⣿⣷⣿⣶⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇⣗⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠈⡿⣷
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⢹⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣆⢰⣆⣄⣰⣤⣼⡏⠀⠀⠀⠠⠀⠸⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠄⠘⠀⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢁⣤⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠻⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⠑⣀⣿⣿⣿⣿⣿⣻⡟⠻⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠟⣿⣹⡿⣿⡙⠉⠀⠀⢀⡀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠉⠉⡤⠄⠈⢇⣘⣿⠅⠂⢛⠀⠀⡙⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠁⠀⢰⡘⢃⠉⠀⠀⠍⠀⡰⣀⠂⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⢀⠆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠀⠈⠉⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠓⠠⠄⠀⠤⠦⠖⠀⠠⠀⠠⠄⠂⠄⠀⠿⠀⠆⠀⠄⠰⠀⠂⠀⠀⠀⠰⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⠀⠈⠡⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⠀⠈⠁⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⣴⠥⠀⠰
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠠⢄⣒⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠂

⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣆⠄⢀⠀⠀⠀⢰⠀⣰⠀⠀⠀⢰
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠏⠾⠈⠳⠁⠼⠸⠇⠘⠠⠿⠑⠼⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⣀⡀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠱⡀⠈⠁⣀⣼⢀⡀⠜⠀⠀⠀⠀⠀⠀⠇⢀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠀⠀⠀⠀⠀⠀⠈⠉⡋⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣀⠄⠐⠀⢠⠀⠐⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠠
⢀⡀⠀⢰⢁⡈⢀⡶⢾⣹⡾⡿⠂⠀⠀⠄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢐⣒⠀⠂
⠈⠻⠀⠈⠁⠠⢭⣙⣤⣄⣘⣭⡼⠶⠞⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠐⠪⢙⠒⠀⠤⠤⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠉⠂⠒⠒⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⡀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⡔⠁⡀⠀⠀⠄⠀⠁⢀⠌⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠁⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢈⡠⠔⠊⠁⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⢀⡤⠤⠤⣶⡌⢥⠈⡢⣠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠈⢣⡀⠶⣅⠙⠦⠌⣠⢆⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠐⣤⠐⠟⠁⢉⠉⠁⠂⠀⠊⣉⠢⡄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠈⡄⢠⡀⠀⠘⠀⠀⠀⠄⠀⡓⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⢀⠀⠀⠀⠀⠀⢰⠁⠀⠰⠋⠙⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡩⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢈⢀⠀⠊⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠠⠄⠲⠎⠀⠀⠀⠀⠀⠀⠘⠆⠀⠓⠄⡠⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠠⠤⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠐⠠⠀⠀⢀⠀⠀⠓⡀⠀⠀⠀⠦⠐⠊⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠂⣉⣆⠀⢀⠄⠰⢤⡀⠀⠀⠸⡀⠀⠀⠀⠐⢰⢔⠋⢤⡄⢀⡹⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡢⠐⡉⠠⢨⡗⠁⢀⠴⠈⠀⠀⠀⠀⠳⡀⠀⠀⡀⠀⢀⡤⠐⠁⢠⡇⠈⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⠋⠐⠁⠀⢠⠔⠁⢰⠁⠠⠂⠀⠀⠀⠀⠀⢱⡀⠀⠑⠒⡁⠈⠑⣀⠈⠦⣼⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠂⠀⠀⠀⠀⠀⠈⠀⠀⠄⠠⠈⠀⢀⠊⢀⡔⠀⠀⠇⠀⠀⠀⠀⠀⢀⠊⠉⠈⠁⠘⠂⠙⠁⠁⠀⠀⢹⠹⠒⠠⢤⣀⡀⠀⠀⠀⠣⠀⠀⠀⡀⠀⡈
⠀⠀⠀⠀⠀⠀⣀⠐⠀⢁⠀⠠⠾⠛⠰⠛⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⡀⠂⠀⠂⠒⠒⠀⠀⠀⠀⠀⠈⣆⢀⠀⠀⠘⠉⠙⠦⣄⠁⠐⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠑⢄⡀⡀⠀⠀⢠⠔⠊⠀⠀⠀⠀⠀⡴⠁⠀⠑⣀⠀⠒⠒⠀⠀⠀⠀⠀⠀⢈⠇⠑⢄⠀⢀⠀⠀⠙⠦⣀
⠀⡀⠀⠀⠀⢀⡀⠀⠀⠃⠀⠀⠰⠛⠒⣀⠴⠃⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⢱⠤⣉⣉⣁⣀⣤⠴⣶⡒⠃⠀⠀⠀⠀⠀⠁⠲⣞⠀⠈⠲⢤⣄⡀
⠀⠁⠀⠔⠀⠀⠀⠀⡈⠀⠀⠀⠀⢠⠄⠀⠀⠆⠀⠀⠀⣠⠎⠉⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠙⠢⣴⠂⠀⠀⠀⠀⠀⠀⠙⠲⠶⠒⠚⣛⣳⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⢀⠔⡁⢦⡀⢴⠀⠀⠀⠀⢦⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢻⡄
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠖⠁⠀⢿⣳⣤⢴⠀⠀⠀⠀⠸⡆⠀⠜⢀⠎⠀⠀⠀⠀⠀⠀⠰⠉⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠈⢢⠀⢸⠟⠁⢸⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⣿⡇⠈⠙⠦⣄⡀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠸⡰⠋⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠟⡇⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⠴⠁⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢠⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡲⡴⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⠾⣽⠛⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠚⠹⣇⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠒⣚⣋⣉⣉⡭⠭⠵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡤⠖⠋⢉⣠⠤⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡴⠚⠉⠀⣀⣀⣤⠤⠤⠶⠖⠒⠒⠛⣛⣛⣶⠶
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠔⣛⣉⣁⣀⣀⡠⠤⠶⠶⣿⣉⠉⠉⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠉⠉⠉⠀⠀⠀⠉⢻⣏⠉⠉⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢹⡀
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⣠⠔⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠐⠂⠀⠀⢀⣀⣤⠴⠒⠉⠀⠀⠀⠀⢠⡇
⠀⠀⠀⣠⡤⠖⣋⠅⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢿⠒⠒⠲⠖⠒⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠘⠷⠖⠚⢥⡤⠤⠶⠛⢳⣤⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠚⠛⠉⠙⠛⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠶⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠒⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢠⠁⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡷⠦⠤⢾⠧⢴⠚⠻⠛⠶⠤⠤⠤⢤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠠⠀⡸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠳⢤⡤⠃⠀⡎⠠⠀⡤⠄⠰⣄⣠⡀⠀⠙⣆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⡀⣤⠔⢀⣀⢴⠾⠀⠙⠀⠓⡘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡷⡇⠘⠛⠋⡀⢀⠀⢀⠀⠀⣅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⡼⠂⠀⠘⢛⠘⢻⡀⢋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⡇⠳⡄⠀⠈⠈⠁⠉⠉⠉⠁⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠇⠙⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡀⠀⠀⡀⠀⠀⡇⠘⢄⡀⠀⠀⡠⠻⣀⣀⡠⠏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢄⠀⠀⠑⠢⠄⣀⠀⠀⠀⠀⣾⠈⠭⠍⠩⠍⠁⠀⡇⠀⣀⢀⢉⣉⣀⡀⠀⣀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠠⡀⠀⠀⠀⠉⠁⠐⠾⠿⢀⣟⡀⠘⠓⠀⣠⠀⠀⢩⡄⣬⠩⠉⠁⠁⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣀⣀⡑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠛⠧⠰⠴⠶⠶⠶⣦
⠀⢀⡴⠛⠛⠙⠋⠁⠉⠙⢶⣄⠀⠀⣿⠶⠛⠋⠉⠉⠉⠙⠲⢬⡒⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡌⠀⠀⣀⣠⡀⢸
⢰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠷⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⠉⠉⠙⠂⠚⠉⠉⢀⠀⢻⠋⠀⠀⠈⢋⣉⡡⠼
⡏⠀⠀⠀⠀⣰⠋⠉⠙⢳⣶⡿⠘⠁⠀⣠⠖⠓⠉⠙⢦⡀⠀⢀⣀⡿⠷⠤⠴⠆⠀⠲⠒⠴⠶⠦⠤⢤⣴⡟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⡀
⠃⠀⠀⠀⠀⠇⠀⠰⠊⣸⡏⠀⠀⠀⠀⠻⣀⡀⣀⡤⠞⠋⠉⠀⠀⢀⣀⣀⠤⠤⠤⠤⠤⠄⣀⣀⡀⠀⠀⠉⠙⠒⠤⣀⠀⠀⠀⢀⡴⣋⡿⠷⢦⡀
⠰⡄⠀⠀⠀⠈⠓⠲⠟⠛⢧⠀⠀⠀⠀⠀⡼⠛⠁⠀⣀⡤⠖⠊⠉⢈⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠉⠒⠦⢄⡀⠀⠀⠑⠢⣴⣩⡟⣹⠤⠖⣶⠟⢧⣀
⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⢀⡷⢦⣀⡰⠋⠀⠀⡠⠊⡥⢤⣀⡔⠊⠉⠙⣿⠯⢤⣞⣿⡾⣻⣏⢉⡵⣶⢦⣀⠀⠉⠒⣄⠀⠀⠈⠻⣿⡧⣶⠋⢉⣽⠛⢯⣧
⠀⠀⠀⠀⠉⠑⠒⠒⠒⠚⠉⠀⠀⡟⠀⠀⢀⠞⠀⢀⡴⠾⠿⡆⠀⠀⣴⠃⠀⢠⠋⠉⠉⢉⠏⠀⠀⠘⡟⠋⠑⢦⠀⠀⢳⡀⠀⠀⠹⣿⣛⣻⣋⣉⣩⠟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡎⠀⣰⢻⠀⠀⠀⣸⣏⣉⣽⣇⡴⡞⠀⢀⡰⠛⢳⡤⡤⣾⠥⠤⢄⡠⢳⡄⠀⢷⠀⠀⠀⢋⣸⣏⣋⡽⠉⠙⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇⢠⣏⠙⠤⡤⠴⠛⠌⢻⡶⣏⠀⠈⠉⣁⣹⣀⣸⣄⠑⣷⠀⠀⠘⡷⠋⡿⣄⢸⠀⠀⠀⢸⣇⣀⣝⠹⢓⡶⠧⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢯⠹⡏⠹⢶⣿⣻⣦⠀⣀⣽⠛⢋⣩⣿⣏⣿⢯⢤⠀⣑⡛⡶⡖⣾⣁⣼⣾⣇⡎⠀⠀⠀⣾⣻⣏⡉⣹⢿⢦⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣀⣀⣈⣿⣷⡧⠴⢟⡛⣛⣉⠉⣉⡀⠘⠻⠾⠀⠈⠁⣼⣼⣭⡟⠓⢻⣁⠀⠀⣸⠟⠀⠀⠀⣼⠉⠑⣴⡙⠙⢳⣾⣁⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⢉⣩⢭⣭⣵⠶⠒⠛⢿⣍⠁⢽⠁⠀⠈⠛⡿⠟⠉⠒⠒⠚⡄⠀⠀⣷⠤⠿⣯⡷⠛⠁⠀⠀⣠⣞⠛⢳⢶⠋⢉⠟⢧⣀⡝⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠒⢄⡀⠀⠈⠑⠂⠕⣶⣿⠞⢣⣜⠄⠒⠒⢲⣧⣄⡴⠥⠔⠊⠁⠀⢀⣠⣴⠿⡁⢈⠷⣅⢀⡽⠧⢤⣀⠟⠦⣤⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠄⠀⣀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠀⠀⢀⣀⠠⠤⠒⢊⠉⠉⠑⣤⡜⢧⣀⣼⢋⠛⢣⡴⠛⣆⡠⠊⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠒⠒⠀⠉⠉⣁⡀⠀⠀⠀⢮⣋⠆⠀⠈⠣⣤⣏⡽⣻⡔⠛⢧⡠⠚⢷⡤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣈⡼⢟⠻⢳⠀⣡⠦⡄⠀⠀⠀⠀⠟⠉⣀⡡⠋⠉⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣩⠟⠓⠋⠉⠁⠀⠛⠴⠀⠀⠀⠀⠀⠱⣴⠿⠧⣄⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣴⣿⣷⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣄⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⣴⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⣼⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠻⣷⡄
⠀⠀⠀⠀⢀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣠⣿⣇⣀
⠀⠀⠀⢀⣿⡟⠻⠿⠛⠛⢛⡛⠛⠛⠛⠛⢛⡛⠛⠻⢿⡟⠛⠻⡿⠛⠛⠛⠿⠿⠛⠛⠛⠿⣿⠿⠟⠛⣛⠛⣿⣧
⠀⠀⠀⢸⣿⡇⠀⠀⢀⠀⠘⠋⢀⣀⠀⠀⠙⠃⢀⡀⠀⠀⡀⠀⠀⠀⢰⡆⠀⠀⢀⡀⠀⠀⠀⣠⡀⠈⠛⠀⣿⣿
⠀⠀⠀⠸⣿⣇⣀⣀⣛⣃⣀⣀⣈⣁⣀⣀⣀⣀⣙⣃⣀⣘⣛⣀⣀⡀⢀⣀⣀⣀⣙⣋⣀⣀⣀⣉⣁⣀⣀⠀⣿⡯
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠟⠛⠻⠟⠛⠛⠛⠛⠛⠿⠛⠻⠿⠟⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡶⠀⠀⠀⠀⠀⠀⣼⠿⡔⠈⠠⣴⠖⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣅⡀⠀⠀⠀⠀⠠⡄⠈⠀⠀⠁⠀⣠⢚⣦⠩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣤⠀⠀⠐⠁⠠⠴⣧⡆⢆⠻⠿⠯⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣶⡖⠰⠊⠀⠈⠄⣀⣂⡀⠀⣠⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠻⠿⠛⠛⠛⠓⠒⠒⢲⠖⠒⠛⠛⠛⠃⠈⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠂⢿⡇⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣦⠀⠀⠀⠈⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠸⣿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣿⣇⠀⡆
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠠⢆⣿⣿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠘⢿⣿⣿⣿⣿⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⠛⠀⠲⣾⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡆⠀⠹⣿⣿⡿⠿⣮⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⡆⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠀⠀⠆⠀⢀⠀⠀⠀⠀⠀⠈⠉⢸⣿⡇⢀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡀⠀⢀⣻⣧⣀⠀⠀⠂⠀⣜⣀⣀⠀⢀⠀⠀⠠⠀⠘⠀⠀⠀⣠⣤⣀⣤⣼⡿⠑⡄⠀⠀⠀⠀⠀⠀⠘⠧
⠀⠀⠀⠀⠈⠁⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⣤⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠿⢀⣀⠟⡆⠀⠀⠀⠀⠀⠠⠘⠻⠗⣉⡽⠀⠀⠀⠀⠱⠐⠘⠀⣠⠂⠀⠀⠀⠀⠀⡄⠶⠻⣷⡦⡆
⠀⠀⠀⠀⠀⠒⠤⠷⠿⠷⡀⠀⠀⠀⠀⢒⡷⢶⣾⠋⠁⠀⠀⠀⠀⣄⡤⣾⡯⡥⣢⠀⠀⠀⠀⢀⡠⠤⠠⠽⠵⢷⠂
⠀⠀⠀⠀⣇⠀⡄⠀⢐⣄⢇⠀⠀⠀⢐⢏⢖⢒⣻⣹⡴⢴⠀⠀⠸⣿⣷⣌⢢⠤⠂⠀⠁⠀⠀⠸⢚⡒⢶⡴⠒⠞
⠀⠀⠀⠐⠉⠷⠤⣤⡾⠋⠈⠀⠀⢰⣭⡙⠋⠙⠛⠛⠛⢋⣧⠀⠀⠛⠙⡿⢿⠞⠗⠈⠁⠀⠀⢉⣭⢉⣁⣀⠄⣿
⠀⠀⠀⠀⠠⠀⢤⣀⠀⣱⠐⠀⠀⠘⠣⠶⠤⣀⠀⣤⡤⠶⠃⠀⠈⠚⠤⠤⣀⡡⠀⠊⠁⠀⠀⠰⢄⡀⢀⣉⣉⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⣀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠈⣅⡈⣀⣤⣤⠗⠀⠀⠀⠲⢶⠄⠲⠝⢠⡟⠀⠀⠀⠀⠀⡶⠠⣤⡂⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠄⡀⠀⠀⠀⠂⠥⠜⠷⢝⣁⢶⠀⠀⣠⣄⡩⣥⣿⢂⠶⠆⠀⢰⣞⢭⣀⡈⢻⣧⡉⣬⣦⣠⠤
⠀⠀⠀⠀⠂⠁⠀⠚⣶⠆⢂⠼⠃⠀⠀⠑⠀⠤⠤⠴⠒⠁⠀⠀⢨⡾⠷⡼⢓⠷⠞⠀⠀⠈⣡⣌⡉⢉⣻⡯⠽⠛⣊
⠀⠀⠀⣏⡙⠊⠳⠐⠿⢛⡡⠖⠉⠀⠀⢠⡄⠠⢤⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠉⠁⠚⠋⠑⠉⣉⡁
⠀⠀⠀⠀⠈⠉⠁⠁⠒⠋⠀⠀⠀⠀⠀⠈⠀⠁⠐⠋⠁⠀⠀⠀⠀⠀⠐⠒⠘⠋⠀⠀⠀⠀⠀⠉⠛⠛⠒⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⠒⠶⣤⡀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁⣠⣄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⡷
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠰⠋⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠊⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡛⠃⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣁⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠿⠀⠈⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⣤⠶⠛⠉⠉⠉⠙⠳⠦⠤⠤⣀⡀⠀⢀⣀⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠾⣿⠟⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣉⡿⠋
⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠠⣄⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⡤⠾⠿⠶⠶⠶⠶⢶⣶⣶⣶⣶⡶⠶⠿⠿⣶⣶⣶⠆
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠛⢶⣤⣄⣀⣀⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣶⠶⣶⣶⠾⠟⢛⣋⣭⣤⡾⠷⠶⢚⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢻⣆⠀⠀⠀⠀⠀⠈⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⣰⠟⠀⠀⠀⠀⠁⠀⢐⣲⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠛⠛⠛⠛⠛⠻⣿⣤⣶⣿⣆⣠⣄⠀⠀⠀⠀⠀⢠⣿⠃⣰⡟⠀⠀⢀⣴⠞⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣦⣄⡀⠠⢾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⡏⢠⣿⠁⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠈⠉⠛⠻⢿⣿⣦⣄⡀⢁⣾⡟⢀⣾⠇⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡟⠀⣸⡏⠀⠀⢀⣿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⡟⠀⠀⠀⣸⣿
⠀⠀⠠⡀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣼⣿⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠀⠀⢀⣾⠟⠙⠿⣶⣄⢀⣾⠟⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⡀⣰⡿⠃⠀⠀⠀⠈⠙⠿⠏⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠹⣿⣿⠿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⢀⠀⠑⠀⠀⠀⠀⢠⡿⠻⠿⣷⣶⣤⣀⡀⠙⣿⣧⠀⠉⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠁
⠀⠠⠈⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠉⠛⠿⢿⣷⣿⣿⣧⠀⠀⠀⠈⠙⠻⠿⣷⣦⣤⣴⣿⠟⠁⡀
⠀⠀⠐⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⡏⠙⠁
⠀⠀⠄⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠈⠠⠀⢀⠀⣿⣿⠀⠀⣀⣤⡶⢞⣋⣭⢽⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣿⣿⡿⠟⠛⠉⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠹⣷⡀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⣀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠷⠛⠂⠞⠐⠃⠐⠐⠃⠉⠘⠃⠘⠀⠀⠟⠃⠛⠋⠛⠃⠰⠧⠃⠚⠴⠇⠸⠃⠜⠋⠀⠘⠀⠘⠛⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⢋⣍⢉⡉⠓⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣀⣠⣤⣤⣀⡀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⡴⣫⣅⢄⣰⠠⠈⠝⢦⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡞⢼⣊⠊⠜⠇⠁⠈⠁⠦⢹⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠓⠱⡠⠁⡅⠂⡀⠀⠈⢠⠷⠶⣄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠉⠉⠉⠉⠉⠉⠉⠙⠓⠲⠤⣄⡀⠀⠀⠀⣇⠀⠀⠀⢳⡄⠔⠈⣈⢄⠐⠊⠌⣀⣼⠟⠛⠁⡞
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⣀⣤⠤⠴⠶⠶⣶⣶⣶⣤⣄⣀⠀⠀⠉⠒⢤⡀⢸⣄⠀⠀⠀⠙⠶⣼⣂⡪⣁⣮⡵⠛⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⡠⠊⠁⠀⢀⣤⣶⣿⡛⠓⢦⣤⡄⠸⣷⢄⠐⢈⢻⡈⠙⠲⣄⠀⠘⢉⡻⣿⣦⡀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⡴⠃
⠀⠀⠀⣠⠊⠀⠀⣠⣶⡏⠀⣹⠛⠻⠶⢿⣤⡇⠀⠙⢮⡧⠄⠁⠑⢦⡀⠀⠙⢦⡈⠁⠈⢻⣿⡶⢄⣀⡀⠰⠆⠖⠀⣀⣠⠔⠋
⠀⠀⡴⠁⠀⢀⣼⠿⣽⡧⢤⣿⠦⣦⡄⢀⡍⠛⢓⢆⠈⢷⡂⢡⡀⠠⠙⡆⠀⠀⠑⣄⠀⠀⠹⡇⠀⠈⠉⠉⠉⠉⠉⠁
⠀⡸⠁⠀⢠⡏⣸⣠⡇⢀⣠⡃⢀⣼⣽⣷⠲⡀⠄⠉⠛⠶⣿⣿⣬⡤⠀⢹⣦⡀⠀⠘⣆⠀⠀⠹⡀
⢰⠃⠀⢀⣿⣧⠋⠉⣹⠁⠀⣿⣏⠀⣼⣃⠀⠈⠻⢧⣙⣶⢤⡘⠻⢦⡀⠀⠀⠻⡄⠀⠘⣆⠀⠀⢧⠀⠀⠀⠀⣀⡤⠔⠒⠋⠉⠉⠉⠉⠓⠒⠤⣄
⡾⠀⠀⣸⡿⠻⣤⠼⢻⠶⠚⠋⠉⠉⠙⠛⠛⠦⢤⣀⠀⠉⠛⢿⣴⠄⣗⠄⠒⠀⢹⣄⣀⣿⠀⠀⠸⣄⣀⡴⣪⠅⠀⠀⣀⣀⣤⠴⠚⠉⣹⠀⢠⠶⡝⠦⡀
⡇⠀⠀⣿⣇⡀⣸⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣠⠄⠈⠙⠶⣽⣦⡂⡄⠀⠳⡽⠟⢦⡀⠀⡿⠋⠘⠁⣠⠖⢋⡽⠋⠁⠀⠀⢠⠟⢿⠏⠀⢹⠀⠈⢦
⡇⠀⠀⣿⠉⢿⠋⠀⠀⠀⠀⣠⠴⠖⠒⠒⢦⠀⠀⠂⢸⡙⠂⠀⠀⠀⠀⠈⢷⡘⡆⠀⢻⣿⣫⠇⠀⡇⠀⣠⣾⡵⠚⠁⠀⠀⠀⠀⢀⠏⢠⠏⠀⠀⠸⡇⠀⠀⢳
⢷⠀⠀⢿⠙⣿⡀⠀⠀⠀⡾⢱⠀⠀⠄⢂⢘⣇⡴⠒⠚⡛⠛⠉⠓⢦⠀⠀⠀⢹⡄⠠⠀⣿⠃⠀⢰⠁⢸⡟⠁⠀⠀⠀⠀⠀⠀⢀⡎⢀⠏⠀⠀⠀⠀⣷⡀
⠸⡄⠀⠸⡇⠜⣧⠀⠀⠀⢷⡈⠀⡀⠆⢀⣸⠟⢀⡤⠥⠦⣤⡀⠀⠀⢳⣀⠀⠸⣌⣅⣠⡟⠀⠀⡜⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⡼⢀⡞⠀⠀⠀⠀⠀⢹⣇
⠀⢱⡀⠀⠙⣦⣈⣦⡀⠀⠀⠈⠓⢳⡿⠛⠁⣰⠯⡀⠀⠐⠠⠙⣆⠀⠀⢹⣆⠂⡈⣹⠟⠀⠀⣰⠁⠘⣯⣿⣇⠀⠀⠀⠀⠀⡜⢁⡜⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⢳⡀⠀⠘⢖⡐⡛⡄⠀⠀⠀⢸⠀⠀⠀⢿⡣⠐⠠⠐⢀⠂⡿⠀⠀⠈⢷⣬⣶⠋⠀⠀⣰⣷⠀⠀⢹⣽⣿⡀⠀⠀⠀⣰⢗⡾⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠙⢄⢀⡈⠳⣄⠿⢦⣀⣀⣼⣧⠀⠀⠈⠳⣤⣁⣵⣥⠖⠁⠀⠀⠀⠈⢿⠁⠀⢀⠞⠁⠙⡆⠀⠀⢻⣷⣹⡄⠀⢰⡏⡼⠁⠀⠀⠀⠀⠀⠀⢀⣤⠏⠀⢰
⠀⠀⠀⠀⠀⠻⢵⡆⠈⠑⠦⣈⠉⠀⠈⠣⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⣀⠔⠁⠀⠀⠀⠘⢆⠀⠀⠙⣧⣇⢠⣯⣥⠃⠀⠀⣀⣀⣠⣴⡾⠟⡡⠆⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠉⠑⠲⠤⠶⢭⣽⠷⠶⠶⠦⠤⠤⠖⠊⣉⡥⠒⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⠿⠿⢾⣷⣶⣶⣿⠧⠴⠚⠉⠀⠀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢤⣀⣀⣀⣀⣀⣀⣀⣠⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⠀⠈⠁⠀⠀⠀⠀⣀⣠⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣮⣄⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⡻⠿⣿⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⣿⣄⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣹⣿⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠉⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡁⠸⢿⣿⣿⣿⣢⡇⠀⣤⣤⣥⣬⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⡿⣦⣴⢸⣿⣿⣧⠻⣦⣟⠛⠉⢛⣻⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏⢻⣷⢿⡖⢐⣶⣷⣿⣿⠄⣿⣿⣻⠀⣐⡈⠀⠀⠋⢋⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣷⣿⣯⣻⣶⠟⣿⣿⣷⣿⣿⣷⣿⣿⣿⡄⢩⣿⢿⠟⣫⡿⣷⣿⣿⣷⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⡛⠂⣩⡽⣝⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⢹⣦⣝⣷⣶⢏⣴⡿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣾⠟⠀⠖⡺⠙⣷⣬⠋⣍⠓⢿⣿⠟⠞⠑⠛⠟⢿⣿⣿⣿⣿⣿⣿⢻⡿⠁⠻⡇⠀⠍⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣤⣥⠀⠀⠈⠷⢆⣸⡇⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⠿⢿⣿⡟⠋⠀⢀⣾⣧⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣮⣉⠸⠿⠏⠠⢾⣿⣆⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠳⢶⣬⣙⣿⣦⣿⣁⢿⣬⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠱⣟⣲⣤⢌⣹⡶⠿⡟⣻⡯⢵⣟⣶⣴⣖⣸⠦⡀⠀⠀⠀⠀⠀⡘⣿⣿⢻⣿⣿⣯⣿⣿⣋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⢿⡏⡧⢸⠿⠿⠋⢛⣿⣶⣿⢛⡿⣧⣽⣾⠀⠁⠂⠀⠒⢸⣫⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣦⣄⡀⠂
⠀⠀⠀⠀⠈⢀⣀⠈⢳⡀⠘⢷⣼⣮⣿⣀⣛⣟⣥⠼⠟⢿⢿⣿⣿⣳⣄⠀⠤⢾⣿⣿⣿⣿⢟⣵⣿⠋⠀⠀⠉⠻⣿⠿⠛
⠀⠀⠀⠀⠀⣶⠒⠀⠀⠉⠳⢄⣈⡙⠛⠷⠦⣤⣄⣀⣀⣦⣾⣦⣬⣷⣤⣶⡶⠿⠟⠛⣩⣴⠾⠋⠀⠀⠀⠀⠀⠀⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢶⣦⣤⣤⣬⣍⣉⣉⣉⣉⣁⣤⣤⣴⡶⠿⠛⠉⠀⠀⠀⣀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠒⠖⠒⣀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⡠⠔⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠈⠻⠿⣿⣿⡿⠿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⣀⣀⠀⠀⢀⠁
⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠀⠀⠀⣿⣿⣷⡎
⠂⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣶⣵⣿⣿⠀⠀⠀⠀⠹⣿⣿⠇
⠈⡄⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⠿⣿⣿⣿⣦⠀⠀⠀⠀⠉⢿
⠀⠈⢄⠀⠀⠀⠀⠁⠂⠤⢀⠀⠀⠀⠀⠀⡀⠠⠐⡸⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⡄⠠⢀⡀⠀⠀⠀⠀⠀⠈⠐⠠⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠑⡀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈
⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠁⣀⡀⠀⠀⠀⠀⣀⡀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠉⠒⠠⠄⠀⣀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠂⠁
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡀⠈
⠀⠀⠀⠀⠀⠀⡀⢤⠙⠧⠀⠢⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⣀⣾⣄⡀⢠⠇⢈⣴⠿⣦⠀⡀⠀⠀⠀⠈⣾⣿⣿⣿⣿⣇⢄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠁
⢀⠶⢴⡧⢻⣿⣿⣿⣀⡘⠻⢷⣿⠃⠀⠀⠀⠀⠁⠙⠛⠻⠿⡿⠙⣻⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⣾⣷⢸⡘⡉⠋⠻⣟⢷⣶⣷⠀⠠⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠
⠀⠀⠈⣿⣽⣿⣿⡄⠂⠈⠀⠙⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣨⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⢻⡏⣹⠀⠀⠁⠀⠀⠀⡐⡁⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣀⢔⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄⢀
⠀⠀⠀⡼⠁⠙⡀⠀⠀⣀⣄⣀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣬⡧⠀⠀⠀⠀⠀⠀⠀⣸⠛⠋⠁⠁
⠀⠀⠀⠀⠀⠀⢁⣴⣾⣿⣿⣷⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⡄⣼⣿⢁⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢻⢶⣦⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⡀⠀⠀⠀⠤⡆⢿⣿⣥⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⢀⣿⣇⡀
⠀⠀⠀⠀⠘⠊⠟⢿⢿⣿⣿⣿⡙⢿⣿⣿⣟⣒⣋⡀⢀⣘⣧⢸⣿⣿⡄⠀⠀⠀⣿⢿⡍⠀⠀⠀⠀⠀⠀⢸⣝⣿⣿⣿⣶⠂⢀⣾⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣿⣿⣿⣶⣾⣯⣽⣿⣶⣿⣿⣿⣿⣇⠻⣿⣿⡤⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣶⣾⡿⣟⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⡻⣿⣷⢆⣀⡄⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡍⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠹⠟⠻⠿⢿⣿⣿⣿⣿⣷⣦⣤⣬⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⢿⡋⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠙⠀⠀⠛⠉⠙⠋⠛⠉⠙⠋⠋⠉
⠀⠀⠀⠀⢰⠂⠆⠀⠀⠀⠐⠐⠀⠀⠀⢰⠄⢠⢰⣄⣄⢠⠀⠀⠂⠄⣀⡄⠀⡄⣄⠀⢀⢦⠀⡀⠀⡀⢠⢀⡄⠠⠀⠀⠀⠀⠀⢠⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢷⡈⣻⠷⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡤⠾⠟⠁⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠿⠦⣤⣌⣉⠛⠶⣦⣄⠀⠀⠀⢀⣠⣤⡤⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠈⠙⠻⢶⣄⠉⠛⠶⠶⢿⠁⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠻⢦⣼⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠲⠶⠒⠒⠒⠒⠲⠶⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡾⠃⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠶⠶⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣤⣄⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⡟
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠢⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⢆⠀⠀⠀⠐⠠⠀⠀⠄⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠰
⠀⠀⠀⠀⠀⠈⠐⠀⠀⠄⠀⠀⠠⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⠲⠀⠂⠀⠄⢠⠄⠂⠊⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣊⠠⠁⠀⠀⠀⠐⢘⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠡⢉⠀⢀⡀⠀⠀⠀⡀⠈⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡈⠀⠆⡈⠂⠄⡁⠀⢔⠀⡑⠀⡐⡂⠘⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠒⢄
⠀⠀⠀⠈⡄⣆⢱⠀⠀⠀⠀⠀⠀⡄⠁⠀⢠⠀⠈⠀⠈⠚⠋⠄⠉⢉⠁⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠃⢱⠘⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠂⠐⠃⠀⠚⠉⠉⠉⡉⠉⠉⠉⠐⠀⠀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⠀⠌⠀⠀⠀⠊⠀⠀⠀⢀⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠄⠤⠒⠂⠀⠀⢠⠏⠉⠉⡁⠀⠀⠀⠠⠀⠈⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡘⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⠰⠀⠒⠂⢀⣶⠚⠈⠉⠉⠉⠉⠉⠉⠂⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⡞⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠂⠠⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢷⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠠⠬⠙⠶⠤⠤⠤⠤⠤⠤⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠀⢄⠀⠀⡀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⠄
⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠉⠤⠀⠀⢀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠈⠘⠆⠀⠀⠀⡿⡄⣶⠆⣶⢠⡆⢸⠃⣄⡶⢠⠆⡶⠰⡂⠀⠀⠀⠀⠀⠀⠈⡀⠀⠔⠄⠀⠀⠀⠀⠀⡀
⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⡀⠃⠀⠁⡀⠀⢈⠉⠈⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠢⠄⠀⠘⠛⠛⠂
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠁⠉⠀⠉⠁⠋⠉⠈⠙⠈⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠐⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⡀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣺⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠠⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠛⠒⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣻⣛⣟⣫⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣽⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⣽⠉⠉⠳⣆⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⣰⢶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⡞⢻⣼⡆⢀⡤⠶⠶⠶⠿⢦⣀⡀⢙⣻⡿⠷⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣛⠻⣦⣀⠀⠀⠛⣫⡾⢛⡛⢷⣄⣀⣀⣀⠀⣉⣉⣉⣁⣀⣀⣀⣀⣀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠀⠠⠍⠛⣻⠋⠁⣀⠀⢀⡀⣿⣛⡛⢿⣀⡿⠶⠶⠾⠟⠛⣛⣋⣭⡴⠛⣦
⠀⠀⠀⠀⠀⢀⣤⣼⡇⠀⠉⠀⠀⢠⠀⠀⣀⠀⠀⢠⡌⠀⣿⢭⣝⠛⠛⠛⠛⠛⠛⠛⠋⣉⣉⣠⣤⠶⢿
⠀⠀⠀⠀⠀⣿⣚⠉⠻⣦⣭⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⡾⠃⢀⣨⡟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⣀⡄⣸
⠀⠀⠀⠀⠀⡏⠉⠛⠓⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠖⠛⠋⠉⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣡⡴⠋
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣠⡟⠲⠶⠶⠶⠶⠚⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠈⠙⠳⠶⢦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⠶⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠳⢤⠄⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁⠳⣤⡄⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡞⢧⡀⠑⠦⠤⠤⠖⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢧⡈⠓⠦⠤⠤⠖⠀⡀⢳⣄
⠀⠀⠀⠀⠀⠀⣏⠣⣀⠉⠒⠦⠤⠤⠒⠊⢁⠜⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠣⣀⠙⠒⠶⠶⠶⠖⠋⣁⡜⣸
⠀⠀⠀⠀⠀⠀⢹⡳⠬⣉⣓⣒⣒⣒⣒⣋⠥⠖⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡳⠬⣍⣓⣒⣒⣒⣚⣫⡥⢞⡟
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠉⠉⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠴⠖⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣯⣸⣿⣮⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣮⡻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢟⢳⣮⡻⣷⣾⣿⡿⣿⣮⡻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⢿⣯⣿⡿⣿⣶⣾⡿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢣⢺⣯⣛⠿⣷⣶⣶⠿⢟⣿⡇⡜⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢘⣷⣽⣻⣿⠿⠿⠿⣿⣟⣯⣾⡃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣘⡷⣽⣛⡿⠿⠷⠾⠿⣟⣫⠾⣃⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣾⣯⣽⣿⣿⣿⣿⣽⣷⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡻⣶⣯⣭⣿⣿⣿⣭⣽⣶⢟⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣭⣻⢿⣿⣿⣿⡿⣟⣥⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣼⠶⠌⠄⢰⣿⣿⠋⠋⠙⠂⠤⠉⠄
⠀⠀⠀⢠⣤⠤⠤⠤⠤⢤⠝⠹⡐⡲⢲⡴⢰⡾⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⣙⣗⣒⣻⣾⠷⠿⣻⣿⣣⣁⣼⣾⠟⠀⠀⠀⠂⠄⣡⠑⠐⠰⠂⠂⠂⡤⠰⠀⠒⢠⠋⠁
⠀⠀⠀⢀⡀⠉⠉⠙⠻⠟⠛⠉⠋⠉⡅⠀⠀⢀⠀⠀⠀⠠⠶⠉⠁⠁⠀⠚⠀⠤⠦⠝⠖⠊⠉⣁
⠀⠀⠀⠀⠉⠓⠀⠴⠒⠒⠀⠤⢂⠀⠊⠀⠀⠀⠀⠀⠀⠶⣿⣶⣦⣤⣀⣤⣶⣶⢶⣲⣶⣶⣶⣿⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠤⠤⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠢⠄⣐⣄⠀⠉⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣠⡀⠀⠀⠙⠳⢌⢧⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⡶⠁⠀⣠⠀⡇⠀⢀
⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠑⡄⠀⠒⠀⠀⠵⣦⢹⡀⠓⠒⠤⠤⠤⠤⠤⠄⠒⣒⣿⠿⠋⠁⠀⠀⡜⠀⡄⠀⡇⢐⡡⠃⢸
⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠠⠤⠌⣣⣧⠘⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⡜⠀⢸⠆⢀⡖⠋⢀⠀⡀
⠀⠀⡆⠀⠀⠀⠀⠀⠤⣠⢀⠀⠀⠀⠀⠀⠁⠒⠀⠄⢄⡙⣦⣧⣤⡶⢖⣾⠟⠉⠀⠀⠀⠀⠀⢠⠁⠃⠀⣄⡠⠎⠶⠋⠁⡠⠁
⠀⣠⢧⠀⠀⠀⠀⠀⠀⠀⠙⠳⣂⢀⠀⠀⠀⠁⠀⠀⠀⠈⠹⣗⣯⠶⠋⠁⠀⠀⠀⠀⠀⠀⠠⠃⢀⣤⠜⠋⠀⠀⣀⢤⡞⠁⡀
⢸⠁⠸⣧⡀⠀⠀⠀⠐⠄⠀⢀⠀⠀⠙⠀⠒⠀⠐⠤⠴⠖⠚⠉⠀⠀⠒⠀⠀⡀⠀⠀⢀⠔⠁⠈⢁⠀⠀⠁⠀⠀⢀⠜⠑⠀⠠
⢸⣄⡀⠺⠿⣷⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⣄⠀⠀⡘
⣶⠃⢰⣄⣠⣀⠈⠹⠉⣉⡛⢿⠟⣶⣦⣤⣄⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣤⣤⣤⢤⣦⣤⣤⣤⣤⣤⡤⣶⡞⠉⠈⠀⢀⠔⠉⣴
⣿⠐⢸⡇⠀⠈⠉⠒⠢⢤⣀⠀⠀⠛⣊⣉⣉⠹⠟⣟⣿⢿⢿⠟⣿⡿⠧⠟⢠⢉⡛⠛⢓⣀⣀⣬⣥⣥⣦⡠⠤⠔⠋⠁⠀⢘⢹
⣿⣄⢾⣿⡀⠀⠀⠀⠀⠀⢀⠀⠉⢙⣿⣿⣿⣦⠠⢄⡀⢀⣀⣀⢠⡀⢀⢀⣀⠀⠤⢴⣾⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⡄⠀⢸⢹
⠐⢠⡈⣿⣷⡄⠠⣤⠄⠀⢿⢻⠁⠀⢯⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡋⠉⢯⣿⣿⣆⠀⠀⠀⠀⠀⠨⡼
⠀⠀⠙⠾⣿⣿⣦⣿⠁⠀⠹⠀⠀⠀⠈⢦⣿⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢼⣿⢿⣿⣷⣄⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠈⠛⠻⢦⣀⠀⠀⠄⢿⣄⠈⢿⣜⠿⣇⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠽⢿⣇⡀⠻⣇⣾⣻⡧⡀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠾⠓⠋⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣿⣶⣿⣷⣿⣿⣶⣷⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣇⣀⠀⠀⠀⠀⠀⠘⣆⣤⡆⠀⠀⠀⠀⢀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣠⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢨⣷⡆⣀⢤⣿⡀⢰⣶⠶⠞⠏⠉⢸⣇⢀⣀⡀⠀⢻⡏⠶⣾⣠⡀⠀⠀⠀⢀⣼⣟⣿⡿⣦⣄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣦⣿⣄⣁⣠⡿⠟⠛⢿⣷⣋⣙⣓⣼⠟⠛⠛⠛⠛⠛⢿⡾⠛⣛⢿⣟⡛⢛⢿⣿⣿⢿⣿⣿⣻⣿⠶⣦⣤⣁
⠀⠀⠀⢀⣠⠴⠚⠛⠛⠋⢩⡿⠛⣳⣶⡄⠀⠀⠀⠈⠉⣉⣁⠉⠅⢠⡾⢉⣙⠓⢦⡈⣿⣟⡿⠿⠛⠁⠼⢿⣿⣿⣿⣿⣿⠿⡁⠐⣢⣯⡙⠛⠻⠶⣤⣀
⠀⡠⠚⠉⠀⠀⠀⢀⣀⣴⣾⡇⠀⠛⢻⣇⡀⠀⠀⠀⣼⠉⢿⢷⠀⠛⣿⡒⠢⠭⠈⠻⣎⡁⠀⠀⢀⣤⣴⣤⣌⣉⠉⠁⠀⠀⢀⣴⢿⠋⢷⣤⡀⠀⠀⠹⠻⣶⣤
⣾⡀⠀⠀⠀⠀⠀⠈⠛⢯⣙⠿⠗⠺⠛⣩⠟⠀⢠⣿⠿⠤⠤⠾⡳⠂⠀⠉⠙⠓⠲⠛⠉⠀⠀⠀⢿⣿⣻⣿⢿⣿⠀⠀⡀⣾⡟⢁⡉⢜⠀⣿⠇⠀⠀⠀⠀⢤⣿
⠘⠻⢶⣄⡀⠀⠀⠀⠀⠀⠉⠓⠶⣶⠊⠀⠀⢀⣿⣽⣦⡀⢀⡜⠁⣠⣾⣿⣿⣦⡀⠀⠀⠀⠾⢷⣾⣿⣯⣧⣿⣿⠀⠀⠠⠿⠷⠾⠿⠿⠛⠁⠀⠀⣀⣤⡶⠞⠋
⠀⠀⠀⠈⠙⠳⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⣻⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⢀⣈⣤⣶⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⢦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⢋⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⣤⣤⣤⡶⠶⠟⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣛⣛⣒⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣛⣛⣛⣋⣉⣉⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⡻⠿⣿⡟⣿⢟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣆⣘⣿⠀⣀⣀⣀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⡿⠋⠉⠉⠉⠉⠙⠛⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠉⠂⠀⠀⠐⠂⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠉⣽⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⡷⠶⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⡦⡟⢀⠀⠈⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣷⣷⣜⣿⠀⠀⣄⠀⣷⠈⡦⣀⠀⠀⠀⠀⠀⠀⣤⡿⠏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⣶⣧⢈⠻⠋⢻⣁⠉⠓⠌⠘⠀⠉⠀⣶⢶⣶⣶⠿⠛⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣦⣀⣌⢛⡀⠀⡤⢸⣦⡈⠀⣿⠃⠁⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘⢋⣉⣾⣿⣿⣿⣿⣿⣿⣦⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡝⢻⣿⣿⣿⣿⣿⡟⣽⡿⢄⡘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⣿⣷⠀⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⠿⠃⡀⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⢀⣼⣿⣿⢿⣿⣿⣿⣿⣿⣿⡟⠋⠀⢠⢰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⢸⣿⢿⡏⠛⠃⢀⣀⣀⣀⣀⣿⣿⣿⣾⣿⣿⣿⡿⢿⣿⣇⣴⣶⡚⣸⣟⣛⣦⣤⣄⣤⣤⣿⣶⣶⣶⣿⣿⣿⣏⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⢿⣿⣿⣿⣶⣿⣿⣿⣉⣀⣤⣿⣯⣿⣿⠛⠋⠉⠁⠀⠁⠀⠈⠉⠉⢀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡟⠉⠛⠛⠻⠟⠋⠀⠋⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢀⣟⢻⣷⡦⠉⠠⣀⣗⣒⣂⣐⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⣰⡈⠙⠳⣿⣿⣿⠟⢛⣉⣉⣩⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣶⣿⣿⣿⣿⣿⣿⡟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿
⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣏⣿⡿⡟⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣧⣱⢿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠛⣿⣿⣿⣿⣿⣷⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⠋⠹⠻⠿⠿⣿⠿⠿⠉
⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠛⠛⠻⠿⠿⠿⠟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠓
⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡁⠀⠀⠀⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠙⣦⠀⠀⠀⠀⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⣧⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣿⣆⠀⠀⠹⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⢠⣿⣿⣷⣄⠀⠘⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⣿⣿⣿⣿⣿⣆⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣆⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡄⠈⠻⢿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠶⠖⠒⠛⠓⠒⠲⠦⠤⣀⡀
⠀⠀⠀⠀⠀⢀⡴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⡦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄
⠀⠀⠀⠀⣠⠿⢶⡿⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠐⠀⠀⠁⠀⠀⠀⠀⠀⠩⣹⠿⠿⠛⠶⣤⣄⢀⣂⢰⣤⢀⣀⣴⢽⣷⣶⣄
⠀⠀⠀⠌⠀⢎⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠈⠘⠛⢿⣿⡿⠟⠃⠀⠈⠀⠙⠳⣄
⠀⠀⠀⠓⢄⣽⣤⣤⣄⣤⣄⣀⠀⡀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠈⠙⠻⡦⣔⣀⣄⣀⣤⣁⢆
⠀⠀⠀⠀⠈⣿⠉⢿⡏⠉⠉⠉⠛⠛⠶⢶⣶⣦⠤⠀⠀⠀⣂⡼⠛⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⢦⡄⠠⠀⠀⠘⠟⠉⢻⣻⠽⠋
⠀⠀⠀⠀⠀⠘⠿⡟⣿⠄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣮⣉⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠈⠉⠋⠀⠀⣀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⡏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢧⣄⣀⡀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢶⣤⣤⣐⠀⠀⠉⠀⠀⠒⢤⢀⣀⡀⠀⡀⡠⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠬⣐⣢⣴⣟⣀⣀⣠⡬⣁⣀⡤⠴⠚⠋⠁
⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠙⠩⢟⠛⠋⡹⠟⠋⠉⣉⠉⠉⠉⠉⠉⣉⡆
⠀⠸⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠈⠀⠀⠀⠀⠁⠀⠀⠠⠤⠾⢿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠈⠉⠉⠛⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⣠⠔⠛⠳⢶⣖⠒⠲⠶⠦⠤⣤⣀⡀⠀⠉⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡠⠊⣱⠋⠙⢦⣀⠉⠓⢦⣄⠀⠀⠀⠀⠉⠓⠦⣄⡀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡠⠋⢠⠞⢡⡶⠒⠶⢬⡙⢆⡀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠞⣁⡴⢋⡤⣤⣍⣒⠶⠶⠗⠀⣹⠀⢠⠞⠉⣷⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣰⢇⣀⠻⠤⢌⡢⣄⡀⠉⢹⡆⡤⠞⢁⡴⠃⡠⣞⣅⣄⣤⣤⣀⣀⣀⣀⣘⣆⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠓⠤⣈⡉⠒⢄⡙⠦⣍⠙⢋⡴⢃⡴⠋⣠⠞⢡⡏⡿⣠⠤⣄⣀⣤⣤⣉⣉⡉⣹⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣭⡓⠦⣈⠓⢮⣛⠋⣰⠋⣠⠞⠁⠀⢸⢠⡇⢹⠀⢠⠖⣲⣦⠀⣼⢁⡟⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢠⠞⣡⡾⠁⢠⣀⠙⠲⢬⣿⣡⠞⠁⠀⠀⠀⡟⢸⠀⡏⠀⠈⠳⠽⠛⠁⣿⢸⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣿⠸⢹⠃⠀⠸⡄⠙⠶⢤⣉⡉⠀⠀⣀⡀⢀⡇⣿⠀⡇⠀⣠⢶⣦⣄⠀⡇⡿⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡘⢦⣸⣄⠀⠀⠹⢤⡀⠲⠤⠍⠉⢉⣼⠇⢸⠁⡇⢰⠃⠀⠘⢤⠿⠞⠸⡇⡇⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠙⠉⠳⣦⣀⠀⠉⠓⠒⠚⠋⠉⣀⡄⢸⢀⣇⡘⠒⠲⠤⠴⠦⠤⠴⠇⡇⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠻⣏⠉⠓⠒⠒⡚⠉⡹⠃⠸⠯⢭⣭⣍⣉⣉⣉⣙⣛⡛⢙⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠙⠲⠬⠯⠷⠿⠷⠤⠤⠤⠤⠤⠤⠴⠞⠛⠉⠉⢉⣩⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⢠⡑⢀⢀⡀⠀⢀⡞⢻⣷⣶⣖⣣⣿⣶⢷⣶⢀⢀⠎⠉⠀⠀⠀⠉⡆
⠀⠀⠀⠀⢀⡤⢞⡠⢤⡀⠀⠀⠀⠀⡠⢇⡀⣾⣶⡄⠀⠀⠀⠀⢰⠖⣾⢫⢽⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⣇⡄⠤⠀⠀⠐⡇
⠀⠀⠀⣠⢏⣓⢁⣀⣀⠧⡄⠀⠀⣴⠣⡀⠀⠀⣿⡗⣢⠀⢀⣖⢫⣉⣉⣉⡼⢒⡄⠀⠀⢀⣴⣶⣶⣷⢄⡀⠀⠀⠀⠡⣀⠀⠀⣀⠔⠁
⠀⠀⠀⠻⢁⣁⡀⠀⢉⣩⠅⠀⠀⠣⢀⣉⠀⠈⣉⡠⠜⠀⠀⠦⣈⣉⠉⢉⣉⠤⠃⠀⠀⡏⠉⠉⠉⠛⠘⠛⢲⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠏⠀⠥⠒⠲⠤⠤⠾⠗⢄⢀⠤⠴⡧⠤⢄
⠀⠀⠀⠀⡠⠄⢁⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⢠⠊⠈⠀⠀⠈⢑⠀⠈⠓⠂⠀⠀⠀⠀⠐⠒⠈⠉⠂⠀⠁⠀⠊
⠀⠀⠀⠀⡱⠎⢟⠵⠉⢢⠀⠀⠀⠀⠀⠀⢀⢆⣀⠷⣄⠀⠀⠀⠀⠈⣇⠐⠒⠒⠂⠿
⠀⠰⣶⠶⠷⠴⠶⠶⠶⠿⢶⡶⠄⠀⣄⠲⣿⣤⣤⣤⣿⣶⣦⣄⠀⠀⡆⠀⠀⠀⢀⣼⣤⣦⣤⣄⠀⠀⠀⢀⣤⣶⠎⣯⡯⣳⠶⢢⣄
⠀⠀⠈⠢⠍⣉⣉⣉⣉⣁⠀⠬⠆⠀⠈⠳⠌⣉⣉⣉⣉⣁⠀⠬⠇⠀⠇⠀⠠⣒⣺⣧⣭⣿⣭⣼⣷⣠⡀⠀⠩⢍⡙⠻⠒⠛⠈⢉⣁⠱
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠼⠦⠄⠀⣀⣁⣀⠀⠤⠜⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣀⡀⣆⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣔⢶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠏⠀⠀⠐⠀⠀⠈⢡⠀⠀⠀⠀⠀⢀⣤⣒⣞⣤⠃⠀⠀⠀⠀⠀⣀⡀⡠⠀⣄⡀⣤⣂⡀⠀⠀⠀⠀⠀⠀⡀⠠⡋⢱⠄⢀
⠀⠀⢺⡵⠦⣤⣤⣤⡤⠐⠭⠦⡀⠀⠻⡟⠻⠿⡿⠿⠿⠿⠽⠿⡂⠀⠘⠲⡷⠷⣴⣿⣷⢴⠿⠤⣬⣟⠲⠀⠰⡽⢿⣷⣾⣿⣾⡿⠧⠦⡀
⠀⠀⠀⠑⠂⠠⠤⠤⠤⠄⠀⠐⠃⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠙⠂⠀⠠⠤⠤⠄⠀⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡿⠿⠛⢉⣡⣶⣾⣿⣿⣿⡆⢠
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⢀⣠⡴⠟⠉⠁⠈⣉⣉⣉⣀⠈⠂
⠀⠀⠀⠀⠛⠻⠿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⢉⣩⠴⠶⠶⣶⣶⣶⣾⣿⠟⢁⣠⣤⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠀⡀⢢⠀⣶⡿⠛⠛⠿⢿⣶⣦⣤⣴⣶⠾⠋⠁⠀⠀⠀⣀⣠⣭⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠸⡄⠁⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠟⠛⠛⢿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢦⣴⠟⢷⣄
⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⢀⣀⣤⠴⠖⠛⠋⣉⣁⣀⣀⣉⣙⠻⣶⣄⡀⠀⠀⣼⠟⠀⣰⠏⢀⣼⣏
⠀⠀⠀⠀⣠⣾⠿⠋⠀⢀⣴⠞⠋⠁⣀⣤⠶⠞⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⢿⣦⡼⠋⢀⡾⢃⣴⠋⠀⢙
⠀⠀⢀⣼⡿⠋⠀⢀⡴⠋⠀⠀⣠⠞⠉⠀⠀⠀⢀⣠⡴⠶⠛⠛⠋⠉⠁⢀⣀⣼⣿⣶⢋⣴⠟⢁⣤⠖⠻
⠀⢀⣾⣿⠁⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⢻⣿⡥⠖⠋⣁⣴⠶
⠀⣼⣿⠇⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⢻⣧⠴⠛⠁
⢠⣿⡿⠀⣰⠏⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⢸⣿⡇⢀⡏⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⢿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⠀⠀⢸⣧
⠈⣿⡇⠀⡴⣲⣴⡄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⣷⠀⡇⡅⢹⡇⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⢸⣿⣼⢃⢧⣟⣿⢣⠋⣰⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣿⠃
⠀⡾⣿⣿⣸⣸⣾⡼⣇⣴⣿⢗⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⢠⣿⠏
⢸⡃⡟⢿⣧⣹⣭⣶⣗⣛⡥⠊⣀⠀⢀⣤⣀⣀⣀⣤⣤⣤⣴⠶⠾⠛⠛⠉⠀⠀⠀⣠⣿⠏
⠈⠧⣑⣊⣥⣾⣿⣲⣏⡉⠿⠭⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⡿⠃
⣴⡖⢚⡳⣳⣿⣿⣕⢪⣽⣟⣛⣶⠦⣄⡀⠀⠀⠈⠉⠉⠙⠛⠛⠛⢛⣻⣿⣿⡿⠋
⠀⠉⠉⠿⠵⢋⣿⡆⢯⣛⢛⣛⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣴⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠿⠊⠀⠀⠉⠉⠉⠀⠉⠛⠛⠻⠿⠿⠿⠿⠛⠛⠋⠉