Lying Emojis & Symbols

⠀⠀⠀⠀⠀⠄
⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠀⠀⠠⢄⣈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⡀⠁
⠐⠀⠁⠀⠀⠀⠀⠈⠋⠉⠁⠀⠹⣤⡀⠀⠀⡈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠨⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣡⢠⣄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢤⣤⡤⠌⠿⡳⣼⣷⣬⣷⣷⣾⣴⣧⣪
⣷⣿⣿⡏⡠⡀⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠓⢾⣿⡻⢽⣿⣹⣿⣿⣿⣿⣷⣾⣟⣿
⣿⣿⣏⣼⣾⢋⠔⣡⣴⣖⣥⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠩⠵⢖⣚⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿
⣿⣿⣿⣿⣷⣵⣿⣿⣿⣯⣵⣾⣿⣭⣷⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣶⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⠿⠿⢿⣿⣿⣟⣛⣉⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⢁⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⣿⡛⠙⢧⡀⠙⣯⡹⣤⡀⠙⠻⣿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⣼⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣧⡙⢷⡈⠻⣆⠀⠳⣄⢌⠳⣞⢷⣔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠏⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣝⢛⢛⣆⠛⢆⡘⠓⢆⠊⠂⠁⠈⠋⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠁⢠⠊⠉⠉⠀⠀⢀⡹⣿⠋⣿
⣿⣷⣧⠘⠷⣄⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⠠⠤⣤⠀⡀⠁⠀⠙
⠈⠁⠙⠦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠃⠀⠠⠊⠁⠈⢉⣉⣉⣉⣉⠉⡩⠃⢀⣿⡧⠀⠸⢏⡐⡛⠆⠀⠀⠈⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠏⠀⢠⠁⠞⠛⠋⠉⠉⠉⠉⠉⡙⠁⣰⣿⣿⣿⣦⠀⠀⣧⡀⠀⠐⠂⠈⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠰⠃⢠⠃⠀⠀⢀⣀⣀⣀⣤⣀⡐⠀⡴⠉⣉⣉⣹⡇⠀⠀⣻⠇⠐⢀⡀⠀⠀⠀⢸⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢠⠈⠀⠙⠛⠉⠉⠀⠀⠀⠴⢀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⣼⠿⢄⣠⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠛⠀⣀⣀⣠⡤⠤⠴⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢑⡛⠉⢉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⡞⠀⠀⠀⠀⠗⠀⠉⠉⠁⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠰⠒⣋⣩⣭⡇
⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⢰⠖⠒⠒⣂⡀⠉⣴⣶⣿⣥⣤⣄⣀
⠀⠀⠀⠀⠀⢀⡁⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⣿⣷⡀⠹⡿⠿⠿⠟⠉⠉⠀⠛⠁⠀⠀⠒⠀⠐
⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠠⢡⠀⠀⠀⢀⠤⠀⠀⡹⠄⠚⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠈⠁⠀⢀⣠⣄⣀⣸⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠶⣦⡠⡙⣦⠳⠸⣇⢂⣧⡑⡄⠘⢀⡆⢆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣛⠲⢌⣼⡁⠀⡀⢙⢦⠈⡷⠀⢹⠀⢸⡇⡁⡆⣨⠃⡎⣰⠚⢕⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⡭⢭⢽⣌⣅⢁⠤⠌⣦⣿⠀⠃⢁⡾⡔⢸⠃⠳⢡⢣⠇⠠⢁⡴⣋⣠⠤⢽⣦⣄
⠀⠀⠀⠀⠀⠀⢿⣋⠞⠰⣻⠞⠛⠸⣨⢿⣈⣷⡄⠙⠇⠔⣰⠋⢸⠁⡞⠰⣀⣉⣀⣭⢅⣉⣀⣀⣉⠛⢶⣄⠈⢇⠂
⠀⠀⠀⠀⠀⠀⠀⢡⣦⣇⢫⢾⡘⢧⠀⡿⣏⣧⣽⠀⢂⡼⠁⠆⣼⢐⡵⠛⠉⠀⠀⠀⣉⣠⣀⣉⡉⠻⣦⡈⡑⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢘⣸⠻⠩⣴⣾⡀⠀⠑⠘⠓⠀⠃⠏⣰⡸⠈⠀⢋⣴⠾⠛⣋⠭⠀⠤⠤⣄⣉⠙⣧⢸⣷⠃⢷⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣮⡻⣷⣉⣙⠇⣀⠀⠆⠀⣆⣤⠞⠳⣶⠂⣠⢞⡤⠚⣋⠥⠀⠤⡀⢀⠀⣙⡇⣹⢛⣟⡄⣾
⠀⠀⠀⠀⠀⠀⠀⠰⣥⢋⠉⠉⠙⠳⣤⡀⠋⠴⡍⢙⡦⠞⠁⠊⣅⢄⣒⣚⡥⠖⡻⠢⣃⠞⣸⢟⣼⣣⢏⡞⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣧⠚⠲⠶⠛⠲⡄⠐⡆⢰⣈⡯⠰⡆⢿⡗⢸⡇⢠⡂⠞⣰⠊⢐⠋⠑⢉⣺⠣⠚⠁
⠀⠀⠀⠀⢀⣶⣶⣦⡴⣇⠠⡏⠀⣂⢠⠠⢀⡍⠻⠸⣿⣄⡁⠘⠤⠚⠗⠛⠒⠛⠁⠚⠓⠂⠚⠉
⠀⠀⠀⠀⠀⠻⢿⣿⣴⣿⢡⠉⢠⡶⠀⡄⢻⡷⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠘⠀⠛⠓⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢘⣿⡿⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⣿⡿⣀⡄⠀⢀⠞⠛⠛⠷⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣶⣄⣠⠟⠚⢿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⠊⠊⠀⠀⠀⠀⠀⠀⠀⠁⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠫⠙⠹⣿⡏⠉⠑⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⢀⠚⡏⡧⡀⠀⠀⠑⢄⡀⣨⣀⡀⠀⣂⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠈⠓⠭⢁⣘⡁⢈⠢⠀⣐⡜⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⠄⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠛⠁

⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠲⢄
⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠂⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡤⣴⠆⢐⡞⣧⠀⢀⡤⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠲⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣏⠀⠀⠀⠀⢴⢛⣿⡏⠙⡭⠏⢙⠈⡿⠉⠀⢸⡔⠒⠒⠒⡆⢀⣀⡠⢤⡆⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢿⠀⠀⠀⣼⡇⠋⢨⡁⠀⠀⠉⠀⣾⠥⠦⠯⠩⣏⠀⢀⣠⠅⠤⠤⠤⠜⣠⡀⠀⠀⣠⡔⠁⠀⠱⣄⠀⠀⠀⢸
⠀⢀⡠⠤⣼⠀⢀⣠⣿⣙⡏⣈⣥⡄⠀⠀⠀⢹⠴⡚⢋⠭⢈⡗⠛⠳⣆⠃⠀⠀⠀⠀⡗⠲⠴⣅⣻⡀⠀⠀⠈⠳⠀⠀⠀⡠⠔⠒⠢⢄
⢰⠏⠀⠀⣸⠖⠉⠀⠸⣿⣾⣿⣽⣉⣼⣀⡴⡁⠜⠀⣀⠀⣸⠀⠀⠨⠇⢠⠀⠀⠀⠰⠇⠀⠀⠀⠈⠉⠉⠉⠒⠤⣾⠤⢠⡧⢊⡇⠀⠈⠇
⡟⠀⢠⡀⢀⡀⠀⠐⠋⠠⠋⠙⠋⠉⠁⠈⢯⠀⠀⠀⠀⠀⠉⠋⠀⠀⡇⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢎⠀⣀⡠⠊
⡇⠀⠀⠈⢉⠟⠶⣄⡀⠀⠀⠀⠀⠀⢀⡀⠬⠳⠦⣀⣀⠀⠀⠀⠀⢸⢀⣀⣀⣀⠀⠋⠛⠉⠉⠉⠁⠈⠉⠑⠢⢄⡀⠀⠀⢀⠎⠈⠁⠀⡠
⡇⠀⠀⠀⢰⠀⠀⠀⠉⠉⠑⠒⠛⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⡎⠻⣅⡠⢊⠁
⢣⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠞
⠈⠳⡒⠒⠻⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⡇⠀⢰⠃
⠀⠀⣧⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡇⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠐⠂
⠀⠀⠀⢠
⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⡀⠀⠀⠀⠈⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⠦⡄⠀⠉⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢧⣑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⢑⠖⠉⠉⡞⠘⡀⠀⠀⠀⡜
⠀⠀⠀⢀⠔⠒⠒⠒⠚⠣⠃⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⣸⣤⣀⢼⠁⠀⢀⠇⢠⠛⠀⠀⠰
⠀⠀⠀⠘⠤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠤⠤⢤⠄⠀⠓⠆⠁⠀⠸⣄⠀⠀⠱⠄⢤⠄
⠀⠀⠀⠀⣴⣢⠐⣺⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡠⠚⠒⠁⠁⠀⠀⡤⠎
⠀⠀⠀⠐⠙⠒⠈⠪⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡐⠳⠤⠴⢄⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⣾⢀⣿⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⡴⣶⣶⠺⣽⣠⠴⣯⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠼⣜⣄⠠⢠⠋⠧⠬⣵⣃⡇⠰⢦⣿⠱⡷⢹⠷⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣊⠑⣂⡀⢀⡳⠤⡄⡠⣃⡙⣇⢛⣤⡁⢼⠲⠍⠁⠲⡉⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡰⠃⣭⡑⣤⣌⠛⠋⢸⡆⠀⠀⠀⢁⠡⠤⣪⢕⠡⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⡍⠁⣛⠋⡌⣁⣾⡀⠀⠫⠭⠅⠀⢄⡒⠶⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⡤⢾⣿⣇⣨⡻⣋⡅⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢘⣇⠀⠀⠀⠈⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠀⠤⠬⠿⠵⠷⠀⠀⠀⠾⢮⣄⣂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠹⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢂⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢱⠐⠉⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡀⠈⠂⢠⠄⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠠⠐⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡌⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡀⠳⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠃⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡤⠀⢈⣉⠿⠇
⠀⠀⠀⠀⢀⠮⠁⠴⠄⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡄⢂⢈⣀⡠⠔⢊⠔⠊
⢀⠄⠐⠀⠃⠀⠀⠀⠀⠀⢆⠀⠀⠘⠠⢉⠂⢄⡀⠀⢀⣀⣀⣀⡀⡠⠔⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⠈⢉⠉
⠀⠀⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠸⣀⠀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡤⠀⠠⠃
⣤⣤⣤⣤⣤⣭⣤⣤⣭⣭⣥⣬⣴⣤⣤⣤⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣳⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⣵⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠉⡿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣌⢿⣤⠆⠀⠁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣛⣻⡏⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠿⣿⣷⣶⣾⠛⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⠃⠇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⢻
⠀⠀⣀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘
⠐⠛⣿⣿⡛⣆
⠀⠠⡟⢸⡇⠈
⠀⠀⠀⠘⠇


⠄⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣴⠟⣦⠶⡆
⣼⠛⣿⠻⣆⠀⠀⢀⡤⢀⣴⠞⠉⣴⣯⡿⢰⣋⠿⢳⣤⣀
⠁⠀⡟⠀⠙⡆⢀⣾⠁⡘⢿⣆⠀⢿⡿⠁⠀⠺⠆⢈⣭⡍⢣
⠀⠀⠀⠀⠀⠀⠘⢿⡄⢷⠀⢙⡇⠀⠀⣰⠆⣆⠛⣶⣼⣇⠈⢧
⠀⠀⠀⠀⠀⠈⠳⣄⠻⣼⠃⣼⠁⢀⣶⠋⢸⣯⣈⣁⠀⠀⠀⠀⠱⡄
⠀⢀⣀⣈⣙⠛⠦⣌⠣⠘⠀⡇⠀⡄⣛⠈⠉⠀⢁⡀⠊⠁⠀⠀⠀⣐⣆
⠀⠀⣀⣤⠍⠓⠤⢄⣉⠀⠀⢳⡀⠿⠘⠷⣄⡘⡀⠆⢠⣤⣀⣐⣀⡿⠃
⢀⣸⣭⠶⠒⣚⣻⣶⠶⢮⠁⠀⠈⠀⠰⢤⣀⣙⡘⠆⠀⠙⠛⠋⡹⠃
⣾⢱⣯⣾⣟⣋⣥⣴⠾⢋⣤⠶⠞⣋⣥⣤⣤⠍⠀⠀⠀⠀⠙⠋
⡟⢸⡿⢙⣿⣿⠉⢥⣾⠟⠁⣠⣾⠟⢉⡼⠁⢀⣤⠶⠒⠛⣫⠖⠀⢀⣠⣴⠀⠀⣀⣀⣤⠶⢂⡀⠀⢀⡀⠀⠀⢀⣤⠖⠛⠁⣀⣀⣀
⢷⠛⠃⣸⡟⣿⡇⣠⣾⣶⡿⠟⠁⣴⡿⢁⣼⡟⠁⠀⢀⡞⠁⣠⡾⢛⡿⠁⡴⢊⣽⠋⣠⡾⢋⡴⢾⣯⠀⠀⣴⡟⠀⢀⣴⠛⠛⠛⠛
⡼⢷⠄⠻⠠⣿⠀⣿⡏⢰⣿⠛⣾⣿⣴⡿⠋⠀⠀⣠⡿⠁⢠⡿⣡⡿⠁⠎⢀⣾⠃⣸⣏⡴⠋⢀⣾⠃⠀⣰⣿⠀⣰⡿⠁⣀⣠⣴⠶
⣤⣶⠎⢢⡀⣧⢸⣿⠀⣾⡇⢠⡿⣿⡉⢀⣴⣾⠿⠛⢀⣠⣴⡿⠋⠀⠀⢠⣿⠃⣴⡿⠁⠀⢠⡾⠃⣠⣾⠟⢁⣾⣟⣴⠟⢩⣴⡶⠶
⣡⣴⠞⣤⠱⣌⠈⢻⡄⢿⠀⢸⡇⢸⣇⠈⣿⣄⠀⠺⣿⡛⢩⣴⣾⢃⣴⣿⣣⣾⣏⣀⣠⣴⣿⣴⣾⣿⣷⣾⣿⣿⣿⣥⣀⣀⣠⣴⡶
⢋⣴⣾⠟⠀⠈⠳⢦⣽⣆⠃⠀⠀⠀⢿⡀⠈⢻⣦⠀⠙⣿⣿⣿⡙⠻⣿⡝⠿⠿⢿⣦⠉⠹⣿⣿⣿⣿⣾⣿⣿⣿⣾⣷⣤⣉⡉⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣄⠀⡀⠶⠀⢲⣤⠂⠑⠂⣄⢀⡠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠐⠉⠈⠸⡀⠀⠦⠀⣰⠁⠂⠈⠁⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢋⠐⡀⠄⣰⠁⠀⠀⠐⢳⣄⢠⢀⢠⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡄⢣⣤⣍⠐⠀⠀⠀⢀⠤⢤⢉⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡁⠀⠘⠀⠑⠚⡆⠀⠀⠀⢞⠀⠁⠈⢈⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⢄⡰⠒⢚⠴⢹⡀⠠⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠢⠷⠬⠂⢫⢈⣏⢌⠇⠏⣇⠀⢻⣧⣆⡂⠀⠲⢄⣀⠴⠀⠀⣰⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⢠⣄⠀⠈⢅⢛⡘⠁⠛⢈⡌⡀⣶⠶⠃⡄⠀⢿⣿⣷⡀⠀⢀⣅⡀⢀⣰⡟⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣄⠻⢛⣀⠾⣥⠀⠢⠹⣟⠸⢼⡅⠇⠛⢈⡐⢿⡐⠆⠠⢿⣿⣿⣅⡈⣁⠀⠈⠁⠀⠀⣠⠎⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠺⢣⣮⠉⢁⡀⢠⡄⢄⢻⢣⡆⣅⢶⡘⠇⠛⣆⠱⠀⢥⠀⣌⠻⣿⠿⣯⡀⠀⠀⠀⠋⢁⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡎⢡⣷⢈⣩⣔⠦⡙⣦⡙⢮⠈⢡⡅⢈⡈⢡⣦⢸⠀⠓⠃⠀⣀⠢⢄⠈⠓⢦⠁⠀⠘⢡⠖⠋⢀⠈⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣴⣦⢉⡈⠛⣋⢠⣶⣌⠛⠀⠤⠀⠙⠂⢻⣌⠛⣀⡐⡎⠂⠐⠆⠀⠀⠙⠢⡀⠀⠠⠔⠁⠠⠔⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣝⢳⣯⢸⡟⢀⡙⠁⣡⣭⡁⠲⣄⣄⠘⣟⣦⠙⠀⣹⡇⠀⠉⠂⠀⠠⢌⠢⠄⠉⠀⢀⠀⢠⠄⢠⠆⢀⢅⠄
⠀⠀⠀⠀⠀⠀⡀⢤⣧⡛⠆⣩⣦⢰⡿⢡⣦⠉⣛⡃⢰⣸⣽⡆⢌⠹⠆⢦⣧⣰⠐⠟⣀⣐⡂⠀⢀⠀⠀⠀⡙⠀⠈⠀⠁⠀⠀⣀⠠⠤⠄
⠀⠀⠀⣠⠐⢛⡳⢼⣿⣍⣷⠿⢣⣶⣠⡌⠡⠌⠭⠍⠼⠿⠿⠿⠞⠇⢴⡌⣯⢱⡆⢆⡁⠐⠂⣀⡘⢇⢰⣾⣷⣿⣄⣤⣴⣤⡾⢁⠀⠄
⠀⠀⢀⡵⠿⠶⣥⡾⣿⣿⣿⣶⣦⣻⣿⣧⡀⠂⠀⢀⠀⠀⠀⠀⠀⠄⣈⣡⣨⡁⢶⠈⠴⢢⠈⠱⠀⠴⢉⡛⠿⣿⣿⣿⣿⡋⣧⡄⠠⠀⠀⠠
⠀⠀⠸⣧⣄⠛⠛⠂⠠⣤⣈⡉⠉⠉⠙⠛⠛⠛⠛⡛⠛⠛⠛⠲⠒⠛⠛⠛⠛⠇⠯⢘⠶⠉⣬⡃⣋⠰⠆⡤⠠⠄⠝⠛⠻⠷⡌⠂⠡⠀⠀⠂
⠀⠀⠀⠀⠙⠛⠿⢾⣿⣿⣿⣤⣾⣇⡀⣸⣿⣤⣾⣥⣾⡟⢁⣤⣤⣤⡀⠀⠀⠀⠀⠺⠁⠠⣾⣀⢩⠔⡃⠆⠐⠰⠀⠀⠀⠀⠜⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠈⠉⠉⠓⠤⠽⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⡤⢬⣀⣀⡀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢥⣢⣄⡐⣢⠄⠃

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⣀⠠⠤⡄⡀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡧⢀⡐⢴⠁⡇⢀⡌⡀⢰⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⣓⠦⢰⡇⢅⠌⠦⠘⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢗⣾⠝⢿⣷⠀⠀⡾⠯⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢟⣤⡀⣅⠀⠀⠀⠀⠈⠀⠀⢷⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣔⣷⣄⠀⠀⠈⠁⢀⠀⢠⠵⠉⠧⢸⠁⣘⠆⠤⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡤⠶⠄⡶⢈⡠⠊⠀⠸⡀⠀⠘⡀⡪⢱⡞⠐⠋⡐⠢⠖⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡟⢿⠒⣄⠄⠋⠠⠄⠂⠐⠨⠀⠓⢈⣐⣿⢘⡑⢎⣅⡬⠹⢨⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢡⡀⠈⢁⠦⡾⠐⠶⠂⠁⢀⠸⠀⢇⢉⠲⡸⠃⡨⢠⡌⣵⡦⢾⢘⡑⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠮⣍⣀⠁⢨⠀⢾⠇⢋⣈⡁⠈⠁⠃⠁⠔⡈⣸⣷⢏⡘⢋⣅⠰⠃⡋⣿⢇⣸⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠙⠌⣍⠳⣠⣷⣿⡇⢈⡉⠙⠀⠚⠃⠃⠒⣣⣼⡷⢋⠁⠟⢸⠇⣯⠟⣴⠾⣭⠁
⠀⠀⠀⠀⢠⣀⣸⠁⠈⢎⠢⡀⡿⠟⠋⠀⠀⠂⡉⢤⢠⠐⢃⡈⣩⣾⣟⡉⠵⠸⣦⣾⣴⠆⣡⢶⣷⡄
⠀⠀⠀⠀⠈⣽⠃⠀⠀⣢⠅⠈⠀⠀⠄⠀⢄⡁⠀⠀⠀⠀⠉⠡⠙⠃⡀⠡⣐⢉⣻⣬⣑⢽⣞⣚⡿⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠸⠀⠀⠀⠀⠁⠦⠐⠿⣧⣵⣤⡤⠤⠤⠄⠉⠂⠂⠘⠁⠈⠉⠉⠉⠉⠉⠁
⠀⠀⣴⡀⠀⠀⠀⣾⣦⢄⡀⢀
⠀⠀⠛⠾⠦⠚⠂⠉⠉⠉⠁⠈


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⢀
⠀⡀⠀⠀⠀⠠⠤⢬⣷⣿⣿⣟⡒
⣼⣿⣶⣢⣿⣿⠔⠂⠰⢿⡿⠛⠻
⣿⣿⡟⢻⢿⣿⣶⣦⠀⠈⢁⣾⣦⠀⠀⠀⠀⣀
⣇⠁⠈⠁⠀⠘⣿⠿⠇⠀⠸⠿⠻⠗⠀⠀⠾⡿⠖
⠋⣠⣄⣰⣦⣷⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣤⣤⣤⣿⣦⣶⣤⣉⣀⣀⢀⢀⠀⡀
⠿⢿⢿⣿⣿⡛⠿⢿⣿⢿⣯⣿⣹⣿⣿⡟⠿⢿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠂⠀⠀⠀⣀⣀⠀⠐⠀⠀⠐⠶⢶
⠛⢶⣾⣷⡛⠟⠛⠒⠶⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠐⢂⡈⠁⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣶⣦⡀⠀⠤⠬
⡀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⢑⣀⣀⣀⣠⣤⣤⠀⠀⠠⠬⠭⠿⠿⠿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⢿

⢀⣴⣾⣿⣿⠟⠁⠀⢀⣷⠎⣉⣹⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠱⠂⠀⢠⣄⣶⣦⡀
⣾⡿⡟⠁⠁⡄⢀⡴⠋⠘⠒⠏⠻⣧⡄⠐⠠⢤⡀⠀⠀⠀⠀⠀⠀⣰⣿⠈⠻⣳⠈⠂⠀⢠⠽⣿⣿⡟
⠟⠁⠀⠀⢛⡷⠛⠁⢈⢠⣿⠂⠀⠈⠀⠀⠀⠈⢿⡀⠀⠀⠀⢀⣼⠟⠃⠀⢠⠟⠂⠀⠅⠈⠳⢽⣿⣿⢡
⠀⠀⢰⣤⠟⠁⠀⠰⢀⣤⡅⠀⠀⠀⠈⠢⠀⠄⠀⢻⡄⠀⢀⠾⠁⠀⠀⠀⠸⠃⢀⡄⢶⡶⠀⠸⣝⡿⢿
⠀⢀⠞⠁⠀⠀⠤⠀⠼⠏⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⠇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣦⠀⠀⢻⣿⣿
⠀⠃⠀⠀⠀⠀⣄⣼⡖⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠆⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⠀⡀⠀⠀⣀⣤⡇⠀⠀⠀⠹
⠀⠀⠠⢤⠔⠀⠰⠿⢴⠤⠤⢤⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⢀⡴⢋⠀⠠⡄
⠀⠀⠀⠸⠀⢀⣢⡖⠀⠑⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⣉⠙⠝⢦⠀⠐⠄⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⢟
⠀⠀⠀⠀⢸⣷⠏⠀⠙⠳⣦⣌⠓⢄⡀⠑⠠⠄⠠⠀⠤⠤⣒⣮⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⢠⣄⣉⠀⠀⠀⠀⠀⠉⠛⠶⠾⣶⣶⣶⣶⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⣲
⠐⣰⣀⠀⣻⣧⣤⡢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡟⠀⣀
⣄⣿⠉⠂⢻⢟⠅⢤⡰⠄⢬⡅⠀⠄⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀⠘⠀⣀⠀⢖⡛⠯
⣻⡷⠃⣻⣋⣔⢂⡴⠷⠚⡉⠀⠋⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠒⠀⠀⠀⠤⠓⠒⠂⠀⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⣦⣄⣹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠇⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡿⢿⣾⣿⣿⡿⠿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠛⠋⠉⠁⠀⠀⢸⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡎⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣾⣿⣿⣿⣿⠂⠰⣿⣿⣿⣿⣿⣿⣿⣿⠛⣒⢒⡘⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣦⣤⠼⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣦⣴⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠸⣿⣿⣿⣟⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣿⣿⣟⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡏⠀⠀⠻⠋⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠈⠉⠁⠀⠀⠀⠠⢄⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠁⠀⠉⠉⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣴⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣛⣿⣿⣿⣿⣿⣿⣧⠀⢀⣾⣿⣿⣿⣿⣿⣟⣋⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢛⣿⣿⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⣿⣿⣿⡻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠈⠉⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⠛⢳⡤⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⣴⣿⡷⢿⣿⣶⠟⠀⠀⣠⡾⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠿⠁⠀⢀⣼⠟⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠋⠀⠀⠘⠃⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⣰⣏⣀⣠⡖⠀⠀⠁⠀⠀⠀⣰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠰⣿⣿⣿⣿⡇⢰⡖⠀⠀⠀⣰⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣷⡿⠀⠀⠀⢰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⢠⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⠁⠀⠀⣠⡾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠾⠛⠉⠀⢀⠄⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⢀⣤⠶⠋⢨⣧⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣴⠞⠋⠁⠀⣠⡴⠞⠋⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠀⠀⢀⣴⠟⠃⠀⠀⠀⣠⣴⠟⠋⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣋⣀⠀⠀⠀⠙⠁⠀⠀⠀⣠⡾⠋⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠙⠛⠃⠀⠀⠠⣤⣴⣟⡀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠐⠶⣬⣳⣿⡇⣰⣀⣤⣀⡀
⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣟⡡⠤⢤⡀
⢀⡀⠀⠰⠏⠉⡼⠋⣾⠿⣿⠓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣾⣧⡀⠐⠀⠀⠀⢰⠛⡄⠙⠆⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣥⣶⡶⠤⡄⠈⠀⠁
⢿⣿⣿⣿⣷⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣷⣶⣥⢱⣷⠦⣿⣶⣠⠀⢰⣿⣦⠀⣰⣿⣷⣄
⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣤⣀⣋⣈⢁⣉⣋⣈⡆⢀⣀⣀⣀⠀⠠
⠙⠻⣷⣮⣭⠩⡟⣛⠛⠿⠿⠯⠿⢿⣽⣛⡛⠛⠛⠙⠋⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⣿⣿
⠚⠢⠙⣏⠙⢻⠛⠻⠶⢤⣈⣓⠀⠀⠀⠉⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠐⠒⠲⠂⡿⢀⡗⠶⠄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠿⠤⢤⣀⣼⡇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠶⣤⡀
⣴⣾⠿⠿⢿⣷⣦⣤⡀⠀⠀⠀⠀⠙⠃
⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠈⠀⠀⠀⠀⢀⣀⡀⠀⠠⠾⠿⠋⠂⠀⠉⠙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣉⢹⡆⠀⠀⠀⠘⢯⣿⠿⠓⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⣠⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠐⠆⠀⠀⢠⡿⠁⠀⠈⠃⠀⠀⠀⠀⠈⠱⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢳
⠀⠀⠀⠴⠝⡂⠀⠠⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⠂⠁⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠣⣄⡤⢤⣀⣈⠛⢛⡿⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢌⠢⠀⠀⢀⣴⣦⡤⠤⠶⠒⠒⠒⠒⠚⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠆⢀⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠙⠀⠀⠀⠔⠊⣡⠀⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣴⣶⠶⠚
⠀⠀⠀⠀⠀⠀⢠⠍⠃⠀⠀⠀⢠⡴⠛⠛⠿⠿⠷⠤⠤⠶⠶⠶⠶⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠾⠒⠒⠒⠚⡿⠛⠁
⠀⠀⠀⠀⠠⣀⣀⡀⠤⠔⠊
⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠘⢿⣿⣿⡿⢿⣟⢹⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⡄
⠀⠀⠀⣀⡀⠈⢻⣿⣷⠂⠙⢿⡿⢻⣿⣿⠛⢿⣟⣻⣿⣿⣿⡏
⠀⣶⣄⠈⠙⢶⣾⣿⣿⣶⣶⣾⢡⣾⣿⣧⣈⣀⣴⣿⣿⣿⣿⣷⣶⡒⢂⡀
⠀⠉⢻⠦⣀⡀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣭⡵⠿⠻⣿⣿⣿⣿⣿⣿⣿⣆
⢿⣿⣿⣿⡿⠛⠃⣬⣿⡿⠋⣉⣉⣉⣉⣙⣿⣿⣿⣶⣾⣿⣿⡿⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣴⣿⣿⣿⣿⣿⣾⣿⣿⣤⣾⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣶⣶⣾
⠀⣠⣼⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣷⣦⣤⣴⣾⣧⢀
⣴⣟⡀⣰⡿⠟⢋⣿⠿⣿⣿⣿⠛⣿⣿⡿⠀⣿⣿⣿⣿⠟⠀⠀⢀⣀⡀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠻⣿⡏⠙⣿⣿⠿⣿⣿⢿⣿
⢿⣿⣿⠟⠁⠀⢺⣿⡀⣿⣿⣿⡆⢻⣿⣟⠀⠉⠉⠻⡿⠀⠀⢴⠾⣏⣻⣧⣸⡿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⢿⣿⣴⣿⣿⣷⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⣿⣾⡟⢿⣿⠇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣭⣵⣽⣿⣿⣿⠿⠻⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣩⣾⣿⣿⣯⣙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⡆⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⣽⣽⡿⢿⡟⣦⠀⡀⠀⣀⠀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠹⣿⠁⠀⠙⠛⠀⢹⣿⡟⢠⣿⡟⢩⣿⠟⢿⣿⣿⣿⣿⡟⠻⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⣿⡿⣿⠛⢿⠁⠀⠀⢹⠟⠉⢻⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠹⠇⣶⣾⣿⣷⡤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠖⠚⠛⠻⣿⡟⡿⢿⠿⠄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣹⣴⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠀⢇⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⢿⡿⠻⠀⢰⣿⠖⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⣴⣾⢒⣶⣄⡀⠤⠌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠇⠀⡇⠀⠀⠈⢃⣴⡆⠀⠀⡄⣠⡀⠀⠀⢀⠀⣼⠿⠃⢻⠁⠀⠀⣤⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠀⢃⢀⡀⣧⠀⠀⠀⡀⢀⣀⠀⣰⣿⡿⠇⠀⠀⢸⠀⣿⠀⠀⠈⢳⣿⣷⣿⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⣿⡄⣄⣠⣤⣀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢰⠀⠀⣀⣠⣿⣷⡀⠀⣀⢿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠲⢤⡤⠤⠀⣠⣤⣤⣀⣄⣀⡀⠀⠉⣹⣿⣤⣿⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠙⠢⢼⣉⡉⠙⠛⠻⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢠⡴⠊⠉⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣶⣴⣷⣦⣦⡀⠈⠙⠻⠿⠿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣇⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠀⠀⢠⣤⣤⣾⣿⣿⣷⣶⣿⣿⡿⠿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣤⡤⠸⣿⠒⢺⣿⣿⣿⣿⡿⠋⠉⠉⠀⠀⢀⣤⣤⣴⣶⣿⣷⣾⣶⣶⣿⣿⣿⣯⣍⣉⣍⢡⣶⣷⣦⣶⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢸⣿⠀⢀⡀⠈⣿⣆⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣋⣩⣩⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣤⣤⡆⣿⣿⣟⣉⣈⣉⣻⣿⣿⣦⡀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣾⣿⣿⣿⣿
⣶⣤⣀⣉⠉⠤⠄⠀⠀⠀⣀⢠⣆⠀⣿⣿⣿⣿⣃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡿⢟⡛⠉⠿⠾⠟⠛⠛⠛⠉⡀⢠
⣿⣶⣶⣶⣤⣶⣶⣤⣤⣾⠿⢻⣿⡄⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷⣾⣇⣀⣀⣀⠀⠀⣀⣤⣶⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣻⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⡇⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣏⣀⡀⢰⣿⠋⣽⢿⣿
⣿⣟⠛⢿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⡿⠛⠛⣛⣿⣿⣟⣿⣎⠋⣨⣿⠈⣩
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠠⢀⣠⣽⣿⣿⣿⣷⣿⣧⣼⣿⣼⣿
⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⣶⣶⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⡀⠀⠈⠙⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⣿⣿⠿⠫⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠙⢛⣷⣾⣿⣿⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣶⣬⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠹⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣬⣽⣿⣷⣤⣤⣎⣻⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠯⢿⣿⣯⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣟⣿⣿⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡟⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡷⠒⢧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣄⠀⣠⠧⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠀⠀⠀⠀⠀⠀⠀⡎⢳⠈⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢿⡸⠋⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢉⣄⠈⠑⠤⡀⠀⠀⢀⠞⡟⠀⡜⠁⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⡡⠔⠒⠉⠀⠑⠄⡠⠬⠷⠴⠯⠜⢀⠎⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⢹⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⣀⣀⠤⠒⠢⢄⣀⣀⠤⠐⠒⠤⣀⣀⡠⠔⠒⠤⣀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠⠤⠤⣀⠀⢀⣠⠤⠤⣀⠀⠀⣀⠤⠤⢄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠶⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣷⣤⣶⣦⣭⣽⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠋⢉⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣟⢻⣿⣿⣿⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣾⣡⣤⣶⣿⡿⠿⠿⠛⢛⡹⠯⠉⠛⠋⠛⢻⡏⠉⠀⠀⠈⠙⢧⣄⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣾⣿⣿⢟⣉⣁⣠⣤⣴⣶⢊⡀⢀⠀⣀⡤⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠉⠷⣄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠧⣀⠀⠀⠀⢀⡀⢀⡀⠀⣀⣀⠀⡀⠈⠳⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠰⢋⠿⠞⠉⠀⠀⠈⠿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠁⠀⠐⠙⣦⠀⠀⠀⠀⠈⠙⠉⠉⠙⠛⠛⠷⠀⠘⢦⣶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⡿⠿⠷⣶⣯⣙⣳⣄⣀⣀⣾⡄⠒⠂⠀⠐⠀⠀⢢⣤⣤⣵⠆⠀⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣤⣤⣤⣦⣤⣼⣶⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠉⠀⠉⠉⠉⢉⣽⣿⣿⠿⣯⣶⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣷⡤⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⣿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⡄⠀⢿⣿⡀⠀⣀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣎⠳⣌⣿⣿⣿⣿⠋⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣾⣿⣿⣿⣧⣀⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣿⣿⣿⣿⣏⡉⠀⠀⣀⣀⠋⠀⠉⠉⠙⠻⣾⣑⣀⣹⣿⠀⡀⠀⠀⢀⠜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣷⣶⣍⡸⢿⣄⠀⠀⠀⠀⢀⣨⣿⠿⣧⣤⣷⡄⢂⣸⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣟⡷⣶⣶⣿⡿⠿⡿⢧⣤⣽⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢋⣉⠛⠷⢶⣤⣍⡓⠶⠄⠀⠈⠙⠛⠚⠟⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⢩⣽⠄⠀⠀⠀⠈⠉⠙⠓⠂⠒⠲⠠⠤⠤⢨⣝⠒⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣭⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⣈⣙⣿⣿⣦⡄⠀⢀⣀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠐⠬⢭⣹⣿⠿⠿⠛⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣶⠦⠤⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠿⠛⠛⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⡀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⢄⢂⠠⠀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠡⡆⢺⠆⢈⣿⣷⡦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣷⣿⠀⢸⣿⣿⣿⠿⣿⣕⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠠⢆⣹⡏⡀⠀⠙⢿⠟⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠂⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠉⠚⡁⢤⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣀⡤⠚⢍⠉⠉⠁⢶⠢⠤⣀⡀⠀⠈⠢⠍⠾⢤
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⡿⠇⢹⠀⠀⠀⠱⡀⠀⢸⠀⠀⠀⠈⠑⠒⠤⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⡆⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠣⡀⠀⠀⢸⡄⠈⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣠⠾⠗⣸⠋⠁⢀⢩⡟
⠀⠀⠀⠀⠀⠀⢃⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢒⡪⠥⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⠮⡑⠒⠤⠤⠤⠄⠒⠛⢋⣢⣴⠎⢉⠽⡓⣂⣁⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠁⠒⠒⠒⠒⠲⢾⠿⠟⢁⢔⠁⠀⠀⠀⢀⡀⠀⢀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢠⣀⣀⠀⠀⣀⡀⠤⠒⠉⠀⠀⠂⠤⠒⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⣀⣀⣀⣀⣀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠙⠀⣠⠖⠋⠉⠀⠀⠀⠉⠛⢦⡀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣠⠖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣀⣀⡀
⠀⣠⠴⠶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠶⠶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠙⠲⠤⢤⣤⡤⠴⠞⢉⠀⠀⠈⠉⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠳⢶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⡀
⠀⡇⠀⠀⠀⠀⠀⠠⠤⠤⠀⢀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣭⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⣀⣠⡿⢏⠉⠈⠉⠉⢳⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⡇⢀⣰⣿⣿⠁⠀⠘⣶⢶⠆⠀⢸⠳⣄⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⢀⡠⠊⠀⡞⠍⠀⠈⢁⣀⠤⠀⡴⠋⠀⠀⢸⣀⡤⠷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⢀⡴⠋⠀⠀⣼⠉⠒⢲⠤⢠⣤⠤⠊⠀⠀⠀⠀⠹⡍⠀⣀⣤⠿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣧⣴⠾⣏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣠⠄⠹⡞⠉⠁⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠳⢶⣤⡀⠀⠀⢸⡇
⠀⡟⢳⡄⠈⠓⠶⠦⠤⠤⠤⣬⡙⠂⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠈⠉⠛⠋⠀⠀⠈⠉⠀⠉⠀⠀⠀⠈⠙⠛⠲⢾⡇
⠀⡇⠀⠙⢦⡀⠀⠀⠀⢀⣠⠴⠛⢆⠀⠀⠒⠊⠁⠀⠀⠀⠀⣀⣤⠴⠶⠛⠦⣄⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⣰⢦⣶⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠙⢦⣤⠶⠋⠁⣀⡤⠞⠳⡄⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⡆⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠸⡎⠀⠛⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠳⣄⡴⠚⠉⠀⠀⠀⠙⢶⣄⠀⡰⠋⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢉⠿⠛⠦⣄⡀⠀⠀⠉⠓⢤⣀⢸⠇⢀⠀⣼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠉⠻⣇⠀⠀⠀⠀⠀⠀⠘⢷⢶⣤⡀⣰⣿⣤⡀⠀⠀⠉⠛⠓⠶⠶⠤⢬⡟⠀⠋⣰⠏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⣀⠀⠌⠳⢤⡀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⢀⣤⣷⣌⠙⠛⠋⠈⠳⠶⠦⠤⠤⠤⠤⢤⣤⡀⠀⠀⢰⡏⠳⢤⡀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠻⣆⠀⣼⠋⠉⠲⡟⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠟⠁⠀⠀⠉⠲⢄⡀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠈⠻⠿⣄⠀⠀⠱⣄⣰⠎⠙⠉⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠈⣧⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠘⣇⠀⠀⣿⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠈⠳⢦⣼⡀⠀⠀⢹⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠈⠹⢦⣤⡼⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠳⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠐⠀⠀⠀⢸⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⡤⢠⠤⣤⠄⣤⣼⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠰⠊⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠆⠀⠀⠀⠙⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠟⣿⡟⢻⣿⡟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠈⢻⢧⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣦⣖⣀⠀⠀⠀⠀⠀⡀⠸⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⢿⣿⣿⡦⠀⠀⠀⢤⣰⣿⣤⣿⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠈⠉⠀⠀⠀⡤⢶⣾⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠔⣿⣶⡄⢠⣟⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⢠⣀⣀⣬⣋⣷⣾⣿⠛⠃⢿⠛⢨⣅⡈⠾⡿⣿
⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠒⠒⠓⠒⠒⠻⠿⠿⠿⠾⠶⠿⠿⢻⣿⣛⡿⣿⣿⡿⣯⣿⢿⣿⣃⣠⣾⣿⣽
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⢁⢘⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣺⣿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠟⠋⠕⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⡠⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⣀⣀⡀⢀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡄⠀⠤⠠⠤⠈⠀⠈⠀⠀⢀⠀⢀
⠆⠀⠀⠉⠈⠀⠠⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠀⢄
⠈⡀⠀⠀⠀⠈⠀⢂⠀⠀⠀⠀⠀⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠑⠀⠀⠀⠀⠑
⠀⠐⠀⠀⠀⠀⠀⢀⠤⡄⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠠⡄⠀⠀⠀⠢⡀⠀⠈⠢
⠀⠀⠡⠀⠀⠀⠀⠁⢀⠃⠀⠀⠀⠒⢀⠀⠀⠀⠀⢀⠠⠀⡀⠄⠀⠀⠀⠁⠱⡀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢄⠀⠀⠐⢄
⠀⠀⠀⠂⠀⠠⠀⢀⠆⠐⠀⠀⠀⠁⠄⡆⣀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢈⡄⠀⠀⠤⠀⠒⠀⠐
⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠓
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⡀⠀⣀⠀⠀⠄⠐⠀⠂
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⣿⣿⡿⠃⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣾⣿⣿⠟⠀⠀⣀⣴⡿⠟⠁⢀⣠⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⣿⢏⡀⣡⣾⡿⣋⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣿⣿⣮⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⠿⠿⣿⡿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢴⡃
⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⢠⠋⠀⠡⣐⣉⠀⠀⠀⠄⠀⠀⡎
⠀⠀⣀⠀⠋⠀⠀⠀⢫⡛⢀⠉⠀⠀⠀⠀⠃
⡠⠊⠀⠀⠐⡀⠀⠀⠠⠅⢈⡀⠀⠀⠀⠀⠀⠢⢄⠠⢄⡀
⠁⠀⠀⠀⠀⢣⠀⠀⠠⠐⠉⠓⠀⠀⡱⠀⠀⠀⠈⡄⢂⢡
⢀⠀⠀⠀⡀⠄⠷⠋⠀⠀⠈⢆⠀⡜⠀⠈⠒⠤⠴⠃⠈⢺
⠨⣀⠔⠻⣃⡀⢀⣀⠀⡀⠀⠌⢠⡇⠀⠐⢀⠀⢀⠤⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠉⢮⠢⣘⣼⠃⠀⠀⡀⠐⠁⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠁⠀⡄⠠⠀⠀⠱⢄⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠊⠀⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠊⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠣⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠉⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠐⢄⠐⠤⣀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠙⣦⣈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⢀⡠⠖⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣶⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣻⣻⣟⣙⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠀⠀⠀⠀⠀⢻⣟⠃
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣤⣶⣶⣶⣶⣶⣶⣿⣧
⢸⡟⣷⣦⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⣀⣀⣤⣤⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⠁
⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠁⠀⠀⠀⠀⠰⣿⠿⣿⣿⠟⠛⠛⠻⠛⠿⣿⣿⣶⣄⡀
⢸⣧⣼⡿⠛⠉⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠷
⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡴⠶⠀⠀⢀⣠⣴⠶⠛⠃
⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣥⣴⣶⣶⣶⣿⡿⠏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣧⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀
⠀⠛⠛⠛⠛⠛⠋⠉⠛⠀⠉⠛⠟⠋⠉⠀⠈⠃⠀⠀⠀⠛⠛⠿⣿⣿⣛⣛⣿⣿⡿⢿⣿⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠁⠀⠀⠉⠙⠛⠛⠶⠶
⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠛
⠀⠀⠀⣹⣧⣄⣤⣤⣤⣤⣶⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢶⣄⠀⠀⣠⣴⣶⡄
⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⡀
⠀⠉⠉⣿⣿⡿⠟⠻⠿⠿⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⠛⠶
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⡀⠀⠀⠀⠸⣧⣿⠟⠁⠀⠀⠉
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠃⠀⠀⠀⠈⠁
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⡿⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠳⠲⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⡀⢠⠀⣄⢀⡀⢠⣗⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⣠⠟⠀⢀⣠⠴⠚⠉⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣶⡈⠉⠀⠀⠀⠀⠈⢃⣽⣄⠀⠀⠀⠀⡾⠁⠀⡠⣸⠋⠀⢸⡏⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⡀⢸⡓⠒⠒⠲⢷⠤⠂⢅⡏⠀⠀⠈⢧⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⢀⠀⠀⠀⠀⠀⠁⠞⠁⠀⠀⠀⠀⢁⠀⠘⢧⠀⠀⠀⠘⡇⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠑⢤⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢹⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠈⡇⢠⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠈⠐⢀⠘⡓⠂⠀⠙⠢⣄⡀⠀⢻⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠂⠀⠂⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠉⠓⠺⠆⠀⠀⠀⠀⢧
⠀⠀⠀⢀⣠⠤⢤⠴⢤⣀⣀⣀⣀⣀⣿⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⢰⠋⠀⠀⢴⠀⠤⠽⠁⠀⠀⠀⠈⠀⠀⠀⣸⣀⣱⡄⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠨⣇⣀⣓⣤⣦⣤⣤⣄⣀⣀⠀⠀⢠⠔⠂⠉⠁⠀⠀⠀⠀⠀⠀⠈⢴⣄⣀⣀⣀⣤⣤⣤⣤⣤⠤⠤⠤⠤⠶⠶⠖⠋
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠛⠦⠼⠴⠚⠛⠋⠉⠉⠉⠛⠓⠶⢤⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠀⠀⢶⣀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠘⢿⡄⠀⢠⠔⣩⣵⡾⠛⠁⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⢠⣴⣶⡀⠀⢹⣿⣿⣇⠀⠀⢻⣿⡰⣡⣾⡿⠏⠀⠀⠀⢀⠀⠀⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢀⣼⣿⣷⣿⣿⠀⠸⣿⡿⠏⠀⠀⢿⣿⣽⢏⣩⣤⡶⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣨⣿⣝⠛⠉⠉⠀⠀⠀⠀⣠⣤⣴⣶⠀⠀⠀⠀⠀⠐⣠⡄⢚⡭
⠀⠀⠀⠀⢀⡀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⠟⣳⠀⠀⠀⢠⢖⣭⣄⡁
⠀⠀⠀⠀⡀⠀⠉⠒⢤⣈⠉⠉⠁⠀⠀⠀⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⣶⣿⣿⠆⠁
⠀⠉⠐⠲⠤⣁⠒⠭⠒⠬⣑⠦⣀⠀⠀⠀⠹⣧⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⠼⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠳⠤⠁⠊⢝⡢⣕⠢⡀⠀⠹⣧⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⢀⠀⠀⠀⠐⠀⢀⣴⣿⣿⡏⠀⠀⡤⣰⡇⠀⠀⢸⠿
⠀⠀⠀⠀⠰⡉⠂⠀⠀⣠⡤⣤⣈⠪⣳⢬⡑⣄⠁⠀⠀⠀⢻⣿⣿⠟⠁⢠⠀⠄⠲⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⢴⣷⠖
⠀⠀⠀⠀⠀⠁⠀⢠⡞⡠⡠⠀⠈⢳⠈⢢⡑⣌⠳⡀⠀⠀⢸⡟⠁⠀⠀⣀⠰⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠠⠆⠀⠀⠀⠈⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣤⣤⡄⠀⠡⠱⠇⠈⠂⣸⠀⠀⠙⡌⠁⠱⠀⠀⢨⠃⠀⠀⠘⠃⠀⠀⠀⣠⣿⡿⠉⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡦⠀⠀⠉⠁
⠀⠀⠀⠈⠻⠳⠏⠀⠀⠁⠠⠴⠛⠁⠀⠀⠀⠘⡄⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣠⠖⠁⢀⣀⣀⠀⠀⠀⢸⡦⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠀⠊⠀⣠⣶⣯⣥⣿⣆⣸⠃⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠐⠂⠠⠀⠌⠀⠀⠀⢀⣠⣴⡆⢀⠀⠀⠈⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠐⠀⠀⠆⠀⠀⠀⠀⠀⠀⢼⠟⠋⠁⠀⠈⠁⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠶⠶⠶⠶⢦⣤⣤⣀⠀⠈⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣿⣶⣶⣤⣄⣤⣤⣤⣤⣤⣴⣤⣿⣦⣤⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠘⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⡽⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⢀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠘⣿⣿⠂
⠀⠀⠀⠀⢸⣷⡄⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣿⣏⢽⣯⣿⢿⡏⢻⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠘⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣾⣿⣿⢸⣿⣿⣾⡇⣸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢘⣿⣟⡿⣸⣿⡿⠿⣷⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⣿⣿⣿⣿⣧⣹⣿⣟⣿⣃⣬⣵⣾⡇⠀⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⠀⠀⠀⣿⢋⣍⣻⢋⣝⢿⣿⣿⡟⢩⣯⢹⡇⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⠿⠛⠋⠀⠀⠀⠀⣿⠸⣿⡟⣸⣿⠈⣿⣿⡇⣘⣿⣿⡇⠀⠀⠀⠙⠛⠿⠟
⠀⠀⠀⠀⢻⣶⣦⣤⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡌⠃⣿⣿⠀⣿⣿⡇⡥⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣠⣶⣶⡗
⠀⠀⠀⠀⠀⠙⠿⣿⡷⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡇⠀⢻⣿⢰⢹⣿⠇⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣾⣌⣫⣿⣘⣋⣤⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⡴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣆⢠⣿⡇⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠿⠈⠿⠇⠸⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣦⣤⣤⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠟⠛⠉⠙⠛⠿⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠻⠶⠀⢀⣶⡀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠴⡦⠤⣤⣙⣿⣧⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⣶⡄⠀⠀⠀⠀⠀⠀⢁⢠⣀⣹⣿⣯⣥⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⢀⡔⣻⠷⠶⠀⠀⠀⠀⠀⠀⠚⠟⢁⡼⠙⠋⠁⢋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣰⣃⠀⢰⡎⠀⠀⠰⠀⠀⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠈⠀⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠘⣦⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠴⠆⠀⠀⠀⣄⣴⣀⢀⣤⣠⣤⣻⣿⣏⠜⣋⡿⣷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⣄⣄⣤⣌⣄⣂⣩⣍⣀⣨⠓⠂⢠⣶⣼⡿⠿⠛⢉⡉⢩⣥⣼⣾⡿⣿⣿⣵
⠿⠛⠛⠛⢒⡀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠛⠉⠋⠹⠟⠓⠛⠤⠐⠪⠉⡿⠦⠶⠉⠥⠵⠶⣛⠉⢋⢹⣿⣿⣿⣿
⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂

⠀⠀⣀⠀⠴⠒⠂⠀⠉⠉
⠁

⠀⠀⠀⠀⢀⣤⠤⣶⣶⣤⣙⣿⣿⡾⠟⠉⠉⠁
⠀⠀⠀⠀⠋⠀⢠⡶⠛⠋⣠⣿⣇⠚⣿⡿⡷⠶⡄
⠀⠀⠠⡀⠀⠀⠀⠀⣠⠎⠘⡟⠁⣦⡈⠙⠳⣄
⠀⠀⠀⣱⣶⠂⢀⣾⠁⠀⢀⠀⠀⠸⣧⠀⠀⠈
⠙⢳⣦⣿⣧⣐⣿⡀⠀⣄⣏⡀⠀⠀⢿
⠞⠛⢻⡿⠿⢭⠀⠈⢻⣿⡯⠤⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⢀⣰⠇⡁⠀⠈⠃⢰⣷⣎⣧⣀⠈⣠⡄⠈⠛⠂⠀⠀⠀⣰⡧
⣿⣿⢿⡷⢿⣶⣤⢾⣿⣿⣿⣿⣿⣿⣿⡤⣀⣾⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠛⠛⠟⠿⠉⠉⠉⢉⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠚⠛⠛⠛
⠿⠿⠖⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⠉⠉⠉⠉⠁⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⡠⢀⡴⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠀⠠
⣱⣿⣷⢿⣃⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠂⠐⠠⠴⠶⠶⠶⠤⢶⠤⣤⣄⣀⣀
⣿⣿⣵⡌⣿⣿⣿⣿⣦⡴⠂⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠦⢤
⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡦⠄⠄⠠⠶⠆⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠛⠶⣤⡀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠑⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣦⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠂⠀⠀⠀⣹⣿⣤⠀⣴⠃⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣦⢠⣤⣤⡀⠀⡴⠂⢀⣠⣤⣴⣶⣶⣦⡀⠙⠂⠈⠻⣿⠀⢹⡄⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡾⣿⣿⣿⡷⠓⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣧⣿⣄⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠿⣿⡿⠃⠀⠀⣈⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⢹⡵⣶⣶⡜⢷⠀⢿⣿⢿⣿⣿⡿⠟⠁⠀⣠⡈⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣷⢿⣿⣿⣦⠀⠀⠀⣈⠀⠀⠀⣀⣀⣚⣛⣁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢺⣿⣿⣿⡇⠀⠰⢡⣾⡿⣿⡟⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⠉⠉⠙⠁⠀⠀⢈⢻⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⡆⣰⢰⣠⣾⢿⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢻⡟⡟⢿⠹⣝⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣷⡿⣾⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁


⡀⠀⠀⢀⣴⣶⣶⢶⡐⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣄⣀
⠀⡳⠘⠸⣿⣯⡤⠾⢿⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠛⢿⣷⡄⠁⠘⢰⠂
⠀⠀⠀⠀⠿⣶⣦⡄⠺⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⣠⠉⠙⠛⠿⠃⠐⠀⡇
⠀⠀⠙⠂⣄⠀⠹⣆⢀⡜⠓⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⠉⠁⠠⠔⠁⠀⠀⠴⠁
⠀⢀⢀⠈⢿⣷⣄⣍⣙⡟⢘⣟⢚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⢰⠞⢷⠤⡔⠏⠠⠀⡀⠁⠀⡀
⠀⠀⠀⠑⠈⠈⢟⠋⣹⡋⣹⣏⠙⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⣽⣙⣗⢺⠛⠟⠀⢀⡜⠘⠴⠪⠂
⠀⠀⠀⠀⠀⠈⢡⢁⠘⠋⣽⡅⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⢉⡟⠙⠷⠟⠀⢊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠲⣍⠀⠀⠄⠐⢤⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⡀⠀⠀⠘⠛⠟⡡⢂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡄⠉⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠈⠘⠃⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⠀⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄⠀⠀⠀⡇⠀⠞⠁
⠀⠀⠀⠀⢀⡠⠴⠒⠚⡯⡀⠐⠒⠒⠒⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠞⠉⠉⠑⡄⢠⡤⠤
⠀⠀⢠⠞⠉⠀⠀⣀⠈⠀⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⠘⢆⡀⠀⣠⠃
⠀⢰⠃⠀⠀⢀⠜⠁⠀⠀⠀⠀⠁⠀⠀⢀⡠⠤⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠉⠁⠉⠲⠄
⢀⠇⠀⠀⠀⠋⠀⠀⠀⠀⣀⡤⠤⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⠓⠒⠢⡤⠔⠒⠒⠒⠒⢹⣧⠀⠀⢀⡠⠖⠚⠉⠉⠉⣑⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣿⠀⢰⠋⠀⠀⢀⣠⣶⣿⣿⣿⣧⣷⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢀⠘⠃⠀⠀⠞⡳⣤⠞⠛⠛⢿⣿⣿⣽⠅⠙⠿⣿⣹⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⢫⠆
⠀⠀⠀⠀⠀⠀⠀⡖⠛⠛⢉⡟⢛⣝⡦⢀⡈⠙⣛⣉⣑⣶⠶⠟⠹⠤⠶⠖⠒⠒⠲⢤⡤⠤⠤⣤⣀⣀⣤⣿⢻⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⣸⡕⠿⠭⢤⣒⠯⢍⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⣤⠏⠉⠉⠉⢹⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⣇⠈⠉⠉⠁⠀⠀⠈⠑⠒⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⣀⡼⠿⠟⠛⠛⠛⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠚⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡨⠏⠒⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣡⡎⠀⠀⠀⠀⠀⠀⠈⡦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠒⠈⠉⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠀⠀⠀⠀⢀⡀⠀⠦⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠤⠀⢀⠀⠀⠀⠀⢀⡀⠄⠚⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢄⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⠞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠐⠚⠉⠀⢠⠔⠊⠁⠀⠀⠁⢀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠊⠁⡦⠐⠁
⠀⠀⠀⢀⡀⠤⠐⠂⠹⢧⣀⠀⣀⠴⠂⠒⠈⣀⡄⠠⠖⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠒⠈⠁⠉⠁⣀⠠⠔⢪⢁
⢀⠂⠁⠀⠐⠀⠀⠀⠀⠀⠉⢉⡀⠠⠴⠚⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂⠈⠒⠂⠒⠈⠉⠀⢀⠀⠀⠀⠠⠐⠊⠁⠈⠑⠠⠈⢁⠆⠀⠀⠀⠀⣰⠈⠀⣰
⢨⢢⢄⠀⠀⠀⠀⠀⠀⡰⠀⠈⠀⠀⢀⣀⢀⡤⠄⠒⠈⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⡀⠄⠀⠂⠈⠉⠀⠀⠀⠀⠀⠂⠀⠈⠁⠢
⡆⡃⠀⠑⠠⡀⠀⣠⠊⠀⠀⡌⠉⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠄⠀⣀⡁⠄⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠨⠢⡀⠀⠈⠲⢌⠳⠶⠞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢀⡀⠤⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⡆
⠀⠀⠈⠢⠐⢄⠀⠀⠙⠢⡀⠀⠀⠀⠀⢀⣀⠤⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠐⢕⢦⡀⠀⠀⠁⡖⡘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠁⠨⡢⢄⡠⡔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡙⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⣤⣄⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣴⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣟⣛⣉⣉⣉⣉⣉⣉⣛⣛⣛⣛⠛⠿⠿⠿⠟⠁⣠⣾⣿⠟⠁
⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣶⣾⣿⠟⠁
⠙⠛⠛⠛⣿⣿⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⡇
⣀⣀⣀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡔⢠⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣿⢿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⣄⣙⣛⣛⣩⡭⠍⢻⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡙⠷⣤⡴⠶⢖⣲⣦⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠠⠼⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣛⣼⣿⣿⣿⣿
⠀⠀⠘⠛⢷⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠟⠛⠀⠀⠀⣿⣿⣿⣤⣌⢉⡉⠉⠉⠉⠉⠉⠉⠉⠉⢡⣤⣿⣿
⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠘⠷⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣌⣁⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡡⣻⣿⣶⣿⣯⣵⣖⡂⢤⣄⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠈⠁⠚⠴⡱⣙⡟⠛⢿⣿⣿⣷⣀⠝⠉⢒⠿⢭⣷⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⠈⡻⠿⢦⣤⣙⣿⣿⣿⠀⢀⠎⠀⡼⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀⠀⠈⢖⢠⡀⠀⠀⠙⠛⢻⠟⠋⠁⠉⠀⠘⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡰⢡⠀⢀⢿⠓⠋⠂⡴⢂⣡⠷⠒⠁⠀⠀⢀⣴⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢎⠜⡰⡡⢠⡖⢺⠀⠀⢀⣊⡹⠃⠀⠀⠀⢠⠔⡝⢹⠘
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠑⡀⠒⠄⡀⠀⡠⢣⣪⢊⢴⣿⠞⢷⣮⢔⣠⠾⡟⠁⣠⠄⠚⢡⢎⠌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠡⢁⠀⠀⠈⠲⢵⠿⢷⣧⢤⣷⣄⣠⣧⣀⣷⡄⠀⡠⠋⠀⣰⢃⠮⢄⣀⡤⢓⠄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢃⠀⠀⠀⠀⠀⠢⣼⣧⣀⣹⡏⠙⣟⠁⠻⠇⠀⠁⢀⡼⢣⠊⡩⢖⡡⡎⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢻⠉⢻⡟⠻⣿⣦⣴⡟⠁⡠⡻⣡⠗⡫⠔⠁⠀⠣⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠈⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣦⣾⡆⢀⣼⡡⠋⣠⢎⡔⣱⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢩⡗⠄⠀⠀⠀⠀⠀⢸⠛⢙⡿⠋⣀⡴⢛⠝⡔⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⣌⠆⠀⠈⢎⠑⠫⣒⣦⣤⠀⠈⠒⢁⣠⢞⡩⢒⢏⡜
⠀⠀⠀⠀⠀⠀⡄⠀⠰⣯⠣⡀⠄⢊⠆⠀⠀⠑⠪⢝⣦⠖⡫⠔⠉⢠⢊⠎
⠀⠀⠀⠀⠀⡠⠁⠀⣾⡫⠤⠒⠪⣑⠢⣀⠀⠀⠀⠸⠕⠊⠀⠀⢠⢣⠊
⡤⠐⠆⠘⠉⠀⠀⢴⠀⠀⠀⠀⠀⠀⠉⠢⢕⠢⢄⠀⠀⠀⠀⡰⢡⠃
⠉⠀⠒⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢍⡢⢄⡴⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠶⠿⠽⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⠛⠉⠁⣿⣦⣄⡀⠀⠀⠀⠠⣀⠀⠀⢀⣴⣶⡆
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠬⠛⠈⠀⠀⠀⠀⠀⠁⠲⠞⠯⠟
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠶⠶⠤⢤⣤⣀⣠⣤⣠⢶⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠧⣌⣥⠀⠀⣏⣩⠿
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠙⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠉⠲⣶⣞⣀⠀⠀⣤⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠙⠛⠛⠁⠙⢿⠏⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠾⣦⡀⠀⠈⠉⠓⣄⠀⠀⠀⠀⣠⣴⢿⣦
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠙⠻⠷⣦⣴⣦⣄⣠⣔⣚⣉⣁⣀⣀⣀⣈⠻⢤⣄⣀⣤⣼⡀⠀⠀⢠⣝⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢦⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⡀⠀⠐⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⠤⣀⣀⠀⣀⣀⣀⣠⡤⠤⠤⣤⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⣀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠠⠔⠲⢤⡀
⠀⠀⢇⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠉⠐⠒⠒⠢⢄⡈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⢉⣁⣁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢣⠀⠀⠀⠀⠀⠀⡠⠊⢀⡤⠒⠋⠉⠁⠈⠉⠙⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡀⠀⠀⠀⢀⠜⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢁⣀⡀⢀⠎⢠⠊⠀⣠⡴⠖⠛⠋⠉⠉⠉⠉⠙⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣩⡜⠀⠀⠀⠀⠁⠀⠗⠴⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠠⣄⠀⠀⠀⠀⠀⠀⠛⠶⣄⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⢠⠋⠌⢀⣀⡈⠁⠀⠀⢀⠔⠋⠑⢆⠈⠂⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠒⠢⢄⣣
⠀⠀⠀⠀⠀⣠⠇⠀⠀⣾⣯⡴⠀⠀⠀⠁⢠⣴⠄⠀⡀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⡜⠁⠀⠐⠀⠈⢉⠃⠀⠀⠄⠀⣿⣿⡶⠀⡆⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⠈⡆
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠘⠀⣨⠝⠒⠒⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢰⠧⢄⣸
⠀⠀⠀⢸⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄
⠀⠀⠀⠈⢧⠀⠀⢏⢓⠲⠦⠤⢤⣤⡤⡤⢤⠶⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⣰⠃
⠀⠀⠀⠀⠈⠳⡀⠘⢆⠀⠖⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠒⠊⡇
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠙⠒⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⣤⣀⣀⣀⡤⠴⠊⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣠⠞⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⡿⠀⠀⠀⡸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⡾⠣⠤⠤⠞⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠙⢦⣄⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⣤⣴⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣼⣾⣿⣿⣿⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣶⣾⣧⢄⡀⠀⣸⣭⣽⣿⣿⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣥⡀⠘⢻⢿⡿⣿⢿⡏⠿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⣿⣿⣿⢿⡟⢧⠀⠀⠈⠘⠁⢹⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢻⠋⣇⠁⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣠⠄⠔⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢿⣄⠇⠀⡰⢈⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⣦⣀⠔⢐⠠⢤⡾⢏⡦⠟⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄⠀⠀⠀⠐⠊⣩⠈⣓⢿⡞⠁⠀⠤⣁⠖⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠚⠛⢻⣏⣽⠟⠋⠉⡙⡦⠾⣿⠛⢓⠃⠀⡀⠞⢷⡞⣻⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠉⠀⠀⢂⡀⠠⠽⢗⣋⣻⡿⠂⢐⣀⣠⣊⣴⣿⣧⣬⠤⠾⢳⢿⡟⠁⣸⡿⠋⠀⢀⣿⠻⠾⢿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠐⠂⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⢽⡁⣀⣠⣾⣸⣧⣼⡿⠒⠀⠘⣿⠟⢁⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠋⠉⠉⠁⠉⠑⠛⠾⠓⠾⠿⠳⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣟⠛⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣯⣙⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⠌⠃⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡈⢋⠁⠐⠀⠀⢀⠄⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⡀⠀⠁⡠⠈⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠁⠀⡠⠊⢀⡴⠋⠀⠀⠀⡂⠀⣇⠀⠀⠀⠀⠀⠀⠀⠲⡓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠠⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠐⠲⣎⢀⡴⢫⡖⠀⠀⠀⠀⠉⠀⠱⡀⠀⠀⠀⠀⣖⣂⣦⣈⣵
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⡿⠁⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠂⠁⠳⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⣐⠀⠊⠀⠀⠀⠀⠀⡠⠀⠰⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠠⢤⡊⢀⣀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠉⠉⠢⡀⠀⠔⠉⠀⠁⠀⠀⠀⢤⠎⠀⠀⠀⠉⠤⡆⠐
⠀⠀⠔⠁⠀⠀⠀⢀⣤⠤⠊⠀⠀⠀⠉⠁⠀⠀⠀⠙⢄⠀⠀⠠⠀⠉⠂⠂⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠐⠆⢤⣶⠖⠠⠘⠛⠁⠁
⠠⠂⠀⠀⠀⡀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⢀⠀⠀⠀⠀⠋⠩⠄⠀⠀⠠⠜
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠦⣀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠄⠉⠀⠀⡀
⠈⠒⠀⠠⠄⠀⢀⣀⣀⣀⣠⡄⠀⠐⠂⠁⠉⠉⠐⠒⠂⠀⠮⠀⢀⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡠⠊⠁⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⣀⠀⣰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣉⣹⢟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠈⠇⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⠉⡿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀⠀⣸⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠙⠒⢀⣤⠝⠛⠤⠈⠉⠛⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣀⡈⠉⠹⠏⠛⠛⡗⠢⣄⠀⢀⠐⠚⠤⠼⠯⠐⠉⣠⠼⠁⠀⠀⢀⡆⠳⠆⣸⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⣀⠤⠴⠊⢠⠀⡉⠀⠀⣠⢀⠈⠉⠳⠲⣼⡆⠢⠀⠀⣤⠴⠒⠀⠀⠀⠀⢀⡠⠂⢉⣅⡀⠠⢁⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠂⠊⡉⠩⢍⣦⣄⡲⢤⠌⣠⠋⡈⠻⠖⠋⠀⠀⠀⣀⠀⠃⣻⠦⢤⡟⠃⠀⠐⠶⠤⠶⠖⠉⠀⠀⡀⠀⠀⠀⠎
⠀⢀⠄⠀⠀⠀⠀⠒⠄⡒⠤⡉⠐⠶⠖⠢⢲⠈⠁⡌⡜⠇⠀⠀⠀⠀⠈⠃⠀⠓⢽⡦⠾⡐⢿⣇⣤⣽⣓⠢⢀⠀⠀⠀⠀⢀⠋⠉⠃⠀⠰⠓⠳
⠀⠁⠀⠀⠀⠀⠠⠐⠂⠀⠀⡀⡄⠀⠀⠰⠌⠁⢴⡼⡙⠀⠀⠀⠀⠀⠀⠀⡀⡈⠁⠐⠒⠛⠛⠉⠀⠀⠀⠀⠀⠙⠢⣀⠔⠛⢀⡠⠶⢚⣛⢯⣁⣦⣤
⣾⡄⠀⠁⠀⣠⠖⠐⠈⠀⠰⠀⠀⠀⠀⣰⠤⠄⠀⠀⠀⠁⠀⠂⠂⠀⠈⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠎⠀⠠⠾⣅⣤⡄⠠⠜⠛⠛⠛⣿⡄
⠀⠀⠀⡇⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠄⠆⠈⠒⢲⠦⠤⣄⣀⠀⠀⠀⣀⡀⠤⣐⡒⠲⠶⢶⡶⢶⣶⣦⣴⡏⠀⠀⠐⠉⠉⠀⠙⠀⠀⠀⠁⠈⢻⣳
⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠐⠂⡐⠢⢠⠄⠤⢤⣤⠀⠐⢄⡉⠙⠒⠀⠉⠭⠴⠒⠚⠉⢉⣁⡿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣍⣉⠋⢬⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⠈⠓⢦⡈⠁⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⣀⣀⠤⠤⠒⠒⠊⠉⠉⠹⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡇⠀⠀⠀⠈⠀⡀⠀⢀⠔⠒⠛⠉⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⣀⢀⠆⠀⡾⢶⣄⠀⠀⠀⠀⢀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢼⣃⣦⠀⠀⠀⢸⠃⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡟⠚⠛⠁⠀⠀⠀⠐⠲⠦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣤⣤⣀⡈⠀⠈⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠿⠷⠶⠖⠛⠁⠀⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣀⡟⠡⠘⠷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⣭⣻⠙
⠀⠶⠁⠀⡄⢸⡇⠀⠀⣤⠀⠀⠀⠀⢀⡼⢁⣉⠉⣝⣧⠛⢓⣒⣛⡀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠉⠛⠳⣾⡏
⠀⠀⠀⠀⠇⠸⠃⠀⠀⣛⠀⡀⠀⣀⢾⡇⠀⣿⠀⠏⡇⠀⠀⠉⠻⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⣰⣅⢰⡆⢀⡄⠀⠀⣀⠄
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢩⠀⠁⠐⣿⣾⡁⢸⣿⢂⣰⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣰⣲⠢⣴⠟⠋⢿⡝⡟⠅⠀⢀⠉
⠀⠀⠀⢈⡇⠀⠀⠀⠀⡀⠀⠀⢀⣿⣿⣇⡀⣿⣼⣧⠘⠀⡠⠀⠀⠀⠀⠀⣦⠀⠈⠀⠈⠀⠀⢸⣿⣿⣧⣤⣌⢧⣷⠀⢀⡬⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠀⠀⠐⠿⠿⢿⡇⡘⠿⠿⠄⠀⠀⠀⢀⠀⠀⠀⣿⡿⠀⡴⠃⠀⠀⠹⣿⣿⣿⣿⣏⠀⠉
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⠀⠠⠘⣷⣦⠀⠀⠲⣶⣤⠄⠀⠀⠋⠋⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠈⢽⡿⠎⡁⠀⢿
⠀⠀⠀⠀⠀⠀⡿⠂⠀⡀⠀⣠⣾⣂⣿⡧⠘⣄⠀⢻⣿⡇⠀⠀⢀⣠⣴⢶⣀⢿⣦⡀⠀⠀⠀⠀⠀⠀⠩⠽⣧⠉⠀⠀⡇⠀⢀⡀⢰⠅⠀⢦⣴
⠀⠀⠀⠀⠀⢰⣟⠂⢠⡄⡋⠻⢿⣯⣤⢄⣐⣛⣢⣼⣿⡷⠦⢄⠀⠛⠛⢻⡿⠿⠻⠧⠦⠤⠤⠀⠀⠀⠀⢰⣿⠓⠶⢶⣉⣿⣛⡇⠀⠐⠀⠀⠠
⠀⠠⢴⡆⠀⡀⠀⢙⣿⡆⡇⠀⠀⠀⠙⢿⣉⠴⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠙⠲⠰⠶⠶⠶⠶⠶⠶⠆⠆⠀⠀⠈⠁⠀⠀⠀⢈⡇⠀⠀⣄
⠀⠀⠀⠀⠈⢿⡿⣿⣏⠁⢸⡿⢻⡿⣏⣸⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣿⢿⣿⠋⠹⣻⣿⡿⠋⠀⠀⠀⠘⠿⠇⡠
⠀⠀⠀⠀⠀⠀⠀⠈⠚⠃⠸⠀⠀⠀⠉⠈⠉⠁⠈⠉⠀⠀⠉⠋⠁⠁⠀⠁⠀⠉⠉⠀⠉⠀⠈⠁⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠡







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢀⣿⣧⣌⡛⠿⣿⣿⣷⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⢠⣾⣿⣿⣿⣿⣶⣄⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠖
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠃⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⠟⠋⠁⠀⠀⠀⣦⣄⣰⣿⣄⣤⣦
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠻⠟⠁⠀⠀⠀⠠⣤⣶⡟⠋⠉⠉⠉⠙⢿⣶⣤
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡏⠀⠀⠀⠀⠀⠀⠀⢻⣇⡀
⠀⠀⠰⠿⠟⠛⠛⠛⠿⠿⣿⡿⠀⣾⠿⠛⠉⠁⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⣿⣄⠀⠀⠀⠀⢀⣴⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢷⣶⡾⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣄⠈⠻⣿⣿⣿⡿⢟⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠈⠁⠀⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⢻⣿⣿⣿⣿⣿⣷⡄⠀⠀⣠⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣄⠀⠹⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠈⠻⣿⣿⣷⣄⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⣿⣿⣿⡟⠋⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠛⣿⣿⣿⣏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆⣳⣏⡄⠀⠀⠀⠀⠀⢰⣾⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⣿⠉⠃⠀⢠⣠⣞⢸⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠸⣿⡿⠊⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣂⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⢡⡞⠁⣭⣤⠀⢠⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⣿⣟⡁⠀⠀⠀⠘⠛⠗⠀⠀⠀⠈⠁⠐⣿⣯⣠⡄⣿⡟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠀⢀⠠⠄⠀⠀⣠⠶⠂⠀⠀⠀⠀⠹⣿⡇⢻⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⣀⣀⡴⡀⣿⣅⢤⡄⠀⠀⠻⠃⢘⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡋⠁⠘⣉⣻⡻⢿⠘⡟⠋⣁⣤⢤⣶⣷⢸⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠠⠄⠀⠈⠹⠗⠷⠍⠀⠔⠂⠀⠼⣿⣦⢶⣿⣿⣧⡾⠦⢱⣤⢴⣿⡿⣿⡏⢸⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⠀⠐⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⠀⠀⠤⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⡀
⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢄
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠢⡀
⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠐⠤⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠀⠒⠒⠤⡀
⠀⠁⠀⠀⢠⠒⠀⠀⠀⠀⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠒⠛⠒⠀⠀⠀⠐⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠉⠀⠁⠀⡀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢚⠀⠂⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠔⠁⠂⡄⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠨⠀⠀⠄⢠⡄⠀⠀⠀⠤⠈⠩⠐⠂⠅⣀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢠⠋⠒⠤⡀⠀⠀⡠⠒⠁⠀⢀⠊⠀⢰⠀⡇⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠀⢀⣸⣀⡁⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣀⠠⠼⠤⠤⠀⠂⠐⠶⡤⠠⠄⠀⣀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⣀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢄⡀⠀⠀⠁⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣽⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢀⣶⣷⡄⠀⢀⣠⠀⢀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣾⣿⡓⠀⠻⡏⠃⣄⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠃⢁⣴⣦⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡿⣹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠙⠛⠛⠛⠉⠩⠭⠿⠿⠟⠛
⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠻⠿⠗⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⢿⣷⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠂⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢧⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⠤⠚⠒⠒⠒⠒⠀⠀⠐⠒⣒⣒⣒⠒⠒⠒⠒⠒⠶⠶⠤⠤⠤⠤⣤⣄⣀⡀
⠀⣀⣀⡠⠤⠔⠒⠋⠉⠑⠒⠒⠂⠀⠀⠀⠠⠆⠀⠀⣠⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀
⠈⠀⠀⠚⠒⠓⣀⣀⣀⣠⣤⠤⠤⠶⠖⠒⠒⠒⢒⠞⠁⠀⣼⠷⡖⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⢠⡶⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢠⡞⠁⠀⠈⠢⡤⠤⠤⠤⠄⠄⠐⠒⠒⡰⢮⡉⢧⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠤⠤⢤⠤⠤⠽⠶⠾⣏⠀⠀⠀⠀⣴⢿⠒⠖⠤⢤⡀⠀⠀⠀⡇⢸⠙⢦⡙⠢
⠀⠀⢀⡟⠉⠀⠀⣴⣶⢚⠉⠸⢹⠃⢧⠀⠀⠀⠀⠀⠈⢦⡀⣠⠞⠁⣸⣤⣤⠤⢤⣯⣩⣭⣭⣥⡼⠀⡸⠀⢀
⠀⠀⠘⣇⠀⠀⠐⣿⣿⡆⠁⢀⠀⣀⣾⠀⠀⠀⠀⢠⠤⠴⢿⡃⠀⣴⢋⣀⣀⣀⣀⡷⠶⠒⠒⠒⠶⠾⠃⠀⠸
⠀⠀⠀⠘⢆⠀⠀⠙⠛⠿⠟⠛⠋⠉⠙⠳⠦⣤⣤⣤⣤⣤⣾⣇⡾⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡧
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣟⣁⣀⣀⣠⣤⣤⠤⠴⠶⠶⠖⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⣿⢀⣀⣀⣤⣤⣤⠴⠶⠶⠒⠛⠛⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣶⣤⠀⢀⣴⣾⠿⡿⣿⣦⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⣿⠏⠀⠀⠀⠦⠹⣿⡆⠀⣾⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⡀⣿⣧⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄⠀⢻⣿⠀⠀⠀⠀⠠⢁⣿⡏⠀⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⢿⣷⣤⣉⣤⣤⣾⡟⠁⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠛⣛⠛⠛⠁⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⣼⡿⠀⠀⣿⡇⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠰⣿⠇⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⡿⢿⣿⣿⠛⠿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⢿⣿⠋⠀⠀⠙⣿⣿⣦⣄⠈⠙⠛⠿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢻⣿⡀⠀⠀⣰⣿⠏⠙⣿⣷⠄⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⠻⠿⠿⠿⠿⣋⣴⣾⠿⠋⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠻⣿⣦⠀⠀⠀⣀⣠⣴⡿⠟⠻⢿⣶⣶⣶⣾⠿⠛⠁⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠿⠿⢿⣿⣶⣶⣿⣿⣶⣶⣾⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢾⣦⠀⠀⣠⣤⣤⣿⡏⠀⠀⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠸⣿⣷⣤⣬⣭⣭⣭⣥⣴⣶⣶⣶⣿⣷⣤⣤⣤⣤⣤⣶⣾⡿⠛⠁⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠈⠉⠉⠉⠉⠉⠉⠉⣉⣁⣀⣈⣉⣭⣭⣭⣽⣿⣿⣿⣶⣶⣾⣿⡿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣶⣾⡿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠘⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠛⠛⠉⠉⠉⠛⠛⠛⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠒⠢⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⡠⠶⠦⣄⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⡇⠀⠀⠀⠀⠙⠒⠠⠀⠀⣠⠞⠛⠀⢠⡀⠀⡇⢀⣠⠄⠀⠀⠘⢧
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢀⣤⣤⣤⣻⣾⣷⣿⣥⣤⣄⡀⠀⠈⣧
⢀⡟⠀⠀⠀⢠⡀⡴⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⡴⠶⢿⡿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠘
⣸⠃⢚⣛⣳⣾⣿⣿⣿⣶⠶⠄⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉⠉⠉⠉⠙⠒⠤⣴⠟⣿⣿⣿⣿⡿⢿⡌⠀⠀⡀
⣿⠀⠠⠯⣵⣿⣿⣿⣿⣯⡙⠂⠀⣀⣀⣴⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠽⣯⣸⣿⠈⣷⢀⣱⣤⣄
⡇⠀⠀⢸⢻⣿⠿⣿⠻⣆⠙⣢⠚⠁⠀⠀⠀⠙⢦⡀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠄⣿⣟⠀⠈⠻⢿⡿⠛⠂
⣇⠀⠀⢀⣾⣿⠀⢻⣴⢻⠉⠁⠀⠀⠀⠀⠀⣤⣴⣿⣯⡉⠉⠉⠓⢤⡀⠀⠀⠀⠀⣻⡏⠣⡀⠀⠀⣇⠀⠀⣀⣴
⣿⠀⠀⢀⣿⠁⠀⠰⣇⣀⣀⣀⣀⣀⣀⣀⣀⣩⢿⣿⢿⠃⠀⠀⠀⠀⢳⡀⠀⠀⠀⣸⣧⣀⠙⠀⠀⢸⣴⣾⣿⣿
⢸⣿⣿⣿⣿⣄⠠⡶⠿⠿⠟⣻⣿⣿⣿⠟⢿⣇⠈⣿⠀⠀⠀⠀⠀⠀⠀⢁⣤⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⢿⣿⣿⠉⣷⣾⡏⠠⡆⣿⣤⠈⣿⣦⣿⣀⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠘⢧⠉⠉⠁⠀⠀⠀⠉⠉⠀⠉⠉⠁⠀⠀⠀⠈⠉⠁⠈⠉⠁⠀⢀⣀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⢳⣶⣶⣶⣶⠿⠿⠟⠛⣛⣓⣦⣄⡀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠋
⠀⠀⠀⠀⠀⠙⢿⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣷⣷⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢠⣿⣦⣍⡛⠿⣿⣿⣷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠏⣰⣿⣿⣿⣿⣿⣦⣌⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⡿⠟⠋⠁⠀⣤⣀⣴⣄⣠⡄
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣷⡄⠙⠋⠀⠀⠀⢠⣴⠟⠋⠉⠉⠙⢷⣤
⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⢠⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠈⣷⣤
⠀⠀⠉⠁⠀⠀⠉⠉⠙⠁⠛⠁⠀⠀⠀⢠⣶⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⡀⠀⠀⠀⠀⣰⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠷⣶⡶⢾⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⡈⠻⠿⡿⢟⣥⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⢸⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣦⡀⠹⣿⣿⣿⣿⣷⣠⣾⣿⣿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠻⣿⣿⣄⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣁⣀⣀⣈⣻⣿⣷⣄⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣻⣿⣿⣧⡀⠀⠀⠠⣰⣴⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠟⡷⣄⠈⠀⠀⠶⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⢡⣏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⡄⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢀⢫⣤⣷⣿⣿⡥⣂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢣⣤⡾⣶⢶⢟⣾⡿⢾⣿⣿⣿⣟⣯⣿⣿⣽⡟
⠷⠖⠒⠂⠀⠀⠀⠀⠀⠀⠉⠓⠛⠻⠛⠚⠓⠒⠁⠒⠚⠚⠐⠋⠛⠉⠉⣉⣉⡉⠑⢋⣩⠭⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣤⡴⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉
⣀⣀⣤⠤⠤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣸⠉⠉⢣⣀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣒⣒⣒⣒⣒⣒⠒⠂⠜⠢⡴⠫⠐⠒⣒⣒⣒⣒⣒⣒⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⠸⡀⠀⢪⠀⠀⢧⠀⠐⣇⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⣀⣀⣈⣀⣀⣉⠀⠀⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠋⠉⠉⠉⠉⣩⠤⠤⠤⠤⠀⠈⡇⡎⠁⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠚⠁⠙⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣤⢄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡤⢤⠤⠃⠀⠀⠀⠀⠀⠀⠘⠦⡤⢤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣼⣦⡄⠀⠀⠀⠀⠀⠀⠀⣤⣧⣼⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⢀⠀⠀⠀⠄⡀⡀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠀⠈⠉⠉⠀⠐⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠄⠰⠄⠈⡄⠁⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠐⠁⢈⣠⠤⠤⠄⣤⣤⠒⠜⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢠⠠⠤⠐⠒⠛⠁⠀⠀⠀⠀⠈⠠⠈⠁⠁⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⢀⠄⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡⠀⠄⠊
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠗⠀⠀⠀⢀⠴⠋⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⠃⠐⠀⠀⢀
⠀⠀⢀⡄⠀⠀⠀⠅⠀⢰⠶⠓⠊⠉⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⢀⠠⠐⠁⠀⠈⠀⠀⠡⠀⠀⠀⠀⠈⠀⠀⠀⢤
⠰⣔⠊⠀⣀⡀⠀⢠⠀⠀⠏⠢⡀⠈⠀⢠⣤⠀⠀⡀⢣⡀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢠⠎⠰⡌⠉⠀⠀⠘⠐⠂⠱⣀⠈⠐⠀⠀⠄⠄⣁⠀⠔⠩⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠊
⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠘⠅⠐⡀⣿⠄⡴⠛⡂⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⡐⢎⡃⠀⠀⠀⠀⠀⠀⡂⠠⠀⡀⠀⠀⠀⢀⠀⠀⠀⠉
⠀⠀⠀⠀⣠⠴⠒⠖⠚⠓⠒⠤⣀⠀⠀⠀⣆⡑⢨⡹⠄⠀⠀⠂⠀⠀⡅
⠀⢀⠔⠋⠀⠀⠀⢀⣀⠀⠀⠀⠈⠛⣆⡉⡍⠙⡷⠀⠈⠏⢀⠀⠈⠀⣰⠀⢸⠕⢀⡼
⢀⠉⠀⠀⠀⠈⠉⠀⠀⠸⠕⢂⡀⢀⠈⢧⣰⠀⡦⠈⠆⢘⢀⠀⠀⠀⠀⠶⠦⣄⠠⠄
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢛⡀⠀⠰⡆⠃⠑⢈⠃⠌⠀⠀⠀⠀⠀⢀⠀⠉
⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠁⠀⠀⡇⠀⠀⡰⣆⠀⠀⠀⢀⡀⣀⠔
⢘⡀⠀⠀⢄⠀⠀⠀⠀⠀⠈⣀⠄⠀⠀⢰⣁⠤⠾⠇⠒⠂⠈⠉⠀⡀⠠⠄
⠀⠻⡀⠀⠀⠁⠀⠀⢀⠀⠰⠁⠀⠀⣠⠋⠀⠄⠀⠀⠂⠀⠁⠈
⠀⠀⠉⠢⣄⡀⠀⠀⠀⠂⠀⣡⡤⠚⠁
⣤⣤⣤⣤⣤⣬⣵⣶⣶⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠊⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠠⡪⢐⠤⠂⢁⣤⣶⣾⣷⣾⣽⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠁⠀⠀⠀⠀⠀⢀⡴⠁⠀⢀⠀⡪⠂⢁⣴⣾⣿⡿⢟⣫⣽⣾⣿⣿⣿⣿⠋⠉⠿⣿⣿⣧⡀⠘⠿⣿⣿⡉⠉⢿⣿⣿⣿⢶⣤⣄⡀
⠀⠀⠀⠀⠀⢠⠋⠀⠀⡠⡠⠀⢀⣴⣿⡿⢛⣽⣾⣿⣿⡿⠋⠘⠻⠿⠿⢶⣤⣤⣀⣉⣉⠛⠷⢤⣈⡉⠻⢦⣄⠙⢿⣷⡀⠹⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠁⠀⢀⠎⢀⢀⣴⣿⠟⣫⣾⣿⣿⠟⣿⣿⣷⣦⣤⡤⠤⠤⠴⠶⠤⠤⢭⣭⣙⣓⠶⢬⣝⡳⢦⣌⠓⢤⡈⠻⣄⠈⢿⣧⠀⢻
⠀⠀⠀⠀⠀⠀⠠⠃⠀⢠⡾⠛⢡⣾⣿⣿⣿⡏⠀⠈⣉⣡⣤⠤⠶⠖⣒⣒⠛⠁⢐⣒⣒⡀⠉⠍⡓⠦⢍⡳⢬⣛⢦⣝⠢⡈⠳⣄⠙⣧⡈
⠀⠀⠀⠀⠀⢀⠁⠀⠡⠞⠁⣴⣿⣿⡿⣿⣿⣿⠿⠛⣋⡡⠴⣒⠭⠅⢒⣒⣒⣒⣒⣒⣂⠩⠍⣒⠦⢅⡂⠍⡓⢬⡳⢍⡳⣌⠢⡈⢣⡈⢳
⠀⠀⠀⠀⢀⠂⠠⠀⠙⠀⡼⣻⣿⣿⠃⠈⠉⣠⡴⢞⣩⢖⡫⠖⡊⠉⣀⣀⣰⠾⢧⣔⣒⡯⠝⣒⠤⣀⠈⠒⢌⠒⢍⠢⡙⢮⡳⣜⢦⡙⣆
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣸⠃⣿⡿⣿⣶⣶⠟⢉⡴⢫⢔⡩⢂⠅⣢⠭⢓⠂⠀⠀⠀⠀⠀⠈⠉⠒⠭⡲⢅⠀⠀⠈⠂⠱⣌⢦⡳⣝⢦⠳⡌
⠀⠀⠀⠁⠀⢸⠀⠀⢀⡏⢰⢻⡇⠻⠛⠁⣰⠟⡴⣣⢊⠔⡵⢊⠔⣪⠕⣊⣭⠥⠴⠬⠬⢵⣶⣬⣢⣬⣔⠁⠠⠀⠀⠀⠈⢢⠱⡜⢎⢳⡙
⠀⢠⠀⠀⠀⣜⠀⠀⢸⠇⠈⠸⣧⣀⣠⡾⢃⡞⣰⣳⢣⠎⡴⢡⠞⡴⠊⣡⡶⠛⢉⣉⣉⣑⣒⢭⣛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠱⡘⡈⢇⢳
⠀⢸⠀⠀⠀⣿⠀⠀⢸⠀⠀⡄⢿⣿⠟⠀⡾⢠⢧⡇⡎⡼⢱⠇⣾⠁⠸⣿⡀⠀⣿⡟⠁⠀⣉⡛⠿⣷⣬⠻⣿⣷⡀⠀⠀⠀⠀⢳⠀⠈⣆
⠀⠸⠀⠀⠀⢹⠀⠀⢸⡄⢀⢸⡄⠀⢀⣼⠇⣼⢸⣿⡇⡇⢸⡀⠻⣷⣄⠙⠷⣄⠘⣷⠀⢸⣿⠛⠳⢮⣻⡷⡽⣿⡖⠀⠀⠀⠀⠀⢃⠀⢸
⠀⠀⡆⠀⠀⢸⠀⠀⠐⣧⠈⣦⠻⣷⣾⡿⠀⣿⠸⡏⢧⠹⣌⢷⣤⣀⣉⠛⢦⡙⢦⢹⡇⢸⡇⠀⣠⣤⡹⣿⡝⡽⡑⠀⠀⠀⠀⠀⠘⡀
⡇⠀⢣⠀⠀⠀⢇⠀⠠⣱⣶⣿⣷⣌⠉⠀⢠⣿⠀⢿⡈⢧⡙⢶⣤⣉⠙⠻⣆⢹⡘⣇⣷⢸⡇⢸⣿⠟⢿⡎⣧⠘⡷⠀⠀⠀⠀⠀⠀⠇
⢹⡀⡈⡄⠀⠀⠈⠆⠱⢻⣯⡹⣽⣿⡷⣦⣾⣿⠀⠸⣷⡀⠻⣦⡀⠙⣿⡄⢹⠀⡇⡿⣿⢸⠁⣿⠁⢀⡀⠹⡈⠀⢻⠀⠀⠀⠀⢠⠀⢀
⡆⢧⠠⠘⡄⠀⠀⠈⢄⠉⣿⣿⣬⡻⣿⡶⣤⣀⣀⣰⣿⠇⠀⣹⡷⠀⣼⢃⡟⣸⢡⢇⡏⡏⣸⠃⣰⣿⣿⡆⠃⠀⢸⠀⠀⠀⠂⢸⠀⢸
⡹⡌⢇⢡⡙⡄⠀⠀⠀⠠⠀⠙⢿⣷⢶⣽⣒⠯⠭⠭⣥⠤⠶⢛⣡⠞⣡⢎⡴⣡⢏⡞⡘⣰⠃⣼⠛⠁⠈⣿⢠⠀⣿⠀⠀⢠⠀⠘
⢳⡙⣎⢧⡱⡌⢆⠀⠀⠄⢀⡀⠀⢉⠑⠯⠓⠯⢟⣒⣒⣚⣋⠩⠔⡪⠔⣫⠞⡡⢊⢌⡴⣡⠞⢁⣠⣶⡆⣿⡼⢠⡇⠀⠀⠸⠀⡊
⢦⠳⣜⢦⡳⣌⢦⡑⢄⠀⡑⢍⡢⣄⡉⠒⠤⣄⣀⡀⠀⠀⠀⠤⣅⣒⠭⢒⠡⢒⡵⣫⠞⢁⣴⠿⠛⢿⣿⣿⡇⡼⠀⠀⢀⠇⢠⠁
⡌⢳⡙⢦⡙⣌⠳⣝⢦⡑⢌⠒⢍⡒⠭⣓⠦⢤⣌⣉⡍⠉⠭⠭⣑⣒⡩⢵⣪⡵⣚⡥⠞⠋⣀⣀⠀⣾⣿⣟⡼⠁⠀⢀⡞⠀⠆
⠹⣆⠙⢦⠙⢎⡑⢮⡳⢭⡲⢬⡂⠬⢑⠲⠭⣅⣒⡠⠬⠭⠭⠥⠴⣒⣺⠭⠗⣊⣩⣴⠾⠿⣿⣿⣿⣿⣿⡟⠁⣠⡛⡜⠄⠌
⣆⠘⢷⡀⠳⣄⠙⢦⡙⠲⣍⡲⢭⣑⠲⠬⣅⣒⡀⠠⠬⠭⠭⠤⠴⠶⠖⠚⢛⣉⣁⣀⠀⢀⣿⣿⣿⡿⢋⣤⡼⢋⢎⢄⠈⠀⠀⡀
⣿⣆⣘⣿⣦⡈⠳⣦⡉⠳⣤⡙⠳⢬⣉⡓⠶⢬⣭⣉⣛⡛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣧⣿⣿⡿⣋⣴⣶⣟⡵⡡⡢⠀⠀⠀⡔
⠌⠻⣿⢿⣿⣷⡀⠙⣿⣦⣀⠙⢷⣦⣈⠙⠓⠶⣤⣄⣉⠉⠙⠛⢿⣿⣷⣦⣠⣼⣿⣿⡿⣟⣭⣾⣿⡿⡫⡊⠊⠀⠀⠀⢠⠎
⡀⠀⠀⠑⠭⡙⠛⠿⢿⣿⣿⣆⣀⣿⣿⣿⣦⠀⣈⣿⣿⣿⣆⣠⣾⣿⣿⣿⣿⣟⣿⣷⣾⣿⣿⡟⢕⡨⣪⢖⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠯⠛⠙⠠⠒⠥⠪⠐⠁⠀⠴⠋⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠙⠿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⢤⡀⠀⠀⠀⠀⠀⣠⣤⡤⠀⠀⣾⡿⠃⠀⠀⠀⣤⣶⣿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⢞⣝⢳⡹⡆⠀⠀⠀⠙⠛⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠿⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⢮⣭⠼⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶
⠀⠀⠀⠀⠀⠀⢰⡿⠀⡀⠈⠓⠒⠛⠁⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠠⠀⠠⠖⠀⠀⢀⣷⣦⠀⣠⣟⣿⠟
⠀⠀⠀⠀⠀⢠⣶⠄⠀⠘⠓⠀⠺⠶⠀⠀⠀⠀⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠛⠉⠁
⠀⠀⢀⣄⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠂⠀⠈⠑⠛⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠈⠛⠷⣴⣿⣄⣀⠀⠀⢀⠀⠀⠀⠀⠀⣤⠀⠘⠁⠀⠀⠀⠀⣰⣆⢀⣴⣦⠀⠀⢈⡟⠁
⠀⢠⡀⠀⢠⣬⡿⡋⠉⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠻⣿⡿⣠⣴⣶⣤⡀
⠀⠹⠏⡀⠀⠀⣠⣶⣶⣔⣿⢿⣷⡄⠀⠀⢀⡶⠀⠀⠋⠀⣰⡿⣿⣿⠏⣀⣤⣞⣻⣿⣿⡿⠟⠂
⠀⠀⣿⠇⠀⠀⠀⠉⠉⠁⠊⠀⢉⣙⡀⠀⢤⠀⡀⢀⠀⢀⡿⠞⠏⠃⠁⣠⠤⢌⡉⠉⠁
⠀⠀⠁⠀⣰⣄⡀⠀⠀⠀⢲⣶⣿⣿⣿⠓⠀⠀⠀⠈⠀⢈⣀⣀⡀⠀⢸⢝⠽⣱⣳
⠠⢾⣿⠶⠾⠿⣿⡀⢀⣀⣀⡉⠛⠋⠉⣥⣶⣾⣣⣴⣾⣿⣿⣿⣿⣦⡈⠳⠭⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⢰⣿⣷⣼⡿⠛⠁⠀⠉⠙⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⣶⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⣾⣞⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⣛⣿⣯⣟⣛⠀⠀⠀⠀⢀⣀⣢⣶⡞⠁⠀⠀⠀⠊⠹⢿⣿⡟⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⢿⠃⠈⠁⠀⠀⠀⠈⣩⣿⣯⣽⡛⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠈⠀⠀⠀⠀⠀⠀⠀⠟⠹⣿⡏⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣸⣿⡂⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⣤⡴⢿⠧⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠱⠀⠀⢠⣾⣗⣤⣴⣾⣷⣖⣴⢿⡷⢿⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⣀⣀⣤⠀⣀⣀⣤⣿⣟⣭⣭⣤⣾⣿⣿⣿⣾⣿⣿⣶⣿⣿⡿
⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠤⣄⣀⣤⣤⣄⣐⣀⣄⣒⣀⣬⣤⣬⣥⣵⣼⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣭⣿⣟⠛⠻⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠙⠉⣉⠍⠀⠁⠁⠙⠛⠛⠓⢙⠋⠋⠈⠑⠛⠋⡐⠿⠟⠙⠿⠷⠭⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠴⠆⠀⠀⣀⡁⠀⠉⠒⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⡆⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣠⣤⡶⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⢸⣿⠁⢀⣿⣿⠇⣀⣀⣀⡀⣀⣴⣿⣇⠀⠻⡟⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠞⣏⢰⡿⠋⠀⡀⢱⣤⣿⣄⡁⠈⠻⣿⣦⢺⣿⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣶⡾⠃⠀⢀⠀⢎⠘⢻⣿⠉⢫⠄⢀⠜⢿⣯⣿⠋⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠯⠀⢀⠠⣤⠤⣼⡆⢸⡇⢀⣶⢴⣇⣠⠄⢹⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⢉⣘⣀⣸⣷⣸⡇⣿⠁⣸⠟⠛⣆⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⠾⠚⢻⠿⣋⣿⠉⠋⢀⢻⣟⣿⣿⢧⣽⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣨⠞⠁⠀⡺⠂⠀⣿⡟⠚⣡⠿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠠⠀⢘⣉⡄⠀⠈⠉⠁⠀⠘⠙⣧⣿⡏⢠⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡀⠟⠀⠀⢠⠞⢷⣶⣄⠀⢿⣿⣡⣯⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢠⣿⡧⠀⠀⠀⠘⣿⣿⣿⡟⠀⣸⢯⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢾⣿⣿⠁⢪⣿⣷⣄⠀⢀⣰⣿⣿⡟⠆⠀⢏⣸⣿⣷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⣿⢏⡟⠀⢾⠋⢿⣿⣽⠶⣦⣄⣤⣤⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣾⠏⣾⠁⠀⠀⠠⢀⣻⣿⡷⣄⣩⣻⣻⣿⠙⢴⣿⣿⣿⠟⣸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⢸⣿⢀⡟⠀⠀⠀⠉⠉⠑⣻⣿⣿⣻⠟⣿⣏⣼⣼⣿⡿⠍⠁⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⠸⠇⠀⢐⠀⢀⠀⠀⠁⠈⠿⠿⠴⠟⢿⣿⣿⠛⠀⠀⠠⣐⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⢻⣞⠀⢀⣬⣁⣙⡃⠤⡶⢠⡀⠀⠀⠘⠀⠁⠀⢀⡀⠴⣶⣺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⣸⠃⠀⡀⠀⠛⠛⠻⢿⣿⣿⣿⣦⡀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⠋⢩⠀⣠⠂⠀⠀⠀⠀⠀⠙⠿⣿⣿⣄⣰⣼⣿⣿⣿⣿⠟⠉⠀⠈⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣵⡄⠀⠀⠐⠀⢀⣣⠀⢈⠉⠉⠉⠉⠉⠹⡟⠁⢠⠀⠀⠄⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⣶⣆⣀⣈⣋⣀⣴⣶⠦⡤⠀⠀⠠⠀⢀⡏⠀⠀⣴⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠾⠿⠷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠠⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠥⡲⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⣰⠁⣀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⡷⢺⡟⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠒⢤⢰⠀⠀⠀⠀⠀⠘⠆⠀⠀⡌⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣻⠁⠀⠀⠀⠀⡇⠂⠀⠀⡇⠀⠀⠀⠀⠀⣯⣴⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠁⡄⠀⢸⠀⠀⠀⠀⠀⠐⣻⡟⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⢰⣿⣾⠇⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣟⣀⡀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⡚⣄⣤⣁⣤⠠⠐⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠁⠂⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⠀⡟
⠀⠀⠄⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⢄⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠉⠉⠉⢹⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⣇⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢰⠃
⠀⢸⠀⠀⢠⣶⢆⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⢸⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⠀⠀⠀⡼
⠀⢸⠀⠀⠈⠱⠛⠁⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠤⢤⣤⢤⡄⠀⠀⡇⠀⠈⡇⠀⠀⠀⢸⠿⠿⠟⡃⠀⠀⠀⢠⠇
⠀⢸⡀⡀⠀⠀⠠⠤⠤⢼⠀⠀⠀⠀⡇⠀⣶⡎⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢹⡀⠀⠀⢸⣀⣒⣘⡇⠀⠀⠀⡼
⠀⠈⣇⠀⠀⠀⠀⠀⠀⡿⢄⠀⠀⠀⠑⠠⠟⢢⣀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠐⠚⠉⠑⠄⠀⠀⠀⣇⠀⠀⢸⣿⣿⣾⡇⠀⠀⢰⠃
⠀⠀⠘⠀⠤⠤⠤⠤⠖⠁⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⣶⣶⣾⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢸⣒⣚⣻⡆⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⢀⣾⣋⢿⠿⠅⠀⠠⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢀⡎⠀⠀⠀⠲⠦⠤⣌⣀⣉⣢⣄⣀⡠⠼⠦⠤⠴⠦⠤⠤⠤⠤⠀⠉⢁⢄⠢⠎⢀⡤⠚
⠀⠀⠀⠀⠀⠀⣟⣃⣚⣦⠶⡟⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠒⠋⠁
⢆⠐⠐⡀⠢⣀⠀⠀⠀⠈⠉⠘⠒⠤⢄⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁
⠀⠀⠐⠥⠂⠈⠁⠂⠤⣀⢀⡀⠀⠀⠀⠀⠉⠣⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠐⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⡠⠔⠁⠀⠀⢀⣀⠀⠤⠤⠤⠤⠄⠀⠘⠂⠒⠒⠒⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠰⠖⠆⠰⠀⠠⠤⢀⣀⠀⠀⣠⡄⢤⠄⢤⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣦⣿⣿⣋⣀⣤⣤⣤⣤⣧⣤⣤⣠⣀⣀⣀⡀⠈⠑⠓⠿⡞⢿⡛⣷⠺⣶⠷⠾⡄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⡶⣤⣀⠀⣝⣯⣙⣏⣙⣟⢻⡛⣄
⠠⡤⠄⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣟⣿⣿⣿⢏⠿⠀⠙⣷⣿⣿⣿⣩⣭⣽⣏⣹⣉⣆
⢂⠣⠀⠀⠀⠈⠙⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣴⣶⣮⣶⣴⢟⠟⠋⢻⣿⣴⣤⠼⣦⣽⣌⣦
⠈⢦⠠⠉⠈⠁⠀⠈⠿⣆⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣯⡷⠣⢊⣤⣾⣿⣿⣿⣷⡾⣦⠬⣤⣱⡀
⠀⠀⢢⠐⠀⠆⠀⠤⠠⣼⣷⣤⡀⠀⠀⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⢁⣴⣷⠯⠂⠀⣈⠀⠀⠻⣿⣿⡿⣿⠿⣦⠤⣦⡼⣄
⠀⠀⠀⠉⠉⠛⠳⣿⡛⠛⠋⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣯⣄⣀⣀⣀⣻⣆⠀⣚⣛⣭⡾⠿⡦⠼⢦⠤⢦⡬⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠤⠀⠀⠀⠀⠀⠈⢉⣠⣶⣶⠾⠿⠛⠛⠛⠉⠭⢒⡩⠔⠊⠀⠀⢡⠿⣖⣻⣖⢻⣖⠚⢶⠖⢶⡖⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣾⣻⣿⠿⠿⠉⠀⠀⠀⠠⠤⠤⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠦
⠀⠀⠀⠀⠀⠀⢠⢆⣄⠸⣿⣧⣀⠀⣠⣮⣂⣤⣾⠟
⠀⠀⠀⢀⣠⣤⣬⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣷⡶⠶⠖
⠀⠀⠈⠉⢉⣉⣉⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⡀
⠀⠀⠀⣊⣉⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣄⡀⠀⠉⠉
⢀⣴⠿⠿⠟⠟⢛⣩⣽⡿⢟⡿⠻⣿⣿⣿⠿⣿⣾⣯⡕⠄
⠀⠀⠀⠀⠀⣰⣿⠟⠉⠀⣸⡇⠀⠙⢽⣿⣷⣌⠙⠋⠁
⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⣿⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⣴⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠉⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⣿⣿⣿⡿⠿⠋⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠻⠟⠋⠉⠀⢀⣶⣿⣿⡷⠀⠀⣴⣶⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣄⡀⠀⠀⠀⠿⣹⣿⣇⣀⣠⡿⠟⢀⣼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⢾⣿⣿⣶⣄⡀⠀⣿⣿⣿⣿⣷⣤⣀⣾⣿⡏⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠙⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠀⠀⠀⠀⠈⣻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡛⠛⠃
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣿⣿⣶⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠲⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣀⣠⣾⣿⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⢿⣿⣿⣿⣿⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠉⠋⠉⠉⠉⠉⠙⠻⠿⢿⣿⣿⣾⡶⠶⣾⢾⡿⣿⡿⡿⣿⣆⠀⠀⠉⠉⠛⠛⠛⠛⠁
⠀⠀⢀⣴⣾⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⠉⠻⢿⣿⣿⣿⣆
⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⡴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠄⠀⠺⠀⠀⠄⠀⠀⠀⠀⠀⠰⢀
⠀⠀⠀⢀⠐⠃⠀⠀⠈⠀⠀⠆⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⢣⡌⠀⠀⠀⠀⠀⠀⢰⠀⠀⣀⠀⠀⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡏⠀⠺⠟⠀⣴⣿⣿⠃⠀⠀⣤⣦
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢩⣁⡠⠀⢸⣏⠀⠀⠀⠀⠀⠠⢶⠀⠀⠀⠀⢤⡄
⠀⠀⠀⡧⠀⠀⠀⠐⠆⠚⠠⢀⠀⢻⠀⠛⣾⢿⣷⠦⣄⣀⠄⠀⠘⣳⣶⣶⠂⠀⠀⠀⠙⠟⠿⡟⠁⢀
⠀⠀⢰⠁⢣⠀⠀⢀⣴⠃⢀⠀⠀⠀⠀⠈⠀⠈⠀⢻⡿⠯⡖⠀⢘⠏⠈⠏⠁⠀⠀⠄⠀⢀⡄⠀⢰⣾⡦
⠀⢀⠘⠀⠘⠄⠀⠘⢙⡁⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⡅⠀⢠⠀⠁
⠀⠀⠀⠈⠀⢣⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢘⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⡰⠇⠰⠚⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠇⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣶⣶⠶⢶⣶⣦⣤⣤⣤⣤⡤⣤⣤⣤⣤⣶⣶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⡿⢿⢿⣿⠋⠈⠛⠝⠛⠉⢉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠐⢠⠔⠀⠀⠀⣤⣶⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠙⠓⠟⢉⣙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠦⠤
⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠲⠤⠄⠠⠀⠀⠀⠀⠈⠓⠣⣖⡀
⠀⠀⠀⠦⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠐
⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠐⠚⠂⠐⠂⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠈⠁⠀⠀⢀⣤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣲⣦⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣠⣴⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠠⣤⢶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠙⢿⣿⣿⣿⣿⣿⡿⢿⣳⣄⡀
⠀⠀⠀⠀⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠛⠛⠋⠉⠉⢁⣨⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⡿⠿⠛⠙⠁