Jack in the box Emojis & Symbols

⠀⠀⠀⠀⠀⡠⠠⣦⣤⣌⡉⠁⠐⠢⡀⠀⠀⠀⠀⠒⢂⣀⣀⡀
⠀⠀⠀⠐⠀⠛⠷⠃⠀⠈⠳⠀⠀⠀⠿⣆⠀⠀⢠⠚⠁⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⣐⡰⠶⠾⠶⢤⣄⠀⢀⡥⠒⠉⠛⣮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⢉⠥⢤⣐⡶⢋⠙⠺⣿⣷⠏⠀⠀⠀⠀⢄⣦⠅⠡
⠀⠀⢀⠐⣁⣠⣤⣀⣎⡅⠃⠀⢡⡌⠁⠰⢆⢀⢸⡏⢀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠨⠠
⠀⠀⣠⡞⠁⠀⠀⠀⠙⠷⣄⠰⡄⠀⠥⠞⢀⠃⢸⠃⠐⠀⠀⠀⠀⠀⣀⢠⡆⠘⣁⣍⡀
⡀⢐⡏⠱⠀⠀⠀⠀⠀⠀⠈⠢⣈⠑⠂⠒⣁⣴⡿⣗⡤⠤⠤⠐⢂⠉⠑⡘⣧⣀⡥⠶⡁
⠑⠁⠉⠁⠀⠀⠀⡀⠀⠀⠀⢴⣹⢋⡟⣯⢻⣌⡟⠺⠣⠠⠀⠁⠴⢁⣰⠴⠋⠀⠈⠉⠁
⠀⡄⠠⡀⠀⣀⡀⠬⠐⠈⠀⠀⠀⠉⠘⠁⢳⠀⠔⡑⠀⠐⣴⠿⠚⠉⠠⠚⠁
⣀⠬⠶⠈⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠳⡫⠈⠂⠀⡪⣺⡄
⢨⡋⣉⢀⠇⠀⣀⣤⠴⠒⣯⠀⠀⠀⠀⠀⡠⢗⢔⠑⢈⠨⣃⡃
⠠⡐⠥⠂⠀⡍⠀⠀⠀⠀⣷⣢⡤⡀⢀⣄⢌⣇⣐⣥⣵⢞⣉⣀⣀⣀⡀
⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢙⠿⠋⠙⠉⠘⠣⣌⠾⣶⠶⠶⢤⣬⡍⣑⠲⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠓⠲⡄⠀⠀⠀⠛⢿⣅⣟⠤⠀⠀⢀⢣⡯⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠺⢦⠴⠐⠀⢶⣿⣩⠟⠊⠅⠀⠀⣸⢠⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡉⢀⠤⠀⣤⣝⣿⣀⣠⡂⠀⡄⡯⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣐⣼⣛⣁⣀⣀⣠⣙⣿⠿⠿⣷⣿⡿⣼⣵⣪⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣚⣻⡿⠦⠤⣿⣀⣀⡀⣈⣿⣶⡾⠿⢟⣻⣿⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡶⠤⢬⣭⣁⡀⠛⠿⣭⣾⣵⠾⢛⠏⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⡀⣀⠶⠀⠉⠉⠓⢼⣯⠈⣀⡨⣼⣸⣾⣁⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡇⠑⡄⠀⢩⠝⠀⠘⢸⡿⡀⣸⣾⠿⢿⡏⣟⣠⠤⠽⢖⡂⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠷⣴⣦⣉⣲⣄⠀⠀⢸⡏⢐⣹⣿⣿⣿⠿⠾⠷⣶⣶⡼⠂⠒⠂⠀⠰⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠢⠬⢁⣚⣛⠛⢸⡟⣻⡷⠚⠛⣟⠛⢶⡦⠬⢉⡀⢊⣈⡻⡯⠩⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡙⠿⢿⣟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢶⣿⣉⡉⠉⠛⢷⣄⣿⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣼⠏⠉⠻⣦⠀⠀⢻⡇⣀⣴⠾⠛⠛⠛⠷⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣿⠀⠀⠀⠸⠏⠁⢀⠀⣠⠶⠛⢿⡏⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠛⢿⣤⡀⠀⠀⠀⠀⠉⠛⠛⢷⣄⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣿⣦⣈⠛⢶⣄⠀⢀⣀⣤⣄⡀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⢶⣄⠀⠈⠙⢷⣤⡙⠻⣿⠉⠁⣉⣿⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠈⢿⡄⠀⠙⣷⡀⠀⠀⠈⠛⠿⠋⠀⢸⡏⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣈⣛⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⣿⣟⠉⠉⠙⣷⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠙⣿⡉⠉⠉⠉⠙⣷⡀⠀⣿⠙⠻⠶⠾⢋⣾⠋⠉⠉⠉⢉⣿⠋
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠈⠻⣄⢻⣆⠀⠀⣰⡟⠁⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⢹⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣤⣤⣤⣤⣼⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡻⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⡶⠶⡄⠀⣀⣤⡄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⢰⣶⡖⢰⣤⣄⣸⣋⣥⣴⢾⣿⣦⡈⢿⣧⣿⡏⣸⡧
⠀⠀⠀⠀⠀⢠⣺⡷⠘⠻⠇⣿⡿⠋⣈⣽⣏⣠⣎⡈⠛⢿⠎⠀⠀⠈⠙⣴⣧
⠀⠀⠀⠀⠀⡛⠃⠀⠀⠀⠀⠘⢃⣼⣿⠟⢉⣭⣿⡿⠷⢬⠀⠀⠀⠀⠀⠈⣾
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠘⡿⠁⣴⡿⠋⠀⠀⠀⠐⠶⡁⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠠⣪⡤⠀⠀⠀⠀⠀⠀⠀⣾⡿⡡⠂⠀⠀⢀⣠⣘⢳⣀⣒⢲⢆⣩⢰⡄⡀⢀⠀⡀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠀⠀⠀⠀⢉⣰⡗⣠⣤⣄⢻⣥⣿⢂⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⣷⡆⠹⣄⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣗⢿⣶⣿⢣⠿⠀⡘⠋⢠⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠃⠈⠀⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡩⠽⠄⠊⠉⢀⠐⣶⡟⣿⣿⣿⣿⣿⣿⣿⠿⡿⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡔⢀⡄⣠⣄⢻⣿⣼⣿⣿⢻⠋⣿⡿⠿⣿⣻⡏⠁⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣻⡿⣘⣿⣾⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣾⡏⢀⡃⣠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢹⣿⣿⣿⣿⣿⣿⣷⣶⣿⣍⣿⣿⣿⣃⣾⠳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⢻⣯⡿⠋⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡄⢸⠟⠻⠫⠉⠿⢟⡿⢿⡛⠛⠛⣛⡛⡇⠀⣀⣠⠂⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⢠⣈⠛⡄⠀⣇⣹⡇⠐⠂⠀⠨⠉⡥⠈⣟⡿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⠈⠀⠀⠈⠟⡿⡀⠀⠀⠀⠀⠀⠀⠿⣷⣿⠄⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠀⠀⠀⠀⣴⣻⡇⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⣿⠀⠀⠀⣰⠡⢤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⢀⠤⡤⠀⢠⢮⣀⣀⣀⣹⡁⢀⠴⠄⠀⡿⣿⣻⣄⣠⣶⣿⢶⠋⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⢶⣶⣄⣀⣀⣀⡀⠀⠀⠀⠀⠈⢒⣻⠂⠉⠹⢀⣿⡿⣛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⡁⠐⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣄⣈⣠⢦⡯⠾⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⢻⣛⠋⠘⢿⣓⠛⠛⠉⠋⠀⠚⠉⠛⢉⡍⠉⠉⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠛⠛⠻⣷⣦⡀⠀⠐⢄⠀⠀⠐⠈⠁⠀⠈⠁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣀⠀⠀⠀⠀⠀⡈⠻⠿⠦⣀⠀⠲⠀⠀⠀⢀⡴⠀⢀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⢠⡀⠀⢀⡾⠇⠐⠈⠁⠀⠀⠓⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠠⠀⣿⣆⠀⠀⠀⠠⠄⠉⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⢸⠏⠀⡀⠠⠒⢢⣇⣠⣴⣿⣿⡿⠛⠉⠋⠙⠛⠿⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠓⠀⠀⠀⡀⠀⣹⡿⠃⡀⣄⠈⠙⠀⠀⠀⠘⡷⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠢⠀⠐⢀⠁⠀⠀⠄⡀⠀⠈⠀⠀⠋⠠⠀⠀⠙⠶⠂⠀⠀⠀⢈⣀⢠⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢄⡤⠀⣄⣶⣍⡴⠀⢀⡀⡉⠁⠀⢠⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⢊⡽⠤⢆⣻⢶⡀⠃⠀⣾⢛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⡄⢈⠀⠀⠀⡷⠉⠀⠀⠐⠁⠔⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡸⣯⡂⠌⠀⠉⠀⠠⣴⠆⠀⠦⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢛⠿⢶⣶⡾⠟⢁⠠⠊⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠊⣠⡁⠀⠀⠂⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠠⠀⠀⠀⠀⢀⠔⠒⠒⠒⢤⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⡀⢤⣔⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠦⢒⡈⠁⠀⠀⠀⢀⠋⢉⡐⠴⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⣤⡄⣀⡀⠉⠁⣲⢖⡉⠉⢀⣀⣠⣤⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⢹⢸⠉⠉⣀⣿⡥⢼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⢸⠀⠀⠿⠟⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡂⠤⠀⣀⠀⢸⢸⢀⣀⠠⠄⢐⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠈⠹⠎⠀⠒⠀⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⡟⠉⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣰⠋⠀⢰⡏⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠉⠉⠉⠉⠙⠓⠶⠒⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⡶⠂⠒⠶⢤⡀⢀⡔⢀⡈⠉⠹⠹⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡁⠀⠐⠷⡾⣠⠇⠉⢓⣦⣀⣀⣀⣠⡎⠹⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣯⡄⠉⠒⠒⠒⠚⠁⠀⠀⢯⠀⢀⡀⠀⠀⢷⠀⢸⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡖⠒⡶⠟⣛⣉⠽⠛⠉⠓⣛⠀⣀⣄⡀⢀⢯⣉⠉⠀⠀⢻⣄⣤⠼⠦⠤⠝
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠒⣟⠀⠀⠱⡀⠀⠀⢀⡈⠉⠁⠀⠙⠋⠪⣀⠅⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣬⢯⡄⣠⠷⠖⠊⠉⣷⠀⠀⠀⠀⠀⠀⢈⠦⢄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡙⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠘⢠⠊⠀⠀⠀⢹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣶⠒⠚⠋⢹⣇⠀⠀⠀⠀⠀⠀⢘⠁⡢⠀⠀⠸⣧⡍⠙⠓⢒⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⠓⠦⢤⣉⡛⠓⠲⠶⠶⠦⠤⠿⢶⣒⣺⡿⠽⠖⠚⠋⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠈⠉⠛⠲⢤⡴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⡀⠀⣰⠿⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡿⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⣍⠙⠋⠀⢧⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡟⠀⠙⣦⡤⠤⢤⡶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⡆⠀⠀⠈⠙⢦⣄⢸⠀⢀⣤⠴⠞⠋⠁⠀⠀⠀⠀⣰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣴⠚⠁⢸⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠋⠙⢆⢹⠀⠀⢸⠀⠀⠀⠀⢸⡇⢀⣤⢦⣤⣄⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⠀⠀⢸⠀⠀⠀⠀⣸⠶⠋⠀⠀⠀⠈⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⢸⠀⠀⠀⢀⣠⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣼⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⡀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⠚⠁⢀⠔⠉⣀⠔⠂⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠈⠁⠈⡀⠉⣲⡅⢀⣤⣁⢠⡾⠐⠒⠒⠚⠲⠌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠥⠤⠤⢀⡠⠚⣁⠠⠀⠀⢠⠀⠀⠀⠑⠢⡔⠒⠒⠒⠠⠍⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠌⠀⣠⠉⠣⡀⢀⠂⠀⠀⣀⠔⢳⠈⢲⠒⠒⠒⠒⠂⠽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠾⢌⠒⠒⠛⡴⡄⠰⣉⡀⣠⢞⡀⠀⡆⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡗⢦⠀⠀⠰⡀⠸⣄⢱⣀⡤⠍⢁⡀⠠⠅⠒⣩⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉⣀⠀⠀⠀⢉⠑⠢⢵⣄⠈⠓⠤⡴⠒⢢⣉⡩⠔⠊⡉⢀⠜⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣁⠀⢠⠔⠊⠀⠐⠒⠂⢳⠦⠄⣆⡀⠀⣠⢁⡀⠤⢗⡓⠒⡲⠂⠉⠀⠪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣏⡠⠄⠽⠂⠀⠂⠒⡖⠁⠀⡔⠁⠀⠘⠁⠸⡃⠠⢄⡀⠙⣄⡀⠀⠀⢠⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠧⠐⠒⠂⡝⠀⠀⡜⠀⣀⠀⡇⠀⠀⠈⠓⣤⡀⠙⣍⡜⢦⢰⣌⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠃⠀⠊⠀⡇⠀⠀⠀⠀⡇⠈⠒⢼⡇⠀⣿⠈⠃⡌⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠸⠀⠠⠁⠀⠇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡘⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠊⠁⢻⣀⠴⢲⣄⠸⡰⠦⢄⣀⠀⡆⠀⠀⠀⠀⠀⢰⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠠⡼⠠⠻⢅⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠂⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠛⣶⢤⣭⣵⣎⡄⣀⣀⠀⠀⠀⠀⠀⠀⡎⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠐⠂⣬⠐⠘⠣⣈⠚⠉⠐⢀⠍⠉⠉⠉⠉⡳⠖⢂⣺⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠁⠒⠠⠭⢈⡀⠛⠯⠍⣁⠀⣀⠤⢂⡩⠔⠊⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠉⠐⠂⠤⠀⢀⡀⠀⠉⠀⠒⠠⡄⠊⢁⠠⠐⠈⡇⢸⢀⡠⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠄⠂⠀⠂⢄⠉⠑⢲⠀⠀⡇⠈⠀⢀⣀⣀⡇⡜⠰⠧⢄⠀⣀⣊⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⡂⠀⢸⠀⠀⡇⠀⠀⠐⠀⢤⡉⣁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠄⡀⠀⢀⠀⠁⠀⢸⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠀⠐⠂⠤⢄⣀⠀⠀⢸⠀⠀⡇⠀⢀⡠⠒⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠒⠠⠤⢀⣀⡀⠀⠈⠉⠀⠀⡇⠈⢁⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠄⠧⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⣶⣶⣾⡿⠃⣄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⢿⡟⠀⠀⠀⠀⠀⠑⠀⢀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⣄⣈⣨⡀⠀⠘⠀⠀⠀⠀⣸⣿⣇⡀⣈⣥⣀⠀⠀⠀⠀⠐⠀⠀⢈⣔⡄
⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⡏⠁⠳⠴⣶⠀⠀⠀⢸⢿⡏⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⡴
⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⣀⠆⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⡇⠽⠹⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠥⠇
⠀⠀⠀⠀⠀⠀⠙⢾⠤⠤⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠫⠩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠀⠀⠀⠀⠀⠀⠘⠇⢠⣠⡾⠖⡓⠲⠃⢤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠢⣼⠆
⠀⠀⠀⠀⠀⠀⢀⣈⣷⣂⣀⣠⡀⠀⠀⠀⠀⠈⠋⠙⠷⢾⠐⢀⣤⣤⣤⡤⠀⠀⠀⠈⠝⠫⣷⣾⠂⣤⣀⡤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣟⣿⣿⠁⠀⠀⠀⠀⠀⢀⣤⣽⣿⣄⠀⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⡛⠟⣻⡛⠋
⠀⠀⠀⠀⠀⠀⢸⡏⠘⠿⠿⢻⡇⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⣀⣀⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠞⠉⡳⡦⣤⣤⠠⠛⠶⠤⣀⠀⠀⠀⠀⣀⡠⠴⠯⢅⣀⣀⣤⡒⠻⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⠋⠻⡉⠢⡀⠀⠈⠳⡄⣠⠊⠁⠀⢠⠔⢊⡟⢯⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢹⡀⠀⠀⡸⠁⠀⠀⢰⠃⠀⡜⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢠⠇⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⣇⣀⠸⢀⡀⠀⣠⡃⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠳⠎⠈⠿⠋⠘⠾⠁⠹⠞⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠠⠤⠤⠄⠄⠠⢤⣤⢤⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠂⠂⠈⠉⠉⠁⠀⠐⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠋⠀⢠⣦⠀⠀⢰⡆⠀⠙⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠐⠒⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠰⣦⣤⣠⣤⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⢈⣉⣉⡁⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⡴⠒⠚⢳⠚⡷⠲⡤⣤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⡟⠁⠰⠶⢸⠀⣿⠀⡇⣸⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣸⠁⣖⡢⠀⢰⠀⣿⠀⡇⣿⢹⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⢤⡀⠀⠀⠀⣀⣼⣁⣰⣿⣄⣀⣀⣰⣻⣄⣿⣠⣇⣿⣸⣧⣀⣱⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⣤⣤⣄⣤⢿⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠖⠂⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⡇⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⡄⠀⠀⢸⠀⠀⠀⠀⠀⢀⡜⠀⢳⡀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠂⠠⣖⠒⠊⠉⠀⠀⠀⠉⠑⠒⣲⠄⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡜⠀⠀⣀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡧⠔⠋⠉⠙⠲⢬⡄⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠙⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠒⠲⠤⠴⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡖⠋⠀⠀⠀⠀⠀⣀⣀⣠⠤⠶⠋⠉⠁⢀⡀⠀⠈⢧⡀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⡴⠞⠋⠀⠀⠀⠀⠀⠀⣠⠚⢡⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⢧⡀⠀⠀⡤⠃
⠀⠀⠀⠀⠀⠀⠳⢦⣄⠀⠀⠀⠀⠀⢰⠃⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⢹⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⠀⠀⠘⣦⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠙⢋⠀⠀⠀⢘⣇⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡄⣾⠁⠀⠀⠋⡃⠀⠀⠀⢀⣀⠀⠀⠸⠀⢀⡴⠋⠉⢻⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠹⡄⠀⠀⠀⠓⠀⠀⣾⠋⠙⢷⡀⢀⡴⠋⢠⡶⠀⣼⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢀⣴⠷⢤⣄⣀⡀⠀⠀⢻⣤⣤⣾⠗⠋⡀⣠⡞⠁⣸⠇⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢷⡀⠰⣤⣉⠉⠉⠃⠀⢉⡉⠠⠐⣨⡾⠟⠀⣰⣿⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⠀⠛⢦⡈⠙⠿⢷⣶⣶⣤⣶⡶⠟⠩⠂⠀⣴⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠳⠿⠶⣤⡀⠀⠀⠉⠀⠀⠀⠀⢠⡞⢁⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⡉⣳⣶⢤⣤⣤⣴⠾⠛⠊⠉⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣁⠤⠖⠋⢉⣀⣤⡴⠖⠋⢉⣉⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⣁⡤⠶⠚⠋⣉⡠⠴⠖⠛⠉⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠤⠐⠊⠉⢁⣀⡀⠤⠔⠒⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡤⠒⠚⠉⠁⠀⡀⠀⠠⠤⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠐⠚⠀⠀⠀⠀⢀⣀⣀⣀⣴⡛⠛⠛⠓⠒⠶⠶⠦⢤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠒⠒⠀⠉⠉⠉⢀⣀⣀⣀⣀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣈⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣎⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⣿⣿⣿⣿⡿⠒⠒⠀⠀⠠⠤⠤⠤⠤⠤⠤⢤⣾⣿⣶⣶⣶⣦⣤⣤⣤⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠿⠿⠿⠿⣿⣧⣌⣁⣉⣉⣁⠀⠐⠒⠒⠒⠒⠂⣻⣿⣿⣿⣿⠿⠟⠛⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢠⣤⣤⣄⣀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠓⠲⢶⡟⠛⠉⠁⠀⣀⣤⣴⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⣿⣿⣿⠟⠿⠿⠿⠿⢿⣿⣿⣷⣶⣶⣶⣶⡄⠀⣸⠃⢠⣶⣾⠟⠋⠹⣿⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⣿⠀⣸⣿⠏⣰⢂⠠⣿⢀⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢠⡏⠀⣿⡇⠀⣧⣈⠟⡟⢩⠉⠉⠤⢤⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⡇⢠⣿⠀⠀⠀⠀⣰⡇⠙⠀⠀⠀⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠇⠀⣾⠀⢸⣿⣤⣀⣀⣼⣿⠠⠀⠀⠀⠀⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⠀⠀⡟⠀⣼⣿⣿⡿⠟⠋⣁⠄⠀⠀⠀⠀⠈⢶⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠀⢸⡇⠀⠛⠉⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠧⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⠶⠶⠦⢤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⢸⠃⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢀⡠⠂⠁⣀⠤⠹⠖⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠐⠢⣤⠃⠀⣠⠊⠀⣴⠩⠤⠤⠤⢦⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠉⣁⠖⠒⠉⠍⠉⢉⠋⠙⠑⠢⣄⡀⠤⠤⠤⢈⣑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠉⠉⠀⠉⢉⠞⠀⠰⠳⡀⠀⠀⠌⠀⠀⠀⢀⢤⠙⢢⡤⠤⠤⠤⠍⣚⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠆⢀⡔⢣⣀⣘⡆⣘⠀⣀⠄⠛⠁⢸⠄⠀⢃⠀⠀⠀⠀⠀⠨⠛⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠐⡀⠘⡍⠢⡀⠀⠰⠯⠇⠈⠒⠆⣒⡡⡆⠀⢸⠀⠀⠀⠀⠀⠐⠄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠃⠙⢤⣀⡀⠣⠰⠑⢄⡓⠴⢋⠐⠒⣄⡠⠁⣀⠼⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡖⠀⠀⠀⢰⡀⠉⠑⠛⢆⡀⠉⠑⠏⠈⠱⠐⡖⠋⠁⠀⣠⠚⠀⠀⢀⡠⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⣤⠀⠰⣎⠁⠀⠀⠉⠉⢈⠝⠒⢲⠤⠄⣀⡼⢠⡄⠒⠋⠄⣉⡩⠋⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡧⠔⠒⣺⠁⠁⠉⢉⡹⠁⠀⣠⠃⢈⠀⡌⠀⠀⢍⡐⠂⠄⣀⢱⣤⡀⠀⠀⢶⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠒⠈⠉⡸⠁⠀⢠⠃⢀⡀⠀⡇⠀⠀⠀⠉⠶⢄⠀⢣⣠⠛⢆⢰⣌⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠸⠀⠐⠂⠀⡇⠀⠀⠀⠀⠻⠀⠙⠢⣿⠀⠈⡟⠈⠁⡆⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠇⠀⢎⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠋⠙⣆⣠⡶⢢⡀⢰⡠⠤⣀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡰⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣅⠥⠬⠿⡁⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⢠⠁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠢⡬⣑⣒⣢⡞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⢐⠂⠭⠿⡀⠻⠼⣛⣉⠙⡓⠛⠛⠳⠒⠲⣤⠤⠀⣌⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠂⠤⠉⣁⣒⠂⠼⢱⣶⡄⠐⠚⠁⠀⢀⡀⠔⢊⡠⠔⠊⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠄⣀⠀⠀⠈⠁⠐⠂⠤⠌⣁⠒⠊⡡⠔⠊⢁⡠⢴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⠉⢁⠒⠂⠤⢀⡀⠀⠀⠍⢠⠐⠊⠁⠀⢸⠀⡟⢸⡋⠉⠉⠉⣹⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠠⠊⠀⠀⠀⠈⠂⠀⠀⢸⠀⠀⡆⢸⠀⢠⠒⠂⠾⠴⢀⠏⠑⠒⠒⠒⠚⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⢸⠀⠀⡇⢸⠀⠀⠉⠐⠒⠰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⢀⡑⠠⠀⠀⠀⠂⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⢀⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⠀⠉⠐⠒⠠⠄⣀⠀⣘⠀⠀⠂⢸⣀⠔⠊⢁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠒⠠⠤⢀⡀⠀⠀⠁⠀⠀⠃⠈⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠂⠶⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠲⣀⡔⠒⣄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⠤⠒⢂⠶⢏⠽⠀⠀⠀⠀⡠⠖⠘⣋⡡⠔⠀⠀⠀⠀⠈⢻⠉⠐⠺⣄
⠀⠀⡠⢄⣠⡪⠴⠒⠒⠚⢄⣀⡀⠀⠀⢠⠎⠀⡠⢺⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠐⡀⠢
⡔⠈⠀⢨⠇⡄⠀⢀⠀⢠⠀⢱⠹⡄⠀⢜⣠⠊⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠇⠀⠱⡀
⣏⣀⢀⡔⠓⠋⠸⠥⠇⠉⠉⢠⣷⠃⠀⡸⠁⠀⠀⡜⡰⢲⠀⠀⠀⠀⠀⡴⢳⠀⠈⠲⢄⢸⠀⠀⢣
⠀⠉⠉⡄⠀⠓⠢⢤⠚⢅⣐⣃⠀⠀⢠⠇⠀⢀⠞⠀⠀⠀⠀⢤⣠⠆⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⢼⣀⣀
⠀⠀⠀⠈⠓⠢⢤⡜⠳⣄⣀⣠⠇⠀⠸⣀⡴⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠈⠉⠑⠦⡀
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⡠⠊⢫⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢦⣜⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡇⠀⢀⡠⢤⠀⠀⠀⡇⠀⡜⢯⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⠎⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠘⠲⣇⠤⠜⠀⣀⡀⠉⠒⡷⠒⠓⠯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⣟⠁⠀⠀⢞⠃⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣮⡠⠟⠀⠀⠘⠦⣄⡠⠀⠉⠉⠛⣻⡘⠀⡶⠚⠉⠉⠁⠀⠀⠈⠢⣄⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⠶⢤⠤⠦⡀⠀⠀⠀⢀⡤⢺⢃⢁⠔⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⡶⠚⠒⠒⠻⣀⡠⠔⠉⢀⣀⠜⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢔⠉⡦⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡇⠀⠀⠀⢠⣧⣀⡠⢴⠏⠀⢸⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠋⠉⠉⠀⠀⠀⡼⢤⠀⢸⠀⣤⠤⠤⡌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠊⠀⣸⡄⠈⠒⠊⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣯⣹⡁⠙⢲⠔⠴⢶⣶⣒⣲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠉⠙⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠤⡄⡤⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠉⡹⠊⠉⠛⠓⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⣇⣀⣀⣀⠤⠤⠞
⣿⣿⠟⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠙⠂⠀⠠⡄⠀⠑⡀⠀⠀⠀⠀⠀⠀⡁⡀⠀⠀⢲⡐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢂⠀⠈⠂⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠈⡈⠁⠀⣀⠀⠒⠀⠐⢠⠀⠀⠀⠀⠉⠚⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠍⡒⠀⠀⠀⠠⠀⠀⢀⡅⢀⡠⠂⠢⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠀⠀⠀⢀⣀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠤⡀⠂⢀⡴⠉⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠠⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⡁⠄⠀⠀⠀⠀⠀⠖⠀⠀⡈⠤⠄⢒⠀⢠⡀⡐⠀⠠⢤⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⢀⠂⠈⠰⠂⠄⠀⣄⠀⠀⢀⠂⠀⠐⠘⠁⠀⠀⠀⠉⠁⠐⠄⠛⡀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⡁⠀⠀⠀⢉⠆⠀⠍⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣢⠖⠈⠤⡀⠀⢀⠀⢈⠁⠀⠲⠒⠒⠂⠄⠢⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⣄⠀⠀⠀⠄⠀⠀⠀⣀⠀⠸⠨⣆⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢱⣦⣄⣀⣀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠱⢄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⡯⠁⠄⠐⢉⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠑⢤⡀⠤⠜⡀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡒⠄⡈⠉⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⠀⠀⡰⠒⠃⠠⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢤⠠⡤⠤⠤⠴⠦⠁⢐⡚⣀⠄⣠⣄⡀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⣎⡁⠀⣄⢀⠀⠄⠈⣦⣤⡀⠀⠀⠀⣐⡢⢩⣍⣭⣭⣴⣷⣤⣀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠻⠒⠚⠁⠀⠀⠀⡀⡤⡀⢹⠉⣉⠭⠭⡉⠉⠉⠉⠉⢉⣉⡉⠉⠉⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠈⣉⡛⠠⢤⢸⠀⣾⡕⠜⠀⠀⠀⡞⢡⣤⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⣷⠀⠓⠀⠚⡠⠔⠢⣄⠘⠬⠽⡣⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠠⠝⠄⠒⠢⡐⡅⢾⢟⠌⠆⠀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⠸⠿⡇⡰⠈⠁⠁⠊⠀⢸⠰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠉⠉⠜⠁⠀⠀⠀⠀⠀⠀⠑⠖⠃

⠀⠀⠀⠀⠀⡠⠄⣂⡈⠉⠉⠁⠐⠠⢀
⢀⡀⠀⣴⠿⠛⠛⠛⠿⢿⣶⣤⡀⠀⠀⠑⢄⠀⠀⢀⠠⠒⠉⠀⠀⠈⠁⠂⢄⡀
⡎⡀⠀⠀⠂⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠣⡔⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠂
⠙⠆⠮⠗⠁⠀⠀⠀⣀⠤⠐⠊⠉⠉⠑⠂⠄⡀⢃⠀⠀⠀⢀⣴⠏⠀⡑⢤⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡀⠀⢀⡾⠟⠈⠁⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⡠⢲⢄⢸⣿⣆⠀⠀⠀⠀⢱⡔⠊⠀⠀⠀⣀⣠⣤⣴⣶⣶⣤⣤⣀⠀⠑⡄
⠀⠀⠀⠀⠀⠁⠀⠠⠊⠀⠀⢠⣿⠋⠀⠁⠀⡠⠀⢄⠀⠀⣀⣴⣿⣿⣿⡿⠟⠛⡟⢛⠛⠿⣷⣦⠘⡀
⠀⠀⠀⠀⠐⠀⠠⠁⠀⠀⡐⠈⠃⠀⠀⣧⡘⠀⠀⡈⠓⠉⠛⣿⣿⡟⠁⠀⡤⠠⢧⠂⠀⠀⠈⠹⣧⡁
⠀⠀⠀⠀⠈⣀⡃⠀⠀⢀⠇⢄⡀⠀⠀⠀⠀⠀⠀⠤⠁⠀⢰⡿⠋⠀⡁⢸⠦⣀⠀⠇⠀⠀⠀⠀⠛⠉⠐
⠀⠀⠀⠠⠊⠉⠑⠀⠀⠎⠀⠀⠀⠁⠒⠠⠀⠀⡀⠀⠀⠀⠈⠀⡐⠈⠀⠀⠙⠛⠊⠀⠀⠀⠀⠠⣄⠄⣤
⠀⠀⠀⠰⣔⠠⣲⠄⠘⢀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠈⠁⠒⣶⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠈⠀⠁⡠⠃⠀⠀⠀⠀⠀⠀⢠⠐⠐⢢⠀⠀⢰⡿⠿⡀
⠀⠀⠀⠀⠀⠀⢰⢁⣤⠀⢰⠉⠀⠈⠑⡄⠀⠀⠀⠀⠀⠿⠁⣱⢈
⠀⠀⠀⠀⠀⠀⠘⠈⣿⡧⡘⠤⣀⢀⡠⠂⠀⠀⢀⠀⠀⠀⠈⢀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⠘⣷⣌⡐⠂⠀⠀⠀⢂⣶⡿⠁⠀⡰⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⢟⠛⠛⢿⣿⠿⠛⠁⢀⠠⠊⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡀⠄⠊⠁⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠁⣠⠖⠛⠛⠓⠲⢤⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⣨⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⠵⠶⠒⠒⠿⠥⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣛⣉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠉⣉⣛⠶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠀⠀⠀⠈⠉⠙⠒⠲⠤⣤⣤⠶⠗⠒⠋⠉⠁⠀⠀⠀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⡟⠒⠂⠤⢤⣀⣀⠴⣿⡛⣦⣀⣀⠤⠤⣴⡖⠿⣽⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⣿⢰⡅⠀⠀⢀⣀⣿⡶⢶⢾⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡿⢎⡇⠀⠀⡯⢾⣿⠇⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⣧⠸⡇⠀⠀⠈⠋⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢇⣙⡒⠦⠤⢄⣀⡀⠀⣿⠨⡇⣀⣀⠤⠤⠖⢒⣋⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⠭⣭⣹⣇⣭⠤⠴⠒⠚⠉⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⣼⣿⣿⣿⡟⠀⠀⠀⠀⣐⣶⣤⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠈⣻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⡂⠀⠀⠀⠹⣿⣧⡀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⣿⣿⠃⠈⠉⠁⠀⠀⠀⠀⠹⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣶⣤⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠘⣿⣿⡆⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⢹⣿⣿⣄⣀⣁⣐⣀⣁⣀⣀⡀⠀⠀⣠⣾⣿⣿⠟⢉⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡰⠛⠿⣿⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣍⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣉⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡿⠿⢛⣇⡀⢤⣶⡆⣶⡦⢀
⠀⠀⠀⠀⠀⠀⣠⡄⢿⣿⢁⣿⣶⠤⣧⣶⡾⠛⠿⣷⣄⢻⠗⠿⢧⣾⢷⡀
⠀⠀⠀⠀⠀⣾⠽⠃⠈⠉⠸⣿⠁⣠⣾⡷⠴⣷⣤⡈⠻⠃⠀⠀⠀⠀⢿⣃
⠀⠀⠀⠀⢠⡍⠀⠀⠀⠀⠀⠀⣼⡿⠋⣠⣶⠿⠛⠛⢛⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⡷⠄⠀⠀⠀⠀⠀⠘⢁⣼⡿⡁⠀⠀⠀⠀⢹⣄⠀⠀⣀⠨⡏
⠀⠀⠀⠀⠈⠟⡋⠀⠀⠀⠀⠀⠀⠸⠏⣴⡁⠀⠀⢠⠖⣶⡘⣦⣤⣘⣑⣷⣽⣤⣤⣦⣴⣰⣒⠆⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣏⣴⣟⣻⡞⣻⠛⢰⡿⠛⣿⣿⣿⣿⣟⣿⣿⣿⣿⡯⠀⠟⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠌⠿⢻⢈⡉⠔⠂⢀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠠⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣊⠉⠡⢀⠀⣴⣦⢻⣷⣿⣿⢿⣿⣿⣿⡯⣿⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣼⣷⠻⣷⣼⣿⣿⣿⣧⣾⣴⣿⣾⣼⣿⣿⠆⠀⠀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⢹⣿⣿⣿⣿⣿⣿⣯⣹⣿⠻⣿⣿⡿⢠⡧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⡿⠛⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⡿⠿⠿⠛⢿⠿⡿⣿⠛⠟⠛⠻⢻⠁⠀⠀⡨⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⡄⡔⠒⢀⠀⣆⢻⡇⠈⠂⠀⢘⢋⡅⢑⣺⡿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠈⠓⠀⠀⠈⠿⣷⡀⠀⠀⠀⠀⠀⠀⡿⣷⣧⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⣔⣿⡁⠀⠀⠀⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠠⡀⢀⠀⠀⠀⠀⠄⠀⠀⠀⢀⢿⢿⡇⠀⠀⢀⡆⠤⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇⡄⠦⠀⠠⠾⣄⣀⣀⣹⡅⢀⠶⠀⢸⡿⣟⣯⣀⣴⣾⠷⡎⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠛⠷⣷⣦⣀⣀⣀⣀⣀⡀⠀⠀⢈⣚⡳⠈⠉⠃⠤⡿⣛⣹⡓
⠀⠀⠀⠀⠀⠀⠀⠀⣐⣳⣾⣇⠐⠂⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⣄⠁⢴⣾⠷⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⠍⠉⠛⠿⠍⠉⠀⠉⠀⠉⠁⠙⠠⠆⠉⠁⠒⠐⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠤⣤⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠦⣀⡴⠋⠁⠀⠀⠀⠀⠙⢦⠀⠂
⠀⠀⠀⠀⢀⣠⡎⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⣄⣀⡀⠀⠈⡇
⠀⢀⠃⢰⠋⠘⡀⠀⢀⠴⠲⠂⠀⠀⠀⠀⠉⠀⠈⢻⣇⣀⡃
⠀⠈⠀⢻⠀⢠⠟⠛⢻⠀⣠⠴⠒⠚⠋⣉⣉⠓⢦⣸⡏⠀⠙⡆
⠀⠀⢀⡔⠓⢺⠦⢤⣾⠏⢠⢖⡕⣄⢸⢡⡬⢣⠀⢹⡳⢦⠞⠁⠀⠄
⠀⠀⠈⠳⠥⠊⣼⣉⡏⣠⠼⢤⡔⠉⠈⢳⡴⢛⠒⡌⣇⡸⠀⠂
⠀⠀⠀⠀⠀⠀⢧⣈⣷⡇⠶⣀⠑⢤⣤⠜⣠⠞⢠⠃⣯⣄⣀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠈⠂⠀⠈⢳⡹⢄⡈⠉⠒⠒⠉⢁⡴⠋⡸⠧⠄⣾⣀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢤⣉⣉⡑⠒⣉⣩⡴⡎⠓⠆⢀⠇⠀⠈⠉⣹⢋⠝⠋⢉⣹⡦⡄⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⢠⠋⡹⠉⠏⠓⠀⢉⡠⠔⠁⠀⠀⠀⠀⡇⢸⠀⠀⢸⡭⠥⠭⠭⠥⠤⠤⣬⢩⢻
⠀⠀⢀⠀⠀⠀⠀⠀⣠⠎⢹⣀⣀⡠⠤⠤⠔⠋⠉⠀⠀⠀⠀⢀⡠⠔⠛⠺⡄⠀⢈⡇⠀⠀⠀⠀⠀⢠⢃⣯⠂
⠀⠀⠀⠊⣁⣀⡶⡞⠁⠀⠀⢈⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠙⠲⠊⠀⠀⠀⠀⠀⣠⠁⣼⠇
⠀⠀⠀⠛⠤⡄⢧⠙⢄⣠⠖⠁⠀⠈⡇⠀⠀⠒⠒⠒⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠏
⠀⠀⢠⠀⠀⡇⠀⠙⡏⠀⠀⠀⠀⠀⢧⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡼⡎
⠀⠀⠀⠂⠀⢧⣀⠞⠀⠀⠀⠀⠀⠀⠻⡀⠈⠀⠀⠀⠀⠈⠀⠀⠀⣹⢀⠀⠀⠀⠀⠀⠀⢀⠎⡘⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣇⠀⠁⠀⠀⠈⣀⡀⠀⠀⣹⣬⠭⠍⣁⣀⣀⡀⠊⡘⡜
⠀⠀⠀⠀⡴⠦⠔⣀⣒⠿⠭⢁⣐⢯⡯⣟⡀⠀⠀⠉⠀⠀⠀⢀⣀⣹⠿⠿⠁⢐⣚⠤⠍⠙⢻
⠀⠀⠀⠀⡇⠖⠂⣠⣤⠤⢉⣉⣛⠛⠙⠓⠚⠯⠭⣭⣩⡭⣙⠩⠕⠒⣐⠡⠭⠒⠒⠃⣭⢩⢸
⠀⠀⠀⠀⡇⡇⠘⠖⢦⢰⠟⢧⡠⣛⡛⣥⠲⠖⠒⠠⡄⢹⠠⠒⢨⢁⣠⠀⠀⣀⠠⣚⡰⠀⢸
⠀⠀⠀⠀⡇⡄⢎⠙⠊⡼⣠⢸⡐⣥⢭⡏⡈⢩⠽⢀⡇⢸⠀⣤⡃⢀⣂⢠⠏⡤⠿⠦⣄⠀⢸
⠀⠀⠀⠀⡇⡇⠀⠙⡛⡿⠹⡭⡓⡶⠾⢷⡟⣦⡇⢸⡇⢸⠀⠃⠐⠆⠀⠸⣖⣓⠖⡆⢹⣀⣸⡀
⠀⠀⠀⠀⡇⡇⠀⠴⠷⠷⢿⡤⢇⣩⣃⡖⣙⢹⢸⡇⡇⢸⠀⠀⠀⡀⠀⠀⣈⠤⡀⣇⣸⠭⠭⣭⡽⢳⠄
⠀⠀⠀⠀⡇⡇⠀⢸⠈⠉⠤⣿⢁⣄⢹⡟⢏⡾⠶⡇⡇⢸⢸⠀⠊⠃⠀⠈⠒⣀⡹⠊⠉⡰⢸⠉⠛⠊
⠀⠀⠀⢠⡁⠓⠦⠤⣧⣭⣤⠾⢬⣁⣼⢋⣠⠠⡎⠀⡇⢸⢸⠄⠣⡌⠦⠀⣉⡤⠥⢒⣨⠥⠚
⠀⠀⠀⠀⠀⠉⠁⠒⠒⠦⢤⣉⣑⡒⠲⠤⢤⢓⣚⡀⡇⢸⢸⣀⡤⠖⢂⣩⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠲⠤⢬⣅⣸⣀⠥⠔⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠲⣄⢀⡤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⢤⣤⡖⢓⡆⠀⠀⠀⠀⠀⠀⣠⠤⢾⠣⠄⣀⡄⠈⠉⠃⠒⠚⠻⡖⠾⠭⣡⣦
⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⢀⡞⠀⠙⠋⠀⠀⠀⠀⢀⡴⠋⠀⢀⡷⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⢓⠦⡀
⠀⠀⣠⠒⢢⣴⡯⠚⠋⠉⠉⠉⠉⢢⡴⣤⡀⠀⠀⢀⠏⠀⢀⠔⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⡄⠈⢢
⡴⠋⠁⠀⢠⡅⣠⠀⠀⢀⠀⠀⢤⠀⠹⡀⣵⠀⠀⢸⢀⡴⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⠀⠈⢷⡀
⡟⠀⠀⠀⣠⠓⠾⠂⢼⣁⡷⠘⠛⠛⠀⡗⡾⠀⠀⢈⠏⠀⠀⠀⣨⠃⡴⣦⠀⠀⠀⠀⠀⠀⢀⡶⢦⠀⠈⠳⢄⡀⢸⡀⠀⠈⣇
⠙⠚⠓⢾⠁⠀⠤⣄⡀⠀⣀⣴⠀⢀⡜⠉⠁⠀⠀⡞⠀⠀⠀⣰⠏⠘⠁⠙⠀⠀⠀⢀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠙⠺⣧⣀⡀⢹
⠀⠀⠀⠘⢦⡀⠀⠀⠉⠻⡅⠘⠲⠔⠒⣤⠀⠀⢰⡇⠀⢀⡴⠁⠀⠀⠀⠀⠀⠘⠧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢹⡖⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠉⠙⠓⣲⠞⠉⠲⢤⣀⡤⠋⠀⠀⠘⣇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⠃⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠳⣄⣟⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⢀⢷⡊⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢠⠟⠈⠙⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠹⣆⣀⡔⠋⠉⣹⠀⠀⠀⠘⢦⣀⣏⣀⣳⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠴⠃⠀⠀⢀⣤⠀⠈⢧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⠒⠋⠁⣠⠴⢦⡀⠀⠈⢻⡉⠉⠉⠙⠲⠦⣤⣀⣀⣀⡀⣀⠀⠀⢀⣀⣀⣠⣤⠴⠖⠋⢳⡀⠀⠀⠀⠘⣄⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣧⡤⠞⠁⠀⠀⠀⠙⢦⣤⠤⠀⠀⠀⠈⠉⢉⣥⠉⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣲⣤⣤⠤⢤⣀⠀⠀⠀⠀⠀⠀⣠⢖⡟⡸⢀⡴⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⣦⣀⣵⣤⠤⠽⣦⠀⠀⣀⡤⠊⠀⠈⢀⡿⠋⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⢤⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡏⠀⠀⠀⠀⠀⢸⠴⠊⠁⠀⢀⡶⠖⢩⡄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠉⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣄⣷⠀⠀⠀⣀⣀⣾⠷⠤⠴⠚⣹⠁⠀⢸⡇⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠁⠀⠀⠀⠀⠀⢠⠿⢤⠀⠈⡇⠀⣤⠤⠤⢤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠋⠀⠀⣷⠀⠘⠢⠴⠋⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡴⠶⡞⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠿⣾⡧⠤⠤⢼⡥⠬⠭⠽⠿⠶⠦⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⢹⡆⡟⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡄⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⢻⡴⠒⠺⠷⣇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣚⣧⣤⣴⣖⣒⣒⣒⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠎⠉⠉⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠂⠉⠉⠉⠉⠁⠲⢄⡀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠑⢄⠀⠈⡀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⡰⠏⠀⠤⠂⡟⠠⠁⠀⠘⢆⠀⠀⠀⠀⠀⠠⡀⠐⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⣀⣀⠌⡠⠖⠀⠀⠈⢀⠃⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡇⠠⢀⡀
⡈⣿⠏⢀⡟⠀⢀⣤⡖⢲⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠈⢢⠀⠀⢸⠀⠀⠀⠐⢄
⢳⡏⠀⡸⡃⠀⣸⡟⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢳⡀⠀⡦⠠⠄⣀⠀⢣⡀
⠀⠉⠉⠉⠀⠀⠈⠓⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⠀⠀⠀⠑⣄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣾⣷⠀⠀⢰⣿⡆⠁⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣄⣥⡀⠀⠀⠀⠈⢠⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠉⠉⢀⠀⡄⠉⠀⢀⡠⠘⠻⠿⠀⣂⠴⣻⣦⠀⠀⣀⡌⠀⣼⡇⠀⣿⡤⠀⠀⠀⠀⠱⣸⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⣤⡤⠤⠤⣤⣤⣀⣀⡀⠀⠋⠁⠀⠻⣟⠴⢋⣽⡆⢀⡹⠁⠀⣿⢠⠀⠀⠀⠀⠐⣷⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣁⠄⠂⠀⠉⠁⠀⠀⠀⠉⠉⠒⠂⠤⢀⡀⠈⠻⠋⠀⠉⠓⠒⠛⠁⠀⠀⠀⠀⠘⠸⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤⠤⠤⠤⠤⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠄⠀⠀⠀⠂⠈⠀⠤⠄⠀⠀⠐⠒⠀⠂⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠴⠆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠠⠤⠤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠢⡌⠀⠀⠀⠀⠀⣀⣠⣄⣶⢂⣤⣄⣀⠀⠤⠤⠐⠚⠁⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢴⡀⠀⠀⠀⠀⡉⢁⣀⣀⠀⠀⠀⠀⠤⠄⠰⠖⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠒⠉⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠐⠠⠤⢀⣀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠃⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠠⡭⣦⣀⠀⠀⠀⠀⣀⣀⡤⠄⠒⠂⠉⠁⠀⠀⠀⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡬⠤⠴⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂⠀⠉⠀⠈⠥⠄⠀⢒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠣⠴⠀⠒⠀⠀⠀⠰⠐⠒⠋⠀⠁⠀⠀⢀⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠲⡀⠀⠀⠀⢰⠁⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠈⢉⠀⣀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠀⡠⣄⣈⡄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡴⠈⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⣀⠠⠄⠖⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠤⠄⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣹⣟⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆⠁⠀⠀⠀⠀⠀⣀⠀⡴⠂⠛⠉⠉⠉⠙⣆⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⢠⠴⠛⠉⠀⠀⠀⠀⢀⡔⢋⡈⠁⠀⠀⠀⠀⠀⠑⠀⠀⠘⣄⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⡎⠀⠀⠃⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠈⢣⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⣳⠀⠰⣿⡇⠀⠀⠀⠀⠀⠈⢋⠀⠀⠀⣸⣀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⢰⡇⠀⠀⠉⡅⠀⠀⣀⣤⣀⠀⠘⠀⣠⠞⠁⠈⡇⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢣⡀⠀⠀⠁⠀⢰⡏⠀⢹⣇⡠⠞⠁⣼⠃⢰⠇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⢸⡋⢙⡒⠦⠤⠤⠀⠹⠶⠟⠋⡠⣢⡾⠃⢠⡟⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠳⣌⠙⢶⣦⣄⣀⣀⣒⣀⣤⠾⠋⠀⢠⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⢬⣷⣤⣈⠉⠉⠛⠛⠉⠁⠀⢀⡴⠋⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠓⢶⣤⣤⣤⣤⣴⠾⠗⠊⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢟⣩⠤⠞⠉⣁⣠⡤⠖⠛⣉⣻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⣤⠴⠖⢛⣉⡤⠴⠚⠋⠉⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠂⠉⢁⣀⠤⠄⠒⠒⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠊⢉⣉⠀⠠⠤⠀⠐⠂⠀⢸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⣀⡀⠠⠤⠤⠄⠒⠒⠒⠫⣄⠉⠉⠉⠉⠛⠛⠒⠒⠶⠶⠤⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠤⠤⠤⠤⠤⠐⠒⠒⠒⠂⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡧⠤⠉⠉⠉⢉⣉⣉⣉⣉⣉⣼⣿⣤⣤⣤⣄⣀⣀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠛⠿⠿⠿⢿⣿⣿⣤⣀⡐⠒⠒⠂⠀⠠⠤⠤⠤⢹⣿⣿⣿⣿⠿⠟⠛⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣤⣤⣄⣀⣀⣀⡀⠀⠉⠉⠉⠉⠛⠒⠒⠒⠶⡾⠛⠉⠁⢀⣀⣤⣴⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⠛⠛⠻⠿⠿⠿⠿⣿⣶⣶⣶⣶⠀⢠⡇⢠⣶⣿⠛⠋⢻⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠀⢸⠇⢸⡿⠋⡴⢡⣰⣇⣭⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠀⣼⡇⠀⠳⠄⢹⠃⡀⠀⠀⠸⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡟⠀⣿⠁⠀⠀⢰⡿⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⣿⣷⣦⣤⣤⣤⣄⣀⣀⣰⣾⣿⠀⢸⡇⢰⣿⣿⣶⣾⠿⠃⡀⠀⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠇⠀⢸⠀⠸⠟⠋⢉⡠⠖⠋⠀⠀⠀⠀⠀⠈⠹⣋⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠒⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡙⠿⢿⣟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢶⣿⣉⡉⠉⠛⢷⣄⣿⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣼⠏⠉⠻⣦⠀⠀⢻⡇⣀⣴⠾⠛⠛⠛⠷⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣿⠀⠀⠀⠸⠏⠁⢀⠀⣠⠶⠛⢿⡏⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠛⢿⣤⡀⠀⠀⠀⠀⠉⠛⠛⢷⣄⠈⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣿⣦⣈⠛⢶⣄⠀⢀⣀⣤⣄⡀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⢶⣄⠀⠈⠙⢷⣤⡙⠻⣿⠉⠁⣉⣿⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠈⢿⡄⠀⠙⣷⡀⠀⠀⠈⠛⠿⠋⠀⢸⡏⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣈⣛⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⣿⣟⠉⠉⠙⣷⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠙⣿⡉⠉⠉⠉⠙⣷⡀⠀⣿⠙⠻⠶⠾⢋⣾⠋⠉⠉⠉⢉⣿⠋
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠈⠻⣄⢻⣆⠀⠀⣰⡟⠁⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⢹⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣤⣤⣤⣤⣼⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣠⠞⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⡴⠋⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⢤⣄⠀⡾⠀⠀⠀⢸⡇⠀⣀⡤⠴⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣄⡀⠀⠀⢀⣨⡽⠷⡶⠶⠶⢶⡷⢾⣅⡀⠀⠀⢀⣠⣽⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⡇⠀⠀⠙⡶⠞⠙⣇⢀⣀⣧⣤⣤⣤⣇⡀⠈⡏⠓⢶⡋⠀⠀⢸⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⢧⣠⠴⠛⠉⠁⠀⠀⠀⠀⠀⠉⠛⠧⣤⣼⠇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⡇⠀⠓⢦⣤⡀⠀⠀⣠⣴⡖⠃⠀⢻⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡇⠀⠉⣉⡁⠀⠀⠀⠀⢈⡉⠁⠀⢸⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠉⣿⢻⠻⡏⠉⠁⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⢦⠀⢠⠞⠙⠲⣤⣀⡙⠒⠚⢁⣀⣤⠞⠉⠳⡄⠀⡴⢷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⣸⠗⠿⣄⣠⡾⠁⠈⡟⠉⠉⣯⠀⠈⢷⣄⣠⠿⠲⣇⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠲⢤⣴⢷⣄⡴⢷⣀⣠⠿⢤⣠⣾⣀⡤⠶⠚⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣠⢬⡁⠀⠀⠀⠹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⠣⡤⠏⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠤⣴⡶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣛⣛⣾⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢀⡴⢻⡇⣤⣶⣒⣳⣤⠤⠞⠉⠀⣠⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠒⢲⡟⠒⠚⠓⠒⠛⠛⠒⠛⠒⠒⠒⣾⠓⠒⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⡾⠀⠳⣤⣀⣀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢧⡀⠀⠀⠀⢀⡴⠋⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡟⢀⣀⣀⠸⡆⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠟⠋⠁⠈⠛⠃⠀⠀⣿⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠋

⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠈⠉⠉⠉⠉⠐⠒⢤⡀
⠀⠀⠀⠀⢀⣴⣿⣶⣶⣿⣿⣶⣶⣤⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣀⠴⠆⠀⠀⠐⠦⠄⣀
⠠⠖⠈⠈⠻⡉⠁⠀⠀⠀⠈⠉⠛⢿⣿⣷⣄⠀⠀⠀⠈⠳⡀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⣇⢠⡀⢀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡄⠀⠀⠀⠑⡴⠁⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠁
⠘⠷⣠⣯⠷⠁⠀⠀⠀⠀⠀⣀⠤⠐⠒⠉⠉⠉⠙⠒⠠⢄⠀⠘⡀⠀⠀⠀⠀⢠⣾⠏⠀⠈⠑⣤⣀⣀⣀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡇⠀⠀⠀⢠⣿⡿⠒⠂⠉⠁⠀⠀⠀⠀⠈⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⣠⠀⢶⣦⡀⠀⠀⠀⠀⠀⠹⡀⠀⡠⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡠⠚⢩⠒⡆⣾⣿⠷⢄⠀⠀⠀⠀⠀⢱⠊⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠊⠀⠀⠸⠀⣿⡿⠁⠀⠈⡆⢀⡴⠂⠲⡌⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠟⠋⠙⡏⢉⠛⠛⠿⣿⣦⡀⢡
⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⠀⠀⠀⢰⠀⠛⠁⠀⠀⢰⡀⡎⠀⠀⠀⡇⡷⠚⠛⢿⣿⣿⣿⠟⠁⠀⢀⡀⣠⣅⠎⠀⠀⠀⠈⠻⣿⡌
⠀⠀⠀⠀⠀⠀⠇⠀⡆⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⡘⠀⠇⠀⠀⣸⣿⡿⠓⠲⡄⢰⣥⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⠿⠖⠤
⠀⠀⠀⠀⠀⢀⡴⢦⡄⠀⠀⠀⡼⠁⠑⠢⠄⣀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠠⡿⠋⠀⢀⠔⠃⠸⣏⣖⣆⣠⠃⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠰⠁⠀⠀⠈⢣⠀⡘⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢶⣘⠠⣞
⠀⠀⠀⠀⢠⣔⡆⠔⣢⡜⢀⠃⣀⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⢶⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠉⠃⠘⠋⠀⢸⠀⠁⠀⠀⠈⠆⠀⠀⠀⢀⡀⠄⣀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⡠⠄⠐⠠⢄⡀⠃⠀⠀⠈⢣⠀⠀⣸⡟⡡⢌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣶⣇⠀⢸⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠟⠁⡠⢾⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢹⣿⣦⡈⠢⢀⣀⣀⡠⠞⠀⠀⠀⠀⣀⠀⠀⠄⠈⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢿⣦⡉⠒⠤⠤⠀⠀⠠⠤⠄⣶⣾⠟⠀⠀⢠⡄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⢿⣿⣿⣶⣦⣦⣤⣶⣶⣶⠟⠁⠀⠀⡠⠋⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⠓⠤⠀⠀⠽⠿⠛⠉⠀⢀⡠⠔⠊⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⡀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠓⠈⠁⢀⡤⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠉⢳⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⣀⣠⠴⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠶⠚⠛⠉⠉⠉⠙⠒⠲⡾⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣏⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠈⣉⣉⣷⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠉⠉⠛⠓⠶⠦⢤⣄⣀⠀⢀⣠⣾⠥⠶⠒⠛⠋⠉⠁⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⡶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣨⢿⢯⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⡶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⠀⠀⠈⠙⠛⠲⠶⠴⠞⣿⢸⢸⡟⠦⠴⠶⠚⠛⢱⡿⠓⠚⠳⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⡇⠀⠀⠀⢀⣀⣸⣿⡞⢳⠛⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⡇⠀⠀⠀⡟⣴⣛⣻⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢘⢸⡇⠀⠀⠀⠱⠾⠟⠉⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣸⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⣤⣍⣙⡓⠲⠶⢤⣤⣀⣀⣻⢸⢸⣇⣀⣤⡤⠶⠖⢛⣉⣩⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠦⢬⣍⣉⣸⣀⣉⣥⠤⠶⠒⠛⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣀⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡤⠤⣄⡀⠀⣠⠔⠚⠉⠀⠀⠀⠉⠳⣄⠈⠢
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠉⠹⠇⠀⠀⠀⢀⠀⠀⠀⠀⠘⡆
⠀⠀⢠⠀⢀⠴⢻⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠠⠞⠒⢶⣥⠀⠀⡇
⢀⠀⠇⢠⡏⠀⢸⣄⣀⣠⠞⠉⠉⠀⠀⢀⣀⣀⣀⣀⠀⠀⢻⣧⠤⢧⡀
⠀⢰⠀⠈⡇⠀⢱⠋⠉⢹⡆⡠⠔⠚⠉⠉⠁⣠⣤⡈⠙⢦⣸⠃⠀⠀⣿
⠀⠘⠀⣤⠚⠙⢿⠦⢤⣾⡟⠀⡴⢩⡑⣆⢸⠁⣶⠜⡆⠀⢻⡻⢶⠞⠁⠘⠠
⠀⠀⠀⠛⠬⠴⢃⣞⡉⣿⢀⣠⢧⣀⡞⠉⠈⠑⡤⠞⠓⢲⡘⡇⣸⠃⠐
⠀⠀⠀⠀⠀⠀⠘⣇⠈⣇⡞⢠⡄⠈⢧⣀⣀⣰⠃⣰⠏⢸⢃⣏⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡀⠣⡀⠈⠛⢿⠱⣄⠈⠓⠦⠬⠭⠴⠚⠁⣰⠏⢸⣉⡉⣹⠂⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⣈⠓⠢⠤⢤⣤⠤⠴⠚⣁⠴⣏⠀⠀⣿⠑⠒⠲⠦⢤⡴⢢⡤⠴⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠳⠶⡤⢤⡤⡴⢶⠛⠹⡄⠀⢁⡴⠃⠀⠀⠀⠀⡏⣰⠃⠀⢠⡶⠾⠽⠦⠤⠤⠤⠤⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠞⠀⠚⠀⠁⠈⢀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⣿⠛⠛⠒⠒⠒⠒⠒⢲⠆⡾⡜
⠀⠀⠀⠄⢀⠀⠀⠀⠀⣠⠞⠀⢧⣀⡠⠔⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⠉⠙⣆⠀⠀⡽⠀⠀⠀⠀⠀⠀⢠⠊⣼⡿⠁
⠀⠀⠀⠐⢁⣀⣠⠖⡎⠁⠀⠀⠀⣀⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⠓⠒⠁⠀⠀⠀⠀⠀⢠⠆⢰⣯⠁
⠀⠀⠀⠘⠣⢤⠘⣆⠙⢤⣀⡴⠊⠁⠀⠈⡇⠀⠀⠐⠂⠒⠒⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠿⠁
⠀⠀⠀⡀⠀⢸⠇⠈⠓⡶⠋⠀⠀⠀⠀⠈⢇⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⠣⠁
⠀⠀⠀⠆⠀⠘⡆⠀⣰⠃⠀⠀⠀⠀⠀⠀⣾⠀⠠⠤⠤⠤⠤⠄⠒⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⢣⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⠒⠒⠒⠀⠤⠄⠀⢀⡞⢠⢣⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⢴⠲⢲⣾⡯⢿⠀⠀⠀⠤⠤⠤⠐⠂⠀⠀⠀⡿⠭⣭⠍⠒⣒⡒⠦⠤⢤⣧⠃
⠀⠀⠀⠀⢸⠛⠛⠂⠶⠬⠍⠓⣆⣤⣌⣓⣯⡿⢤⣀⣁⣤⣤⣤⣤⣤⣤⣤⠤⠟⢛⣉⣀⠨⠤⢒⣚⣩⣍⢸
⠀⠀⠀⠀⢸⠐⠋⢁⠤⠶⠶⠀⣤⠤⠍⣉⣈⡙⠋⠑⠒⠒⠶⢴⣒⠪⠀⠒⢋⡃⠩⠐⠒⠚⠉⠉⢠⠆⢰⢸
⠀⠀⠀⠀⢸⠀⠀⠘⠒⢲⡀⡟⠉⣆⠔⣩⡝⢦⠟⡟⢒⠒⠠⡄⢸⠀⠎⢉⣄⣀⣤⠀⠀⣀⡠⣖⣁⠜⢀⢸
⠀⠀⠀⠀⢸⠀⠀⢏⠹⠞⢰⠃⡄⣿⠸⣅⣭⡿⠀⠉⣩⠷⢸⠃⢸⠀⣂⡘⠀⣁⡇⢠⡏⣠⠼⠦⢤⡀⢈⢸
⠀⠀⠀⠀⢸⠀⠀⠈⠙⡾⢫⠟⣯⣿⢦⣤⠴⣧⡼⠦⢵⠄⢸⠀⢸⠀⡏⠉⢰⠉⠁⢸⡤⠷⡶⢦⠀⡇⢸⢸
⠀⠀⠀⠀⢸⠀⠀⠀⢀⣧⣼⣰⣎⢀⡆⢓⠆⣤⠟⠞⡏⢰⠸⡀⢸⠀⡇⠀⠀⠀⠀⠀⠙⢋⠀⢸⠀⣿⣛⣽⠓⢲⠦⡄
⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠶⠈⢙⡤⠒⠚⢷⣧⣼⢦⣧⣸⠀⡇⢸⠀⡇⢀⢴⠀⠀⠰⣆⠁⠲⣜⠓⠯⣬⢭⠭⢿⣤⠇
⠀⠀⠀⠀⢸⢐⡄⠀⠈⡇⠤⠍⠙⣧⠸⠗⢸⣧⠈⠞⢉⡽⠂⡇⢸⠀⡇⢀⠀⢆⠀⠀⠤⠚⠲⠁⣀⠤⠇⣶
⠀⠀⠀⠀⠺⠤⣉⣉⡓⠚⠿⠶⢏⣙⡲⠖⠻⠥⠔⢆⠙⣄⠀⡇⢸⠀⡇⠈⠑⠒⢛⡠⠤⠒⢊⣡⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠦⢤⣬⣉⣙⡒⠲⠮⠭⣭⣀⡇⢸⠀⣧⠴⠚⢋⣡⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠶⠤⣥⣬⣆⠤⠖⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣄⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠀⠀⠀⠉⠻⣷⣄⠀⠀⢀⠔⠁⠀⠀⣀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠂⠀⠀⡠⠀⠂⠉⠉⠁⠀⢈⠀⠀⢀⣾⡁⠄⠆⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢠⢀⣦⠀⠀⠀⢁⡀⠚⠁⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⠋⠁⠀⠄⢀⡆⣀⣴⣿⣿⠿⠛⠛⠛⠻⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠂⠀⠀⠁⠈⣿⠟⠁⣠⡀⠑⠀⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⠁⠔⠀⠈⠒⠂⠀⠀⠀⣥⠄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠶⢀⢦⣿⣵⢃⢀⡄⠡⡀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠕⠓⠚⢮⡛⠆⠀⢠⠏⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡐⠄⠀⠘⠀⢀⡀⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⣤⣤⣤⣤⠟⠁⠰⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠌⡀⢉⠉⠀⠀⠁⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⡄⠒⠒⠠⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠤⢤⣘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠊⠅⢀⡀⠀⢀⣔⡁⠨⠐⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡶⠀⠤⠀⠠⢿⢫⠀⠀⢤⡴⢶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⢠⣼⡟⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢸⠀⠀⠈⠉⢀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠤⠐⢺⠐⠀⠄⠑⠂⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⣀⣀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠀⠀⠀⠈⠓⢤⠀⠀⢂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠲⠄
⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡀⡤⠤⢀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠰⠋⡠⠐⠀⠌⡐⠀⠀⠈⠂⠀⠀⠀⠀⠀⠂⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠠⣴⡖⢺⡄⠉⠀⢀⣀⠰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⡀⠀⠀⢣⠉⠒⠀⡀
⣧⡟⠁⣾⠇⠀⣾⠏⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⢤⠀⠈⣀⢀⠀⠈⠢
⠈⠓⠶⠒⠀⠈⠻⣤⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣂⠀⡇⠀⠈⠰⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⣴⣦⠁⠀⠀⣤⣄⠀⠰⠀⠀⠀⠀⠀⠀⡰⠉⡀⣃⠀⠀⠀⠈⠣⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠈⠛⠁⡀⢀⠉⠁⠀⠀⠘⠻⠿⠀⣂⡔⣶⡀⠀⠀⡀⠁⣰⡏⢉⣷⡄⠀⠀⠀⠹⣌⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⠬⠤⢄⣀⣁⣀⡀⠀⠊⠁⠘⢿⡣⠔⣹⣇⠀⡹⠁⠈⡟⡀⠀⠀⠀⠀⣿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢉⡠⠐⠀⠉⠉⠀⠀⠈⠉⠁⠒⠢⠄⣀⠀⠹⠋⠀⠉⠓⠚⠋⠀⠀⠀⠀⠀⠻⠦⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠒⠒⠂⠐⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠑⠠⠄⠂⣁⠤⠀⠒⠀⠀⠀⠈⠁⠈⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠈⠉⠒⠒⠂⠀⠤⠤⠤⠀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢄⠀⠀⠀⠀⣠⠄⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡁⠀⠄⣘⠁⠀⠀⠀⠀⢀⣀⢠⡤⣀⡀⠀⣀⣀⣀⠤⠜⢁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡀⠀⠉⠉⢉⠉⣉⣉⡉⣉⠉⠀⠠⠄⠠⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠐⠶⠄⣄⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⠆⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠉⠙⠒⠠⠴⣦⡄⠀⠀⠀⣀⠠⠤⠒⠒⠈⠁⠀⠀⠀⠀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠄⠠⠄⠓⢒⡉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⢸⠀⡀⢀⡀⠀⠀⡈⢀⣀⡄⠸⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢠⠆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣄⠀⠀⠀⠀⡰⠃⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠐⠒⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⡆⠀⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢢⡘⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠐⠉⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣽⣿⣷⣞⣨⣿⣹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠂⠀⠀⠀⠠⠐⢢
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⠋⠃⢢⠳⠀⠀⠀⡄⠀⠀⠀⠀⢠⣿⣿⠉⠈⠹⠃⠀⠀⠀⠀⠀⠀⠀⠂⠘⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣤⣬⣼⠆⣆⠀⢈⡀⠀⠀⠀⢸⣿⣵⢶⣴⣶⣶⣶⠀⠀⠀⠀⠀⠈⠀⠀⢠⣾⡼⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠍⠉⠉⠀⠉⣇⠀⠘⠓⠻⡇⠀⠀⠀⢸⡟⣿⢸⣿⢽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠞
⠀⠀⠀⠀⠀⠀⠀⡀⢺⡋⠀⢠⡴⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢹⠘⠎⠸⢟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢾⣧⠤⠴⠖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠘⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢶⡒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣸⣀⡱⣤⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣘⣉⠃⠀⠀⠀⠀⠀⠀⠀⠈⢠⣪⢾⡧⢁⡕⡂⠁⠀⠙⠀⠀⠀⠀⠀⠀⠠⠤⡤⢤⡀⣙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣬⣥⣤⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠇⣁⣴⣾⣿⡿⠖⠂⠀⠀⠀⠀⠈⠈⢹⣿⡇⠰⠷⣴⠶
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡷⣾⣿⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣯⣿⣽⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣉⠿⢙⣿⠙⢉
⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⠿⠿⡟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣾⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠿⠖⠛

⠀⠀⠘⠈⠁⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢲⠀⡤⣤⣤⣤⣤⡀⠀⠰⠦⡀⠀⠀⠀⠀⠂⡇
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⡠⠚⠄⠐⠁⢈⣿⡟⠅⠀⠀⢤⠀⠀⠂⠄⠈⠀⠄⠁
⠀⠀⠀⠐⠄⠀⠁⢀⠀⠀⠀⠀⠀⠀⢰⣄⠒⠁⠀⠀⠀⠒⠐⠈⢻⣷⣦⡀⠀⠘⣇⣀⠀⠡⠈⠀⡄
⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠶⠆⠀⠀⠀⢀⣾⡟⠝⠃⠑⢤⣼⣿⡇⠀⠈⠀⡀
⠀⠀⠀⠀⠁⠀⠄⣀⠀⠤⠤⠀⠀⢠⠉⢸⣿⣦⣄⣀⣀⣤⣶⡿⢿⣷⢹⠄⠀⠀⠙⡛⠛⠄⠀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣄⠀⠐⢦⠀⢈⣹⣇⢹⣿⣻⢿⣿⠁⢈⣿⡿⠒⠀⠂⢀⣀⣤⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⡴⠿⠛⠹⠈⠁⣿⡈⠉⠀⠈⢿⡇⠀⣠⣾⣿⣿⣿⡇⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠡⠀⠘⣿⡅⠀⠀⢐⣿⣿⡀⠛⠉⠈⣿⣯⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⢳⡶⣿⣤⣀⠀⠸⣿⣿⡅⠀⠀⠀⣿⡟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠬⣯⡙⠶⣭⣿⠿⣿⣿⡿⣧⣤⣤⣤⣭⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣖⠒⠷⣆⡙⠶⣮⣐⢯⡒⠬⡙⠒⠿⠏⠉⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣚⠛⠲⢬⣙⣻⣾⣟⢿⣿⣵⣟⠶⠀⠈⠐⠀⠀⠂⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠒⠛⠒⠠⢤⣍⣙⣯⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡭⠝⠛⠛⠛⠛⠻⠿⠛⠛⠿⢿⣿⠀⠀⠀⠀⠀⠀⠠⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠛⠒⠒⠒⠶⠶⠒⠲⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠈⠁⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠠⡀⢠⠤⣔⣂⣀⡉⠉⠉⠐⠂⢄⠀⠀⠀⠀⠀⠀⢖⠒
⠀⠀⠀⠀⠇⠀⢧⣖⡼⠀⠀⠉⢿⠀⠀⠀⠠⠷⣄⠀⠀⠀⢀⠴⠊⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⡀⢀⣩⣽⣤⡀⠀⠋⠀⠀⢠⢐⡠⠬⣤⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡭⠔⠒⠚⡉⣉⣩⣙⢿⣦⡐⣤⠞⠁⠀⠀⠀⠝⠛⠢⠠⠠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⣴⡏⢀⠔⢓⠧⣭⡟⠡⠤⠈⠻⣿⣿⠃⠀⠀⠀⠀⠀⠢⣽⠖⠑⠘⠀⠀⠀⢀⡀
⠀⠀⠀⡠⠊⣄⣠⣴⣤⡰⣤⡅⠁⠀⠀⣤⡄⠀⠘⠗⡠⡀⢸⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⠀⡀⠐⠬⠲
⠀⠀⢠⢡⠞⠁⠀⠀⠀⠈⠛⢧⣂⠠⣆⠄⠰⣥⡼⠀⡹⠁⣸⠁⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃⠴⢋⣸⠁
⠀⠀⣼⠛⢢⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠓⠤⠄⠤⠖⢁⣠⣿⡦⣀⢀⣀⣀⡠⠄⠒⠋⠄⢼⡄⠀⣉⣉⡍⠃
⢂⡀⠛⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣶⡤⡤⡴⢾⡟⢷⣮⣻⡟⡠⠀⠠⢂⢪⠣⠁⡘⣜⡟⠋⠓⠢⠝
⠀⠀⠀⠀⠀⠀⠀⠠⢐⠢⠀⡠⠔⠘⠶⠫⡼⢇⡿⣼⠻⠾⢃⠌⠔⠐⡁⢡⡢⣣⡡⠖⢛⢍
⠀⢰⡈⠢⣀⡀⠤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⡠⢢⢊⠈⠔⡰⡿⠋⠁⠀⠈⠂⠁
⣠⠀⠜⠂⢨⡇⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠑⢞⠔⡡⡡⠠⠪⢊⣷
⢈⡍⢛⡁⢈⡇⠀⠀⢠⣠⡤⠖⡮⡂⠀⠀⠀⠀⠀⢀⠭⣊⢔⢔⠕⡁⡝⠘⡇⠀⡂
⠀⠉⣜⣡⠇⠀⠓⠉⠁⠀⠀⠀⡯⡠⣀⡀⠀⠀⡀⠘⢺⡡⡠⠢⢎⣮⠞⠋⠀⠀⠁
⠠⡑⠌⠁⠀⠈⠁⠀⠀⠀⠄⠁⠙⠛⠛⠵⠷⢾⣷⠷⠿⡾⠛⢻⡝⠣⢂⣭⣉⣍⣒⣒⣒⣲⠤⢤⣀⡀⢄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢀⣔⡋⢁⢁⠀⠄⠄⣨⡵⡎⡻⡿⠉⠉⠙⠛⢻⠃⡶⠨⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠀⢉⠗⠀⠈⠀⠀⠈⠙⣷⣼⡏⢓⡀⠀⠀⠈⢰⢕⠪⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠛⠢⡶⠖⠚⠀⢾⡟⣯⡼⢋⠁⠄⠀⠀⠇⣎⢕⠾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢎⠁⢀⡀⠀⣀⣨⣻⣗⡂⠤⡄⠀⢀⢰⡫⢔⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣲⠶⠋⠀⠁⠀⠀⠙⣿⣿⣿⣿⣯⣿⣶⡊⣼⠨⣑⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣟⣚⢩⢉⣿⠛⠛⠛⠋⠻⣿⢟⣋⣭⣽⣿⣿⣿⡾⡟⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠚⠻⠩⠭⢛⣓⡲⠦⢦⣢⡶⢞⡯⡿⣛⣹⣡⢖⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡟⠻⠓⠲⠶⡤⣤⣉⣉⠓⣶⣿⠖⡟⠙⡵⢃⢘⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⣀⣀⣠⠳⡀⠀⠀⠉⠃⣿⣽⡌⢢⣪⢪⣼⣨⣿⣯⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠈⢣⡀⠀⢉⡱⠂⠀⠐⢸⣿⡆⡡⢧⣿⢿⢿⣿⢹⣇⢤⣤⣭⣶⠒⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣄⣼⡔⠓⢌⡄⠀⠀⠀⢸⡷⢨⠾⣓⣽⣶⢿⣿⣮⣯⣧⣠⢦⢀⡯⠥⠤⠤⠭⠅⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣍⣉⠓⠚⠯⢝⣖⣦⣤⢰⣿⣴⣟⣋⣽⣷⣿⣷⣒⣒⡊⠛⠛⠛⢁⠤⠤⣤⡤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠂⠬⢁⣸⣏⣶⠛⢛⣂⣶⣶⣆⠬⡭⣭⠉⠁⠦⠀⠒⠒⠚⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⢉⠟⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣰⠃⠀⢰⠇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡖⠉⠀⠀⠀⠉⠉⠒⠚⢋⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠽⠋⠉⠙⠲⢤⡀⡞⠡⡌⡁⢘⠙⡧⣄⡀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢇⡀⠀⠓⢟⡴⠃⢈⣩⠧⠤⠤⠤⢶⠁⠛⣛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣽⡶⢀⠈⠉⠉⠉⠀⣀⣈⣧⠀⠠⡀⠀⠈⡇⠀⠷⡃
⠀⠀⠀⠀⠀⠀⠀⠻⡍⣹⡋⠒⢤⠖⠋⠀⠈⠩⣀⡤⠶⢄⣰⠳⠤⡄⠀⠀⢹⡶⠒⠋⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡋⠹⢥⡄⠀⢣⢀⣀⡤⣄⠀⠀⠀⠀⠀⠑⢶⠃⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⢷⡠⠏⠉⠁⠀⢸⡄⠀⠀⠀⠀⠀⢰⠉⡂⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⣀⣀⣠⡤⡇⠀⠀⠀⠀⠀⠀⢹⢄⡀⠀⠀⢷⠶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠯⣄⣀⠘⠷⣤⣄⣀⣀⣀⡀⢱⣤⣃⣀⣠⣼⣟⣀⣤⠤⠿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠙⠓⠦⢤⣀⡉⣉⣉⣭⡭⠷⠖⠛⠉⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣴⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡶⠦⠼⠁⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡾⠙⢧⡀⣀⣀⣀⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆⠀⠀⠘⠢⣄⠀⢸⠀⠀⢀⣠⡤⠾⠁⠀⠈⠉⠉⣩⠏⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⢀⡼⠛⢸⠀⠘⠻⣤⣀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⠖⠲⣄⢸⠀⠀⢸⠀⠀⠀⠀⢹⠁⠀⣠⣄⣀⢸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠻⡄⠀⢸⠀⠀⠀⠀⣿⡴⠛⠁⠈⠉⠙⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⢸⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣼⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⣀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣶⣿⠉⣷⣄⠀⠀⠀⠀⡧⣿⢯⠷⡾⡿⢭⠂⠀⠀⠀⢾⡟⠉⠉⠉⣯⡆⠀⠀⠀⢀⣂⣂⣉⠈
⠀⠀⣿⣤⣿⣻⣏⣸⡆⠀⠀⠀⢈⣿⣯⣤⣼⣿⣁⠀⠀⠀⠀⡱⠆⢸⡷⠀⠶⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠹⢧⣼⣏⣨⠟⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠻⠀⠀⠀⠀⢾⣧⣀⣀⣠⣟⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⣰⠁⠀⢘⡄⠀⠀⠀⢸⠙⠋⠋⢳⠃⠋⠋⢻⡇⠀⠀⣼⣛⣛⣛⣻⡄⠀⠀⠀⡠⠴⠟⠙⠋⠹⢶⣄⡀
⠀⠀⡼⡌⠀⠀⠀⡿⣆⠀⠀⢸⢀⣀⡀⣰⡀⣀⣀⣸⡇⠀⠀⠀⠙⣤⡜⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠘⠚⠒⠒⠒⠒⠚⠚⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠠⠊⠀⠈⠢⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣀⣀⢀⡀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⣀⡠⠤⣲⣷⠀⠀⠀⣴⠤⣤⢦⡤⢶⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠻⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⣭⣶⣾⣿⣿⣿⠀⠀⠀⢹⠀⣿⢸⡇⣾⠀⠀⠀⠛⢷⣾⢫⡙⣶⠟⠃
⠀⣴⡇⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠋⡟⠻⢿⣿⣿⣿⠀⠀⠀⢸⡆⣿⢸⡇⡿⠀⠀⠀⠀⢀⣿⡿⢿⣧
⠀⠉⠉⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠈⠉⠀⠀⠀⠘⠧⠿⠼⠶⠇⠀⠀⠀⠀⠚⠁⠀⠀⠙

⠀⠀⢘⣆⢠⣾⠆⠛⠋⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠠⣾⣆⣀⣀⠀⠀⠀⠀⠸⠿⠃
⠀⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠀⠀⠀⢀⡤⡞⠿⠗⠀⠀⢀⡾⠿⠂⡼⠿⠆⠀⠀⣾⠁⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠀⣀
⠀⠀⠀⢀⡴⣦⠀⠰⣞⡇⠀⠀⣵⠏⠀⠀⠀⠈⠙⠁⣤⠀⠀⠀⡜⠁⠀⡼⠁⠀⠀⠀⠠⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣹
⠀⠸⣿⠞⠚⠁⠀⠈⠃⠁⢀⡞⣽⠶⢶⡄⠀⢀⣤⣴⠃⠀⠀⠀⠉⠙⠚⠁⣀⠀⠀⠀⠀⠀⠀⡤⢄⣰⣩⠆⠀⠀⢠⣶⣦⠀⢰⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠸⣆⠧⢁⣸⠇⠀⠈⠙⠃⠀⡠⠤⡀⠀⠀⢀⣴⣏⠀⠀⢀⡤⣄⠀⠙⠊⠉⠀⠀⠀⣠⠏⠈⠁⠀⠈⠁
⠀⠀⠀⠯⠵⠫⠖⠃⠀⠀⠀⠈⣙⠛⠁⠀⢀⠀⠀⠀⠀⢻⠉⠀⠀⠀⠻⣴⠋⠀⠀⣸⣼⠃⠀⢀⣶⠀⠀⠀⣀⠀⢀⣴⣄⠀⢠⣖⡦
⡀⠀⠀⣀⠀⣀⡴⠛⢦⡀⠀⠀⠉⠋⠀⢠⠏⠉⢹⠃⠀⠸⠀⠀⢀⣀⠞⠁⠀⣰⢞⣽⣄⡀⠀⢘⣧⠀⠀⣼⠻⠃⡼⠉⠁⠀⠸⡇
⠃⠀⠀⠻⠟⢡⣤⣠⠜⠻⣄⠀⠀⠀⣀⡼⢀⣄⡟⠀⠀⠀⣠⣄⠻⡟⠀⠀⠰⡇⠏⠀⢨⡇⠀⠘⠀⠀⠰⢿⣗⣺⠃⠀⣼⠃⠀⠃⠠
⠃⠀⠀⠀⠀⠈⠛⡁⠀⣠⠞⠁⠀⠘⠿⠃⠈⠛⠁⠀⣴⡆⠻⣅⠀⣹⣆⠀⠀⢙⠲⠶⠛⠁⠀⡤⣤⠀⠀⢠⣄⠀⠀⣸⣧⠀⠀⠀⢰
⠀⣠⣖⠁⠀⠀⠘⠿⠋⢁⡀⢠⢞⡆⠀⠀⠀⠀⠀⠀⠹⣄⢀⡼⢾⡥⠞⠁⠀⠙⠓⠁⠀⡔⢠⡟⠁⠀⠀⢸⡸⠃⠀⠈⠉
⠈⠁⠋⣀⣀⠀⠀⠀⠀⠸⠿⠋⠉⠀⠀⠀⣠⢾⡅⠀⠀⠈⠯⠖⠋⠀⠀⠀⠀⢀⠀⠀⠀⠈⣉⠀⠀⠀⠰⣉⠇⠀⢠⠏⠙⢲⠆
⢀⡀⠈⠓⢮⣇⠀⠀⠀⠀⢀⣠⣴⣦⠀⠀⠀⠉⠀⢠⡤⠤⣤⠀⠀⠀⣀⣤⡞⠉⢳⡀⠀⠸⡯⠭⢽⠀⠀⢀⣤⣴⠋⣀⣠⠏
⣻⠏⠀⠀⠸⡿⠃⠀⠀⠐⠋⠀⠈⠉⢠⣶⡄⠀⠀⢸⠋⠉⣿⠀⠀⢺⡁⠀⣳⡀⢀⣷⡄⠀⣷⣦⢸⡇⠀⠀⠉⠁⠈⠛⠋⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⢠⣶⡆⢸⠉⠀⢀⣤⡿⠰⣿⠇⠀⠀⠀⢳⡐⠿⠃⠈⠛⠁⠀⢀⠀⠀⠿⠗⠀⠀⠀⠀⡞⣇⠀⠀⠉⠋
⠀⠀⡀⠀⠀⠀⠀⠘⡷⢄⠀⢸⡥⠤⣾⠀⠀⠀⠉⠁⠀⠀⣤⣆⠄⠀⠿⠗⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣤⡀⢈⠽⡄
⠀⣷⣿⠄⠀⠀⠀⠀⠛⠃⠀⠈⠛⠉⠉⠀⠀⠀⢀⣀⠀⠘⠱⠋⠀⠀⣤⡄⠀⠺⣝⣦⠀⢠⣶⡇⠀⢠⣶⡄⢸⠋⠁⠈⠊⠁⠀⣄
⠀⠉⠈⠀⠀⠀⢠⡒⡄⠀⠀⠀⢠⠖⣛⣛⡶⠾⡥⠾⠃⠀⠀⢠⣦⠀⡟⠁⠀⠀⠀⣿⡆⠀⠉⠀⠀⢸⣉⣤⣿⠀⠀⠀⠀⠀⣿⠋
⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⢀⣀⢿⠈⠁⢛⣷⠀⠀⠀⡄⠀⠀⢸⡁⠀⢻⠀⠀⠀⠀⠈⠀⠀⣤⣄⠀⠘⠒⠒⠋⠀⠀⠀⠀⠟⠙
⠐⣿⡟⠀⠀⠀⠘⠃⠀⠀⠈⠫⠘⠛⠲⠛⠁⠀⠀⣼⣥⡀⠀⠘⣧⠴⠞⠀⠀⣀⡀⠀⠀⠀⣿⣉⠀⠀⠀⢀⡄⠀⠀⢠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠛⠒⡆⠀⠀⠀⠀⠀⠀⠀⠸⢧⡞⠀⢠⣦⡀⠀⠀⠀⡴⠛⠛⠀⠀⠀⠸⠜⠀⠀⣰⡟⠁⠀⠀⣾⠀⠀⡇
⢠⣶⡄⠀⠀⠀⢰⠇⣀⣸⠃⠀⠀⢀⣀⠸⣿⠃⠀⠞⠀⠀⣀⡯⠇⠀⠀⠘⠁⠀⠀⠰⣖⢦⠀⠀⠀⠸⠉⠃⠀⠀⠀⢿⣶⠀⣇
⠋⠉⠁⠀⠀⠿⠟⠀⠻⠛⠀⣠⣤⠸⣟⠀⠘⣦⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢀
⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠙⣏⠀⢈⣶⣚⡭⠇⠀⢠⠤⣄⡴⣻⠆⠀⢰⠯⣟⣷⠀⠈⠛⡁⠀⠀⢸⡭⠃⠀⠀⢀⡞⠓⢦⣤⣀⠈
⠀⢀⡤⠖⢿⠉⠈⢧⠀⠀⠀⠀⠈⢿⣭⠞⠉⠀⣀⠀⠀⠉⠁⠉⠁⢀⣀⡏⣠⣤⡏⠀⠀⣿⣿⠀⠀⠸⡇⠀⢠⣄⡾⢀⣄⡾⠀⢉⡟
⣀⣈⣧⣀⣈⣧⣀⣨⣇⣀⣀⣀⣀⣀⣀⣀⣀⣼⣻⣇⣀⣀⣀⣀⣀⣻⣟⣀⣈⣉⣀⣀⣀⣘⣀⣀⣀⣀⣁⣀⣈⣛⣁⣘⣛⣁⣀⣼⣁
⣿⣿⠋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣤⠤⣄⡀⠀⠀⠀⠀⢠⣺⣭⠷⣠⣚⣯⡷⡄⠀⠀⠀⢀⡤⡤⠤⠤⢤⢤⡀⠀⠀⠀⠀⡀⡀⠄⠄⠀⠂
⠀⣼⠛⣿⣤⡾⠻⡆⠀⠀⠀⠘⠯⣯⠝⠉⢯⣿⠽⠁⠀⠀⠀⠠⣛⠀⢠⣀⠀⣛⡇⠀⠀⠀⢰⣶⣶⣦⣦⣦⣤
⠀⣿⠟⢻⡾⠷⣶⡇⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣴⠀⠀⠀⠀⠐⣭⠀⠘⠉⢨⣭⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠈⠳⠾⠷⠴⠋⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠉⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠛⠛⠿⠿⠿⢿⡟

⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⣬⠀⠀⠀⣣⠀⠀⠀⠀⡇⠁⠁⠁⠑⠁⠁⠁⠁⡇⠀⠀⠠⣯⣭⣭⣭⣽⠄⠀⠀⠀⡤⣶⣿⣁⠍⠠⠿⠷
⢀⣞⡇⠀⠀⠀⢨⢳⡀⠀⠀⡇⡀⡀⡀⡠⡀⡀⡀⡠⡇⠀⠀⠀⠀⢑⣴⡋⠀⠀⠀⠀⠀⢺⣿⣿⣿
⠘⠚⠒⠒⠒⠒⠚⠓⠃⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠔⠁⠀⠉⠢⠀⠀⠀⠀⠓⠒⠒⠂⠉⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣶⠀⠀⠀⢠⡬⣥⡤⣤⡬⣦⠀⠀⠀⠀⠀⠀⢰⣧
⢿⣿⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⢰⣖⣫⣥⣶⣾⣿⣿⠀⠀⠀⠘⡇⢸⡇⣸⠀⣿⠀⠀⠀⠲⣶⣶⠟⡟⢶⣶⠶
⣠⣿⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠘⠻⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⣷⢸⡇⣿⠀⡏⠀⠀⠀⠀⠀⣽⣾⣿⣾⡁
⠛⠛⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠈⠓⠒⠋⠉⠙⠻⠀⠀⠀⠀⣿⣸⣇⣿⣰⡇⠀⠀⠀⠀⠰⠟⠁⠈⠙⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠇⠀⠀⣼⣇⠀⠰⠀⠖⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢑⠚⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⣠⣠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠘⠃⢀⠀⠀⠰⣴⣶⣼⣷⣷⣿⡻⣿⣯⠀⠂⠶⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⡀⣴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠻⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠟⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢀⡜⠛⣿⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⠀⢠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢀⣴⠋⠘⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠂⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⣤⣶⣶⣶⣷⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣼⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡿⣿⡿⣻⣿⣦⣄⣀⣀⣀⣀⣀⣤⣶⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⡿⠿⢿⣿⣖⣀⣀⣤⣼⣿⣿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢼⢸⡇⣿⣧⠑⣽⣦⣈⣉⣙⣛⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⣿⡛⠟⣉⣽⣻⣿⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠹⣏⠻⣶⣦⣭⣭⣽⡿⢟⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⣷⡻⠛⢋⡽⠋⢠⡾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠇⠀⠈⠀⠀⠈⣉⣉⣥⠴⣊⣽⡿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠃⠘⢷⡄⠉⠀⣰⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢩⣤⣾⡿⠋⠀⠻⣿⣿⣧⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠙⠛⠻⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢹⣿⣿⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡄⠀⠀⠀⠀⡇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⢀⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣸⡇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣠⠟⠀⣸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⢰⡏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⡿⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢠⣿⠃⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⣿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠿⠃⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⢿⣿⣧⠻⣿⠿⠆


⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠉⢹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣮⡻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠋⠀⠀⣠⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣿⡌⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⢀⣀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣯⣠⣶⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⢈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⣿⣿⣿⠟⠁
⠀⠀⠀⢀⣠⣤⣤⣤⣤⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣴⣿⡿⠋
⠀⢀⣾⣿⡿⠿⠟⠿⢿⣿⣿⣿⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⣠⣾⣿⣿⣧⣀⣼⣿⡿⠋
⠀⠸⣿⣿⣄⡀⠀⠀⣀⣼⣿⡟⠀⢀⣴⣿⡿⠿⠿⠿⣿⣿⣿⡟⠁⢙⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠟⠁⢠⣿⣿⠋⠀⠀⣠⣾⣿⠟⢿⣿⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⣀⣠⣤⣴⣿⡿⠃⢀⣠⣾⣿⡟⠁⢀⣴⣿⣿⣿⡀⠀⠀⣴⣶⣾⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠟⠛⠉⠀⣴⣿⡿⠛⢿⣿⣶⣿⡿⠋⢸⣿⣷⠀⠀⣿⣿⡏⠉⢉⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⡇
⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⣠⣿⡿⠋⠀⠀⣸⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⣿⣿⡇
⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠁⣀⣤⣾⣿⠟⠀⠀⠀⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢸⣿⡏⠀⠀⠀⠀⠰⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠘⣿⣷⠀⠀⠀⠀⠀⠈⠻⣿⣶⡄⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣿⣿⡇
⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣀⣀⣀⡀⣿⣿⡇⠀⠀⠀⢀⣾⣿⡿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣄⣀⣀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⢺⣿⣯⡁⠀⠀⣉⣿⣿⠇⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣾⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠈⠀⠈⠠⠀⠄⠠⠀⠀⠀⠀⢀⣤⡀⡀⠠⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⢢⡀⠈⠁⡀⠀⠈⠀⠠⡌⡙⢿⡛⢿⡀⢙⣷⠀⢰⣦⡼⢷⣀
⠀⠀⠐⡆⠀⠀⢨⠁⠀⠀⠀⣦⣦⣄⡀⠀⠀⠂⢥⠰⠌⡛⣁⠀⠼⠏⣾⡿⠛⠉⠀⠞
⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⠰⠫⢻⢻⠛⢦⡣⠀⠀⠀⢡⠁⠀⠀⠀⢆⠋⠀⡀⠔⢐⣶⣤⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠡⡄⠎⠁⠀⡀⠸⠦⠀⠀⡄⠈⢃⠀⠂⢁⢣⠂⠁⠂⠀⣀
⠀⠀⠀⠂⣈⠀⠂⠀⠀⠀⠀⠀⠀⠢⠀⠘⠀⡀⠄⠋⠢⢠⠸⡦⠜⠀⠀⠀⠐⠃⠄⠀⠀⠈⣡⠆⢔⠀⠀⠂
⠀⠀⠀⡾⠁⠀⣴⡏⠐⠂⢤⣤⣀⣂⡙⠀⠴⣤⡈⠠⢖⠚⡠⠅⢂⠄⠀⠂⠨⠄⠄⣠⠔⢾⡇⢴⠂⠐
⠀⠀⠀⠀⢶⣤⣿⡇⠀⠀⠈⠛⢻⠋⠀⠄⠀⡀⠈⣂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠒⢺⠇
⠀⠀⠀⢀⣰⣿⡏⠀⡓⠊⠐⢀⣈⠀⠠⠎⠀⠀⠨⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠐⠀⠄⣁⣴⡿
⠀⠀⠀⠈⡿⠋⠀⠁⠀⠀⠀⢀⠉⢠⡄⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⠀⠙⠂⠰⢿⣽⠇
⠀⠀⠀⢷⠔⠀⢀⣤⣶⡎⣤⠜⠁⠀⢀⠄⣀⡂⣀⠀⠀⠒⠀⠀⠀⡀⠀⢀⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠨⠄⠀⢸⣏⣌⣁⠙⢶⠀⠀⢸⠀⢇⠀⠀⠀⣠⠧⠀⠀⠐⢃⠀⠘⠠⣁⡀
⠀⠀⠐⠁⠀⠀⡀⠉⣱⠏⠀⠰⠀⠀⠈⡇⠈⠃⠌⠁⣀⠀⠀⠀⠐⣠⢦⡤⠚⠻⢟⡀⠖⡤⠀⠀⡠
⠀⠀⠀⠀⠐⢾⣶⣴⣿⠁⠀⠀⠀⣤⣴⣷⣄⡀⠀⠀⠅⠀⢂⣠⠶⢹⣥⣂⢐⡘⠃⠁⠾⠷⢂⣠⡀
⠀⠀⠀⠶⠀⠀⠻⣿⣃⣀⣀⣀⠈⢨⠟⠉⠉⠍⠈⠀⠸⢂⠯⠀⠩⠀⡟⠟⠅⠈⢢⠲⢟⠛⠋⠉⠀⣠⠀⡀
⠀⠀⠐⠀⢰⠀⠀⠈⡿⠏⠛⠉⠀⢀⡀⠀⡘⢆⠛⠠⠠⢹⣥⢤⠴⣎⠁⠈⢃⡔⠁⠁⡀⣘⡂⣴⡞⠛
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⣰⣾⣿⣗⠀⠄⠈⠦⠤⠀⣸⢕⠥⡊⠀⠑⡤⠋⣀⣤⠅⣬⣭⠃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⡻⠷⠠⣤⣤⣤⣾⡃⠁⢀⠄⠠⡀⡏⠀⠀⠋⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠛⠉⠢⠁⡡⠠⢴⣶⣶⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣬⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢋⡿⣟⠲⣄⠀⡈⠢⣄⠀⠀⠀⠀⠀⠀⠈⣉⣽⢿⣰⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠤⣩⡇⢿⡘⡜⢦⠈⠢⣈⠳⢄⠀⠘⣻⡶⢫⣵⣿⣿⠟⠵⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡶⠟⣫⡵⠞⠁⢸⡄⠀⢻⡌⢿⡀⠈⢹⣖⡛⢲⣿⣰⣿⣿⠟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣴⣿⣋⡀⠀⠀⢀⣻⡄⠀⠳⣄⣉⣒⠈⣹⡷⢫⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⣿⣿⣷⡮⠀⠀⠩⢲⣿⣶⣦⣤⣭⣿⣿⣿⢃⣾⣿⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢡⣿⣆⠀⣀⣀⠀⠀⠀⠀⢸⢃⣽⣿⠃⣼⡟⣿⣿⡜⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣎⠻⠍⠉⠀⢀⣰⡦⢺⡿⠁⣸⣿⢱⣿⣇⠇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣿⣜⢿⣿⣦⣀⣤⣴⡿⢋⣴⠏⠀⢰⣿⢿⠘⣿⣏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⢻⠘⢷⣿⣿⣿⢹⢏⣴⣻⠃⠀⢠⣿⡷⠚⢣⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡟⢰⡄⢣⡀⢨⡿⣡⣏⣾⣳⠃⠀⢠⣿⣿⠁⢠⠞⢹⡟⡳⣕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⢀⡿⣿⠀⠇⢀⣴⢿⣿⣿⠃⠀⢀⣾⡟⢸⠀⠀⠀⠀⣷⠘⣎⠳⣍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⣿⣠⠾⢋⣘⣸⣿⠃⠀⢀⣾⣿⠁⡼⠀⠀⠀⠀⢸⣧⡈⢳⢬⡳⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⠟⣉⡤⠖⠋⠉⣹⢇⠄⢀⣾⡿⠻⠷⣧⣀⣀⣠⣤⣾⣿⣷⣄⠑⢌⠚⢦⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⣓⠟⠁⠀⠀⠀⣰⢃⠎⣰⣿⡿⠁⠀⠈⠲⣝⢿⣿⣿⣿⣿⣿⣾⡳⣤⣙⣶⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠞⠁⠀⠀⠀⠀⣰⢏⡞⣰⣿⡿⠁⠀⠀⠀⠀⠙⣦⠹⣿⣿⣿⣿⡿⣟⢦⡹⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⡀⠀⠀⠀⠀⣰⢏⡞⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡆⢿⣿⣿⡟⠁⢻⣟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⠀⠀⠀⠀⣰⢏⣞⣼⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣗⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣰⢏⣾⣾⢏⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⣴⢯⡟⣾⢃⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠈⠁⠈⠉



⣿⢿⠿⡿⢿⢟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣛⣛⣻⣿⣛⣛⣻⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠁⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⣦⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⣧⣄⠀⡀⠀⢀⣶⣿
⠀⠀⠀⢠⣴⣾⣿⠿⠛⠿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⣾⣿⣯⠋⠉⡁⠈⠹⣿⣿⠃⠀⠀⢀⠔⠛⠻⣆⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠀⣿
⠀⠀⠀⠘⢿⣿⡄⠐⠛⠆⢦⡿⠋⠀⠀⠀⠊⢠⡀⢠⣻⠀⠀⢠⣴⡞⠉⠙⠛⠙⠃⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠟⠁
⠀⠀⠀⠀⠀⠉⣷⢉⢦⣼⣥⠀⠀⠀⠀⠘⠀⠀⢤⠿⣸⠀⠀⢺⣿⡇⠀⠀⠀⠀⢀⣿⣿⠆⠀⣀⡤⠤⠤⢦⣄⠀⠀⠀⠀⠠⠖⣨⡿⠉⠟⠃⠀⠈
⠀⠀⠀⠀⢠⣿⣿⣿⣆⠺⣴⣶⣶⣦⡀⠀⠀⠐⠒⠁⠇⠀⠀⠀⠻⠇⢀⡀⣀⠀⠸⠿⠋⠀⣼⠏⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣬⣥⡤⠚⣞⣶
⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠝⣻⣿⣹⣵⢲⡀⠲⣶⡆⠀⢀⡀⠀⡃⠰⡀⣒⠀⡄⠀⠈⠀⡀⣿⠢⢤⠀⢠⠤⠀⠟⠁⠀⣿⡇⣿⣿⢷⡶⣿⣿
⠀⢸⣿⠟⢋⢓⣾⣿⣿⣧⣤⣿⠟⠋⠁⠸⣿⣦⣽⣏⡀⠘⣷⡀⠀⢀⠀⠭⠀⠀⠖⠀⡠⠻⣜⣷⣾⣤⡄⠀⠀⠀⠀⢰⢁⣼⣋⠿⣤⡽⠿⢾⡇
⠀⠈⠛⠳⣤⣿⣿⣿⣻⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⢷⡤⣀⡙⢸⣋⣡⣀⡔⠁⠀⠈⢿⣿⣟⡛⠀⠰⠀⠀⢸⣾⣿⣿⡒⢂⣠⣐⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠁⠀⣄⣄⣰⢿⣿⣽⣿⡿⠿⠋⠉⠀⠀⠀⠙⠁⠁⠀⠀⠀⠀⠺⢿⣻⡿⠿⠀⠀⠐⠈⠁⠀⢹⠿⣿⣿⣿⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡽⠖⠚⡎⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠹⣿⣭⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠖⠶⢦⣴⠶⠾⣷⣤⠶⠶⣶⠛⠉⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⣀⡀⠀⠀⠛⠀⠀⠉⠀⠀⠀⠈⠃⠀⠀⠀⢨⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣛⣿⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⣠⡴⠶⠟⠛⠛⠋⠉⠉⠉⠛⠛⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡶⠿⠟⢛⣿⡟⣉⣉⣉⢻⣯⡉⠉⠙⠛⠿⣷⣤⡀⠀⠀⢸⣇⠀⠀⠻⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⠿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⡿⠛⠉⠁⠀⠀⠀⢸⡧⢠⣤⣨⣿⣿⣽⡇⠀⠀⠀⠀⠈⠙⣿⣆⠀⠛⢩⡇⠀⠀⠀⠉⠻⣏⠉⠉⠉⠉⢩⣿⠟⠁
⠀⠀⠀⠀⠀⣰⡾⠋⠉⠀⠀⠉⠳⣦⠀⠀⠀⠀⠀⠈⠷⣤⣀⣀⣀⣤⠿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧⣠⠞⠁⠀⠀⠀⠀⠀⠹⣇⠀⠀⣰⡟⠁
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠙⣉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢠⡿
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⡖⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣼⣿⣶⣦⡀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠻⣄⠀⠀⠚⠉⠀⠈⢿⣷⠀⠀⠀⠀⠀⠈⣇⢸⡁
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⣿⠀⣷⡀⢹⡆⠀⢀⣄⣀⣀⣾⡟⠀⠀⠀⠀⢀⣴⠟⠋⠛⢷⣆
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⣀⣤⣾⣷⣶⢾⣿⡟⠛⠋⠁⠀⠀⠀⠀⢠⡟⠀⡿⡇⠀⣿⠀⠈⠛⢻⣿⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⠛⠉⠀⠀⠀⠀⢹⣧⠀⣴⠒⠶⠶⠶⠶⠋⢀⣼⢢⡇⠀⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⣿⠀⣀
⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢷⡀⢦⣄⣀⣀⣴⠾⢅⣸⠃⣸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣠⣿⣷⣿⣿
⡀⠹⣧⡀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⢦⣉⠩⣭⣦⡴⠞⣁⣴⣯⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⠿⠛⣹⡿⠁
⠻⢷⣬⣹⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉⠛⠶⠶⠶⠟⢫⡟⠈⢻⣆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⣷⣴⣾⠟⠁
⠀⢶⣬⡽⢿⣿⠿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢻⡆⠀⢰⣿⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣤⣄⣀⣀⣀⣀⣀⣠⣿⣿⠟⠲⢦⣄⣠⡴⠶⢾⣧⡴⠖⠀⠀⠘⣿⣀⣸⣿⣤⣤⣤⣶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⣹⡿⣿⠀⠀⣀⣬⡏⢀⠀⢠⣿⣀⡀⠀⠀⠀⣿⠋⠉⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⡀⠈⠁⢈⣷⣬⣭⣿⠋⢉⡿⠿⢶⣾⡟⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠘⣷⡀⣠⡾⢻⣤⢻⡉⡖⣻⠃⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠘⠛⠋⠀⠀⢻⣸⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣸⣇⣀⣤⣤⣤⣤⣤⣤⣶⣶⣶⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⣿⠋⠉⠉⠉⠉⠉⠉⠉⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⠦⣤⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⡧⠴⠶⢶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣰⣀⣠⣤⡌⠙⡆⠀⠀⠀⠀⢠⠟⢠⣤⣀⣀⣇⣀⣼⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣽⣭⣥⣤⣤⣤⣤⣼⣷⣿⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄⣠⣤⣤⣤⠶⠶⠶⠶⠶⣤⢤⣮⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⣠⣴⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⣙⣻⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠿⠿⠛⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⡠⠂⢰⣾⠟⠁⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⡟⢀⣴⡟⡱⢂⡴⠁⠀⠀⠆⢀⠀⠀⣀⣀⡀⠀⠀⠈⠄⠀⢀⣤⣶⣾⣷⣦⠀⡀⠀⠀⠴⠏
⠀⠀⠀⠀⣠⡾⢰⣾⡟⠉⠀⡞⠀⠀⠀⢸⠀⢀⠀⠠⣿⣿⣿⠂⢀⣤⣸⡴⣿⠟⠛⠛⠛⠿⡀⠈⢆
⠀⠀⠀⢰⡏⢀⣿⡟⠀⠀⣸⠃⠀⠀⠀⠀⢆⠸⡄⠀⠈⠙⠃⡀⠙⣻⣿⣔⢁⡀⠀⠐⣰⣶⡜⣿⣿⠆
⠀⢀⡴⣿⠀⣼⣿⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⢱⣌⠢⣄⡀⠀⢀⣠⣿⣟⣿⡿⠃⠀⠀⢋⡀⠀⢹⣿⠀⠀⠻⣿⠟
⢠⠎⠀⣿⢠⣿⣷⡿⢁⣀⣤⣴⣧⣤⣤⣤⣀⣰⡼⢻⣲⣶⣿⣿⣿⣿⣿⣿⣇⠀⠁⢀⣀⡤⢀⣿⡏⢰⠀⠀⠀⠀⡀
⡎⠀⠀⣿⣸⣿⡿⠁⠀⣠⣏⣤⣴⣿⣿⣿⣿⣿⣷⣾⣯⣽⣿⣿⣿⣿⣻⢸⣿⣦⣍⣉⣉⡠⣾⣿⡇⠀⣷⠀⠋⠀⠁⠶⠶⠿⠻⠶⣤⣀
⠻⣄⠀⠘⣿⣿⣇⠀⣼⠋⠉⣡⠞⣫⡽⣿⢿⡏⠈⠁⠀⠈⠛⢿⠿⣿⣏⣼⣿⣿⣿⡿⡉⡀⣿⣿⠿⢤⢻⡼⠸⠓⢢⡀⠀⠀⠀⠀⠈⠻⣷⡄⠄
⠀⠙⢷⡄⢹⣿⣿⢠⡏⠀⠘⠁⣼⣇⣸⡟⣼⣷⣄⠀⠀⠀⠀⢸⡳⣿⠇⣈⣿⠕⠻⡄⠀⠀⢹⣿⣿⣿⣇⡷⡋⠀⠀⠹⠶⢤⣤⡴⠀⡄⠹⣿⣾
⠀⠀⠀⣿⢾⣿⣿⣼⠁⠀⢀⣀⣿⣿⣯⣿⣿⣟⠽⠷⣄⠀⠀⢸⣧⣿⣵⡟⠃⠀⠀⠉⠀⡇⢈⡀⡿⣿⣿⣷⣌⠢⠀⠀⠀⠀⠐⢶⡖⢼⡀⣿⣿
⠀⠀⠀⢸⠘⣿⣿⣿⠀⣰⣿⣿⣿⣿⣿⢿⣿⣿⣷⣦⣿⣷⣶⣿⣿⡇⡟⠀⣴⡀⠀⠀⠰⣦⡠⠟⡛⣴⣿⣿⡟⠀⠀⠁⠀⠀⠀⠀⠹⣄⣿⣿⣿
⠀⠀⠀⠘⣇⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠗⠀⣿⣿⣦⣀⢀⡝⠀⠀⢹⡎⣿⣿⡃⠀⠀⣴⣶⣾⣷⡄⠀⣽⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣆⠈⠻⣿⣿⣟⣯⠟⣹⣧⠀⠀⣿⣿⣿⡿⠟⠁⣀⣴⣯⣰⣿⡟⠀⠀⣴⣿⣿⣿⣿⣧⠚⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣴⣿⣿⣿⠀⠀⠟⠋⠉⢀⣴⠿⢋⠟⠁⢻⣿⣷⠀⣴⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠀⠀⢀⣴⡟⠃⠀⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠻⡿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣽⣿⣿⣿⣾⣿⣿⣧⣽⣶⣾⡟⠿⠗⡀⠀⢠⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣻⣿⣿⣿⡉⠀⣠⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⡀⠀⠠⠎⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣟⠙⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠑⠻⠋⠀⠀⣼⡟⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⡐⡀⠀⢠⣾⣿⣧⣿⣿⣿⣿⣿⠟⣫⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⣾⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠗⠀⣴⣿⣿⣿⣿⣿⣟⣽⣿⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⠇⢸⠉⡟⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⠀⠀⠀⠇⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣮⣻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢦⡙⢿⣿⣮⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⣷⢿⣎⢿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⣛⣃⣼⣿⣼⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠰⠀⠐⠀⠆⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠄⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠘⠁
⠀⠀⡈⠀⠀⠀⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⡠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠠⠂
⠀⠀⠐⠂⡀⣄⣤⡄⢠⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢴⡄⠀⠀⠀⣲⠂
⠀⠀⠀⠀⡀⠈⠐⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠉⡔⡁
⠀⠀⠀⠀⠁⠖⣀⠶⢄⡀⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠆⢸⡆⠀⡼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⢀⠉⠐⠒
⠀⠀⠀⠀⠀⠂⠀⠙⠉⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⡆⢰⠁⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠐⠒⠠⢄⠄⠀⠈
⠀⢰⠋⠃⡇⣔⠀⡀⡀⢤⢠⠀⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠆⠀⠙⠂⠀⠉⠁⠀⠻⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⠀⡀⠀⠛⠛⣤⣀⠀⣀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠠⠀⢀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⢆⠀⢀⠈⠉⡀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⣦⡀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢿⠀⡀⡞⠀⠉⢠⠸⢃⠗⢠⣀⡴⠀⠀⠀⠀⠀⢸⣷⠒⠋⢸⣧⠈⠒⣲⣿⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠘⢀⡀⠈⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⢐⣞⠠⠯⠬⠅⠡⠌⠋⢅⣲⠚⣄⡠⡴⠃⠀⠀⠀⠀⠀⢻⠙⠀⠀⠀⠀⠀⠀⠈⠀⢣⡀
⣧⣄⣀⠀⡀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠠⠠⠤⠀⠀⠀⠀⢀⠈⠉⠚⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠄⠠⠠⠀⠀⠄⠀⠱
⠙⢿⣿⣤⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⠁⢂⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣌⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⣄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠻⠛⠉⠉⠛⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠛⠂⠀⡀⠴⠆⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣿⣇
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡁⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠙⠦⣤⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡄⠀⢀⣀⣀⣀⠀⣤⣄⡀
⠀⠈⠉⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠙⣟⠁⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢰⣿⡆⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⢀⠀⠀⢼⣿⣿⣿⣿⡄⢸⣿⣇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⡏⠀⠀⠷⣄⣀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠈⣿⡟⠀⢸⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⠃⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠀⠀⠉⠉⠉⠁⠈⠉⠁⠀⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣴⣀⡁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣽⣿⣿⠿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢟⣿⣟⣹⣿⣿⣻⣿⡤⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣿⣯⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣳⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣏⡛⣛⣽⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠈⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠤⠀⠄⠀⢀⠀⢠⡄⠤⠄⠒⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⢠⣄⣀⣠⣴⣶⣷⣞⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠋⠁⠀⠀⠀⠁⠀⠀⣠⠊
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡄⠤⠛⠉⠀⠀⠀⠀⠀⠈⢧⣄⣀⡀⠀⢀⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠶⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠭⠀⠀⠀⠶⠶⠶⠶⠴⠦⠀⠀⠀⠀⠀⣤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣦⣞⣠
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠐⠒⠒⠒⠒⠒⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⣀⣰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣖⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⠓⠀⠀⢠⡄⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡄⠀⠠⠄⠀⠀⣀⡀⠸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⢀⡄⠀⠀⠉⠁⠜⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣬⠁⢀⢚⡁⠄⠀⠀⠀⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⡞⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⣀⡀⠀⠀⠉⠁⠓⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣴⣿⣿⣿⣷⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⡿⠋⢉⡙⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡇⠀⢿⣷⣄⣿⡿⣼⣿⡿⣫⣍⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⡇⠄⠀⠈⠉⠙⠟⠙⠻⢧⣟⣛⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⠀⠀⠀⠀⣄⠀⠀⣠⠌⠉⠉⠉⠁⠀⢀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⡠⠟⠻⡟⠋⠀⠀⠀⠀⠀⢀⠰⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢰⡇⠀⣁⠀⠀⢠⣀⠀⠀⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⠀⠘⠿⣷⣿⣷⣶⠿⠉⢲⡤⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⣠⡞⣹⡿⣿⣷⣶⣤⡲⢁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⠘⠿⠛⠛⠓⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⣀⣤⣿⣄⢀⠀⠀⠀⠀⢀⠄⠀⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡘⢿⠻⢿⡿⠿⣿⠇⠀⠀⠚⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠀⢀⣀⣀⣀⣄⣀⡀⠀⡠⠁⠀⠀⠀⡤⠀⠠⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠛⠻⣇⣾⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⢷⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⣷⣶⣾⣇⣀⣤⣴⡶⠶⠶⠶⠶⠶⢶⣤⣄⣀
⠀⠀⠀⢠⣾⠟⢉⣠⣴⣶⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⣿⣥⣶⣿⣿⣿⣿⣿⣿⡿⠁⣀⣤⣶⣶⡿⠿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠙⢷⣦
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⣄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠉⣹⡟⠉⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠈⢿⡄
⢀⣀⡏⠂⠀⢰⡟⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠈⣿⡄
⠿⠿⠃⠀⠀⣾⠃⠀⠀⢀⣿⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠸⣿⡆⠀⠀⢸⣧⣴⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣿⡇⠀⠀⢸⣿⡿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣶⣤⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄⠀⣾⠏⠀⣿⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢻⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢹⢹⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⠀⢠⠇⠀⠀⢀⢜⣥
⠀⠀⠀⠀⠀⠘⣿⡀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣾⡾⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⢸⣇⣴⠋⠀⠀⠀⠘⢌⣡
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠙⠻⢿⣿⣿⣿⣿⠀⠀⠙⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢋⣴⠿⣿⠁⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⣰⡟⠁⠀⣹⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠀⢀⣾⡿⣿⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠋⢹⣇⠀⠘⣿⠀⢹⡀⠀⠀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⢠⡄⠀⠀⢿⣄⠀⠈⠁⠸⣇⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠙⠿⢶⣦⣤⣿⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢾⢳⡄⠀⠀⠀⠀⣠⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠋⠀⠀⠀⠀⢰⣿⠿⠟⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⢼⣿⣿⣿⡟⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠻⠟⠋⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⡀⢀⡆⢀⣠⣶⣿⣿⣿⣿⡿⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣬⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠙⠛⠁⠀⠀⠉⠛⢿⣯⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⢻⣿⡿⠋⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⢀⣿⠟⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠒⡈⢁⣈⣰⣿⡄⠒⠂⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣸⣿⡿⠻⣿⣿⣿⣿⣆⡐⠊⠀⠀⠀⢀⠀⠄⠘⣿⣿⣄⠁⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⢰⣿⡿⠁⠀⠈⠛⢿⣿⣿⣿⣦⣆⠀⠀⠀⠐⠀⠀⢹⣿⣿⣷⣄⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⢸⣿⠈⠻⣿⣦⡀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⡀⠀⢸⣿⠀⠀⠈⠻⢿⣦⣄⠀⠀⠈⠙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣿⣿⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿



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

⠀⠀⣄⡀⢠⡀⠀⣠⡄⠀⠀⠀⠀⠀⣶⡦⢀⠀⠀⠀⠀⠀⢀⣤⣾⣦⣄⠀⠀⠀⠀⠠⠐⠒⣄⡒⠀⠄
⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⣼⣿⡗⣲⡄⠀⠀⠀⠀⠛⠛⢻⠛⠛⠃⠀⠀⠀⢃⢠⡦⢐⣴⡆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⠀⠀⠀⠀⠀⣐⣀⣸⣀⣐⡀⠀⠀⠀⠨⢑⡨⠤⢂⡰⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⣾⡟⠛⠛⠛⣿⡆⠀⠀⠀⢸⡏⠉⠉⢹⡇⠀⠀⠀⠀⣴⡺⡽⣖⢄⠠⠀⠀⠀⢀⣤⣤⡄⣤⡤⣄
⠀⠀⡟⠀⠰⡇⠀⠙⡇⠀⠀⠀⢸⡷⠤⠤⢼⡇⠀⠀⠀⢸⢞⢎⢏⣎⣾⠀⠀⠀⠀⢸⣿⣿⣷⣿⣤⣿
⠀⠀⣿⣲⣴⢳⣴⣾⡇⠀⠀⠀⢸⣇⠀⠀⢸⡇⠀⠀⠀⠈⣯⢗⣷⣮⣎⠀⠀⠀⠀⠻⠟⠃⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉

⠀⠀⣦⣬⣀⣀⣥⣴⡄⠀⠀⠀⢠⣶⣾⣷⠂⠀⠀⠀⠀⠀⠀⢀⣒⣒⡀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠱⢆⣾⣿⠀⠀⠀⠀⢀⣿⡿⣿⣷⠄⠀⠀⠀⢸⣿⡏⣄⡌⣿⣿
⠀⠀⠿⠿⢿⡿⡿⠿⠃⠀⠀⠀⠈⠀⣾⣿⠿⠃⠀⠀⠀⠀⣼⣿⣷⣶⣶⣆⠀⠀⠀⠀⠉⢡⣟⣣⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠃⠀⠀⠀⠀⢠⠖⢻⠛⠲⡄⠀⠀⠀⠀⠾⠮⠴⠧⠷⠇
⠀⠀⠀⢸⡁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣹⠶⠀⠀⠀⠀⠀⠀⡟⠑⣾⣷⠊⣻⠀⠀⠀⢠⠀⢴⠄⢴⠄⢠
⠀⠀⣀⣀⣙⣲⣒⣋⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⠾⣅⣨⠶⠃⠀⠀⠀⠈⢶⣾⣶⣾⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠤⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢰⠁⣘⠄⣀⠛⠄⡀⠂⠌⠀⢀⣀⣀⣀⠀⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠌⡘⢀⡏⠔⠂⠚⠂⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⡰⠤⠌⠴⠃⣘⠴⡄⠀⡤⣤⠒⣶⣒⢆⣜⣡⣌⡹⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁⡊⠀⠀⠀⠀⠀⣯⣱⡇⠀⠁⢉⣉⣉⢉⡁⢤⢄⠀⢤⠐⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠂⠌⣑⠦⠄⠀⠒⠚⢶⢯⡇⠀⣭⡿⣼⣛⠾⣽⢃⠎⣐⠢⠌⡁⠀⠉
⠀⠀⠀⠀⠀⠈⡐⠠⡎⠁⠀⣀⡤⣴⡖⣶⣸⢿⡇⠀⣿⣽⣳⣯⣿⢣⣏⡲⢠⠋⠴⠅⠀⡦⢀
⠀⠀⠀⠀⠀⠀⢀⠂⠴⠀⠀⠯⠓⣋⣩⣭⢯⣿⡇⠀⣿⣿⣟⡾⣭⣛⡾⣱⠃⡌⣑⠇⠀⡵
⠀⠀⠀⠀⠀⡀⠂⠌⡰⢇⠀⢰⢏⡟⣼⣟⡿⣽⡇⠀⣿⣿⣿⣽⢧⣟⡳⢧⣝⢰⣁⠄⠀⣇⠂
⠀⠀⠀⠀⠀⢀⠀⠠⢁⠺⡀⠀⢿⡽⠗⠛⠛⠋⠁⠀⣿⣟⡿⣯⣟⡶⠋⠁⠀⠀⠀⠀⠀⡏⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠂⡘⢃⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡾⣹⠿⣼⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⢙⡀⠀⡆⠀⠀⠀⠀⠀⢀⡟⡳⢯⡟⠰⢦⡀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣱⠀⠸⡤⢤⣀⡠⢴⠛⠤⢁⠒⡬⢁⠆⡌⠃⠎⡐⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⢇⠂⠬⣑⠂⡌⠑⠎⡜⠠⢌⡐⠠⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢸⡌⠠⢀⡘⠠⢁⠒⠠⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣭⣽⣶⡿⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣄⢠⡄⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⣼⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠀⣀⣀⣠⣀⡀⠀⠀⣼⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠉⠉⣩⣀⣤⣾⣿⢿⡋⠁⠈⠁⠀⠀⠉⠀⠈⢹⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⣿⣿⠋⠩⠀⠀⠀⠀⢴⣤⣿⣥⣔⡶⡄⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠈⣷⡀⠀⠀⠀⠐⣴⣿⣿⣿⣿⣿⣿⣞⡆⠀⠀⠀⢠⡄⠀⠲⠿⠿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣠⢫⣢⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⢿⣛⢿⣵⡀⠠⢤⠙⢿⣿⣿⣿⠿⢿⢏⡀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣿⡿⡆⠀⠀⠀⠈⡻⠿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠒⠦⠀⠀⠤⠀⠨⣵⡭⢿⣟⠷⠆⠀⠀⠀⠀⠉⠀⠀⢦⠀⠀⠀⠀⠛⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⣤⣄⡉⠀⠐⠁⠀⣀⣀⠀⠀⡀⠀⠀⠀⠘⡂⠰⠆⠀⠀⠀⠀⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣿⢁⠀⣤⣛⠀⠉⢹⣧⠘⠃⠀⠀⠙⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣓⣯⣼⣿⣿⡇⢲⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⢿⣿⣧⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠛⠉⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣃⣿⣿⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠘⢿⣿⡷⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣨⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠉⢾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⡏⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⣀⣀⣄⣉⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡄⣤⢠⢶⡀⡄⡄⢴⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠌⠀⢸⠿⢻⢸⢸⡇⣿⠇⢸⢸⡇⠸⠓⡀⠰⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠢⠔⠀⢁⠀⠀⠆⠈⠠⠀⠠⠉⠁⠒⢠
⠀⠠⣐⣦⣤⣄⡢⡌⠁⡀⠀⠘⠀⠘⠈⠛⠁⠘⠀⠚⠈⠓⠀⠀⠀⠁⠀⠀⠈⠀⠀⠃⠐⢤⠀⠈⠂⠀⠀⡠⠃⠈⠇⠀⠀⢐⡀⣥⠀⣀⠴
⠀⠌⠛⠛⠋⠉⠀⠃⠀⠈⡆⠀⠈⠄⠀⢀⠄⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠆⡠⠃⠀⣬⠆⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⢀⢐⡀⣀
⠀⠄⠀⢿⣿⡗⠀⡬⠀⢂⣀⠀⠀⡼⠰⠄⠀⡀⣀⠀⠀⠃⠀⠀⠠⠞⠀⠀⠀⢀⠑⠐⡂⠡⠄⣀⠀⠀⠣⠀⡔⠉⠢⡀⠀⡰⠊⠁⠀⠀⣄⠑⡄
⠀⠈⠂⠠⠭⠀⠊⠀⡈⣤⣴⣶⣶⣶⣶⣦⣤⠀⠀⠈⠀⠃⠀⠀⢀⢀⡀⠀⠀⠀⡄⠙⠤⢄⠀⣻⠈⢈⠁⠸⠀⠀⠀⠓⡀⣄⠀⢁⠀⠌⠠⢀⠇
⠀⢀⠄⠠⠠⡀⠀⠀⡇⣿⣿⣷⣿⣟⣿⣿⣿⠀⠀⠀⡆⢰⠀⠀⠙⠺⠹⠈⠠⠀⠰⠀⠀⢛⠒⠚⠀⠂⡃⠘⡐⠀⠀⣈⠃⠈⠐⠀⠀⠀⠀⠁
⠀⠊⠀⠀⠀⠈⢂⠀⡇⣿⣿⡟⡿⣿⡿⢿⣿⠀⠀⠀⠆⠀⣤⠆⠂⠐⠒⠒⠒⠒⡖⠀⠀⠸⠀⠀⠀⠨⠁⠀⢱⠀⠀⡄⠀⠀⢠⢴⣯⡀⠀⠈⠢
⠰⣅⣀⣀⢀⣀⠃⠀⠇⠛⠿⠿⠿⠿⠿⠟⠣⠀⠀⢀⠀⠀⠉⠀⢀⡠⠄⠐⠀⠤⡀⠀⠀⠀⠆⠀⠀⡠⠀⠀⠈⠒⠒⠁⠀⠀⠁⠀⠈⢿⣄⠐⡰
⠀⢽⠔⠛⢸⡇⠀⠀⠸⠋⠉⠉⢉⣉⡉⠉⡈⠉⠻⠅⡠⠐⡠⢤⣌⡰⢄⠀⠀⡠⠄⢹⠒⠄⡀⠀⠀⡴⠒⠒⢠⡀⠀⠐⠄⠀⠐⢀⠀⠸⠿⠋
⠀⢺⢤⡄⢾⠀⠀⠀⠀⢸⠉⠀⠀⣠⡤⠀⠃⠀⢀⡐⠀⠠⡀⠀⠀⠀⢀⠀⢸⠀⠀⣘⠢⠱⠈⢄⠀⠀⠀⠒⡆⣶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠙⠲⠧⠟⠂⠀⠀⠀⠈⡄⠀⠠⡤⠛⠀⠸⠀⠑⢄⠀⠖⡱⠆⢩⠀⢊⠐⢚⡁⢎⠠⢊⠜⢀⠒⠀⠢⠄⠤⠬⠤⠤⠤⠊
⠀⠀⠀⢀⣤⣶⣶⣦⣀⠀⢃⠀⠀⠶⠐⠂⠀⡆⠀⠀⠑⠠⢀⠀⠀⠉⠉⠀⠀⠀⠀⢁⠠⠐⠁⠀⠀⠀⠀⢀⠠⢀⠀⠀⣿⣻⠏⣿⡟⣿⡏⡉⣿
⠀⠀⣴⢟⣭⡶⣶⣭⡻⣆⠘⠒⠒⠈⠉⠉⠀⠀⠀⠀⢀⣀⡤⡀⠀⠀⠀⠀⠀⢀⢠⠶⠤⠤⠀⢄⠀⠀⢠⠁⠀⠀⢑⠀⠟⠛⠛⠃⡀⠈⠛⠛⠛
⠀⠰⠿⠟⠁⠀⠈⠙⢳⢿⠀⢠⣴⣶⣶⣶⣶⣶⣶⡄⠈⢥⠜⠁⣀⡤⡄⠀⠼⢁⣄⡀⠀⠘⠃⠃⠀⠀⢸⠀⠀⠐⠘⡄⠀⠀⢘⠀⠉⠈⠈
⠀⢃⣀⡀⠀⠀⠀⠀⣄⣠⠀⢸⡏⠁⠸⠀⡏⠠⢸⡇⢠⣀⢶⡄⢈⣀⠀⠀⠄⠈⠛⢀⠔⠒⠤⣀⡸⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄
⠀⠀⢹⣷⣄⠀⢀⣼⣿⠀⠀⢸⣇⠀⡀⡄⠃⠰⢺⡇⠈⠁⠀⢠⣼⣙⠃⠀⠈⠒⠒⠁⠀⣀⠀⠀⢀⡀⠀⠑⢰⡶⠒⠁⢐⠀⠀⠀⠐⠈⠀⠐
⠐⠂⠘⠋⡻⠀⠘⢋⡙⠀⠀⢸⣿⢾⣷⣿⣾⣤⣾⣿⠀⠀⢀⠀⠈⡉⠀⠀⠀⠀⠀⠀⢺⠁⠋⠙⠛⠛⣿⣆⠘⠷⠇⠀⠺⣶⣿⣛⣓⣚⣻⣷⣾
⠀⠈⣠⣉⠤⡀⠀⠀⣈⡄⠀⣰⣿⠀⡇⡆⢀⠘⢸⣿⠀⢰⠓⠲⠾⠶⢾⠄⢄⠀⠀⠀⠇⠀⣘⢀⡀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢀⠿⠏⢀⡠⣿⣿⠿⠷⠀⠉⢿⣶⡷⠶⠿⠶⠾⠟⠀⢸⠀⠀⠄⠠⠀⠀⠀⠀⢀⡞⠋⡧⠰⠎⠀⡤⠢⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⠅⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⡀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⢠⠆⠌⠀⠸⡄⠀⡀⠀⠀⠀⠃⠀⢸⠀⠂⠀⠞⣀⣀⡀⠙⠀⠈⠀⠁
⠀⢀⡀⠤⠤⠤⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⠀⠀⢀⣀⡀⠀⠑⠀⠀⠀⠊⠀⠀⠰⠀⠘⠒⠁⠀⠀⠀⠧⠴⠃⠐⠀⠀⢼⣿⣿⣿⡗⠀⡠⠀⠐⠄
⠀⠂⠀⠀⠀⠀⠀⣳⠀⠀⡖⠃⠀⣀⠀⠀⠑⡄⢸⠀⠀⠉⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠄⠀⠀⡄⠀⠀⠙⡍⠉⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠇⠀⠈⠿⢿⠀⠀⠃⠈⡀⠀⠀⠀⠀⠀⡥⠐⠭⠂⠀⠀⠀⢀⠄⠠⣀⠀⠀⣀⠀⢀⠇⠀⠀⢠⠇
⠀⠀⠀⠀⢶⠂⠀⢸⡀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⠇⠀⠀⢀⠔⡨⢔⠁⠀⢀⠀⠀⠐⣵⡿⢷⡌⡆⠀⠜⠀⠣⡀⠀⢔⠺⠂⡄⠀⢸⠀⠘⠃
⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠟⠣⠞⠃⠈⠀⢀⠀⠀⠈⠁⠀⠸⠀⠀⠀⠠⡊⠀⠙⠾⠋⠈⠀⢸⡀⠀⠀⡄⠀⠀⠀⣤⡆⠀⠈⡄
⠀⠑⠚⠚⠛⠛⠛⠋⠀⠀⠑⠆⠀⠀⠀⠀⠀⠐⠃⠀⠉⠑⠒⠒⠢⠤⠆⠀⠀⠀⠈⠢⠠⠊⠀⠀⠀⠈⠑⠀⠊⠁⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠈⠉⠀⠒⠢⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⡿⠟⠛⠻⣿⣷⣄⣀⣀⣤⣴⣤⣄⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢸⣿⠁⢠⣶⣶⢸⣿⣿⣿⡿⠛⠛⠻⣿⣧⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢹⠃⠀⠀⠙⠷⢦⣝⣉⣼⠿⣿⣿⢰⣿⡧⠀⢸⣿⡆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣾⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠈⠙⠶⣭⣥⣴⣿⡿⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠼⣄⠀⠀⠀⠀⠀⠈⠣⣄⣀⣀⠴⠒⠄⠀⠀⠀⠀⠀⠀⠸⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣆⠀⠀⠀⠀⣀⠠⠄⠒⠒⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠌⠣⠙⣆⠀⠀⢀⡔⠛⢿⣶⣶⣶⣿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⢠⠴⠋⠀⠀⠀⠉⠉⠉⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣴⡿⠃⢸⠀⠀⠀⠀⢠⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⢢⣹⣍⣉⣀⣠⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡆⠇⠀⠀⠀⠁⠀⠀⠀⠀⡰⠁⠀⣠⡄⢄⡞⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⣾⣿⣿⡆⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢁⣀⣀⣀⠀⢀⡠⠀⠐⠛⠿⣿⣿⣿⣿⠇⠉⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠓⢷⣁⢀⣀⣿⣿⣿⡿⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠿⠿⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟⠛⠉⠁⠀⠀⠈⠙⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢀⠀⠀⢀⡔⠁⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠉⠄⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⡄⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣀⣠⡴⠖⠋⠀⠀⠠⠀⠀⡀⢀⠈⠁⠀⢀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠄⠀⠀⠀⠀⣰⡟⣁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠉⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠐⠀⠀⣄⠀⠁⠈⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠁⠀⠀⠉⠉⠃⠀⠁⣀⣀⣀⣀⣀⣤⢶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⡾⠋⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠛⢿⣿⡋⡁⡬⠟⠿⠿⢿⣿⣿⣿⠃
⠀⠰⠋⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠹⡏⡏⠐⠦⠐⠂⢀⡝⠙⡃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⠟⠉⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣰⣿⣷⡄⢸⢷⠆⣀⣤⢂⡼⠷⠀⣗
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠾⠁⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⠟⠃⠀⠀⣼⣿⣿⣿⣿⠀⠘⣾⠋⠁⠁⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠙⠋⢿⣿⣿⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣀⠸⣿⣿⣿⣿⣟⢠⣤⠖⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣄⠀⠈⠠⠀⢀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠈⠛⠿⠛⢡⠞⡁⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⢻⢀⡀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠛⠛⠋⠉⠉⠉⠉⠛⠻⢿⣧⣤⣂⡤⡀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠈⠉⠉⠘⠒⠒⠀⠐⣂⠭⢀⡀⠙⠋⣋⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣽⣿⣿⣿⣿⣦⡀⠀⢀⠀⣠⢾⡛⢿⣷⣶⣶⣶⣶⣶⣶⡿⠋⠀⠈⠀⢀⣴⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⣻⣿⣤⣼⣁⣰⣿⣿⡟⠭⠐⠂⠁⠀⠀⠀⢀⣴⣿⣷⣿⣿⣧⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣴⠿⠛⠉⠀⠀⠀⢀⣴⣿⣿⠿⠟⠛⠙⠋⠙⢻⠿⠟⠛⠛⠉⠭⠍⠀⠀⢀⣀⣀⣀⡤⣤⣴⣿⣛⣿⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⠁⠀⠀⠀⠠⠁⠀⠀⠀⣦⣴⣶⣶⣶⣶⣾⣿⣻⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣷⣿⡁⠉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⢋⣭⣭⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢠⣆⡘⣿⡟⠛⠛⢻⠋⠉⠁⠲⣾⠆⡉⢁⣼⣿⣿⣿⠿⠋⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⣿⣿⣾⣿⡀⠀⠘⣇⠀⠀⠐⢦⣨⣿⣿⣿⣿⡿⢯⠀⣡⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⣷⢿⣻⣧⡄⠀⠘⡄⠀⣀⡼⣫⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡌⢳⣄⠀⠀⣿⣿⣿⣿⣿⣄⠀⢘⡶⣫⡾⣻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡞⣄⠻⣷⣼⣿⣿⣿⣿⣾⣿⣷⣮⠟⠛⢾⣯⠋⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠡⠆⠈⠛⠿⣿⣿⣿⣿⣽⣿⣿⠟⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⣀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠚⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠎⠁⠉⣀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⣿⣿⠛⠂⠀⠀⠀⠀⣠⡠⢦⣀⣤⡔⢷⣄⣠⡄⠂⠀⠉⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⢃⡀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣿⣾⣷⢤⣤⡔⠒⠁⠙⣈⡿⠡⠀⠀⠩⡅⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⣬⠀⠊⡷⢀⠀⠀⠀⠀⣠⣿⡟⣸⡟⣿⡇⠀⣿⣿⡀⠀⣀⣼⡼⠗⠀⢀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠴⠊⡱⠀⡀⠀⢀⡼⣻⡿⣰⡿⣸⣿⢿⣏⠈⢿⣿⢾⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠁⠀⠀⠀⠘⠋⢴⡿⠓⠿⠟⠻⠿⠷⣄⡟⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠠⣄⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⢠⠀⡂⠀⠀⠀⣀⣤⣠⡈⡆⠀⠀⠀⠀⢰⠀⡄⠀⠀⢀⣴⠿⢿⣦⠀⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⠻⠏⣿⡇⠀⠀⠀⣈⠀⠶⠀⠀⠀⣿⣿⠛⣿⡆⠀⠀⠐⣻⡏⠁⠁⠀⠀⢰⡇⢸⢀⡇⠀⠀⠘⣿⣤⣴⡿⠀⠀⠀⠀⠋⠠⠋⠀⠀⠀⣿⣫⣿⣻⡇
⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣒⢴⣲⠀⠀⠀⣠⣴⡞⢿⡆⠀⠀⠀⠉⠸⢃⡀⠀⠀⣾⢷⣶⡶⣶⠀⠀⢠⡤⡤⢠⢤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⣄⣄⠀⠀⠀⠀⠴⣷⡄
⠀⠀⢶⣿⣾⣿⠀⠀⠀⣷⣴⠿⠟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⡆⡇⢠⣾⠀⠀⠰⣷⠀⠀⣾⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⢰⣿⡇



⠀⠰⠶⠆⠰⠆⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⣀⣠⣞⣟⡷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠⠤⠤⠤⠤⢤
⠀⢀⡀⠰⢆⡰⠆⠀⠀⢠⣄⢠⣤⠀⠀⠈⢿⡾⠾⠁⠁⠀⠀⣶⣦⣶⣤⣤⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢺⣧⣼⠇⠀⠀⠀⢸⢡⡆⡟⠀⠀⢸⠹⠀⠇⠳⢸
⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠁

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣐⡿⢣⣤⡄⠀⠀⠀⠄⠛⣿⡆⠀⠀⣀⣀⣤⣄⡀⠀⠀⠠⠀⠀⢂⠀⠀⠀⠀⠀⢠⣮⠛⠀⠀⢠⠊⡉⣳⣀⡀⠀⠀⠀⢰⣦⡤⠀⠀⠀⣄⣰⡀⡄
⠀⠘⠿⢣⣾⠍⠀⠀⠘⣿⣦⡄⠈⠀⠀⠀⣿⡏⣶⢎⣿⠀⠀⠀⠱⣖⣻⡆⠀⠀⠀⡴⠟⠁⠀⠀⠀⠘⢄⣀⠜⠉⠁⠀⠀⠛⢻⣿⠟⠀⠀⠘⡘⢤⢼⡘
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠈


⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⣼⣿⣿⣧⠀⠀⢸⢾⡯⡭⡿⠇⠀⢀⣾⣶⣶⣾⣷⠀⠀⠠⠤⢤⠤⡄⠀⠀⢠⣼⣟⡷⣤⠀⠀⠀⠀⡖⠿⠀⠀⠀⠀⠀⡠⠚⠁⡀⠀⠀⣤⣶⠀⠀⡀
⠀⠀⠘⠾⠷⠃⠀⠀⠈⠀⠀⠁⠁⠀⠀⠈⠻⠟⠉⠻⠟⠀⠀⠠⠤⠦⠹⠄⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠶⠀⠚⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠛⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠐⠠⠢⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⢸⣿⡇⠈⠁⠀⠀⢰⣋⣟⣹⣏⡇⠀⠀⠘⠁⠀⠁⠀⠀⠀⠀⡏⠀⢹⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠦⢀⢀⠂
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠁


⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣿⣿⣼⣷⣶⣧⣼⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣷⣶⣾⣿

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣾⣤⣦⣤⡀
⠀⠀⠀⠀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⢻⣿⣿⣿⣿⠁⠈⠈⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠛⣿⣿⡟⠐⢤⡀⣤⠤⠂⠀⠛⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠘⣿⡇⠀⠘⠀⠹⠂⠀⠀⠀⠈⢹⡿⣻⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⠤⠂⠙⡄⠀⠀⠀⣸⣱⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢀⠾⠶⢻⣕⣶⠖⠀⠀⠀⣠⣿⡿⠛⠋⠁
⠀⠀⠀⠀⡠⠆⢙⡿⠶⣼⣿⡷⠃⠀⣠⣾⣿⣿⣿⣶⣶⣶⠒⠂
⠀⠀⢀⡴⠓⣲⣾⣷⣤⠌⣿⣥⣴⢎⣙⠋⣿⢿⡿⠟⢁⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⣰⣿⣿⣿⣾⡟⠀⡧⠈⠉⠀⠙⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠊⡹⠀⣠⠊⠇
⠀⢠⣾⣿⡿⣿⠿⠿⠁⠀⡼⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠃⣴⠁⡎⠀⡠⢲⠂
⣴⣿⡿⠋⠙⢤⡀⠀⠤⠰⣇⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠸⣷⠇⡘⣠⠎⡰⠃
⠉⠉⠠⠀⠀⢸⠷⣀⣀⣤⣿⠀⠀⠀⠐⣦⣄⠀⠈⠉⠒⠒⠀⠀⠘⣧⠀⠀⠀⠰⠀⡀⢄⠀⢠⠃⠀⢀⠀⠛⢁⠌⣀⡠
⠀⠀⠀⢢⡀⠘⠀⠀⡉⠺⢿⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢁⠱⡄⠑⠎⠀⣰⣄⠒⠀⠀⢉⡩⠖⠋

⠀⠀⠀⠸⡄⢰⠀⠀⢨⣁⣾⡆⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⡟⣿⡀⠀⠀⠀⢆⠸⡀⠀⠀⢹⡿⠀⠀⣰⠃
⠀⠀⠀⠀⡇⢸⠀⡄⢄⡈⠀⢧⣀⣀⡀⠤⢞⡤⠀⠀⠀⠀⠀⢀⡼⠀⠘⣧⠀⠀⠔⠛⣲⢧⠀⢀⠘⠁⢀⡜⠁
⠀⠀⠀⠀⡇⠸⠀⠘⠂⠀⠀⢸⠧⠤⠤⠀⠁⠀⠐⠂⢐⠶⠾⣿⠃⠀⠀⠘⡇⠀⠀⠀⠀⠄⠀⢈⣷⡖⠉
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢀⣾⣷⠶⠂⠀⠀⠀⠀⠀⠀⠠⠚⣿⡀⠀⠀⠀⠘⢦⣤⣠⣬⣷⡽⠟⠉
⠀⠀⠀⠀⠀⠈⠳⠤⣀⡴⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠳⣄⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⣷⡄⠀⠀⠐⠶⠀⠈⢏⠛⢦⡀⢸⣧⣀⠀⠀⠀⢰⡀⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢹⣿⡆⢀⣀⠀⠤⠤⠼⠿⠶⣇⢸⣎⠻⣆⠀⠀⢸⢷⡸⡿⢦⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⠀⡀⠤⠄⠒⠒⠘⡖⠚⠺⡿⢿⣿⠀⡀⢸⠀⢻⢃⠀⠃⢸⢿⣄⢀⡀⢀⡀
⠶⠦⣶⣶⠞⠛⠛⠓⠢⢀⡴⠊⢸⣿⣿⡉⢀⠤⠾⠂⠉⣁⣩⡅⠈⣇⠀⢻⢽⣿⣿⠀⠚⠈⣏⠉⢹⣀⢹⡆⠀⠛⢸⡆
⠀⠀⠀⠈⠉⠓⠢⠤⣠⣋⠀⡠⢺⣿⣿⠛⢧⣴⣶⣾⣥⣈⠿⠁⠒⡧⠠⠉⡐⠄⣺⡐⢨⣾⡷⠀⢺⠂⠘⣥⣐⠛⣻⠴⣆
⠀⠀⠀⠀⢶⠆⠀⢰⢣⣿⣿⣇⠸⣿⣇⢀⣿⡟⢧⡞⠙⢻⡆⠸⠿⠷⠀⠀⠀⢁⣼⠀⠉⡩⢐⣬⣽⢴⣶⣍⣿⠛⠁⠀⠈⢷⡀
⠀⠀⠀⠀⠀⢀⣤⣇⠀⢻⠿⠟⡼⠙⢿⣧⡙⡿⠥⠈⠲⣼⠃⠐⢾⡿⠀⠀⠘⠛⠋⠐⠍⡀⠿⠿⠋⠀⠉⠁⢠⡄⠠⣦⣤⠖⠁⢴
⠀⠀⠀⠀⠀⢸⣿⣿⠏⣻⠶⢤⣇⣀⢸⠉⠻⠷⠖⠒⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠜⢵⣶⠎⠈⢀
⠀⠀⠀⠀⠀⣼⣿⣿⠀⣿⠀⠀⡇⠉⢙⠓⠶⣤⣀⠀⣶⣷⣀⠟⠀⠀⠀⢀⡀⣀⢀⡀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⡞⢉⡟
⠀⠀⠀⠀⠀⣿⣿⡏⠀⢸⢀⡴⣷⣶⣼⡀⠰⣿⣿⡆⠀⠉⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣧⠉⡄
⠀⠀⠀⠀⢠⣿⣿⣧⠀⠈⢿⣴⣶⡀⠙⣷⠀⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣸⣿⢿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢀⡀
⠀⠀⠀⠀⢸⣿⣿⠘⡆⠀⠘⡄⠀⢱⡀⠘⢆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣶⣶⣞⢲⡖⠀⠀⠀⠀⠀⢠⡆⢀⣀⡿⣶⡆⣾⡏
⠀⠀⠀⠀⠈⢉⡉⠙⠛⣖⠒⠻⡦⠤⠷⣤⣌⣢⠀⠸⣤⡀⣀⠀⠀⠀⠀⢸⡇⢿⡿⣿⢩⠿⣧⠀⠀⠀⠀⠀⠈⠁⡘⠻⢿⣤⠁⠉
⠀⠀⠀⠀⠀⠛⠃⢀⣀⡈⢶⣦⠘⣆⠀⢿⣿⡷⠑⣔⠿⠳⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢆⠌⠲⡄⠠⡤⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠨⠿⠶⠈⠻⣃⡠⠷⡌⢉⡱⣶⣾⡧⡀⠈⣢⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠖⠙⣦⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠢⣀⡼⠫⡀⢈⣻⣁⠈⣿⢋⣉⠫⡉⠀⠀⠀⣴⠇⠀⢰⠗⣼⠇⠜⣿⡠⠀⣨⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿⣧⠖⠉⢢⡠⠚⠻⣿⠿⠓⢷⠾⠛⢄⠀⠑⢄⠀⠀⡄⠀⠀⢢⠜⣄⡠⠊⢣⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣼⠟⢁⣤⡊⠁⠈⠢⡀⢈⡷⣶⣾⣷⢤⡀⠑⠢⣠⣷⢆⡘⡤⢊⠔⠱⡈⢣⣾⡗⠙⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢺⣧⠹⣷⡄⣀⠴⠊⠓⢄⠈⠙⠯⡀⠈⠢⣀⠹⠫⡀⢉⢘⣅⣠⡾⠋⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⢻⣧⡜⣿⣄⠀⠀⠀⠀⣙⡦⡖⣿⣿⣷⡌⠑⢧⣈⢔⣴⡿⠋⠀⠀⠰⣤⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡘⢿⣦⣠⠔⠊⢁⣀⠈⠛⢍⣩⠗⢆⡘⣷⡿⠋⠀⠀⠘⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠚⠉⠀⠀⣤⣈⠙⠋⢠⣾⠛⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢰⡘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣧⣀⠀⢘⣷⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢳⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠟⠛⠛⠓⠀⠀⠀⢀⠀⠀⠀⠘⠛⠛⠛⠿⠿⠿⠆⠀⠀⠀⠀⢸⡇⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢸⡇⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠂⠀⠀⣾⣿⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⢸⣷⠀⠀⠀⠀⠘⣿⣷⢀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⢀⣿⣿⣿⠃⠀⠀⢰⣿⣿⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠈⣿⣇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠁⠀⠀⢠⣿⣿⣿⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠹⣿⣆⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⢠⣾⣿⣿⣇⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡹⣿⣷⡀⠀⠀⢀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡄⠀⢀⣴⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⣄⡘⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⣠⣴⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⡿⠿⠿⣶⣤⣤⣀⣀⣀⣤⣤⣶⠿⠟⠿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠑⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠒⢦⡀⠈⢁⡔⢁⢍⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢨⣀⡠⠶⠖⠚⣛⣛⣉⣉⣉⢭⡽⠃
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⣇⠐⡩⢦⡘⠒⣡⡈⠢⡀⡰⣡⠔⣒⣒⢠⠃⠀⠀⠀⠀⠀⠰⠯⣥⠶⡖⠉⠙⢉⠁⠀⠈⢸⡄⠀⡇
⠀⠀⠀⠀⣶⡋⢻⡆⠙⠲⠤⢤⠔⡡⣊⡴⣛⣳⣄⠉⢶⠄⠈⠳⡇⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠠⠈⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⢎⡴⣫⣾⢟⣵⣿⣦⡀⠀⠀⡤⠜⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠠⠄⣿⠀⡇⠀⠀⢸⠃⠀⠀⠀⠃⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⢔⣹⠿⣋⣴⡷⡄⠀⠘⢆⠀⠻⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⣀⣀⣰⣦⠦⠤⠚⠛⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠻⡥⣪⢟⣭⡾⢟⣦⠀⠀⢳⣀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣉⣉⣉⠥⠤⠤⠒⠒⠚⠋⠻⣧⣤⣤⣤
⠀⠀⠀⡠⠂⠀⠀⠐⠄⠄⠀⠾⠿⠀⠀⠀⠘⢷⠟⡫⣴⡿⡫⣣⡀⠀⠉⠀⢱⣀⡀⠀⠤⠠⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⢀⢀⠠⠇⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⠈⠉⡂⠀⠻⣊⢕⡪⣞⣭⣷⣂⠀⠀⠀⢷⢓⠲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿
⡎⠀⠀⠀⠀⠀⠀⠀⠸⣴⡀⠀⠀⠀⠀⢠⠇⠤⠤⢽⣯⡪⢟⣫⣶⣷⠂⠠⣶⢸⠀⠀⠀⢔⣢⠒⣼⡂⣀⠀⠀⢀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠈⠀⠀⢒⢀⡀⠀⠀⠀⠈⠙⢶⣤⡀⠤⡼⢦⠤⣤⣃⠘⣾⣟⠟⣋⢼⡧⠤⠒⠉⠀⢰⠒⣤⠿⣋⡡⠄⠈⢷⣾⣷⣄⠈⠉⠏⠀⠙⢯⡁
⠀⠀⠀⠈⠁⠙⠦⣄⣀⠀⣰⡪⠛⢧⢠⠇⠈⠲⠃⠈⠉⢙⣶⣿⣿⡉⠀⠀⢀⡤⠒⠉⠱⡜⢧⠀⠐⠂⢀⡼⠈⢿⣿⣆⠎⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡹⠒⠒⠒⠢⣤⠀⣾⣿⣿⠋⣡⠴⠊⠁⠀⠀⠀⢀⣵⢈⡧⢴⠊⡿⣦⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠘⠂⠤⠼⠁⠀⠀⠀⠀⢹⠀⢻⡿⠗⠉⠀⠀⠀⣀⠴⠒⠉⠉⠈⢋⠀⠛⠃⠈⠁⠀⠀⡟⣠⡆⠀⠀⠀⠀⠀⡟⠫⠯⠃
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠒⠉⠀⠀⢀⡤⠖⠛⠁⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠰⣟⠄⠃⠀⠀⠀⠀⠀⠃⠀⢀⣤
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠴⠚⠉⠓⠦⡄⡀⠀⠀⢰⣿⢋⠽⢻⣾⣧⡀⠀⣀⠀⠄⠀⠈⠀⠀⠀⠀⢸⣶⣤⡀⠉
⠀⠀⠀⠀⡄⠀⣀⡀⠀⣬⠀⠀⠹⡆⠀⠀⠀⠈⠳⣍⣉⠀⠀⠀⠀⠀⢄⠈⠳⠀⠀⠸⣷⣝⠊⣠⣿⡴⠋⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷
⠀⠀⠀⠀⣷⠈⣿⣇⠀⠙⠧⣄⠼⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣶⣶⣬⣉⠀⠀⠃⠀⠀⠈⠓⠶⠿⠿⠅⠀⠰⠆⣉⠢⣄⠀⠀⠀⠀⢠⡿⠿⠟⠁
⠀⠀⠀⠀⢸⡆⢹⣜⡄⠀⠀⠀⠀⠀⠀⢀⠔⠉⢱⣶⣿⣿⣿⣿⣿⠟⠉⠛⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠳⢄⠀⠀⡜
⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⢀⣿⠿⠿⠻⣿⣿⣄⣀⣀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠀⠒⠒⠾⢷⣈⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠉⣀⡤⠔⠊⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣄⣠⣤⡶⠖⠉⠉⠀⠁⠢⣀⠀⠀⠠⡖⠒⠢⣝⢆⠀⠀⢱⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⠰⠋⠈⠀⠙⢦⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⣉⣀⣀⠤⠤⠀⢠⣤⣈⡑⢦⡔⠋⠀⠀⠀⠀⢣⠀⠀⠀⠠⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⢣⠀⠀⠀⠀⡀⡇⠀⠘⢿⣿⣿⠿⠋⠀⠀⠹⡄⠀⡰⠁⠀⢸⡿⠋⠁⠀⡍⠓⠢⠤⠤⠴⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠑⠤⠤⠤⠜⠁⠀⠀⠈⢟⠁⣀⠤⠤⠤⣄⠘⣴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠠⡄⠀⠀⠀⢀⠏⠒⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⡠⠤⡄⠀⠀⠈⢎⠃⠒⠒⠒⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠓⠤⠂⠀⣀⣠⣼⠀⠾⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣤⣴⣾⡿⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣹⣛⣩⣙⣙⣟⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀⠀⢀⠤⠒⡤⡀⠀⢀⡠⠤⣀⠀⢀⡀
⠀⠀⢸⠏⢳⣝⣿⣦⣙⣷⣌⠻⣿⣕⡄⠈⣁⠤⠐⠒⠉⠁⢡⡁⡇⠀⠘⠻⠅⠀⡓⢓⣈⣅⢡⠔⠡⡄⢖⡄⠱⡈⠉
⠀⠀⡾⢄⣀⣦⣤⢀⣀⠀⠈⠉⠉⠉⠁⢂⢀⠀⡀⠀⠀⠀⠘⠞⡇⢸⠲⠤⣔⣒⠽⣼⣧⠔⠉⡇⠉⡥⡠⣒⢂⠇
⠀⢠⠁⠉⠙⠃⠶⠾⣛⣩⣵⡶⠦⣤⡄⠀⠉⡏⡝⠉⣀⡒⠒⠓⠁⠘⠚⡲⠢⢌⡉⠑⠺⣿⠒⠸⢦⣭⡡⠴⠊⢀⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠫⠃⠀⢀⠑⠃⠀⠉⣁⢀⡀⠀⠀⢠⠃⠀⠀⠀⠉⠡⠀⠀⠀⠀⠀⠈⢳⠀⠛⠃
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠽⢿⠃⡤⠂⠉⠀⢀⠈⠉⠢⣌⠒⡦⠤⣀⡀⠀⠀⢀⠀⠀⠀⢀⠎⠉⠁⡹
⠀⢀⣍⠁⠒⠠⠤⢁⣀⠀⠀⠀⠀⡇⠀⠀⢀⠎⠀⠀⢆⠀⠸⠁⢀⡠⠈⢣⠈⠁⡒⢩⡝⡒⠤⢌⡀⠀⣜⡀⠀⢰⠁
⠀⠘⠉⣀⡤⠔⠊⠠⣀⣈⠉⠒⠒⠀⠀⣠⡿⠀⣀⣀⠠⠆⣘⠂⠁⢀⡀⠈⡆⠹⣛⣀⣉⠐⠚⠓⢡⣌⠀⠈⠲⠃⣤
⠀⢠⠊⡀⠀⠐⠑⠂⠀⠌⢳⠀⠲⠖⠛⠁⢠⡀⠀⢀⣠⢴⡀⠺⡁⡡⠁⢀⠇⡴⠇⠤⢈⠓⡄⠀⡠⡔⠂⠀⢲⣄⠉
⠀⢹⣦⠤⣀⣁⣀⣀⣀⣀⣼⠀⢠⠿⠴⣄⢄⢢⡀⠉⠒⠰⠇⠀⠑⠁⢀⠜⠀⣇⢐⡒⢡⡅⡇⢔⠑⠒⠒⣂⡭⡊
⠀⠸⣿⠀⣿⠀⣿⠇⣿⢸⠇⢀⡖⢔⣒⠨⡉⠃⠙⠢⢄⣀⣀⣀⡠⠔⠃⠰⠀⣿⢨⠬⢸⡇⢱⠀⢷⢋⣴⢟⣴⢷
⠀⠀⣿⡄⣿⠀⣿⢰⡏⡞⠀⡾⢁⡨⠙⠃⡉⠀⡀⠖⠠⢀⠶⠰⣖⠀⠉⣿⠀⢫⠃⠫⠄⡧⢸⠀⠘⣟⣵⠟⣡⠾
⠀⣤⠙⠓⠻⠶⠿⠾⠓⠁⣸⠐⢌⠛⣣⡐⠀⡾⠂⡖⠄⡘⠄⠀⣿⠆⠀⢻⡆⠘⢄⡀⣓⣠⠚⢠⡄⠘⣥⢞⣥⡾
⠀⣉⣤⣀⣠⣄⣠⣒⣂⣄⡈⠐⠀⠀⡰⡠⠊⠀⠠⠛⠂⠌⣃⣄⢩⣇⣀⣸⣇⡀⠀⣍⠉⠀⠀⢠⡀⠀⠀⠉⠀⢤
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⠀⠘⢠⣤⢄⠀⠀⠀⠀⠉⢉⠾⣿⣴⣟⣻⢆⢀⣸⣧⠰⠆⢠⣀⣀⡀⠀⠀⣠
⠀⠉⠢⠠⠠⠂⠒⠤⠤⠔⠊⠘⠛⠁⠘⠠⠀⠀⠈⠑⡇⢀⠔⠁⢦⢆⠀⣤⠈⣿⠀⠀⢿⡄⢰⣽⡃⠀⠀⠀⡀⣹⠢
⠀⠐⠂⠒⠂⡆⠀⠀⠀⢀⠀⠀⠀⠐⠈⠐⠐⠉⠁⠀⠉⠁⡀⢴⠟⢢⠠⠁⠀⠶⡇⠀⢹⡇⢰⣏⢁⣀⠀⢤⣠⠀⣽
⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠐⠦⡀⠀⠀⠀⠀⠐⢿⢶⠶⠥⢐⠡⠀⠠⠱⠑⡀⢸⣥⡀⠸⠗⣀⡙⢦⠑⣷⡺⢃⡴⠁
⠀⢰⠀⠀⠐⠀⣸⣿⣷⣿⣬⣀⣵⠄⠃⠀⠀⡄⠈⢢⠣⠀⠀⠰⢐⡂⡂⠱⠝⠀⠀⣠⠚⠁⠐⡵⡌⢻⣟⢻⣟⠐⠗
⠀⢸⣄⡠⠂⠀⠸⣄⡜⢛⣻⣿⠏⠀⠈⢀⣀⡇⠀⠀⢣⠑⡈⠚⣉⠡⠌⣁⠂⠀⢸⡠⠉⢀⢀⡤⣧⠋⡹⢧⡑⢷⡀
⠀⢸⠻⣄⠀⠀⠀⠈⠉⠸⠋⠁⠀⠀⠀⣠⠟⡇⠰⣿⡇⠛⠊⠀⢈⢦⣤⣤⠚⢥⠺⠓⠭⠔⡉⠀⢡⡜⠁⠀⠳⣾
⠀⠈⠓⠚⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠒⠂⠁⠀⠈⠙⠂⠀⠘⠋⠀⠊⠉⠉⠉⠁⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠏⠉⠙⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⠃⠉⠀⠘⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣂⠀⠀⠀⢀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⡗⠶⠖⣿⣬⣿⡄⠀⠀⠀⠀⠀⠀⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⠿⠃⠐⠒⠒⠀⠿⣿⣷⣤⣀⣀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣤⣴⣶⣾⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⣿⣿⣿⡆⢰⣿⡏⠉⠛⠻⠿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠉⢙⣋⣛⡛⠻⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⣿⠃⠀⠀⠀⣀⣀⣀⠈⠉
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣦⣄⠀⢤⠀⠀⠀⠀⠀⠘⢋⣤⣶⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠚⠏⢿⣿⠏⠟⠚⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠐⠚⠛⠒⠂⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷

⣷⣷⣿⡟⠁⣠⡶⣿⣿⣿⣿⣮⣷⡍⠻⣿⣶⠶⣖⣻⣻⣛⣰⡆⠦⣬⢈⡛⢲⢤⡙⠑⠆
⣿⣿⠏⢀⠞⡙⠂⠘⢹⣿⣿⣿⣿⣿⣿⣮⣻⣿⠁⣇⠉⠈⠹⠙⢋⠗⡶⣍⡑⠦⠛
⣿⡟⠀⡞⠰⠀⠀⠖⢙⠟⠛⢯⡿⣿⣿⣿⣿⣿⠿⠛⠿⠛⠙⠟⡶⢦⣤⣀⠛⠓⠲⢄⠀⠀⠀⠀⠀⢀⣤
⢭⠇⠀⣧⠁⠀⢻⠎⠁⠀⠀⣾⡿⢿⣾⣿⢿⣿⣯⡙⠿⢦⡘⠷⣷⣿⣭⣇⠀⠀⠀⠀⢣⠀⢀⣠⢾⣿⣽
⣧⣾⡀⢲⡀⡐⠀⠀⢰⣿⡆⢀⣴⣾⣿⣕⢌⣿⠟⣮⡢⡈⠟⢶⣽⠿⣿⣿⣿⣤⠀⠀⢠⠐⢿⣿⣿⣿⣿
⣷⣿⣇⠀⠛⠀⣠⡤⡀⠉⣰⠟⣵⣯⣙⣿⢿⣟⠻⣿⣺⡯⡢⣈⠻⣷⣜⠿⣿⣿⣷⣤⠜⠃⣀⢿⣿⣿⣿
⣿⣿⣿⣷⠀⠀⠻⠷⠇⣼⡇⣽⣿⣿⣻⣼⡻⣿⣷⣝⣻⡮⡪⡻⣅⣍⠙⢳⣞⠻⣿⣿⡄⠀⠉⣈⣿⣿⣿
⣿⣿⣿⡇⣠⡖⢦⠀⣸⠿⠗⣿⡿⠛⠛⣿⡿⠿⢿⣟⣛⣿⣛⠊⠪⡳⠕⢌⠹⣷⣮⡉⠙⢄⠀⢀⣸⣿⣿
⣿⣿⣿⠀⠻⣿⠿⠃⣿⠃⠘⣿⣁⣴⣾⡟⢂⣴⡌⣿⠿⠿⢿⣯⣦⡁⠀⠈⠳⣌⠿⡿⠀⠀⠀⠀⢼⣿⣿
⠿⣿⣿⠀⠀⡠⢄⠀⢿⡀⠀⢻⣿⣿⣿⣷⣿⣿⣷⢸⠀⠀⢀⣿⣿⣼⡢⡙⣆⠈⠀⠀⠀⠀⠀⠀⡀⣿⣿
⣷⡿⢿⡄⠸⣷⡾⠂⠸⣧⠀⠈⣿⣿⣦⢿⣿⣿⣿⣤⣷⣤⣾⣿⣿⣿⣿⣮⡺⣷⠄⠀⠀⠀⠀⠀⠑⢎⡛
⡟⣷⣾⣇⠀⠀⣠⠤⠀⢻⣇⠊⠈⢿⣿⣷⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢮⠀⠀⠀⠀⠀⠀⣇⣀⠑
⣟⡿⣿⣿⡄⠐⣿⣷⣶⠀⠁⠀⢀⡀⠀⠙⠿⣿⣾⣿⣿⣿⣄⣀⣽⣿⣿⣷⣤⣼⡇⠀⠀⠀⠀⣨⡃⡝⢴
⣿⣿⣿⣿⣿⣄⠈⠉⠁⠀⢀⣾⣯⣀⣡⠀⠀⠀⢀⣩⡿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⢀⡪⣻⣮⡓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣻⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠿⠿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠐⣤⣴⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⡀⠂⢈⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣷⣄⣀⣹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣅⢉⣝⣥⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣁⠚⢿⣦⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⡉⠻⣿⠎⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠛⢻⡷⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡍⠉⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣏⣹⣧⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⣤⣼⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣤⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣉⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣯⣾⣿⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⡵⣇⣈⣤⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠐⠀⣻⣷⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⡗⣠⣼⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣩⣁⡤⠿⠏⡧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠃⣿⣿⣿⣿⣿⣿⡇⠀⠈⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠃⠀⠈⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣿⣿⣿⣿⣿⣿⡇⠀⣤⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣾⡴⠂⠿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠛⣛⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⠻⢿⠟⠛⠛⠛⠋⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠙⠻⢶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⠷⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠾⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠻⠶⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠻⢶⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣤⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣹⣧⣤⠀⠀⠀⠀⢀⣸⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣄⣀⣀⣀⠈⢺⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀
⠀⠀⠀⢤⣴⣧⠄⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⣾⡅
⠀⠀⠀⠘⠉⢃⣴⣿⣿⣿⣦⣤⡶⠞⠋⠁⠀⠀⢀⠀⠆⠀⠂⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣔⣾⡯⢤⡿⡝⠛
⠀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⣷⠿⣿⢾⡇⠙⠁⠀⢰
⠀⢿⣿⣿⣿⣿⡿⠛⠿⠿⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⣀⣀⠔⠈⠈⠐⡌⠧⠉⠁⠀⠀⠀⡠⠃
⠀⠀⠛⠿⠿⠋⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⣣⣤⡀⠸⠟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠾⣶⡶⢶⠿⣶⢶⡶⠶⢾⡿⢼⡿⠛⢷⡄⠀⠀⣠⡎⠀⠀⠀⠀⠀⢈⠠⠠⠃
⠀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⣿⢀⡈⡀⣸⢠⣧⠈⠱⣯⣼⡇⢸⢁⡿⠛⠃⠀⠛⠋⠀⢀⡀⠀⠀⠀⠠⠒⠒⢢
⠀⣀⣠⡄⠀⠀⡀⠀⠀⠀⠐⠂⠀⣿⢸⣿⡇⢹⠈⠋⢸⣷⣄⠙⡇⢸⡀⢿⢶⡄⠀⠀⠀⠀⠈⠡⠀⠘⠁⠠⠟⠒⢲
⠀⠘⠿⢁⡄⢦⠙⠀⠀⠀⠀⠀⠀⢿⡼⠛⠷⠾⢷⣴⠟⢷⣤⠾⢷⡾⠷⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡭⣅
⠀⢀⠔⢤⠙⠀⡀⠀⠀⠀⢰⡆⣾⠀⠀⢰⡆⣶⢠⣶⠀⠀⣶⠀⡷⠀⠀⣶⠀⣶⢰⡇⠀⠀⠀⠠⠀⠀⠀⠀⠔⠠⢬
⠀⠈⠙⠀⠀⠀⠀⡠⠀⠀⠸⠇⠻⠀⠀⠸⠇⠿⠘⠟⠀⠀⠻⠐⠇⠀⠀⠛⠀⠛⠘⠃⠀⠀⠀⠀⣠⣄⠀⠠⠖⠒⢦
⢰⠆⠀⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠠⠧⠤⠤
⢀⡚⠀⠖⠄⠀⡄⠀⠀⠀⢀⣠⣄⣠⣾⣶⠀⢠⠀⠀⢀⠀⠰⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣄⣈⣤
⠀⠉⠉⠲⠒⡑⣉⠤⠖⠦⢺⣿⣿⣿⣟⡟⢀⣾⣿⡖⠈⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢡⣙⢻⢒⡀⠀⠀⠙⠿⣿⠏⠀⠀⠈⠹⠀⠀⠀⠀⠛⠀⡿⠀⠀⠀⢀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠙⣿⣿⣿⣶⣿⡶⠄⠀⠀⠀⣾⣿⣿⣿⣿⣷⣀⡀
⠀⠀⠀⠀⠀⢰⡏⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⢁⣴⣿⣶⣾⣷⣤⣀⠀⠀⠀⠠⣶⣶⣤⣤⣶⣦⡄
⠀⠀⠀⠀⣠⣾⣷⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⢁⣿⣿⡿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⣿⣿⣿⣿⠏
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠀⢸⣿⣿⣿⣿⣿⠿⣿⡿⠋⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣴
⠀⠀⠀⣾⡿⠛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡆⠀⢿⣿⣿⣿⡏⠀⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣶⣤⣴⡾⠛⠁
⠀⠀⢀⡿⠀⠀⣿⡟⠛⠛⢹⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⡍⠀⢀⡼⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⣸⡇⠀⠀⣿⣧⡀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠉⠁⠀⢀⣿⡏⠁⠀⢺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢀⣿⠃⢻⣿⠉⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢾⠏⠀⣴⠟⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡀⠀⣅⠀⠀⣧⡂⠀⠀⢀⠀⡀⡀⠀⡂⠀⠀⢸⢀⠠⠠⢠⠀⡄⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠈⠀⠁⠀⠀⠁⠁⠈⠈⠀⠀⠀⠁⠀⠁⠁⠀⠈⠈⠁⠀⠈⠀⠁⠁⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⠀⠀⣠⣤⠄⠀⠀⠀⠀⠀⣤⣤⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⢰⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⣀⠈⠛⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⠎⢻⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡤⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠋⣏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄
⠀⠀⠂⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⡄⠀⠀⠀⢠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢸⣇⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢻⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠙⠦⡀⠿⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢻⡏⠀⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠛⠛⠛⠻⠗⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣤⣾⣿⣷⡀
⠀⠀⠀⠀⠀⣤⣀⣀⣰⣿⠷⠶⠿⠿⢿⡏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⣿⠿⠿⢿⣿⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠈⠙⣻⣿⣿⣿⣿⣿⢶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⢿⣿⠏⠉⠈⠁⠉⠉⠛⠿⠇
⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠀⣤⣽⣿⣧⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣯
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢴⠆⢶⠆⢠⣾⣷⣶⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡀
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⣾⠇⠀⢹⡇
⠀⠀⠀⠀⠀⠈⠉⢻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠰⣦⠈⠻⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⢀⣤⠄⣠⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢤⣶⣾⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⢀⠔⢹⠑⢄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣘⡿⠿⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠂
⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⣀⣀⡀
⠀⠀⠀⠀⢹⡿⣿⣿⢿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠯⠿⠿⠽⠽⠽⠿⠿⠯⠯⠯⠿⠿⠽⠽⠽⠽⠿⠇
⠀⠀⠀⠀⢸⡁⢀⣀⢈⣿⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣼⣿⣼⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣶⣶⣶⢲⢲⢲⣶⣶⡖⡖⣶⣶⣶⣶⢲⣲⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣾⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡟⠀⢿⣺⣿⣯⣱⣶⡶⠐⢒⣠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢬⣽⣿⣿⣿⡇⢘⣟⣿⣿⣿⣿⣯⣶⠟⣫⣥⣶⣶⣮⢑⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣋⣥⠀⠨⠿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⡋⠈⠻⢍⡉⠉⠑⠻⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠞⠋⢁⡤⢶⠖⠛⠉⠩⢛⠝⠉⠿⠟⣿⠿⡟⢯⠛⢷⡉⠓⢦⡀⠐⠥⣀⢀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢋⡤⠚⠃⠀⠀⠁⠀⠠⠎⠀⠁⠀⠀⠈⢀⠉⠀⠱⡀⠑⠄⠈⠀⠀⠙⠆⠀⠉⠀⠙⢦⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡵⠋⠀⠀⠀⠀⠠⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠀⠀⢂⠀⠀⠀⠘⣦⠀⠀⢮⠀⠹⡅⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠠⠀⠀⠀⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⣿⠀⠈⢇⠀⠹⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣷⣤⣤⣬⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠄⠀⠀⢆⣠⣾⣿⣿⡄⠀⠈⠃⠀⢻⠇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⣀⣤⣾⣿⣿⣿⣿⠹⣶⠀⢠⠀⠈⣶⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢊⡂⠀⡂⠀⢮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⠟⠃⠀⢻⡇⠘⡆⠀⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⡇⠀⠈⣷⣌⣉⠛⠛⠛⢿⣿⣿⠟⠉⢀⡀⠀⠀⠀⠈⠻⢿⣿⣿⠿⣛⠡⠂⠀⣀⣾⡏⠀⢻⠄⠘⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡆⠀⠃⢦⡀⣹⠀⠀⢸⡇⠈⣿⠁⠻⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⢿⡏⠉⠁⠀⣰⣶⣿⠋⣷⠀⢸⡄⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⢠⠸⡿⣷⣤⢀⡘⡇⠀⠛⠀⠸⡆⠀⠀⢧⠀⠀⠰⠀⠀⠀⣼⣀⣸⣦⣼⣿⣿⠇⠀⣿⠀⠘⡃⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠈⠄⠙⢻⣿⣿⣿⣿⣶⣤⣶⣦⣧⣤⣶⣎⣠⣦⣴⣶⣷⣾⣿⣿⣿⣿⡟⠜⢃⠀⠀⣿⠀⢠⣥⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠘⣧⠀⠈⠀⠉⠿⢿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻⠀⠀⣹⠀⢀⣛⠀⣼⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢻⣆⣸⡀⠀⠀⠈⢛⠣⠷⣏⡛⣿⠿⢿⣿⣿⣿⡿⠿⠟⠛⠰⠀⠀⠀⠀⠸⠀⠼⡋⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣾⣷⡀⠀⠀⠘⢧⠀⠯⠁⠼⠋⠛⠁⠀⢉⠁⠀⠀⠀⠄⠀⢨⠀⢰⡃⣠⣗⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⡄⢆⠀⢻⣦⠀⣧⠀⣼⠀⢀⠀⢸⡆⠀⠀⣲⠆⢠⣿⢠⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⣧⣘⣿⣧⣻⣧⡹⣷⡀⡄⣼⡇⢀⣾⣿⣠⣿⣷⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⠷⠾⠿⠿⠛⠟⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢠⣶⡀
⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡷⠃⣿⣿⣿⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⢸⣼⡇⠀⠀⢰⣄
⣿⣿⣿⣿⣿⣦⣶⣶⣿⣿⠋⣀⣾⣿⣿⠟⢹⣿⣿⡟⢙⣿⡇⠀⠀⠀⠀⠀⠀⡟⠉⢹⡀⠀⣴⣿⡋⠀⠀⢠⣼⠈⠀⢰⣶
⣿⣿⣿⣿⠟⢻⣏⣁⣿⡇⢸⣥⣦⠘⣿⣷⣿⣿⣿⠗⣿⣿⡇⠀⠀⠀⠀⢀⣤⡇⠰⡿⠇⢸⣯⡞⡟⢻⡄⠈⠉⠀⠀⣾⡇
⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡮⢿⣹⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢙⡦⢷⠞⠁⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣘⣹⣿⣿⠉⣹⣏⣀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠻⣿⡾⠀⠀⣾⡷⢰⠿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⠿⠟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣇⣀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡴⠏⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣶⣦⣄⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣿⣿⡿⠋⠀⠀⣀⡈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠔⢋⠉⢺⠃⠀⠐⣿⣿⣇⠀⠀⣾⣿⣿⡆⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⢿⣿⣿⣿⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠁⠑⣿⠀⠀⠀⠙⢿⣿⣷⣦⣝⣿⣛⣱⣿⣿⣿⣿⠋⣴⣾⣶⡄⠀⠈⣿⣿⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢻⠟⠛⠿⣿⣿⣄⠻⣿⡿⠃⠀⢀⣿⣿⣿⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠛⠿⢶⣶⣶⣶⣿⣿⡿⠋⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠰⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⣀⠈⣿⣷⣶⣤⣶⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠳⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⡴⠋⡤⠛⠛⠈⠙⢏⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⡟⠀⢰⠮⠡⠀⠀⠀⠘⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⠔⢇⡀⠀⢀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢰⣿⣇⠀⠘⢢⣄⠀⠀⠀⢀⣷⣤⠞⠉⠉⠉⠙⣶⠚⠋⠁⠀⠀⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠈⢿⣿⣶⣶⠊⠀⠑⣤⣴⣾⣿⠃⠙⠦⡤⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⡱⠁⠀⠀⠀⢻⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⣀⠤⡞⠁⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⠁⠀⡇⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⣀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠿⠤⠤⡀
⠀⠀⠀⠀⠀⡴⠚⠶⡄⠀⠀⣠⡴⠞⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠚⣉⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⢸⡁⠀⠰⠹⣤⠞⠁⠀⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠎⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠘⣇⠀⠀⡆⠘⠆⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣆⠀⠸⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⠤⣒⠚⢲⣦⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠘⠦⣀⣸⠤⠴⠖⠋⠉⠉⠉⠉⠉⠙⠋⠒⠖⠲⠖⢶⠏⣀⠤⠒⠉⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠤⠤⠀⠀⠀⡀⠄⠀⠀⠀⠀⠄⢀⠀⠄⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠐⠂⠐⠒⠀⢀⠐⠀⠀⠈⠐⠠⡀⠀⢀⣀⣤⣤⠀⢈⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠂⠐⠠
⠀⠀⠀⠀⠀⠒⡄⠄⡰⡋⠄⠂⠀⠀⡀⠀⠀⢠⡾⣶⣄⡀⠀⠘⡈⠛⠻⣿⣿⣷⣶⡀⠀⠀⡀⠀⠀⠀⠀⢐⠁⠀⠈⠈⠀⠀⡀
⠀⠀⠠⠀⠀⡈⠀⠀⢁⠀⠄⠀⢡⠀⠡⠄⠠⣈⠀⠀⠀⡄⠀⢁⠑⠀⠀⠈⠉⠁⢻⣿⡀⠊⠀⠂⢀⠀⠀⡜⢂⠀⠀⢀⡂⠀⢄
⠀⠀⠀⠀⠈⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⠒⠀⢎⠙⣧⣀⠀⢀⠌⢣⠀⠀⠀⠀⢀⣸⣿⠇⠀⠀⠀⠂⠈⠞⠀⠄⠠⠀⠒⠉⠀⠄⠢
⠀⠀⠀⠀⠈⢃⠁⠀⠀⢡⠀⢠⠀⠐⠀⠈⡈⠀⢠⡈⠻⣿⠀⡀⢀⠋⠀⠡⡈⠛⠉⠉⠀⠈⠀⣘⣁⠜⢀⠀⠀⠀⡇⢄⠐⠀⡀
⠀⠀⠀⠀⠂⢈⢂⠱⣿⣿⣆⠀⠄⠀⠁⡀⠀⠠⠀⠕⣄⠙⡆⠁⡀⠇⢀⠀⠈⠀⠀⡠⠋⠙⣠⢠⡀⠀⠄⢀⠠⢀⣇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣠⠈⣷⡙⠿⠿⡦⠐⠀⠀⠀⠀⠀⢀⣼⠟⣠⣿⠂⠀⠀⡌⠐⠠⠀⢀⡂⠀⠀⢄⠶⢀⣀⡴⠁⠀⠀⠀⡀⠀⢀⠐⠰
⠀⠀⢀⠀⢀⠀⢤⡜⢹⣄⠲⣤⠠⣄⣀⣠⡥⠶⠟⣡⡔⠙⢿⣤⣤⡌⠀⠑⠠⠀⠐⡆⠀⠀⡘⠐⣼⠁⢺⡈⠀⠀⠴⠁
⠀⠀⠂⠀⣴⠰⣿⣇⠻⢿⣧⣘⢂⣿⣟⣿⠟⠒⠓⠈⡙⢢⡰⠟⢅⣠⢶⡷⢠⠀⣰⠁⢁⢀⠃⠼⠳⠞⠉⣿⡶⡃⠀⡀⢠⡠⠀⠀⡀
⠀⠀⠐⠀⠘⣧⡙⠿⣿⣿⠿⢫⡜⣿⠟⠁⠀⠀⠀⠀⠀⠂⠔⠁⣠⠁⡸⠃⡢⠔⠁⠀⠀⠒⠌⠀⠀⠀⣠⡿⠁⢊⠕⠄⠅⠀⠠⠁
⠀⡄⠀⠐⠄⠹⢿⣶⣶⡶⠾⢿⡷⠀⠀⠀⠀⠀⠠⠊⠄⠀⠀⠀⠌⠉⠀⠊⠐⠀⠀⢀⠀⠈⠀⠀⡠⣾⡟⠁⠀⠈⠐⢂⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⣴⡆⣧⡈⠃⠀⠀⠀⠀⠀⠀⠈⠁⢂⠀⠱⠀⠀⠂⠀⠀⢀⠁⠀⠰⢰⣷⡌⡄⠀⡔⡀⠞⠁⠀⠘⣷⡀⡀
⠀⠀⠀⠀⠀⠀⣿⡿⠓⠄⠙⠛⠋⣡⣄⠀⠀⠀⠀⠨⠀⠀⠀⠐⠀⣰⡀⠀⠀⢄⠀⠀⠁⠀⠀⠈⣭⡇⢸⡰⢧⣴⣉⢂⠀⣰⡿⠃⠁
⠀⠀⠀⠀⢀⠀⡛⠁⠀⠖⢢⢻⡿⠛⠋⠁⢄⠈⠂⢀⠒⠄⠀⠀⠦⠐⠈⢤⡀⠀⠀⠀⠀⠁⠀⣸⡿⢁⠇⠘⡦⡙⣇⣠⢸⠉
⠀⠀⠀⠀⠀⡐⢤⣴⡊⠀⢀⡌⠀⠀⢀⡀⠀⠐⢢⡤⠓⠀⠁⡅⡈⠉⠂⢔⠙⠒⠦⡄⠀⠦⢮⠩⠔⢍⠃⠠⣌⣷⢨⠄⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠸⢋⡵⠋⡰⠀⠀⠄⡛⠋⠀⠀⢁⠀⠁⠀⠀⠛⠀⠀⡷⠄⠰⡀⢀⡁⠀⠀⢀⢀⡀⠐⢄⡔⠛⢶⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠆⠈⠣⣀⠀⠀⠀⢀⠁⣶⡞⠀⠀⡀⠀⠀⠀⠠⠄⠂⡱⠁⠀⣘⠼⠀⠙⣄⠔⠃⠂⠁⡜⡀⢀⠀⠂⡄
⠀⠀⠀⠆⠀⡉⢐⠂⡆⠀⠠⢁⣾⣿⣶⡾⣿⠸⣿⠁⠀⠀⠘⠿⢀⠀⠀⠀⢨⡀⠀⠠⢉⠇⠀⠙⠏⣤⣤⡤⢊⣼⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠂⡑⠘⡃⡇⠀⢀⠸⢻⡿⠛⠛⣿⣆⢻⣷⠄⠀⠈⠠⠀⠀⣤⣄⠀⣼⣦⣄⠁⠀⠠⠒⠦⢩⣭⣶⡿⠁⠠⠁⠀⠀⠈
⠀⠀⠈⠈⠀⢀⠜⠡⠈⡀⠀⠂⠹⠵⠾⠃⣾⣿⣦⡀⠨⣁⣶⣶⣶⡿⠟⢁⣼⣿⣿⣿⣷⣦⡈⠀⠀⠊⠈⢉⠀⢠
⠀⠀⠈⠂⠠⠄⠀⠁⠀⠐⠦⣄⢀⠀⠀⠄⠙⠋⣹⣿⡶⠤⠍⠩⠄⠀⠀⠀⠈⡣⠊⠉⠻⢿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠈⠀⠒⠀⠠⠀⠀⠀⠈⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣭⣭⣛⣛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⢹⡟⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣶⠀⠀⢰⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣷⠄⠀⠄⢀⣀⠀⠀⠀⠀⠸⢯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠋⠉⠉⠁⠀⠀⠀⠀⠛⠙⠓⢶⣤⣄⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢟⣳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⡵⠄⠀⡄⢠⣶⠾⠿⠀⠀⠀⠀⠀⠀⢠⣯⣡⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣀⠀⢲⡄⢠⠀⠀⠀⠀⠀⢀⡀⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⣿⣷⠆⠀⠰⠆⠀⠀⠀⠀⢀⢖⣥⣶⣶⣶⣮⡂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⣠⣶⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢃⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⠴⠶⢤⣾⠟⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⢡⣾⣿⣆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠀⢠⢙⣛⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⠀⠀⠀⠀⠚⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⢀⠂⢀⡄⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠘⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⢶⡆⠀⠀⠘⠿⠋⠉⠁⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣸⣷⠀⠀⠀⠀⠀⠈⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⣧⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣦⣼⣿⣶⣤⣿⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⣿⡆⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣯⣰⣿⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠏⠁⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⢻⡿⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠈⠋⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⣶⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣾⣦⣄⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⠀⠀⠀⠀⠀⢀⡤⠒⠀⠐⠒⠦⢼⡆⠄⠴⠶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄
⠀⢶⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠀⠀⠀⡏⢷⠤⠖⢊⣅⠀⠀⠀⢀⣠⠞⠉⠙⠁⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠉⢿⣆⢦⣴⣶⣷⣶⣦⣂⡶⢠⡄⠀⠀⠈⠻⣷⣄⡀⠀⠈⠓⠄⠐⢈⡉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⢰⣏⡸⠛⠋⠙⣛⠻⠷⣾⣿⣿⣃⡀⠀⠀⠈⠻⣍⠐⠂⠀⢤⡼⠉⠙⢿⡆⠀⠐⠂⠀⠀⣤⣤⣤⣤⣤⡴⠿⠂
⠀⠀⠀⣾⠈⠻⣦⡀⠀⠀⠉⠒⢮⡛⢿⣿⣿⣆⠀⠀⠀⠀⠈⠐⠀⠸⡇⠀⣀⣠⡟⠀⠀⠀⠀⠀⠉⠉⠀⢀⠔
⠀⠀⡘⡛⠂⠀⠈⠻⣆⠀⠀⠀⠀⠙⢦⠙⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⠃
⠀⢠⠁⠀⠀⠀⣀⡴⢻⡿⣆⠀⠀⠀⠈⢷⠘⣿⣷⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡍⠀⠘⠓⢶⡄
⠀⣾⣄⡤⣾⡟⠉⠀⠀⣿⢹⡄⠀⠀⠀⠀⢷⠘⣿⣿⡀⠀⠀⠀⠀⠀⣀⠞⠘⠿⢿⣿⡏⢠⣀⣀⣺⠋⠁
⢮⣿⣿⢿⣿⠀⠀⠀⠀⣿⡆⣿⠀⠀⠀⠀⠈⡄⢹⡏⠓⠀⠀⠀⠀⣿⣴⢆⠀⢀⡀⣮⠁⣼⠋⠉⠉⢦
⠀⠉⠁⠘⠋⠀⠀⠀⠀⣿⡇⢹⡆⠀⠀⠀⠀⠉⠈⣷⡀⠀⠀⠀⠀⠀⠀⠑⠀⠋⡠⠉⠉⠉⠒⢄⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡞⠇⠀⡈⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⡈⠀⠀⢠⠁⠀⠀⠀⠀⠀⠈⠂⠄⡀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠐⠀⠉⠐⠂⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⢀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢢
⢀⣴⣾⣭⣒⠤⣀⠀⣾⠏⠐⠂⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⣠⠟⠉⠁⠉⡙⠒⡤⣄⠀⠀⠀⠀⠀⠀⠁⠺⣓⣄
⠘⢿⣿⡿⣿⣿⣦⣝⣿⣄⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠁⣤⡸⣿⣮⣻⣶⣤⣤⣄⡀⠀⠀⠀⠑⠅⠠⣀
⠀⠈⠻⣿⣎⢿⣿⣿⡟⣧⡄⢈⡥⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠈⠀⠀⠰⠀⠠⡄⢰⡞⣿⣷⣿⡌⠻⣿⣿⣷⣄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠘⢿⣷⣝⠯⣴⡉⣵⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠐⠆⠀⠲⣀⣀⣈⣿⣿⣿⣼⣻⠀⢿⣿⣿⣿⣷⣄⠀⢀
⠀⠀⠀⠀⠈⢻⡟⠲⢱⣉⣀⠈⠻⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢠⡄⠁⢙⣫⣿⣍⡟⣷⣸⠀⠸⣿⣿⣿⣿⣿⣦⡼
⠀⠀⠀⠀⠀⠈⠑⢬⠮⣭⣿⡇⠀⠀⠈⠙⠛⠻⠿⠾⠶⠶⠿⠾⠶⢾⣷⡀⠀⠠⠀⠀⢰⣏⣽⣯⣯⠇⡇⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠹⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠘⠿⠿⠸⠀⠸⡀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠈⠀⠂⣤⠀⣤⣕⣤⣀⣀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣦⡾⢦⡈⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡚⣤⡗⣿⣿⣷⣦⠙⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠺⠿⣿⣿⡄⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠹⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠙⠛⠷⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶
⣿⣿⣥⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠤⠤⢤⠀⠀⠀⠀⠀⠈⡔⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠈⠀⠀⠀⠈⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣤⠇⣀⣀⣿⡔⠉⠀⠀⠀⠀⠀⠀⠀⢍⠓⠦⣄⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠢⡀
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⣼⠃⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⢰⠃⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠈⣇
⢀⡏⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⡇⢹⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠸
⢸⠁⠀⠀⠀⠀⢀⡏⣿⣿⣷⠀⠀⠀⠀⠸⣿⣿⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣿⡇⠀⠀⠀⢸⣿⣿⡀
⡟⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣇⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⢰⠀⠀⠀⠐⠂⠀⠀⠀⣼⡿⠁⠀⠀⠀⣸⣿⡿⣇
⡇⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⠀⠀⠀⠈⠂⠙⠳⢦⣤⠴⠂⠀⠀⠀⠀⠐⢦⣴⣾⡿⠃⠀⠀⠀⣰⣿⣿⡇⠹
⡇⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⣿⣿⣿⡏
⡇⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠇⢀⠄
⢡⢰⠀⠀⠀⠀⢹⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣄⡀⠀⠀⢀⣠⣤⣴⣤⣤⣤⣿⣿⣿⣿⣿⡏⠀⡸
⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢠⠇
⠀⢇⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠉
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠉⠉⠉⠉⠉⠛⠻⠿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠠⣀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⣀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⡄⡀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠞⠁
⠀⠀⠀⠀⠀⠈⠳⢤⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⣀⣀⣀⣤⠤⠴⢆⣀⡀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⢀⣠⠞⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣷⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣯⣴⣤⣿⣭⣾⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦
⠀⠀⠀⠀⠀⢾⣮⣧⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠛⡟
⠀⠀⠀⠀⠀⠙⠛⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣿⣿⣇⠈⣻⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣧⠙⣿⡛⢠⣿⣿⣿⠀⠀⠀⠀⠀⣱⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⢻⣾⣿⣷⣿⣿⣿⣿⣇⣀⣠⣄⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣽⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣥⠀⠈⠻⠿⠿⢿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠉⢿⣿⣿⣿⣿⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢰⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⢠⣤⣤⣞⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠄⠀⠀⠀⠀⠉⠛⠻⠿⠆⠀⠀⠀⠸⠿⠿⠷⠦⠤⠤


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⡇⣿⠀⠀⠀⠀⢾⣿⣿⣦⡄⠀⠀⢰⣾⣷⣿⣶⠀⠀⠐⡀⠐⢆⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣴⣤⣤⣄⣤⡀⠀⠀⡀⠠⠤⢀⡀⠀⠀⠀⠀⣠⣶⡄⠀⠀⢰⡖⣒⣒⢺⣶
⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠆⠀⠀⣠⣾⣿⣿⠿⠀⠀⢸⡇⠧⡼⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠓⠀⠀⠘⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠋⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⡀⣄
⠀⠀⠀⠀⠀⠀⢸⣼⣶⣦⣦⡆⠀⠀⠀⠶⠾⠄⠀⠀⠀⣔⣻⢾⣳⡀⠀⠀⠀⢸⡛⠟⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣻⠞⠁⠀⠀⠀⣰⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿