Chameleon Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡠⠤⠔⠐⠒⠐⠀⠈⠂⠀⡀⠀⠀⠰⣠⠴⡴⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠄⠀⠮⠃⢇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠒⢄⣀⣀⠁⠁⠀⠐⢚
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠁⠀⢤⠗⢶⠁⠈⠀⢀⠀⠂⠀⠀⠀⠀⠄⠀⠈⠀⠂⠀⠤⠠⠠⠀⠐⣠⠟
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠠⠀⠘⠓⠏⠁⠁⠀⠂⠒⣄⡤⣀⠀⠠⠔⠊⠁⠀⠀⠀⠀⣀⡠⠖⠉⠄
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠀⠀⡚⠀⠀⢀⠀⠒⠒⠒⠊⠁⠀⠀⠀⢈⣦⣄⠃
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠠⠤⠀⢈⡁⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠉⠛⠛⣤⣼⠒⠆
⠀⠀⠀⠀⡅⠀⠈⠁⡀⡀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠈⢂⡈⠃⠂⠐⠂⠦⠠⠀⡀⠐⢁⠀⠀⠀⠰⠀⠨⠂
⠀⠀⠀⠀⡂⠀⢀⠌⠐⠈⡦⡇⠠⠔⠃⠐⠁⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢀⠄⡈⠓
⠀⠀⠀⠀⡇⠀⠀⢧⣑⣊⡴⠁⠀⠀⠀⠀⠀⠀⡀⠂⠀⠉⠐⠀⠄⠀⠀⠈⠂⠀⠐⠨⢤⠀⠀⠂
⠀⠀⠀⠀⢁⠀⠀⡄⠉⠀⡠⠂⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠘⠢⠄⠂⠀⠁⠀⠀⠀⡕⠙⣦⠐
⠀⠀⠀⠀⠈⡀⠀⠀⠀⡀⠃⠀⠂⠀⠀⠆⠀⠀⡰⠀⠒⠐⠢⣀⠀⠀⠑⡄⠀⠀⠈⠀⠀⠸⠊⠁
⠀⠀⠀⠀⠀⠡⠀⠀⠉⢅⠀⠀⠀⠀⠐⠀⠀⡈⠀⡴⠒⠐⢄⠀⠒⠀⠀⠸⠀⠤⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠄⠈⢆⢌⡡⠀⡆⢀⡃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠤⢀⠠⠐⠀⠌⠁⠀⠸⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡀⠀⠀⠀⠉⠠⠀⡄⢀⠄⠔⠉⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠁⣀⣀⠠⠤⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠞⠐⠉⠀⠈⡆⠀⠀⠀⠀⠐⠀⢠⠆⠒⠐⠂⠀⢤⠄⣁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠂⠀⠀⢀⡀⠤⠐⢚⡇⠀⢈⡀⠐⢹⣄⡏⠠⠀⠀⠀⠀⣜⡒⠏⠬⠴⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⠚⢤⠰⠚⢋⠀⠄⠀⠆⢸⡇⡇⠀⠀⠀⣸⠥⡇⠀⠀⠀⠀⣰⢇⢸⠀⠀⠀⠀⣀⠞⡤⢀⠁
⠀⠀⠀⠀⠀⠀⠀⡄⡾⠋⠛⢲⡀⠀⢆⠓⠚⠒⠃⠉⠉⠉⣇⢃⠀⠀⠀⡇⢉⠻⡄⠀⢀⢰⠅⡎⡘⡄⠀⢠⠌⢍⣱⠇⠀⠐⠄⠂
⠀⠀⠀⠀⡠⠄⠉⡈⢧⣀⣈⣼⠁⡐⠨⢦⢤⢤⣄⣀⡀⠀⣷⠈⢣⡀⠀⣃⠠⡍⢡⠀⠀⣍⠠⣴⢢⡇⢀⡗⢆⠎⡟⠀⠀⠀⣠⠄⡀
⠀⠀⠀⢠⢤⣬⣦⣸⣥⣭⣭⠤⣀⡀⠀⢛⠺⢤⡃⠸⠀⠀⡟⠀⠀⠱⡀⢸⠸⣇⡊⡇⢀⠆⡒⢸⠂⣧⢎⢸⠠⢘⠃⣠⢴⢚⢍⣡
⠀⠀⠀⠀⢄⠀⠠⠘⠂⠀⠀⠀⠀⠉⠓⠙⣦⡀⠀⠉⠀⢰⠇⠀⠸⠀⢺⠌⣰⢻⡒⣦⠞⣘⠀⠨⡇⠌⠠⠔⣾⡾⠯⣐⠕⣊⠖⠁
⠀⠀⠀⠀⠀⠀⠢⢍⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⣦⡀⡰⢏⠀⠀⠀⠇⠀⣼⠃⡀⣓⡈⣈⡇⡀⣰⠂⠀⢀⡼⣖⠥⢻⢯⠿⠁⠀⢀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠈⠈⠀⠯⡡⢞⠓⠚⠂⠀⠀⡐⠀⠉⠀⠁⠀⠀⠀⠀⢀⠃⠀⠀⠌⠛⠢⣠⡣⠻⡙⠉⠩⠇⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⠀⠤⣀⠀⠀⡐⠀⠠⠁⢀⠔⠀⠀⠔⠁⣀⢀⠀⠀⠀⠀⠀⡀⠁⠀⡠⠊⠀⠨⠀⢰⠓⡻⠏⡬⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠂⠃⠲⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈⠀⠁⠉⡉⣁⠈⠏⣵⢋⣀⣀⡰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠈⠁⠰⠁⠠⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢥⡠⢀⠀⠐⠀⠁⠰⠆⣀⣀⣀⣀⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢯⣀⠄⢀⣄⠙⠂⠀⠀⠀⡄⠹⡄⠀⢠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⠀⠘⠁⣈⠑⡆⠀⠀⠦⠴⠄⠱⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠁⠈⠚⠂⠐⠁⣀⠀⢻⡉⢓⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠋⡠⡆⠀⠄⠀⡍⣢⠄⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠒⠀⠀⠁⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣻⣛⣿⣫⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣭⣽⣿⣽⣿⣿⣿⣽⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⣿⣿⣿⣿⣶⢲⣲⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⡗⣪⣽⣿⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⢟⢿⣅⣿⡟⣜⢠⠇⢎⡧⢆⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⣟⢷⣿⣯⣿⣿⣿⣿⣾⣟⡛⠓⢮⣀⣿⠐⣂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣯⣯⣿⣿⣿⣷⢿⣟⣿⣿⣿⣿⣻⣈⣶⣇⠰⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠞⣿⣿⣿⣿⣾⡷⡇⠾⠸⣓⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣻⣿⣾⡿⣻⣹⣿⣿⣿⣿⣷⡄⠴⠏⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣟⣿⢷⣿⡯⣿⣟⢿⣯⠸⣉⣺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣤⣴⣿⣟⣹⣯⣯⣫⣿⣿⣿⣿⠿⣇⡞⢻⣶⢏⣕⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠐⠂⢻⡿⣿⣶⣿⣿⣿⡿⠛⢻⣟⣿⢿⣿⣷⣿⣿⡼⣿⠉⡩⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡼⢃⣓⣼⣿⣿⣣⣾⣿⣿⣄⠀⠀⠻⣿⣿⣾⣿⣿⣽⣾⣿⣿⣖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡏⣾⠟⢵⡏⣿⣿⣿⢯⣯⣿⣦⣴⣿⣿⣿⠻⡹⣻⡿⣿⠲⣖⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⡞⠀⣸⣿⣧⡘⣿⣇⢹⣿⣁⣈⡐⣓⣛⢽⣿⠽⠤⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢳⣤⣿⣿⣇⣽⣿⣿⣯⡽⠃⠁⠘⢿⣿⢿⡭⢚⢳⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⢻⣿⣿⣿⣿⣿⡹⡇⠀⠀⠀⠀⠛⠿⢫⣬⡕⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣹⣿⣷⣮⠿⣫⣿⣦⣄⠀⠀⠀⠈⠙⣽⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⣿⢰⣇⣿⣿⣿⣵⣿⢿⣿⣦⠀⠀⠀⠀⢹⣟⠷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢸⣿⣼⣿⣗⣘⣻⣷⣽⣿⣆⠀⠀⠀⠀⣡⣞⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡳⢄⡽⠿⠿⢿⢿⣯⣽⣿⣷⣦⣤⣼⣭⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⣿⣹⣾⢿⣿⣛⠿⣿⣛⡹⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡅⠉⠉⠀⠀⠀⠀⠙⢿⣯⣿⣿⣿⢟⢙⣷⢟⣛⣻⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢻⡩⣮⣾⣥⣥⣼⣿⣴⣼⡉⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣤⣤⣤⣴⣴⣵⣷⠾⠛⠯⠻⣽⣭⣿⣿⡿⢿⣿⣶⣶⣼⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⣿⠛⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣶⣲⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣼⣿⢥⡏⢉⣿⣛⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡟⡻⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢷⣟⣃⡌⢛⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣦⣤⣎⣶⣛⣿⣿⡟⣵⣿⣿⣿⣮⣭⣛⢯⣿⣧⣴⠾⠉⠉⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⢿⣻⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣾⣞⡻⣿⢷⣶⡟⠀⡌⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣣⣚⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣍⡿⢿⣷⡶⠟⡸⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⡿⠛⠿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣷⠈⣻⣷⣠⣴⠼⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣷⣎⢼⣿⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⡇⢠⣽⣿⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣿⣦⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⢣⣴⣿⣿⣿⣷⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣯⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣻⣿⣥⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⣤⣦⣿⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡟⣶⣴⣿⣿⣿⣿⣿⣻⣦⣙⣿⣿⣿⣷⠉⣿⡿⢫⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠟⠋⠀⠻⠿⠿⣿⣿⣿⣿⣿⡿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⡇⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣿⣯⣿⣾⠿⠿⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣿⠀⠀⠀⠠⣧⡀⠀⠀⠀⢀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠙⢿⣶⣶⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣎⢷⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢽⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢟⣍⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣽⣿⣦⣤⣄⣀⣀⣀⣀⣀⣤⣴⡾⣿⣹⣷⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣶⣶⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢱⡄⢠⣄⣀⢀⣤⡀⣠⣠⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠈⠈⠁⠁⠁⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠿⠛⠿⡟⠛⠻⠿⣟⠿⢻⣻⠟⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣶⣿⣿⣷⣶⣿⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⢀⡿⢿⡶⠶⢄⣀⡀⠀⣀⡀⠀⠀⢠⠠⠆⣀⣒⢁⠣⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⠤⣤⣶⣿⣄⣀⣸⣋⣠⣴⣧⢀⠄⠀⡮⠐⢢⡩⢴⣾⣷⣨⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⡹⢋⣽⠾⠛⠙⠋⣹⠇⠀⠀⠀⢀⣔⠃⠜⢻⠛⢭⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⡚⠛⣿⡿⡟⣿⣿⡿⠊⠡⠾⠃⠀⣠⠖⣰⢧⣞⣸⡠⠲⠊⠈⠑⠷⠃⠋⡋⠁⠡⠀⠀⡀⠐
⠀⠀⠀⠀⣠⣴⣦⣤⣶⣶⣤⣄⣠⡀⠌⢁⣀⡟⠁⠄⠘⢦⠆⠄⠚⣥⢎⠑⠿⣿⣿⡿⠄⡠⠀⠠⠐⣇⠆⠙⠓⠶⡒⠁
⠀⠀⠀⡰⠟⣛⡛⢻⣿⣿⣿⠛⣿⣿⣾⠟⠁⢀⣡⣶⣶⣦⣤⣾⡋⠀⠳⠀⢀⡅⢁⠀⠀⠀⠀⠠⠸⢏⡉⠒⠄⡀⠈
⠀⠀⢐⠑⡁⣤⣤⡲⡀⢿⢇⡆⣹⣿⣟⠂⠄⡛⠿⠿⣛⢩⠡⡐⡖⠣⠤⣀⢠⠖⠃⠦⠐⠀⠉⢉⡀⠀⠁⠀⠀⠀⠉⠚
⠀⠀⣼⣰⣱⠛⠟⡧⠗⢽⡞⠁⠉⣹⣿⣶⣄⣛⠂⠀⠀⠀⠀⠋⢺⠀⠦⠶⣨⠀⠀⡤⠖⠰⠀⠉⠐⢧
⠀⠀⡏⠂⠀⢣⠚⠁⠻⢿⡄⢢⠑⣿⡿⢿⣿⣧⡄⠚⠟⠢⣿⡄⢀⡋⠍⢂⣸⡷⠋⠠
⠀⠀⣇⠀⣀⣎⠀⢀⡀⠚⠻⣄⡷⠀⠀⢹⣿⣿⣧⣄⡄⢀⣨⡟⡏⣡⢮⠡⢹⠁⡴⠁⠐
⠀⠀⢰⠼⣾⠙⠔⢂⣺⣖⢶⡡⢇⣠⠟⠋⡉⠉⠋⠀⠉⡯⢅⠮⠝⠈⣈⡰⠛⠉⠉⠁⠃⠀⡀
⠀⠀⠀⢁⣩⠘⡁⠻⡿⢏⢲⡏⠻⡋⢚⢆⠧⣤⣄⠀⣠⣆⠀⠀⠀⡟⢧⡷⠶⠦⠒⠒⠒⠚⠉
⠀⠀⠀⠈⢘⢥⣁⢃⡵⠶⢔⡀⠀⢄⡠⠘⠀⢸⣿⣿⣥⡔⠡⠂⠀⠁⡬⡟⠒
⠀⠀⠀⠀⠈⢄⡀⠓⡴⠀⠀⢨⣴⣦⡀⢀⠐⠂⠘⢿⡿⠇⠈⠉⠂⡠⠔⢈⠢⣄⢀
⠀⠀⠀⠀⠀⠀⠻⣿⠃⠄⡀⠤⣿⣿⠫⠤⠹⠤⠰⣾⡏⠠⢄⣀⠦⠀⡄⠊⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠵⡧⠞⢀⠁⡠⡆⠀⠃⠘⠁⠁⡞⠀⠈⠀⡁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠲⠋⠤⠐⠢⠂⠙⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠁⠐⢃⣿⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠃

⠀⠀⠀⠀⠀⣠⠔⠒⠲⣤⣀⣤⠤⣶⣾⡇⣀⣤⣀⣠⡀⢀⡀
⠀⠀⢀⢀⡘⠁⢐⣠⣴⣟⡷⠾⢾⣿⣿⠏⣻⣥⠛⣹⣏⡹⢕⣶⣆⣀⡀
⣠⣽⣿⣋⣛⣛⣿⡿⣟⠫⡴⢂⣾⣿⣿⡘⡊⢿⡥⣸⣿⡗⠸⣿⣕⠛⣟⣶⣦
⠙⣿⣿⢿⣻⣿⣿⣿⣿⣿⣾⣿⢿⣿⣇⠁⡉⣻⣧⡟⠸⡯⣽⢿⣿⣩⣾⣯⣭⣷⣶
⠀⠙⢿⡡⠈⣛⢿⣿⣽⣿⣿⣿⣿⣿⣗⡖⢛⣻⣿⠁⢰⡿⠉⠞⣗⣰⢿⣟⢂⣴⣿⣶
⠀⠀⠀⠉⠒⢢⣤⡉⠛⠚⠭⠭⠭⢼⣏⠀⢻⣿⠛⠲⣞⡋⢱⣼⡟⢣⣸⠇⢿⣿⣧⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠳⠦⣤⣅⠈⣴⠏⠀⠘⡿⣶⣤⣈⠙⠶⣄⠐⠋⢁⢤⡶⠟⣻⠿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣟⡀⠐⠮⠙⠿⣷⣦⣜⣷⣦⣀⡀⠀⢰⡯⠀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡦⠀⠲⠄⠸⡆⣿⣿⣯⡛⠿⣿⣾⡇⠀⢸⡇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣷⠀⠀⠀⠐⠇⣿⡹⣿⣿⣄⣠⠿⣀⠀⣼⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢤⣜⣀⡏⠀⠀⠠⢾⣼⣿⡿⡿⠛⣯⣥⣾⣿⠟⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣯⡙⢲⣬⡓⢶⣄⠠⠠⣾⣿⠟⠀⠀⠾⠿⠟⠉⣿⣶⣶⣶⠂⠀⠀⠀⠉
⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠶⠾⠋⠀⠀⠙⠛⢶⣌⠳⢤⡀⠀⠀⠀⠀⠀⠀⢀⡼⠛⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣉⠓⠶⠤⠤⠴⠒⢉⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣶⣶⣶⣿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠒⠢⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⠒⠒⠙⠒⠊⠒⠤⣀⣀⠇⠀⠀⠀⠔⠊⠀⠈⠉⠓⠦⢄
⠀⠀⠀⠀⠀⠀⠀⡔⠒⠊⠀⢀⣀⣠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⡀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⢠⠞⠁⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠒⠚⠀⠀⠀⡀⠘⡄
⠀⠀⠀⠀⡴⠁⡠⠊⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡜⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⢠⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⡸⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠊⠉
⠀⠐⡅⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠙⠂⠈⠉⠉⠉⠉⠉⠉⢳
⠀⠀⠑⢄⠀⠀⠀⠑⠦⣀⠀⠀⢀⠄⠀⢰⠄⠒⠑⢄⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠠⠤⠤⠤⠴⠃
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠉⠟⠁⠁⠐⠁⠀⡀⡠⠬⢦⠀⣈⡏⠉⠈⠓⠚⠁
⠀⠀⠀⠀⢳⠀⣀⠀⠐⠠⡀⠀⠀⢠⣖⠺⠤⠔⠁⠀⠀⠉⠉
⠀⠀⠀⡠⠞⠁⠀⠀⢀⠴⡺⠂⠀⠁⠈⠳⡀
⠀⠀⢸⠀⠀⠀⣠⠊⠀⢱⠁⡰⠒⢄⠀⠀⢱
⠀⠀⠘⢄⠔⠋⠘⣄⠀⠀⢀⠁⠀⠀⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠁⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠤⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⢲⡖⠦⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢦⡶⠉⠹⣿⡀⠀⠀⠘⢷⡦⠀⠀⣶⡈⠙⢲⢤⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⠀⢀⣴⠚⡭⣷⣿⡆⠀⠀⠐⠟⡐⢀⠀⠠⡷⠐⢀⠺⣷⣀⠸⣧⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣾⡿⢻⣾⣿⣴⡋⠰⣶⢳⢾⡻⢧⡀⣼⣧⣤⠄⢻⣷⠀⠀⠆⣶⡈⠩⢠⠯⢿⡂⡀⣈⠣⡀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⣀⢀⣄
⠀⠀⠀⠀⠀⣿⣿⣹⣿⣿⣿⣿⣿⣷⣭⣾⣿⣷⠵⡀⢊⣿⠿⢦⡄⡉⠰⣀⠀⠙⠃⠂⣸⣗⣾⣧⡵⠻⣧⣜⣦⠀⠀⠀⠀⠀⠀⠐⠄⡀⢠⣞⡷⠾⢶⣟⢂
⠀⠀⠀⠀⣰⣿⢿⣿⡿⣉⢿⣿⣿⡿⠿⣙⡛⣿⡧⠿⠋⠁⠀⢠⡇⢅⡀⠀⠄⡀⠀⠀⠨⠫⣼⣿⣿⠋⠀⣬⣿⡳⡀⠀⠀⠀⠀⠀⠀⠙⣎⣿⠹⣶⠀⢻⣽⡇
⠀⠀⣠⣾⡛⢃⣼⣟⡥⠿⠀⢻⢿⡟⣹⠉⠉⠀⠀⢀⣼⠃⡀⣸⣧⣯⣼⠦⠀⣠⠀⢰⣴⣦⣛⡏⠟⢠⣾⣿⣿⢃⣿⡄⠀⠀⠀⠀⠀⠀⣟⡵⢌⣨⠀⢸⣯⣿
⠀⢠⡿⠛⢿⡟⢛⠫⠛⠀⢀⡞⢙⣯⣿⠙⢴⣀⣾⣿⣿⠇⡩⣼⣿⣿⣿⣾⣿⣥⠀⠱⡈⡠⢯⡝⠀⢸⣿⣿⣿⣿⣿⡹⡄⠀⠀⠀⠀⠀⠈⠛⠛⠁⢀⣸⣿⣿
⢀⣿⣧⣾⣿⢛⣤⣶⠒⢀⣠⣴⢟⣽⣿⣷⠟⠉⠁⠀⠀⠅⠀⣾⠉⠉⠙⠚⠙⠛⠻⣿⣷⣦⡗⠀⢠⢿⣿⣿⣿⣿⡿⣶⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿
⣸⣷⣿⣛⡷⡋⣞⣽⡿⡿⢟⣡⣼⣿⣿⣿⠀⠀⠀⠀⢀⠇⠙⢏⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⠁⢠⣫⢿⣿⣿⣿⣿⣷⣿⢻⡞⠦⣀⣀⠀⢀⠀⠀⠴⣿⣿⣿⢇
⣿⣿⡿⣻⣷⣿⣻⣿⣧⠾⠟⠋⣉⣾⣿⣿⠀⠀⠀⠀⠁⢦⡀⠉⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⣴⣟⠾⣿⣿⣿⣿⣿⣿⣼⠋⠸⢷⢍⢰⣭⣶⡷⢸⣤⠟⠁
⠙⠋⠁⠾⠚⠛⠉⠉⠀⠀⠀⠀⠙⣿⣿⠇⠀⠀⠀⠀⠀⠀⠱⠙⡀⠀⠀⠀⠀⠀⠀⠁⠉⢀⠀⢿⡿⠟⠚⠻⠿⠿⠿⣿⣿⣯⣧⣷⣾⣽⠼⠿⠛⠋
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠶⠓⠡⠤⠤⠤⢤⡀⠀⠀⠀⠸⠀⠎⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠁⠠⠈⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣔⣦⣾⣮⣄⣀⠀⠀⠀⠀⠀⠈⠙⠰
⠀⠀⠄⢠⣔⣦⣁⣠⡀⡾⠁⠀⠀⠀⠘⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠉⠛⢶⡠⠤⢴⣆
⠀⠀⠰⡗⠁⠀⠀⠀⠑⠑⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠈
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣶⠷⠂
⠀⠀⠀⠀⠈⠳⠶⢴⠲⣤⡤⡴⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡶⡗⢻⣿⡾⣷⣦⣢⣄⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣽⣿⣟⠛⠙⢋⣠⣼⣏⡁⣀⣈⡀⢹⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⣉⣿⠿⠟⠲⠾⠃⠍⡿⠋⠛⣿⡿⣷⣿⠁⠙⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⠉⢺⣿⡶⠜⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠀⠠⡿⣾⡟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢾⣄⡀⠌⠉⠀⠀⠀⢀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠦⡀⣼⣿⠀⡵⠀⢙⡆⠙⠛⢫⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠇⠀⢐⡗⠁⠀⡀⠀⠂⠀⠈⠁⠄⠀⠀⠀⠀⡀⠀⠀⠀⠙⣿⡷⢰⡇⠀⠀⠀⢂⣴⣿⣿⠌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡔⠋⢀⡠⠢⠁⠀⠀⠀⠀⠀⠀⠀⢀⣬⣾⢧⠀⠀⠀⠰⣿⣷⣼⠀⠀⠀⠀⠸⣿⣿⣷⣶⣶⠈⣧⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠄⠀⠤⠬⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⠖⠛⠁⠀⠀⠀⣠⣷⣿⣿⣿⣦⣤⣤⣤⣴⠈⣛⡄⠙⠛⠀⢈⣩⠉⠹⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠠⣤⡓⠂⠀⠀⠀⠀⠀⠀⠤⠄⢋⠁⠀⠀⠀⣀⣐⣦⣿⣿⠏⠟⠛⣫⠌⡁⣀⠀⠈⠉⠛⠗⢲⠦⣾⣿⣇⣠⡾⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⣠⠃⢀⡔⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠦⣤⡴⠶⠾⠻⠛⠗⠶⣶⣦⡤⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠇⠀⡴⠃⡈⣀⣀⣀⠀⠀⠀⠠⣸⢷⣤⣄⡛⠚⢐⡶⡀⣠⢀⠀⠀⠀⠀⠀⣶⣶⣾⡷⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡞⠂⡨⢱⣴⡴⠟⠋⠉⠻⢦⣀⢳⣼⣿⣿⣿⣿⣿⣾⡤⠇⠁⢚⣷⣷⣤⡀⠀⠀⢀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⠁⣰⣷⡿⠋⠀⢀⡀⣀⠀⠀⠉⠻⣿⣾⣿⣿⠿⠋⣁⣀⣤⣶⡿⠟⠟⢉⠙⢦⣱⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠜⠁⠀⠀⢢⣼⣷⠷⠿⢄⠀⠀⠈⠉⠋⠁⠀⣰⣿⣿⡏⢀⠄⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠁⠀⠀⠀⠀⠘⠁⠈⣀⣴⣧⣶⡔⠀⠀⠀⠀⠀⠀⠉⢧⣶⣶⣶⠟⢡⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⠋⠈⣩⣷⣶⣤⣀⠀⠀⠀⣀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠠⡾⠄⠀⠀⠀⠄⡐⢤⣩⣿⡟⢁⡴⠟⠛⠛⠛⠛⠿⢿⣿⠖⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢥⠀⠀⠀⠀⠀⠀⠀⢽⣿⡴⠋⠀⠀⠀⢠⡀⠀⠂⠀⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⠀⠀⠀⠀⠀⠀⠞⣿⠃⠀⠀⡠⠚⠉⠉⠻⣄⠀⠈⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠂⠀⠐⣿⠀⠀⠀⡇⢀⠀⠀⠆⢹⡇⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣀⠀⠀⠐⠆⠀⠙⣯⠀⠀⠙⠦⠤⠒⢀⡾⠂⠀⠐⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣤⠀⠀⠀⠔⡀⠘⠻⢶⣷⣴⣤⡾⠏⠁⠀⠀⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡾⡄⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡴⡄⠉⠀⠛⠀⠀⠀⠀⢤⣹⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣾⣷⣾⣤⣤⣶⣶⡶⠿⠋⠁

⣀⣀⣀⢀⣀⣀⣀⣀⢀⡀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢀⣠⣴⠶⣾⣟⣛⣷⢶⣶⠀⠀⠀⣠⢠⣤⡆⣤⣤⣤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢛⣽⢋⣴⣟⢡⣬⣛⣛⣛⣋⡅⣼⠏⠀⣠⡾⠋⠾⠛⠺⠿⠿⢫⣴⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣛⣚⢛⠁⣤⣄⠉⠛⢳⠮⢭⣭⣭⠾⠋⢀⣤⠶⠒⢛⣋⣉⣙⣷⣶⣶⣬⣍⡩⢶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠰⣟⣛⣛⣋⠹⣆⡉⠁⢀⡾⣸⣿⣿⠶⠞⢣⣾⣋⣀⣠⣶⡿⠿⠛⠛⠛⠛⠿⠿⣿⣿⣶⣍⠉⣭⣭⣄
⠀⠀⠀⠀⠀⢶⣭⣛⠿⢿⣿⣮⣽⣛⣯⡾⠟⣻⢅⠄⠴⢛⣛⣛⣛⣋⡁⠐⠛⠻⠿⣯⣍⣛⠳⢦⣌⡙⢿⣿⣮⡛⠿⠧
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣤⣭⣭⣥⢖⣼⢏⣿⢀⣾⠿⠛⣛⣩⠉⠁⠰⣾⣷⣶⣤⣍⡛⢿⣶⣌⡛⢦⡈⢻⣷⡜⢷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢮⣭⡛⠿⣷⣬⣵⡾⠏⣈⣵⣾⣿⠟⣡⣾⣿⣦⣌⠙⠿⣿⣿⣿⣶⣌⠻⣿⣦⡻⣆⠙⣿⣆⢩⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⢴⣶⣶⠸⣿⣿⣿⣧⣈⡛⠛⠛⠿⠿⢿⣶⣤⡉⣍⡛⠻⠷⣮⠻⣿⣿⣧⠘⣿⡄⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣌⠛⠿⠿⠿⠿⠿⠿⠿⠶⠶⢦⣭⡻⠘⣿⣦⠰⣶⣶⡌⢿⣿⣧⠸⣷⠰⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠷⠿⣿⣿⣿⣷⣶⣶⢀⣼⣿⡿⢀⣿⣿⣿⡌⢻⣿⡆⢻⠀⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠏⣼⡿⣫⠀⢸⣿⣿⡿⠛⠈⣿⣿⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⢯⣾⢡⠇⢼⣿⣿⠇⣾⣧⠸⣿⡄⠰⣮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢋⣿⠧⢋⣤⢸⣿⣿⡄⣿⣿⠆⣿⡇⣤⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢄⣾⣿⠀⣾⣿⣇⢿⣿⡇⣿⢰⡆⢿⢁⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣵⡿⢿⣿⠀⢹⣿⣿⡘⣿⡷⠀⢼⣷⠀⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠾⣡⣥⣤⣤⣭⣭⣴⣶⣶⣿⣶⣄⡚⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣾⣿⣿⣿⠛⠟⠻⠾⠿⠿⠿⠿⠿⣿⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢸⣿⠇⣾⣿⣿⡿⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣼⣿⣿⢯⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣿⢯⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣯⡉⣨⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣀⠀⠀⠀⢸⣿⣿⠁⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣋⣭⣭⣭⡛⢷⣄⠀⣼⣿⣇⡟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⣊⣭⣭⣭⡙⣿⡆⢿⠀⣿⡿⣸⡇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣿⢋⠀⠉⣿⢸⡿⡼⣸⣿⢣⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣇⠸⣿⣘⠦⠌⢁⣾⠃⣠⡿⣣⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⣿⣷⣌⡉⠠⠚⠋⠡⠞⠋⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣙⠻⠿⠿⠶⠶⢛⣡⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠶⠶⠾⠛⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣯⣄⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡿⢿⣿⡇⠺⠟⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠉⠀⠀⠀⠈⢁⣴⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠟⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠛⠛⠿⠈⣉⣀⣘⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣤⠤⠴⠾⣷⡎⢠⣿⡿⣫⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣿⣿⣿⣿⡻⣿⣦⡉⠓⠶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠛⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢛⡋⠳⣭⣿⡿⠃
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠆⣿⣿⣿⣏⢭⠙⠿⢿⣿⡶⣿⡇⠀⠘⠋
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣀⠀⢹⣿⣿⣷⣬⣿⣿⣦⣭⣁⡤⠗
⠀⠀⠀⠀⢸⡿⠛⣿⣿⣶⢀⣽⣿⣿⡿⣯⠈⢯⣿⠉⠁
⠀⠀⠀⠀⢸⣿⣤⣽⣿⣿⣾⠁⣾⠋⢠⡾⣷⡆⠁
⠀⠀⠀⠀⠸⣿⣿⡿⣿⣿⣟⠀⣷⠄⣈⠀⠀⠓
⠀⠀⠀⠀⠀⠹⣿⣷⣦⣽⣿⠰⢸⣦⣇
⠀⠀⠀⠀⠀⣠⠾⠿⣿⣿⣿⣷⣾⣿⢛
⠀⠀⠀⣠⡀⠀⠴⣄⣹⣿⣿⣿⣿⣿⣿⣿⡦⣄⡀
⠀⠀⠐⢦⡁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠑⠦⠀⠀⠘⢿⣿⡁⠀⠈⠙⠻⠟⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠗⠠⠄⠈⣿⣷⡀⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠘⠦⠄⡀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣏⠈⠙⠃⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⢷⣄⠀⠀⠀⣀⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⢻⠦⢰⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠿⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⢀⣤⠾⠋⠀⢀⣠⠴⠂⠘⠁⠲⡄⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢁⣈⣽⣿⣿⠿⠶⠶⢶⣏⡁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠷⠛⠋⠉⢰⠏⢀⣴⠖⠲⠦⣌⠻⣄⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠁⣰⢿⠀⣴⣶⣄⠈⣆⠻⠀⠀⠶⣄⠀⣸⣇⣠⣴⠶⠟⠛⣻⡟⠛⠛⠛⣷⣶⣤⣀
⠀⠀⠀⠀⠀⢸⣧⣤⡘⠛⠁⠀⠀⠀⠀⢹⣞⣄⡛⠿⠛⣠⡿⠀⠀⠀⠀⠈⢷⡿⠛⠉⢿⡀⠀⣰⠏⠀⠀⠀⢠⡟⠀⠈⢙⡷⣦⡀
⠀⠀⠀⠀⠀⠀⣿⡈⠙⠦⣄⠀⠀⠀⠙⢦⣙⠛⠵⠶⠿⢛⡡⠃⠰⣄⠀⠀⠈⣷⠀⠀⢼⡷⠞⠁⠀⠀⠀⣰⠟⠀⢀⣠⠟⠀⠈⠻⣄
⠀⠀⠀⢀⣄⠀⠘⣧⡀⠀⠈⠓⠦⣄⠀⠀⠉⠛⠓⠒⠛⠋⠀⢀⣴⣿⠆⠀⢠⡿⠀⢦⣀⠀⠀⠀⠀⠀⠘⠷⠶⠾⠛⠁⠀⠀⠀⠀⣹⣧
⠀⠀⠀⠘⣿⣧⡀⠈⢷⣄⠀⠀⠀⠀⠙⠓⠲⠆⠀⠀⠤⠴⠾⠋⠙⠃⠀⠀⠀⣠⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣠⠞⠋⠹⣇
⠀⠀⠀⠀⠹⣎⢻⣄⠀⠙⢷⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣤⠂⠀⢀⠀⠿⣄⡀⠀⠀⠈⠛⣦⡀⢠⡾⠋⠉⠉⠉⠓⠢⠝⠲⠤⣤⣿⡆
⠀⠀⠀⠀⠀⠹⣦⠙⢷⣄⠀⠉⠻⢦⣄⡀⠀⠉⠛⠛⠒⠚⠛⠋⠁⢀⣴⠏⠀⠀⠈⠙⠳⢦⣄⣀⣼⣿⡿⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⠳⣦⣀⠀⠈⠙⠛⠶⢶⣤⣤⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⢀⡼⠋⢁⣿⠃⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⢀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠙⠻⢶⣤⣄⡀⠈⢷⡀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡶⠋⣿⠀⠀⢀⡾⠁⢻⡄⠀⣠⡴⠚⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠉⠛⠻⠿⣧⣤⣤⣄⣼⠛⣳⡶⠦⣤⣴⠏⣠⣾⠯⢤⣴⡇⢀⣤⠟⠁⠀⠀⢷⠀⠈⠛⠶⢦⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣀⣤⣤⣤⣤⣤⣄⣀⣀⠀⠈⣿⠈⣿⠛⠻⣿⠀⠀⠀⢰⣯⣤⣶⡄⠀⠀⢿⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⡾⠋⠁⠀⠀⠀⠀⠉⠉⠙⠻⣿⣥⣤⡿⣷⡈⢁⣴⠟⣠⣿⠁⢀⣼⠇⣠⠀⠈⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠚⠉⠻⣟⠋⠁⢠⡟⢀⣼⣇⣈⣿⠀⠀⢠⣦⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣤⣄⡈⠙⠛⠛⠖⠋⠉⠉⠉⢻⡄⠀⠀⢳⡀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⡀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠉⠲⢤⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠉⣩⣿⣆⠀⣻⡆⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣄⠙⠛⠁⣠⡿⠁⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠻⣿⠛⠛⠋⠀⠀⠀⢀⣠⡾⠋⣤⣤⣀⡀⢀⣀⣠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⠙⠲⠶⠶⠶⠞⠋⠁⠀⠀⣿⠁⠉⠉⢉⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⣀⣴⢦⡀⠀⠀⠀⠀⠀⠸⣧⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⢿⣷⣦⣤⡶⠶⠟⠋⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣎⣤⣤⣿⣄⣴⣀⣤⣦⣡⣼⣞⣱⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⠤⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠒⠊⠁⠈⣿⣿⣿⡗⠀⠀⠀⣻⣿⠙⢦⣀⢀⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣯⣿⣿⣿⣄⠀⢰⡏⠛⡿⠁⠀⠀⣠⣿⠃⠀⣸⣿⢯⠀⠀⠈⠘⠛⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠃⠐⠉⠉⠙⡇⠈⠂⠈⠀⡀⠁⠀⢀⡔⠣⠂⠀⠀⠛⠃⠎⠀⠰⠀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣺⣿⣇⠀⠀⠀⠀⠹⡄⠂⠀⢀⠀⠀⠀⢸⠄⡘⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⠿⣯⣨⣦⣄⠀⠀⠀⠈⢢⣠⣸⡄⠀⠀⢨⢀⣴⡆⠀⠀⠔⠠⠇⠀⠀⠀⠀⠈⡘⠃⠐⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⢿⣿⡿⠗⠀⠑⢆⡀⣿⣿⠃⠀⠀⠀⢻⡿⠋⠀⠀⢰⠀⠐⠀⠠⠄⠀⠀⠈⠥⠄⠤⠤⠄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⢐⡇⣿⠃⠀⠀⠀⢤⡟⠀⠀⢀⡴⠧⠤⡴⠦⠒⠒⠶⠦⠤⠤⠤⠤⠴⠚⠁
⠀⠀⠀⠀⢠⢤⣀⡀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠠⡟⠛⠒⠒⠒⠒⠻⡄⠀⠈⢣⡻⣄⠀⠑⠦⡀
⠀⠀⠀⠀⢸⠀⠀⠉⠑⣲⠏⠀⠀⠀⠀⢀⣀⡤⠔⠊⣿⠁⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠱⢬⣙⣢⡄⢈⡓⠦⠤⠤⠤⠴⠶⣦
⠀⠀⠀⠀⠈⠓⠲⠦⣴⠃⠀⠀⠀⢀⠔⠻⠥⣤⣀⣠⠇⡀⣻⣄⠀⠣⠤⣄⠀⠀⠀⠀⣀⣀⣙⣦⠀⠀⢺⠛⢁⣪⠤⣤⠤⠤⠤⠴⠶⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡰⠣⢤⣤⣁⣀⡀⠹⠜⠀⠀⡇⠀⢰⠉⠉⠉⠩⠭⠍⠀⢀⣀⣸⣇⣀⣼⠋⠉⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠗⠋⠉⠈⠉⠣⣍⠉⠙⠛⠛⢧⣀⣸⠉⠉⠛⠓⠚⠛⠉⠉⠁
⠀⠂⠀⠀⠀⠀⠀⢸⠀⠂⠀⡎⠀⡞⡩⢀⠙⡄⠘⡆⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢡⠀⠳⠥⠂⢀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠢⠀⠑⠀⠤⠐⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⣀⣠⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢈⡄⢿⡅
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢋⣿⡟⣷⢀⣀⠀⠀⠀⠀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣛⡛⠋⡺⣧⣿⣶⣶⠦⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⡜⣾⢿⡿⢣⣾⢿⡷⠻⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⢜⢿⣷⣶⣶⣶⣦⣝⡢⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⣔⣦⠳⣍⡭⢹⣿⣯⣭⡻⣮⣷⣠⣄⠀⣰⣎⣩⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣹⣦⡂⠀⣸⡯⠀⠀⠉⠒⠿⠿⠿⠃⣸⡏⠻⠾⢳⣿⡟⠁⢬⣱⡄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⣀⣤⡬⠿⢯⣄⠛⣃⡈⠃⠀⠀⠀⢩⡡⣍⡙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠃⠙⠾⠛⢾⡟⢿⣆⠀⠀⠀⠸⠙⣎⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢇⠀⠀⠀⠀⠈⠛⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠭⠇⢀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠸⡌⠑⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠹⠀⠻⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⠤⠤⠒⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠂⠀⠀⣠⣶⣿⣟⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣾⣟⣯⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣟⣻⣿⣿⣿⣦⢀⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢯⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡾⣿⣿⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⣠⣶⡆⣸⣿⢷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣷⣾⣿⣿⣿⡿⣫⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣈⣵⣿⣿⣿⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠟⠛⠛⢛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⢸⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠻⢿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣟⣻⣟⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⢿⣷⣿⣾⣷⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢤⡟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⡿⠋⠉⠉⠉⠛⢿⣿⡿⣦⠀⠀⠀⠀⠀⠀⢀⣾⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣮⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⢐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⣹⢦⣀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠈⠛⢿⣿⣍⡳⣦⣀⠀⠀⠀⠻⣿⡿⣿⣶⣿⣿⠟
⠀⠀⠀⠀⠀⣴⣻⠟⠛⠛⠿⣿⣷⣄⠀⠀⠀⠈⠛⠷⣧⣿⡓⣦⡀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⢨⢿⡏⠀⠀⠀⠀⠘⢿⢻⣆⠀⠀⠀⠀⠀⠈⠙⢷⣿⣟⣦
⠀⠀⠀⠀⠀⢾⣳⣀⠀⠀⠀⠀⠘⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⢻⡿⢽⣇
⠀⠀⠀⠀⠀⠈⠙⢽⣶⣤⡀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣷⡞⠛
⠀⠀⠀⠀⢤⣄⠀⠀⠉⠻⢿⣶⡄⠀⠹⡿⣷⡄⠀⠀⠀⠀⣠⣾⣿⣯⠅
⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠹⣿⡄⠀⠹⣟⣛⢶⢶⣶⢾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢀⣿⠁⠀⠀⠀⠙⠚⠮⠼⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⣰⣲⣆⡒⠒⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣹⡝⡋⠭⠰⠦⠰⠄⡌⢀⡃⠀⢮⣙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠺⣏⣷⣎⣤⣉⣀⠀⠀⠀⠀⠐⠈⠅⢙⠂⣈⢢⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣒⣲⠭⠓⢀⡼⣙⣧⡽⣯⡞⠋⠀⡠⠙⢢⠀⠀⠀⠀⠀⠀⠙⢄⡜⣎⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡖⢻⠷⠚⠉⣠⡤⢒⡏⣾⡉⣿⡗⢋⡀⠀⠀⣿⡞⣦⣕⣤⣀⡀⠀⢀⡀⢆⡤⠛⡭⡇
⠀⠀⠀⠀⠀⢠⣾⣍⡗⠁⣰⡶⠋⠁⢀⡾⡰⠿⣿⢯⠻⢷⣨⠃⠀⠰⡻⣧⣼⣿⠒⣿⣽⢙⣳⣆⣾⠟⣽⠃
⠀⠀⠀⠀⢠⡿⣥⡏⢀⡞⠁⠀⠀⠀⣾⢿⢿⣷⣿⡏⠉⠉⠀⠀⠀⠀⠙⠦⣛⣩⡛⡿⣛⣻⡿⣫⣷⠟⠁
⠀⠀⠀⠀⣼⡞⢻⡄⠸⣇⣀⣀⣠⢞⣷⣷⣴⢯⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉
⠀⠀⠀⠀⢹⡛⢿⢛⣷⣶⠞⢴⣾⣏⣹⠿⢛⡜⠁
⠀⠀⠀⠀⠀⠳⣬⣟⠻⠿⠷⡿⠛⠛⣱⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠿⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⢬⡔⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣫⠿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⠏⠀⠀⠈⢁⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢿⣟⣃⣄⢠⣆⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣾⣿⡸⢯⢷⣏⢹⠸⣿⠻⢿⣿⠻⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣯⢿⣻⣛⣿⣷⣾⣟⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢿⣿⣿⢟⣯⣭⡝⠻⣯⣾⣏⣀⣀⣖⡃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡇⣾⣷⡟⠚⠋⠁⠀⠘⢯⡟⠷⠿⠛⠾⠯⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢿⣳⣶⡾⢯⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⢀⣾⣦⣿⡾⣋⣼⣐⣛⣿⣿⣷⠎⣳
⠀⠀⣠⣶⠟⠉⠁⠀⠉⣧⠀⠠⢄⣆⣸⣹⡇⠀⢹⣿⣾⣿⠋⠀⠀⣿⡹⠁
⠀⠰⡟⠁⠀⠀⠐⠦⠴⠋⠀⢋⠺⢷⠿⠟⠁⢀⣾⣈⣵⠏⠀⠀⠀⣿⡷⣶⢦⡡⠄
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣼⡉⣹⠏⠀⠀⠀⠀⠈⠉⠙⡟⣝⠂
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣽⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁
⠀⠈⢻⣿⣦⣀⠀⠀⠀⠀⣀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠉⠻⠿⣿⣿⣯⣹⣥⠿⠟⠋



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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣤⣤⣶⡿⢿⣻⣿⣿⣿⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣀⣴⣴⣶⢿⡾⣷⡟⠙⡿⠿⣿⡇⡙⣿⣿⣿⣯⣿⣿⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡞⢛⣿⣯⣽⣿⣿⣿⠥⢄⣀⡙⠙⠠⢰⠶⣾⣷⣿⢉⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡴⣋⣭⢿⣿⣿⣿⠿⠿⠾⠿⠿⣿⣿⣷⣶⢉⣦⣦⡼⠟⠋⣽⣿⡶⢿
⠀⠀⠀⠀⢀⠔⣽⣿⡿⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⠌⠉⠀⠀⠀⠀⠘⠿⢧⣤⡄
⠀⠀⠀⢀⣾⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢿⡧⠄
⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠋
⠀⠀⠀⠸⢹⡇
⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢈⣡⠘⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⠇⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠘⡇⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠁⠁⠀⠠⠀⠀⡀⢀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡼⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠃⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠠⠤⠄⡀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠆⠀⠀⢀⠐⢀⠤⠤⡀⠃⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠄⠀⠀⠠⠄⠀⠠⠀⠀⢀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⢀⠡⡀⠋⠀⠀⠂⠀⠉⠀⠀⠉⠂⠀⠐⠠⠔⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠄⠀⠀⠀⠀⠒⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠁
⠀⠀⠀⠀⠀⠀⢠⣗⣷⡄⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢢⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣄⠀⠱⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⠀⠁⠀⠂⠀⠀⠀⠈
⢀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠐⠀⠁⠀⠀⠀⠀⡀⣀⠀⠀⠀⠣⣀⠀⠈⠄⠀⠄⠀⠁⠉⠁⠐⠘⠃⠉⠁⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠
⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⢨⡅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠁⡠⠂
⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⡔⠈⠒⠲⠂⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⢈⣀⠤⠒⠉
⠀⠀⠀⠀⠉⠈⠈⠁⠀⠀⠀⠁⠃⠀⠤⠄⠤⠀⠊⠁⠀⠀⠀⠐⠂⠠⠠⠜⠉⠀⠐⠀⡐⠒⠒⠒⠂⠒⠐⠒⠒⠂⠒⠂⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⡷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⣠⠶⠶⣶⠿⠿⣶⠴⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⢄⠀⠀⠀⠀⠀⢠
⠀⣿⠰⠷⣿⡘⢛⣧⣚⣂⣿⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⡅⠧⠴⠇⠀⠤⠒⠔⠀⠲⠁⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠈⠛⠛⠉⠛⠛⠛⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠀⠀⠀⠀⠀⢂⡠⠐⠁⠀⠰⠃⠄⢸⠤⣄⠀⠘⡀
⠀⠀⢀⡀⠀⠀⠀⠀⡄⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠡⣀⠬⠈⠁
⠀⡠⢸⠱⡀⠀⠀⠆⠜⠂⠀⠀⢄⠀⠀⠀⠀⠀⢆⣤⣲⢄⡤⡄⢠⠔⢶⡀⣐⠀⡌⠀⠀⠀⣶⡆⠀⠀⠀⢄⠀⢀⠀⠀⠀⡀
⠀⡆⢸⠀⢸⠈⢠⢸⠀⠀⠀⠈⠈⠇⠐⠂⠀⠀⠀⠯⠤⠞⢇⡠⠾⣄⡬⠃⠉⠊⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠈
⠀⠀⠺⠀⠀⠀⠄⠀⠢⡀⠀⠂⠐⠀⢁⠀⠀⡀⠀⠀⡄⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡄⠀⡀
⠀⠀⠀⠄⠀⠀⠀⠉⠅⠐⠒⣤⠉⠀⠀⠀⢸⡆⠈⠉⠀⠀⠴⠤⠇⢰⠀⢄⡙⡄⠀⢄⠀⠈⠂⠰⠖⠊⠁⠀⣀⣇⣀⣀
⠀⢠⠀⠀⠀⠀⢀⡀⢀⠀⡀⠀⠀⢠⠒⠢⣸⡇⠉⠀⠀⡀⠀⠀⠀⣇⠀⠀⠇⠹⠀⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀
⠎⡈⢃⠬⠀⠐⡴⠄⠀⠀⠁⠂⠀⠁⠀⠀⠈⠉⠐⠂⠀⠉⠒⠖⠀⠘⢶⠤⠔⠁⠀⠀⢰⠋⠀⠀⠀⠀⠨⢀⠀⠂⠀⢀⣬⠀⠠
⠀⠀⢁⡠⠤⠄⠀⠤⣄⡀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠈⠋
⠀⣔⡻⣿⠄⡤⠐⣫⣽⣿⣦⡀⠀⠀⠀⠀⠉⢬⡃⢠⡗⠜⡄⠀⠀⠀⠂⠀⠀⠀⠀⢂⡀⠀⠀⠀⢀⠆
⢰⠹⣻⣄⠀⠀⠾⢛⡛⢻⣿⡇⠀⠀⠐⠁⠀⠉⠐⠀⠹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠇⢸⠒⠁
⢸⣿⢿⠅⠀⠀⣴⣿⣿⣮⡝⡇⠀⠀⠤⡀⠀⠀⠀⠀⢀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⣿⣷
⠀⢧⣤⣴⣄⠀⠈⠉⢹⣿⠱⠃⠀⠀⠀⠃⠀⢂⠀⠠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠈⠋⠉⠋
⠀⠈⠙⠿⢿⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠒⢒⡓⠂⠀⠀⠀⢠⠨⣄⠡
⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠄⠭⠌⠉⠁⠁⠠⠂⠀⠀⣀⠴⠒⠀⠀⠘⡇⠀⠀⢠⠟⣛⢻⠋⢻⣷⠀⠀⠀⠈⠁⠔
⠀⠠⡀⠄⠐⠠⠎⠈⡆⠄⠘⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠠⡞⠁⠄⠀⠀⠀⣀⠇⠀⠀⠘⢷⣤⣾⢶⣶⡿⣦⡄⠀⠂⠀⢆
⠀⠀⠀⠒⠀⣀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠀⠞⠁⢀⡿⠑⠀⠀⠀⢀⣸⣧⣙⣹⣡⣜⡿⠀⠀⠀⠈
⠀⠀⠀⠀⣊⣀⠤⠴⢚⠓⠀⠀⠀⠀⠂⠀⠤⠀⠐⠁⡀⠀⠀⢐⠁⣀⡴⠋⠀⠌⠀⠀⢠⣿⢉⢸⡏⡏⢬⣿⡇
⠀⠰⠂⠈⠁⠀⡠⠔⠀⠀⠀⠀⠈⠁⠁⠇⠨⢨⠠⡀⡃⠀⠀⠸⠚⠁⠒⠀⠀⠀⠀⠀⠀⠛⠻⠟⠛⠻⠿⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⣴⣦⣿⢿⣻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣤⣶⣶⣷⣶⠾⢿⣿⣿⣿⡇⠿⣿⣿⣿⣿⣿⣿⣻⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠦⢾⣷⣲⣿⣿⣟⣛⣋⠈⠁⠻⠿⡄⡘⣤⣬⣿⣷⣷⠛⣿⣿⡿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠺⣿⣯⣷⢶⣿⣿⣿⣽⣿⣿⣿⣯⣭⣝⠒⣄⣡⡂⣳⣼⠿⣿⣿⣌⣿⠋
⠀⠀⠀⠀⠀⠀⡠⣾⣿⣶⣩⡽⠿⠛⠛⠉⠉⠉⠉⠉⠉⠛⠿⣿⡟⢠⠿⠟⠛⠉⠀⠀⢿⣿⡏⡙⠁
⠀⠀⠀⠀⢀⠎⣼⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣇⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠧
⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⣾⠃
⠀⠀⠀⠠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉
⠀⠀⠀⠀⣇⣿
⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⣠⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⢀⣤⣶⣾⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣰⣿⣿⠿⠛⠛⢿⣿⣿⣶⣶⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣀⠀⠀⣿⣿⣇⣠⣤⣶⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢻⣿⣿⠉⠁⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣄⠀⠀⢿⣿⣧⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⢀⣰⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠈⠉⠻⣷⡄⠈⢻⣿⣇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⢸⣷⠀⢸⣿⣿⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⣸⡟⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣷⣦⣤⣾⣯⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⡴⠞⠉⠁⠈⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠘

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣰⣖⡶⡐⠒⠒⢠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣘⡿⣛⠹⠅⠶⠠⠆⢨⢁⡘⠀⢰⣉⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠔⣗⣯⣞⣥⣋⣀⠀⠀⠒⠂⠀⠂⠥⢘⡃⢀⡑⣬⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣔⣲⠩⠇⢐⡱⣃⣏⡸⢯⡼⠋⠁⠀⠉⠲⡀⠀⠀⠀⠀⠀⠈⢢⣀⢿⣉⢿
⠀⠀⠀⠀⠀⠀⠀⣀⡴⢻⡷⠞⠉⢁⡤⠖⡾⣵⠋⣿⣧⢛⡁⠀⠀⢸⣖⢳⣜⣦⣀⡀⠀⠀⢀⢰⢠⠔⢫⢼
⠀⠀⠀⠀⠀⢀⣾⣏⡿⠋⢠⡤⠛⠉⠀⣼⣱⠿⣿⣏⡳⢷⣂⠇⠀⠘⡽⣿⣬⣿⡗⢻⣭⡍⣓⣆⣰⡿⢫⡟
⠀⠀⠀⠀⢀⣿⣡⡟⢀⡸⠋⠀⠀⠀⣼⢹⣿⣷⣿⡟⠉⠋⠀⠀⠀⠀⠈⠲⣝⣋⣛⢿⢟⣛⠿⢟⣽⡶⠊
⠀⠀⠀⠀⣼⡖⢻⣄⠸⣇⠀⢀⣠⢜⣿⣿⣤⣟⣥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⢻⡟⢿⠟⣳⣤⡾⢤⣶⣏⣉⡿⠟⡹⠃
⠀⠀⠀⠀⠀⠳⣬⣛⠿⠿⡷⡾⠛⠛⣡⣾⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠿⠿⠟⠛⠉

⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠈⠈⠀⠀⠀⢀⠠⠐⠈⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠀⠠⠂⠁⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠁⠄⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢂⡀⠀⠀⣀⠀⠀⠀⣄⡤⠀⢀⣾⠀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠠⠀⠀⢀⡇⠠⠴⣧⡔⠂⠘⣅⣆⠿⠻⣛⡛⠛⣇⣄⢲⠶⠳⠢⣆⡀⠀⡇⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠤⠴⠋⠀⢩⡾⠁⠀⠀⢠⠟⠉⠙⢿⣷⡿⠾⣿⣟⣿⣄⠀⠀⠀⠑⢦⣳⣄⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡆⠀⢀⠔⠉⠀⠀⠀⡰⠃⠀⠀⠀⡸⢛⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠈⠛⠧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⢀⡼⠁⠀⠀⠀⡠⢁⠒⠳⣄⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣶⡄⠀⠀⠀⡴⠋⠀⠀⠀⠀⡰⠁⠇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠉⠻⣷⣴⡁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠉⠁⢀⡠⠞⠀⠀⠀⠀⣠⠎⠀⠀⣈⠦⣁⠀⡀⠀⠑⣄⠀⠀⡀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡀⠐⠾⠲⠀⠀⣰⠃⠀⠀⠀⡃⠀⠈⠳⡀⠀⠀⠈⠑⠾⣁⡠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⡤⠴⢟⣅⠃⠀⠠⠞⠛⠒⠤⣀⣨⡳⣴⠶⠴⠀⠀⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⢒⣀⣩⣿⡟⠻⡓⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠈⠄⠒⠛⢋⠿⡿⢿⡟⠀⢇⠠⠐⢛⡈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣴⢏⡀⠀⠀⠀⠀⠑⠖⠁⣀⣵⠀⠀⠀⠀⠀⠀⣠⡠⣛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⡼⠀⠀⠀⠀⠿⢖⡠⠒⠃⠈⠀⠀⠀⠀⠀⠀⢈⡟⠉⡛⠡⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠁⠀⡊⠀⠀⠀⠀⡀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠿⢿⠦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⡄⢀⠊⠉⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠠⠏⡣⠀⠋⡉⠛⣆⠀⠀⠀⠀⠀⠀⠂⢘⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠉⠉⠣⢄⠀⠀⠀⠀⠀⠀⠈⠢⠤⠄⢤⠀⠀⡰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠐⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⢠⡴⠁⠈⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠃⠀⡀⢏⢂⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡰⠀⠀⠘⠀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠀⡀⠙⠦⣀⠀⠀⠀⠀⠣⠠⠞⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀⠐⠃⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⡄⢈⣢⣄⣀⣀⠀⢤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠄⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠠⣄
⠀⠀⠀⢀⡄⠠⠆⡲⣦⡄⠙⠢⢴⣄⣀⠄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠠⣀⣠⡦⠄⠉⢠⣴⢆⠰⠄⢀⡀
⠀⠴⠒⡾⠁⠀⠙⠁⠀⠄⠀⠈⠂⠀⠉⠂⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠀⠐⠁⠀⠠⠀⠈⠋⠀⠈⢷⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⢤⠂⠀⠀⠀⠠⠤⢤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡊⠀⠨⠀⠀⠀⡶⠲⠓⡧⠄⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢩⠈⠂⢢⡼⠒⠂⢴⠨⠛⠚⡓⠀⠀⠠⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⡈⢆⢢⠜⠁⠀⠀⠀⠀⠈⠚⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⠅⠢⠌⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠐⢈⣉⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠠⠤⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠠⠐⠂⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠂⠬⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⠐⠂⠩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⣁⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠂⠠⠈⠩⡄
⠀⠀⠀⠀⠀⣀⢴⠶⣺⡉⢭⣣⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠡⠐⣀⠠⢇
⠀⠀⠀⢀⣼⡋⡐⢉⡄⠋⠁⠀⠀⠈⠘⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠂⢅⠐⢺
⠀⠀⢀⡖⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠔⠲⣈⡛
⠀⠀⢸⣍⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡇⠀⠀⠀⠀⠀⠀⠀⡟⡐⢊⡑⢈⡇
⠀⠀⢸⡄⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⣰⠢⠌⠤⠑⣼
⠀⠀⠘⠦⢈⡁⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢍⠡⡉⢄⡱⠃
⠀⠀⠀⠱⡂⠠⠀⠜⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡚⠋⠠⣈⠂⠠⡜⠁
⠀⠀⠀⠀⠙⢀⠒⢈⠥⢋⠗⠶⢤⡄⢠⣤⣤⡤⠤⡒⢲⡉⠇⡐⠨⣀⡂⠠⣨⠎
⠀⠀⠀⠀⠀⠀⠰⢆⡘⠀⠄⡀⢂⠄⡁⡐⢀⠆⡁⢌⠰⠀⠢⢈⠐⢄⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠆⣡⢈⡄⢣⠘⡜⢀⠀⡆⡐⣀⡧⠬⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠈⠉⠉⠉
⠀⠀⣀
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣄⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣆⣻⣷⣷⣾⣷⣿⣾⣷
⣿⡿⣡⡶⡠⣂⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⡽⡿⣿⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⣿⣾⣿⣿⣶⣿⣿⣶⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠿⡉⠫⡽⣙⠛⢿⣷⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣝⢧⠙⢄⠙⢆⠙⢜⣧⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠋⠁⢹⣿⣿⣿⣿⣿⣿
⣯⣍⢷⡍⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠘⠁⠀⠀⠀⠘⠋⢻
⠙⠛⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣁⣤⠤⠤⠤⢤⠈⣠⣿⣀⠘⢱⠙⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠀⠀⠀⠀⣀⣀⣀⣀⠀⡔⠛⢛⣟⠀⢸⡇⠀⡀⠀⠀⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⡀⠀⠙⠉⠁⠀⢀⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⢿⣣⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠠⠀⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠤⠄⠐⣒⠨⣽⠁
⠀⠀⠀⠂⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢯⣤⣴⣦⠘⣿⣿⣶⠶⠂⠠⠀⠀⡄
⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠥⠀⠀⡀⢀⣀⠄⠃⠀⠀⢀⣠

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

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

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡰⣰⠃⠀⠳⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢀⠠⣔⠤⠂⠐
⠀⠀⠀⠀⠀⠚⠁⠖⠀⠀⠀⠀⠙⠄⠈⢨⠆⡤⠶⠄⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠈⠉⠀⠀⠠⠄⠀⢠⠀⡀
⠀⠀⠠⣀⠀⠀⠀⢀⣿⡆⠀⠀⣰⡆⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⢉⠀⠀⠀⠀⣴⡐⠀⠀⠳⡽⣦
⠀⠀⠀⠈⡁⠀⠀⠀⠋⠀⠀⠂⠛⠁⠀⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠁⠁⠀⠁⢹⣧⠀⠀⠀⠻⠇⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠻⠇⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠉⠀⠃⠠⠄⠀⠀⠀⠀⢀⣈⠀⣠
⠀⠀⠀⠀⠀⠀⠉⠲⠠⠤⡴⠀⢀⠀⠈⠐⠻⠂⠀⠀⠀⠀⠀⠀⠰⠛⠉⠈⠉⠻⣦⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠐⠤⠦
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠘⢦⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠂⠀⠀⠄⢠⠀⢁⡠⣄⡀⠀⠈⠾
⠀⠀⠀⠘⠚⠁⠀⠸⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⡈⠅⡆⠅⢆⢱⠀⠀⠺⠤⠖
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠁⢀⠨⠐⡀⡵
⠀⠀⠀⠀⠀⠀⠄⠣⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠘⠰⠖⠘⠒⠉
⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⢠⠀⢀⣠⡄
⠀⠀⠀⠀⠉⠉⡄⡀⠈⢀⠀⠱⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠈⡄⠀⡀⠐⢀
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⡇⢻⠀⠀⣾⡆⠀⠀⠀⢰⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⢐⡇⢸⠀⠀⡗⣇⠀⢠⣄⢸⡀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣄⠀⠀⡿⢿⡆
⠀⢸⠁⢸⡄⢸⡇⢹⠀⣸⢻⡀⣇⠀⠀⠀⢀⠄⠬⠁⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⡼⠀⠀⠀⠈⣷⡀⠀⡇⢸⡇
⠀⣸⠀⠈⡇⣸⠁⠸⡇⣿⠸⡇⠘⣆⠀⠀⣈⣄⠀⡔⠛⡀⠀⠀⠀⠀⠈⠀⠀⡅⡰⠁⠀⠀⠀⣀⣿⢻⡶⠇⢸⠃
⠁⡿⠀⠐⣧⡿⠀⠀⢿⡟⠀⣇⠀⠈⠳⣈⣣⣤⠀⠲⣀⣙⠄⠀⠀⠀⠀⠀⣠⠞⠀⣠⡧⠀⠀⠘⣿⢀⣿⠀⢸
⣐⣇⣀⣀⣿⣧⣤⣶⣟⣷⡿⢿⣀⠀⠀⠈⣛⠦⣄⣀⣀⠀⠀⣀⣀⡤⠴⠋⠁⠀⠀⣹⣇⡿⢻⣿⣿⠛⣿⠀⢺⡄⠸
⣿⠾⠟⠛⠛⠻⠿⠿⢭⣄⣐⣀⣭⠷⠚⠋⠉⠛⢦⣀⠉⠉⠉⠉⢠⡶⠀⠀⣴⣶⣶⣿⣏⣿⣸⣧⡿⠶⠿⠿⢛⢟⢻
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠶⠤⣤⣤⣭⣳⣦⣤⣤⣽⣿⠤⠶⠿⠗⢚⠻⠩⢁⡀⢠⣤⣤⣴⣦⠀⠀⣾

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣶⣤⠠⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣌⠿⣿⡷⠐⣿⣿⠿⠷⢢⡀⠀⠀⢀⣤⡎⣿⣷⣮⣻⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡘⣿⣿⠿⣅⢰⡇⢠⣬⡿⠜⣿⣿⡿⢋⣽⡏⢾⣷⡾⣢⠗⢼⢱⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⡶⡈⠂⠁⠘⠚⠁⣶⢾⣿⠷⣙⢿⣿⣖⣴⡆⠁⠀⢸⢫⡷⢇⣀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⡿⢠⡉⡁⠀⠀⠀⠀⠘⠉⠁⠘⢿⣟⣫⣬⣿⡏⢸⣿⡿⢳⣝⣯⣩⣩⠥⢅⣂
⠀⠀⠀⠀⠀⠀⢸⡇⢍⣉⣼⣥⣬⣡⡀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⠛⣿⡅⠻⠗⣼⣿⣿⠉⠋⠴⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠇⣼⣿⣿⣿⠇⣿⣾⡶⢤⣤⣄⣤⣶⣒⠿⣿⣟⣬⠏⠁⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣮⡉⢡⣺⣷⡿⢿⣷⣮⡍⣻⣿⣿⢂⡊⢉⠁⠀⠀⣠⣽⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡈⠋⠉⠉⠏⢿⡛⠁⠘⠻⠛⠻⠇⠀⠀⠀⠀⢨⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣴⠒⣄⠀⠀⠀⠀⠀⣠⣤⣤⣤⣶⠶⡶⢖⠒⠤⣤⣄⡀
⠀⠀⠀⠀⣀⣠⣖⣉⠉⠈⠆⣠⣤⢶⡟⠀⠀⢻⣵⢀⡂⡆⠀⠀⠀⠀⠩⠉⠙⢶⡤⣀
⠀⠀⢠⡞⣽⣿⡕⢆⠉⠀⠀⢿⣷⡾⠀⠀⠀⢸⠣⠈⢠⠁⠀⠀⠀⠀⠀⣀⣄⡎⠀⠀⠉⣦⣄
⠀⠀⣾⡁⠧⣍⣠⠎⠀⠀⠀⠸⣿⡽⣄⠤⠤⢚⣚⠓⡞⠀⠀⠀⠀⠀⣾⠇⡄⠉⠑⠺⢳⣾⢩⠷⢄
⠀⢠⠤⠉⠀⠀⢀⡀⣄⣀⠀⠀⢹⣷⣤⡤⠞⣿⢃⡸⡃⡁⠀⠀⠀⡀⠺⠀⣽⠙⠲⣴⣾⣧⡇⠀⠀⠱⡄
⠀⠸⣿⠿⠿⠛⠛⠉⠁⠈⠁⢀⣵⣯⠏⠀⡜⢡⡞⠠⢀⠁⠀⠀⠀⠀⠘⡇⢻⠀⠀⠛⠉⠉⠉⠳⣄⢀⢹⡄
⠀⠀⠀⠉⠛⠒⠲⠖⠒⠚⠛⠛⠉⡉⣒⣾⢀⣞⣤⣤⣌⠀⢀⢀⢄⣞⠭⡇⢸⠁⠀⠀⠀⠀⠀⠀⠈⢿⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣖⣠⡥⠋⠀⠀⠀⠈⠉⠁⠀⢀⠒⠆⠾⢸⠀⠀⠀⢀⣤⡤⣦⢤⠘⣇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⢀⣏⡟⣩⡍⠳⠇⣿⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢾⡷⣫⡜⣷⠏⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣟⣛⣿⡿⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣴⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠿⠿⠿⠿⣿⣿⡿⠟⠋⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡽⠟⠛⠛⣯⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣾⣿⣿⣿⣁⣀⣀⣠⣤⣴⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣵⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣤⣶⣶⣶⣦⣠⣾⣿⣿⠿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⣾⡏⠀⠀⢀⣽⣿⣿⡋⠁⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⣘⣧⣠⣶⣿⣿⠟⠹⣿⣶⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣟⢿⡟⠁⠀⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠒⠢⠤⢄⣀⡂⠀⢠⠁⢠⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠓⠧⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⡄⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⣠⠔⠒⠲⠴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⢦⡠⠊⠀⠈⠀⡀
⠀⠀⠀⠁⢀⠀⠀⠐⡁⡀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢠⠖⠛⠓⢤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣀⡀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠈⠉⠛⠓⠒⠒⠊⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⣄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠒⠤⠄⠀⠤⠄⠚⠁⠀⠀⠈⢷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠐⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠙⢆⡀⠀⠀⠀⠤⠞⠁⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣠⠞⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡇⠀⠀⠀⠀⠀⢀⠄⢀⡤⠊⠁⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢌⠛⠶⣤⣀⣀⠀⠀⠀⢨⠃⠀⠑⣤⣀⠤⠞⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⢄⠀⠈⠉⠙⠛⠛⠛⠓⠒⠈⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⡌⠷⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠑⢌⠀⠀⡀⠈⠀⣼⣧⡀⢀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢜⠢⠀⠌⡙⠳⣆⢢⠁⠈⠌⠃⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠙⠑⠀⠀⠁⡲⠐⠓⡷⠒⠐⠀⠠⣀
⠀⠀⠀⠢⢬⡐⠢⣩⣘⡓⠒⣧⣢⣤⡉⢘⡃⠀⠀⠐⠆⣈⠀⠤⠀⠀⠀⠀⠆⠀⠀⠐⠆⣤⠟⠀⣀⠈⢀⠀⢠⡅⠀⣰⠆⠠⠇⠀⢀⡀⠀⣸⠙⡂⢄
⠀⠀⠀⠀⠀⠠⠉⠀⡀⠉⠉⠀⠀⠈⠉⡄⠀⠀⠀⠀⠀⠀⠚⠀⠀⠑⢄⠀⠀⠈⢠⡀⠸⣧⠀⣴⣿⣿⡆⣀⣦⡤⠖⠛⠓⠒⠓⠐⠨⠄⠴⠁⠈⠁⠾⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠿⢥⡀⡀⡀⠰⢞⠹⣷⣮⡃⣤⠀⠻⢤⠐⠂⢁⡃⢸⣄⣻⣿⣷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢗⠠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓⠻⢖⣋⣈⣳⠤⠤⠤⠯⠥⠤⠽⢷⣿⣿⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣠⠀⡘⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣌⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⠀⠀⣴⠀⠀⠀⠀⠀⠀⢰⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣰⡏⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⢦⣁⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠈⠢⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠇⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⢧⡀⢀⠀⠊⣀⡼⠟⠛⢶⣀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡭⣶⣿⠏⠀⠀⠀⠀⡿⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

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


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⢰⣧⢀⣤⢰⣟⣄⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡃⣼⣿⡧⠿⢃⡿⠟⣡⣶⡎⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣻⣿⣷⣶⣿⠍⠛⠿⠋⠻⣿⢣⡄⠈⠃⣾⣿⣿⠇⡚⢛⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠳⡙⠛⠿⢋⠈⢻⣿⣿⣷⣦⣿⣿⢰⣆⠹⠛⣡⡖⢀⣿⣿⣥⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣷⣀⣾⡇⢸⣷⡄⠙⠿⣿⣿⡿⡟⣹⣟⠀⣾⣿⠗⣿⣿⠿⡣⠾⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⡟⢷⣶⣷⣤⠿⠃⢈⣻⣟⢸⣧⢀⣠⡆⢻⡅⠉⢀⣿⡯⣠⡆⠀⢨⣴⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢾⣧⡝⠄⣙⠿⠟⢷⡛⠿⣿⣿⠟⢧⣸⣿⠇⢸⣿⣴⣿⣥⣴⣻⡁⠀⣼⣻⣟⣋⣀⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⡟⣒⡛⢦⣄⣀⣿⣆⣀⠻⣿⣾⣿⡔⣦⣼⣿⢋⣀⣠⣾⣭⣷⠿⢋⣷⡶⠀⠸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠿⢋⣾⣿⡿⠉⣽⡿⠉⠙⠛⢿⣷⣦⣿⣧⣼⣿⣿⡿⠛⠉⠀⠘⠻⠗⠻⣿⣿⣆⠀⠙⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠛⢛⣿⣿⣿⠿⢷⣦⣤⣀⣀⣤⣤⣤⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢹⣶⣷⣾⣿⣿⣿⢻⣥⠴⠆⠀⠀⠉⠛⢿⣿⣿⣿⣿⢦⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣤⣷⣾⣿⣿⣿⣿⣿⡿⠇⣠⣤⣤⡀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠸⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠚⠉⠉⠛⠿⠿⠶⠆⠀⠀⠈⠻⣆⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⢲⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠈⠓⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⣾⢡⣿⣿⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⠀⠀⠀⠟⠛⣿⣿⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠃⠙⠉⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠈⠁⠙⠻⢿⣟⣉⡈⠉⠛⠛⠙⢛⣋⣹⣿⠟⠛⠵⠂⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⣿⡏⢻⡏⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣇

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣯⣍⢴⠾⣿⣡⡿⡛⣿⠿⣿⣿⠻⣿⣿⠙⠖⠿⠸⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⣿⣽⣿⣿⣏⣠⣾⣿⣿⣿⣿⣷⣴⣧⣙⣠⣿⣿⣇⣀⠀⡀⣀⠒⠒
⠀⠀⠀⠀⢀⠀⣀⣴⣿⠿⠟⢋⣿⣿⣯⣿⣿⣿⣿⣿⡟⠛⠋⣉⣉⣻⣾⠻⣿⣿⠛⠛⠻⠿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⣱⣿⢟⣽⣦⣼⣿⣯⣟⣛⣙⣿⣧⣾⠿⠝⠊⠈⢁⣉⣻⣿⣛⠻⠛⠳⣤⡀⠀⢠⢞⡤⠖⠒⠂
⠀⠀⠀⠀⣰⡿⣷⠞⠙⠛⢿⣏⣀⣀⠉⣿⡿⣿⣧⠴⠶⠖⠉⠉⠁⠀⠈⠉⠉⠳⠦⣤⡤⠤⣄⣀⡁
⠀⠀⠀⠀⣿⣿⡏⠀⢀⡴⠛⣉⡉⡛⢿⣆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⢿⣿⡇⠀⢸⡅⠀⠀⢸⠀⠈⣿⡄⢀⡀⢀
⠀⠀⠀⠀⠘⢿⣿⣆⡀⠑⠒⠒⠋⢀⣼⡟
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣶⣲⣿⠽⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣀⣤⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠟⠑⠀⠀⠂⠋⠀⡗⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⢿⣻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⡀⢤⣳⣿⠏⢐⣂⣀⢀⡀⣈⢮⣯⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⢹⣿⣿⢷⣯⣭⣷⣿⡿⠏⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠄
⠀⠀⠀⣠⡴⢉⠔⣿⣽⣯⡆⠈⠻⣿⣿⣷⣾⠿⠃⠀⠠⣌⣠⡼⠿⣿⠿⢿⣿⡀⣬⣯⣿⣿⣿⠛⠁⠀⠀⣠⣾⣿⣿⣿⣿⡿⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⡾⠋⡔⠁⢸⡷⣿⠁⠀⠀⠀⠀⠈⠀⠀⣰⣴⣿⡿⢿⣿⣧⠀⠑⣤⡶⣿⣷⣿⣆⠚⣿⠋⠀⠀⢀⠞⣿⣿⣿⡟⠛⠋⢀⣠⣤⣿⢇⣼⣿⣿⠏
⠀⠀⣿⣔⣆⢰⣟⣿⡟⠀⠀⠀⠀⣀⣠⣴⢿⣿⣿⣀⣀⣀⣈⡁⠃⠠⠘⣤⡿⠋⣿⣷⣿⠛⠃⠀⠀⠀⠀⠉⠉⠁⠀⣀⣀⣈⣁⣤⣷⣿⣿⠿⠋
⠀⠀⠫⢽⣛⣾⠿⠋⠀⣠⣤⠴⣿⣫⣽⡿⣿⣿⣽⣯⣭⣭⣭⠙⣶⣿⠟⠋⠀⠀⠙⠿⠿⡾⠶⠦⠀⠀⣀⠴⠾⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⡶⣟⣤⣴⣿⣿⣷⣾⠿⠿⠻⠿⠿⣿⡏⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⢊⣁⣤⠰⣶⣦⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣰⠋⣹⣿⣿⣿⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣴⢦⡤⠀⠀⠀⠀⢀⡠⠔⠋⢀⣴⡆⣵⣼⣿⣸⣷⣿⣿⣿⣿⣿⣶⣦⣀⣀⣀⣠⣤⣤
⠀⠀⠀⢀⣾⣽⣾⣱⠾⠋⠀⠀⣀⣤⣠⣴⣤⠀⠀⠀⠀⠀⠈⠿⠉⣿⡆⠀⠀⠀⠊⠁⠀⢀⣴⡞⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⢀⠏⣼⣿⠟⠁⠀⠀⣠⡸⣿⣿⠞⣻⠇⣾⣿⠦⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠐⠛⠻⠒⠋⢻⣟⡷⢿⣯⡿⢿⡿⠋⠉⠀⠈⠁
⠀⠀⣼⣼⣷⠏⠀⠀⠀⢰⣿⣷⣶⢆⠁⠛⢼⡾⠁⣿⡿⢦⣤⢤⡖⠿⣧⣿⣗⡀⠀⠀⠀⠀⠀⠘⠉⠉⠛⠛⠛⡿⠇
⠀⠀⣿⣿⡿⠀⠀⠀⠀⡿⠙⢟⡁⠦⠀⠀⠀⠀⠿⠿⣃⣻⣿⣜⣛⠳⠒⣌⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠄⢠⡀
⠀⠀⢻⣿⣷⠀⠀⠀⠀⠡⣿⣿⣿⢳⣆⡀⠀⠀⠀⠀⣹⣿⣧⣿⡇⣦⣤⣵⣿⣀⢀⡀⠀⠀⠀⠀⢸⢃⠀⠀⡀⠀⠀⠀⠀⠀⡀⠦⣅⡞⡁⠀⢀
⠀⠀⠈⢧⣻⣆⠀⠀⠀⠘⠷⠈⣰⣿⣿⣝⡶⢤⣶⡸⣿⣿⠈⠿⠘⠋⣵⣿⢻⡶⠟⢍⡄⠀⠐⠂⠿⠛⢄⣁⠑⢘⠓⢀⠂⣼⣹⣶⠿⠆⠙⠁
⠀⠀⠀⠀⠻⢿⣦⣄⠀⠀⠀⠀⠁⠀⠫⡿⠋⠸⣿⣃⡘⠋⠁⠀⣀⢠⣶⠍⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⠋⠁⠈⠀⢠⠕⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⡀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢠⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡞⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢞⠢⠄⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠂⠀⡔⠠⠠⠺⠂⠀⠢⡈⠂
⠀⣀⡀⡀⠀⠀⡌⠀⠀⡰⠁⠐⠒⠒⠒⠒⠒⠶⠒⠆⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠐⣇⠀⢘⠁⠈⠙⠀⠀⠀⠐⠄⠀⡠⠃⢄⣠⠦⡀
⢡⠀⠀⠀⠀⠀⠰⡀⠐⠀⠀⠀⠀⠀⠀⠀⠰⢤⠀⠀⠀⠀⠀⠉⠐⢺⠀⠀⠤⠀⠀⠀⠘⠕⢀⠀⠒⠠⡀⠀⠀⠀⠀⠀⠀⡀⠶⡄⠀⠈⢢
⠀⢆⠀⠀⠀⠀⠀⠀⠀⢒⠒⠒⠂⠀⠀⠀⢤⡌⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⡉⠐⠀⠈⠢⡀⠀⠑⢄
⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⣄⣀⠀⠀⠀⠀⠀⠀⠹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢀⣀⠀⠁⠀⢀⡠⡇⠀⠀⠀⠣⡀
⠀⠀⠘⡀⠀⠀⠀⠀⢀⠔⠀⢸⠀⠀⠀⠀⢀⣀⠐⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠠⠤⠐⡆⠉⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠈⢢⠀⠀⠀⠈⢦
⠀⠀⠀⠡⠀⠀⠀⢠⠂⠀⢀⠦⠀⠀⠀⣀⢢⠠⠵⠀⠀⠀⠀⢀⠠⠄⠚⠒⢲⠀⠉⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠑
⠀⠀⠀⠀⢃⠀⠀⠆⠀⢠⠂⠐⠃⠈⢁⡀⠤⠤⢄⣃⣠⡔⠊⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢀⣈⡦⠠⠤⠀⠀⠒⠚⠒⠀⠉⠉
⠀⠀⠀⠀⠈⠆⠀⠢⠀⠀⠐⠂⠘⡇⠀⠀⠀⠀⠀⠈⠀⠱⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣀⡀⠀⠠⠤⠄⠚⠒⠀⠉⠉
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠤⠄⠐⠂⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡀⠤⠵⠀⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡤⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⣿⡿⢻⣷⠈⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⢺⡀⠀⠀⣿⠟⠁⠘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⢸⡇⠀⣀⠀⠀⡀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣀⠀⠀⠠⣿⣿⣿⡿⠀⠸⢇⣀⣿⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⢿⣿⣿⣧⢰⡿⠞⠛⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣶⠀⢸⣿⣿⣇⠘⠁⣠⠖⠈⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⣿⡤⠀⠻⣿⣿⣶⣶⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣴⣿⣿⣿⣧⣿⣸⣤⣾⠿⠋⠉⠁⣾⠿⠿⠃⠀⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣼⡟⠃⣿⠇⠐⠟⠃⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠈⢻⣿⣷⣾⣄⢁⡀⠲⠀⠿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣇⠀⣱⡟⠃⠉⠉⢙⣤⠜⢺⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡿⠷⠾⣷⣆⣹⣧⣼⣿⣿⣿⡿⠿⠟⠛⠃
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠘⠟⠙⢣⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⢸⡇⢠⣾⡅
⠀⠳⣄⡀⢸⣿⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⢸⣿⣾⠟⠁⢀⡄⠀⣠⡆
⠀⠀⣿⡇⠘⣿⡁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣀⣽⣧⡴⠞⠋⢀⣶⡟
⠀⠀⠉⢁⡴⠈⠉⠀⠀⠁⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠻⣿⣿⣿⣿⡟⠉⠁⣠⠶⣠⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠑⣄⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣆⠈⠻⣿⣿⣷⣮⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⠃⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⣿⡇⠀⣹⠿⠟⠋⠉⠀⠀⡀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠃⠀⠀⣶⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡜⢷⢀⣿⠀⠀⢠⡤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⢁⣴⡗⠉⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠉⠀⠀⠀⠠⣤⡀⠀⢸⣿⣿⠉⢿⣿⣯⡘⠻⣧⣴⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⣨⢻⣶⣌⣿⠃⠀⠀⠹⠁⣠⡾⢛⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣿⣿⣠⣆⠀⠀⠿⠄⢹⣿⡏⠀⠆⠀⠀⣴⣟⣵⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣆⡀⠀⣤⡀⢻⣿⠀⢺⡆⠀⣼⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠻⣿⠉⠈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⡟⠉⠉⠉⠉⠀⠹⣿⡾⣿⣇⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⢰⣦⣤⣄⠀⠈⠁⢸⣿⣤⣿⠇⠀⠀⢀⣤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣰⠁⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⢻⣿⣿⣷⣤⡀⠈⠻⣿⠇⢀⡀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡇⢰⡿⠻⣿⣿⡿⢡⠀⣿⢠⡜⢣⣾⠏⠀⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠺⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠜⠁⠀⠘⢿⠇⢸⡄⠸⣿⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⠈⠁⠀⣴⠁⠈⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⠀⠀⢠⠆⠀⢀⣤⡀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠙⢧⡀⢰⣈⣷⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⣾⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣶⣼⣿⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠀⠀⠀⠺⣿⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣏⠈⣿⣿⣇⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⠈⣤⠀⢀⣴⡀⣿⣧⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⡀⢃⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⢷⣄⡹⡀⢸⣿⠇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠘⡷⠗⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣤⠀⠀⠀⣉⠛⣿⠈⠻⠀⢈⡟⠀⠀⠀⠀⠿⠃⢠⠀⣀⠀⠀⠾⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣤⣘⠿⣤⡄⠀⠀⠸⠟⠀⠀⠀⠀⠀⡂⠁⢠⣿⡆⠘⣀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠿⣿⣷⣽⣿⣴⡀⢰⡆⠀⠀⠀⢠⣾⣿⣿⠀⢻⣿⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠈⠻⣿⣿⣿⠀⢸⡇⢠⡆⢠⡿⠋⠈⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠈⠻⣿⣤⣤⡆⢀⣶⠏⠀⣶⣾⣿⡀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠈⠙⣿⠁⠈⣡⡄⠘⣿⡇⢀⠁⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠸⡀⢀⣿⡇⢸⣿⣇⡈⣡⡭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⢰⣧⠘⣿⡇⠈⢻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣄⠀⢸⣿⠀⠸⠇⠀⠏⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢿⣿⣿⣿⣿⢈⣿⠀⠀⠀⠀⢸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢻⣿⣇⠈⠸⠿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⢹⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣼⣿⡀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢈⠃⠀⠰
⠀⠀⠀⠀⠀⢠⠀⣠⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠄⢺⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡴⠊⠁
⠀⠀⠀⠀⣰⠏⠀⠆⠀⠀⠀⠀⠀⠀⠀⣠⠾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣤⡿⠏
⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠨⣗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⢡⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣀⣿⠿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣦⣿⣯⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⢷⣦⣄⡁
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣧
⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠘⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡤⣶⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⡀
⠀⠀⠀⠨⡻⢐⠀⠄⠀⡀⢀⡀
⠀⠀⠀⠀⠈⠂⠔⢀⡀⡄⠺⠷⣆⡀⢤⠦⠠⢠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠒⠂⠀⠀⠀⢄⠤⠠⠤⠠⢄⣄⣀⡀
⠀⠀⢀⠀⢄⡀⠀⣤⣀⣦⢀⠀⠄⣿⠘⠂⢀⡀⠄⠀⠀⠀⠐⢃⠀⠀⢀⡀⢠⡄⠂⠰⠀⠀⢠⠀⢘⡀⢀⡉⠀⠁⠁⢲⠤⡀
⠀⠀⠀⠈⠉⠐⠚⠙⠒⠉⠙⠛⠂⠉⠀⠀⠀⠈⠡⠄⠀⠤⡀⠀⠐⢀⡀⢼⡏⢠⣼⣦⡄⢀⣉⣠⠼⠧⠨⠆⠠⢃⣠⠟⠐⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠾⣅⣀⠀⠠⡖⢳⣦⡗⣤⠀⠦⡄⠀⢁⡂⣳⡘⣿⣿⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡛⠛⣋⣥⣽⠒⠀⠐⠙⠠⠞⠳⣿⣷⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⠈⠁⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣤⠒⢋⡅⠀⠘⠁⠀⢃⠀⢨⡍⠲⣤⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠀⠰⡖⢛⣍⣉⣛⠓⣶⣾⡂⡰⠈⠙⠋⢀⢀⢀⡀⠙⠉⠈⢆⢀⣷⣶⠖⢛⣁⣠⡁⢠⠄⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⡙⠃⣌⡓⡄⠀⠀⠀⣀⠅⠨⣀⠂⠀⠀⠠⠞⣁⠜⢋⠉⠁⠌⠉
⠀⠀⠀⠀⠀⠀⡤⠶⣤⣀⢰⡾⡀⠀⠀⠙⠒⠦⣥⣶⣆⠀⠀⡄⠀⠀⢠⠀⠀⣠⣶⣬⡴⠲⢋⠀⠀⠀⢳⡄⢀⣾⠶⢤
⠀⠀⠀⠀⠀⠘⠀⢀⣈⣳⡚⠀⠙⠀⢃⠀⠀⢠⠈⠉⠻⢶⣠⡑⠆⠰⢊⢄⡶⠟⠉⠁⣤⠀⠀⡘⠀⠏⠀⢇⣾⣁⡀⠀⢃
⠀⠀⠀⣠⠞⠋⠈⠉⠉⢍⢳⣄⣳⡿⢤⣲⡌⠛⡀⠈⠀⢤⠉⠪⠋⠙⠕⠉⡠⠀⠀⢠⠙⢡⣖⡠⢿⣞⣠⣞⡍⠉⠉⢁⠘⠳⣄
⠀⢀⡞⠑⣶⠞⠣⠀⡀⠀⠚⠀⠻⢷⣤⠞⠻⣆⠀⠐⢤⡆⠀⠀⠀⠀⠀⠀⢰⡤⠂⠀⣰⠟⠳⣤⡼⠟⠀⠓⠀⢀⠀⡜⠳⣶⠎⢳⡀
⠀⠎⠑⠚⠁⠄⠠⢹⣍⠓⠤⠄⠀⠀⠙⠹⢶⡄⠀⠁⡄⠀⣦⡬⠀⠀⢧⣴⠆⢠⠜⠀⢠⡶⠛⠏⠀⠀⠠⠤⠞⣩⡏⠄⠀⠈⠓⠋⠱
⠘⠳⠞⠀⠀⠐⣤⣶⠈⠙⠦⣴⠶⠦⢤⠇⠀⠧⠲⣄⣸⢦⡀⠀⣀⣄⠀⢀⡴⠇⣠⠖⡬⠁⠘⢤⠤⠶⢦⠴⠊⢁⣶⣤⠆⠀⠀⠳⠞
⡷⠞⠀⠀⠀⠀⠁⠉⡠⠚⢉⡋⠺⢔⠲⢤⣤⣄⣀⠈⢷⣆⠈⢧⠉⠉⡼⠅⢠⡿⠁⢈⣠⣤⡤⠖⡢⠟⢙⡉⠓⢄⠉⠈⠀⠀⠀⠀⢱
⡇⠁⠀⣠⠀⠀⠀⠀⢀⠿⣶⡶⠲⣤⣉⠢⡀⠁⢤⡀⢈⡀⣡⡈⠧⠼⠁⣤⢀⡀⢀⡄⠈⢁⠔⢉⣤⠖⢶⣶⠾⡄⠀⠀⠀⠀⣤
⢧⠀⠀⣭⠃⢀⠄⠀⠎⣸⣿⠁⣴⡈⢻⣶⡌⠲⣖⠻⡉⠀⠀⠁⠀⠀⠀⠀⠀⢉⠟⣰⠖⢡⣶⡟⢁⣶⠈⣿⣇⠱⠀⠠⡂⠐⣽
⠸⡀⠀⠀⠀⠀⢈⠀⡆⣿⣿⠰⣿⡟⠸⣿⣿⣆⠈⠢⡜⢀⠀⣰⠃⠈⣄⠀⣀⢯⡴⠃⣰⣿⣿⡇⢺⣿⡇⣿⣿⢰⠀⡅⠀⠀⠀⠀⢀
⠀⢳⡀⠐⠀⢀⡂⠀⢁⢾⣿⡄⢿⠁⣼⣿⣿⠏⢳⢤⡄⢈⠋⠁⠀⠀⠈⠙⡁⢠⡤⡞⠹⣿⣿⣧⠀⣿⢠⣿⡿⠐⠀⢐⠀⠀⠐⢀⡞
⠀⠈⢧⠀⠀⣠⠃⠀⠉⢢⣙⡿⣤⣶⣾⣿⡟⠚⠁⣀⣴⡟⠿⠁⠀⠀⠈⠾⢿⣦⣀⠈⠓⢻⣿⣿⣶⣤⢾⣋⡔⠉⠀⠈⣄⠂⠀⡼⠁
⠀⠀⠀⠳⠄⠱⣄⡀⢀⠀⠉⠛⠶⠾⠿⠤⠴⠿⠛⢉⣴⠃⢴⠀⠀⠀⠀⡆⠘⣦⡉⠛⠿⠧⠤⠿⠿⠴⠛⠉⠀⠀⠀⣠⠎⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⡅⠈⢢⠀⠀⠀⠀⠀⢀⠀⠀⠀⠟⠛⠀⠰⠀⠀⠀⠀⠆⠀⠛⠻⠀⠀⠀⡀⠀⠀⠀⠀⠀⡴⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠙⠈⢂⠀⠙⠦⣄⠀⠀⠀⠀⠀⢠⠴⠦⣤⣴⠇⠀⠀⠐⣆⣠⠴⠶⡄⠀⠀⠀⠀⠀⣀⠴⠋⠀⡐⠁⠋
⠀⠀⠀⠀⠀⠂⠀⠓⠀⠑⣄⠄⠀⠙⠓⠖⢠⠄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠛⠁⠀⠀⠈⠠⡄⠲⠖⠋⠀⠀⣠⠊⠀⠊⠀⢐
⠀⠀⠀⠀⠀⠐⡎⠀⠑⠀⠉⢣⡀⠀⠀⠀⠛⠆⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠐⠛⠀⠀⠀⠀⠐⠁⠀⠈⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠘⡄⠈⠃⠀⡀⠙⣄⠄⠀⠈⡓⠄⠀⢸⣷⡀⠀⠀⠀⠀⢀⣾⡇⠀⠠⢚⠁⠀⠀⣤⠋⢀⠀⠐⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠐⠃⠀⢄⠈⠳⣤⠀⠘⠂⠀⠈⣿⣷⠀⠀⠀⠀⣾⣿⠁⠀⠀⠋⠀⣤⠞⠁⣠⠂⠘⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡑⡌⠀⡁⠙⢳⡀⠆⠀⠀⠈⢋⠀⠀⠀⠠⠉⠁⠀⠀⠠⢀⡜⠋⢈⡄⢡⣊⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢈⠢⡙⢦⡁⠉⠛⢦⡈⢂⠀⠀⠀⠀⠀⠀⡐⢁⡴⠛⠉⢀⡠⢋⠴⡁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠘⠢⡉⠢⣀⠈⠉⠚⠷⢤⣄⣠⡤⠶⠓⠉⠀⣀⠔⢋⠄⠃⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢈⠒⢌⡙⠢⣀⠀⠀⠀⠀⠀⠀⣀⡔⢋⡡⠒⡁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠐⠉⠒⠤⣉⠒⠤⠤⠒⣋⡤⠒⠉⠒⣦⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⡄⠀⠁⠀⠀⠉⠀⢀⣠⡴⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣶⣤⣤⣶⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠒⠉⢹⣿⣿⠃⠉⢹⣷⠲⣄⣀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⠿⣿⠷⡀⠹⠉⠏⠀⢀⠾⠃⢠⡿⠳⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆⠀⠀⠘⡆⠀⠀⠂⠀⠀⠇⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠻⢿⣽⣤⡀⠀⠈⢢⣴⡆⠀⠀⣤⣶⠀⠀⠁⠇⠀⠀⠀⠁⡂⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⠻⠛⠉⠀⠙⡌⡿⠁⠀⢀⡽⠋⠀⢀⣂⣀⣀⣀⣀⣀⣀⣀⣀⡴⠃
⠀⠀⠀⣠⢀⡀⠀⢠⠎⠀⠀⠀⠀⠀⢀⣠⠀⠀⡟⠇⠀⠐⠒⡇⠀⠺⡯⡀⠃⢄
⠀⠀⠀⢱⣀⡉⢱⠋⠀⠀⢀⡤⠔⠊⢹⠁⡄⠀⡇⠀⠀⠀⠀⠈⠢⡀⠑⢬⣷⣄⠝⠒⠒⠒⠒⡲
⠀⠀⠀⠀⠈⢩⠃⠀⠀⡴⣁⡙⡙⠒⣯⠂⢿⠀⠨⠑⠒⠲⠒⠒⠒⢻⡄⣸⡤⠚⠻⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠸⠀⠀⡸⠒⡙⢉⠛⡍⠙⠒⠻⣄⡜⠚⠒⠒⠛⠊⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠁⢈⣐⠂⠃⢸
⠀⠀⠀⠀⠀⠈⢆⠅⠐⠄⠁⠔⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠠⠄⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣠⣴⠶⠎⢘⣉⣿⣟⣛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡿⠿⢿⣿⣟⠁⠂⠾⢟⡋⠉⠁⠁⡈⠁⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⣯⣴⠟⠉⠻⠒⠂⠀⠀⠀⠙⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠶⡿⠿⠿⠿⣷⣬⣾⣿⣿⡟⢻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣵⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣷⡆⢀⡴⣿⠗⡈⢿⣶⣦⣾⣶⣼⣿⡟⢿⣿⣷⣄⠀⠀⠀⠀⠀⠙⣿⣿⣩⣮⣩⡆
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣀⣿⣴⠻⠞⠋⠈⠉⠙⠛⣿⣿⣯⠻⣿⣿⣿⣿⣷⣤⣀⠀⠀⣀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⢧⣿⣯⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⢨⣝⢛⣿⣿⠙⣻⣿⣿⣿⣿⣿⣿⣯⣾⡏
⠀⠀⠀⠀⠀⣿⣿⣾⣿⣼⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣛⣷⣾⣿⡿⢿⣿⡟⠛⠳⠟⠉
⠀⠀⠀⠀⠀⣿⣏⠻⣿⣥⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠎⠙⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣯⣜⢿⣿⣧
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣮⡛⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣎⢻⣿⣷⣌⣙⢶⣶⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣶⣾⣿⣷⣯⣷⡎⢉⣿⢛⣷⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠿⣿⣿⣯⣬⣤⣽⣟⠾⢿⣿⣛⣿⣿⣿⣿⣿⣷⢿⡟⣿⣿⣿⣿⣽⣿⣿⣷⣤⣍⢸⣿⣿⡟⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡑⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣼⣿⢻⣿⣿⣿⣿⣿⣿⡿⣹⣿⠈⢿⣿⣿⣷⣮⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⢿⣫⣽⣿⣿⣭⣭⡭⠅⠠⠿⠟⢿⣷⣎⠲⣿⣿⣿⣿⣿⠿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠛⠿⠿⠿⠋⠐⠒⠂⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣿⣿⡿⣿⣿⡆⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣷⢻⣷⣿⣟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⡆⣿⣿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⢿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣻⡿⢃⣤⣴⣶⣦⣽⡿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣿⣵⣶⠀⣿⣿⡟⣼
⠀⠀⠀⠀⠀⠀⣢⣿⡎⠙⢁⣤⡿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣝⣫⡿⠿⢿⣬⠿⠿⢟⣶⠶⢶⡄⠰⠶⠾⣯⣴⣿⡿⠇⠀⠈⠻⣿⣿
⠀⠀⠀⢀⠀⣿⣋⣿⣷⠶⢦⣤⡔⢻⣿⣶⣿⣷⣿⡿⣷⣄⣹⣛⡿⣿⣿⣿⠿⣿⣾⣿⣶⣶⣶⣿⣿⣿⢿⣿⣿⠟⣺⣿⣯⣽⣶⣶⡀⠀⠀⠉
⠀⠀⢠⣿⡇⢿⣿⣏⣤⡾⠿⠛⣡⣼⡷⠟⠛⠛⠛⠻⠛⠻⠿⠿⢿⣿⣿⣟⣤⣿⣿⠿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣾⣿⣷⣮⠛⠛⢠⣤⡶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⡿⢿⣷⣦⣶⣿⣿⣿⣿⠈⠁⢹⣿⡟⣛⣻⣿⣿⡿⠟⠁
⠀⠀⣩⣾⣿⣷⣤⡀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆⠙⠫⣭⣤⡤⠼⣿⠿⠿⠛⠉⠁
⠀⢸⣿⣿⣿⣯⣽⣧⣽⡇
⠀⠀⣏⣿⣿⣿⣿⣿⡍⢷
⠀⠀⣹⡛⢿⣿⡟⢛⣰⠘⣷⣤⡀
⠀⠀⠘⢿⣦⣿⣿⣞⣋⣽⣿⣟⠻⡿⠷⣴⣤⣤⣤⣤⣠⣤⣤⣤⣠⣄⢠⣤⡀⢠⣤⣤⣄⣀⣀
⠀⠀⠀⠈⢿⡿⣿⣾⣿⣿⣭⣛⠻⣿⣿⠿⣿⣤⣤⣼⣿⡿⠻⣛⣇⣸⣷⣦⣄⢺⣿⣿⣿⡄⣄⣠⣿⣿⣶⡶⢠⣄
⠀⠀⠀⠀⠀⠙⢷⣹⣿⣿⣿⣿⡿⣫⡟⣿⣧⣴⣤⣭⣿⣧⣿⣿⣭⣿⣯⣭⣽⣿⠏⠛⣡⣼⢿⣿⠿⣿⣧⣤⣾⣿⣿⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⣿⣽⣾⣭⣤⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⣶⣿⣿⣯⣻⣷⣿⣿⣏⣰⣷⣿⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⣿⣿⣿⣿⣛⣿⣿⢏⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣧⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡶⢫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠆⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣼⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠿⢌⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡰⡇⢠⡟
⠀⠀⠀⠀⠀⢀⢀⣀⣤⡤⠴⠒⠲⠶⠶⠾⠶⠶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⣬⠛⣦⠞⠁
⠀⠀⠀⣠⣤⡾⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⠟⠃⠀⣩⡾⠋
⠀⠔⠋⠈⠛⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠭⣭⣛⣛⣛⠛⠛⠿⠟⠛⢛⣛⣉⣡⣤⡤⠞⠂⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠛⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡀⣢⠖⣨⣄⠀⠀⢀⣈⡅⠲⣶⢀⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠒⠐⠓⠾⢛⠄⠀⠀⣀⠂⣂⠀⠀⠁⡚⠻⠐⠒⠒⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢦⣈⠖⡀⠀⠓⠪⢴⣄⠀⢀⠀⠀⠀⣠⡶⠔⠚⢀⠀⠸⣀⡴⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠐⠲⡷⣀⣴⢀⣄⠖⠀⠈⠙⠺⠂⠔⠊⠁⠀⠳⣠⡀⣦⣀⣿⠖⠂⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡱⠟⢂⣀⡈⠉⠻⣶⣛⠇⠀⠰⠄⠀⠀⠀⠀⠖⠀⠰⣛⣶⠞⠉⢁⣀⡀⠛⢾⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠀⠠⣄⡓⠌⣡⢤⡀⠉⢳⣄⠢⡈⠋⣀⠘⢃⠔⣠⠴⠉⠀⢠⣤⠡⢚⣥⡄⠀⠡⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠉⠀⢀⣉⠒⠔⠠⢤⠈⠻⠀⢻⠉⡙⠀⠟⠁⣤⠴⠢⠒⣉⣀⠀⠈⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡇⠀⠀⠔⣽⠏⡜⢶⣔⠠⡴⠒⠂⠀⠁⠀⠁⠒⢎⡄⢢⡴⢫⠙⣿⠱⠀⠀⢸⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⢀⣿⡰⡿⢸⣿⣦⠑⠖⠀⠀⠀⠀⠀⠰⠊⣰⣿⡇⢻⡇⣿⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⢀⠀⠀⣙⣧⣡⣼⣿⠇⠊⣡⣶⠀⠀⠀⣶⣌⠉⠸⣿⣷⣌⣼⣏⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠱⡄⠀⠈⠙⠊⠿⠀⠛⢉⣽⠀⠆⠀⠀⠂⣧⡉⠛⠀⠿⠃⠊⠁⠀⢠⠎⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠐⢄⡀⠀⠀⠀⠨⢀⣈⡀⠀⠠⣀⡀⠄⠀⠀⠀⠀⡠⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠠⡄⠈⠓⢂⡄⠀⠀⠉⠀⠀⠀⠉⠀⡀⠀⡄⠒⠁⢀⡄⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠄⠀⢰⣄⠀⠀⠀⢠⡆⠀⠰⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠱⣄⠈⠀⠈⣿⠆⠀⠀⣿⠃⠀⠉⢠⠎⠁⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⢀⠂⠈⠒⢦⡀⠈⠀⠀⠀⠀⠀⡴⠒⠁⠔⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠈⠀⢀⡁⠓⠆⠤⠐⠚⠉⡀⠀⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⠐⠀⡀⠤⠀⠤⢀⠀⠂⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⡀⠀⢀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠢⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣀⠜⠀⠀⠀⢀⡁⠀⠀⡠⠐⠀⠀⡤⠤⢄⠀⠀⠀⢀⠤⠠⠤⠤⠤⠤⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢠⡆⠀⠀⠀⠀⠉⠑⢄⡌⠀⠀⠀⠀⠈⠒⠈⣐⠢⢄⠛⠐⠒⠒⠤⡀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡄⠀⡗⠂⠉⠀⠈⣄⠀⠀⠓⠤⣦⣀⡙⡄⢀⡴⠃⠀⠀⡘⠑⠉⡾⠀⠀⠀⠀⠀⠈⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⢣⠀⠀⠀⠀⠘⢆⠀⠀⠀⣀⣄⠈⣏⠡⠄⠀⠀⡰⠁⠀⢠⠁⢠⠃⢰⠀⢀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠣⣀⣔⠙⢧⡿⣇⡠⠌⠦⠺⡅⡰⠀⡌⢀⡶⠀⡼⠋⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠛⠲⠼⢤⡆⠀⢸⠀⣀⡠⠤⠐⠊⢁⡨⠏⠲⠊⢹⠉⠀⠀⠀⠀⠈⠑⣴⠀⠀⡱⠞⢤⡄⣀⣁
⠀⠀⡠⠒⢲⠄⢀⠤⣀⠀⠠⠒⠀⠙⣆⣋⢉⡀⠤⣴⡂⠉⠀⢠⠒⠒⠀⠉⠉⠉⡖⠤⠤⢄⣀⠥⢄⠜⠀⠀⠀⠀⠀⣀⠤⢤⠄⠒⢤
⠀⠀⠀⠉⠁⠨⢄⡠⠤⢃⡴⠢⡤⢔⣺⡋⢠⡊⠉⠀⣑⠀⠀⠀⠉⠒⡄⠀⠀⡔⠁⠀⠀⠀⠀⣤⡤⠷⠖⢣⡀⠀⠘⡄⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⡀⠀⠀⣏⣹⠁⠀⠙⠤⠜⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠤⢰⣞⣅⠀⠀⠑⠒⠢⠤⠤⠜
⠀⠀⠀⢀⣠⠖⠒⠌⠗⢨⠤⣤⠞⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⠦⡠⠔⠉⠐⠒⠊⠁⢰⠁⠀⠈⢢
⠀⠀⠀⢹⠀⢀⠏⠾⠸⢿⠖⠊⠀⠀⠀⢀⠖⠉⠀⠀⡉⠁⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢃
⠀⠀⠀⠑⠤⠃⠀⠀⠀⠌⠀⠀⢀⠤⠤⠆⠀⡤⠜⠂⠁⠚⠢⡂⠀⠀⠀⠀⠀⡤⠚⠂⠉⠒⠢⡄⠀⠔⠁⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠃⠀⠀⠀⡰⠁⣠⣾⣿⣷⡄⠘⡄⠀⠀⠀⡜⠀⣴⣾⣿⣶⡄⠙⡄⠓⠄⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⢄⠈⠢⣀⠀⠀⣃⠀⢿⣿⣿⣿⡿⠀⡇⠀⡀⠀⡄⠀⣿⣿⣿⣿⡇⠀⡧⢀⠀⠀⠱⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠱⡄⠀⠀⠀⠘⢆⡈⠙⠛⠋⢀⠔⠀⠋⠀⠁⠘⢄⠈⠛⠛⠋⣀⠜⠀⠀⠉⠂⠔⠀⢠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠑⠦⣅⣀⡠⢺⣂⠀⡈⠑⢆⡈⠁⠀⠀⠀⠀⠀⠀⡜⠉⠁⠂⠈⠁⣀⣀⡫⠒⠤⠤⠒⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⣀⡀⠀⠀⠀⡐⠊⣅⠀⠀⠈⠑⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠶⣂⠀⠀⡠⠄⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠙⢦⣰⡬⠿⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠿⠑⣆⠎⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⢫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠤⠄⡄⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠈⡄⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⢄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠒⠉⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠧⠤⠀⠀⠀⠤⠴⠓⠂⠉⠁

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

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠿⣿⣷⣦⣀
⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄
⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⡀
⠀⠀⠀⠹⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣄⡀
⠀⠀⠀⠀⠙⠿⣿⣟⡿⠷⠶⠒⣛⣛⣉⢩⣍⣉⡙⠻⢿⡶⣤⣀⣾⣷
⠀⠀⠀⠀⠀⠀⠈⠛⠩⠭⠭⠭⠥⠖⠒⠂⠀⠚⠿⣷⣦⡈⠪⡙⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣢⣶⣄⣀⡀⠀⢈⣹⣏⠢⡈⢢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠛⠻⠿⢿⣷⡿⠿⠿⢷⡙⢄⠙⢮⡻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣖⣢⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠱⡀⠹⢦⡙⠦⣄⣀⠀⠀⠀⠰⣦⡀⠀⣀⣠⡴⠟⢋⣩⡴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠢⡀⠙⠳⣤⣍⣙⣛⣓⣒⣛⣿⡿⠟⣉⡤⠖⣋⣵⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣈⠓⠤⣀⣈⣉⣉⣉⣉⣩⠤⠒⢋⣡⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⣭⣭⣭⣭⣥⣶⣶⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠠




⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣟⣿⣿⠿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠤⠄⣀⢀⣀⣀⠀⠀⠀⢠⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣼⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⠁⠁⠄⠢⠈⠐⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠖⠛⠛⢦⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠠⠤⠄⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⡲⠶⠂⢈⠂⢕⠠⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠈⠻⠀⠀⠀⠈⡛⠶⠞⠁⠀⠀⠑⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠈⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡀⣠⠏⠀⠀⠈⣷⢀⠀⠀⠐⠊⠁⠀⠀⠀⠀⣀⡀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠛⠁⠀⠀⠀⣰⣿⠏⢆⠘⠢⠄⣀⣲⣶⠯⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⠞⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡷⠋⠀⠀⣀⣠⣶⣿⣿⠟⠒⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣧⠀⠀⠘⣿⢿⣿⡟⢻
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠔⠉⠀⠙⠶⠶⢤⣴⣤⣤⣄⣁⣰⣵⣹⣛⣧⠀⠀⢇⠈⣻⣁⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠿⠛⠉⠀⣑⣤⣀⣀⡀⠠⡈⠛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡴⢶⣌⢳⣶⣤⣤⠀⠀⢀⡤⣋⠉⣁⣀⠤⠒⣈⣩⡤⠿⠟⠛⠉⠑⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠀⠀⡌⢩⡿⠟⠉⠫⡀⠈⠉⠀⠀⠀⠀⠨⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣛⡟⠁⠀⠀⢀⡠⠿⣷⣾⡿⠛⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⢹⠁⠀⠀⡠⠊⣠⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠹⡆⠀⣼⡧⠞⠛⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢷⡜⢁⣀⡤⠤⣤⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠹⣿⣿⠁⠀⠀⢹⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠈⠙⠦⠤⠴⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠊⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⠀⠀⠄⠐⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢵⠆⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠂⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠌⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⢠⠏⠀⠄⣀⠤⠀⠀⠀⠀⠀⠀⠀⡝
⠀⠀⠀⠀⢠⠏⠀⠀⠸⡠⣤⠄⣠⡀⠘⡀⠀⠘⡄
⠀⢀⡤⠐⠀⡀⠀⠀⠀⠱⡅⠰⠀⠀⠀⠀⠈⠈⢁
⡔⠁⠀⠀⠀⢡⠀⠀⠀⠐⠐⢠⣆⠀⠀⣀⠀⠀⠀⠑⢤⠐⠒⣄
⡆⠀⠀⠀⠀⠀⢧⠀⠀⠠⠀⠁⡜⠀⠈⣠⠇⠀⠀⠀⠈⡆⢡⠸
⢀⠀⠀⠀⢀⠠⠘⠞⠁⠀⠀⠀⢇⠀⢰⠁⠀⠙⠒⠤⠴⠃⠀⢹⡀
⠈⣄⡴⠺⣟⠀⠂⠀⡠⠀⠀⠀⢬⠠⣸⠀⠀⠂⠀⠀⠀⡠⠔⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊
⠀⠋⠀⠀⠀⠉⠉⠁⠈⠑⢄⢢⡘⣄⡏⠀⠀⠀⠐⠔⠁⠀⠈⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡙⠛⠁⢀⠀⠠⠁⠈⢂⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⠚⠐⠁⠀⠐⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠄⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⠀⠜⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠂⠀⠈⠳⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠤⠤⠔⠂⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠢⡀⠁⡀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡘⠢⣀⠑⢆⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠙⠳⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⢀⣠⠖⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣭⣽⣿⣯⣽⣯⣭⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠒⠶⠙⠲⠒⠲⡦⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡶⠀⠁⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣤⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⡀⠀⠂⢠⠤⢅⣈⡀⠁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠈⣀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢠⡥⣒⣢⡶⠧⠴⠔⠂⠴⠏⠷⠆⠀⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⢸⠆⠀⣀⢀⠀⠶⠄⣓⠛⠉⠓⠀⠙⢫⡕⡄⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⡴⢤⠶⠮⠊⠁⡲⠌⠉⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡨⠁⠈⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣟⣏⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣽⣽⣿⣯⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛
⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠴⣦⢴⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠂⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠋⠄⡿⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠂⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⠐⠒⠀⣩⢉⠉⠫⠝⢁⢐⠒⠢⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⢘⠶⠂⠀⠀⣁⠀⢖⡰⠉⠘⠀⡠⠂⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢰⠁⠀⢀⣀⣠⠄⢰⡤⠰⠁⠴⢠⠀⢀⢀⠠⠀⢀⣈⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⠃⠀⠀⠀⠀⠀⠈⠒⠊⣁⠀⢠⢤⡔⠀⠀⠀⠀⠘⠤⠥⠐⠒⠈⠀⠐⠦⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣮⡀⢀⡀⠀⠀⠀⠀⠀⠐⢀⠠⠔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⠟⢰⠁⠈⠑⠒⠤⡀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢢⠀⠀⠀⠀⣔⣁⣵⡀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠘⡓⠢⢤⣀⣘⠐⢮⠀⠸⣖⣤⢀⡀⢰⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠊⠀⢄⣐⣠⠴⠟⠛⠓⠒⠈⠉⢉⠅⢰⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠐⢄⠀⠈⠉⠛⠂⠤⣀⡈⠛⢖⠀⡂⣠⢀⠐⣂⠔⡨⠔⡡⠔⡠⠂⠔⠡⢊⠔⡡⢐⠵⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠊⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡣⡀⠀⠀⠀⠀⠀⠈⠑⠢⣿⡰⢙⠟⣈⢔⡨⢐⡡⡒⠕⡈⢐⠡⢊⡴⢚⠗⠁⠀⠀⢃⠀⠀⠀⠀⠀⠀⡐⠁⡘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⠐⠡⢐⠅⡊⢔⠡⢤⢁⠾⢃⠗⠁⠀⠀⢠⠃⠀⠑⠄⠀⠀⢀⠊⡀⠀⠁⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠒⠀⠁⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠔⡩⢒⠡⡢⢃⠡⡸⠋⢀⠀⠀⠀⠞⠀⠀⡗⣐⢀⠔⠁⡄⠁⠀⢰⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⣂⡀⠀⠠⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡔⠛⠩⡒⢅⡞⢁⠂⠀⢱⠀⠀⠘⠤⡤⣴⡯⠁⠀⡁⡇⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢒⠒⠀⠀⠡⠤⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⠊⣲⠋⠀⠈⢢⣴⠋⠀⠀⠀⣀⠔⠁⠰⠀⠀⠀⠁⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠐⠒⣂⣀⡉⠉⠤⠕⢄⠠⠀⠀⠂⠀⠀⠀⢻⣾⠃⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠘⠀⢠⠀⠂⣤⠖⠋⢀⢌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡐⠒⠀⠤⠀⢈⣐⡻⢦⡀⠀⠄⡀⠀⢸⠇⠀⠀⠄⠂⢀⡠⠂⠡⢀⠁⠂⠀⡄⠀⣨⠴⠋⠁⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡤⠞⠋⠉⠉⠉⠙⠛⠓⠤⠦⠤⢠⣮⡈⣠⡴⠾⠓⠚⠉⠙⠲⡔⢘⣠⠶⠛⠉⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡺⡆⠀⡁⡏⠉⠁⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡪⠜⠀⠀⢀⡀⠐⠈⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠖⠤⠤⠤⠄⠀⠂⠉⠀⠀⠀⠀⠀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠃⢁⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠌⠀⡀⡀⠀⡀⠀⠀⢀⠀⠄⠃⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠀⠒⠒⠀⠀⠤⠤⠠⠤⠤⠀⠤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⢀⠀⠀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠠⠐⠈⠄⢂⠁⠐⡈⠀⡂⠅⠂⠁⠐⠀⠐⠈⠐⠠⢀⠀⠊⠔⠠⠂⠅⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠈

⠀⠀⡀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢿⣆⠀⢸⣆⠘⠋⣀⠀⠀⠀⣷⡆⢀⠀⢠⣄⠀⠰⠀⠀⠀⢀⠘⠇⠀⠀⠀⢠⣄⠁⣀⠀⠀⠠⣤⣁⣼⠂⠀⠀⢀⠛⢀⠂
⠀⠀⠈⠻⠆⠈⡛⠀⠾⠋⠀⠀⠀⠈⠁⡿⠁⠀⠁⠲⠀⠀⠀⢀⡘⢧⡶⠂⠀⠀⠀⣉⠾⠋⠀⠀⠀⣬⡙⠁⣀⠀⠀⠈⣈⠺⠈
⠀⠀⢶⣦⡀⣰⠆⢀⣴⡶⠀⠀⠀⠀⠃⠀⠀⠀⠰⡀⢀⡀⠀⠈⣿⣆⣴⡇⠀⠀⠈⢻⣧⡶⠀⠀⠀⠉⠛⣿⡟⠀⠀⠀⠉⠲
⠀⠀⠀⠉⠀⡿⠀⠈⠁⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⡹⠈⠁⠀⠀⠈⠉⠋⠀⠀⠀⠸⣇⣡⡄⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢈⡙⡔⠗
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⠰⣶⡄⢠⡄⠀⠰⢾⣶⢄⢿⠄⠀⠀⠸⣶⣾⣯⡔⠀⠀⢀⡀⠀⡄⣧⡀⠀⢰⣿⠀⠀⡀⠀⠀⢠⡄⠀⠀⡀
⠀⠀⢈⡄⠈⠀⠃⣀⠀⠀⢠⣌⠃⢸⠇⢀⡀⠀⠺⠃⠙⢇⣴⡆⠀⠀⠁⠀⠀⠀⠰⣌⢰⣧⡿⠆⠀⢠⣜⣡⠀⠀⠙⠳⠀⠘⠃⠀⢚⡁
⠀⠀⢀⠀⡄⠀⠀⣨⠀⢰⣬⡛⠀⡆⢰⣿⠃⠀⠀⠀⠶⢌⣾⠀⠀⠀⣠⠀⠀⠀⢳⣶⡧⠍⠀⠀⠀⢦⡟⢋⠀⠀⢠⣤⣦⣶⣤⠛⠉⠁
⠀⠀⠉⠀⠀⠐⠀⠀⠀⠘⠿⣷⠀⠄⠈⣁⣀⠀⠀⠀⠀⠀⠁⠀⠐⢶⣿⣶⠆⠀⢸⠃⠀⠀⠀⠀⠠⣤⣷⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠐⠛⠛⠛⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠁

⣿⠿⢿⢿⡿⠿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿

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

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

⠀⢀⣴⡶⠒⣶⣖⠢⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠛⠿⠛⠋⠁⠉⠉⠉⠓⠾⣷⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠲⡀⠀⠀⠀⣤⣶⠀⠀⠀⢀⡟⡽⠀⠀⠀⠀⠀⠀⢀⡀⠈⢿⣹⡄
⠀⠀⠀⠀⣀⣠⠄⢤⣤⣀⠀⠀⠀⠈⢳⡿⡄⠀⢸⣿⡿⠀⢀⡴⣫⠜⠀⠀⠀⠀⠀⠀⢰⣿⣿⠄⠘⣧⡇
⠀⠀⣠⣎⡴⠛⢋⡉⠛⢤⣷⡄⠀⠀⠈⣧⡧⠀⡾⣾⠁⠀⢸⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⣦⣼⢧⠇
⠀⢰⢉⡟⠀⣾⡝⠛⢳⡄⣧⣸⠀⠀⠀⣟⢹⠀⡷⣯⠀⠀⠈⠲⢍⣓⠢⢤⣀⠀⠀⠀⠀⢀⣼⣶⣭⡛⡄
⠀⢸⢻⡀⠘⣟⣆⡀⠘⣡⠛⡟⠀⠀⢠⠟⡏⠀⠹⣛⣦⡀⠀⠀⠀⠈⠉⠓⠦⡑⡆⠀⠀⣸⢴⠃⣠⠇⡄
⠀⠘⢾⣧⡀⠈⠻⠯⠽⠟⠋⠀⠀⣠⣿⡴⠀⠀⠀⠙⠳⡽⢆⠀⠀⠀⠀⠀⣠⢇⡍⠀⣠⠼⣾⢿⣵⠊
⠀⠀⠘⢦⣿⣤⣀⡀⡀⡀⣀⡤⢾⡧⠊⠀⠀⠀⠀⠀⠀⣿⢻⠀⢀⡠⢖⣫⠵⠋⢠⢊⡴⠋⠉⠛⢯⡷⢆
⠀⠀⠀⠀⠈⠓⠾⠻⠍⠽⠻⠒⠉⠀⠀⠀⠀⠀⠀⣠⣾⣹⠏⠀⠎⡴⠋⠀⠀⠀⠘⣌⠣⣄⡀⠀⠀⢹⠞
⠀⢀⣤⣶⡶⣶⣒⣶⣲⡦⢤⣀⡀⠀⠀⠀⠀⢠⢞⡷⠛⠁⠀⠀⢄⢳⣄⠀⠀⠀⠀⠀⠙⢲⣭⣝⣒⢯⡿
⠀⠘⠿⠿⠋⠉⠉⠉⠉⠉⠛⠶⣟⣲⣄⠀⠀⢞⣻⡁⠀⠀⠀⠀⠈⠓⠮⣝⠲⡀⠀⣴⣛⡿⠶⠛⠉⣧⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠳⡀⠈⠫⣽⡒⣦⣄⣀⠀⠀⠀⠈⢳⢹⢸⣤⣏⠀⠀⣠⠴⣣⠏
⠀⠀⠀⠀⢀⣠⢴⣶⣶⠦⣄⡀⠀⠀⠈⢿⣱⡀⠀⠀⠉⠛⠺⣯⣷⡄⠀⣠⠟⡜⠀⠓⢾⣛⣿⢵⡊⠁
⠀⠀⠀⡰⣯⡽⠋⣉⣉⡙⢧⡽⡄⠀⠀⠘⣧⣇⠀⠀⠀⠀⠀⣨⢧⡗⢰⣣⠞⠀⠀⠀⠀⡏⢿⠺⣏⣷⣄
⠀⠀⢰⢷⡏⢀⣎⡖⠉⢰⡄⣷⣿⠀⠀⠀⡷⡟⠀⢀⣠⠴⣞⡿⠞⠁⢡⣿⠀⠀⠀⠀⠀⠘⢮⡳⢄⣿⣾
⠀⠀⢸⢺⡇⠈⣏⢧⣀⣈⣴⢣⠏⠀⠀⣸⢳⠇⣴⣯⠿⠚⠉⠀⠀⠀⠈⠜⡄⠀⠀⠀⠀⠀⠀⣹⣦⢻⠃
⠀⠀⠈⢏⣷⣄⠈⠑⠚⠓⠚⠁⠀⢀⣴⣻⠎⠘⣷⡇⠀⠀⠀⠀⠀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠞⠉⣹⠸
⠀⠀⠀⠀⠳⢽⡳⣤⣄⣀⣠⣤⡶⣻⠜⠁⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⠃
⠀⠀⠀⠀⠀⠀⠉⠑⠚⠓⠛⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢠⣶⠿⠃⠸⢶⣤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢡⣧⢀⠀⠀⠈⢋⢀⣉⣀⢀⣤⣀
⠀⠀⡄⠀⠀⠀⠀⠀⠐⢁⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠘⠋⠈⠛⠛⠈⠉⠉⠀⠈⢀⣛
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⡠⠔⠛⠉⠀⠀⡄⠀⠀⠀⠾⢉⡡⠉⠘⠂⠒⠐⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⡀⢀⠀⠀⠄⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡄
⠀⠀⢀⡀⠀⠀⠂⠀⠀⡸⠄⠀⠀⠀⣠⡐⠇⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠚⠣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢰⡟⣹⠆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠇⢳⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠶⠚⠂⠉⠛⠿⠿⠭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠠⠁⠀⠀⢹⡀⠀⡼⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠳⢤⠀⠀⠀⠸⠄⠐⠐⠀⠀⠃⠰⠃⢯⠀⠀⡰⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠃⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠆⠈⠖⠀⠀⠐⠀⠀⠀⠀⡀⠐⠲⠀⠀⠀⠀⢀⠀⠀⡀⠘⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⠀⠀⠀⠀⠠⢠⡆⠈⠘⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠘⠀⠀⠉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⢒⠒⠒⠀⠀⠀⠀⠀⠃⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠢⣀⡤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆
⠀⠀⠀⠀⠀⠘⠛⠓⠒⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠀⡄⠀⠋⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠉⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢷⡂⠀⢐⠄⠐⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠘⠂⠀⠀⠀⠀⢈⠗⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠴⠀⠰⠀⠀⠀⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠊⠀⠀⠀⢀⠠⠊⠀⠀⣠⢔⣅⠁⠀⠀⣠⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢀⡤⡪⠖⠁⠈⠑⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠒⠚⠉⠒⠠⠤⠐⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢶⡀⠀⠀⠀⢸⠟⠀⠀⢳⣿⣿⣿⣦⡀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣾⣿⣦⠊⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡀⠀⢶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠠⣤⣹⣿⣿⡆⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⢰⡀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣶⠖⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠁⢀⠀⠠⢈⣷⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣿⣇⠀⠐⣿
⣸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠞⠃⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧⠀⠀⠀⢀⣼⣿⣿⣷⣄⡈⠙⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⠄⢻⣿
⣿⡆⠀⢠⣧⠀⠀⡀⠀⠀⠀⢨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢠⣿⠿⠛⣻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠈⠛⠉⠙⣿⣿⠉⠉⠉⣽⣿⣿⡤⣾⣿
⣿⡗⠀⣸⣿⡀⢰⡇⠀⠀⢀⣸⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⢾⣿⣿⠶⣿⣿
⣿⣿⡀⣻⣿⡃⣼⣧⠀⠀⠠⣽⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⠀⠘⢿⣿⣿⡿⣿⣿
⣿⣿⣽⣿⣿⣿⣿⣿⠄⠀⠴⣾⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⣤⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣭⣿⣿⣿⣿⣿⡃⢀⣤⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙
⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⠻⣿⣥⣤⠄⠀⢀⣠⣤⠴⠆⠶⠠⠦⠠⠤⠀⠤
⣿⣿⣿⣿⣶⣦⣤⣄⠀⠀⠀⠀⠈⠉⠀⠻⢿⣿⣷⣾⣽⣶⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿
⣿⡿⣿⣿⢿⣿⠿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿
⣉⣠⡄⠀⠀⣀⣠⡤⠤⠤⠠⠤⠠⠤⠀⠤⠄⠄⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⢿
⣽⣤⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠉⠃⠙⢿⣿⣿⣮
⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⠿⣿⣉⣠⣄⠀⠀⣀⣠⣤⡤⠤⠠⠤⠠⠤⠀⠤⠄⠄
⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠈⠙⠈⠻⣿⣿⣶⣭⣽⣥⣴⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿
⣿⣿⣿⣿⣿⣿⡿⠎⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿
⣋⣁⡈⠁⠀⢀⣀⣠⡄⣤⠠⣄⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⣿
⣻⣥⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠚⠋⠙⢿⣿⣷⣬

⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠊⢑⡲⠦⠤⣒⠉⠉⠉⠉⠒⠢⣄
⠀⠀⠀⠀⠀⢠⠋⠀⠠⠒⠀⡀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⣎⠻⢖⣶⢤⣤⢀⡠⠀⠀⠀⠀⡠⠖⠒⠒⠦⡄⠀⠈⠆
⠀⠀⠀⠀⣸⠀⠉⠉⠀⠓⠷⠚⢀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⢆⠀⠸
⠀⠀⠀⡔⠁⡄⢀⠀⠀⠀⠀⠐⡹⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⢸⡀⠘⠡⠏⠀⡠⡲⠒⠚⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡄
⠀⠀⠀⠝⡒⠤⠤⠜⢠⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⡴⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢦
⠀⠀⠐⠌⠀⠉⠒⠊⠁⠀⠀⠀⢀⠔⠉⠉⠀⠉⢉⡱⠶⠋⠀⡠⠋⣀⣀⠤⣄⣀⡀⠀⡀⠀⠀⠀⢣
⠘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡴⠚⠁⢀⡠⠔⠉⠉⠛⠢⣄⡠⠔⠚⠉⠉⠉⠉⠳⣄⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢠⠏⠀⠀⡔⠛⠒⠦⢄⡀⣠⠖⠉⠀⠀⢀⣀⣤⣀⡀⠀⠸⡄⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⠀⠀⠀⢇⠀⠀⣀⠴⠋⠀⠀⣀⣴⠊⠁⠀⠀⠀⠛⠀⢠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠉⠉⠀⠀⢀⡤⠚⠁⠀⠉⠒⠤⠤⣤⡤⠴⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣵⣶⣶⣴⣶⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣏⡶⠀⢠⡀⠀⡶⡀⠀⠀⠀⣠⠤⣠⠤⠤⡀⢀⡎⡔⠒⡍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡠⠎⠒⢹⡇⠀⠀⠀⡜⠀⠈⠢⡀⢸⠁⠀⠀⠀⠀⠸⠤⡓⢬⠬⠔⠃⣀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠉⠀⠀⠘⠄⡀⠈⣧⠀⢀⠜⠀⠀⠀⠀⣈⡯⠁⠀⠀⠀⠀⠀⠀⡅⢀⡠⠐⠈⠀⠀⢠⠀⡠⠠⣄⣀
⠀⠀⠀⠀⠀⠀⢀⠄⠄⠰⠄⢀⠕⠀⠀⠀⠀⠀⠀⠈⡆⢯⡀⠈⡹⠃⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⢀⡱⠤⠙⠂⠀⠀⠀⠘⡀⠐⢀⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠞⠀⠀⢀⠜⠨⡀⠀⠀⠀⠀⠀⠀⠀⢣⢸⣡⣎⡁⡀⠀⠀⠀⠀⣤⡒⠁⠀⠀⠀⠀⣸⠒⢦⣀⠀⠀⠀⠘⡓⣃⢴⠀⠈⠈⠀⢀⢴⠆
⠀⠀⠀⠀⠀⢇⣀⠤⠠⠃⠀⠀⠈⡇⠀⠀⠀⢀⣀⡤⠊⢈⣤⠤⠔⠓⠚⠀⠉⠉⠉⠉⣱⠚⠲⠲⡋⠡⣐⡆⢠⠃⠀⢀⣀⢹⡀⢨⠓⠂⢠⣴⡞⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢟⠐⠂⠤⢈⡒⠀⠒⠙⣌⠶⠊⠉⠀⠀⠀⠀⢀⡤⠐⠒⠒⠊⠁⠀⢸⠀⠘⠦⠤⠤⠼⣄⠀⢠⡈⠛⠣⡆⢀⣠⠞⠉⠉⡂
⠀⠀⠀⢠⠉⠁⠈⠂⠀⠈⣆⠀⣀⠞⠉⠉⡲⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⢄⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⢀⠽⢃⡘⠒⣤⣶⠿⠛⠂⢹⠃⠃⢀
⠀⠀⠐⢀⠠⠄⠁⢟⠐⠂⠸⣏⢀⡠⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠘⢆⢑⡊⢸⡁⠈⣓⠀⢀⠃⠈⢆⠀⠈⣢
⠀⠀⠀⠀⠀⠀⣄⠈⢇⢀⡰⢁⠊⠄⠌⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢨⡀⠀⠀⢀⣀⡀⠀⢀⡠⠋⢁⠔⠒⠳⡄⠀⢙⠎⠀⠀⠀⣌⠐⠁
⠀⠀⠀⢠⣨⡻⠸⠶⣌⡏⠀⢀⠷⠴⠤⠖⠀⠀⠀⢀⠠⠤⢄⣀⡀⠀⠀⠀⣀⡴⠃⠀⠀⠙⠲⠒⠉⠀⠈⠉⠁⠀⠀⡇⠀⠀⠀⠘⣆⡌⠀⢀⡴⠲⠸⢏⣣⡀
⠀⠀⠀⠈⠢⡅⠀⠦⡷⠘⣄⠎⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠘⡄⠀⠈⡧⠇⠀⡬⠊
⠀⠀⠀⠀⠀⢱⠈⡏⠀⠀⡘⠀⠀⠀⣀⠤⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠰⠀⠀⠈⡍⢠⠁
⠀⠀⠀⠀⠀⠀⢃⠘⢆⣀⡃⠀⠀⡜⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡘⠦⣀⠀⠀⠀⠀⠀⡇⣠⠞⢀⠎
⠀⠀⠀⠀⠀⠀⠈⢢⡀⢸⠣⡀⠀⢧⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠉⠢⡀⠀⠀⢙⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⠀⠙⠦⡀⠉⠒⠠⡄⠀⠐⠓⢦⠤⢤⠔⠣⠀⠀⠀⡀⠀⠀⠀⠀⠀⠜⠒⣤⠤⡴⠚⢢⠔⠓⢦⡀⠀⠈⡆⠀⢸⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢱⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢃⠀⡰⠀⠀⢀⣀⣀⣀⣀⣀⡠⠊⠀⠀⠀⠑⡀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠈⠑⠒⠈⠑⠢⢄⣀⠀⠀⠀⠀⠉⠁⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠢⠄⢀⣀⠤⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⠈⡄⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣡⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠊⢁⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠒⠒⠀⠀⠐⠒⠒⠛⠉⠉

⣷⣶⣶⣷⣶⡷⢾⣾⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⡶⠿⢷⠶⠶⢶⣶⡶⠾⠶⢾