Royalty Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⣂⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣣⣾⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⢀⠆
⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣽⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⡎⡻⣿⣿⣿⣶⣶⠶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⣿⣿⣿⠿⣿⣿⣷⢸⣿⣿⣿⣷⡝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⠏⠀⣸⣿⡿⣼⣿⣿⣿⣛⣋⣈⢻⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡷⢠⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣈⠉⠁⣰⣾⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄
⠀⠀⢀⣠⣴⣾⣿⡟⢫⣍⣙⣿⢿⣿⣿⣷⡼⠛⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣟⣿⣍⠛⢻⣿⢿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢠⣾⣿⡟⠉⠀⣼⣿⣿⣿⠿⢿⣷⣾⣭⣭⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣮⢻⡆⠊⠙⠀⠀⡿⠸⢹⣿⣿⡿⣿⣿⢻⣿⣿⣿
⢻⣿⣯⠁⣠⣾⣿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠜⠁⠤⠟⣿⣿⡇⣿⡏⣼⣿⣿⣿
⡞⢡⡟⢠⡛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⢹⡿⣿⢠⡿⢹⡏⠁
⠻⠞⠀⠀⢹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠩⣿⣿⣿⣎⠻⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠃⠑⠝⠦⠓⠈⠷⠄
⠀⠀⠀⠀⢸⡋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣶⡶⢦
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣯⣿⣿⠿⢿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⡏⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⡗⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⡏⢸⠹⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⣴⣆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢠⣶⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⡖⣦⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠵⣯⣼⣿⣿⢸⣧⡴⢞⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⢻⣶⣿⣿⡿⣟⠈⢷⠄⠙⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣉⣿⣿⠿⠋⣵⠄⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣴⠟⣵⣧⣼⡿⢋⡅⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⠀⠀⠀⠀⢀⡾⢫⣾⣿⠟⠉⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⣠⡟⣱⣿⡿⣁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣰⡟⣼⡿⢫⡄⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣟⢱⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣧⠀⢰⡟⣼⠟⣥⡌⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣋⣣⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⣤⣿⢱⠏⣦⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⣿⣿⠀⠈⢻⣿⣿⣦⣼⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡴⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠛⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢈⣿⣶⡄⠈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣟⢛⡿⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣷⣦⣤⣟⢷⣭⡻⢶⣦⣤⣤⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡋⣻⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⢻⣿⣿⣷⣬⡛⢶⣭⣛⠿⣍⣽⣿⠛⣿⠟⣿⠟⣿⣏⣹⣧⠼⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠈⠙⠻⠶⣮⣭⣽⣛⣓⣛⣛⣫⣭⣭⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡃⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣻⡏⢻⡾⠿⣒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⣿⣹⣿⣰⡿⢆⢰⡆⢀⣀⠀⢀⣄⣷⢟⣼⣡⣿⠀⠀⣤⣀⢻⣇⣙⣗⣶⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡟⢏⣿⡇⣿⣿⣿⣵⣇⢹⠃⠸⡿⢿⡿⢿⠃⣡⣞⣿⢿⣰⣷⡝⣿⣾⣿⣿⢦⣸⠎⣿⢿⣶⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢿⡇⡟⣿⣇⣼⣿⡿⠙⠇⣿⣿⢻⡞⣧⣄⠀⠸⡿⢸⡧⠱⢟⡉⣸⣿⡿⣿⣿⣿⣟⠻⣦⡙⠿⣽⣼⣿⢻⠿⠟⠓
⠀⠀⠀⠰⠶⠶⠛⠡⠾⠦⠽⠿⣿⣿⠿⠷⠦⠀⠋⠿⠾⠿⠿⠿⠾⠷⠿⠦⠶⠿⠍⠁⠀⠉⠛⠉⠉⠁⠘⠓⠶⠿⠧⠬⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⢤⣄⡤⠤⠴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⡤⢂⣌⡫⠥⠄⠠⠶⠖⠲⠶⠄⠀⠒⠰⠚⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠶⡀⠀⢁⢐⣒⣠⣶⣠⣴⡾⠶⠟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⡄⠀⡄
⠀⠀⠀⠀⠀⣰⣧⣾⣄⣴⣵⣿⣤⣼⣿⡿⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠦⣀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠛⠛⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⢀⡂⠀⠀⠀⢀⠄⠀⠀⢀⣀⣼⣷⣶⣶⣶⣿⣶⣶⣶⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡠⠀⠈⠐⠀⠀⣠⣾⣋⣽⠟⠛⢛⣩⠋⣩⣉⡙⢻⣿⣻⣿⡾⣿⣶⣆⡀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⡠⣀⣄⣴⣿⢿⣛⣏⣵⣿⣼⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⢻⣿⣷⣞⢪⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠤⢠⣀⣜⣻⣯⣟⣋⣵⣾⣿⣿⣿⣿⡿⠟⢟⣿⢟⣹⣿⣻⣛⠛⢿⣿⣿⣯⣾⣿⣯⣛⣿⣿⣯⣝⡠⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⣨⢫⡜⠛⣼⣿⢿⢿⣾⣿⢿⠇⡐⢷⣼⣷⣿⣿⣿⡛⠉⠉⠒⡿⠉⠻⣿⣿⣿⣿⣻⡯⣩⣿⣿⣍⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⣹⡼⢈⣼⠟⠋⠁⢸⣿⣿⣾⣤⣽⣿⣿⣿⣿⣿⣿⡅⠀⠀⢠⣷⡶⠀⠹⡟⠿⣿⣿⣿⣿⣧⣭⣹⣷⠉⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠀⠸⠃⢒⣴⡟⣠⣾⡃⠀⠀⠀⣼⣿⣿⢙⣽⣿⣿⣿⣿⣿⣿⣿⣿⡗⠶⣿⣿⣗⠂⢤⡇⠀⠙⠻⣿⣿⣿⣿⣿⠞⠁⢀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⣀⠀⠔⢛⣯⣼⣿⣿⡇⠀⢀⠀⢸⣿⣿⣿⣠⣽⣿⡿⣿⣿⣿⣿⣿⣷⣶⣿⣿⠗⠀⣸⠇⠀⢉⣫⣿⣿⣿⣿⣟⠛⢋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⣤⣼⣿⣿⣿⣼⣯⣆⡨⠆⠈⣿⣿⣟⡋⢽⣿⣷⣭⣙⣻⣿⣿⣿⣿⣿⡿⠂⣬⡟⠀⠀⣠⣿⡿⣫⠿⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠛⠻⠿⣿⣿⣿⣿⣓⠅⢀⠈⠿⣿⣿⣾⣿⣹⣿⣿⣿⣿⣿⣿⢿⢫⣴⣾⢟⣁⣴⣾⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠙⠻⣿⣾⣧⣄⣀⡈⠻⢿⣿⣿⣷⣄⣯⣿⣾⣾⣶⣿⣿⣷⣞⡯⢿⠛⠛⠀⠉⠀⠀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⢶⠶⠶⢷⣿⣿⣿⣿⠿⣿⡿⣿⣿⡛⢯⠄⠒⠉⠁⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠠



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣞⣛⣻⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣟⣛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡯⠤⠼⣧⣿⣿⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣿⣿⣼⠧⠤⢼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡾⠛⠛⠯⣵⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⣿⠵⠛⠛⢷⣶⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠦⠤⠤⠬⢦⡀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢀⡴⠥⠤⠤⠶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣀⣀⣀⣀⣙⣦⣀⠀⠙⢷⣄⠀⠄⢠⠀⣠⡾⠋⠀⣀⡴⣋⣀⣀⣀⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠙⢧⣾⣿⣼⠋⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣈⣼⡟⢹⣷⣥⣄⣤⣤⣶⣶⣶⣶⣶⡶⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠃⠉⢋⠭⠉⠉⠉⡈⢿⣿⣼⣿⢁⠈⠉⠉⠤⣚⠉⠉⠛⠳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠒⠀⠀⡠⠎⠀⢀⡤⠊⡀⠋⣿⣿⠉⢀⠁⠀⡀⠀⠘⢤⡀⠀⠐⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠔⠂⢀⣄⣠⠤⠲⠋⠀⠐⢀⠂⣿⣿⡀⠀⠣⠀⠈⢓⠠⢄⡠⠀⠀⠂⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⠊⠀⡌⠀⣴⠃⢀⠎⢀⢂⠎⢸⣿⠐⠔⠀⠱⡀⠈⢦⠀⠱⡀⠀⢦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣞⠀⣠⠁⢀⠎⠀⡠⠂⠀⠘⠏⠀⠈⢦⠀⠐⡄⠀⣃⠀⣱⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣾⣤⡾⠁⠀⠀⠀⠀⠀⠀⠈⠳⣤⣼⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⢛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢻⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠇⠀⠀⠉⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠿⢻⠟⠀⠉⠋⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⢻⣿⣿⣿
⣿⣿⣿⢿⡿⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠛⠁⠀⠀⠀⠀⠀⠀⠹⠿⣿
⣿⣿⢥⠟⠓⠠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣏⠛⠓⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢳⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠹⠟⠛⠻⠛⠉
⠘⠂
⡌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠒⣯⣹⣧⣼⣏⣽⠛⣷⢦⣄
⠀⠀⠀⠀⠀⠀⣼⢙⣻⠏⠀⠀⠀⠀⠉⠙⠳⢼⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣏⣩⠷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣏⣉⣽⠛⢲⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣤⣤⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⢉⠄⠂⣒⣦⡹⢻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡽⣼⢱⢁⡔⡢⢀⠽⣓⠼⠿⢷⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⡇⣧⠫⢸⠁⢁⢇⡞⡁⢀⡤⠒⡂⠀⣉⠽⢋⡭⢛⣳⡶⠶⣄
⠀⠀⠀⠀⠀⠀⣴⠋⢋⠀⡾⠸⡇⠹⠀⢈⠄⠜⠀⡾⡡⢎⢠⠞⡵⠋⡤⢂⡭⣶⣫⣤⣄⠈⢉⡱⣦⡀
⠀⠀⠀⠀⠀⣸⢯⢶⠃⢠⣧⠀⡇⠀⢸⢸⠀⡰⢰⢳⢠⢃⠃⡞⠀⣸⣡⡏⡸⣛⣭⣶⣿⣯⡂⢐⡭⢷⡄
⠀⠀⢀⡤⠶⢃⣾⠈⢀⢻⡿⢠⢷⡇⠁⡰⣸⠁⠞⡟⡈⡜⢰⡇⣰⠃⡿⣝⣿⣫⠵⣫⡥⠌⢩⣿⣿⣛⠻⢷⣦
⠀⠀⢸⡀⠀⠸⣿⢰⢿⢸⣅⣉⣾⡔⡇⣇⣻⠀⣼⢣⢣⡇⣼⠃⡏⣼⢿⡀⠸⢵⢀⣶⠰⢖⢼⢻⢁⢨⡓⠀⡽⣷⣿⣧
⠀⠀⠀⠙⠶⠦⠿⣾⣾⠿⠟⠛⠙⣷⡟⢹⠋⡼⡇⣿⢸⢣⡏⠀⣿⢸⣤⠏⢰⢠⡙⣉⣷⢀⣸⠀⢎⣯⣵⢦⣿⣉⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣯⣞⠀⠀⣿⠸⡸⡇⠀⣾⢜⣿⠂⣎⢲⠅⠀⠴⢀⠲⠹⡜⠳⡬⢩⢠⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣦⣀⣸⣶⡁⠙⣦⠻⡼⣯⠀⢿⠈⡨⡆⠲⡄⢨⣠⠃⠈⢧⠎⡬⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡤⠞⠋⣛⣷⣤⣬⣧⣙⣾⣷⣄⣡⢹⡦⢤⣜⣯⣀⣀⡀⢊⡴⣣⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢰⠇⠊⠑⠊⠉⠙⢧⡉⠀⠈⠙⣿⢸⢷⡋⠉⣹⣼⣡⣸⣽⠾⠋⢼⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⠆⡄⠀⠤⠀⠀⣀⣹⡆⠀⠀⠘⡿⠶⠿⢿⣰⡿⣏⢉⡖⡩⢡⠴⠞⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢆⢀⡐⣄⣦⡿⠛⠁⠀⠀⠀⠙⠓⠶⠶⠞⠀⢿⣼⣸⠀⡄⠔⠋⠭⠽⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣇⣀⠀⠀⡀⡄⠀⢙⠉⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡾⢿⣤⣦⣏⣠⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠏⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠞⠛⠿⢛⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣤⡖⠂⠈⣿⣿⡃⠀⠀⣿⣿⣄⣀⡰⠋⢀⡠⣴⣶⣺⢟⣫⣽⠿⢛⣫⣽⠶⠿⠟⠛⠓⠒⠤⣄⡀
⠀⠀⢀⠔⠉⣿⣿⣧⣀⠠⠿⠟⠓⠒⠒⢚⠟⠋⢉⡒⢴⣫⣾⡿⠋⣱⣿⡟⣠⣾⡟⠉⠀⣀⣤⣶⣾⡿⠿⠿⠿⢿⣶⣄
⠀⢠⣿⡀⠀⢨⠟⠋⠀⠀⠀⠀⠀⠀⢀⣎⡤⢚⣁⡈⢿⣿⡟⠀⣴⡿⢋⣼⣿⠏⠀⣠⣾⡿⠛⠉⠀⠀⢀⣠⣤⣤⣤⣬⣑⣄
⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⡛⢿⣡⠟⠉⣹⣶⣻⣇⢸⡟⠀⣼⣿⠏⠀⣴⣿⠏⠀⠀⢀⣤⣾⡿⠛⠉⠀⣀⣀⣉⡙⢳⣄
⠀⡀⠈⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡞⢃⡶⠿⠿⠿⣿⠇⢸⡇⢰⣿⡿⠀⣼⣿⡏⠀⣴⢟⣩⠾⣫⠄⣠⣴⣿⠿⠛⠛⠛⢿⣟⣳⡀
⠀⢧⣄⣠⣼⣧⣀⡀⠀⠀⢀⣀⣴⣿⣿⣷⣮⠔⣀⣈⣻⣿⣇⣿⠁⣾⢹⡇⠀⢿⣹⡇⣸⣧⣿⡁⠚⣡⣞⣭⠭⠀⠀⠀⠀⠀⠀⠹⢿⣿⣤⡀
⠀⠈⠿⡿⠿⠃⠈⣿⣿⡇⠈⢹⣿⣿⣿⣿⣿⡾⣿⡿⢿⣿⣿⣿⠀⣿⢸⡇⡇⢸⣯⡿⡛⠀⡈⠛⣻⣿⣭⣥⣤⣶⣾⣿⣿⣿⣷⣦⡀⠉⠙⢿⡀
⠀⠀⠀⠈⠒⠠⠴⠿⠿⠣⠀⠞⠋⠉⠻⠉⡟⠀⢠⠂⠀⠀⠘⣿⣇⢻⡄⢿⣿⡞⣵⠏⣽⠿⣤⡾⢋⡽⣻⡿⠛⠉⠉⠉⠉⠉⠉⠙⢷⣄⣀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣇⡠⢘⡆⢰⡆⢠⡿⠿⣿⣿⣾⣿⢸⡏⣸⠁⣠⢀⣴⢏⡼⠋⣠⣤⣶⣶⣶⣤⣤⣄⠀⠈⠛⢿⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠟⠙⠎⠋⠛⠁⠀⠀⠙⠻⣿⣿⡇⠁⣾⡁⣾⣿⣾⠶⠛⠉⠁⠀⠀⠀⠉⢻⣿⣦⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣿⣿⢛⣻⣁⣠⠶⠛⠛⠛⠻⣶⡀⠀⠟⠛⢿⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⣿⣉⠍⠳⠚⢛⣛⠳⢦⣼⣟⠻⢷⣄⠀⢿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⢿⣿⡄⠛⢫⠔⢠⠦⠤⣍⠳⠶⠦⡍⢳⡄⣻⣄⣾⣧⡾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⡇⠰⢶⣠⡐⢤⠐⡦⠈⡉⠙⢦⡀⠈⢉⠹⣯⡹⣟⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⣰⢾⡶⠢⣅⣴⡦⢀⡁⠠⠛⡅⠈⡰⡆⢸⡶⠛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢹⠇⢸⣏⢸⣿⡴⠏⣶⠃⠀⠉⠀⣶⡄⠲⠄⣤⢀⣶⣞⣋⡉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⡀⢿⢸⣿⣿⠁⠙⣆⢠⠤⠞⣼⡄⣚⣤⢠⢠⣿⡟⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡙⠎⠀⢻⢁⣭⣙⡻⢤⣤⣌⣿⡾⠿⢃⣼⠃⢰⣇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡉⡾⡬⠁⠈⠹⡏⣠⠟⣫⡤⠶⠛⠁⢀⣾⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣄⠷⣶⣦⣵⢾⠡⠆⠱⠶⢶⣀⣴⣿⣯⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠛⠷⣦⣬⡥⢠⣴⡟⢛⣿⣿⡿⢛⡁⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡻⠦⣀⣀⣠⣴⠿⣿⣇⡼⠋⡽⢃⣉⠙⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠈⠛⠀⢉⡿⣤⠇⠈⠁⠈⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢸⣶⣹⣆⣷⣰⡈⣆⠜⢱⠇⠀⡀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠘⠀⠀⡸⠀⠀⠃⢠⠀⠈⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠁⠀⠓⠘⢂⠑⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠐⡤⠀⠈⠄⠀⢰⡄⢈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⣥⢷⡌⣠⣾⢁⡟⢻⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠊⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠠⠄⠀⠠⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣉⡆⢠⡴⢦⠒⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠄⣿⣈⠃⢵⣊⠘⣶⣛⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣎⣾⣂⢳⣚⣜⠷⠆⢩⣿⣛⠇
⠀⠀⠀⠀⠀⠀⠀⡏⣎⣠⣬⣉⣾⣫⡟⠃⠁
⠀⠀⠀⠀⠀⠀⡼⣯⠟⣩⠟⡷⠾⢭⣳⡄
⠀⠀⠀⠀⠀⢰⡇⣏⢘⣥⠖⣀⣦⣌⣉⣳⡀
⠀⠀⠀⠀⠀⢘⣧⡇⡟⣱⠎⢉⡰⢓⣛⣩⠿⠦⣀
⠀⠀⠀⠀⠀⡼⢸⣯⠞⡏⣴⡟⣠⠞⡿⣶⣒⡺⠾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿
⠀⠀⠀⠀⠀⡷⠘⣟⣸⠰⢹⢠⢫⡞⣵⡏⣶⢖⣒⣒⣚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠏
⠀⠀⠀⠀⠀⡇⡸⠹⣻⠀⢺⡏⣼⣀⡞⣰⡇⣾⢏⢹⢶⣿⡿⣞⠻⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣯⡩⡄
⠀⠀⠀⠀⢠⠁⠡⢨⠿⣄⣿⡗⢳⣿⣷⡏⣷⣟⡌⣾⢿⡿⢠⢃⠾⠟⡤⠿⣶⡀⠀⠀⠀⠀⠀⠙⢤⡾⢦⡀
⠀⢀⣀⡀⡤⠀⢡⠏⠀⠀⠹⣿⣼⣿⣿⣇⡇⣿⣼⣿⣿⠃⡼⢉⣼⡿⡨⡗⣺⣿⡄⠀⠀⠀⠀⠀⠀⠑⢞⣸⣆
⠀⣫⡐⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⢫⠇⡇⣧⠀⠇⡍⣼⢁⡇⢳⣧⡞⢛⣶⡄⠀⠀⠀⠀⠀⠀⣿⢼
⠀⠈⠁⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⢦⣽⢾⣇⠀⢉⣛⡗⠤⣿⠿⢿⣾⠉⢻⡄⠀⠀⠀⠀⢀⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⠬⠜⢌⡫⢁⣈⢭⣳⣮⡽⠟⠉⠛⠶⣎⣻⣧⣨⣳⡤⣀⡠⠾⣬⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠦⠤⢬⣛⠚⠁⡐⡛⠅⠀⠀⠀⠀⠀⠀⠈⠳⢍⣓⠲⠿⠦⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠢⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠁⠊⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⢣⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠤⠤⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠻⣿⣿⣶⣄⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠛⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⠀⠙⢿⣿⣿⣶⣶⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣶⣶⣿⣿⡿⠋⠀⣴⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢀⣴⣿⠟⠉⠉⢉⣠⠶⣿⣿⣿⣿⣿⣿⣿⣿⠶⣄⡉⠉⠉⠻⣿⣦⡀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇⣴⣿⡟⠁⠀⠀⠐⠋⣴⡿⣛⣉⡉⠉⠉⢉⣙⣛⢿⣦⠙⠀⠀⠀⠈⠻⣿⣦⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⣰⠃⠀⢀⢞⣵⡿⢋⣴⣾⣿⣿⣷⣦⡙⢿⣮⡳⡀⠀⠈⣆⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡏⠀⠀⣿⠀⠀⢀⣿⡏⢠⣿⠏⠁⠀⠀⠈⠹⣿⡄⢹⣿⡀⠀⠀⣿⠀⠀⢹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣴⣾⣿⠟⣿⣿⠀⠀⠀⠻⠿⠿⠾⣿⡇⣸⠃⠀⠀⠀⠀⠀⠀⠘⡇⢸⣿⠷⠿⠿⠟⠀⠀⠀⣿⣿⠻⣿⣷⣦
⠀⠀⠀⢀⣾⣿⣿⡏⢸⣿⡇⢀⡆⠀⠀⣤⣶⣦⣤⣄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⣤⣴⣶⣤⠀⠀⢰⡀⢸⣿⡇⢹⣿⣿⣷⡀
⠀⠀⠀⣼⠏⣿⡿⠀⣿⣿⠁⣼⣗⠀⠀⢻⡟⢧⣛⣹⣿⣦⠀⠀⠀⠀⠀⠀⣴⣿⣏⣛⡸⢻⡟⠀⠀⣻⣧⠈⣿⣿⠀⢿⣿⠹⣧
⠀⠀⠀⠋⢸⣿⡇⠀⣿⣿⡀⣿⣟⡄⣀⣤⣭⣆⡈⡉⣿⡏⠀⠀⠀⠀⠀⠈⢹⣿⢉⢁⣰⣭⣤⣀⢠⣻⣿⢀⣿⣿⠀⢸⣿⡇⠘
⠀⠀⠀⠀⢸⣿⡇⠀⢹⡇⠁⣿⣿⡶⠀⠀⠠⠾⠋⠡⢻⠇⠀⠀⠀⠀⠀⠀⠸⡟⠎⠙⠷⠄⠀⠀⢶⣿⣿⠈⢸⡏⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠘⣿⡄⠸⣿⣧⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣠⣼⣿⠇⢠⣿⠃⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⢻⡿⡄⠙⣿⡏⠁⢀⡀⠀⢸⣧⡖⠀⠀⠀⠀⠀⠀⢲⣼⡇⠀⢀⡀⠈⢹⣿⠋⠠⢾⡟⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠘⣿⣧⣿⡀⠈⢷⡀⠀⠈⠃⢠⡞⠀⠀⠘⣿⣧⣀⣀⣀⣀⣀⣀⣼⣿⠃⠀⠀⢳⡄⠘⠁⠀⢀⡿⠁⢀⣿⣼⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠈⢿⣆⡀⠀⣾⠛⠶⢤⣀⠀⠉⠛⠿⣿⣿⠿⠛⠉⠀⣀⡤⠶⠛⣷⠀⢀⣰⡿⠁⠀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠙⢿⣦⣿⣷⣤⣀⣀⡀⠀⠀⠀⢹⡏⠀⠀⠀⢀⣀⣀⣤⣾⣿⣴⡿⠋⠀⠀⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢻⡘⣿⣷⣄⠀⡀⠈⠙⢿⣤⣀⣀⣤⠄⠀⢠⣿⣿⡄⠀⠠⣤⣀⣠⣤⡿⠋⠁⢀⠀⣠⣾⣿⢃⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣷⡄⠀⠈⠙⠿⣷⣶⣶⡾⠿⠟⠻⠿⢷⣶⣶⣾⠿⠋⠁⠀⢠⣾⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣴⣿⣯⠖⠋⠁⠀⠀⠈⠑⠲⣽⣿⣦⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣬⣭⣝⣫⣭⣥⣶⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⢛⡢⠀⠀⠀⣠⠭⣭⣾⣭⣄⡀⠀⠀⠀⣈⣋⡳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣬⣻⠇⠀⠴⠶⢒⣰⣶⣀⠚⠻⠾⠟⢻⣩⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⣡⡶⠟⢫⡾⠛⣩⡻⠯⠙⣶⠘⡳⠄⠻⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢮⡾⠋⠊⠀⢠⣶⠾⠿⡹⠾⢷⡆⠀⠙⡈⢦⡀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠁⣼⡇⡄⡴⠒⣠⠞⢻⣆⠰⠆⣠⢸⣧⠀⠻⣄⠛⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢲⣿⠁⣼⣸⣇⡛⠞⢸⠇⠂⠠⠳⠀⠑⢛⣸⡇⢼⡀⢿⡆⢠⡧
⠀⠀⠀⠀⠀⠀⠀⢰⡏⢘⣿⡇⠏⢰⢾⣱⢴⡌⠀⠀⠀⠁⣰⡦⣾⠳⡇⢸⢁⣾⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣻⡧⢈⣻⣧⢣⣀⠛⠄⠈⠁⠀⠀⠀⠀⠹⠄⢁⠞⣀⡈⢨⣿⠇⠚⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣷⡍⢻⣿⡜⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣾⠟⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡉⠙⠳⣮⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠴⠻⡁⣼⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⣀⢠⡑⠀⠐⢄⠀⢀⠔⠃⠔⢠⠚⠉⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣧⣅⡀⠀⣡⡁⠀⣀⣤⣾⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⣀⠉⠉⠙⠋⢙⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢶⠞⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⡶⢶⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⡤⣴⣌⣛⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠘⠲⠶⣭⡙⠳⣄⠀⣠⠄⠊⣩⣵⡶⢴⣾⣿⣿⣤⣤⣌⡉⠒⠠⣀⣠⠀⢀⡀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣆⠀⠘⣻⣶⡾⠳⠀⢀⣚⣉⣉⣠⡄⢀⢀⡤⢤⣤⣬⣭⣐⡰⠚⠁⢠⡿⠋⠐⣻⣶⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢿⣿⣯⣿⣿⠿⠶⠶⠛⠛⠛⠉⣀⣴⣿⣿⣷⣄⣄⡉⠉⢉⣛⠀⠀⠘⢁⣴⣿⣿⡿⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣿⣿⢟⣀⣠⣿⠿⣻⣥⣾⣿⠟⠛⢻⡿⣯⣍⠙⢿⣷⣤⠙⠿⣷⣦⠈⠻⣿⠿⠀⢮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣉⣀⣾⡿⢛⡥⠆⡰⠟⠛⢁⣀⣴⣟⣿⣌⣩⣅⡀⠙⠋⠀⢳⣌⠹⣧⠀⠀⢴⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⣹⡿⠛⠀⠋⠀⠀⠀⣠⣾⣿⣿⣿⡧⣿⠿⠿⣿⣿⣦⠀⠙⠀⠈⠧⠙⢷⣄⠀⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣠⣾⡿⠁⢠⣿⡗⠀⠀⠀⣉⣉⡅⠀⣀⣠⣿⣄⠀⠀⠀⣍⠁⢀⡀⢸⣷⡀⠀⠰⣄⠐⢌⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢰⣿⡿⠃⢀⢸⣿⡇⢰⡇⢸⢛⠉⢀⡾⠟⠁⢹⣟⢧⡄⢹⣻⡆⣾⡇⣾⣿⡇⡀⠀⠘⢷⡄⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⣿⣿⠀⠀⣾⠀⣽⡇⢴⣿⡆⠛⠀⣿⠇⣾⠠⢸⡘⣷⣇⠈⣇⣿⡿⡀⢸⣧⠀⣸⡄⠀⠘⣷⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠸⣿⣷⠀⢠⣿⡾⡿⣿⢶⠶⣦⡀⠀⢏⣴⡃⡄⠈⢁⡜⡀⠀⢸⣧⢶⡾⡿⢿⡆⢻⣧⠀⠀⣿⡄⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣻⣿⡆⢸⡏⠠⣷⡿⢮⡳⣤⣿⡆⡩⠘⠁⠀⠈⠀⠸⣀⣴⣦⣤⢞⡵⢷⣿⢠⠘⣿⠀⣴⣿⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠸⣿⣿⣇⢸⡇⠀⠻⣤⡈⠻⠆⣹⡷⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠸⠟⢃⣴⠏⡼⠀⡇⢸⣿⣿⡿⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠠⣌⠙⣷⠘⣧⣤⣄⠘⢓⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⢀⣛⡃⢀⣀⣰⠁⢸⣿⣿⠃⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢻⣿⣮⡀⣿⡟⢟⠚⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠌⠹⠋⢻⣿⣄⣼⣿⡇⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠘⢿⣿⣷⣮⡓⠀⣦⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡆⠀⢀⣼⣿⣿⠏⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⡀⠀⠉⠛⠛⢿⣶⣯⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⣾⠿⢛⡡⠀⣠⣾⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⡙⠀⠀⠀⠀⠀⢞⡿⣿⡱⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣸⣏⡋⠐⣊⣥⣶⡿⢛⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⠦⣀⠀⠀⠐⣦⡙⠲⠄⠈⠳⣤⡀⠀⠀⣠⠞⠁⠠⠔⢛⣥⡿⠋⠉⢉⣀⣠⡔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠶⠿⢿⣦⣿⣝⠳⠤⠤⠀⠈⠳⠤⠜⠁⠀⠠⠤⠖⣫⣾⡷⠿⠿⠛⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣷⣦⣀⣀⣠⣾⣤⣀⣠⣤⣤⣶⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠉⠙⠉⠉⠙⠛⠉⠁⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣤⣤⠀⠀⠀⠀⢀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣧⣴⣿⡷⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣄⣀⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢻⣿⣆⡤⠱⢿⠷⠸⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠿⠂⡁⢘⠏⠶⢸⠲⡏⠈⠾⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⣤⣤⣤⠤⣤⠤⢴⣶⡶⣶⡶⡟⢿⣇⣸⠠⣇⢾⣎⢰⡒⣣⡴⢤⠄⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣞⠻⢽⡴⢃⡼⢡⠋⡾⠀⠏⠀⡜⢹⠀⡛⠃⣿⠈⢹⠟⠀⠙⢮⢱⣬⣩⣿⣄⢚⡺⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠞⠁⡉⢰⠘⠃⠛⢲⡏⠘⣡⡇⣠⢰⡇⠘⠀⡗⠀⠹⠠⠶⠀⢃⠀⠈⣦⠹⡍⢸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡼⢰⠃⣿⢠⢰⠀⣸⢧⢰⣿⠁⠁⣸⡇⠀⡆⠀⡆⠀⠀⡂⠀⢇⠐⠀⢀⠁⠘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡃⡞⣄⢸⣼⣸⣄⡇⣸⠸⠋⡾⢸⢹⠀⣼⠁⢀⢷⡀⣸⠃⡇⠘⠀⢧⢈⢳⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡷⣇⢹⣎⢹⡏⢛⣃⠿⣷⣔⢇⣾⡀⢠⡇⠀⣾⣀⣡⣻⣆⡀⠀⢰⠨⠖⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠃⢻⣦⣿⣎⣿⡯⡏⠀⠀⠀⠉⠙⠪⠿⠯⠿⠿⠼⠛⠋⠩⣅⠀⠀⢸⠂⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡛⠷⣌⢳⣼⣿⠞⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢿⢀
⠀⠀⠀⠀⠀⠀⢰⢿⡃⠀⣴⣭⡟⢓⣶⣿⡛⠏⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⠀⠠⣾
⠀⠀⠀⠀⠀⠀⠘⠿⣫⠿⡰⠅⢢⠊⠘⣿⠻⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣞⡻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠠⡀⠀⠀⠶⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠒⡛⣣⣀⣄⡞⢛⣛⡛⠀⠀⠠⠄
⢢⡀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣤⣒⣧⣽⠶⡞⠻⠒⠒⣛⠛⠓⠰⠶⠶⠶⠦⠤⠖⠊⠉⠉⠉⠉⠁⠛⠋
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠀⠀⠈⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠀⠀⠁⡀⠀⠠⡔⠚
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠇⡇⠀⠀⠐⠀⠓⢠⣦⢡⠁⠤
⢀⡐⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠆⢃⡀⠀⠀⠈⠀⠀⠀⠀⣀⡠⢀
⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⣀⣠⣤⠴⠂⠋⠁⠱⠈⡈⠀⠁⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢀
⠻⢛⣀⠀⠀⠀⢀⡀⣤⣠⣼⠀⠨⢛⢏⣇⠇⢋⠄⢁⠄⠀⡐⠀⠇⡀⠀⠒⠄⠀⠀⡸⡄⡀⣐⡀⠦⠒⠊⠉⠁⠀⠈
⠀⠛⠿⠸⣿⡄⠿⠓⠉⢡⡃⠀⠀⠀⠀⠀⢠⠉⣸⢸⡃⢰⡇⢰⠀⠳⣔⡄⢤⠄⠂⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠊⠀⢢⣆⠀⣄⣧⠼⠇⠿⠁⠤⠷⠦⠷⠙⠊⠉
⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⢸⢠⣽⠼⠈⠁
⠀⠀⠀⠀⢀⠰⣴⡴⠮⠿⠛⠋
⠀⠀⠀⠀⠈⠁
⠀⢀⠀⠀⠠
⣶⣶⠿⣶⠶⠶⣶⢶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣄⢀⡀⣠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⢀⣤⣼⡧⠐⠼⠅⠂⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⢾⢏⡞⠱⣫⡾⢻⣝⡽⢡⢣⣶⠒⠏⢿⢀⠎⠻⠀⠰⠊⠋⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣳⣾⣦⠃⡎⡞⠀⠀⡏⠀⢸⠿⠱⣡⢻⠃⠀⡀⠸⡝⠷⡀⠲⡖⢀⠧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⣱⢧⡟⣏⠠⠁⢡⠀⠰⠁⡆⠘⢠⠐⢱⠏⠀⠀⠀⠁⢳⣔⣉⠀⠀⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡻⢞⣿⠘⡄⡴⠃⢀⡔⣠⢁⢠⡇⠀⡇⢀⠀⠀⡀⠈⡄⠘⢦⣄⡜⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢹⠓⢧⢸⠀⡗⠁⠀⡏⢸⢡⠎⢸⠀⡇⢳⠀⡆⠀⠀⠀⠉⠓⢄⠈⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣩⠧⠀⠪⠧⢆⡇⠴⢖⠃⣺⣊⢀⠘⢞⣔⣈⢰⡻⡀⠄⠀⠀⠂⣠⠇
⣠⡀⠀⠀⠀⠀⠀⠀⣬⠉⠦⠀⠀⠐⠉⠀⡆⠀⠩⣡⠋⠉⠈⠀⠀⠀⠀⠁⠉⠘⠄⠀⠐⡇⠤⠖
⡿⠿⡄⠀⠀⠀⠀⣸⠃⢈⡢⠆⠀⠀⠀⢀⢇⠩⠭⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡗
⢠⣴⣷⣄⠀⣀⠔⣿⡖⠀⠚⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠙⠅⡀⣿⣦⠤⠊⠀⠀⠀⠀⡀⡀⠀⠀⠀⠈⠂⠀⠈⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡐⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢿⠀⠓⣀⠀⢰⡤⣤⣀⠀⢀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣞⠳⢺⡭⣝⠎⢿⡉⠑⢠⡟⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⡿⠃⠛⠇⠰⣾⡷⢟⢰⡘⠓⠀⠁
⠀⠀⠀⠀⠀⡠⠔⠒⢒⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠳⠄⠀⠀⠀⠀⢐⡃⣼⣹⠏⣂
⠀⠀⠀⠀⢰⡁⠀⢰⣿⡿⠋⠀⠈⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡸⣄⠀⠀⠔⠀⣁⠉⠁⠋⡠⠃
⠀⠀⠀⠀⠀⠙⠒⠛⠻⢧⡀⣠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢧⠨⠱⢁⠀⠠⡖⠀⢀⡰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠋⠉⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⢸⡦⣀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⡇⢰⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⣾⡉⣿⠸⡿⣿⡙⢾⢣⠉⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣦⡜⠀⠀⠀⠀⠀⠀⢀⢴⣾⣯⠹⣧⠹⣇⠘⡆⢁⠘⣿⡿⣌⠓⢄⢨⠑⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠻⠁⠀⠀⠀⢀⣴⢸⣿⢸⣿⠻⡄⠙⣇⢹⡄⢀⡾⠐⣮⢳⡈⣧⠈⠆⠐⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣶⣦⡠⠁⠀⠀⢀⡄⠻⣿⠸⣿⡄⣿⡆⢻⡄⢿⡄⠃⢸⢇⣂⣬⣤⣓⠈⢆⠀⡇⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠟⠁⠀⠀⡔⢻⣷⡀⢻⡄⠿⢀⡸⣷⠀⢿⡈⠃⠀⠀⠈⡟⠉⠉⠉⠑⠀⢣⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠿⣿⣷⠊⠀⠀⢀⠾⣿⣆⠙⢧⠀⠙⠀⠈⣷⡙⣧⠈⠃⠀⠀⠀⢠⢃⣠⣤⣄⠘⢦⣸⣷⠇
⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⠇⠀⠀⢠⣾⣆⠈⠻⠆⠀⠈⠀⠒⠤⢿⣧⡈⠁⠀⠀⢀⠂⢸⠶⠄⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⣼⠿⠿⡞⠀⠀⢠⣷⡈⠉⠳⢦⠀⠀⠈⠓⠶⣤⣌⣻⡷⡄⠀⠀⠂⠀⠈⣿⠛⠋⠉⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⣇⣠⣤⡇⠀⠀⣶⡉⠛⠷⠦⠀⠀⠀⠴⠶⢶⣬⣛⢿⡷⠀⡠⠀⠀⠀⡼⠻⡟⠛⠉⠷⢶⣿⡇
⠀⠀⠀⠀⠀⠀⢹⠋⠉⢇⠀⠀⣏⠻⠷⠦⠄⠀⠀⠐⠒⠶⢦⣍⡻⢿⠗⠈⠀⠀⠀⠈⠀⠀⢳⣶⠶⠆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡾⠿⠿⡄⠀⣿⢷⣦⣀⠀⠀⠀⢀⣠⣠⣄⠙⢿⡞⠀⢀⡀⠂⠀⠀⠀⠀⣸⠶⢖⣂⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣇⣠⣶⡿⢶⣿⣶⣤⡉⠀⠀⠀⠀⣈⡙⠻⣿⣾⣴⣾⣯⣄⣀⣀⣀⣀⠀⢉⣤⣭⡥⠀⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⢸⣿⠉⠉⠛⠁⠀⠀⠀⣠⣾⠋⣩⡟⠛⠛⠛⢛⡛⠛⠿⣿⣷⠸⡉⠁⠀⠀⠈⠻⣏⣙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠳⣟⠛⠀⢀⣀⠠⠽⣿⣀⣏⣀⣀⠀⠛⠚⣧⢙⡆⠉⠀⠀⠙⢦⣀⡀⠀⠑⣎⣢⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠖⣶⣤⣄⣀⣀⠀⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢲⡶⣆⣤⠤⣴⣤⣤⣠⣾⣯⣽⠷⠶⣏⡉⠉⠛⠛⠛⣦⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣾⣫⣴⣿⣿⣋⡴⣊⣿⣅⣙⡹⠟⢁⣐⣶⣿⣿⣿⡂⢠⠄⠋⠀⣀⠙⠱
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⣿⣿⣿⣿⣿⣿⣯⣾⣿⡈⠛⠋⢠⠼⠿⢟⣬⣗⠲⠦⠌⠫⠠⠺⣿⡏⣠⡀
⠀⠀⠀⠀⠀⠀⠀⡸⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⢐⠆⠤⠖⣋⢙⣿⣉⡛⠓⠖⠢⣄⠘⢿⣇⠙⠁⢀
⠀⠀⠀⠀⠀⠀⢀⢣⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⠃⣀⢋⣽⣿⣿⡿⢯⡀⠀⢀⣦⠠⣀⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣸⣿⢸⢠⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⡷⢦⠀⣴⢿⣿⣽⡄⢀⠶⠟⣺⡉⡆⠹⣇
⠀⠀⠀⠀⠀⠀⣿⡇⢸⢸⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣶⣴⣾⡄⠋⠈⡇⣠⢾⣿⡿⠷⢿⡄⢹⡆⢀
⠀⠀⠀⠀⠀⠀⢿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣟⣿⣿⠇⠀⣠⣼⣿⢘⣫⡼⠃⢈⡇⢸⡇⠀⢨
⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⣿⢹⣿⣿⡀⢿⣿⢿⣿⣿⠇⠀⠀⠀⣿⡷⢬⡀⠐⠈⢹⠇⣼⠇⠀⠎⣰
⠀⠀⠀⠀⠀⠀⠸⣟⣿⡿⣿⡿⣿⣿⢸⣿⣿⣷⠈⠿⣾⣿⣿⠃⣽⣆⠀⠉⡃⣤⠙⠦⢛⣵⡾⡿⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣹⣿⢹⡿⢸⣿⣿⡿⠧⠀⠠⣿⣿⣯⣭⣡⠖⠑⠉⠀⣠⣴⡟⠁⠀⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣿⠀⣿⣟⠚⡿⣿⣦⠠⠄⢀⣬⣥⣀⣀⣰⣿⣿⣿⠤⢐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣸⢿⡿⣿⣿⣷⣿⣆⠙⣿⣿⡿⠝⣿⣋⣾⠀⠀⠀⠀⠀⠘⠙⠟⠛⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣟⢿⣾⣿⣿⠹⣿⣆⣿⣉⣷⡎⠈⠿⣿⣷⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣬⣛⢷⣿⣿⣿⠀⠹⣿⣿⣿⣿⣿⣇⡀⢙⡿⠟⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣬⣉⣻⣿⣿⡏⠀⠀⠀⢻⣿⣿⡎⠽⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠃
⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠻⣿⠃⠀⠀⠀⠈⣿⣿⡗⠓⠉⣷⣶⣄⠀⠀⠐⠉⢠⣤⣾⡏⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠘⢧⣤⣶⣷⣏⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⢰⣦⣼⠤⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢈⡟⠉⠙⢿⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠄⣀⠤⠟
⠀⠀⠀⠀⠀⠀⣼⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢠⡉
⠀⡀⠀⠀⠀⠀⠿⠿⣿⣶⣿⡹⠆⠀⠀⠀⠀⠀⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠇
⢾⣿⣦⣀⣠⣶⣤⠀⡛⣘⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠈⠻⣿⣿⣿⠿⠋⠀⠉⣵⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⣀⡁⠀⠀⠀⠀⣿⣿⣿⣴⢦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠠
⠀⠀⠈⢿⣿⣿⡟⠄⠀⢠⠃⢻⣿⡇⠀⠀⠀⡀⡈⠰⢶⠁⠀⠀⠀⢈⣠⠷⡆⠠⠤⠖⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠈⠉⠁⠀⠀⠠⠄⠁⠄⠀⠀⠀⠀⠘⣀⣋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣧⠀⠀⠀⢀⡀⢰⠇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠰⡆⢽⢿⢀⣀⡀⢸⠅⡿⠀⡿⠀⠀⠀⢀⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣆⢠⣀⡄⢻⢸⣿⣸⣿⣯⣾⢰⠃⢰⣇⣤⣦⢄⣾⡟⣠
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡙⣟⣎⣻⣾⣿⣤⡏⣿⣿⣿⢷⡎⣶⣼⣿⣿⣯⢞⢟⡼⠁⠀⠀⠀⢀⡄
⠀⠀⢀⣶⣄⡀⢀⣤⢆⣷⣜⣎⢻⣿⣿⡜⡇⢸⣿⡟⡿⢠⡇⣸⣿⣿⠏⣼⣾⣶⣷⡶⣠⣴⣯⠞
⠀⠀⠀⡙⠻⣝⡮⣝⣿⣿⣿⡻⣇⠹⣿⠇⢣⠸⠉⠀⠃⠸⠁⠜⣿⢃⡞⣽⣿⣿⣯⢞⡵⢋⣡⠆
⠀⠀⠀⠉⠓⢦⣙⠮⣙⠿⣿⣿⡊⢦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠸⣿⠟⠫⢖⣥⣶⣩⣀
⠀⠀⠀⢀⣼⣶⣿⣿⣟⡦⣌⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤⣾⣿⣿⡿⢛⣀⣤⣴⣶
⠶⢤⣶⣶⣬⣽⣿⣿⣿⣿⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠿⠿⣛⣻⠭⠿⠚⣋⣉
⠀⠰⠦⠤⢭⣭⣉⡛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣏⡍⠁
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣿⣿⣿⣿⠿⠿
⣀⣀⣠⣬⠭⢽⣿⣿⣿⣿⡻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⣶⠦⢶⡗⠞⠯⠽⢿⣿⣿
⠹⠶⠛⣛⣋⡭⣵⣿⣽⣶⣷⣦⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢿⣿⣿⣿⣿⣏⡛⠒⠲
⠀⠀⠈⠉⠀⢲⣿⣿⣿⡿⣋⣠⢖⣡⣀⢄⠀⠀⠀⠀⠀⠀⠀⡄⠠⣄⣬⣝⢦⣬⣛⠿⣛⠿⠇
⠀⠀⠀⠀⠀⣈⡵⣾⠵⣪⣿⣵⣿⣿⢫⢎⣰⣴⣀⡄⡇⢲⣠⣜⢦⠘⢿⣿⣷⣿⣦⣝⠻⣽⣲⣄⣀
⠀⠀⠀⠴⣿⠷⢋⣥⠞⣹⣻⣿⣿⢷⣯⣾⣿⣼⣿⣧⢷⠘⣿⣿⣯⣿⢮⢿⡿⣿⡷⠈⠳⠆⠉⠛⠉
⠀⠀⠀⠀⠁⠀⠉⠀⠀⠘⠟⣴⢋⡾⣿⣿⢇⣿⣿⣿⣿⢦⢻⣿⣿⡿⣦⠳⣵⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡟⠀⠻⡿⢸⢼⣿⡟⢻⠸⡿⡜⡿⠛⠈⢧⡙⢮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠀⠀⠀⣿⠈⠁⠀⢸⡄⢷⣷⠀⠀⠀⠀⠀⠈⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⡲⣏⢭⡉⢿⣄⢻⡄⢹⡇⢠⢹⡇⢹⡃⢲⡦⡤⣤⣤⣤⠖⣲⣿⠕⣠⢋⣽⠶⢂⠤⠤⣔⣼⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣍⢻⡀⠙⠜⡌⢻⡄⣿⠀⡇⢸⡇⢸⡇⡟⡼⠀⣸⣱⣷⠃⣼⠃⡼⢱⡇⠐⠁⣾⠇⡴⠋⢀⠘⣿⠇⠠⠞⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣄⢹⢈⡇⢰⠀⢣⢈⣇⢸⡄⡇⠸⡇⠀⠿⢻⠁⠀⠃⡏⢻⠀⢿⠀⣇⢸⡿⠀⡜⠁⢸⠁⣰⡇⠀⠛⠇⠜⠺⢆⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⡼⢊⠈⠇⢸⠇⠀⠊⣿⡸⡇⢷⡄⠃⢠⠄⢸⠀⢠⠀⣹⢸⣤⠸⢰⡘⣄⢳⠀⠀⠀⢻⠀⢻⠁⡄⣾⠀⡎⠀⠀⠀⡈⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡛⢻⡷⡈⢾⠀⣾⠀⡄⢧⠹⡀⠑⠈⣧⢦⡏⠄⢠⠀⢸⠇⠃⡾⢸⡇⠈⣷⠘⣧⠀⠀⢳⠀⡄⢚⣇⡁⣬⣷⣀⠚⠦⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠿⢾⠛⢌⠘⢧⣇⢠⡇⠈⢧⡑⠀⠀⣿⠺⠀⢠⡿⣡⢟⠆⡼⠁⢀⠇⠀⢹⡇⣿⡆⠀⠘⣧⣉⠎⠀⠙⠏⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⣼⣤⣼⢙⡚⠯⣀⠿⢼⠁⠐⠦⣽⣶⠦⣸⠋⠐⠛⠷⠿⠾⣜⠀⠀⠈⠆⠀⠈⠧⣏⢷⡀⠠⡈⠏
⠀⠀⠀⠀⠀⠀⢸⡟⠈⡇⠀⢙⠖⠁⠀⠀⢉⣿⠷⢶⣌⢑⠊⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⢤⣴⢶⡣⡈⢀⠉⠀⠀⠘⡀
⠀⠀⠀⠀⠀⢀⡜⢻⠟⠀⠐⠁⠀⠀⠀⠀⡘⢷⣤⣀⡨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠃⠀⠙⠲⣙⠀⠀⠀⠣⡀
⠀⠀⠀⠾⢿⠧⢿⡏⠀⡀⠀⠀⠀⠀⠀⢾⣙⠲⢍⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠄⠀⠀⠙⠣⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⢨⡇⠀⢄⡀⠀⠀⠀⠀⠈⢣⡉⠓⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢧⣀⠀⠀⠀⢸⢀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢽⣀⠀⣉⠆⠀⠀⠀⠀⠀⠙⠤⣀⣉⣙⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⠨⠇⠀⠀⠀⠉⠒⠃

⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⣿⢟⣯⣶⡶⢿⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢙⣿⣶⣌⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠊⠁⢸⠟⣡⠟⠉⣠⠀⠐⣛⣷⡮⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣰⡏⢱⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣾⣾⡿⣿⣿⢇⡜⡀⠘⠀⠁⠀⡎⣿⣷⢆⣉⣉⣠⠄⢈⣙⠳⣠⣾⠆
⠀⠀⠀⢀⣴⢯⣹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⡾⣿⠋⢰⢋⡿⣾⢃⣇⠀⠀⣦⠀⢹⠞⠁⣩⢶⡒⠒⡄⣰⠶⢭⠈⢿⡼
⠀⠀⣴⣛⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⢻⡇⠇⢸⡆⠀⡜⠁⢻⣆⢻⡆⢷⢻⡦⠀⠊⠀⡘⡉⢉⠹⠆⠹⠳⢦⠁⢸⡀
⠀⠀⣧⣤⣿⡀⠀⠀⠀⠀⣀⣤⢶⡏⣿⢃⣼⣿⢹⣿⠈⠀⣿⠀⠈⠁⢀⠀⠀⠀⢻⣿⣿⣼⣬⡅⠀⣴⠀⡃⢖⣲⠉⠂⠀⢠⡀⠙⠊⢿⡆
⠀⠀⠸⣟⣩⡷⡤⡴⣶⡛⣿⣥⠿⢟⣿⣿⣻⡏⡘⣜⢧⠀⠀⣆⢀⡾⢁⣴⠎⠀⠈⠉⣙⣿⣿⣖⢤⠻⢦⠑⠀⠀⠀⠀⠀⠉⠁⢾⡡⣾⣻⣆⡀
⠀⠀⠀⠈⠳⠼⠥⣧⠿⠟⠉⠀⡴⣻⠏⣿⣿⠈⣧⡈⠈⡇⣸⣯⢻⠁⡞⣡⡴⢋⠀⠈⠉⠉⠉⠉⠑⣖⠦⡀⡄⠀⠲⠀⢠⠆⡄⠊⢰⠳⠟⣿⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢋⠀⠉⠘⣧⠈⢻⣆⡿⡿⠀⢸⣆⣾⡏⢰⡏⢰⣶⣶⠃⠀⣀⢀⠈⣷⣿⣜⢳⡴⢶⡦⠾⣅⣸⢹⠀⢸⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠋⢀⠀⡀⢸⡆⡆⣿⠀⠁⠀⢀⡨⡻⣿⡟⠛⠛⠛⠓⠢⠀⠈⡇⣙⣈⢱⣘⡯⣁⣹⡀⠀⠀⠙⢾⠀⠀⠀⠋⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢴⡷⢣⢠⡇⢸⠇⣧⠇⢀⣠⠜⣋⣠⣾⣿⠀⠀⠀⠀⠀⠀⠳⠗⠣⠾⠷⠞⠫⡁⠀⣩⠃⠀⠀⠀⡄⡄⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡼⣡⡞⢁⡏⡼⠀⠘⣸⣿⢊⠀⠰⣤⣀⣀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠞⠁⠀⡀⢤⢀⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠃⣿⣷⡼⣷⣄⣀⡴⣋⡜⠆⣷⣤⣘⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⢠⣄⠈⣀⠡⣸
⠀⠀⠀⠀⠀⠀⣠⠔⠙⠏⢛⠟⠉⠉⠉⣩⠋⠀⠈⠙⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⠾⠓⠋
⠀⠀⠀⠀⢀⡆⢀⠄⠀⢠⠏⠸⠿⣴⠦⠃
⠀⠀⠀⠀⠈⢷⣊⣔⣠⠎
⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⢀⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⠂⡀⠄
⠀⡀⠠⠶⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⣿⠼⡄
⠛⠱⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣷⠕⠿⣆⣦⣤⣄
⠀⠒⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⡰⣦⣿⡟⣟⠋
⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡶⢷⠉⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡎⣀⡏⣿⣿⣾⣾⣷⣾⣷⡴⠂⠴⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⠀⠶⠮⠁⠀⠀⢻⣿⣿⣿⠛⣻⣿⣿⠃⠀⠀⡉⠉⣴⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠚⠏⠁⣿⣇⣶⣿⢷⣿⠝⠀⠀⠀⠁⠀⠀⠈⠉⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⢈⣻⡆⠀⣀⠉⠀⠀⠉⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠐⠀⠀⠀⠒⠊⠥⣦⣾⠅⠀⠀⠀⠠⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣆
⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⡀⠙⣹⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠁


⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡃⠶⠌⠙⣀⢁⠸⠀⢿⡿⠷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢈⡭⣧⠈⠍⢉⡸⠂⢰⠀⠒⠁⣰⢸⠇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢡⣄⠈⠁⠘⢁⠏⠀⠀⢀⢹⡎⠂⢘⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠉⠉⠒⢀⡇⠈⢸⢁⢰⡘⣶⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⢊⠙⡖⢤⡞⠀⠀⠘⠘⣇⠃⡟⢘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⢁⣸⡷⡈⡎⠀⠀⠀⠀⣿⠀⠀⡏⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⢸⡟⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠙⣇⣄⠑⣌⠂⠀⠀⠀⠀⣷⢡⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢹⠻⠆⣧⢸⡛⡄⠹⠄⠀⠀⠀⠀⡏⠈⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣤⠴⣮⣉⠖⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣾⠁⣿⠀⡔⢶⡈⠀⠛⠻⡄⢆⠀⠀⠀⠀⢀⡟⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠲⣏⣹⠦⠄⠒⠉⠀⠀⠀⠀⢀⢤⡴⣶⢋⣽⠋⣿⠀⣿⠀⣿⠀⡘⡄⢻⡄⢄⢦⡀⠀⠀⠀⠀⠀⡞⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣎⣉⡻⠊⠁⠀⠀⠀⠀⠀⠀⣠⢾⢃⡎⠸⣰⠃⡎⢸⠀⢂⡞⠱⢸⠛⢂⠻⣇⠀⠇⠈⠀⠙⠀⠀⠀⠀⠀⢠⠟⢃
⠀⠀⠀⠀⠀⢠⠋⣡⡝⠀⠀⠀⠀⠀⠀⣀⢤⡞⣇⣌⠈⣿⠀⣿⠀⢣⡇⠀⣼⣰⠀⠀⡄⠸⡆⠈⢳⡄⢠⠀⠀⠀⠀⠀⠀⣰⠟⣣⠟
⠀⠀⠀⠀⠀⠈⢶⡄⢸⠖⢢⡤⠰⣖⢿⣤⡎⢠⢹⡎⠆⣿⠀⠙⢆⢸⣰⠆⠇⠹⣀⠟⣔⣄⡙⢦⡈⠃⡘⠀⠀⠀⢀⣦⠞⣡⣚⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠚⠦⠤⠔⠒⠉⢀⡎⠁⢸⡀⠅⣆⠻⡄⠀⠀⠀⠛⠇⠂⠀⠹⡆⣌⡉⠑⠄⠻⠐⡇⠀⣠⠶⠋⠐⠊⣩⡅⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⢧⢠⡀⢹⡄⠻⣄⠈⢂⠀⠀⠀⠀⠀⠀⠑⢦⠀⠉⠁⠀⠀⢀⡗⠋⠁⠀⡄⠘⢂⡴⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠰⣌⠈⠳⣄⠙⠂⠈⠁⠀⣠⠤⠖⠒⠒⠀⠤⠜⣂⠀⠉⠓⠂⠀⡇⠀⠀⢰⠉⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⢀⣀⣀⡀⠀⣀⡉⠴⠋⠀⠀⠀⡀⠠⠬⠶⡾⠛⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣠⠊⠁⠀⠀⠈⠀⠹⣄⡀⠀⠀⠀⣀⡈⠤⠒⠉⠀⠀⢃⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠉⢇⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠠⣄⠀⠀⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠎⢀⡄⣄⢠⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⣸⢦⢠⡃⠠⡤⡄⢼⢢
⠀⠀⠀⠀⠀⠀⠸⣀⣶⡆⣬⡇⣸⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⢐⡶⢘⣾⣿⠆⠀⠀⠀⠀⠈⠢⣀⣸⣼⣷⢷⠷⠼⠧⠻⠚⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠁⠙⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠠⡀⢤⢀⡀⣀⣀⣀⢠⠀⣄⢤⠤⠄⠀⠤⡤⡄⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠋⣭⡝⠀⠄⡐⠉⠀⠘⠘⡆⢷⢰⠛⡜⢠⢳⠘⢇⠳⡀⢠⠰⡙⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠠⣄⡀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢠⡌⠀⡏⠈⢧⠘⡇⠢⠸⡔⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠨⣱⠀⠘⠃⠀⠀⠐⢀⠀⠀⠀⠄⡃⠘⣼⠀⠳⠀⡸⠀⢧⢠⢠⠌⢢⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣠⡜⠁⠦⠁⠆⠀⠀⠀⠈⠀⢀⡆⡀⣿⡄⢹⠃⢤⠀⠧⣈⡝⠼⢩⣾⠘⠈⢞⡄⠀⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣘⠁⠰⠀⠀⠀⠸⡘⣿⣱⠃⢆⣌⣀⣀⠀⠁⠀⠐⠛⣨⢟⠀⠀⠐⠋⢰⠄⠦⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠀⠀⠀⠀⠋⠉⠘⠃⠀⠋⠁⢋⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣫⣍⣹⣝⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣴⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⣠⢰⣏⠽⡍⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣆
⠀⠀⠀⠀⠐⣟⠄⡀⢰⡤⠛⠉⠀⢰⣿⣿⣿⣿⣿⢟⡿⡿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⡙⣻⢛⠓⣾⣿⣟⡶⠄
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣧⡈⡧⠄⢿⢠⡼⢯⡹⢻⣿⣿⣿⢿⠀⠯⢹⣼⣷⣾⣉⢤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⣿⢾⠿⠿⣽⣷⠚⠷⠷⠾⠷⣀⣿⣿⣿⠀⠀⠐⢋⠻⣭⠏⠰⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣧⣛⡪⠛⠁⠀⣗⢲⣬⠇⠀⠀⠀⠀⠀⠈⡈⣿⣿⠉⠲⢏⣨⡕⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠉⠀⠀⠀⠀⡿⣐⡎⠀⠀⠀⠀⠀⠀⠀⣄⣿⡏⠀⠀⠀⠈⠀⢤⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠹⣧⠀⠀⠀⠀⠀⠀⠁⢤⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⢠⢖⣲⢖⣰⠒⢲⡦⢤⠤⡤⠴⠆⢶⣴⣤⠤⠀⢀
⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⢀⡀⢶⡀⢿⢀⢸⢣⠃⡏⡞⣡⢺⠁⣾⠊⠀⠾⠀⢡⠀⠀⠀⠀⠐⣶⢋⠁⣱⡄
⠀⠀⠀⠀⠀⢸⣫⢀⡀⣴⡨⠛⠉⠀⡰⣄⢹⢸⢠⡷⢡⢀⢧⡆⠃⢠⠇⡄⠁⠀⠀⠀⠀⠀⢀⡌⠁⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠉⠚⠉⠀⠀⠀⠀⢠⡀⣇⠟⡐⢉⣄⡘⠄⠸⡇⣠⢻⣂⠕⣀⠀⠀⡀⠀⠀⢏⡃⣿⣠⡄⡀⢤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡐⢲⡵⡟⠽⢭⣱⡖⠺⠼⠦⠻⢄⡄⠀⠀⠈⠈⠀⠂⠸⢫⡉⠋⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣌⣃⡹⠊⠀⢸⠀⢤⣬⠀⠀⠀⠀⠀⠀⢃⠀⠀⡌⠑⠴⢀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠊⠁⠀⠀⠀⢰⠇⢠⠇⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠈⠙⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠈⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠠⠴⠜


⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⣄⡀⢀⡤⡠⠤⠴⠶⣿⠖⠲⠤⠤⡀⠀⡀⠀⢀⡤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠈⠻⡇⠉⠀⠀⠤⠄⠬⠿⢍⡁⠤⠀⠀⠀⣉⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⠀⠀⠐⠒⢋⣩⡭⢷⣿⡿⢭⣝⠓⠒⠀⠀⠐⣣⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠿⠁⢰⡞⠑⢋⣭⣴⠆⣷⠤⣦⣬⡑⠈⣷⠀⠈⣿⢀⡕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⢃⣤⠀⠀⠠⠖⢢⠶⠛⠶⣄⠲⠄⠀⢀⣄⠐⣄⢻⡄⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢡⡟⡄⡌⣛⠰⠆⠀⠀⣇⠀⠀⠀⡸⠀⠀⠴⢈⡛⡁⢶⢦⡙⡆⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆⠳⡏⣽⠛⢇⣂⡄⠁⠀⠀⠀⠀⣀⡐⡺⠻⡇⠟⢸⣼⡇⢀⠰⡄⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣿⠀⠀⡅⠻⡀⠁⠙⠟⠀⠀⠀⠀⠀⠿⠁⠈⣠⠇⠀⠀⢹⣷⢸⠀⠐⡈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⢽⣧⣶⡿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⣷⣦⣾⠛⠘⠀⠀⢡⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠹⣿⣄⠲⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⢀⣼⣿⠋⣠⠎⠀⠀⢸⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠠⠈⠉⠙⠻⢍⡀⠀⠶⣄⠀⢠⡴⠀⢂⣩⠿⠋⠋⠐⠈⠀⠀⠀⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡀⠀⠘⢷⣆⡀⠀⠀⣲⠁⠀⢀⣰⡞⢠⣤⠔⠀⠀⣠⡀⠀⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠳⣄⠀⠈⢻⡶⠖⠛⠛⠛⠒⠶⠏⠀⠋⠁⠀⠀⢠⡿⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⢧⠀⠀⠳⠤⢀⢀⡠⠤⠀⢀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠈⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠘⣧⠀⠀⠀⠳⣄⠀⣠⡶⠋⠀⠀⡀⢀⡾⠃⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⣆⠀⠈⠳⣄⠀⠀⠈⠀⠁⠀⢀⡤⢊⡴⠋⠀⠀⠀⠀⢀⣞⣠⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡄⠀⠀⠈⢣⠀⠀⠙⢦⡀⠀⠀⣴⢞⣩⠴⠋⠀⠀⠀⢀⡆⢀⠊⠛⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⢳⡄⠀⠀⠙⣦⠀⢠⠋⠀⠀⠀⠀⠀⣠⠟⠀⠄⢀⣩⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠢⠀⠀⠀⠳⣄⠀⠀⠸⠄⠀⠀⠀⠀⣀⣴⠞⠃⠀⡸⠀⠈⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⡈⠁⠀⠀⠀⠀⠀⣶⠟⠋⠁⠀⠀⠈⠀⠀⠾⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣔⠠⣄⠀⠀⠁⠀⣀⡠⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠾⠿⣰⠏⠁⠀⠀⠀⠀⢀⡘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠄⡀⠰⡤⠖⢻⡵⠂⠀⠀⠀⠀⠀⠘⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠠⡄⢀⠅⠀⠀⠀⠀⠀⠀⠒⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠐⠠⠀⠀⠄⠀⠀⣀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠐⠿⠒⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣄⣸⡇⠀⣠⠀⠀⣤⠀⢠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠚⠛⠛⠋⠑⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⡿⠀⢠⣿⡟⢀⣴⡶⢲⡆⣾⡶⠛⣿⡇⣿⠏⣀⡿⠃⠀⣼⣷⠀⠀⣴⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢱⣿⠇⢠⣿⣿⢃⣿⡟⢀⣾⢳⣿⣵⠞⠉⢠⣿⣿⠉⠀⠀⢸⣿⠃⣠⢾⣿⠇
⠀⠀⠀⢠⣤⠀⢀⡾⠁⣾⡿⢠⡏⣿⡏⠘⣿⡛⣋⡴⣻⡟⢿⡀⠀⢸⡿⣿⠀⠀⢸⡎⠛⠛⠃⣾⡟
⠀⠀⠀⠸⣯⡠⠟⠀⠀⢿⣷⠏⢸⣿⠃⠀⡈⠉⠉⠀⣿⡇⠘⣇⠀⠺⠇⠘⢦⣤⡼⢣⡤⠀⣼⡿⢡⣎⡗
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠘⠿⣤⠾⠁⠀⠀⠀⠉⠁⠀⠘⠳⠤⠾⠀⠀⠀⠀⢿⣀⣼⡿⠁⣞⡷⣞⠄
⠀⠀⠀⠀⢀⡴⢻⣿⠀⠀⣿⡇⠀⢀⣠⣴⠖⠛⣷⠰⣿⡇⢀⣴⠞⠓⣦⡀⠀⣶⡆⢀⣭⣭⣤⣤⣤⣤⣶⣶⡦
⠀⠀⠀⢰⠟⠀⢸⣿⠠⠾⠻⣤⣶⠋⣿⣿⠀⣠⡿⢡⣤⠀⣿⡏⠀⠀⠛⠃⠀⣿⣇⡤⣤⡀⠀⠀⣿⡇
⠀⠀⠀⣿⠀⠀⢸⣿⠀⣠⠞⠙⣿⡄⣿⡷⡞⠁⠀⢸⣿⠀⣿⡇⠠⠶⣶⡶⠂⣿⡏⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠈⠀⠀⢸⣿⣰⠏⠀⢀⣿⠇⣿⡇⢻⡄⠀⠸⠿⠀⠻⣧⣀⣰⣿⠇⠀⣿⡇⠀⣿⠃⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢻⣀⣀⣼⠟⠀⣿⡇⠀⠻⣤⣀⣠⠆⠀⠈⡇⠘⠿⠂⠀⣿⡇⠘⣿⣠⡷⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠦⠠⠁⠀⠀⠀⠀⠀⠀⠉⡏⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠓⡄⠀⠀⠀⠀⠀⠀⢀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠖⠀⠀⠀⠀⠀⠀⠀⢘⣉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠛⢲⠤⢤⣤⢤⠤⠤⠔⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠁⠀⠀⠋⠀⠠⠄⠸⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣶⡆⠀⠀⠀⣶⡄⠀⠱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠦⢀⡀⠀⠀⠀⠀⠀⣀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠘⢬⡉⠭⠏⠀⡰⠉⠃⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠆⠒⠓⠤⠤⠄⡀⠤⢴⢳⢛⡖⢲⢦⣀⢀⣤⡤⡖⠒⡖⡶⠒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⠤⠔⠋⠀⠀⢸⡏⣸⠁⢻⡏⠑⡀⢳⠹⣇⠈⣷⠃⠘⢡⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⡗⠁⠀⢸⠀⢀⡇⡼⠀⢸⠀⢹⡇⢠⠇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠳⠀⠀⠀⢰⠻⢁⡄⠀⡞⠀⡜⢰⠃⠀⢾⠀⠈⡇⢸⠀⢠⡏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠀⠀⠀⠀⣧⠎⠀⠀⣠⠀⠀⠀⠀⠀⢸⠀⣼⠃⢸⠀⠞⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡄⠀⠀⠀⢧⡀⢀⡴⠃⠀⠀⠀⡼⠀⡾⡞⠁⢀⠎⠀⡀⢨⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣑⣄⠀⠀⠀⢷⠋⠠⠔⠚⠛⠻⡧⠞⠋⢱⡀⣁⣤⠞⢀⢾⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠉⠉⠲⠦⣄⣸⠀⢀⣠⣔⡊⣹⢧⣴⡎⠁⠹⡔⠁⢀⡀⢼⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡭⠿⠷⠦⡜⠈⣆⣉⣥⣤⣭⡇⠈⢶⠲⠖⠒⢻⡏⠉⣀⣸⡤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠤⠤⡇⠀⡟⠋⠁⠀⢰⠀⠀⠀⠑⡖⠊⠁⡟⠉⠁⢈⡷⡞⣳⢴⡞⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠴⠶⠦⡄⠀⢸⠛⣛⠓⢸⠀⠀⠀⠀⣙⠉⠁⣏⣉⣉⣨⠁⠈⠛⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠶⠶⠆⡇⠀⢸⠝⠛⠛⢃⠀⣴⠖⠦⣏⢈⣠⣇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢠⠊⠙⠲⣤⠃⢠⢞⠖⠲⠒⡞⢶⠯⠼⠀⣼⠋⡏⡜⠉⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠤⠤⠤⠞⠁⠸⣸⡀⣆⢀⣇⡸⠀⠀⠀⠈⠓⠓⠛⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⢐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢳⣵⢿⠘⣡⠀⢗⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠳⣌⣧⣿⣾⡇⢁⢸⣃⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⢦⡝⣷⢹⡜⣮⣷⡄⠀⠀⡶⢂⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣽⣭⣭⣭⣉⠙⠻⣦⣝⠷⣤⣄⢠⣀⣀⡀⣀⡀⢀⠀⡄⣄⡬⣎⢷⠸⣇⢻⡜⣟⢆⠀⠂⠗⡭⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠛⠻⣿⣿⣦⠈⢿⣿⣄⠉⢿⡏⣻⢃⣿⠁⣿⢠⡇⣿⢷⢻⠸⣇⢹⣆⢻⣜⡆⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⡈⢴⠾⠷⣶⣶⣦⣄⡈⠻⢿⣷⡀⠹⣿⣧⠘⣿⡏⢸⡟⢸⡇⣼⠁⡿⢸⡜⡆⢻⣆⢻⣇⠈⠁
⠀⠀⠀⠀⠀⠀⣺⠀⣤⣤⣤⣈⣉⠛⠿⣶⣆⠙⣿⡄⠸⣿⠀⣿⠁⣿⠃⣾⢀⡟⣸⠇⡌⣧⢻⡌⢻⣆⠙
⠀⠀⠀⠀⠀⣸⣿⠀⠀⢀⣉⣉⣙⡛⠷⡄⠋⡇⢹⡇⠀⣿⢠⣿⠀⣿⠀⡿⢸⡇⠏⣼⣻⡘⢧⡹⡄⠉
⠀⠀⠀⠀⢰⣿⣿⡀⠀⠉⠙⠮⢭⡙⠃⠀⠀⢫⠸⡇⢸⡇⢸⡿⢠⡟⢸⡇⢸⠀⢞⣩⣍⡛⢎⢳⠘
⠀⠀⠀⠀⢼⣿⣿⡇⠀⢨⣉⣛⡓⠄⠀⠀⠀⠀⠀⠃⠘⠇⠸⠁⢸⡇⠸⠆⠘⠄⠐⣶⡖⠭
⠀⠀⠀⠀⠘⣿⣿⠃⢀⣲⣶⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠘⣿⠇⢽⣿⠯⠑⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣯⠁
⠀⠀⠀⠀⠀⠀⠈⠀⢸⣻⡁⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣲⡆⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⣦⠠⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠘⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠺⠏⠓⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠉⣉⣙⠒
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣰⣿⣶⣤⣿⣦⣇⠈⢇⣀⣤⣤⡴⠶⠶⠾⠿⠷⠶⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠠⠾⠟⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣾⣿⣿⣿⣍⠛⠃⠙⠂⠘⠛⠋⠁⠠⢴⣶⣶⣶⠶⠒⣂⠀⢉⡉⢙⣛⣒⡂⠀⠐⠁⠀⠀⠀⠀⠀⣰⡏⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⡛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠞⠉⣀⣠⣴⣶⡿⠟⠛⢶⣦⡈⢉⣽⠿⠶⣤⣀⠀⠀⠀⠐⠾⢿⣿⠋⢠⡇
⠀⠀⠀⠀⢀⣠⣶⣿⣷⡀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⣿⣿⣿⠏⢀⢀⢀⠀⠙⣿⣦⣀⠀⠀⠀⠉⠀⠀⠀⠀⣐⣻⠟⠀⢻⡇
⠀⠀⠀⠀⣛⣿⣿⣿⣿⡿⠃⣿⣿⣿⣿⠷⠀⠀⠀⠠⠶⣻⡿⠀⠀⣠⣤⣤⣶⣬⣿⠈⠣⠶⢶⣯⣝⡛⠛⠻⢷⣄⠀⠠⢿⣿⡟⠀⢠⡿⠁
⠀⠀⣠⣾⣿⣟⠁⠸⣿⡀⠀⠘⣿⡏⠉⠀⣀⠀⢀⣶⡿⠋⠀⠀⠾⠛⣛⣏⠀⠀⠹⡇⠀⠀⠀⡈⠛⠉⠓⠆⠀⢉⢳⡀⠀⠉⣉⣁⣼⡁
⠀⠈⠽⠿⣿⣿⣍⡀⠈⢷⠀⠘⠋⣀⣴⣾⠋⠀⣸⣿⠁⠀⠀⠀⣴⣾⠋⣻⣶⣶⢀⠀⠀⠀⠲⣌⣻⡀⠈⠲⣄⠘⣦⡁⠀⠀⡈⠛⢿⣿⡀
⠀⠀⢠⣾⣿⣿⣯⣄⣀⣀⡀⢀⣾⣿⡿⠁⠀⣶⡿⠁⠀⠀⠀⣀⠀⠀⠀⠈⠈⣿⣼⡇⠀⠀⠘⣌⠛⢷⠀⠀⠹⡆⣌⣿⣆⠀⠘⣆⠀⠹⣧
⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⣡⣿⡿⠋⠀⠀⠀⠉⣰⠂⣴⣿⣦⣿⣇⠀⠀⠀⠘⢿⡿⠁⠀⠀⠀⠘⣧⠈⠀⠀⠀⠀⠿⠃⠀⠠⡀⠘⢷⡄⢿⡆
⠀⠙⠉⢙⣿⣿⣿⠋⢀⣾⣿⡿⠁⠀⣠⣦⠀⣰⣿⡆⠉⠉⢛⣉⣈⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄⠀⢠⣴⣶⣶⠦⠀⢄⢳⡀⢸⣷⠈⢿⡄
⠀⠀⣰⣿⣿⣿⠃⠠⠟⣹⣿⡧⠀⣼⣿⣿⠀⣿⣿⠇⠐⠺⣿⣿⣿⠿⣦⡀⠀⠸⠃⠀⠀⠀⠀⠀⠘⠁⠀⣿⣛⣋⠎⣀⠄⠘⣮⡇⢸⡿⠀⠈⢿⣆
⠀⢠⣿⣿⣿⣿⡆⠀⢠⣿⣟⡀⠰⣿⣿⡟⠀⣿⡟⠀⠰⣄⡘⠷⠶⠾⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⠉⢠⣞⡡⡄⠀⣿⠇⢸⡇⠀⠀⠸⣿⡗
⢀⣿⡿⣻⣿⣿⣷⠀⠘⢹⣿⣃⠀⠹⣿⡇⠀⣿⡇⠀⢀⠈⠙⠒⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠁⠀⣀⣠⣿⠀⢸⠀⠀⠀⠀⣿⣷
⠘⢉⣴⣿⣿⣿⣿⡆⠀⠀⢹⣿⡀⠀⢻⣧⠀⢿⣷⣄⠹⠷⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⢀⣉⣹⡇⠀⢸⡇⠀⠀⠀⣿⡿
⠀⡾⠟⣿⣿⣿⣿⠁⠀⢀⣾⣿⡇⠀⠀⠻⣆⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⡀⠹⣿⡇⢀⢸⡇⠀⠀⠘⣿⡗
⠀⠀⢀⣿⣿⣿⡟⠀⠀⣼⣿⣿⠇⠀⠀⣶⣾⣷⣶⠤⣄⡘⠿⣶⣤⣴⠇⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⣷⡀⠈⠀⢻⡿⠿⠀⢲⣴⣿⠃
⠀⠀⣼⣿⣿⣿⠃⠀⡀⠉⢰⣿⡇⠀⠀⠈⠛⠁⠠⣤⣤⣤⣶⡿⠉⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⣷⠀⠀⣿⠃⢠⣤⣸⡿⠁
⠀⢠⣿⣿⣿⠏⢀⣴⡿⠀⠀⢸⣧⣤⡀⠀⠀⠀⠀⢾⠛⠉⠉⠀⢒⣠⣤⣴⣾⣿⠿⠀⠀⢀⣀⣀⣠⣤⣀⣀⠀⡀⠁⠹⣾⡇⣼⡟⣠⣿⣿⠋
⠀⢸⣿⣿⣿⣾⣿⣿⠇⠀⠀⡼⠟⣿⣧⠀⠀⢀⣤⣤⡄⠀⠀⠀⠉⢉⣩⣉⣻⣿⡆⠹⢾⣭⣤⣉⠉⠉⢉⣤⡶⠏⢀⣴⠿⠃⢹⣇⣿⡿⠃
⠀⠸⣿⣿⣿⠿⣿⣿⣀⣤⣤⣦⠀⠋⣽⠞⠀⠀⠈⣻⣧⠀⠀⠀⢀⣉⣉⢛⠛⠳⠶⠀⠀⠉⠙⠿⣅⣄⠾⠁⠀⠀⠈⢀⡠⠀⣾⣿⡿⠁
⠀⠀⠛⠁⠀⠀⢻⣿⣿⣿⣿⣟⡠⠀⢀⣤⠖⢀⡶⣛⡅⣀⠀⣤⠀⢉⠁⠉⠙⠛⠒⠶⠤⠄⠀⠀⠸⠃⠀⠀⠀⠉⠉⠀⠀⣠⡻⠟
⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣿⢁⣼⣿⠃⠀⠀⠈⠉⠘⠋⠀⣿⣷⣬⣉⣘⣁⣀⠀⠀⠀⠀⢀⣠⣴⣤⣤⡀⠀⢀⣶⠞⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢰⣿⢻⣿⣿⣟⣵⡄⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⢷⣤⣤⣴⣾⣿⠟⠛⠻⢿⢿⣿⠋⠁⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⠊⠻⢟⣵⣷⣤⣤⣄⣀⠈⢱⣿⣿⣿⣿⣿⣿⠘⣿⣿⠟⠻⡷⠀⠀⢰⠋⣸⣿⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣆⢻⣿⣿⣿⣿⣿⣦⣼⣄⣀⣀⣠⣴⣶⣷⣾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠹⣿⣿⣿⣿⡆⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢾⣻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⠘⣿⢯⣉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢧⠀⠀⠉⠛⠷⣶⣶⣤⣤⣤⣤⣤⡶⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⠴⠖⢦⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡏⠀⠀⠀⠀⠙⡆⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣠⡀⡄⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠳⣄⠀⢀⠚⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣦⢾⡇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠶⠶⠶⠾⠛⠛⠛⠛⠶⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠲⠶⣤⡀⠠⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡴⡞⠛⢲⣶⣌⠙⢦⡀⠀⠀⢀⡞⣉⠀⠀⣶⣶⣦⠀⠹⣦⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣾⢼⣿⣿⣿⣿⡿⠀⠀⣿⠀⠀⣿⢰⣿⣷⣾⣿⣿⣿⠀⠀⣿⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠈⠷⠿⠿⠛⠁⠀⠀⣾⠀⠀⣧⠈⠻⢿⣭⡭⠟⠁⠀⠀⣾⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⠘⢧⡀⠀⠀⠀⠀⣀⣼⡏⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⢋⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣨⠁⠀⠀⠉⠛⠶⠶⠟⠋⢠⡇⠀⠀⠀⠈⠻⠦⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⣉⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⣠⠶⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠰⣷⡆⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⢻⠓⠓⣠⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⢉⣧
⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠂⠀⠉⠉⠉⠀⢀⠔⠋⠹⣆
⠀⠀⠈⢻⡈⣷⠀⠀⠀⠀⠀⠀⢠⡶⠛⢦⣈⡙⠛⠒⠲⠦⠶⠖⠊⠉⠙⢲⣦⣤⡴⠚⠁⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠘⡅⢘⠀⠀⠀⠀⠀⢀⡼⠄⠀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⢸⠇⠘⣦⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢷⠀⡇⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣄⣀⣾⡁⠀⢹⣆⣀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠸⡆⣧⠀⣴⠋⠉⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣷⢸⣴⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠸⡀⠛⠀⢀⣤⡶⢿⠀⠀⠀⠀⠒⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⡇⢠⡴⠋⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠳⠋⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠂⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠅⠀⠀⡸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣾⣶⣶⣴⣷⣶⣶
⣿⣛⣻⣛⣛⣛⣛⣃⣙⣛⣿⣛⣛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⣭⣭⣭⣭⣽⣿⣯⣭⣭⣽⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣿⠟⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⣤⣿
⠀⠀⠀⠀⠀⢀⡀⢠⣤⡤⠀⢰⣶⠉⠉⡩⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠰⣾⣏⠀⢸⣿⠧⠠⠼⣿⠷⠮⢡⣖⣽⡿⢋⣵⠟⣩⡶⠛⠉⣉⣤⣤⣤⣬⣶⣤⡀
⠀⣴⡇⠀⡹⠛⠉⠀⠀⠀⠀⣰⣡⢒⣈⢻⣿⡏⢠⡿⢫⣾⡟⠀⣴⡿⠛⠉⠁⢀⣠⣤⣬⣍⣢⡀
⢀⠻⣿⣿⠀⠀⠀⠀⠀⠀⣨⢙⡷⢋⣤⣿⣻⡇⣿⠀⣿⡟⢀⣾⡟⠀⣀⣤⣾⠟⠉⣀⣤⣤⣤⣙⣦⡀
⠰⡀⠀⣸⡄⠀⠀⠀⠀⣀⣿⣾⣡⡋⠉⢭⣿⣀⡟⣸⢿⠃⢸⣿⠀⣞⡵⠋⠕⣣⣾⡛⠋⠀⠀⠙⢷⣷⣄
⠀⠻⡿⠿⠋⢸⣶⡆⠹⣿⣿⣿⣿⣶⣿⢿⣿⣿⡇⣿⢸⣤⢸⣿⡖⠋⡙⠳⣿⣿⣭⣤⣶⣶⣶⣶⣤⠉⠙⢷
⠀⠀⠀⠀⠒⠚⠛⠁⠘⠉⠉⡏⠙⠀⠐⠀⠀⠘⣷⣹⣎⣿⡟⡽⣩⠟⠶⢟⡽⣻⠟⠉⠁⠀⠀⠈⠙⢷⣤⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⣤⡧⣌⠧⡼⠋⠛⠿⣿⣿⡇⢃⡴⢱⣏⣞⣡⠶⠟⠛⠻⢿⣷⡄⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣸⣷⢿⣟⢉⡤⠶⠶⠶⣤⠀⠿⠿⣶⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡏⣿⡩⠙⠒⢯⣟⠶⠼⣟⠳⣆⠘⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⣿⠇⡉⠐⠁⢃⡚⠮⢭⡓⠎⠧⢽⡶⣷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⣩⣤⢩⢊⣌⠁⠁⠰⢆⠘⣠⠰⣬⠷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇⣿⣴⠹⠇⠀⠁⢠⡆⠲⢠⠀⣷⣞⣉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣄⠃⢻⡗⣅⣡⡔⠒⣹⣆⣽⠆⡆⢛⡋⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡐⣸⡐⠛⠲⡽⢻⠾⣛⣩⠾⠃⣼⣶⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠷⣦⣴⠴⠡⠀⠤⢤⣠⣾⣯⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠛⠲⠾⣰⣾⠉⣿⣿⠏⣅⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⠉⠒⠐⡾⠛⢻⡞⡁⠫⠒⢦⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⣴⣆⢠⡁⣀⠘⣼⠁⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢠⠇⠀⡁⢀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⣌⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⣇⢀⣰⢀⠻⠶⠃

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠁
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠡⠀⠀⠀⠀⢹⡀⠀⠀⠐⠂⠀⠀⢀⡀⠀⠸⡋⠁⠋
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠠⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠉⠃⠀⠘⠠⠤⢀⠄
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⢀⡄⠀⠀⠀⢠⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⢠⡞⠃⠀⠀⠤
⠰⠒⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠮⠒⠃⠀⠀⠀⠁⠀⠀⠀⢰⣞⣵⣋⣒⠀⠈
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡔⠊⠙⡐⠀⢰⡗⠞⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡤⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⣶⢖⡶⠶⠶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠙⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠇⢠⡀⠉⠿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠹⡄⠀⡄⠀⠙⢢⣤⣀⡵⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇⠘⢦⡀⠙⢦⡙⠂⣠⣝⢲⣦⡴⢫⣽⣷⣀⡀
⠀⠀⠀⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⠙⣆⠈⢳⡀⠹⣄⡀⠁⠘⢧⣼⣟⣋⣀⣠⠟⠋⠘⢹⡆
⠀⠀⢰⣏⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣤⠤⣴⠶⢶⠒⢻⡟⣿⣏⠁⠸⣇⠘⢿⠀⠘⣦⠀⠃⠀⠹⠛⠀⣤⠸⡏⢸⠁⠀⠀⣠⠤⠤⠟⠁
⠀⠀⠈⢿⣹⢦⣤⣤⣤⠶⢶⡏⢉⣿⣀⠀⡟⠀⣿⡀⢸⡆⠘⡇⠘⣿⠀⣦⠘⣆⠸⡇⣤⢸⡆⢰⡄⢦⠀⠀⢸⠄⣇⣀⣡⠶⠋⠁
⠀⠀⠀⠀⠙⠒⠿⠶⠿⠒⠚⠉⢉⡼⠁⣠⠀⠀⢿⣧⠈⢧⠀⢻⡀⢸⠀⢿⠀⠸⠄⠀⢻⡆⠀⢀⣷⠞⠛⠓⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢿⠀⡼⠻⣆⠀⠘⢿⠀⠘⠧⠈⢃⣸⣆⣸⣇⣀⣀⣤⠤⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠉⠀⣋⡼⠁⡄⢹⡄⠀⠸⠄⣠⠴⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢠⡏⢀⡴⣚⣫⠽⠁⠀⣿⢠⡇⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠘⢷⣸⢤⡇⠀⠀⠀⣀⣡⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠀⣴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢻⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠙⠛⠛⠻⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣬⣙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⢉⣩⣭⣤⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⢀⣤⣾⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⢀⣴⡿⠛⠉⠀⠀⠀⠀⣀⣠⣤⣤⠶⣶⣶⣶⣶⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⣠⠟⠁⠀⠀⠀⣠⡴⠞⠋⠉⠁⢀⣠⠞⠋⢀⣤⣾⣿⣿⡿⢿⣿⣟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⣰⠋⠀⠀⣠⣴⠟⠉⠀⠀⢀⡠⠖⢋⣠⣴⣾⣿⢟⣩⣾⠟⠀⠘⣿⣿⣎⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢰⠃⠀⣠⣾⡟⠁⢀⣠⣔⣮⣵⣶⣿⣿⡿⢟⣫⣴⡿⠛⠁⠀⠀⠀⣿⣿⣹⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠃⢀⣼⣿⣯⣶⣾⣿⣿⣿⣿⣟⣻⣭⣷⣾⠿⠛⠉⠀⣴⠶⠖⠢⠀⣿⣿⡏⢸⡇
⠀⠀⠀⣀⣀⡀⠀⣿⡇⠀⠀⣾⣿⣿⣿⣿⡟⠙⠛⠛⣛⡛⠉⠉⠁⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⣿⣿⡇⠘⠀⡀
⠀⡰⢋⣀⡀⠉⠀⠸⣇⠀⢰⣿⣿⣿⣿⣿⡏⠳⡶⠛⡿⣿⣷⡄⠀⠀⠀⠀⠀⢠⡾⣿⣿⠻⣶⣿⣿⠇⠀⢠⠇
⠀⠀⡊⠀⠁⠀⠀⠀⠹⡄⢸⣿⢿⣿⣿⣿⡇⠀⠙⠄⣳⣟⠋⠀⠀⠀⠀⠀⠀⠈⢒⣿⡛⠐⢹⣿⣿⠀⢀⡾⠀⠀⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⣿⡌⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⢀⡾⠁⠀⢸⡀⣀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠙⠺⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⣼⣿⠏⡰⢫⠇⠀⠀⠘⠛⠃⠀⠀⢸
⠀⠀⢀⠴⠊⠁⢀⣀⡀⠀⠀⠢⡙⣆⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠈⣀⡈⠁⠀⠀⣠⣾⡿⠃⠀⢠⡏⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⢰⣏⠀⠀⠀⠀⠀⠉⠓⢄⠀⠘⠜⠄⠀⢸⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⣀⣴⣾⡿⠋⠀⠀⣠⠏⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠈⠁⣠⠤⠒⠀⠀⠀⠀⠀⢳⠀⠐⠀⠀⣼⣿⡇⣿⣿⡿⠀⠈⠉⠛⣿⣿⣿⡿⠋⠀⠀⠀⠀⠁⠀⢀⡀⠀⠢⢄⣀⣀⡠
⠀⣠⠎⢁⡴⠚⠁⠀⡀⠀⠀⠀⠃⠀⠀⢠⣿⣿⡇⣿⣿⠃⠀⠀⠀⢸⡟⢸⣿⠁⠀⠀⠀⠀⣤⡀⡴⢻⣇⣀⠀⠀⠀⠁
⢰⠃⠀⣾⠁⠀⠀⠚⠃⠀⠀⠀⠀⠀⢠⣾⣿⡿⢰⡿⠁⠀⠀⠀⠀⢻⣇⠀⠙⢦⠀⠀⠀⠀⠣⢘⡀⢋⡉⠉⠳⣄
⡟⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠁⣾⠁⠀⠈⠂⠀⠀⠘⢿⡄⠀⠀⠀⠀⣠⢒⣡⢚⠛⢄⠲⢤⡤⠞⠃
⣇⠀⠀⠀⠙⠻⢶⣶⣶⣶⣾⣿⣿⠿⠛⠁⠀⢠⡇⠀⠀⠀⢰⠀⠀⠀⠀⠻⣦⠀⠀⠀⠉⣽⠁⠉⢸⡄⠀⠀⢷
⠘⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠳⣄⠀⠀⣿⠀⣠⠞⠙⢦⣀⠸
⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠸⠟⠁⠀⠀⠀⠈⠓⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠙⢧⡀
⣠⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣉⣢⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡔⠊⠉⠉⠛⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠑⢢
⠀⠀⠀⣰⠃⠀⣤⣶⣶⣤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⣴⣶⣶⣤⡄⠾⣆
⠀⠀⠀⡇⠀⣾⣿⣿⣿⣿⣿⣆⠀⢦⣀⣀⡠⠤⠴⠖⠒⠶⠾⠽⠯⠭⠖⠒⠒⠦⠤⢤⣀⣠⡴⠃⣰⣿⣿⣿⣿⣿⣷⠀⢹
⠀⠀⠀⠣⠀⣿⣿⣿⣿⣿⡿⡿⡄⢀⡭⠈⢁⠴⠿⠛⠛⣉⣛⣃⣙⣛⣉⠛⠛⠷⢦⡈⠉⣍⡁⢰⢿⢿⣿⣿⣿⣿⣿⠀⠜
⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⠁⠀⠀⠉⠀⠐⢁⣰⣶⠿⠛⠋⣉⠉⣜⣉⠉⠙⠿⣶⣄⡈⠢⠀⠙⠀⠀⠘⠻⣿⣿⣿⣯⡆
⠀⠀⠀⠀⠘⠛⣿⣿⡿⠀⡔⠁⡴⠃⠀⣠⠋⢁⣠⠤⠐⠚⠁⠆⡉⣝⡛⠂⠤⣤⡈⠻⣄⠀⠘⣆⠈⢆⠀⢿⣿⣿⡛⠁
⠀⠀⠀⠀⠀⢸⠋⠀⣠⡞⠀⣼⠁⠀⠀⠀⡴⠋⡄⠐⠉⡟⣡⠀⠀⠛⢻⠙⠂⢠⠙⢢⠈⠀⠀⠈⣧⠀⢷⣄⠀⢙⡇
⠀⠀⠀⠀⠀⠨⠍⢰⡟⠀⢰⣿⡄⢀⣄⣤⡄⢸⡇⠀⣾⠄⠗⠇⠘⠆⠰⢳⠀⢸⡇⢠⣄⣠⡀⢰⣿⡆⠀⢻⡄⢩⠅
⠀⠀⠀⠀⠀⠀⢀⣾⠁⡆⠘⢻⡄⠈⢉⣉⣙⠈⠳⡀⠀⠘⠀⠀⠀⠀⠘⠂⠀⠘⢀⣉⣉⡉⠉⢸⡟⠃⣸⠈⣿⡀
⠀⠀⠀⠀⠀⠀⣾⡗⠐⣿⠀⣼⣟⠛⣿⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⠟⣿⠛⣻⣧⠀⣿⠀⢻⣷
⠀⠀⠀⠀⠈⢐⣿⡇⠀⡟⠀⡏⠙⢧⣈⠓⠶⠺⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠗⠖⠚⣁⡾⠋⢹⠀⣿⠀⢸⣿⡂⠃
⠀⠀⠀⠠⠀⣸⣿⡇⠀⣿⠀⢷⡀⢦⣭⠙⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠀⠈⠋⣭⡴⢠⡞⠀⣿⠀⢸⣿⣇⠀⠄
⠀⠀⠀⠀⢰⣿⣿⠁⠀⢹⢰⠀⠻⣄⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⣠⠟⠀⡆⡏⠀⠈⣿⣿⡆
⠀⠰⠁⠀⠸⣿⣿⡀⠀⣿⡌⣄⣷⣄⠎⠈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣱⡱⣠⡾⣠⢡⣿⠀⢘⣿⣿⠇⠀⠘⡆
⠀⡦⠀⠀⠈⣿⣿⡇⠀⣿⣷⠸⡇⠁⢀⡞⡱⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⡁⠈⣸⠇⣾⣿⠀⢸⣿⣿⠁⠀⠀⢴
⠸⡣⠂⠀⠀⢹⣿⣷⡀⠙⢿⡄⠀⣌⡏⢀⡄⠀⠀⣤⣀⣀⣀⡀⠀⣀⣀⣀⣿⡷⡀⢤⡀⢹⣴⠀⣰⡿⠋⢠⣾⣿⠇⠀⠀⠐⢘
⠐⣿⡁⠀⠀⠀⢻⣿⣷⣤⡀⠹⢦⡈⢱⣿⡁⠘⠀⠹⣤⣄⡈⠙⠋⢉⣤⣤⠟⠃⠃⣘⣿⣮⢗⡴⠏⣠⣴⣿⣿⡟⠀⠀⠀⢈⣾
⠀⠉⠏⠀⠀⠀⠀⠘⠿⢿⠿⣶⣄⡙⢦⣌⣉⡉⠑⠂⠈⠻⢿⣀⣀⡿⠛⠁⠐⠊⢉⣉⣡⡴⢋⣤⣾⢿⣿⠿⠁⠀⠀⠀⠀⠛⠉
⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣇⣉⢉⢉⠁⠀⠀⠀⣸⡇⠀⠀⠀⠉⡉⡉⣉⣼⣿⣿⣿⠛⠀⠀⠀⠀⠀⢀⣴⣿⠂
⠀⠀⠀⠈⠻⣧⢀⡄⠀⠀⡀⠀⠈⠉⠿⢿⣦⣤⣄⣀⣀⡠⠾⣿⡟⠲⢤⣀⣀⣤⣤⣼⣿⡿⠋⣁⣴⠀⠀⠀⢀⡀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣾⣾⡇⠀⠀⢀⠀⠘⢿⣿⣿⣿⠀⢡⣄⣈⣁⣠⡎⠀⣿⣿⣿⣿⠇⢠⡈⠉⠀⢸⣷⣾⣿⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠧⡴⠻⠷⠖⢲⣿⣿⣿⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠟⢷⠾⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⢻⢿⣿⢿⣿⣿⡿⣿⣿⡝⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⣍⠀⠋⠋⠉⠲⠋⠙⢀⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⣀⠀⠀⣀⢀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠘⢏⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠡⠀⠈⠙⠢⢤⣀⠰⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢱⣄⠀⠀⠀⠙⣷⣾⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⣠⡆⢀⠤⠀⣐⣈⡀⠐⢿⡇⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠰⠖⠉⠒⠓⠳⢰⠋⠉⣩⡶⠛⠁⠙⢦⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠃⠀⠀⠀⠀⢨⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠞⠃⠀⣼⠀⠀⠀⠀⠀⡸⠃⢸⡥⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠞⢃⣀⣀⣤⠶⠛⠁⠀⣠⣵⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣴⠖⠲⠤⠔⠒⠒⠒⠐⠂⣀⡤⠖⠛⠉⠉⠁⠀⢀⣀⣠⣤⣿⣷⣟⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠉⠢⣄⡀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢀⣀⣤⣴⣾⡿⢟⡭⠏⠹⡋⣋⢱⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠻⠟⠗⠖⠻⠟⠻⠋⠉⠁⢀⠈⢰⣞⣴⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡦⣤⣄⣠⡄⠀⣀⠀⣀⢀⣀⣀⣁⣘⢠⣂⣠⣤⣼⣾⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⢹⡄⢹⡟⠈⢿⠛⣿⠛⢻⡟⢻⠟⢻⡟⢻⡏⢩⣇⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡎⣧⠀⣿⠀⢸⡀⢿⠀⣸⡁⢸⠃⢼⠀⢺⠇⢺⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⣸⡆⢿⠂⠸⡇⢸⢢⣼⡇⢸⠂⣿⡀⣿⢓⣿⣿⣿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣿⣇⢺⣧⠀⣷⢾⡧⣿⣧⣿⢲⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡸⣿⣿⡧⣿⡎⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣧⣿⣿⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣶⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣾⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢟⡛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠿⠟⠻⠛⠛⠙⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢛⢦⡀⠀⠀⠀⢀⢔⣉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⣸⢱⣿⣷⣤⣄⠀⠴⣧⣾⣿⡇⢧
⠀⠀⠀⠀⠀⠀⠀⡷⠟⠛⢩⢓⢦⢀⢖⣲⡄⠈⣙⣾⣄
⠀⠀⠀⠀⠀⢀⡾⠷⠦⠀⠱⠭⢔⣈⠒⠕⠁⠀⢉⣭⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀
⠀⠀⢄⣀⣀⣿⣟⡓⠂⣀⡄⠀⢻⡿⠃⠀⢦⣄⣠⠤⠴⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⠻⣏⠀⢹⣿⣦⣄
⠀⠀⣀⣀⣈⣇⣉⣉⣽⣭⣥⠀⠀⡇⠀⠨⠥⢾⣖⣒⣛⣹⣋⠉⠉⠀⠀⠀⠀⠀⠀⣰⣍⢙⡷⠚⠉⠉⠛⠻⠿⠟
⠀⠀⠀⠀⠀⢹⣄⡀⠼⢶⠂⠀⣠⢧⣀⣀⣩⠟⠁⢀⣴⣿⣶⣤⣤⣀⡀⠀⠀⠀⠀⣏⠉⡟
⠀⠀⠀⠀⠈⠀⠉⠓⠤⣀⣉⣉⣳⣴⣁⣀⣤⠴⠞⠋⣿⣿⢸⣿⡟⢸⣿⡶⣄⢀⣼⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠉⠁⠀⠀⠀⠀⠀⣿⠇⢸⣿⠁⢸⣿⠁⠈⠉⠙⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⡀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠃⠀⠸⠇⠀⢰⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣶⣟⠷⢹⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣣⢈⣛⡴⢋⣼⢷⡄⣠⠤⠤⠤⣤⣀⣤⡤⢴⡶⣾⣣⡤⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡰⠛⠛⠆⢠⠃⠀⠀⠀⠀⢀⣿⣯⡈⡇⠹⣏⣠⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠥⣦⠴⠃⠀⠀⠀⠀⠀⢸⣥⣠⣠⠃⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢐⡀⠀⠀⠀⣒⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡴⠛⠁⠀⣠⡶⠖⢠⣤⠤⢠⣤⡄⢠⡤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⢠⠟⠀⠀⠈⠀⠀⢠⡾⠋⢀⡴⠋⠁⣴⠟⠁⠀⢼⠁⠈⢀⠐⢁⠀⠀⠋⠀⠀⠀⠑⠀⢀⠈⢄
⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠀⠀⠀⠀⠀⡰⠋⠀⢠⡞⠁⠀⡼⠁⢠⠆⠀⠀⠀⠀⡘⠀⠀⠛⡤⠀⠀⠀⠀⠀⣌⠇⠀⠈⡄⠀⠃
⠸⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⠀⢀⡿⠀⠀⠂⠀⠀⡇⢀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢱⡀⢹⠀⠀⡀⠀⠀⠀⢀⠀⢸⠋⣠⠁⠀⠠
⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠁⠀⠀⠙⠺⠂⠀⠙⠀⠀⡀⠁⠀⠸⠶⠁⠀⡀
⠛⠉⠁⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣤⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠈⠄⠀⠠⠆⡀⠀⢀⣍⡀⠀⢀⠠⠅⠒⠚
⠀⠀⣀⣄⠀⠀⠀⠀⠀⠛⠿⢧⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢾⣿⠟⠁⠀⠀⣀⣄⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠂
⠀⠀⠑⠀⠀⠀⠀⠹⣿⣷⣦⠔⠈⠳⣄⡀⠀⠀⠀⠀⠄⠻⣷⣄⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠻⣶⣤⣀⣤⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⡄⠀⠈⠻⣶⣤⣄⡄⠀⠀⠉⠛⠷⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⡀⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠈⠉⠀⢀⣀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠈⠙⠻⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⡄⠀⣀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣄⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⠒⠂⣐⣶⣶⣶⣲⣶⣤⣤⣤⣤⣤⣤⡴⢖⣶⣾⣿⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⡾⠱⡹⣫⣂⡴⣫⢊⢟⣵⣿⣯⣶⢁⣥⣿⢟⡩⡶⢛⡡⢴⣏⢒⣬⣥⣤⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⡟⠁⡐⡘⡸⢱⢏⡼⡱⡡⢊⢊⣿⢯⢇⡾⢙⡵⠁⡪⠞⣋⣠⢠⠐⠀⠰⣞⢿⡅⣢⡄⣄⢡⠌
⠀⠀⠀⠀⠀⠀⠀⢰⠰⣿⠀⢠⠇⠁⡀⣱⡞⡸⡱⠀⣎⠘⡜⡜⢰⡆⠊⠁⠀⠀⠚⠉⠈⡾⠷⢂⡴⠲⣚⣗⡃⠖⣌⢠⡌
⠀⠀⠀⠀⠀⠀⢠⣾⡇⢹⠀⢸⠀⡇⠀⣿⠁⣿⠃⣼⡇⠀⠃⠁⣿⣤⠀⢰⠃⡀⠀⠀⠔⣡⠞⣽⢀⣨⢙⡟⣍⠀⢘⡅⢤
⠀⠀⠀⠀⠀⠀⣄⠁⣷⠸⢀⠺⡀⡇⠀⣿⢠⣿⡖⢹⢡⠇⠀⣤⡏⣾⠀⠈⠀⡇⠀⠀⢠⡇⡶⡾⠦⢏⠸⠇⠘⢂⢠⣜⣾⠇⠄
⠀⠀⠀⠀⠀⠀⡉⠋⢿⠄⠘⠃⢀⠃⠀⣿⣸⣯⠇⣿⢸⡆⠀⢻⠇⠙⠃⡀⠀⡇⠀⡀⣸⡁⣇⡻⣌⠺⠇⠀⠀⠹⠛⢍⡬
⠀⠀⠀⠀⠀⠀⠙⣿⡌⡜⣄⢧⠘⡄⠀⢱⢸⡾⢀⡟⣏⡇⢠⠸⡄⠀⠀⣇⠠⡱⡄⠄⠉⠛⢥⡬⢔⡀⠀⠀⠀⠀⢀⠈⠁⠁
⠀⠀⠀⠀⠀⠘⠛⣿⠀⢹⢬⣎⢣⡑⠀⢠⡀⠳⡌⣧⠋⣆⢸⠀⠳⡠⠀⠘⠂⠑⠈⢂⠀⠀⣤⣁⡀⣿⠎⠳⡄⠂⠎⣠⡀
⠀⠀⠀⠀⠀⢠⣶⣽⠀⠨⢧⡝⣧⢳⡙⠢⠍⠠⢁⡹⣄⢸⡞⣿⡄⠀⠀⠀⢢⠀⠀⠲⣍⠀⠜⠟⠀⠀⠁⠈⠑⠀⠈⣷⠈⠂
⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⡀⢓⣌⢷⣝⣶⠄⠀⠂⠳⠙⣠⣿⣌⢃⡀⢀⡄⠀⠑⢄⡀⠈⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠠⢤⡍⠀⢈⡉⠘⣟⠲⠷⢷⡇⠀⠀⠀⠀⠈⠉⢷⣬⣝⠘⠁⠤⠒⠰⡍⠃⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣒⣛⠁⢠⠊⣿⣻⣬⣍⣓⡲⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠞⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣉⡉⠀⢄⠱⡷⠤⠶⣍⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⡔⠂
⠀⠀⠀⢀⡘⠟⠀⢾⡮⠥⠤⠀⠲⠾⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠰⠇
⠾⢿⡗⡌⠋⠁⠀⠈⠁⠀⠀⠀⠓⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⢀⣰⣿⣷⣦⠀⠘⠛⠳⣶⣾⠇⠀⣲⣶⣶⣶⣶⣷⣶⣷⡀⠀⢳⡴⠚⠃⠀⣾⣷⣶⣶⣦⠰⣿⣿⣿⣿⣿⣿⣿
⣠⣶⣿⣿⣿⣿⣿⣷⣶⠖⠳⢬⢻⠀⣰⣿⢿⢿⢿⣿⣟⣿⣿⡇⠀⠈⠴⣿⣿⡿⢿⣻⣷⠙⣿⡀⢽⣿⣿⣿⣿⣿⣿
⢿⣿⣿⡟⣿⣷⣇⠸⣧⠘⠀⢨⣧⡀⣿⣷⣿⣿⡿⠁⠈⠻⢿⡇⠀⠀⠁⢸⣿⠇⣼⣿⠄⠀⠹⡇⢸⣿⣿⣿⣿⣿⣿
⠀⣿⣺⣻⣿⠿⣻⣷⣌⡛⠾⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠁⣼⣷⡾⠛⢁⣾⣿⠟⣀⠀⠀⣗⢸⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢻⣿⣿⣽⣿⣿⣿⣷⣌⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣿⣿⣻⣤⣾⠟⠀⢰⠃⢠⣿⣿⣿⣿⣿⣿
⠀⠸⠃⠀⣻⡿⣿⣟⣿⣾⣭⣿⣿⣿⡿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡏⠉⢁⣄⠀⣄⣄⣀⢸⣿⣿⣿⣿⣿⣿
⢀⠀⣤⣬⣭⣽⣿⣿⣶⣿⣿⣿⣿⣿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠙⠻⢁⡼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⠿⣿⡏⠉⠷⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠘⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠛⠁⠀⣿⣇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠹⣷⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢴⣤⣤⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⠇⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣨⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠛⣻⣿⣿⣿⣿⣿⣿⣿⡿⣷⡄⠈⠛⢿⣶⣤⠀⠉⢀⡤⠖⠁⡰⠾⢋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣭⣿⣿⣿⣿⣿⣿⣿⣯⣄⣀⡀⠀⠀⠙⢿⣦⣤⠟⠀⠀⣀⣀⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤⣈⣭⣉⣁⠀⠀⠀⠉⠀⠠⢤⣩⣭⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠂⠀⢀⣤⣶⣿⣦⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠳⡄⠀⠀⠀⠀⢸⠁⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂
⠀⠀⠀⠀⠀⠲⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣌⠢⡀⠀⠀⡾⢶⣤⡀⠀⢸⠀⠀⠀⠀⠀⡠⠔⠈⢠⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣍⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⡇⠚⢉⣁⡀⠈⣢⣞⣀⣠⢤⣁⣀⠀⠣⣀⠠⠐⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠉⠙⠒⠢⠤⠤⢃⡜⠉⠀⠀⠉⠳⣴⠋⠀⠀⠀⠈⢻⡙⠒⢤⡀⠀⠀⢠⠃⠀⠀⠀⣀⠤⠐⠒⠈⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠀⠀⢀⡴⢻⠁⠀⠀⠀⠀⠀⠘⡄⢀⣀⠀⠀⠀⢣⠀⠀⠈⠳⣄⠺⠤⠐⠂⠉⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢠⠎⠀⢸⡀⠀⠀⣖⣆⠀⠀⢳⢰⣿⡆⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠕⢠⠇⠀⠀⠀⢣⡀⠀⢿⡿⠀⠀⣾⡀⠛⠁⠀⣀⠞⠀⠀⠀⠀⠀⡄⢣⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠲⠶⣒⡈⠉⠁⠀⠀⡸⠀⢤⠀⠀⠀⠙⠦⣤⣤⠴⠚⠀⠈⠙⠋⠉⠁⠀⠀⠀⠀⢀⠘⠀⠀⡇⠑⠢⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⡀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⠀⠀⠀⢀⣀⠬⠵⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⢡⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⢸⢠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠸⡀⠀⠀⠸⡀⠀⠒⠀⠤⠤⠠⠤⠀⠐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠖⠉⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⣀⠔⠊⠁⢀⣀⣀⠤⠤⠤⠀⠒⣦⠈⢧⠀⠀⠀⠀⠀⠁⠀⠦⠀⣀⡀⠴⠔⠊⠁⠀⠀⠀⠀⠀⢀⡴⢁⠤⠀⣀⡀⠀⠀⠀⠀⠑⠢⣀
⠀⠰⠞⠒⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠘⢆⠀⠀⠀⠉⠉⠁⠒⠂⠤⠤⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⠠⠄⠒⢌⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢁⡤⢄⣀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠗⠊⠉⠀⠀⠀⠀⡆⠀⠀⠈⠉⠓⣒⠲⠲⠶⠖⠒⣋⡉⠀⠀⠀⠀⡁⠀⠀⠉⠒⠤⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⡴⠋⠀⢇⠀⠀⢀⠜⠀⠙⣄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡴⠋⠀⠀⠀⠘⡀⢀⠎⠀⠀⠀⠈⢢⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡠⠊⠀⠀⠀⠀⠀⠀⣇⠎⠀⠀⠀⠀⠀⠀⠱⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢂⠀⠁⠀⠀⠀⡌⠄⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⢹⠈⠀⠀⠠⠀⠣⢠⣤⠉⡒⣄⡀
⠀⠀⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠈⠀⠈⠣⠀⠀⠀⠀⠂⢈⠈⠀⠁
⡴⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠂⢆⠙⠘⠌⠒⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠐⠡⠅⠁
⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠂⣲⣶⢶⢰⢻⠏⠩⠂⡠⠀⢈⠀⡄⠀⠀⢀⡀⠃⠀⠀⠄⠀⠀⢀⠀⡀⣀⣤⠦⠤⠀⠀⠠
⠉⣠⣶⠄⣄⠀⢠⣶⣘⡷⠗⣿⠀⢸⠑⠣⢿⠂⣆⡞⡰⣾⠀⢀⡆⠀⠃⣆⠀⠀⠉⠀⢀⡠⠚⠦⡦⠖⠂⠉
⠀⠈⠙⠀⠿⠗⠈⠉⠁⠀⡌⣟⠙⠁⠀⠐⢱⢤⠇⣡⠇⣿⠄⢸⡇⡸⡄⢜⡳⣭⠼⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢁⠐⢡⣧⡀⣔⡽⠖⠛⠐⠛⠀⠚⠛⠛⠛⠂⠉
⠀⠀⠀⠀⠀⠀⠀⢲⠀⢀⣇⢸⣠⣼⠜⠃⠁
⠀⠀⠀⠀⠀⣀⣦⣢⠼⠜⠚⠚⠉
⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⢀
⠀⠀⠁

⠀⠀⠀⠀⢀⠀⠈⡀⢄⠐
⠀⠀⠀⠀⠁⢀⡄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠃⠪⣱⣶⠛⠁⠀⠀⠀⠀⠀⠀⠄⠀⠈⠐⠀⠀⠈⢆⠈⠂⠘⠀⠇⠁⢐⠆⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⡀⠀⠢⢁⠀⠀⠀⠀⠈⠈⢦⣆⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣦⡘⢸⡀⡌⠀⢀⠀⡌⠀⠀⠈⠈⢶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠰⡄⠠⣤⡝⡻⠃⠀⢏⣀⠀⡀⠇⡀⠀⢴⠀⢈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⡷⠀⢄⣽⣰⠋⢩⠅⠰⠁⠊⢹⣷⣠⢆⢀⣤⠜⠀⠈⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠩⠿⢋⡡⣶⣢⠀⠈⠻⣆⣀⣐⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠁⠞⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠁⠀⠤⠦⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⠙⢦⡀⠀⠀⠀⣀⠠⣄⣐⡀⣤⣤⣦⡤⢀⣂⣠⠄⡀⠀⠀⠀⢀⡴⠋⡀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⢤⡘⣦⣦⡌⢂⡦⠟⠛⣩⠭⠶⠒⠒⠛⠒⠒⠒⠶⠭⣝⠛⠻⢴⡒⢣⣴⣴⢃⡤⠊⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠠⣔⣤⣽⣿⡿⢛⡁⠖⠉⣁⣴⢖⣒⣶⠶⠟⠻⠶⣶⣒⡲⣦⣈⠉⠢⢈⡛⢿⣿⣯⣤⣠⠄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⣨⣽⠿⣋⡴⢋⣀⠀⢊⡥⠖⠈⢀⣤⡶⠶⠶⠶⣤⡀⠀⠲⢬⡑⠀⣀⣙⢦⡙⠿⣯⣅⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠻⢋⣾⠏⡴⢋⡴⠂⠈⣠⣶⠖⣀⣠⠶⠞⠳⠦⢤⣀⠲⣶⣄⠁⠐⢦⡙⢦⠹⣷⡙⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣿⢟⡞⠐⣿⠀⠠⠊⢡⡔⢀⡅⢠⡜⣋⣙⢣⡄⢠⡀⢢⡌⠑⠄⠀⣿⠂⢳⡻⣿⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣭⠀⣸⠂⠜⠡⠾⠗⠢⠘⠂⠟⡀⠟⠠⠉⠁⠤⠻⢀⡻⠐⠃⠰⠺⠷⠌⠣⠐⣧⠀⣭⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣰⣿⠁⣸⠃⣠⠀⠒⢿⡋⣹⣿⣦⠸⠇⠐⠃⠀⠀⠀⠀⠐⠃⣴⣿⣏⢙⡿⠒⠀⣄⠸⣇⠈⣿⣆⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⣿⡟⢠⡟⢀⠙⠀⠱⣜⢧⣬⣥⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣬⣭⠾⣣⠎⠀⠋⡀⢻⡄⢻⣿⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⣿⠁⣸⡇⢾⠘⢶⣄⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣠⡶⠃⡷⢸⣇⠈⣿⡇⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⣿⡆⣿⣧⠈⢷⣄⢨⣥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣬⡅⣠⡾⠁⣼⣿⢸⣿⡇⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⣿⣇⠸⣿⣦⠈⠻⣧⠁⢫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡝⠈⣼⠟⠁⣴⣿⠇⣸⣿⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠙⣿⡀⠙⠻⢇⣤⡀⠰⠏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠹⠆⢀⣠⡸⠟⠋⢀⣿⠇⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢳⡘⡧⢻⣶⣌⠛⠿⣆⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠿⠛⣡⣶⡟⢼⢃⡞⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣿⢠⡀⠙⢿⣿⣷⡜⠀⠀⠀⠀⠹⣶⣶⣤⣀⣀⣤⣶⣶⠏⠀⠀⠀⠀⢳⣾⣿⠿⠋⢀⡄⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⡈⠻⣷⣦⣬⣽⣛⠀⢀⠀⠀⠀⠀⠉⠻⣿⣿⡟⠉⠀⠀⠀⠀⡀⠐⣛⣯⣥⣴⣾⠟⢁⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣤⣌⠉⠛⠛⣉⠐⠠⠀⠀⠀⠈⠀⠀⣹⣏⠀⠀⠁⠀⠀⠀⠄⠂⣉⠛⠛⠉⣡⣴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠖⠻⠾⣂⣂⣐⣂⠠⠄⠞⠛⠛⠓⠢⠄⢐⣂⠒⣀⠷⠟⠲⠚⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣄⣀⣀⣀⣀⣀⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠃⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⢀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠋⠀⠀⡀⠈⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⠂⠀⣤⡀⠈⠁⠀⢀⣀⠀⡗⢄
⠀⠀⠀⠀⠀⢠⠎⠀⠑⠄⣀⣀⣀⣀⡀⠀⠄⣀⡀⣉⠰⣡⢀⠦⡀
⠀⠀⠀⠀⠰⠁⢄⠐⠃⠀⠀⠀⠒⠀⠀⠠⠤⠀⠀⠀⢀⠤⢊⣀⠈⢆
⠀⠀⠀⠀⡆⠀⢀⠉⠒⠀⠤⠀⠀⠀⠀⠤⠄⠀⠒⠈⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠈⠁⠀⠀⠋⠀⠀⠀⠐⠒⠀⠀⠘⠁⠀⠉⢁⡀⡴⠁
⠀⠀⠀⠀⠀⠸⠈⠉⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠩⠶⠦⠀⠀⠀⠀⠤⠶⠭⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠢⠴⠃⠀⠀⣀⢄⢶⠋⢉⠔⠖⢄⣀⡀⠤⠐⠈⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡤⠠⠤⠄⠀⢀⠔⢠⢂⡆⢀⠊⠘⠀⠱⠱⡤⠄⠒⠁⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⠧⠀⠀⠀⠤⡜⠀⠂⢻⠃⣬⡀⠿⠆⠀⠀⡇⠠⠤⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⢰⠀⠸⠈⣻⠏⢀⠀⡜⡰⠁⠒⠠⢄⣀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⢀⡠⠘⢢⡀⠀⠄⠰⡀⠈⣎⠔⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠁⠀⠀⠀⠈⠑⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⠛⠛⠁⠀⢀⠀⠀⠀⠀⠘⠑⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠙⠛
⠀⡀⣀⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣠⠄⢙⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢂⠀⠀⠈⠁⠀⠀⠀⠀⢠⣀⠄⢙⡇⠀⠀⠀⠐⣶
⠀⠱⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡄⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠓⠇⠀⠀⠀⠻⠲⡰⠦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠇⠀⠀⠀⠻⠲⡰⠶⠀⠀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠢⠤⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠦⠤⠴
⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠂⠈⠁⠀⢀⡀⠠⠀⠀⠀⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠂⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢤⠤⠤⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⡁⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⢀⢤⠤⠤⣄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠋⠁⠀⡋⢠⠂⠂⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⡋⢠⠒⠂⠀⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡠⠃
⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢠⢚⠶⢀⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢰⠾⠶
⠀⠀⠀⠸⣦⣀⢀⣡⠟⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠠⠀⠐⠀⠈⠁⠀⢸⠁⠀⠀⠸⣦⣀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠠⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠘⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠀⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠁⠀⠐⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⢤⣼⠁
⠀⠀⠀⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀
⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⠰⠇⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣡⡰⠆⠀⠀⠀⠀⠟
⠀⠀⠀⠑⠀⠀⠀⠠⣄⡄⠀⠀⠀⡯⣑⠀⣘⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢀⣄⡄⠀⠀⠀⣯⣁⠀⣘⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠐⢦⣀⡀⠁⠀⠀⠀⠈⠈⠘⠁⠀⠀⠤⡄⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠁⠀⠀⠀⠈⠈⠊⠉⠀⠀⠤⡄⠀⠃⢀
⠶⠀⠄⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠄⠀⠀⠃⡀⠠⠍⠉⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠤⠆⠀⡴⠉⣉⡉⠃⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⠤⠇⠀⡴⠉⢉⡉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠄⠀⡅
⡆⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⢶⡽⠒⠀⠀⠀⠀⠘⠀⠤⠰⠊⠀⣠⣀⠀⠂⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⠶⠽⠖⠀⠀⠀⠀⠈⠀⠤⠰⠊⠀⣠⣀
⠃⠀⠀⢸⡀⠃⠀⠃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠸⡄⠉⢰⠇⠀⠀⢸⡀⠃⠀⢃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠸⡔⠉
⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⢰⣿⡇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢿⠟
⠀⠀⠀⠤⠄⠀⢀⣀⠀⢸⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⢰⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⡤⠰⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃
⠛⠀⠘⣰⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣤⡔⢤⣄⣀⡴⠃
⣥⠀⠞⠙⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⣦⣸⣿⣟⣿⠋
⠐⣆⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠢⣄⡀⣿⡿⡻⢛⡋⣠
⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠏⠤⢀⣾⡇⣸⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣶⣦⣀
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡀⠀⢹⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣡⠀⠡⠀⠂⠠⠶⣖⠈⠀⠀⠀⠿⠤⠂⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠇
⢀⠀⣿⠀⠘⠀⠀⠻⡍⢀⣷⣤⣤⠶⠲⣦⠀⠀⠀⠘⣿⣿⣿⠟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠟⢩⡟⣿⠋⡦
⠘⠀⣿⠀⠀⠀⠀⠀⢷⣿⣽⢆⣿⣷⣶⣿⠀⠀⠀⠀⠘⣿⣿⣶⣿⣧⢾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⠏⣤⡿⡙⢱⡇
⣿⡀⣧⡄⢠⣶⣜⢃⡈⢿⣧⣈⣿⢿⣿⡏⠠⠄⠀⠀⠀⠈⠙⣿⣿⠿⠞⠛⠿⠿⠿⠽⠿⠾⠛⠋⠙⢿⣿⣿⣧⢿⣿⣿
⣿⢷⣿⡇⠿⠿⠛⠉⠁⠈⡟⢻⠉⠁⠼⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠈⠻⣯
⡉⢻⣿⣧⡀⢜⡀⠀⠀⠀⡅⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⠧⠀⠈⢆
⠛⠿⡻⠻⠛⠻⠿⠿⠢⢴⡧⢣⠄⢰⡀⠀⠀⢈⣰⣦⣞⡂⣿⢛⣿⣆⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠃⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⢀⣠⣤⣜⣱⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠟⠽⣿⢢⣤⣤⣆⣀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠃
⠀⢰⡆⠀⠀⡀⡠⠴⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⡟⣍⢻⢿⣦⡴⣀⢹⣻⣇⢀⡴⠂⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠉⠁⠙⠉⠋⠉⠙⠟⢷⣸⡇⠁⢈⣀⡉⠀⠀⠀⠉⠙⠛⠳⠶⢄⠀⡠⠇⠀⠀⠈⠐⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠧⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠅⠀⠀⠀⠀⠀⠈⠁
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢠⡆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⣾⡇⠀⠀⣾⡇⠀⢠⣿⠀⠀⢠⣾⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠸⣿⡄⠀⢻⢿⡄⠀⣟⣧⠀⡼⣿⠀⢀⣿⡏⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⢢⣄⠀⠀⠹⣿⣄⠀⢹⣟⠢⡈⣯⠹⢦⡏⠈⠿⠡⣏⡴⠏⡿⠁⣠⣾⡿⠀⠀⣠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⠙⣿⠙⠢⣯⣠⣼⣷⣶⣿⣷⣷⣶⣶⣿⣤⣹⡷⠊⢡⡞⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠐⣦⣄⡀⠀⠈⠾⣯⠉⢻⣦⣾⣿⣿⡿⢿⡿⠟⠏⠿⠙⢻⠿⡿⣿⣿⣷⣾⡉⢠⡴⠋⠀⣀⣴⣾⠋
⠀⠀⠀⠀⠀⠈⠻⠿⣖⡶⠂⢹⣶⣿⡿⠟⠉⠁⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠈⠋⠻⣿⣿⣾⡓⠙⣋⡽⠛⠁⠀⣀⣀
⠀⠀⠈⠻⣷⣦⣤⣤⠌⢻⣤⣿⣿⠿⠀⠀⣴⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣆⠀⠀⠛⣿⣿⣾⠃⠴⠶⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠷⣆⣰⣿⡿⠟⠀⢠⠎⠁⣀⣀⣀⠀⠀⠠⠈⠀⠀⠀⣀⡀⠈⢱⡄⠀⠻⣿⣿⣦⡞⠋⠁⠀⠀⣀⣀⡀
⠀⠘⠻⢿⣶⣶⣖⠋⠉⣿⣿⠟⠀⠀⢈⣾⣟⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣭⣍⠛⢦⠁⠀⠀⠿⣿⣿⡀⣯⣽⡿⠿⠛⠁
⠀⠀⠀⠀⠀⢀⣉⠝⢾⣿⠏⠀⠀⠀⢸⣿⣿⣗⣚⣶⣿⣿⡇⢸⡿⣟⣶⣲⣻⡷⠀⠀⠀⠀⠀⢻⣿⡟⠓⢤⣄⣀⣀⣀⣀
⠐⠒⠿⢿⡿⠿⠯⣤⣸⣿⡇⠀⠀⠀⢨⣿⡋⠛⠛⠋⢹⣿⠇⢸⡇⠈⠁⠉⠉⠀⣀⡀⠂⠀⠀⢹⣿⡧⣾⠓⠛⠟⠛⠋⠉
⠀⠀⠀⠀⠀⣠⡴⢊⠹⣿⣷⡀⠀⢑⠐⢿⣷⡄⠀⢀⣼⠛⣄⠈⠛⡆⠀⠀⢀⣘⡿⠀⠈⠀⣠⣾⣿⣃⣀⡙⣦⣤⣤⣀⡀
⠀⠐⠚⠿⠿⠟⠛⢙⠟⢿⣿⣿⣦⠀⠀⠀⠀⠀⣴⠟⠉⠓⣿⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⢦⡈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣀⣠⣴⣧⡴⢬⣿⣿⣿⡄⠀⠀⠀⠰⣧⣶⡶⠿⠿⠿⠗⣖⣴⡄⠀⠀⠀⠀⢠⣿⣿⡿⠷⠖⠦⣿⣶⣤⣄⡀
⠀⠀⠐⠛⠛⠛⠉⠁⣠⣇⣈⣻⣿⣿⣆⠀⠀⠰⠀⠀⠿⠷⠖⠲⠾⠿⠀⠀⠆⠀⠀⣴⣿⣿⣿⣥⣄⣷⣄⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⢡⡋⢘⣿⣿⣧⡀⠀⠀⠀⣠⡤⡶⠲⠤⡀⠀⠀⠀⣀⣾⣿⣿⣅⡈⣧⠈⠛⠿⢷⣦⡀
⠀⠀⠀⠀⠐⠉⠉⠀⠀⣠⣾⡷⠋⡍⡈⣙⡿⣿⣦⣀⠀⠀⠁⠀⠀⢀⣀⣴⣿⠿⣿⣁⠘⡆⠳⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠀⢠⣗⡞⠉⣧⣄⡼⡏⢙⣿⠛⡛⡿⣏⠉⡿⣄⠄⡇⠙⣦⣧⡀⠀⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣿⠋⠀⢀⣿⡟⠀⢷⣾⠁⢷⣹⡇⢹⣰⠇⠈⣷⣷⠀⠈⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢀⣾⠟⠀⠀⣸⣿⠀⢸⣿⡇⠸⣿⡄⠀⠈⠻⣧⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣿⠃⠀⠸⡿⠀⠀⠙⡇⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠑⠀⠀⠀⠘⠉⠃⠛⠘⠐⠃⠘⠚⠐⠂⠀⠁⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⡀⠈⡄⠀⠀⠠⠀⡠⠆⢀⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠄⠀⠀⠀⠀⠈⠊⠀⠘⠁⠐⠀⠀⠀⢈⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡖⠠⠤⠀⠠⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠢⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⡀⠘⠂⣾⣿⣿⣿⣿⣿⣶⠀⠂⢂⣤⣤⣤⣀⡒⠀⠀⢁⠂⠄⡀⠀⠘⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⡠⠲⣹⣿⣿⣿⣿⣿⡿⣠⡀⣾⣿⣿⣿⣿⣿⣿⣶⢀⡟⠀⠌⠂⡀⠀⠀⠀⡠⢁
⠀⠀⠀⠀⠀⠀⠒⡂⠀⠂⠀⠠⠊⠀⠀⠙⠿⣿⣿⡟⠉⠀⠁⠁⠻⣿⣿⣿⣿⣿⣿⢋⠞⠙⠢⢄⡀⠈⠁⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠠⠈⠀⠀⠉⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠮⠭⠿⠭⠖⠁⠀⠀⠀⠀⠉⠲⢄⣀⣤⠁⠀⣀
⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠁⠋
⠀⠀⠠⢆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠐⢄⠀⡀
⠀⠀⠄⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⢂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⢠⠊⠀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢄⠠⠄
⠀⠀⠠⠀⠁⠀⢀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠡
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠑⠀⢀⠃⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⢀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⢀⠀⢨⠀⠀⠠⠌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⢠⠀⠀⠀⢀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂⢀⠠⠇⠀⠠⠒⡀⠀⠀⠜⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠈⠐⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣓⠂⠄⠀⠀⠀⣀⣀⡴⣘⣑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢛⡷⠐⢸⢻⡷⡙⡋⣴⣿⣿⣗⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠱⢓⠓⣘⠶⠝⢳⣈⠻⣿⠏⣾⣦⣄⠀⠀⠀⠀⠀⣀⣀⣤⡤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡴⠂⠄⢏⠥⢘⠈⢁⠋⠃⢷⡌⢿⡳⣤⣤⠒⠿⣌⡠⡈⠉⠀⠉⠳⡌⠻⣴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡏⠀⣠⠟⠫⡉⠘⠼⣧⡀⡜⡇⢈⠿⣏⠛⠳⠀⠈⠣⠙⣆⠘⣆⢱⡀⢠⠙⡀⣍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠈⠀⠨⡥⢐⠥⠂⣿⢱⠇⠈⢰⠈⣇⢰⡄⠀⠈⣷⠨⡀⠸⡄⠙⠀⣇⢣⢹⣾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠞⡶⠂⢖⣄⢠⣌⣠⡜⣡⠏⡰⠀⢸⢆⠋⠀⡇⢠⡀⠙⠆⢳⡀⢻⡀⡄⢸⢸⠘⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢝⣃⣉⣽⣏⠳⡍⡴⠋⣼⣣⢀⠌⡘⠀⡄⠀⢸⡇⠀⠐⠀⡇⢸⠁⠇⠀⣤⣴⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡑⢸⡧⡰⠟⣁⣠⠂⢀⡷⠁⣧⡿⢠⡆⡟⠀⠁⢀⠆⠰⢹⣲⣯⠶⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⠷⡀⠀⡴⠉⢀⣠⠍⠀⠀⠹⡇⣴⠀⠁⠀⣠⣭⠄⣠⢻⡟⣿⡶⠋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠓⣦⠒⠀⡀⢦⡄⠀⠀⠀⠙⣷⣥⣦⣤⣾⡟⣡⡤⣂⣿⠀⠹⣷⡾⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠿⣿⣦⣝⣾⣠⡄⠀⠀⠀⠛⠋⠉⠈⣿⣿⡆⠈⠹⡷⡀⠀⠘⢷⣷⠋⢰⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠓⠚⠋⣿⠿⠀⠀⠀⢸⠀⠀⠀⠀⣸⣏⠉⣡⢼⠇⠀⠀⠀⠀⠈⠓⠿⢧⣞⣤⣁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠀⠀⣿⠻⠶⠆⠀⡆⠀⠀⠀⢠⣯⡤⠀⠴⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠿⠶⣀⣿⣷⠀⠀⠸⠀⢀⡀⠐⠋⠁⢀⠀⣾⠗⠈
⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣻⣯⣤⣰⣀⡄⠀⡈⠁⠀⣏⡀⢾⣤⣶⣀⡀⣀⣼⣻⠶⠛⠉⠁
⠀⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⢛⣷⣜⡓⢆⣚⡙⠉⠀⠀⠀⠈⠉⠉⠁⠐⠒⠀⠀⠀⠉
⠈⠀⠄⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣶⣶⣦⡀
⠀⠀⠀⢀⠀⠀⠀⢠⣧⣴⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠁⠀⠘⢿⣿⣶⣿⣿⣿⣇⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⠤⠤⣄⣀⣀⣤⣴⡶⣶⣿⣿⠿⠓
⠀⠀⢀⠀⢀⠀⠁⢾⡿⠁⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣭⣧⣶⣶⣶⣯⣿⡿⠿⢿⣭⣄⣀⣀⡀
⠀⠀⠀⠀⠤⡤⣬⡞⠱⠂⢸⣶⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⠿⠛⠛⣻⣿⣿⣿⣿⣿⣯⡤⢶⣿⡿⠿⠛⠛⠁
⠀⠀⢀⠓⠂⢠⡅⡀⠰⠠⠘⣿⣿⠟⠁⠀⠻⠿⣿⣿⣟⣉⣥⣤⣶⣿⣿⡿⠿⢻⣿⣿⣿⠗⠁
⠀⠀⠠⣤⣀⣻⣌⡀⠀⠠⠀⠜⢧⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠛⠛⠁⣠⣵⣾⠿⢋⢀⣤⣤⠭⠤⠤⠤
⠀⠀⠀⠀⢉⣁⣀⣀⣀⣀⣀⣀⣀⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠿⢏⡀⠐⠚⡚⣿⡿⣶⠄⣀⡀
⠀⠀⢩⢿⡟⣛⣿⢿⣿⣻⣿⣿⣿⣿⣼⣿⣶⣶⣶⣶⡶⣶⣶⠶⠶⠲⠒⠒⠻⣿⡿⠪⣷⣿⣶⣶⣼⣫⣗⠢
⠀⠈⠉⠈⠛⠛⠈⠙⠛⣻⣿⡛⡿⠿⣿⣟⣿⡟⣿⠛⡛⠹⠃⠐⠀⢀⣀⣾⣤⠈⡏⣩⣮⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠀⠄⠤⠊⠋⠁⠒⠡⠖⠍⠽⠙⠋⢉⠀⢀⡴⢀⣠⣴⣿⡿⠿⠿⢀⡾⠿⢿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠚⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢸⣺⠵⠛⠋⠉⠉⠁⠂⠀⠀⡀⣰⣁⣠⣶⢾⣿⠟⠛⠃
⠀⠀⠔⠀⠀⣀⠀⠀⠀⠀⢦⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣾⣼⠟⠻⠯⡀
⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡔⠊⠉⠉⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠉⠉⠑⢢
⠀⠀⠀⣰⠃⠀⣴⣶⣶⣦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢁⣴⣶⣶⣦⡄⠾⣆
⠀⠀⠀⡏⠀⣾⣿⣿⣿⣿⣿⣆⠈⢦⣀⣀⡤⠤⠴⠖⠒⠶⠿⠽⣯⠽⠶⠒⠲⠶⠤⣤⣀⣠⡴⠃⣰⣿⣿⣿⣿⣿⣷⠀⢹
⠀⠀⠀⠣⠀⣿⣿⣿⣿⣿⡿⣿⡆⢀⣭⠈⢁⠴⠿⠛⢛⣉⣛⣋⣙⣛⣉⡛⠛⠷⢦⡈⠉⣭⡀⢰⢿⢿⣿⣿⣿⣿⣿⠀⠜
⠀⠀⠀⠀⢱⣽⣿⣿⡿⠿⠃⠀⠀⠉⠀⠐⢁⣰⣶⠿⠛⠋⣉⠉⣜⣉⠉⠙⠿⣶⣄⡈⠢⠀⠙⠀⠀⠘⠿⣿⣿⣿⣯⠎
⠀⠀⠀⠀⠘⢻⣿⣿⡿⠂⡔⠁⡴⠃⠀⣠⠋⢁⣤⠤⠐⠛⠁⠄⡉⣝⡛⠂⠤⣤⡈⠻⣄⠀⠘⣆⠘⢦⠀⢿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⢸⠋⠀⣠⡞⠀⣼⠁⠀⠈⠀⡴⠋⡄⠐⠉⡿⣡⠀⠀⠛⢻⠙⠂⢠⠙⠦⠈⠂⠀⠈⣧⠀⢿⣄⠀⢙⡇
⠀⠀⠀⠀⠀⠸⠏⢰⡟⠀⢸⣿⡄⢀⣄⣠⡄⢸⡇⠀⣾⠄⠗⠇⠘⠆⠰⢷⠀⢸⡇⢠⣄⣠⡀⢰⣿⡆⠀⢻⡄⢨⠇
⠀⠀⠀⠀⠀⠀⢀⣾⠁⣆⠘⢻⡄⠈⢉⣉⣉⠈⠳⡀⠀⠘⠀⠀⠀⠀⠘⠂⠀⠘⢀⣉⣉⡉⠉⢸⡟⠃⣸⠘⣿⡀
⠀⠀⠀⠀⠀⠀⣾⡟⠐⣿⠀⣼⣟⠛⣿⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣽⡟⣿⠛⣻⣧⠀⣿⠂⢻⣷
⠀⠀⠀⠀⠈⢰⣿⡇⠀⡟⠀⡏⠙⢧⣈⠓⠶⢺⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠗⠶⠚⣁⡾⠋⢹⠀⣿⠀⢸⣿⡂⠃
⠀⠀⠀⠰⠀⣸⣿⡇⠀⣷⠀⢷⡄⢦⣭⠙⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⠀⠈⠋⣭⡴⢠⡟⠀⣿⠀⢸⣿⣇⠀⠆
⠀⠀⠀⠀⢰⣿⣿⠁⠀⣹⢰⠀⠹⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⣠⠟⠀⡆⡏⠀⢈⣿⣿⡆
⠀⠰⠃⠀⢸⣿⣿⡀⠀⣿⡌⣄⣶⣄⠂⠈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡱⣠⣾⣠⢡⣿⠀⢘⣿⣿⠇⠀⠘⡆
⠀⡦⠀⠀⠈⣿⣿⡇⠀⣿⣷⠸⡇⠁⢀⡞⡱⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢠⡁⠈⣸⠇⣿⣿⠀⣸⣿⣿⠁⠀⠀⣴
⠸⣫⠂⠀⠀⢹⣿⣷⡀⠙⢿⡄⠀⣄⡏⢀⡄⠀⠀⣤⣀⣀⣀⡀⠀⣀⣀⣀⣿⡷⡀⢤⡀⢹⣴⠁⣰⡿⠋⢠⣾⣿⡏⠀⠀⠀⢸
⠐⣿⡁⠀⠀⠀⢻⣿⣷⣤⡀⠹⢦⡈⢳⣿⣁⠘⠀⠻⣤⣤⡈⠉⠉⢉⣤⣤⠟⠇⠃⢘⣿⣾⢗⡴⠏⣠⣴⣿⣿⡟⠀⠀⠀⢈⣿
⠀⠙⠏⠀⠀⠀⠀⠘⠿⢿⠿⣷⣄⡙⢦⣌⣉⡉⠓⠂⠈⠛⢿⣀⣀⡿⠛⠁⠐⠊⢉⣉⣡⡴⢋⣤⣾⣿⣿⠿⠁⠀⠀⠀⠀⠹⠉
⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣇⣉⢉⢉⠉⠀⠀⠀⣸⡇⠀⠀⠀⠉⡉⡉⣉⣼⣿⣿⣿⠛⠀⠀⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠈⠻⣧⢠⡄⠀⠀⡀⠀⠈⠉⠿⢿⣦⣤⣤⣀⣀⡤⠾⣿⡿⠶⢤⣀⣀⣤⣤⣼⣿⣿⠋⣁⣴⢀⠀⠀⢠⡀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⡇⠀⠀⢀⠀⠸⢿⣿⣿⣿⠀⣡⣄⣘⣁⣠⡎⠀⣿⣿⣿⣿⠏⢠⡈⠉⠀⢸⣷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠧⡴⠻⠿⠖⢲⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠿⣿⠟⢷⠾⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣻⢿⣿⢿⣿⣿⡿⣿⣿⡟⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⣝⠀⠋⠉⠉⠲⠋⠙⢀⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⣀⠀⠀⣀⢀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢸⠀⡏⢠⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠁⣀⣀⣀⡀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠊⠀⠀⠀⠈⢢⠀⢁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢸⠀⠀⣀⡀⠀⡸⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠉⠑⠤⠤⡑⠤⠤⠤⠖⠁⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⢀⠤⠠⢀⠔⠚⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⠢⠖⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠲⢤⠞⠁⣀⠴⠒⠒⠒⠲⢀⡴⠚⠉⠀⠈⠉⠓⢄⠱⠤⠠⠤⣄⠀⠀⡀⠠⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠊⠀⠐⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠁⠉
⠀⠀⠀⢀⡴⠒⠒⠀⠀⢀⡤⣖⠋⠩⡍⠉⠉⠓⠲⣤⡀⢀⠔⠋⠁⠀⠀⠈⠙⠦⣄
⠀⠀⢠⠏⠀⠂⠀⠀⡰⠋⠀⠃⠘⠐⣇⠀⠀⠀⣍⠁⠛⣎⠀⠀⢀⠄⠚⢆⡀⠀⠘⣦
⠀⠀⢸⠀⠀⠀⠀⠰⠁⠀⠀⠐⠐⠉⠁⠀⠀⠀⠀⠓⠀⢸⡀⠀⠈⠢⣉⠔⠁⢀⠀⢸
⠀⠀⠸⡄⠀⠀⠀⢀⠀⠀⣀⠀⡀⠀⠸⡄⡄⠀⡀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⡼
⠀⠀⠀⠙⢦⣄⡀⠸⡄⠀⡆⢂⡑⠀⠀⠣⠂⠈⠁⠀⠀⡼⠀⢤⣀⣀⣘⣑⣠⡶⠚⠁
⠀⠀⠀⠀⠀⠀⣄⠀⡙⢦⡙⠃⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⡀⢰⠀⢈⡭⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠃⡴⠈⣝⠒⠒⠒⠒⠂⡀⠀⠀⠀⠀⢱⠈⢠⡎
⠀⠀⠀⠀⠀⠀⢀⡇⢰⡇⠀⠈⢦⠀⡏⢠⠞⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⡸⠀⢸⠇⠀⠀⢸⠀⢰⠋⠀⠀⠀⠀⠀⠀⢰⠃⢸
⠀⠀⠀⠀⡤⠖⠒⠒⠶⢄⠀⠀⢸⠀⢸⢀⣄⣀⡀⢀⡤⠤⡼⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡟⠀⢸⠀⠀⠀⠉⠳⠄⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠃⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢢⡀⠀⠀⠀⢀⠇⠸⡀⠀⠀⠀⢀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢄⠀⠀⡼⠀⠀⢧⠀⠀⡰⠋⠀⡇
⠀⠀⠀⠀⠀⠀⢻⡒⠤⣀⠀⠀⠸⠀⠀⠈⠳⣠⠃⠀⠀⠘⣤⠎⠀⠀⢠⠁⠀⠀⣀⠤⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠙⠢⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠔⠋⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠉⠉⠉⠑⠒⠤⡀⠀⠀⠀⠀⢀⠎
⠀⠘⠳⣒⠒⠒⠒⠂⠤⠵⠀⠀⣠⠞⡡⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠮⠔⠒⠒⠒⠒⣨⠟⠁
⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⡔⢡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⢘⣢⠀⠀⣸⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣔⡁
⣀⡠⠤⠒⠒⠉⠁⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠒⠒⠤⢄
⠈⠉⠑⠒⠤⢤⣀⡀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⡤⠤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠛⢄
⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠔⠥⠤⠤⠒⠒⠒⡲⠀⠀⠈⠢⣑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢖⠒⠒⠒⠒⠤⠬⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡀⠈⠑⠲⠭⠍⣉⣀⠠⠤⠖⠊⠁⢀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⡤⠒⠉⡇⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠉⠒⢤⣀⠀⢣
⠀⠀⠀⠀⠀⠀⡼⠔⠊⠁⠀⠀⢠⠁⠀⢀⡔⠙⡄⠀⠀⢠⠋⠢⡀⠀⠀⡆⠀⠀⠈⠑⠢⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠋⠀⠀⢱⠀⠀⡎⠀⠀⠙⢆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠱⣤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀⠀⠀⡔⠋⠱⠤⢶⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⠒⣄⠀⠀⢀⠚⠠⠂⢀⠤⣄⠓⠛⢢⡑⢄⣆⠀⡀⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⣄⠀⡴⢶⠲⡒⠒⡖⡚⢲⠲⣆⣠⠙⠚⡗
⠀⠱⡤⡑⠼⠟⢍⡑⣄⡀⠘⠂⠊⠀⢤⡠⣵⠋⠁⢌⣀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠⣍⣘⢃⠀⢣⢸⠀⢰⢀⠀⡇⣿⣁⠀⠘⢆
⢀⠞⠀⠀⠀⠀⣠⠞⠳⠥⠀⠀⠀⢇⣠⠇⢸⡧⣵⠀⢠⠒⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⢀⡀⢹⣤⠂⢸⢀⠀⢸⡄⢠⢇⡿⢹⣱⠒⠋
⢇⠀⢀⡴⢾⡉⠙⠢⣄⣔⡇⠈⡂⢀⠤⣤⠋⠧⣈⣁⡜⠀⠀⠀⠀⢀⡀⠤⡤⠤⠤⢤⣤⣤⣀⡀⠀⠈⠓⢻⠃⣧⠜⠋⠈⠒⠞⠚⠒⠛⠛⠉⠉⠀⠀⠉
⠀⠉⠉⠀⠀⠑⠤⠴⠛⠾⣌⣉⣠⡧⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠠⠀⣌⡶⡷⠷⢶⡀⠤⣍⣣⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⡒⠲⠤⠤⡶⠤⠤⠼⠖⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⠀⡇⠀⠀⢀⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⡩⠤⢕⢤
⠀⠀⠀⠀⠀⡰⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠠⠤⠤⠧⠤⠤⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⠀⡠⢀⠑⠤⠚⠀⣣
⠀⠀⠀⠀⠀⣧⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠒⠁⠀⠢⠴⠃⢀⣀⠸⣄⡆
⠀⠀⠀⠀⠀⢏⠳⣄⠈⠓⠤⣀⡀⠀⠀⢀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⡄⠀⠐⢄⠸⣄⡨⠃⡠⠁
⠀⠀⠀⠀⠀⠘⢦⠀⠑⠢⠤⣀⣉⡉⢉⢉⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⣀⡴⠂⠈⠉⠑⠢⠤⣀⠀⠀⠈⠣⣆⠀⠁⡁⠄⣀⣰⠻⠥⣀
⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡀⠀⠈⣉⡡⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁⠀⠒⠀⠤⣠⠀⠈⠁⠀⠀⠀⠀⣀⠄⠊⠉⡄⠀⠀⠀⠀⠈⠈⡑⢟⡉⢠⠻⡖⠊
⠀⠀⠀⠀⠀⠀⠀⡼⠀⡰⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣎⣁⡘⠊⢇⡀⠀⠀⠀⠂⠀⠀⠀⠉⠓⢂⠔⠊⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠈⠉⢀⠿⣓⣦⣀⠀⠀⢀⣤⣀⠂⠀⢸⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠱⡀
⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⣞⡋⠤⡀⠀⠀⠀⠀⠀⠔⠅⠈⠀⠀⠀⢉⡹⠓⡄⠉⡆⠀⣸⠗⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠱⣄⢑
⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⢬⠑⠒⠒⢮⠒⠉⠓⢦⠒⠒⠲⣤⠐⠀⢠⠀⡾⠊⠀⠀⠀⠀⠀⣠⠔⠛⠉⡡⢈⠩⠛⢦⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣆⠠⣀⣔⠠⠤⢄⣍⣀⠀⣀⣸⠀⡠⠞⠉⠁⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣀⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢏⠠⡀⢀⡴⢋⢀⢔⢹⡏⠩⢑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠚⡟⠀⠰⡉⢁⣼⢅⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠤⣼⠗⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠈⢷⡤⢀⣷⣀⠴⡻⠉⠈⠎⣆⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠚⠉⠙⠢⡈⢆⠀⠈⣷⡖⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣄⣙⣺⣤⡤⠴⡟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠐⠚⠉⠁⢄⠙⢆⠀⠀⠀⢹⠈⣄⡴⠙⠛⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⢆
⠀⠀⠀⠀⡀⠠⣀⡀⠀⠀⠔⠉⢍⡉⠲⣌⠢⠀⠀⠀⠑⠘⣀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠳⡄
⠀⠀⠀⠐⡀⠀⠸⡈⠑⠚⠒⢀⡀⠱⠀⠈⢆⣷⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⡝⠉⠀⡨⠛⢳⠔⠣⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘
⠀⠀⠀⠀⢇⠀⣸⠽⠛⠉

⣿⡆⠀⠀⠀⢻⣿⣿⣇⠀⠀⠘⣿⣿⣇⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⠁⢀⣿⣿⡟⠀⠀⠀⣰⣿⣿⠏⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿
⣿⣿⡀⠀⠀⠈⣿⣿⣿⡀⠀⠀⢻⣿⣿⠀⠀⠘⣿⣿⡇⠀⠀⣾⣿⡏⠀⣼⣿⡟⠀⠀⠀⣰⣿⡿⠃⠀⣠⣾⣿⡿⠋⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠿⠟⠛⠋
⣿⣿⣧⠀⠀⠀⠘⣿⣿⣧⠀⠀⠘⣿⣿⡆⠀⠀⣿⣿⡇⠀⠀⣿⣿⠇⢀⣿⣿⠁⠀⠀⣰⣿⡿⠁⢀⣾⣿⡿⠋⠀⠀⢀⣤⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⣀⣠
⢿⣿⣿⣧⠀⠀⠀⠸⣿⣿⣆⠀⠀⢹⣿⣷⠀⠀⢻⣿⡇⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⢰⣿⡟⠀⣠⣿⡿⠋⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⣀⣤⣴⣾⣿⣿⣿
⠀⢻⣿⣿⣧⠀⠀⠀⠙⣿⣿⡆⠀⠀⢿⣿⡆⠀⠘⣿⣇⠀⠀⣿⣿⠀⣼⣿⠀⠀⢀⣿⡟⠀⣴⣿⠟⠀⠀⣠⣾⡿⠟⠉⠀⠀⣀⣤⣶⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠹⣿⣿⣧⡀⠀⠀⠘⣿⣿⣆⠀⠈⢿⣿⡀⠀⢻⣿⠀⠀⣿⣿⠀⣿⡟⠀⠀⣼⡿⠁⣼⣿⠃⠀⣠⣾⡿⠋⠀⠀⣠⣴⣾⣿⠿⠛⠉⠁
⣄⠀⠀⠈⢿⣿⣷⡄⠀⠀⠈⢿⣿⣆⠀⠈⢿⣷⠀⠘⣿⡇⠀⢹⣿⠀⣿⡇⠀⢠⣿⠃⣸⡿⠁⢀⣾⡿⠋⠀⢀⣴⣾⡿⠟⠉⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿
⣿⣧⡀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠹⣿⣧⡀⠈⢿⣧⠀⠹⣿⡀⠸⣿⠀⣿⡇⠀⢸⡟⢠⡿⠁⣠⡿⠋⠀⣀⣴⡿⠛⠁⠀⠀⣀⣠⣴⣾⣿⣿⡿⠿⠿⠛⠛⠛
⢿⣿⣿⣦⡀⠀⠀⠙⢿⣷⣄⠀⠀⠈⢻⣷⣄⠀⢻⣧⡀⢻⣧⠀⢿⡆⢹⡇⠀⣿⠁⣿⠁⣴⠟⠁⢀⣼⠟⠉⠀⢀⣠⣶⣿⠿⠟⠛⠉⠁⠀⢀⣀⣀⣀⣀⣀
⣀⠉⠻⣿⣿⣦⡀⠀⠀⠙⢿⣷⣄⡀⠀⠙⢿⣦⡀⠙⣷⡄⢻⣆⠸⣇⢸⡇⠀⡟⣸⠇⣼⠏⢀⣴⠟⠁⢀⣤⡾⠟⠋⢉⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣤⡈⠛⢿⣿⣶⣄⠀⠀⠉⠻⢿⣦⣀⠀⠙⠻⣦⡈⠻⣆⠹⣆⢻⡀⣇⠀⡇⡟⣸⠃⣠⠟⢁⣠⠾⢛⣩⣴⡶⠿⠛⠛⠋⠉⠉
⠙⠿⣿⣿⣶⣤⡈⠛⠿⣿⣦⣄⡀⠀⠉⠻⢷⣤⣀⠈⠻⢦⡈⢷⡜⢦⢳⢹⠀⣿⢱⢃⡼⢁⡴⣋⡥⠾⢛⣩⣤⣤⣶⡶⠾⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠉⠛⠿⣿⣷⣦⣄⣉⠛⠿⣷⣦⣄⣀⠈⠙⠳⢦⣄⡙⠲⢌⡪⡣⡷⡀⣻⠁⢊⣴⣿⣪⠵⠖⢛⣉⣉⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛
⣤⣄⡀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣶⣦⣤⣍⣙⠛⠶⠶⢤⣤⣉⣙⡒⠍⠊⠐⣥⣇⡴⠋⠊⢑⣒⣛⣉⣉⣉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣶⣶⣤⣄⣀
⠿⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠒⠒⠒⠛⢛⣛⠊⢉⠴⢛⠉⠒⠶⠶⣤⣭⣭⣉⣉⠙⠛⠛⠷⢶⣶⣤⣄⣀⠀⠀⠉⠙⠻⠿⣿⣿⣶
⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⠶⠶⠶⠶⠒⠛⣉⡥⣒⣾⠏⡔⠀⡇⣦⡢⣍⠲⢍⡙⠻⣯⣛⠷⢶⣤⣀⡀⠉⠛⠻⢿⣷⣦⣄⡀⠀⠀⠉⠻
⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠖⢛⡥⢞⡕⣡⢣⠃⡇⣿⠰⡹⣌⠳⣄⠙⠷⣄⠉⠻⢶⣬⡙⠻⣷⣦⣄⡀⠈⠙⠿⣿⣷⣦⣄
⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⠿⠿⠛⠛⢉⣡⣴⠞⣫⡴⢋⡴⢃⡟⢰⠇⣿⡆⢳⠘⣦⠘⢷⡀⠈⢳⣦⡀⠙⠻⣷⣄⠙⠻⣿⣶⣄⡀⠈⠙⠿⣿⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣤⣤⣶⠿⠛⣉⣴⠞⠋⣠⡟⢁⡾⠁⣾⠀⣿⣿⠘⣧⠘⣧⠀⢻⣦⠀⠙⢿⣆⠀⠈⠻⣿⣦⡈⠙⢿⣿⣦⣄⠀⠈⠻
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠟⠛⢉⣠⣴⡿⠛⠁⣠⡾⠋⢠⡿⠁⢰⡏⠀⣿⣿⡀⢿⡄⠹⣧⠀⠻⣧⡀⠈⢻⣷⡄⠀⠈⢿⣿⣦⠀⠙⢿⣿⣷⣄
⠛⠛⠛⠛⠛⠛⠉⠉⠉⢁⣀⣠⣤⣶⣿⠿⠛⠁⠀⣠⣾⠟⠁⣰⡿⠁⢀⣿⠁⢰⡟⣼⡇⢸⣧⠀⢻⣧⠀⢻⣷⡀⠀⠹⣿⣆⠀⠀⠙⣿⣷⣄⠀⠙⢿⣿⣷
⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⡿⠟⠛⠉⠀⠀⣀⣴⣿⠟⠁⣠⣾⠟⠀⢀⣾⠇⠀⣾⡇⣿⡇⢸⣿⠀⠈⣿⡆⠀⢿⣷⡀⠀⠹⣿⣧⠀⠀⠈⢿⣿⣦⠀⠀⠻⣿
⣿⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⣀⣴⣾⡿⠛⠁⢀⣴⡿⠋⠀⢀⣾⡏⠀⣰⣿⠀⣿⡇⠘⣿⡆⠀⢹⣿⡀⠘⣿⣧⠀⠀⠹⣿⣧⠀⠀⠈⢻⣿⣷⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⠟⠉⠀⣠⣶⣿⠟⠁⠀⣠⣿⡟⠀⢠⣿⡇⢸⣿⠇⢰⣿⡇⠀⠈⣿⣇⠀⢹⣿⡆⠀⠀⢻⣿⣧⠀⠀⠀⢻⣿⣿⡄
⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⡿⠛⠉⠀⢀⣴⣾⣿⠟⠁⠀⢀⣴⣿⠏⠀⢠⣿⡟⠀⣾⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠸⣿⣷⠀⠀⠈⣿⣿⣇⠀⠀⠀⢿⣿⣿⡄
⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⣀⣴⣾⣿⡿⠛⠁⠀⠀⣠⣾⣿⠋⠀⢠⣾⣿⠁⢠⣿⡟⠀⣸⣿⡇⠀⠀⢻⣿⡇⠀⣿⣿⡇⠀⠀⢸⣿⣿⡄⠀⠀⠈⣿⣿⣿
⠛⠛⠉⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠋⠀⠀⠀⣠⣾⣿⡟⠁⠀⣠⣿⣿⠃⠀⣾⣿⠃⠀⣿⣿⠃⠀⠀⢸⣿⡇⠀⣿⣿⣧⠀⠀⠀⣿⣿⣷⠀⠀⠀⠸⣿⣿
⠀⢀⣀⣤⣴⣾⣿⣿⣿⡿⠟⠋⠀⠀⠀⢀⣠⣾⣿⡿⠋⠀⢀⣴⣿⡿⠃⠀⣼⣿⡟⠀⢸⣿⣿⠀⠀⠀⣾⣿⡇⠀⢹⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⣠⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⡟⠁⠀⣰⣿⣿⠁⠀⣾⣿⡟⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⡀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⢀⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡅⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⢦
⠀⠀⠀⠀⠀⠀⢮⣀⣄⣄⡼⠀⠀⠀⠀⠀⠀⡰⠣⠭⢽⡀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠦⡀⠀⠀⠀⠀⢀⠷⠔⠒⢺⡄
⠀⠀⠀⠀⠀⠀⠈⢟⠢⡱⠁⠀⠀⠀⠀⠀⠀⣗⢒⠒⢒⡇⠀⠀⠀⠀⠪⣀⢤⣀⡠⣀⡼⠀⠀⠀⠀⠸⡀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢸⠁⠀⠀⠀⠀⠀⠸⣴⣤⣦⠇⠀⠀⠀⠀⠀⠀⠈⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣱⠦⡀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⡀⠀⠀⠀⠀⢀⡠⠚⣧⣼⣦⠀⠀⠀⠀⠀⡰⠁⠉⠉⡖⠁⠀⠀⠀⠀⠀⡴⠷⠞⢄
⠀⠀⠀⠀⢰⣫⢤⣀⠤⣄⡠⢌⣲⠀⠀⠀⣏⣩⣭⣤⣤⣤⡇⠀⠀⠀⠀⠳⡒⠒⢲⠞⠀⠀⠀⠀⢀⣠⣏⣙⣉⣹⣄⡀
⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠯⠭⠭⠭⠤⠴⠃⠀⠀⠀⠀⠀⢹⢻⠏⠀⠀⠀⠀⠀⠈⠑⠒⣢⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠽⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠤⠶⠭⡉⠑⣄⠀⠀⠀⠀⢰⠣⢢⢔⢮⢣⠀⠀⠀⠀⠀⡔⠉⠀⠈⠱⡄⠀⠀⠀⠀⢸⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⡏⡞⡉⠳⡈⡆⢸⠀⠀⠀⠀⠸⡰⣐⣾⣮⡾⠀⠀⠀⠀⢰⠳⡖⢶⢲⡖⣷⠀⠀⠀⠀⢸⣿⣿⡛⢻⡇
⠀⠀⠀⠀⠀⠱⣱⢌⣒⣁⠷⠊⠀⠀⠀⠀⠀⠉⢲⣿⠃⠀⠀⠀⠀⠀⠈⣇⣧⢸⢸⢧⡏⠀⠀⠀⠀⢸⠋⠛⠻⠿⣯
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡔⠋⠀⢙⠲⡀⠀⠀⠀⠀⠀⠀⢀⣠⡓⢤⠀⠀⠀⠀⠀⠀⡞⡁⡙⡆⠀⠀⠀⠀⠀⠀⣀⣄⣤⣀
⠀⠀⠀⠀⠀⡜⠡⠆⠀⠈⠀⢳⠀⠀⠀⠀⠀⡴⡏⢣⠈⡞⠁⠀⠀⠀⠀⡖⠚⠈⠘⠒⡆⠀⠀⠀⣰⡎⢱⡀⢀⠎⢑⡆
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠙⢀⠎⠀⠀⠀⠐⣮⠣⣸⡤⠋⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢏⠀⠀⠀⠀⠙⢌⡏⠙⠋⢹⣡⠃
⠀⠀⠀⠀⠀⠀⠈⠙⣾⠉⠁⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠋⠳⠬⠇

⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠲⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⢀⢏⣤⣘⡿⠓⠶⣦⡀⠀⠀⠀⣸⢁⣤⡘⠏⠉⢳⡆⠀⠀⠀⢀⣴⠏⢀⢀⣽⣦⣄
⠀⠀⠀⠀⠀⣾⣾⡷⠿⠀⠈⣠⣿⣧⠀⠀⠀⣿⣨⣍⣀⣦⣡⡾⣷⠀⠀⠠⡏⢁⣴⠞⠛⠁⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢻⣿⣿⣿⡇⠀⠀⠈⠉⠀⠉⠻⣿⣷⢿⣷⠀⠀⠛⠛⠁⣠⢤⣄⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢹⣿⠏⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡿⠀⠀⢠⣿⠀⠉⠚⢿⣿⡏⢹⡇
⠀⠀⠀⠀⠀⣴⠀⠀⠀⣰⠟⣱⡿⠋⠀⠀⠀⣰⡆⠀⠀⢰⠏⢠⡿⠁⠀⠀⢿⡇⠀⠀⠀⠀⡟⠀⢸⡇
⠀⠀⠀⠀⢸⢿⣄⠀⠔⣡⡾⠋⠀⠀⠀⠀⠀⢿⡄⠀⠀⠈⣠⡟⠀⠀⠀⠀⡎⢷⡀⠀⠀⡠⠇⢠⡿⠁
⠀⠀⠀⠀⠘⠳⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠳⢬⣤⣤⠾⠋⠀⠀⠀⠀⠀⠘⢤⣉⣁⣀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡤⠶⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣁
⠀⠀⠀⠀⣴⡏⠙⣦⡴⠿⢿⣶⣦⡀⠀⣰⠏⠉⡻⠿⠿⡏⠉⡙⣷⡄⠀⢀⣾⠋⢁⣄⠈⠛⠿⣦⡀
⠀⠀⠀⠀⢹⣿⢺⣿⣿⠦⣾⠟⢿⡇⠀⢿⡀⠀⢰⣶⣶⡌⠲⢶⣿⣿⠀⢸⠁⢸⡟⠋⠛⠛⠶⠈⣷⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⡼⠃⣸⣿⡀⠀⠹⣦⣼⣿⣿⣷⡴⣀⠈⣿⡇⠘⣗⠻⣿⣴⠃⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠈⠻⠿⢹⣿⣡⣾⠁⢹⡇⠀⠀⠀⠻⣿⣿⣿⣟⢉⣴⣿⡇⠀⠙⠺⠋⡽⢿⠋⢠⣀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⣾⠇⠀⠀⠀⠀⠀⢸⣿⠟⢿⣿⣿⠁⠀⣼⠃⠸⡇⣤⣴⣿⠏⢰⡿
⠀⠀⠀⠀⣴⠃⠀⠀⠀⢘⣿⡄⢰⡟⠀⠀⢠⡴⠒⠀⠀⡏⠀⣾⣿⠏⠀⢸⣇⠀⠀⠑⡀⢹⡏⠀⣼⠇
⠀⠀⠀⠀⠸⡄⠰⣄⠀⣼⠟⣰⡟⠁⠀⠀⣿⡁⣦⢀⣼⠃⣴⣿⠏⠀⠀⠸⡉⢲⣀⡀⢀⡼⠃⢰⡟
⠀⠀⠀⠆⠀⢄⣀⣿⣊⣵⠾⠋⠀⠀⠰⠂⢮⣿⣿⣿⣧⣾⠟⠁⠀⠀⠀⠄⠱⣘⠛⠛⠋⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⣴⠚⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣎⢉⡷⣄⡀⠀⠀⠀⠀⠀⠀⡤⠶⢶⣦⣄
⠀⠀⠀⠀⣠⡾⠁⣠⣾⣿⣷⣤⡀⠀⠀⠀⢠⡴⣿⣿⣷⣤⠞⠻⣆⠀⠀⠀⠀⠀⣀⡀⠀⢹⣿⡇
⠀⠀⠀⣾⠃⣤⣄⠉⢸⡏⢁⡄⠹⣦⡀⠀⣿⡹⣿⡟⣿⠄⢠⣶⣿⡄⠀⠀⠀⢾⣭⣿⣦⡀⠻⡏
⠀⠀⠀⢿⡸⣿⣿⠀⣸⣧⣾⣁⠀⣿⣧⠀⠘⢷⣬⡶⢿⣄⣼⢻⣿⡇⠀⠀⠀⣸⣿⣿⣾⣿⣾⡇
⠀⠀⠀⠀⠹⣄⣰⡾⠋⠉⠉⢛⣷⣿⣿⡄⠀⠘⢧⢠⡿⠋⢁⡞⠙⣷⡀⠀⢶⡍⣠⡾⠋⠁⢻⣧
⠀⠀⠀⠀⠀⠙⠋⢀⣠⡴⠚⣉⢽⣿⣿⡇⠀⠀⠀⠛⣁⠴⠋⣠⠀⢹⣧⠀⣰⠟⣾⣇⣸⣷⣼⣿⣆
⠀⠀⠀⠀⢀⡴⠂⣿⠀⣀⣀⣤⡆⢀⠃⣤⠀⠀⠒⢳⡿⢶⣄⠙⠀⠀⣿⡆⠀⠀⣿⠙⠿⣿⣿⣿⣿⡀
⠀⠀⠀⢠⠏⠀⠀⣷⠘⠋⢻⣿⠃⡌⢠⣿⠀⢸⣤⣿⠀⠀⠙⢷⣤⣼⣿⣇⠀⠀⣿⡀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⢼⠀⢀⡄⢻⡀⠀⣸⣿⢀⠆⢸⡟⠀⠚⣿⣿⡄⠀⠀⢠⡀⣺⣿⡟⠀⠀⠹⣧⡀⣿⣿⣿⣿⡇
⠀⠀⠀⡜⣇⠘⣦⣌⡉⢿⣿⢏⡌⣡⡿⠃⠀⠀⡏⠙⠁⣰⡇⠸⣇⣾⣿⠧⢄⠸⣷⣹⡟⡛⣿⣿⣿⠇
⠀⠀⢤⠘⣎⡀⠀⡴⠏⣼⣿⣿⣿⠟⠁⠀⠠⡄⠁⣤⣾⣿⣿⣾⣿⡿⠋⠀⢈⠁⠈⣋⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠈⠙⠲⠶⠾⠟⠛⠋⠁⠀⠀⠀⠀⠀⢤⣾⡿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠐⠐⠸⠟⠋

⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⣀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣸⣿⣿⣿⣿⣿⢿⣿⡏⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⣿⣿⣿⣿⠃⠸⡿⣅⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⢸⣿⣿⣿⣿⠋⠰⠁⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⣀⣤⠄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣸⣿⠛⢿⣿⣿⣿⣿⣥⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠁⠀⠀⠀⠈⠃⠀⠀⠉⠉⠻⢿⣿⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠖
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣶⠄
⣦⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⣾⣿⣿⠷
⣷⣶⣶⣾⣿⣿⣦⣤⣤⣴⣯⣤⣤⣤⣤⣤⣼⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣭⣭⣬⣭⣯⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆⢹⣿⣿⣿⠠⠏⢀⣼⣿⣡⠏⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣿⣿⣿⣿⠃⠀⣾⣿⠘⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢸⡇⢸⣽⣿⡏⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⡠⣾⡟⠀⠀⣿⣿⣿⠏⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣳⠀⡇⠈⣿⣿⠀⠀⢺⣿⡏⠀⠀⠀⠀⠀⠀⣠⠞⣱⡟⠀⠀⠀⣟⠻⠏⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⢇⢂⣿⣿⡄⢀⠀⢸⡆⠀⠀⠀⠀⠀⣰⡏⣾⣿⡅⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⡜⢿⣿⣄⢢⠀⠣⠀⠀⠀⠀⠀⢿⡅⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣻⣷⠘⢮⡻⣿⣦⡱⡄⠁⠀⠀⠀⠀⠸⣧⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠿⠟⠃⠀⠳⣌⠻⣿⣾⢆⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣂⣷⠀⠀⠀⠀⠀⠀⢹⡄⠈⢻⣯⢆⠀⠀⠀⠀⣸⣿⣿⣿⣯⢷⡕⢤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⣼⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡙⠷⣝⡢⡀⠀⠀⠀⠀⠙⢿⣷⠦⢄⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⢀⡜⠀⠀⠀⣿⣟⠀⠀⢀⣾⣿⣿⣿⡿⢻⣿⣿⣷⡈⠹⢮⣦⡀⠀⠀⠀⠀⠙⢄⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠞⠀⠀⠀⢠⣿⡆⠀⢠⣿⣿⣵⣿⣯⣤⣌⢿⣿⣿⣇⣀⣈⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⢠⣿⡟⠀⠀⣾⣯⣾⣿⣿⣷⣄⠙⣮⢿⣿⣿⣿⣿⣿⣷⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⣰⢟⡿⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠘⡎⢿⣿⣿⣿⣿⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠈⢿⣿⣿⡁⠀⠁⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣀⣤⣴⣶⡾⠿⠿⠛⠛⠋⠉⠀⠀⠀⣠⣾⠟⣿⠋⠀⠈⠉⠛⣿⣿⣿⠀⠀⠈⣿⣿⣿⢀⠀⠻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⣋⣙⣻⣤⣶⣲⢶⣦⡀⠘⣿⣿⡇⠀⠀⠘⣿⡇⠀⠀⢠⠈⠉⠈⠙⡄
⠀⠀⠀⠀⠀⣰⠟⢁⣼⣿⣿⠟⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⡇⠀⠀⢠⣿⣶⣷⡀
⠀⢀⠀⠀⣰⡏⢠⣿⣿⠟⠁⠀⠀⠀⠀⣠⣶⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠇⠀⠀⠀⣿⣿⣿⣿⡆
⠀⡘⠀⢰⡟⢠⣿⣿⠃⠀⠀⠀⢀⣠⣾⢟⣩⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠙⣿⣿⣿⣿⡇⢠⡿⠀⠀⠀⠀⠙⠛⣻⠆⠛
⠀⡇⠀⣿⠃⣼⣿⠇⠀⠀⢀⣴⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃⡼⠃⠀⠀⠀⠀⠀⠀⠛
⠀⡇⠀⣿⠀⣿⣿⠀⠀⣰⣿⠟⢡⣾⣿⣿⣿⣿⣿⣿⠟⣁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡇⠀⣿⢸⣿⣿⠀⢸⣿⠏⢀⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⡴⠀⠀⣀⡀⠀⠀⠀⣀⠀⢸
⠀⡇⠀⡏⢸⣿⣿⠀⣼⡇⠀⣸⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⡟⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⢎⣤⠞⠋⠉⢻⣴⣶⣯⢽⣤⡼
⣴⠃⢰⠇⠀⣿⣿⡄⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠇⠀⣠⡾⠟⠛⢻⡄⠘⣿⣿⠏⠁⠀⠀⠀⠈⠉⠉⠀⠀⣿⠏
⠏⠀⠸⠀⠀⢹⣿⡇⣿⠀⣸⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⡿⠀⡾⠋⠀⠀⠀⢸⣿⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⠛⠃⣸⣇⠀⠀⠀⠀⠀⠉⠻⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⣿⣻⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣴⢻⣿⣧⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢀⠈⣜⣾⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠖⡀⠀⠐⠂⠤⡀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠑⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠉⠁⠀⠀⠈⠁⠀⠓⠠⢤⠂⠀⠈⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⡐
⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⡠⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠑⢄⠔
⢀⠔⠈⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⡰⠟⠛⠛⠛⠀⠀⠀⠀⠙⠛⠛⠻⢦⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠃⠂⠉⠁⠢⡀⠀⠀⠌⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⡀⠀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢡⠀⠁⠂⠀⠀⠊⢀
⠀⠀⠀⠀⠀⠱⡀⠈⠀⠀⠀⣤⣾⣿⢫⠃⣰⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⡏⡏⢉⣷⣦⡀⠀⠀⠃⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠈⢻⣏⠇⢠⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⡿⡜⠀⣾⣿⠉⠀⠀⠀⢠⠀⢀⠠⠐⠁
⠀⠀⠀⠀⢀⠠⠄⡇⠀⠀⠀⠀⠈⠻⣄⣾⣿⣿⣿⡿⠋⠀⠀⠈⠿⣿⣿⣹⠁⣼⡿⠃⠀⠀⠀⠀⠸⠉
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⢂
⠀⡔⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠰⣶⣦⡐⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⣠⣶⡞⠀⠀⠀⠀⢀⠂⠀⠡
⠈⢀⠂⠈⠁⠐⠢⠀⢱⡀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⡶⠶⠶⠶⢶⣶⣾⣿⠿⠋⠀⠀⠀⠀⢀⠎⠀⠀⠀⠑⠂⠄⠐
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠉⠛⠡⠤⠀⠀⠠⠤⠘⠋⠁⠀⠀⠀⠀⠀⢠⠮⢀⡀⠀⠀⢀⠠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢀⠤⠀⠀⠂⠁⠉⠐⠂⠀⠠⠤⠤⠄⠠⠀⠐⢅
⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⢄⠀⠀⠀⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠘⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣬⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡠⡔⣶⣴⣶⠒⠆⠀⠒⠔⠀⠈⠀⠀⠎⠈⢡⠤⠠⠀⠄⠀⡸⠁⠊⢀⠀⡀⡀⢀⠤⠀⢚⡶⢄
⠀⠀⠙⣠⢿⡟⢻⡃⠍⢍⡁⢦⣀⣰⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡀⠀⠀⠶⠀⠈⠀⠀⠀⠁⣎⣦⠞⠁⢻⣷
⠀⠀⠀⠾⠼⠥⠞⣀⡾⣆⠳⡌⣿⡼⠀⠀⠀⠀⠇⠈⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⢳⡄⠀⠀⠀⠀⣟⣿⣆⡸⢻⣿⡆
⠀⠀⡆⣷⡆⢰⣿⣿⣁⢈⣳⣿⣿⣷⣤⡰⠀⠀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⣸⠖⢺⣏⡇
⠀⠀⢻⣿⠁⠀⠈⡻⢿⡼⢿⣥⣿⣿⡍⡀⠀⡄⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣆⣹⠀⠀⠀⠀⠀⣿⢙⣿⢡⣴⣾⠁
⠀⠀⠀⠩⣶⣿⣲⣻⡿⠁⠀⠋⢿⣽⣷⠃⠀⠀⠀⠐⣄⣷⠀⠀⠀⠀⢀⠈⢳⣦⡙⠀⠀⠀⢀⠲⣿⣻⡏⠀⣽⡝
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⠆⠀⠀⠈⠂⠳⣤⣄⠂⠈⢷⣶⣝⠀⠀⠀⠀⢀⣱⣿⣿⣷⠶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣴⠀⠀⠀⢐⣻⣿⣿⣧⡀⢰⣮⠉⠐⢄⠀⠀⠖⢿⣿⣿⢿⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢸⣟⣻⣿⣛⠀⠉⠀⠀⠈⢀⠀⢀⡐⣿⠻⣷⣿⢈⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢙⣯⣷⠸⠃⠀⠀⠀⠀⠀⠈⠠⣩⣤⡿⢇⠁⢽⡿⣀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢻⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣸⡆⠈⣼⣬⠙⠀⣰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣠⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠇⣹⡇⠀⠙⠃⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢾⡟⡁⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠜⣶⡿⠀⠀⠀⠀⠈⠓⢼⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡾⠧⠇⢈⣽⠀⠀⠀⠀⠀⠀⠠⠂⠀⢰⣿⡇⠀⠈⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠛⠂⠰⠒⠛⠋⠀⠀⠀⠀⠀⠀⠓⠒⠉⣉⣭⣬⣿⣿⣿⣿⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⠀⠀⢀⢀⣠⣆⣀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠁⡚⠋⢽⣶⣳⠀⡍⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡐⣨⠀⠄⠀⠦⠈⠘⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢸⢰⠑⠶⠻⣄⣧⡄⢸⣧⠟⡷⠘⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠁⠆⠸⠀⠀⠄⠁⣿⡿⠀⠀⠩⣿⠁⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⡀⠖⠂⠘⠉⠇⠀⡀⢀⡷⡆⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡏⢐⡀⠐⡄⠀⠛⣤⠉⡀⠀⠈⠻⣬⠊⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠠⡕⠇⠈⠁⡀⢳⣆⠀⠀⠀⠈⢠⠀⠐⠛⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣁⠰⠺⠀⡗⡆⠀⠀⠑⡀⠙⢿⣄⣄⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠀⠉⠀⠆⡄⠁⠇⠀⠘⢧⡈⠢⡀⠙⢎⡦⣀⠁⣠⠄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠢⢀⠀⠀⢰⠀⢱⢰⠀⠸⡆⠀⠹⣄⠈⠀⠀⠙⢢⡽⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄⣦⠀⠃⠀⠀⣆⢰⣾⡇⠄⠰⡀⠀⠈⠻⣄⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠘⣧⢻⣆⠂⠹⡀⠙⢮⣿⢁⠀⠀⠙⠢⡀⡴⠉⠀⠀⠀⢠⠔⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠳⡄⠸⡆⢇⠰⡀⠣⢃⠈⢧⡈⢄⠀⠀⠀⠈⠀⠀⠀⠀⣠⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⡀⠀⢻⡄⠀⢸⡆⠱⡀⠈⠣⡀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠘⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⡂⠈⠒⢌⠑⠌⠂⣳⠀⢧⠀⠐⣥⠀⠀⠀⠀⠀⡀⠀⠀⡴⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠤⠽⠒⢎⡳⠭⠀⢠⣈⣆⠘⣆⡀⠀⠁⠀⢦⡄⠀⠁⡀⢸⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣊⡑⠶⠦⢍⠳⠌⡖⠈⠿⣤⣀⠱⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣀⣤⠹⠟⠲⠤⢆⠀⠀⠀⠉⠻⣄⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢀⣠⣤⡤⣤⡶⢤⣤⣤⠶⡏⢵⢉⠔⣩⡶⠀⠒⠒⣾⠀⠀⣸⢦⣀⡈⠓⠂⠀⠰⠄⠀⠀⢠⠀⢰⡀
⠀⠘⠄⠰⠏⠁⢯⠀⢇⠀⠳⠘⣏⢸⣯⠄⠀⠀⠉⠁⠀⠀⠁⠀⣿⢁⣪⠍⠉⡄⠀⠀⠀⠀⠀⠈⠣⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠔⠀⠤⠶⠀⢀⠀⡀⠀⠁⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⡀