Chemistry Emojis & Symbols

⠀⠀⠀⢀⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⡷⣧⣤⣝⣶⠃⠀⠀⠀⠀⠀⠀⠀⣀⠛⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠰⠀⣰⣁
⢀⡜⠁⢰⣿⣞⠈⢦⡀⠀⠀⠀⠀⢐⣶⡿⠁⠀⠀⠀⠀⠈⠉⠙⢿⣷⣼⣿⡆⠀⠀⠀⠀⠀⠀⡼⡿⢿⠃⠀⠆
⠮⢴⣎⣀⣉⣀⣱⣆⣳⠀⠀⠀⢦⣾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢴⠦⠄
⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⢆⠀⠀⠀⣄⠀⠀⠀⡠⠋⢹⣿⡟⠑⢄⠀⠀⠀⠀⠀⢀
⠘⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠁⠀⠀⡇⠀⡱⠀⠀⣀⠀⠀⠀⢁⣤⣄⣽⣡⣤⣌⡀⠀⠀⠀⠀⠸⣷⠆
⠀⠈⠻⣿⣿⣿⣿⡏⠀⠀⠀⠈⠑⠁⠈⠲⠀⠀⠁⠉⠀⠀⠀⠺⡿⠋⠀⠙⠿⡟⠀⠀⠀⠀⢀⣤⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠐⠊⠁⠀⠀⠀⠀⠸⠿⣿⡿⠗
⠀⠀⠀⠀⡄
⠀⠀⣶⠀⡄⠀⠔⡀⠀⠀⠀⢀⣘⠁⣤⠀⠀⠀⠴⠀⠀⠀⠀⠀⠐⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣷⠀⡇⠀⠰⠇⠀⠀⠀⠘⠛⢄⣀⠶⠀⡶⠀⡀⠀⠀⠀⠀⠰⠿⠿⢆⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣶⠦
⠀⠐⠛⠂⡃⠀⢰⡆⠀⠀⠀⢀⣤⠜⢻⠂⠘⠛⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣰⣄⣀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠉⢀⡘⠗⠀⠀⠁⠀⠀⠀⠀⢄⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⡆
⠉⠉⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢶⣤⣴⣧⣄⠀⠀⠀⠀⠀⠀⢀⠉⢁⡘⠙⣀⠛⢃⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣆⠀⠀⠀⢀⢞⡟⠉⠉⢙⠲
⠀⢰⣏⣠⣟⣀⡀⠀⠀⠀⠀⠸⠉⠉⠉⠈⠉⠁⠘⡃⠀⠀⠀⠠⣾⣷⡇⠀⣸⣿⠀⠀⠀⣤⠞⠳⠄⠰⠚⠳⡇
⠀⠀⢉⣿⡋⢉⢹⠆⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠂⠂⢿⣿⣿⠿⠃⠀⠀⠀⠠⡀⡀⢸⡀⣠⠀⠄
⠀⠀⠀⠙⠛⢻⡟⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣠⣶⣾⣿⣿⣶⣶⠀⠀⠀⠀⠈⠛⠋⠉⠑⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⢀⣞⢿⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢾⣽⣶⣶⢳⣶⣿⣿⡇⠀⡀⠀⠀⠀⠀⢀⣤⢾⠉⠁⣠⣄⣨⢽⡶⢄⠀⠀⠀⠀⠸⣟⣩⠇⠀⠀⠀⠀⠓⠈⢸⣀⣀⣀⣀⣀⣴⣭⠥⠚
⠀⠀⠀⢰⠀⠀⠀⣷⣭⠉⠉⠀⣘⣿⡇⠀⠉⠀⠀⠀⢠⢾⡵⡏⠀⠸⣿⣿⣿⢿⣧⡈⢳⠀⣄⡀⠀⠈⠁⠀⠀⠀⠀⣠⠶⠛⠋⠿⣌⠀⠀⡤⠀⣠⡀
⠀⠀⠀⠈⠀⠀⠀⡏⠙⠀⠀⠀⢸⣚⡇⠀⠀⠀⠀⠀⣞⣋⣤⡽⠀⠀⠿⢯⣔⣓⣠⠟⠀⡇⠈⠹⠀⠀⠀⢀⣀⡀⣀⠹⡀⠂⠈⠀⢈⠑⢄⠑⠀⠉⠰⠇
⠀⠀⠀⠀⠀⠀⡼⠉⢳⠀⠀⠀⠚⠃⣇⠀⠀⠀⠀⠀⢏⠁⢀⣴⡶⣤⡀⠀⠈⠉⢁⣴⣶⠇⠀⠀⠀⠀⠀⠘⠉⣁⣹⣅⣘⢦⡀⠀⡀⠠⡈⢧
⠀⠀⠀⠀⠀⢸⣡⣶⣷⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⡜⢆⠻⣅⣴⣆⡇⠀⣠⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠀⣀⠭⠛⢷⣤⣤⡤⠎⠲⠄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠈⠂⠈⠓⢬⣻⡛⣃⣀⣿⡿⠟⠁⠀⠀⠀⠀⣀⣠⢠⠄⣷⣄⡀⠉⠄⠀⠨⠀⢱
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⢖⣛⡲⡄⠀⠀⣀⣀⠉⠉⠉⠀⢀⠔⢤⠀⠀⠀⠀⠉⠁⣨⠷⠛⡉⠍⡻⣍⠙⠛⠛⠉⠀⢀
⠀⠀⠀⠀⠀⠻⣿⣛⣛⣻⣿⣿⣿⡿⠿⠋⣾⢼⠦⢇⡇⠀⠘⢥⡾⠃⠀⠀⠀⠈⠲⠛⠀⢠⡚⣦⠀⠊⠁⠂⠁⠀⠀⠀⢈⡇⠀⠀⠀⠟⠻⠰⠆⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠈⠑⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⢀⣀⡀⠃⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⡔⠉⠀⠀⣆⣀⡇⠀⣏⣉⠁⣰⡄⣴⡀⠀⡆⠀⢯⡉⠀⠉⣿⠉⢸⣠⠞⠀⠘⠞⠁⠀⠒⠒⢿⣷⣾⠆
⠀⠀⠀⠀⢠⣀⡄⢠⡀⣤⢀⠀⡀⢀⢉⠉⠀⠸⣀⢀⡀⠐⠇⠀⠸⠀⠻⠤⠀⠃⠛⠁⠃⠀⠇⠀⠰⠛⠀⠀⠙⠀⠘⠀⠉⠀⠀⠂⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠸⠁⠇⠹⠀⠛⠘⠀⠉⣿⠻⠸⡇⣀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠐⠃⢠⡟⠛⡇⠀⠀⢸⠿⢷⠀⠃⠀⢸⠛⠛⡇⠙⠀⠀⠀⠀⠴⠀⣶⢻⡿⠿⠿⠟⠻⢷⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠁⠁⠭⢉⠳⣴⡒⢲⠄⠀⠀⠀⢸⡇⠀⡃⠀⠀⢸⠀⢸⠀⠀⠀⢸⣤⣤⡇⢀⠀⠀⠀⠀⠀⠀⡟⠉⡇⠀⠀⠠⡣⢼
⠀⠀⠀⠀⠀⢊⠘⢀⡎⠀⠀⠀⠀⠀⠀⣡⣴⣸⡖⠋⠀⠀⠀⡇⢰⣟⣿⡇⠀⠀⢸⣀⣸⠀⠀⠀⢨⣿⣿⡇⢸⡀⠀⢠⣄⡄⠀⣧⣄⡗⠀⠀⢀⡧⢸
⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⢀⠠⠄⣂⡥⢖⢛⠽⡇⠀⠀⠀⠛⡟⠛⣿⣿⡿⠛⠛⠛⣿⣿⠛⠛⠛⢻⣿⣿⡟⢻⠋⠀⠈⠿⠁⠀⣇⣤⣧⣤⣶⣾⣶⣾
⠀⠀⠀⠀⠀⣰⢯⢻⣄⡬⠴⠒⠉⠁⠀⢰⢎⡼⠁⠀⠀⠀⠀⡇⠀⣿⣷⣇⠀⠀⠀⣿⣿⠀⠀⣀⣨⣿⣿⣇⣼⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠋⠉⠙⠦⣄⣀⣀⣀⣀⠴⠋⣠⣤⣤⡀⠀⠤⣿⠽⠿⠛⠛⠒⠒⠒⠛⠋⠉⠉⠉⠉⠉⠀⠀⠙⠀⠀⣴⡄⠀⠀⠉⠛⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣉⣩⡀⠀⠀⣇⠻⣪⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣹⠃⠀⢀⣀⡋⠉⣟⡆⠀⠀⡄⢀⡤⠴⠧⣄⣀⣀
⠀⠀⠀⠀⢀⡐⠀⠀⣤⠤⠶⣿⣛⣛⡟⠉⣻⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⢹⠀⠀⠀⠈⢁⡴⠞⠋⠤⣹⢦⢸⠀⠀⠠⢰⠋⣀⠀⡈⠜⣿⠤
⠀⠀⠀⠀⠀⠀⣠⢴⠷⠾⢷⠖⢾⠉⠀⠄⢹⠀⠀⠀⣸⠀⡤⠖⢦⣀⣤⣤⣴⡁⠀⣹⠃⠀⠋⠊⡼⠛⢀⡴⠚⢹⠁⢊⡇⠀⠀⠤⢼⠀⣏⡉⣹⢆⣼⠧
⠀⠀⠀⠀⠀⠀⣧⠘⠒⠖⠋⠀⢸⠀⠄⠀⢸⠀⠀⠀⠘⠲⣇⠀⢠⠟⠉⠁⠀⠙⠋⠁⠀⠀⠀⡼⠁⡰⠃⠀⣀⡇⠠⣿⠀⠀⠀⠐⠘⣆⢹⢣⣽⣼⠟⠄
⠀⠀⠀⠀⠀⠀⡗⠐⠂⠀⠀⠀⢸⠀⠁⡂⢸⠀⠘⣤⡆⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡇⢸⠃⠀⠀⢿⡇⢸⡟⡿⠁⠀⠀⠁⠀⢻⣇⣿⠉⠂
⠀⠀⠀⠀⢀⠀⡇⠀⠒⠀⢀⡀⢸⡀⠀⠤⠈⡇⠀⠙⠀⢀⣴⣾⠇⠀⣠⣶⡆⠀⣀⣀⠀⠀⢸⡇⠈⢳⠤⣀⣀⣹⣋⠉⠀⠀⠀⠀⠀⠀⣼⣼⣿
⠀⠀⠀⠀⠚⠁⡇⠀⠀⠀⠀⠀⠸⡧⠴⠖⠚⠁⠀⢀⣤⣞⠛⠛⠒⠲⣿⡿⠤⠿⠾⠼⠁⠀⠘⣧⠸⣯⡇⠀⠀⠠⣭⢉⡇⠀⠀⠀⡀⣀⣿⣿⣿⠀⠄⠄
⠀⠀⠀⠀⠀⠀⣣⡤⠖⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠁⠹⣿⣷⡀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠷⢶⣦⣤⣤⣤⣼⠟⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣒⣚⣉⣉⣉⣍⣉⣉⣙⣒⣒⣶⡦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠰⢞⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣷⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠓⠶⠤⠭⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠤⠴⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⡶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⣴⡶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⡀⢠⠀⠀⠠⠄⠠⠀⠄⠠⠄⠠⠀⣉⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡾⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠎⠑⠠⢤⣤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠤⠄⠀⠀⠈⢧⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠋⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡱⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡜⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡼⠁⠀⠀⠀⠀⢯⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡜⢦
⠀⠀⠀⠀⠀⠀⣠⢏⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢧⡀
⠀⠀⠀⠀⠀⡰⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢳⡀
⠀⠀⠀⠀⡼⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡱⡄
⠀⠀⢀⠞⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡙⣄
⠀⢠⢎⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣆
⠠⢃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠃
⣦⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠
⢯⠳⢭⣓⠲⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⣋⡵⢫
⠀⠑⠦⣄⣉⠙⠒⠲⠮⠭⢭⣥⣒⣒⣒⣒⣒⣢⣤⠤⠤⠤⠤⠤⠤⠤⠒⠒⣒⣒⣒⣒⣒⣋⣩⣭⡭⠤⠶⠒⠚⣉⣠⠴⠋
⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⠤⠤⣤⣀⣀⣈⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣀⣠⡤⠤⠤⠶⠒⠚⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠘⠁⠀⢈⠂⠀⠀⠀⠀⠺⣿⣿⣿⡿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⠿⠿⠇⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠜⠀⠀⠀⠀⠣⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣷⣷⡄
⠀⠀⠀⣀⣀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⣧⣦⣤⣤⡄⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣿⠀⠀⠈⣿⣿⣼⣿⣾⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⡟⣾⣿⡇⠀⠠⣶⣶⣶⣶⣶⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣠⣿⣿⣿⣄⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠿⣿⣿⣧⣿⡿⠃⠀⠀⠙⠿⢿⠿⠟⠁⠀⠸⢿⣿⣿⣿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⢤⣤⣤⠀⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⠀⠈⠉⢱⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣀⣸⣄⣼⣀⣸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⠀⢰⣤⣼⠀⢰⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢙⣉⣻⠋⢻⣉⣻⠋⡟⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⠿⢿⣿⣿⠿⢿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⢸⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠀⢸⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⠀⢸⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡂⢸⣿⣿⠀⢸⣿⡿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣿⣶⣾⣿⣿⣶⣷⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣶⣶⣿⣷⣶⣾⣿⣶⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⡀⠀⠀⠀⠀⠠⡆⠀⠚⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠁
⠀⠀⠀⠀⠐⠁⣠⠀⠠⡌⠁⠀⠀⠀⠀⢀⣦⠈⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠠⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡂⠉⢂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠐⡐⢀⣀⡀
⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠸⠅⠀⢨⡀⠀⢠⠀⠀⠀⠀⠈⠉⡅⠒⢦
⠀⠀⢀⡄⠴⠥⠤⠤⠤⠤⠴⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⠶⠃⠀⠀⠀⠀⠀⠘⠀⠀⠸⢧⡆
⠀⠀⠂⢵⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⢸⡀⠀⠀⠀⣀⡴⠁⠀⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠈⠉⠀⠀⠀⠂⢀⡀
⠀⠀⠀⠨⠟⠀⠀⠈⠋⠀⠀⠀⠀⠴⠀⠀⠀⠀⠰⠀⢡⠀⠀⠀⠀⠀⠈⠁⠃⠙⠿⠶⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠘⠀⠀⠀⠀⠀⠀⠀⠰⢸⠀⠁⠀⠀⠰⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣤⣤⡤⠀⠀⠀⠀⢰⠁⣇⠀⠄⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢧⠀⠀⠀⠀⠀⠁⠁⣆⡆⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠲⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢈⣉⡱⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⡇⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⢀⢃⠀⠀⠀⡠⠐⠉⠀⡀⠀⠀⢤⠀⠉⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⢆⡔⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⢀⠀⠠⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⢈⠀⠀⠘⡄⠁⠀⠒⠀⠀⠀⠀⠀⠆⠈⡂⡽⠒⢄⠐⠕⡠⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡸⢃⡀⠅⠹⣄⡔⣉⣂⣄⣢⣠⣠⣄⣱⣤⣇⣁⣀⢑⢿⣋⣡⣴
⠀⠀⠀⠀⠀⠀⡐⠠⠊⠬⠾⠤⠲⢵⢬⣔⠷⢦⢹⡉⠑⢁⡬⠆⠄⠀⠀⠸⡬⠐⠒⢔⠀⠀⣊⡴⠇⡆
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠀⢀⡴⢻⠜⠣⡪⡧⢯⡁⢕⣙⠒⠲⣶⠀⢐⣌⡪⣐⣗⡡⠪⠂⢕⢇⠁
⠀⠀⠀⠀⡰⠀⠀⠀⠈⠂⠊⡤⣨⡂⢕⣕⣄⡪⢐⣎⡗⠻⣕⠀⠀⣀⠀⡋⠀⠉⠖⡡⢈⢔⢕⡽⠸
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⢍⠪⣺⣇⢠⡿⡊⣽⡿⣤⣷⡑⠉⢊⠤⡱⢂⢖⠭⡪⡱⢗⠵⡡⠁
⠀⠀⢠⠁⠀⠀⠀⠀⢅⠌⠀⠸⣋⣌⠞⡊⣕⠄⡪⢐⢡⣱⣹⡶⡋⢕⠅⡪⢚⢕⠈⣊⡔⢁⠔⠁
⠀⠀⠈⢔⠄⡀⠀⠀⠀⠀⠀⢨⠢⠐⠍⡪⠐⢅⢨⣲⡋⢇⡗⣬⣪⣕⡥⢪⠄⠵⠊⣉⠔⠁
⠀⠀⠀⠀⠈⠉⠀⠘⠁⠰⠶⠷⠾⠶⠿⠾⠿⠥⠉⠐⠊⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠠⡀⢀⠄⡀⢄⢠⠠⢀⠀⡀⡀⠄⠠⡀⢀⢀⠀⠀⠄⡀⠄⠀⡀⠀⡀⢀⠀⡀⢀⠀⡀⢀⠀⣀
⠀⠀⠀⠀⠁⠈⠀⠀⠁⠈⠐⠁⠊⠀⠑⠁⠊⠂⠅⠊⠐⠀⠐⠑⠨⠀⠅⠊⠔⡩⠂⠕⠨⢂⠕⠪⠐⠡⡪⠠⠃⠔⠔

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢡⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡂⠀⢸
⠀⠀⠀⠀⠀⠀⠐⢖⠚⡞⡒⠒⠒⠒⠒⠒⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠃⠀⠀⠀⠀⡂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⣸⣀⣀⣀⣙⣇⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⡤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡏⡇⠀⠉⣉⡉⢹⡇⠀⠀⠀⠀⠀⠀⣠⠴⠃⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⣤⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⢡⠀⠀⠀⠆⢸⡇⠀⠀⠀⠀⠀⡼⠁⠀⠀⣤⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠘⡄⠈⣁⡁⢸⡇⠀⠀⠀⠀⢰⣇⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠤⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⣇⠀⠀⡃⢸⡇⠀⠀⠀⠀⠀⢿⢆⠀⠘⠝⣀⠺⢠⣿⠇⠀⠀⠀⠀⠀⠀⣿⣇⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣣⣤⣼⣧⣬⣭⣴⣪⠇⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣯⣶⠵⠃⠀⠀⠀⠀⠀⢀⣴⣻⣧⣤⣼⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠛⠛⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣉⣉⣉⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠓⠒⠒⠒⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠒⠒⢺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠠⠬⣏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠚⢻⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠐⠒⢷⣻⡇⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠉⠩⣇⣿⡄⠀⠀⠀⠀⠀⠀⣿⡏⠉⢹⢿⡉⢩⣍⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⢀⣺⡾⣷⠀⠀⠀⠀⠀⠀⠸⡿⡄⠈⠉⢀⣬⠁⣤⠀⠀⠀⠀⠀⢠⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢨⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡇⠀⠀⠀⠀⠀⠀⠹⣝⢄⣀⣀⣉⣀⣉⣀⣀⣀⣀⡴⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⡤
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⠀⣀⣀⡀⢀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⠇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠐⠂⠣⢄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠄⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⡇⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢁⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣼⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⣷⣿⡿⠫⠊
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢻⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠀⠀⠀⠀⠀⠘⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣟⠉⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢿⣿⣿⣿⣏⣿⣏⣿⣿⣿⣿⣿⣿⢯⠇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⡣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡄⠀⠞⠒⠤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣾⣿⣿⣶⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢁⡴⠛⠁⠀⣆⣇⡴⠋⠉⠉⠉⠁⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠁⠀⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠃⢀⠴⠊⢁⡤⠛⠁⠀⠀⠀⠀⠈⠁⠀⢴⢿⠆⠀⠀⣠⠀⡀⠀⠱⡀⠀⠀⠀⠀⢀⠜⠀⠀⠠⣳
⠀⠀⠀⠐⠈⠀⢿⣅⣠⠖⠉⠀⠀⣤⡄⠀⢠⠆⠀⡇⠀⠈⠉⠉⠀⠀⠁⠀⠇⠀⠀⠈⠒⣤⣴⠶⠁⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢠⡄⠈⠀⠀⢺⣀⣞⣀⣀⠀⠀⡄⢀⡀⠀⢠⠀⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣀
⠀⠀⠀⠀⠀⠘⠁⢰⣖⠤⣀⠀⢀⢀⠀⠀⠀⡟⢉⣁⣹⠃⠀⠈⠀⠀⢠⠃⠀⠀⠀⠀⠈⡕⠂⠉⠰⠦⠀⣀⢠⡯⠚
⠀⠀⠀⠀⠀⠀⢀⡮⠉⣻⠚⢁⠀⠂⠂⠀⠀⠙⠋⡼⢉⣹⠂⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣹⡎
⠀⠀⠀⢀⡤⠒⠊⠁⢠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⡹⠋⠉⢱⠄⠀⠀⠑⢄⠀⠀⠀⣀⠇⠀⠀⠀⢠⠀⣽⢒⡲
⠀⠀⠀⣿⣿⣶⣦⣄⠘⡆⠀⢰⡠⡇⡀⢰⠆⠀⠀⠀⢧⠄⠀⠉⠀⠀⠀⠀⠀⡄⠈⠁⠀⠀⣀⡀⠀⠸⡟⠉⠁
⠀⠀⠀⠙⢿⣿⣿⣿⣷⠇⠀⠘⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠘⣄⡜⠀⠊⠀⢀⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⢀⠤⣶⣶⣶⣐⡦⠀⠟⠳⠄⠇⠀⠀⠀⢹⡀⠀⢀⣴⠷⣴⠁
⠀⠀⠀⢀⠠⡞⢶⠀⠀⠀⠀⢾⣁⣀⣀⡤⠆⠀⠀⠈⢺⢻⠟⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⣴⠛⠁⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠚⠋⠿⣅⠀⠀⠀⠀⠀⠀⠀⡞⣷⡏⢹⡄⠀⠀⠀⠀⠀⣤⠀⠀⠘⠂⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠠⠚⢧⠀⠀⠈⢛⣦⣄⣼⠆⠀⡀⠀⠀⠀⠀⢹⢻⣷⠘⣇⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⢾⣉⣁⣹⠀⠀⠘⠉⠈⠀⠀⠠⠈⡼⣿⡄⢿⠀⠋⠉⠹⡇⢱⠆⠀⠀⠀⠀⠀⠀⠘⠬⠜
⠀⠀⠀⢠⣄⠀⠀⢠⠖⠊⠛⢿⡉⠀⠀⠀⡤⠄⢤⠀⡀⠀⢣⢿⣷⣸⡇⠀⠀⠀⡸⢧⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠈⣁⠀⠀⠙⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠯⠚⠀⠀⠀⠘⠃⠀⢱⡄⠀⠰⣮⣭⣭⣵⠂
⠀⠀⠀⠘⠖⢃⣘⡛⠀⠀⠈⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠇⡇⣀⡤⠒⠁⣠⣀⠀⢸⠀⠉⡇
⠀⠀⠀⠀⢐⠛⠉⠉⢹⠛⠉⡇⠀⠀⠀⠀⠀⠀⣀⡀⠤⠶⠶⠤⠤⣀⣀⣀⣀⣩⣤⡄⠀⠑⠋⣠⠜⠀⠀⠡⡀
⠀⠀⠀⠀⠈⡄⠀⠀⢸⠇⠀⢰⠀⠀⠛⠋⠉⠉⠙⠓⠤⠤⠤⠤⠜⠋⠉⠁⠀⠀⠀⠠⣶⠀⠀⣷⣶⣿⣿⣿⡟⢄
⠀⠀⠀⠌⠀⠉⠉⠹⠥⠤⠔⠚⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠒⠒⠒⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠠⠀⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠒⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣄⡀
⠀⠀⠀⠀⡄⠁⢀⠄⢄⠀⠉⠀⠀⠈⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⠉
⠀⠀⠀⡗⡁⠀⢈⠀⢈⠀⠀⡁⢈⠀⠀⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⡌
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠐⠐⡁⢸⢀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⣇⣂⠿⡀
⠀⠀⠀⡇⠐⠄⢠⠀⠀⠀⠀⡇⢰⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣈⠛⠀⠀⠀⠈
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠃⠂⠘⠈
⠀⠀⠠⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠄⠘⠚⠀⠀⠒⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠠⠄⠈⠁⠠⠭⠬⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⡀⢀⣤⠀⣠
⠀⠀⠀⠀⢰⠋⠙⠦⠤⠤⠤⠤⠜⠋⢱⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⠖⠶⣧⠀⢀⡇⠠⡟⣋⡛
⠀⠀⠀⠀⢸⠀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡎⠑⠒⠢⠤⠄⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠨⢐⡂⢸⠁⢙⡀⢠⢣⣛⣸⠆
⠀⠀⠀⠀⢸⢀⣒⣒⣒⣒⣒⣒⣒⣒⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡁⢸⠀⠐⢅⢨⠀⡆⠒⡀
⠀⠀⠀⠀⢸⠀⠒⠒⠒⠒⠒⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⢘⢨⡅⢺⠀⠸⡤⠰⠀⠊⠯⠇
⠀⠀⠀⠀⢸⠀⠨⠭⠭⠭⠭⠍⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⠄⢺⠀⠀⡇⢸⠀⢸⢻⡼
⠀⠀⠀⠀⢸⠀⠨⠭⠭⠭⠭⠍⠉⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⣿⠀⢘⡇⢘⠀⢸⣶⣾⡆
⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠁⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠉⠉⠿⠆⠾⠇⠠⠀⠀⠯⠤⠇

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

⠀⠀⠀⠀⡞⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⢻
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠛⠿⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⡴⠋⠀⢰⡆⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⣠⣾⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⢆
⠀⢸⣁⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢤⣀⣠⣤⣄⡀⢈⣧
⠀⢸⣿⣿⡛⢳⣶⣤⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡀⠈⠗⠀⢠⣬⡁⠈⢧
⠀⠈⢻⣿⣿⣟⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠙⠋⠀⠀⠀⠹⠿⠃⠀⢈
⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⡄
⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⢿⣿⠇⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣒⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣀⡀⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢒⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠻⡇⠄
⠀⠀⠈⢩⣶⣁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡾⠇⠅
⠀⠀⣴⣾⣿⣷⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠭⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠿⡿⢣⠆
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⡏⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠲⣮⡀⣸⡖⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⢈⣆
⠈⠢⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⢦⢴⡷⠛⠳⣷⢤⢮⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢈⣧⠤⢹⡇⠤
⠀⠀⠈⣇⢰⠁⠹⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠫⣀⣨⢻⡄⠚⠂⡼⢧⣀⡸⠗⠀⠀⠀⠀⠀⠀⠠⠤⡷⠤⢵⡁⠀⣠⡧
⠀⠀⠀⢡⢸⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⣏⠿⣏⡏⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⡵⠒⠛⠁
⠀⠀⠀⠸⠇⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠊⠱⡀⡰⠉⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⢒⣀⠑⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣖⡊⠉⠀⣠⡶⠞⠋⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢸⣿⣿⣯⣭⣥⣵⣚⣃⡢⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⡴⣖⡿⠟⠛⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⢸⣾⣿⠟⠉⠀⠀⠀⠀⠉⠉⠀⠒⠒⠒⠶⢤⣭⣶⢶⣾⢿⣿⡶⠋⠀⠀⠀⠀⠀⠀⠀⢠⡎⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⠈⢿⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠶⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠈⠛⢦⣄⣀⠀⢠⠟⠉⢻⣦⠀⠀⠀⠀⠀⠀⣾⣯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⢀⣼⣿⣶⣷⠿⣧⡀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠈⠻⣦⣀⣀⣀⣠⡾⠛⠁⠀⠀⠀⡴⠒⢶⣄⣀⣀⡎⠉⠙⣇
⠀⠀⠀⣠⣶⠟⠱⠁⠘⣧⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⣀⡨⠋⠀⠈⢳⣦⠼⠿⠄⠴⠲⣦
⠀⠀⢀⣿⣏⣿⡓⣲⣶⡾⠂⠒⠛⣻⣿⠀⠀⠀⠀⠀⠀⠰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢾⡁⠀⠀⠘⣧⣀⣼⠇
⠀⠀⠘⣿⣾⣿⣏⣿⣿⣹⣟⠀⣉⣿⡾⠃⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⡇⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠈⠛⠻⠿⠷⠾⠷⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⢶⣦⡄
⠀⠀⠀⠀⠀⣠⢤⣤⢤⡄⣠⠦⢤⣤⠀⣤⠤⡤⢤⠀⣤⠤⢤⣄⢠⡤⢴⣦⡄⠀⠀⠀⠀⠈⠐⠌⢰⡄⠀⠀⠀⠀⠀⠀⡿⠉⠉⢹⠁
⠀⠀⠀⠀⠀⢘⠟⠉⢻⠀⢸⠋⠙⢻⠀⢹⣉⣻⡏⠀⡙⠉⠉⡇⠀⣏⣯⣩⡅⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⠶⣤⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⣀⣼⢸⣿⣶⣿⠀⢸⡀⠀⣼⠀⢸⣿⡇⡇⠀⣷⢤⢠⡇⠀⡇⠀⠁⠀⣷⣄⠀⠀⠀⠀⠀⢹⠟⠟⠛⡏⠀⠀⠀⡇⠀⠀⠸⡀
⠀⠀⠀⣭⡿⡿⣿⠿⠿⠁⣸⣦⣠⡿⠀⢽⣿⣧⠧⠀⣿⣤⣵⠧⠀⣧⣤⣴⠇⢿⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢰⠁⠀⠀⠀⢁
⠀⠀⠀⣈⣿⢡⣽⣽⣯⠭⢭⣯⣥⣽⠻⢭⣥⡴⠤⠤⣤⣼⢤⡷⠤⣼⡿⢿⡿⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⢠⠇⠀⠀⠀⠀⠘⡀
⠀⠀⠀⣁⣿⢸⡟⠃⣿⠀⢸⠋⠉⢹⠀⢸⣿⠃⠀⠀⣿⠁⠀⡇⠀⡇⠀⠈⠁⣿⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⣘⣀⣀⣀⣀⣀⣀⣱
⠀⠀⠀⠀⣿⢸⡇⠀⣹⠀⢸⠀⠀⢸⠀⢸⣿⡀⠀⠀⡿⠀⠀⡇⠀⡇⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⣸⠀⠀⠀⢿⣰⣿⠻⠿⠧⠤⠶⠾⠛⣇
⠀⠀⠀⠀⣯⠘⠁⠀⣿⠀⢸⠀⠀⢸⠀⢸⣿⠁⠀⠀⣿⠀⠀⡇⠀⡇⠀⠀⡄⣿⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠙⠳⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠄⣿⠐⡆⠀⣿⠀⢸⠀⠀⣸⠀⢸⣿⠀⠀⠀⣿⠀⠀⡇⠀⡇⠀⢸⠇⣿⠆⠀⠀⣾⣖⣒⠒⠒⠀⠀⠀⠐⢒⣶⡄⠀⠀⠀⠀⣠⣿⣇
⠀⠀⠀⠖⣿⢸⡇⣠⣿⠀⢸⣀⣠⣿⡀⣸⣉⠀⣄⣀⣿⠀⣠⣇⢀⣇⠀⣰⣄⣿⠂⠀⠀⣿⣿⡏⢉⠉⠉⠉⠉⠉⢿⣿⣧⣶⣶⣴⣊⣻⣿⠟
⠀⠀⠀⡤⣟⠘⡿⠿⣿⣈⣈⡛⠛⠛⢓⡈⠻⣿⣁⣀⡘⠷⠿⣓⡒⢛⣿⣿⣆⣻⣆⠀⠀⢻⣿⣇⠀⡀⠀⠀⠀⣆⣼⡿⠉⠉⠉⠉⠉
⠀⠀⠀⡄⡿⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠙⠒⠛⠛⠛⠛⠛⠛⠛⢿⡷⠀⠀⠀⠙⠿⣿⣷⣦⣴⣾⡿⠋
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠤⣀
⠀⠀⠀⠀⣰⠀⠀⠀⠀⢀⡤⢄⡀⣀⡴⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡩⠾⠖⠒⠒⠀⢴⠄⡤⡄⠠⠶⠶⠆⠀⠷⠇⠤⠼⠃⠐⠂⠸⠥⠿⠢⠆
⠀⠀⢠⣎⣿⡄⠀⢀⠂⢇⠀⠀⠈⡿⠧⡀⠀⢣⠀⠀⠀⠀⢠⣦⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⣠⣤⣤⣄
⠀⠀⠈⠉⠸⣽⠀⠈⠀⠈⣆⡤⢴⠗⠒⠛⠳⣻⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢠⡀⠸⠾⠿⠟⠁⠀⢹⠛⠷⡞⠀⣄
⠀⠀⠀⢠⢿⠀⠀⠀⡰⠊⠈⠣⣼⠀⠠⡶⠀⢸⢦⠀⠀⡇⠀⠀⢂⢀⠀⠀⣯⡤⠴⣿⠀⠀⠈⠁⠀⡶⠀⢸⠀⠀⢸⠀⠠⡇⠀⠈⠀⠀⠘⡷⠋
⠀⠀⠀⠳⢟⡀⠀⠀⢧⣀⡀⠀⠙⠢⣀⠀⠀⣜⡠⢷⠋⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⡇⠀⠀⠀⠀⠀⣷⠀⢸⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⣿⣢⠀⠀⠘⠋⠐⢻⠒⠋⠛⢯⣇⣀⡸⠀⠛⠀⠀⠀⠀⠀⠀⣠⠁⡀⢇⠀⠀⠀⠀⠀⣿⣤⣸⠀⠀⢸⣠⣼⡅⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢱⠉⠀⠀⠀⠀⢀⠸⡄⠀⢠⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠆⠀⠘⢦⡀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣚⣧⠀⠀⠀⠀⠚⠁⠙⠒⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣰⠃⠀⠁⠀⠀⠀⠱⡀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠘⠛⢸⣿⠁
⠀⠀⠀⠀⠀⠉⠈⡷⡀⠀⠀⠀⠀⣀⣤⡴⠼⠞⠚⠉⠀⢃⠀⠀⠀⠀⢧⣀⣀⣂⣀⣀⠠⠴⠁⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠽⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠀⠀⠸⣿⡿⠃⠀⠸⠔⢻⢽⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠶⠖⠛⢇⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹
⠀⠀⠀⠀⠀⠐⠁⠄⠠⡞⠋⠉⠉⠀⠀⢀⠀⠘⡀⠀⠈⠀⠈⡆⠀⠀⡀⢀⢀⠀⠀⢀⠀⣦⣀⠀⠠⠶⠀⢦⡀⡶⡇⣤⠐⠷⠀⠀⠀⢺⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠈⠀⠀⠀⠀⢡⠀⠀⠀⠀⢱⠀⠀⠓⠚⠙⠐⠆⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠁
⠀⠀⠀⢠⣦⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠘⡆⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠭⢠⠀⠀⠀⠀⠀⣇⠟⠧⠀⡋
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢱⠀⠀⠈⡀⠀⠀⠀⠀⢳⠀⢀⡠⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⠀⣀⢧⡀⠀⠀⠉
⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠔⠋⠀⠉⠢⣀⠀⣠⠜⠁⠀⠙⠢⡀
⠀⠀⢀⡠⡼⠶⠀⢣⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀⣸⠄⠀⠀⠀⠀⡠⠠⠀⠀⠀⠤⠤⡔⠁⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠁⢸⠀⠀⠈⢦⠀⠀⠀⢣⢀⡤⠖⠊⠉⠉⠁⠀⠀⢀⡁⠉⠲⠀⠀⣇⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠏⢇⠀⠀⠀⠀⠀⠀⢀⠁⠄
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠑⢄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠴⠃⠈⢣⠉⢈⠆⠀⠀⠀⠱⡀⠀⠀⠀⠀⡴⠀⠘⡄⠀⢀⣀⣀
⠀⠀⠠⠖⢸⠒⠂⠀⠀⠀⠀⠉⠓⠒⠶⠤⣴⠄⠀⠀⠀⢀⠊⠀⢆⠀⠃⣸⠦⠂⠀⠀⠀⠀⢀⠗⠒⢒⡒⠒⠃⠀⢠⠋⠉⠁⢠⡶⣦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠒⠆⢚⠓⣍⠁⠀⠀⢀⢀⠀⠠⠋⠀⢰⢵⣌⡇⠀⠐⠃⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⢸⠀⡀⡀⣀⣀⠀⣇⡀⢀⣘⠁⠀⠀⠀⠀⠄⠉⠁⠀⠀⠀⠂⠀⠀⠀⠘⠽⣛⠁⠀⠀⠀⠘⠃
⠀⠀⠀⢉⠽⠉⠉⠉⠉⠉⣩⡍⠉⠉⠉⠉⢩⢡⡄⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠰⡀⠀⠀⡀⢀⣤⠖⠀⠀⠀⠀⠀⠀⠀⠆⠀⣂⠔⢢⢀⠀⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠉⠀⠹⠀⠀⠀⠀⠀⠀⠀⢀⣩⠂⠤⡽⡓⠉⠁⠀⢴⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⠫⠞⠀⠀⠀⠀⠀⠀⢼⡠⠇⠀⠀⢠⠛⠀⠀⠂⠂⠀⠀⡜⡙⠀⠀⢸⠀⠀⠀⠀⢉⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢭⠀⢁⡀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠒⠀⣾⡄⠀⠀⠃⠀⠀⢀⠄⠀⠀⠀⢸⣀⠠⠀⠀⡌⡤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣶⣀⣖⣤⢀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⢹⠉⣃⠀⠀⢰⠀⠀⠀⢀⡠⠖⢻⠿⠀⠀⠀⢘⠥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠈⡏⠀⠁⢄⢠⠂⡤⢠⠀⠀⠤⠀⣤⣤⠄⠘⠀⠀⠀⡾⠟⠀⠀⠐⠉⠀⠀⢸⠀⠀⠀⡛⡿
⠀⠀⠀⠀⠀⠀⠀⣞⠿⠑⠀⡠⠀⢈⠉⠀⠀⠡⠛⠓⣆⠀⠀⠀⠀⢨⢄⠐⠀⠀⠀⡀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢣⠄⠁⠈⢤⠆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠠⢀⠀⠈⠀⡆⢠⠜⣤⣤⡄⢰⠀⠀⠘⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣀⠁⠀⠀⢀⠰⣶⠄⠀⠶⠾⠶⢾⡣⠄
⠀⠀⠀⠀⣀⣀⠄⠀⢀⡀⠀⠆⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢬⠀⠀⠉⠉⠀⠛⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠴⠊⠀⠀⠀⠀⣩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⢈⠀⠊⠀⠀⠀⠨⣳
⠀⠀⠀⢀⣀⣀⣾⣆⠀⣀⠀⠀⠤⡦⣷⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠈⠀⠉
⠀⠀⠀⠈⠀⠀⣄⣀⣠⣶⣦⣿⠶⢦⣧⣤⠤⣤⣤⡤⠤⡤⡤⠄⠀⠀⢠⣼⡍⣁⣀⡀⣤⣀⡀⠀⣠⡤⠸⣦⣤⣤⣄⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠾⠿⠿⠲⠾⠗⠒⠒⠛⠀⠈⠓⠚⠛⠛⠉⠉⠃⠚⠛⠃⠈⠉⠙⠛⠉⠉⠉⠉⠉⠛⠉⠁⠁

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

⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡠⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠒⠒
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣞⣀⣈⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⢠⡞⠙⠉⠉⠉⠉⠀⠈⠉⠉⠉⢏⠃⠀⠀⠀⠀⠀⠀⠀⠠⠴⠒⠒⠢⠴⠦⢄⣀⣀
⠀⠀⠠⡟⡀⠀⠤⠀⠀⠀⠀⠠⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠑⠒⡄
⠀⠀⠐⠑⡀⠀⠀⠀⠀⠀⠀⠉⠉⢁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⣠⡇
⠀⠀⠀⠈⠊⠢⠄⢀⡀⢀⣀⠀⠀⠁⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠉⢇⠀⢀⠠⠂⢈⠢⠀⢃⠀⢀⠀⠀⠰⣗⠀⠀⡠⠀⠀⠀⢪⡆⠀⠀⢀⡄⠀⢠⡄
⠀⠀⠇⡀⠁⢀⠀⠀⠀⠈⠀⠀⠀⠙⠀⠑⠐⢤⡚⡤⠀⠀⠀⠂⣧⢀⠄⢠⠄⠁⠰⡈⠠
⠀⠀⠀⠀⠀⠈⠈⠁⡇⠈⠀⠀⠁⠐⠈⠐⠀⣀⣉⣃⡄⡀⡞⠀⠈⢇⡀⢸⠈⠀⠸⠁⠸
⠀⠁⠀⡀⠀⠀⠠⠤⠅⠤⢄⠀⡄⠂⠢⠀⠀⠐⠒⠄⠋⠇⠑⠀⠀⠁⠀⠈⠀⠂
⠚⠓⠚⠓⠚⠒⠒⠒⠒⠒⠚⠒⠛⠒⠚⠒⠚⠒⠒⠒⠒⠚⠒⠂⠀⠀⠄⠀⠆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠢⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣂⠉⠩⠍⢁⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠊⠁⠀⠀⠀⠀⠈⢂⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⠀⢀⡀⢀⠀⠀⢀⠀⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠡⡀⠀⠀⠀⠀⡨⢀⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡱⢌⡒⠶⠄⣊⠥⣢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠤⠤⠤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠘⠃⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢘⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠁⠚⠂⠀⠈⠐⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠢⣀⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠠⡂⢰⠀⢀⢄⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⡀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠐⢄⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡋⠈⡐⡊⠉⠁⠀⠈⠀⠴⠄⠙⠊⠁⢙⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠰⣁⡹⠀⠏⠀⠂⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢡⠀⠈⠀⠀⠀⡤⠀⠀⠀⠀⡤⢄⠀⠌⡌⠐⠀⠀⠀⠀⠀⠀⢀⠎⠀⠑⠤⡤⠊⠀⠒⠠⢄⡀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡱⡀⠀⠀⠀⠀⡠⠀⠀⠀⠑⢀⠌⠜⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠱⡀
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⡄⠀⠈⠪⢕⠠⠀⡀⠦⠌⠀⠄⡊⠔⠁⠀⠀⠀⠀⠀⠀⠀⢰⢧⡀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠱
⠀⠀⠀⠀⠀⠘⠂⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠻⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡰⢀⣀⣆⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⢩⡍⣭⢩⣭⡍⡍⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀
⠀⠀⠀⠀⠠⠠⠉⠁⣀⠈⡜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠇⠄⠰⣰⠀⠀⠀⠀⠀⢀⡀⠤⠄⣀⠀⠀⠀⡠⠁⢀⠌
⠀⠀⠀⠠⠡⠁⢀⠀⠈⠀⠘⠌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠔⠁⠑⠤⡀⢠⠎⠁⠀⠀⠀⠀⠈⠢⠊⠀⡠⠊
⠀⠀⠠⠡⠁⠀⠉⠀⠀⠀⠀⠈⠜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡀⠀⡇⠀⠀⢠⠊⠐⡀⠀⠀⣷⠊
⠀⠠⠡⠁⠀⠀⠀⠀⢞⡱⠀⠀⠐⡐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⠀⢀⠀⠀⠙⠃⠀⠀⠈⠂⠊⠀⠀⠀⡅
⢀⢀⠁⠀⠛⠂⠀⠀⠀⠀⠀⡈⠂⠑⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⣧⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣅⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢀⡀⢀⠀⡀⠀⠀⠀⢀
⠀⠉⠀⠀⠀⠀⠰⠶⠀⠀⠀⠘⠄⠺⠃⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠉⠀⠀⠀⡀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠘⢂⡀
⠀⠛⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠈⠀⠀⠀⠀⠀⠉⠀⠁⠈⠀⠁⠀⠉⠀⠀⠠⡰⢤⡞⠁⠀⢦
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠈⠁⠈⠀⠀⠀⠀⠀⢠⠀⠀⠉⠂
⠀⠐⠲⢀⠈⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠈⠃⢀⠘⢤⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠀⣠
⠀⠀⠀⠀⠤⠀⡄⠀⠄⠀⠀⠀⠺⠗⠀⠀⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠁⠀⠂⠀⠐⠂⠓⠀⠀⠻⠉
⠀⠀⠀⣀⣀⠀⠀⠀⠤⠀⠃⠀⠀⡀⠀⠀⠘⠁⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉
⠀⠀⠀⠋⠉⠀⢸⣄⠀⠉⠒⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡂⠀⢔⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠏⠙⠀⠒⠀⠀⠀⠀⠀⠀⠦⠀⠁⠀⡀⠀⠃⠀⠀⠀⠀⣠⠰⢶⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠐⢄⡀⠈⠉⢁⠀⣀⡆⢠⣤⣶⡄⠀⡁⠀⠀⠈⠁⠀⣤⠹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠈⠷⠶⣶⡦⠇⠀⠀⠙⠑⠀⠀⠘⠋⠉⠁⠀⠁⠀⠐⣄⢛⡆⠘⠂⠀⠠⠆⠒⠚⠲⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠢⠀⢀⠀⠀⢠⡀⠀⠈⡄⠀⡀⠀⣰⠃⣹⠶⢤⡦⢸⣿⠷⠈⠁⠀⠀⠀⠀⡀⠐⠂⠐⠒⠓⠂⠀⠰⢀⠸⠂⠘
⠀⠀⠀⠁⠀⠀⠛⠁⠀⠀⠹⢼⡈⠀⠈⠉⡀⠀⠘⠓⢁⡁⠀⠀⠂⠀⠀⠁
⠀⠀⠐⠁⠀⢀⠀⠰⠄⡀⠄⠈⢠⠀⠀⠘⠕⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠰⠁⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠶⠀⠀⠀⠉⠀⡀⠀⢀⡤⠞⠁⠐⠆⠀⣀⠀⡀⠀⠈⠀⠀⠘⠙⠁⠀⠀⠀⠀⠀⠀⠀⢚⢲⡀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠉⠁⠠⠖⠁⡠⠀⠀⠂⡀⠀⠁⠀⠁⠀⠀⠤⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠈⠂⠉⠀⢰⠐⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠐⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀⠀⠀⠁⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⣖⡇⠀⢀⠀⠐
⠀⠀⡀⠁⠀⠀⠀⡏⢧⠄⠀⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⡀⠈⠈⠀⠀⠉⠀⠀⠠⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁
⠈⠠⣄⠀⠀⠀⠀⠠⠐⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠐⠀⠀⠟⠀⠀⠀⠈⠀⢀⡀
⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠘⠀⠒⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢦⠀⢸⡉⡿⠀⣠⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠀⠀⠙⠀⠀⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠃⠀⠀⠀⠈⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡴⠦⣄⣀⣀⡤⠖⠒⠒⠲⢤⣄⣀⣀⣀⡤⠖⠚⠳⣄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣙⠲⢤⣀⣀⠀⠀⠀⣀⣀⣤⠴⢚⣡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠬⣭⣭⣭⡭⠤⠴⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⢠⣿⠷⠀⠀⣄⡀⢀⣠⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠐⠚⠀⡀⠀⠀⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⢀⣀⣀⣠⣤⣄⣀⣀⣠⣤⣠⣤⣤⣤⣄⠄⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⢠⡏⣶⡟⡙⢹⠼⡇⡁⡇⠦⢗⣧⣰⣲⡏⠒⡀⢀⢀⡀⠠⢶⠶⠀⠀⠀⢀
⠀⠀⢲⡄⠀⠀⠀⠀⠰⠁⠀⠐⠊⠛⠶⠿⠛⠿⠾⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⡀⠀⠸
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠠
⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⡳⣇⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⣰⣦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠂⢀
⠀⡄⢁⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⡆⠀⠁
⠀⠀⡄⠠⠌⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠀⠀⠀⠀⠀⢄⣴⣢⠀⠀⠀⠀⠀⠘⠢⠐⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡄⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠘⠻⠟⠁
⠀⠀⠀⠀⣸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⡀
⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦
⠀⣀⡀⠀⣠⣴⠎⠿⣿⡿⠃⢶⣄⡀⢀⣀⠀⢀⣤⡦⠹⢿⣿⠟⠰⣦⣀⠀⣀⡀⠀⣀⣤⡶⠂
⣼⣿⣿⡇⠉⠀⠀⠀⠀⠀⠀⠀⠉⢰⣿⣿⣿⡌⠁⠀⠀⠀⠀⠀⠀⠈⠁⣾⣿⣿⣧⠉⠁
⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣨⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠘⠛⠛⠁⠶⣦⡀⣠⣤⣤⡀⣤⡶⠎⠛⠛⠋⠰⢶⣄⢀⣤⣤⡄⢀⣴⠾⠛⠀⠀⠀⠀⣴⣦⣴⣦⣴⣤⣦⣤⣤⡆
⠀⠀⠀⠀⠀⠈⠰⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⢀⡘⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⢰⠾⠛⠁⠀⠉⠁⠈⠛⠷⢠⣾⣿⣶⡀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠁⠀⠀⢩⣭⣭⣭⣭⣭⣥⣭⣭⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣙⠛⠿⢿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⠀⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣶⣾⣿⣿⣿⣶⣶⣦⣤⣍⣙⠛⠛⠻⠿⠟⠛⠛⢛⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠿⣿⣿⣿⣄⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⠠⣶⡄⣿⣿⣿⣿⣿⣿⣿⠟⢛⠛⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⣥⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠉⠛⠿⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠖⠒⠚⢒⡶⠖⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⠋⠀⠀⢀⡴⠋⠀⢀⣀⣀⡀⠀⠙⢦
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡾⠀⢀⣞⣉⠀⠀⠉⢳⡀⠈⢧
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢰⠇⠀⡞⠈⠉⠉⠙⠓⠦⢷⣀⠸⡆
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡀
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⣇⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠹⣄⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⢤⣀⠀⠈⠳⣄⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⢀⣀⠴⠒⠛⠛⠒⠦⣄⡀⠀⠀⢀⣠⠴⠒⠛⠛⠒⠦⣄⡀⠀⠀⣀⡠⠖⠒⠉⠒⠲⢤⣀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠖⠒⠋⠁⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠚⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⣀⡤⠶⠖⠛⠛⠉⠉⠉⡁⠒⠀⠄⡀⠀⠀⠄⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⣀⠴⣯⢀⠀⠀⠀⠀⠀⠈⠐⠠⣀⠈⠑⠠⣀⠉⠒⢄⣀⣀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠠⠁⠀⠠⠊⢁⠈⠁⠀⠈⢦⡈⠐⠂⠄⠀⠀⠀⠀⠀⠙⠂⠄⡀⠁⠂⠤⢈⠐⠈⡂⠀⠀⠂
⠀⠀⠀⠀⠀⡄⠀⠀⡄⢠⠂⠀⢀⠄⠀⢀⡽⢦⣀⠀⠀⠉⠐⠠⠄⣀⠀⢀⠀⠉⠐⣂⣠⠼⠛⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡷⠁⠀⠠⠊⠀⠠⠈⠀⠀⠀⠀⠀⢀⡐⠂⠀⠀⠀⠀⠐⠘⠛⣉⠴⠶⣭⣾
⠀⠀⠀⠀⠀⠁⠐⠢⡤⢄⠅⠠⠐⠋⠀⠀⠀⠀⢴⠋⡉⣒⡉⠉⠗⠚⠛⠓⠊⠉⠁⠠⠀⠠⠄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⡀⠀⠀⡜⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄⠀⠈⡄
⢀⠀⠠⠘⠀⠀⠔⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈⠄⠀⠘⠄⣀
⠣⢀⣀⡀⠤⠊⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠀⠀⠀⠈⠢⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠠⡁⣰⠂⠀⠤⣄⠺⠆⠀⢀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠠⡱⠁⠈⠂⠄⣀⠀⠈⠙⠪⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢡⠛⠃⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⣞⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⢂⢅⢀⠐⢢⠀⢰⡈⠵⣶⠒⢄⠀⡰⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠒⡞⠉⠀⠀⠀⠀⡄⠱⠞⠹⠀⠀⠈⠳⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀⠀⠀⠀⠀⣉⣉⢏⣉⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⣀⣀⣀⠡⣀⡀⠀⠀⢤⠄⠏⣥⠀⠀⠀⣀⣜⡀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢍⡀⠀⠀⠀⠉⠐⠒⠀⠀⠒⢒⠾⠯⡍⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡈⠑⠒⠤⣀⠀⠀⠀⠀⢀⠪⠁⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣤⡀⡿⠀⢔⠒⣽⠲⣤⣀⣲⠅⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢹⡇⠀⠀⠀⢸⠀⡆⠀⡁⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣸⠇⠀⠀⠀⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⢀⡆⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢲⣶⣾⡽⠄⡀⠈⡀⠏⠀⡇⠀⡟⠀⠀⠀⣀⣀⣤⡠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠄⠠⠤⠴⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠡⠤⠤⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣇⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣽⣳⣒⣛⣯⠁⡏⢨⣭⣷⠶⠾⢽⡋⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡄⠀⠀⠀⣄⡘⠯⣂⠄⠀⣀⠀⢀⡰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠉
⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⣤⣀⡀⠀⠀⢀⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀
⠀⢀⡀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠈⠙⠉⠑⠲⡀⠀⠀⠠⠀⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠐⢻⡁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠠⡆⠀⠠⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠅⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⡦⣀⠄⠀⠘⠛⡀⠀⠀⡤⠤⠶⠶⠶⢷⡤⠴⡏
⠀⠈⠀⠀⠢⡀⠀⠀⠀⠀⠀⠈⠃⠱⠠⠀⠐⠃⠨⠃⠀⠀⠀⣀⣈⣉⣀⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⠡⠋
⠀⠀⠀⠀⣰⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢘⢠⢁⣀⡄⠉⠩⠋⠊⠑⠀⠀⠀⠃⠀⣸⠀⠀⠀⠀⠀⣿
⠀⡤⠄⢠⣃⠀⠒⠂⠀⠀⠴⠶⠄⠀⠀⠀⠠⠤⠌⣀⠀⣀⡀⢀⠀⠀⢀⣆⠤⠬⠤⢬⠂⠉⠒⣐⣀⣀⣰⡿⠤⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠉⠉⠉⠉⠐⠤⡀
⠀⠀⠀⠀⠀⡠⠊⠁⡠⠄⠒⠈⠁⠀⠈⠁⠒⢄⠈⠳⡀
⠀⠀⠀⠀⡜⢀⠔⠁⠀⡀⠤⠀⠒⠂⠠⢄⠀⠀⠑⡄⠘⡄
⠀⠀⠀⡜⠀⠏⠀⢠⠊⢀⠔⠈⠉⠉⠢⡀⠱⡄⠀⠘⡀⠰
⠀⠀⠀⡇⠀⠀⠀⡎⠀⠎⠀⠀⠀⠀⠀⠰⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⢇⠀⢁⠀⠱⡀⠰⡄⠀⠀⠀⡠⠃⢠⠃⠀⠀⢀⠀⠃
⠀⠀⠀⠈⢆⠀⠀⢄⡑⣄⠈⢦⠒⠉⣀⠔⠁⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠑⠢⠄⣀⠘⡄⠀⡗⠉⠀⠀⠀⠀⠀⠀⠘⡀⠈⠄⠀⠀⢠⠤⠤⠤⣤⠀⠀⠀⠀⠀⡤⠤⠤⢤⡀⠀⠀⠀⠀⢠⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠒⠲⠻⠀⠀⠀⠘⡓⠐⠒⠐⠚⠃⠀⠀⠀⡟⠒⠐⠒⠚⠃⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⢀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⢤⣤⣤⣀⣰⠤⠀⠀⠀⠤⣧⣤⣄⣠⡧⠄⠀⠀⠤⢼⣤⣄⣀⡮⠄
⠀⠀⠊⠀⡄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⡄⠀⢀
⠀⠀⠀⠀⠇⠀⠀⠀⠲⡉⠠⠆⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠐⠺⡄
⠀⠀⠤⠠⠄⠀⠀⠀⠀⠸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠠⡇⠀⠀⠀⠀⠀⠸⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⡇⠀⠀⢈⠁⠀⠤⢘⣋⠒⠂⣙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢣⠀⠈⠀⠀⠀⠀⠀⠀⠢⡀⠠
⠘⠉⡀⠂⡜⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠴⠃⡀⠂⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡎⠀⠠⠄⠀⠀⠐⠀⠊⠢
⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠂⠀⠤⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠂⠐⡄⠀⠀⠀⠀⡀⠀⠀⠐⡠⠀⠀⠀⠀⢀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠐⢧⡀⠠⠀⠀⠀⠤⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠈⡀⠀⠀⠀⠀⢰⠄⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠰⠄⠬⠄⠀⠩⠀⠤⠬⠥⠤⠤⠤⠤⠬⠤⠤⢤⡥⠤⠤⠤⠤⠤⡥⠤⠤⣬⠤⠤⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠀⠀⠇⠀⠀⠀⠀⠀⣿⡄⠀⣾⠀⠀⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠀⠀⠆⠀⠀⠀⠀⠀⠿⠃⠀⠸⠀⠀⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⡅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢸⠂⠀⠠⠃⠀⠀⠀⠀⠀⡇⠀⠂⡇⠀⠀⠀⠀⠀⢰⠂⠀⠆
⠀⠀⠀⠀⠀⠀⢀⡤⢒⣪⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣸⠆⠀⠀⠀⠀⠀⣥⠀⣀⠇⠀⠀⠀⠀⠀⢸⡀⠀⣄
⠀⠀⠀⠀⠀⣰⠞⡼⠋⠀⠙⠀⠐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⡀⠀⣀⣿⣼⡁⠀⡠⠈⠉⠉⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⠔⠋⠁⠙⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡤⠶⢽⡆⠀⠀⠀⠀⠀⠀⢠⣾⠓⠒⠒⠒⠦⢌⡳⣄⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⡴⢋⡽⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠶⢠⠀⠀⣀⠉⣮⣱⠞⠳⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⢸⣻⣧⠀⠀⠀⠀⠀⠀⠸⡏⠓⠾⠲⢄⣉⣠⠃⠀⠀⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⢱⣻⡀⠀⠀⠀⠀⠀⠀⣷⡀⢠⣄⡠⠀⠀⠀⣼⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣷⣳⡀⠀⠀⠀⠀⢀⣇⣙⣢⣤⡤⠤⠔⠋⢸⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⠓⣒⡩⡇⠙⠁⠀⠀⠀⢀⡴⢚⠟⢩⠟⢢⡀⣠⠴⢚⡆⢹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠁⠀⣿⢠⣖⣲⣄⠔⠁⠰⣅⡀⢸⣄⣾⢿⠀⢀⡜⠀⠈⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⢼⡅⢸⡇⡀⠀⢠⡞⠁⢸⠈⣸⠀⢓⠟⢀⣀⣀⡧⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⢀⣷⡾⠍⠉⠉⠒⠷⣄⣸⠀⡇⢠⠏⠀⠈⢣⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⡯⣒⣂⠀⠀⠀⠀⠀⠹⣄⡷⠃⠀⢀⠔⠉⠀⣠⠞⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢀⣀⣠⣼⠛⠓⢶⠶⢶⠶⢦⡀⠀⠈⡳⣄⡐⠁⠀⠀⡴⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣰⠃⠀⠀⠀⠳⣼⠀⠀⠙⢦⠚⢁⠎⠉⠳⣄⡼⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⣀⣀⣀⡀⢹⠀⠀⠀⣸⢳⠎⠀⠀⠀⠈⠳⡄⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠥⠤⠤⠤⠭⢾⡇⠀⠰⡇⠋⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⣠⠞⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⢠⠖⠉⠀⠉⠢⠀⠀⠀⡴⠂⠁⠐⠢
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠈⠢⣀⣀⡠⠞⠀⠀⠈⠦⡀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠁⠀⠀⠀⠀⠀⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⡈⠀⠀⡀⠀⠀⠀⠀⡠⠊⠀⡁⠂⠀⠀⠀⠀⠀⠀⣀⠥⢤⡁⠀⠀⠀⠊⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣀⠀⠀⠈⢂⠀⠀⢄⠀⠀⢀⠄
⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠲⡀⠀⠀⠀⠀⠊⠀⠈⢂⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠
⡁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠠
⠀⡠⠠⢀⠀⢇⠀⠀⠀⠡⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠈⠀⠀⠀⠑⠂⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠁⡸
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠑⠢⡀⠀⠁⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⡀⡄⢠⡄⢹⠀⠑⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⠀⡠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠁⠁⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠒⢄⠀⠀⠀⢀⠔⠋⠉⠉⠑⣄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠄⠀⠂⢈⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠔⠈⠁⠢⡀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠳⢄⣀⡤⠊⠀⠀⠀⠘⢄⠀⠁⠀⢀⠜
⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠈⠀⠀⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠀⠀⡀⠀⡀
⣟⣿⣻⣻⣛⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣻⣻⣛⣿⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢰⢶⠷⢶⡶⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡿⠟⠹⠀⠀⠀⠀⠀⣸⡀⠀⠀⢠⠀⠀⠀⠸⠀⢀⣷⡀⠀⢠⣿⣬⣠⣀⡴⡟
⠀⠀⠀⠀⢀⡄⠀⠀⢰⣷⣍⠁⢀⠘⠀⠀⠀⠀⢀⡟⢃⠀⠀⠈⠀⠀⠠⠌⢅⣼⡿⠷⠆⠘⢏⣈⣉⣉⣡⡗
⠀⠓⠃⠀⠀⠁⠀⠀⢸⡟⠀⠿⣿⣆⠀⠀⠀⠀⡘⢠⣞⠞⠀⣀⡀⠀⢀⠀⠀⠙⠁⠀⠀⠀⠀⠉⠙⠃⠈
⠀⣤⡆⠀⠀⠆⠀⠀⢸⡇⢀⠀⠈⠉⣁⠀⠀⣠⣤⠞⠉⡷⠞⢣⡙⠈⠁⠑⠰⠾⠁
⠀⢠⣤⣤⡤⣦⣤⢶⡾⣷⡈⢻⣷⣾⡎⠳⠞⠿⠋⢀⡀⢱⣶⡞⣷⢰⡇⣤⡀⠛⠂⠀⠀⡀
⠀⠘⠋⠹⠇⠀⠹⠀⠈⠿⢁⠀⣿⣻⡧⠐⠃⠘⠘⠃⢠⢨⣿⠇⢸⣿⠀⡆⠿⠳⢦⣄⡘⠃
⠀⠀⠀⢀⣀⣀⣀⣀⠀⣰⠫⢏⡿⣿⣧⣤⣤⣤⣴⠀⠀⠰⠀⠀⢠⡟⢂⡀⠖⠀⠀⠈⠹⠓⢤⣀⠙⠉
⠀⠀⠀⠀⠠⠀⢾⠋⠀⣿⠀⠈⠃⣿⣷⣿⣟⣿⣿⠀⡀⠀⠀⠀⠀⠖⠛⠛⠀⠐⠓⠀⠄⢀⠀⠀⠗
⠀⠀⠀⠀⢀⡌⠆⠀⠀⣿⠂⠀⠀⣿⣿⣜⡟⠋⢸⢠⣼⣄⠀⠄⠀⠀⠀⠀⠀⢆⣤⡀⠀⠈⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠀⠀⡇⢀⣿⣷⠒⣿⣼⡟⡧⠀⠀⠀⠀⠀⠀⠀⢤⡉⠁⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠇⣤⠛⠩⠋⡃⢸⣟⠃⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠸⢄⣧⠀⠀⠘⠁
⠀⠀⠠⠠⠄⠀⠄⠚⠀⠿⠄⠀⢹⢠⠇⠀⠒⠒⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⠀⠀⠀⣀⡀⠘⠛⠁⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣿⣾⠟⣻⣏⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠻⠛⠓⠲⠷⡶⠂⠀⢸⠦⢿⠿⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢁⣁⢀⠀⠀⠁⣁⠀⣤⠶⠀⠀⠀⠄⠠⠐⠒
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡖⠳⡀⠈⣿⡇⠀⠀⡐⠿⠀⠀⠀⣀⠰⠷⢽⡄⠐⢀⢈⣔⣁
⠀⠷⠖⠂⣀⡈⡅⠤⠀⠀⡿⠂⣄⠠⣟⡂⠀⠚⠋⠏⢹⢯⠀⢈⣾⠃⠒⠂⢀⡀⠀⠋⠁⠀⠀⠀⠀⠂
⠀⣤⣠⡸⣰⡇⠀⢠⢠⡤⢷⣁⡋⠀⣿⡗⠃⠀⠀⠁⠘⢆⢙⡎⣻⠇⠀⠀⣶⠀⠀⠀⡤⠀⡄
⠀⠛⠇⠙⠁⠘⣇⣈⣥⣴⢼⡏⡜⠀⠁⠉⠈⠈⡄⠑⣖⣀⡄⣠⣿⠂⠀⠀⣼⠄⠀⠀⡘⠆⠀⠘⠅⠴⠲⠆⢠
⠀⠂⢰⣶⣾⣿⡟⡟⠉⠻⠉⠁⠀⠘⠀⠀⠀⠀⠓⢀⠉⡜⡿⠽⠿⠧⠀⠀⠘⠁⠀⡀⠉⠈⠙⠂⠐⠂⠐⠲⠅
⠀⣿⠸⠟⠋⠛⠃⠁⠀⢸⠛⠋⠛⠛⠋⠉⠉⠉⠉⠹⡏⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⡀⠀⠀⣿⣶⣄
⠀⠁⠀⠀⠀⠀⠀⠒⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠂⡇⠀⠀⠛⠙⠋
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣞⣛⣛⣛⣛⣛⣛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⡀⠀⠀⠀⢿⠣⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⢐⡇
⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠒⣶⠀⠀⣼⣀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠢⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠠⡇⠀⡀⠀⣿⠀⢰⣿⢿⣿⣿⣿⣿⡆⠀⢀⣀⣀⣀⣀⣀⣀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢁⡀⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀⣏⣉⡉⣉⣉⣉⣉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⣿⣼⣿⣽⣿⣿⣿⣿⣿⣿⣧⢸⡇⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠉⠉⠉⠁⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠭⡇
⠀⠀⠀⠀⠀⢀⡼⠇⠰⠆⠀⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡜⠉⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣉⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡇
⠀⠀⠘⣟⣿⣿⣿⣼⣿⣯⣿⣿⣿⣿⣿⡟⠉⠉⠉⠹⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠘⢾⣿⣿⣿⣧⣽⣿⣿⣿⢟⡟⠁⠀⠀⠀⠀⠙⠯⣿⣿⣿⣿⣿⣿⣿⡯⠟⠁⠀⠀⠀⠀⠀⢰⣯⣿⣿⣿⣿⣿⣿⣯⣷⠄
⠀⠀⠀⠀⠀⠙⠺⢽⣿⣿⣿⠽⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠉⠉⠉⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠂⠄⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢰⡁⠀⠀⠀⠤⠠⠤⠐⠈⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠠⠸⠅⣀⠀⠀⠀⠀⠀⡀⣀⠀⠠⠤⠌⠂⣰⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠈⠠⣆⠀⣤⡄⠀⢀⡀⠀⣠⣄⠀⢀⠔⠈⠁⠀⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠌⠀⠩⠡⢀⠋⠈⠂⠈⠁⠐⠁⠀⠀⠒⠈⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠂⠀⠀⠀⠀⢀⠜⢄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⡃⠀⠀⠀⠃⠀⠀⠑⠠⣀⠀⠀⣀⠠⠔⢡⠁⠈⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠄⠠⠀⠠⠀⠠⠀⠀⠀⠃⠀⠀⠀⠐⡀⠀⠀⠀⠐⠠⠀⢐⠀⠀⡠⠃⠇⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⢁⠀⠀⠀⠀⡑⢄⠸⠀⠎⠀⠰⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⢰⠒⠲⠲⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⢂⠀⠀⠀⠡⠀⠑⡇⠀⠀⡄⠀⠀⠀⠠⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡱⠶⡈⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⢂⠀⢀⠀⡐⠀⠀⠀⢀⣆⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣰⢰⠀⠀⠀⢀⡀⡇⠀⠇⣀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⢡⠀⢲⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠉⠏⢸⠀⠠⠤⠰⠀⠀⠀⠀⠈⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⣀⢀⠀⠠⠀⠈⠢⡀
⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠒⠒⠒⠒⡀⠱⡀⠀⠈⢂
⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠒⠒⠂⠐⠀⠁⠀⠀⠀⠑⢄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘

⠀⠀⡀⠀⠀⠀⠀⠑⠒⡎⠀⠀⠀⠀⠀⠀⠘⡆⠀⠴⠁⠀⠀⠠⠀⠀⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢘⠀⠀⠈⠁⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡰⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⠒⠊⠀⠀⠀⠀⢠⠃⠀⠀⠀⢣⠀⠀⠀⠠⠊⠉⠒⠤⡤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢄⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠄⡀⠀⠀⠪⠀⠀⣀⡌⠀⠀⠀⢀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠁⠀⠀⡀⠄⠀⠉⠀⠀⠀⠀⠀⠑⠂⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶
⡀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠥⢀⠀⠀⠀⠖⠁⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠈⠁⠢⠀⠀⠀⡎⠉
⠑⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡡⠀⠀⠢⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⣤⠀⠀⠀⠈⠉
⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠈⠄⠀⡤⠒
⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠵⠒⠉⠉⠉⠲⣄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⠄⠨⡀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠤⠤⠤⠊⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠙⡄⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠒⠐⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠚⠓⠒⢤⡀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠘⢤⡀⠠⠄⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⠉⠁⠉⢢
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢆⠀⢀⠃⠀⠀⠀⡂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⡀⠀⠀⠀⢬
⠀⢠⠔⠒⠢⠀⠀⠀⡠⠒⠒⢧⠀⠀⠀⠀⠀⢀⠜⠉⠉⠑⡄⠀⠈⠀⠀⠀⠀⠀⠱⢄⣀⣀⠴⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⠫⠂⡔⠒⡄⠀⠀⠀⠀⠐⠠⠀⠤⠓
⠀⠀⠄⠀⢠⠃⠀⠀⡄⠀⠀⠀⠱⡄⢀⡀⢠⣜⠀⠀⠀⠀⡇⠀⠀⢀⣄⠢⠀⠀⠀⠀⠀⠀⣀⠴⠖⠲⠦⡴⠁⠀⠀⠑⡄⠀⠀⢘⠒⠁⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⢀⡠⠂⠀⠀⠀⠈⠲⠤⠤⠊⠀⠀⢀⠊⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⡇⠀⠀⠪⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤
⠀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀⡰⠁⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠂⠒⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⢀⠀⠀⣸⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡔⠀⠐⢄⠀⠀⠀⠈⠀⠁⢀⠄⠀⠀⠑⢄⣀⠀⠀⢀⠔⠁⠀⢀⠄
⠀⠀⠀⠉⠉⠀⠀⠀⠠⠒⠀⠀⠀⠈⠀⠀⠈⠀⠀⢄⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⠉⢢⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡱⠀⢀⠤⠤⢀⡤⠀⠀⠀⠀⠀⠀⡔⡄⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢂⠴⠒⢲
⠀⠀⠙⠂⡰⠁⠀⠀⠠⣐⣄⣀⠰⡌⠀⠀⠀⡃⠀⠈⡀⠀⠀⢀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠤⣄⠀⠀⠀⠈⢡⠇⠀⠂⠁⠀⢰⡉⢹
⡀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠑⠤⠊⠀⠀⠀⠈⠒⠀⠀⠀⠀⠇⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠲⡁⠀⠀⠀⠀⠈⠣⠜⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠙
⠁⠀⠀⠀⠈⠁⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠂⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⡳⠔⠦⠆⠀⠀⠀⠀⠀⠀⠖⠁⠀⠉⠢
⢄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠙⣄⠤⠄⣀⡰⠉⠀⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠉⠁⠀⠀⡠⠀⠠⡀⠀⠀⠈⡵⠀⠀⠀⢀⠇⠀⢠
⠀⠉⠁⠀⠀⠀⣃⠀⣀⠀⡐⠁⠀⠀⠀⠙⡄⠀⠀⠀⣁⠀⠀⣀⠒⢂⡀⠀⠀⠀⣠⢞⡉⠀⠀⠈⠓⣤⢊⠉⠐⣄⠀⠀⣇⢀⠀⡈⠀⠀⠀⠈⠂⠠⢴⢊⠀⠀⠈
⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠂⣨⡤⠖⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣭⣧⣤⣚⣡⠤⢀⣀⣀⠀⠀⠀⠀⢀⣠⢴⡶⠾⠗
⠀⠀⠀⠀⠀⠀⣶⡇⠀⠀⠘⣿⡟⠋⠀⠀⠀⠈⠉⠉⠈⠛⠒⢶⣖⣾⣿⡿⠊⠀⠀⠀⠀⠀⠀⣏⢈⡧
⠀⠀⠀⠀⠀⣠⣾⣤⠀⠀⠀⠙⢧⡀⠀⠀⣰⠟⣦⠀⠀⠀⠀⠀⣿⣻⠟⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉
⠀⠀⠀⠀⣠⣿⣿⡿⣧⡀⠀⠀⠀⠉⠙⠛⠁⠀⠘⢧⣄⣀⣀⠼⠛⠁⠀⢠⠔⢦⣀⣀⡔⠈⣯
⠀⠀⣴⣾⣡⣁⣘⡆⢈⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠦⠜⠁⠀⣸⠖⠛⢢⠎⢳⡄
⠀⠀⣿⣳⣷⣾⡿⣥⠄⢸⣿⠇⠀⠀⠀⠀⠙⠿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣹⡆⠀⠈⠓⠚⠁
⠀⠀⠈⠛⠻⠿⠿⠿⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⣶⠶⣶⠆
⠀⠀⠀⠀⢿⠿⠟⠘⡿⠿⡇⢻⢿⣿⠀⡟⠋⡗⢘⣿⣿⠃⠀⠀⠀⠀⠀⣨⣤⣤⣀⠀⠀⡇⠀⢸
⠀⠀⣠⣆⣺⣶⣇⢀⡇⠀⡇⢸⣿⢉⠀⡧⣤⡇⢸⡏⠉⢰⣦⠀⠀⠀⠀⠹⠓⠛⡏⠀⠀⡇⠀⠰
⠀⠀⠺⣿⢻⣟⣃⡴⣷⢶⣣⡾⠿⡞⠀⠓⡶⣃⠘⣷⣾⣋⡇⠀⠀⠀⠀⠸⠀⠀⡇⠀⡰⠀⠀⠀⢀
⠀⠀⢚⡏⣿⠟⡏⠈⡟⠛⡇⢸⣿⠉⠀⡟⠉⡏⢸⠋⠙⢹⡇⠀⠀⠀⠀⢠⠀⠀⡇⢠⠁⢀⠀⠀⠈⠄
⠀⠀⠀⡗⢸⠀⡇⠀⡇⠀⡇⢸⣿⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢀⡼⠀⠀⢧⣾⠛⠛⠠⠒⠟⢳
⠀⠀⡄⡇⣀⠀⡇⢀⡇⠀⡇⢸⡿⢠⠀⡇⠀⡆⢸⠀⢰⢸⣇⠀⠀⣔⣁⣀⡀⠀⠀⡈⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠥⣧⣤⢀⣇⢀⣇⣠⡇⢸⠇⢘⠀⣧⢀⡇⢸⡇⢸⣸⡇⠀⢸⣿⡖⠒⠂⠐⠒⢲⣿⣧⣤⣤⣜⣿⡗
⠀⠀⠴⣏⣹⣿⣷⣀⣙⣛⣛⣈⣻⣏⣀⣙⣿⣓⣈⣻⣿⣎⣧⠀⠈⢿⣧⣠⠀⠀⢰⣴⡟⠉⠉⠉⠁
⠀⠀⠰⠋⠉⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠻⠄⠀⠀⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠖⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⠠⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠑⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠒⠒⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠖⢲⠎⠀⠀⠀⢹⡤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢒⠒⠒⠒⣶⡒⠒⠒⠒⠒⠒⠒⠒⠺⣄⠀⠀⢀⡼⠓⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠰⡻⠆⢤⣄⠿⠛⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷⠚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠐⠂⠀⠀⡔⠂⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⠤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡈⠄⠠⡀⠐⣂⡖⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠪⠴⠁⠈⣦⠟⢣⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣤⣀⣀⣀⣀⣀⣠⣔⣁⣀⡀⠀⠀⠀⠀⢀⡽⠒⠿⡀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠞⠭⠭⠭⠭⠁⠸⣀⠀⣀⠛⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⣸⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡈⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣄⣀⣠⠄⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠨⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣿⡿⣿⣿⣏⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⢿⣿⣧⠈⢿⣿⣆⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣄⠹⣟⣷⡜⢿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣮⣿⣁⠳⣿⣷⡝⢿⣿⣌⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣟⠻⣿⣯⠙⣿⣿⣿⣿⠛⠛⠿⠿⠿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠙⣿⣦⡹⣿⣷⡙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣷⡜⣿⣷⡈⢿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⡤⢄⣀⣀⣀⣾⣿⣿⡌⢿⣿⡈⢻⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⢿⣿⣼⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢻⣷⡝⢿⣷⡝⢻⣿⣿⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⢻⣿⣎⠹⣿⣾⣿⠋
⠀⢀⣤⣦⣤⣀⣀⠀⠀⠀⢀⣼⣿⣿⠹⣿⣆⢹⡿⣶⣿⣿⠏
⠀⣾⣿⣿⣿⣿⣿⣿⣟⣛⢿⣿⣿⣿⣤⣿⣿⣿⣿⣿⠏⠁
⠀⣾⣿⣿⣿⣿⡻⢿⣿⣟⣿⣿⠿⠿⠿⠿⠛⠋⠁
⢀⣿⣿⣿⡏⢿⣿⣄⣹⣿⣿⠃
⣿⣿⡏⢻⣿⣄⣿⣿⣿⠟⠁
⣿⣿⣷⣿⣿⣿⠿⠛⠁
⠟⠛⠛⠉⠉

⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⢀⣆⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠰⢸⠀⠠⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⡢⠀⠈⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡠⠞⠁
⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠁⠂⠈⠀⠀⠀⠀⠘⡂⠀⠂⡀⠤⠄⠀⠀⠠⠄⠀⠀⠤⠁⠪⠀⢀⠃⠀⠀⠃⠀⢨⠀⣠⠾⠀⠄
⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠚⠁⠀⠀⢀⠀⠀⠈⠂⠀⠀⡈⢀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠡⠤
⠀⠀⠀⠠⡀⠀⢀⡀⠀⠐⠒⠒⠒⠒⠚⠂⠧⠀⠀⠂⠀⠁⠀⠀⠀⠄⠀⠐⠂⠀⢀⠀⠀⢀⡇⡤⠄⠀⠀⠀⠀⠀⠀⠰⠐⠂⠖⠒
⠀⠀⠠⠤⠤⠜⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠐⠂⠀⠙⠈⠀⠀⠘⠁⠈⠀⠀⠑⠂⠀⠀⠀⡄⡆⠀⠀⠀⠀⢈⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠑⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⠘⠂⠀⠀⠀⠠⡀⠀⡈⠀⠀⠀⠀⠐
⠀⠠⠐⠐⢊⠁⠀⠂⠐⠀⠘⠋⠉⢩⢍⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠑⠂⠀⠀⠀⠀⠠⠍⠀⠀⢀⡀⠀⠀⠀⠄⢂⡠
⠀⠀⠀⢀⠄⠀⠀⠀⡤⠠⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠓⠒⠀⠀⠂⠀⠀⠀⠀⣡⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠔
⠀⠀⠀⡀⠀⢀⠀⠀⠙⣾⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠒⠚⠀⠠⠀⡤⠀⡐⠁⢨⠓⠀⡉⠂⠀⠀⠀⠀⢀⡄
⠀⠰⡠⠀⠁⣌⠁⠀⡴⠀⠐⠻⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠐⠓⠀⠀⠁⠀⠀⠀⠀⠒⡒⠂⠈⢢
⠀⠀⢉⠠⢄⠊⢀⡄⠀⠀⠀⠀⠠⠟⠣⠤⠤⠤⢤⡦⠀⢠⣖⡀⣔⣂⠀⠀⠀⠀⠀⠠⠀⠀⡦⠄⠄⠀⡇⠀⠀⠀⢨⠉⠀⠀⠊
⠀⠀⠀⠀⠉⠀⠈⠀⠂⠀⠂⢀⣀⠀⠀⠠⠀⠀⠀⠠⠀⢨⠉⡄⡍⠡⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⡀⠀⡀⠀⠀⡀⠁
⠀⠀⠉⠀⠉⠀⠀⠀⡀⠀⠰⠥⠤⠤⠤⠤⠕⠀⠀⠀⠀⠀⣄⠇⢘⡀⠀⠀⠀⢋⠋⠀⠀⠀⠀⠀⠄⢂⠀⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣈⡁⠁⠀⠀⠀⠀⠀⢸⠀⠁⢰⠀⡆⠈⠀⡀
⠀⠀⠀⠀⠀⡘⠀⠀⠁⠀⠀⠀⠒⠒⠐⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⣀⣄⣈⣀⣀⠈⠉⠁⠀⠉⠁⠤

⠀⠀⠀⠀⢀⠠⠐⠀⢈⠉⢉⠀⠀⢄
⠀⠀⠀⠔⠁⠐⠀⠀⠀⠄⢀⠀⠐⠀⠡
⠀⠀⠈⠀⠀⠠⠀⠔⠀⠀⠢⠐⠄⠀⠀⠁
⠀⠀⡄⠀⠀⠇⠘⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠐⠀⠀⠈⢄⠑⡤⠒⣁⠔⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠆⠘⠈⠀⠀⠀⠀⠀⠀⠠⡀⢀⡶⠶⢲⡀⠀⠀⢠⠶⠴⢦⡀⠀⠀⡰⠶⠴⣄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢁⣀⡀⢀⡁⠀⠀⣀⣀⡀⢸⡁⠀⢀⣀⣀⠀⣌
⠀⠀⠁⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢨⠈⢀⠁
⠀⠀⠀⠀⠀⠀⠡⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⠁⠈⡄
⠠⠀⠀⠐⠀⠀⡔⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⢄
⠈⠀⠒⠚⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠠⠀⠁⠀⠀⠈⠦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠑⠤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠠⠀⠀⠄⠀⠀⠀⠆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠉⠀⠉⣍⠉⢉⠉⠉⠉⠉⡍⠉⠉⠉⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠅⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⡠⣤⡊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠄⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠇⠀⡄
⠀⠀⠀⢀⣎⠞⠁⠘⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉
⡄⠀⠀⢚⣛⡂⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣤⣤⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣨⠽⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢶⣶⣾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡄⢐⣶⣀⠀⠀⠀⠀⠀⠀⠘⠳⡌⣿⣿⣿⣟⣛⣥⡀⠀⠀⠀⠀⢶⠒⣶⣶⣶⣶⣶⣶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⢀⣀⣀⣠⢮⢿⡟⠁⠀⠀⠀⠀⠀⠈⠑⠀⠈⠛⠛⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢻⣶⣶⣶⣦⡆⢸⣧⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣿⣿⣷⣾⡇⣻⣿⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣿⣿⣿⠇⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⣀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⡿⠛⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣆⣿⣻⣿⣿⠁⠀⠀⠀⣿⣿⣿⣶⣶⣦⣶⣶⣶⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⡀⢀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣥⣄⣽⣿⣟⣴⠟⣛⣟⣟⣛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⣿⠃
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡏
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟


⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⣀⣀⣀⣀⣠⠔⠂⠉⠉⠉⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠈⢿⠉⠀⠀⠀⠈⠉⠉⠓⠤⡀⠀⠀⣉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠲⡉⠁⡀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠈⡶⠋⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠚⡦⣀⠀⢀⠇⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢰⠃⠈⠑⠊⠀⢳⠈⠓⠤⣀⣀⣤⣀⡀⠀⠀⠀⣀⣀⣀⣀⡠⠽
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠤⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢀⠛⡄⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣭⠥⢤⣄⣀⡴⠚⠉⠉⢹⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠼⡄⠀⡸⢯⡄⢳⠠⠦⠤⣀⠀⢸⠶⠾⡏⠀⠀⣄⠀⢸⠁⢧⡀⠀⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠈⠻⡔⣧⠈⠓⠃⠀⠀⠹⣄⣀⣀⡠⢏⡀⢀⠝⠒⠒⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡶⠦⢤⣤⠤⣤⣤⢤⡴⢮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠑⠢⢷⡁⣢⣇⣀⡕⣸⠀⠀⢰⡒⠛⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠙⠣⡃⠀⢱⣇⣳⣀⠜⢳⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠈⠟⠉⠁⠀⡇⠀⠀⠐⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡤⠞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣿⠉⠒⠶⠤⠤⠤⢶⡺⠋⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠰⢹⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡿⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠠⠇⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠰⢸⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⢀⣠⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⣀⣀⠿⠈⠁⢄⠒⠒⠒⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢤⡤⠤⠤⠴⠗⢶⠒⠒⢻⠋⠁⠀⠀⠀⠀⢸⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠆⠀⠀⠀⠀⠀⣠⠔⠉⠳⡀⠀⠐⢰⢧⣀⣉⣸⡀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠸⡀⠀⠀⣠⠚⠁⠓⣄⡀⠈⠒⢒⣧⢀⣀⣠⠴⢧⣀⠀⠀⠀⠀⣸⠀⠀⠸⣀⠀⠀⠀⠢⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢡⠀⠠⠃⠀⠀⠀⠸⡉⠉⠉⢿⡄⠀⠙⢤⢠⠜⠈⠢⡀⢠⠊⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⠀⠀⠑⠦⠴⠋⢱⠀⠀⠀⠀⠀⠀⢠⠇⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⢀⣀⡠⠔⠃⠀⠘⢄⠀⠀⠀⠀⠀⡴⠀⢠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠲⠤⣀⣀⣀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠛⠿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠏
⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄⡀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠋
⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠉⠉⠀⣹⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⠆







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

⠀⠀⠀⣠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⡷⣥⣦⣵⡞⠁⠀⠀⠀⠀⠀⠀⢠⡉⠟⠀⠀⠀⢰⣿⣿⣿⡿⣿⣆⠀⠀⠀⠀⠀⠘⠆⣠⣃
⢀⠎⢀⢹⣿⢇⠘⢆⠀⠀⠀⠀⠈⣷⣿⠉⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣿⡀⠀⠀⠀⠀⠀⠰⠿⠻⠃⠄⠒
⠳⠼⠧⠤⠤⠤⠽⣮⠇⠀⠀⠐⢾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣷⡄⠀⠀⠀⢀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣀⡀⠀⠀⠀⠀⠐⠛⠂
⢿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠄⠀⡄⠓⢄⡀⠀⠶⠀⠀⠀⠎⠀⢹⣿⠃⠉⢄⠀⠀⠀⠀⠀⣤⡀
⠀⠻⣿⣿⣿⣿⣿⡆⠀⠀⠀⠤⡀⠥⢀⠘⠁⠀⠦⠀⠀⠘⢰⣶⠖⠓⣶⣶⡎⠀⠀⠀⠀⠀⣻⡃
⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠭⣀⠀⢈⠻⠁⠀⠀⠀⠀⣶⣿⣿⣦
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⣀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⣠
⠀⠀⡏⠈⡇⠀⣼⠀⠀⠀⠐⠿⡂⠉⢠⡀⢠⠈⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣤⣀
⠀⠠⠿⠄⡇⠀⣤⠀⠀⠀⠀⢀⣈⢶⠎⠀⢶⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁
⢀⡀⠀⢀⡇⠀⠋⠀⠀⠀⠀⠙⠁⠀⣦⠀⠀⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⢻⡆
⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⢶⣤⣴⣧⡀⠀⠀⠀⠀⠀⠀⠘⠛⣀⠛⠃⠘⠓⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀⠀⠀⠀⡰⡺⠉⠉⢓⠢
⠀⢰⣋⣰⣏⣀⡀⠀⠀⠀⠐⠋⠈⠉⠀⠉⠑⠚⠃⠀⠀⠀⠠⣶⣾⡄⠀⣿⡇⠀⠀⢀⡴⠻⠦⠠⠞⠳⡆
⠀⠀⢩⣟⠉⡉⣿⠀⠀⠀⠀⣀⢀⠀⡀⢀⡀⢀⡃⠀⠀⠀⠀⠠⣿⣿⡿⠟⠁⠀⠀⠀⠄⢀⢀⣇⣠⠀⠄
⠀⠀⠀⠉⠛⣻⠋⠁⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⣴⣿⣾⣿⣷⣶⣆⠀⠀⠀⠀⠉⠋⠈⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢻⣷⡀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣰⣿⠃⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢰⣿⠏⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣤⣀⡀⣀⣠⣴⣶⣶⣾⡿⠁⠀⠀⠈⢿⣷⣶⣶⣦⣄⣀⢀⣀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⠈⠙⠻⠿⠿⠛⠋⠀⢸⣿⠁⠀⠀⠀⠀⠈⣿⡇⠀⠙⠛⠿⠿⠟⠋⠁⠀⠈⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣴⣶⣶⣶⣦⣄⣀⣀⣠⣶⣾⢿⣿⣆⠀⠀⠀⠀⣰⣿⣿⡷⠀⢀⡀⠀⣠⣴⣶⣶⣶⣦⣼⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠈⠻⢿⣶⣶⡿⠟⠁⠀⠀⠀⠙⠋⠀⠙⠉⠀⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡠⠶⠊⠻⠟⠃⠶⡄⣠⣤⡀⡴⠆⠻⠿⠃⠲⡤⣠⣤⡀⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⠀⠀⠀⠐⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣥⡀⠀⠀⠀⠀⠀⠀⠀⣤⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢣⣄⠀⢀⣀⠀⣀⡌⠿⠿⢇⣄⡀⢀⣀⠀⢀⣤⠄⠀⠙⠛⣛⣛⣛⣛⣛⣛⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⣿⡇⠉⠀⠀⠀⠀⠈⠁⣿⣿⡇⠉⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠈⠙⠃⣾⣿⡆⠀⠰⠶⠶⠶⠶⠶⠶⠶⢊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⣾⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣉⣙⡛⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⡆⠛⣛⣛⣛⣛⣛⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⢿⣶⣶⣬⣭⣭⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠿⢿⣿⣅⣢⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣇⠾⢈⣿⣿⣿⣿⣿⢋⣭⡻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣛⣡⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠋⠀⠈⠙⠛⠛⠿⠿⠿⠛⠛⠉

⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦
⣿⣿⣀⣸⣿⣧⣀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣸⣿⠇
⠈⠻⠿⠟⠛⠻⢿⣿⣶⣤⣴⣿⡿⠛⠉⠉⢿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠏⠀⠀⠀⠀⠈⠻⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣿⣿⠛⢻⣿⡆⠀⠀⠀⠀⢀⣿⡿⠀⠀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣀⣀⣠⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣿⣿⣶⣾⣿⣇⠀⠀⠀⠀⢸⣿⠇⠀⢰⣿⡟⠙⣿
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠿⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠉⠉⠙⢿⣷⣤⣴⣶⣿⣿⣄⣠⣾⣿⣿⣾⣿
⠀⠀⣾⡿⠿⢿⣷⡀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⠀⢀⣿⡿⠋⠉⠉⠙⢿⣿⡋⠀⠀⠁
⠀⠀⢿⣷⣤⣾⣿⡁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠉⠛⠙⢿⣿⣄⠀⠀⢀⣰⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⢿⣷⣄⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⡿⠿⠿⣿⣶⣄⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⣹⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠈⢹⣿⣿⠟⣁⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⣀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣰⣿⡿⠃⣼⣿⠟⢿⣿⣤⣤⣴⣿⠟⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠛⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⡀⠀⣠⣾⡿⠋⠀⠀⢻⣿⣦⣾⣿⠛⠛⠻⣿⣦⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⡿
⢀⣤⣶⣦⣤⣶⣿⠿⠛⠿⣿⣾⡿⠋⠀⠀⠀⣴⣿⠟⠉⠉⠁⠀⠀⠀⠈⠉⠉⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⣾⣿⠉⢻⣿⡏⠁⠀⢀⣴⣿⠟⠁⣠⣶⣶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣦⡄⠀⠀⢀⣴⣶⣶⣄⠈⠻⣿⣧⡀
⠹⢿⣷⣿⠟⠁⢀⣴⣿⠟⠁⠀⢰⣿⡏⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠉⣻⣿⣶⣶⣿⣿⠉⢹⣿⡆⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠀⠻⢿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠋⠀⠀⠙⢿⣿⣿⣿⡄⠀⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⢀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⡀⠀⠀⠙⢿⣷⣄
⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⣿⣷⣀⣀⣠⣾⡿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣷⡄⠀⠀⠙⢿⣷⡄
⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣼⣿⠿⠿⢿⣿⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣠⣿⡟⠀⠀⠀⠈⢻⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠙⠛⠛⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠛⠛⠋⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣄⡀⠀⠀⢀⣠⣤⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⢿⣷⣶⣶⣾⣿⠛⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣿⠟⠉⠉⠻⣿⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠏
⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⠛⠙⠛⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⢠⡞⠲⣽⣍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⣼⠀⢀⡬⣅⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⣸⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠤⣤⣤⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⣛⣉⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠋⠉⠀⠀⢀⣨⠿⠖⠒⠲⢦⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⣴⡆⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠹⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠘⠁⢠⡶⠛⠛⠒⠦⣏⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⡆
⠀⠀⠀⠀⢀⡤⠤⢤⣀⣠⢼⡇⠀⠀⠀⠀⠀⠀⣾⣤⣄⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢳⣔⣷⡆⠀⠀⠀⠀⡟⠀⣷⠀⠀⠀⠀⡏⠉⠙⡇
⠀⠀⠀⢰⡏⠀⠀⠀⠀⡏⣰⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠛⢶⣄⠀⢹⡄⠀⠀⠀⠀⠀⢀⣿⠉⠀⠀⠀⢀⡼⠁⠀⣿⠀⠀⢀⣤⡇⠠⡞⠓⢦
⠀⠀⠀⢸⡇⠀⠀⠀⢼⠳⣏⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠙⣧⠀⣻⣄⣀⣀⠀⠀⠈⢹⠷⠤⠤⢖⡯⠤⣄⠀⣿⠀⠀⣯⢠⡇⢠⠼⠧⣾⠁
⠀⠀⠀⠈⣧⠀⠀⠀⠈⠳⣝⡛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠒⠦⣄⠀⠘⣇⠀⠀⠀⠈⠙⢦⣤⠞⠀⠀⠀⠜⠀⢀⡼⢰⠇⠀⠀⠸⣼⡇⢸⠃⠲⡼⠃
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⣠⠖⠛⣉⣉⡉⠀⠀⢀⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡟⠀⠀⠀⢀⣽⡇⠈⢹⣦⠞
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⡇⠀⠞⠓⠛⠉⡀⠀⠉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢠⡞⢹⡇⠀⢸⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢦⣤⣤⠴⠋⠹⡄⢸⡧⠝⣦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣴⠋⠀⢸⡇⠀⣾⠳⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠰⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠷⠛⠁⠉⠀⠀⠀⣠⠞⠁⠀⢀⣴⠟⠁⠀⠀⠸⠦⠤⠟⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⢀⡤⠞⠁⠀⣀⡴⠋⠁⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⡠⡾⠃⢸⡄⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⡾⠀⠀⢀⣠⠶⠋⠀⣀⡤⠚⠉⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⡏⠳⠶⠞⠁⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⡷⠤⢰⠋⣷⠖⠒⠉⠁⠀⠀⣀⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠑⢥⡀⢣⠀⡼⠀⣼⣆⡠⠤⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢺⡀⢸⡀⠇⢠⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⣒⠒⠒⢒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠘⠀⣿⠀⠘⠀⠀⠀⠀⠀⠀⢫⠍⠉⢡⠝⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢹⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⣸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⣸⡀⠀⢸⡄⢠⡋⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠸⠀⠀⠇⠀⠀⠀⠀⣴⠃⠀⠀⠀⠹⡇⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶⠷⠶⠖⠛⠒⠒⠒⠚⠒⠒⠓⠒⠒⠒⡾⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠘⡗⠚⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠰⠶⠤⠤⠤⠤⠤⠤⠶⠋⠉⠉⠉

⠀⠀⠀⠀⠖⠒⠒⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⢲⠂⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⣷⣶⣤⣶⣶⠆⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠄⢄
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠑
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢣⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⣿⣿⠿⠁
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠛⠛⠛⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⣿⣿⡟⣻⣿⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣯⣿⣽⣿⣿⣿⡿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠯⠭⠯⠭⠽⠽⠽⠭⠊⠀⠀⠀⠀⠀⠀⠀⠑⠽⠿⠿⠿⠿⠿⠿⠿⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠲⡤⠤⠄⠀⠰⠮⣁⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠁⠀⠀⠘⡀⠀⠀⠀⠀⠀⠉⠢⣄⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠔⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⢸⠦⣀⣰⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣸⠀⠀⠀⠀⢣⠈⠑⠒⠓⠒⠠⠤⠤⠖⡐⠒⠖⠈
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢋⠀⠀⠀⠀⠘⡄⠀⢀⣐⣂⣀⣀⣠⠴⠒⠒⣷⠆
⠀⠀⠀⠀⠀⠀⠘⠶⡀⡰⢣⡄⢆⡴⠤⣄⢀⠶⠺⡏⠀⣠⠀⡏⠹⡀⠘⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠣⣗⠀⠉⠀⠀⠑⠤⠤⠚⠠⡤⠈⠉⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⠀⢀⣀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠒⠽⡅⢠⣏⣹⢨⠀⠀⡔⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⠳⢅⢀⡯⢵⡤⢚⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⢄⠀⠀⠀⠈⠁⠀⠀⡄⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠙⠒⠤⣀⣀⣀⣀⡿⠋⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠄⡆⠀⠀⠀⠀⠘⡄⠀⠀⠀⡮⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⡆⠀⠀⠀⠀⡜⠀⠀⠀⠂⠇⠀⠀⠀⠀⠀⠱⠤⡀⠴⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡇⠀⠀⠀⠰⢁⠀⠀⠀⢀⣒⣀⣀⠠⡤⠄⠚⠁⠀⢢⠄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃⠀⠀⠀⠀⢀⠜⢫⡉⠁⢀⣌⠀⢂⡇⠀⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⡀⠀⢠⠚⠑⢢⣀⡉⢂⡺⠀⣈⠡⠶⣄⠀⠀⠀⡸⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢇⠉⢉⢇⠀⠐⠂⠒⠈⢣⠀⠂⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠆⢸⡀⠀⠀⠀⠉⠁⠈⡆⠀⠀⠀⣀⠎⠀⡄⠀⠀⠀⠀⡸⠀⡄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠁⠀⠑⠤⣀⣀⣀⣀⠔⠉⠉⠉⠉⠀⠀⠀⠈⠂⠤⠤⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⣠⠖⢉⠗⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡇⠀⠀⠀⣠⠞⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣇⣸⠓⠒⢦⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠉⠉⢀⡴⢁⡴⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⢳
⠀⠀⠀⡴⠒⠲⢄⣀⣀⣀⣀⣀⣀⣀⣞⡀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢶⠋
⠀⠀⢸⠀⠀⠀⣸⣏⣉⣉⣉⣉⣉⣉⣉⣉⣹⠆⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠑⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠖⢺⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡉⠀⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⣻⠆⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣦⠤⠤⣄⡀⢀⡴⢋⡴⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢳⣼⡆⠀⠀⠙⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⡟⣷⣤⠴⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴
⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠈⠀⠴⠂⠉⠒⢊⡉⠕⠀⠢⠢⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠈⠈⠀⠀⠀⠄⠤⠠⠀⠄⡄⠠⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠠⠀⠀⢀⡀⡀⣀⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣍⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠠⡐
⠀⠀⢿⣿⣿⣿⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠥⠎⠃
⠀⠀⠀⠉⠉⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄⢦⠦⡖⢲⣆⠔⠀⡖⠐⠖⠂⠠⢰⠀⠐⢦⠀⠒⠀⠀⠀⠐⠠
⠀⠀⠠⣶⠒⣲⠆⢲⡶⢶⠄⣶⠀⠀⠀⠀⠸⡀⣸⡄⡆⡞⠸⠀⡧⡏⣬⢩⠁⢼⡬⠀⠀⠸⠄⠘⠃⡀⠀⠀⠠
⠀⠀⠂⠘⠃⠋⠀⠈⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠈⠀⠈⠉⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⢀⠀⠄⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣾⣻⣷⣄⠀⠀⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠠⠀⠀⡄⠀⡇
⠀⠴⠾⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣶⢦⡀
⠀⠀⠀⢀⠀⣄⠀⠀⠀⠀⢆⡆⠀⠀⠀⠀⣹⣄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⣠⠞⢵⠀⠈⠢⠈⠓⠤⣄
⠀⠀⢀⠀⠀⢘⣆⠀⠀⣘⣙⣻⡄⠀⠀⠀⡇⢘⣿⠀⠀⠀⠀⢸⣏⠀⠀⠀⠈⢰⠞⠁⠀⠀⠀⠐⢆⠀⠈
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⣈⠉⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⡑⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⣀⢀⡀
⠀⠀⠀⢰⡶⣒⢿⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⡇⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⢸⣿⢿⠿⣿⠀⠀⠉⠉⠘⡇⠀⠀⠀⢐⠶⠖⠀⠀⢠⠄⠀⢤⠀⠀⠀⡇⢸⠈⡇⠀⢀⠔⠋
⠀⠀⠀⠚⠋⠉⠉⠛⠀⠀⠀⢐⠊⠀⠀⠐⠒⠒⠒⠒⠀⠀⠙⠒⠒⠚⠃⠀⠀⠁⠀⠚⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢜⡆⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⡀⢀⣨⡇⠀⠀⣀⠠⢚⡃⠀⠀⠲⠲⢔⠶⡲⢖⠶⡲⢲⠆⠀⠀⠀⢈⠀⠀⣹⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠀⠁⠈⠀⠁⠉⠈⠀⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡤⢤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⡄⠀⠠⡄
⠀⠀⠀⢜⠓⠂⠀⢠⡄⠀⠃⢸⠀⢱⠀⠀⠸⠛⠛⢩⣧⠀⠀⠖⠀⠀⢸⠄⠀⠀⢠⡾⠛⠁⠀⣴⡇⠀⢸⡇
⠀⠀⠀⠘⠦⠀⠠⠞⠁⠀⠀⠸⠀⠀⠄⠐⠶⠒⠒⠚⠲⠆⠀⠰⠤⠤⠞⠀⠠⠊⠈⠀⠀⠀⠈⠀⠁⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣾⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣛⣛⣛⣛⣛⣀⣀⣀⣀⣀⣀⣘⣛⣛⣛⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣴⣿⣿⣦⠹⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⢿⣭⣭⣿⢣⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢱⣶⣶⣶⣶⣶⡶⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⣵⠶⣮⢡⡌⣱⡶⣦⡙⢷⡀⢸⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣾⣿⣧⣹⡆⢰⣟⣴⣿⣷⣼⣷⢸⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣛⣛⣋⣴⣦⣙⣛⣛⣿⣿⣿⢸⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢸⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣟⠑⠛⠛⠛⠛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣐⣛⣛⣛⢛⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⣠⣤⣄⡀⠀⠀⢀⣄⢸⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⠟⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⣾⣿⣿⠸⠿⠿⠿⠿⠿⠿⠿⢡⣿⡿⠃⠾⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠇⣾⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣋⣍⡛⢿⣿⡿⢛⣉⣙⠻⣿⣿⠟⣋⢸⣿⣿⣿⣿⣿⡟⣸⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⢸⣿⣿⣿⣿⣿⠁⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠻⣿⣿⣿⠟⠛⠛⢿⣿⣿⠿⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡄⠀⡞⠒⠤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣠⣾⣿⣿⣶⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⢁⡴⠛⠁⠀⣇⣇⡴⠋⠉⠉⠉⠁⠀⠀⠀⠀⣎⠉⠉⠉⠉⠉⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠛⢀⠴⠊⢁⡤⠟⠁⠀⠀⠀⠀⠈⠁⠀⢸⣿⠇⠀⠀⣠⠀⡀⠀⠹⡀⠀⠀⠀⠀⢀⠜⠀⠀⠰⣳
⠀⠀⠀⠐⠈⠀⢿⣅⣠⠖⠉⠀⠀⣤⡄⠀⢠⠦⢐⡇⠀⠈⠉⠉⠀⠀⠁⠀⢇⠀⠀⠈⠓⣤⣦⠶⠁⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢠⡄⠈⠀⠀⢾⣠⣞⣀⣀⠀⠀⡄⢀⡀⠀⢠⠀⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣀
⠀⠀⠀⠀⠀⠘⠁⢰⣖⠤⣀⠀⢀⢀⠀⠀⠀⡟⢉⣉⣹⠃⠀⠈⠀⠀⢠⠃⠀⠀⠀⠀⠈⡕⠊⠉⠰⠦⡄⣀⢠⡯⠚
⠀⠀⠀⠀⠀⠀⢀⡾⠉⣻⠞⢁⠀⠂⠂⠀⠈⠙⠋⡽⢉⣽⠂⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣹⡎
⠀⠀⠀⢀⡤⠒⠛⠁⢠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⡹⠋⠉⢻⠄⠀⠀⠑⢄⠀⠀⠀⣀⠇⠀⠀⠀⢠⡀⣹⢖⡲
⠀⠀⠀⣿⣿⣶⣦⣄⠘⡆⠀⢸⡠⡇⡀⢰⠆⠀⠀⠀⢧⠄⠀⠉⠀⠀⠀⠀⠀⡅⠈⠁⠀⠀⣀⡀⠀⠸⡟⠉⠁
⠀⠀⠀⠙⢿⣿⣿⣿⣷⠇⠀⠘⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠸⣔⡝⠀⠊⠀⢀⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⢀⠤⣶⣶⣶⣒⡦⠀⠟⠳⠄⠇⠀⠀⠀⢹⡀⠀⢀⣴⠷⣴⠁
⠀⠀⠀⢀⠤⡞⢶⠀⠀⠀⠀⢾⣁⣀⣠⡴⠂⠀⠀⠈⢺⢻⠟⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⣴⠟⠁⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠚⠋⠿⣍⠀⠀⠀⠀⠀⠀⠀⡞⣷⡏⢹⡆⠀⠀⠀⠀⠀⣤⠀⠀⠘⠒⠃⠀⠀⠀⣿
⠀⠀⠀⠀⢠⡚⢧⠀⠀⠘⣛⣦⣄⣼⠇⠀⣀⠀⠀⠀⠀⢱⢻⣷⠸⣇⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⢾⣉⡁⣻⠀⠀⠘⠉⠀⠀⠀⠠⠈⡜⣿⡄⢿⠀⠋⠉⠹⡇⢱⠆⠀⠀⠀⠀⠀⠀⠘⠬⠜
⠀⠀⠀⢠⣄⠀⠀⢨⠖⠊⠛⢿⡉⠀⠀⠀⡤⠄⢤⠀⡀⠀⢣⢿⣷⣸⡇⠀⠀⠀⡸⢧⠀⠀⠀⠈⠈⠈
⠀⠀⠀⠈⣁⠀⠀⠙⠀⠀⢐⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠯⠚⠀⠀⠀⠘⠃⠀⢱⡄⠀⠰⣮⣭⣭⣵⠂
⠀⠀⠀⠘⠒⢃⣘⡟⠀⠀⠈⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡆⡇⣀⡤⠒⠁⣠⣀⠀⢸⠈⢩⡇
⠀⠀⠀⠀⢐⠛⠉⠉⢹⠛⠉⡆⠀⠀⠀⠀⠈⠀⣀⡀⠴⠶⠶⠦⠤⣀⣀⣀⣀⣩⣤⡄⠀⠑⠋⣠⠜⠀⠀⠡⡀
⠀⠀⠀⠀⠈⡄⠀⠀⢸⠇⠀⢸⠀⠀⠛⠉⠉⠉⠙⠳⠦⠤⠥⠤⠼⠋⠉⠀⠀⠀⠀⠠⣶⠀⠀⣷⣶⣿⣿⣿⡟⣄
⠀⠀⠀⠌⠀⠉⠉⠹⠥⠤⠖⠚⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿⠟⣿⠋⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡿⣦⡀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠃⢿⡀⣀⣤⣤⣤⣤⣼⣧⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣤⡄⠸⣇⠈⠙⢦⡀⠀⠀⠙⢷⣤⣀⣀⡈⣻⣦⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠛⠁⠀⠹⡆⠀⠀⠙⢦⡀⠀⠀⢿⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣤⣤⣤⣤⠀⢻⡄⠀⠀⠀⠙⢦⡀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⡿⠿⢿⣿⣿⣧⠀⢻⡄⠀⠀⠀⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⣾⣷⣶⣾⣿⣿⣿⣇⠈⢷⡀⢤⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⣿⣤⣤⣴⣿⣿⣿⣿⡆⠘⣧⠀⢹⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⣿⣧⣤⣤⣼⣿⣿⣿⣿⣿⠀⢸⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⣄⠀⢀⣀⡀⣀⠀⡀⣀⣀⠀⡀⢀⡀⣀⢀⣀⡀⣀⣀⢀⣀⡀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢺⢸⡗⣇⢀⡇⢏⢀⡁⣿⠒⡇⣿⠒⢸⣿⣼⡇⣿⢘⢲⡄⢸⡇⢸⡷⣏⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁



⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⡒⢲⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⢒⠒⣦⠀⠀⠀⠀⠀⣴⠒⠒⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠉⢹⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⡇⢀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⣴⣶⣶⣦⣀⢀⡀⢻⠀⠀⠀⠀⣰⠃⣤⣶⣦⣤⡈⢧⠀⠀⠀⠀⠀⡇⢸⣹⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣿⣿⡁⣹⣿⣿⣿⡇⢸⠀⠀⠀⢰⠇⣼⣿⣀⣸⣿⣷⡈⢧⠀⠀⠀⠀⡇⢸⡿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⢿⣿⣿⣿⣤⠟⣠⠎⠀⠀⠀⡏⢸⣿⣿⣿⣿⣏⣿⣷⠘⡇⠀⠀⠀⡇⢸⣷⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣬⣭⣭⠴⠞⠁⠀⠀⠀⠀⠳⢬⣭⣭⣭⣭⣭⣭⣥⠾⠁⠀⠀⠀⠹⢤⣭⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡉⠩⣭⡽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠉⢩⣭⠗⠀⠀⠀⠀⠀⠺⡍⠉⠉⣭⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠃⠀⠀⠓⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣀⡀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡇⢰⣦⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⣴⣿⣿⣿⣷⣤⣤⡄⢹⠀⠀⠀⠀⡼⢁⣶⣿⣷⣶⣆⢹⡄⠀⠀⠀⠀⡅⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⣿⣿⣿⣿⣿⣿⣿⠇⣼⠀⠀⠀⣼⢁⣾⣿⣿⣿⣿⣿⣆⢳⡀⠀⠀⠀⡇⢸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⠿⢿⠿⠟⣃⡼⠁⠀⠀⠀⣇⠺⣿⣿⣿⣿⢿⡿⠿⢀⡇⠀⠀⠀⣇⠸⠿⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠲⠒⠚⠁⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠒⠋⠀⠀⠀⠀⠈⠓⠖⠛⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠰⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡂⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠴⣄⡀⠀⠐⠂⠀⠈⠁⠡⠐⠤⡀⠀⠀⠀⠀⠠⢀⣀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⣀⡠⠤⠐⠊⠁⢊⠀⠠⣂⠉⢐⠄⠀⠉⠑⠦⣄⡀⠀⠁⠂⠀⠀⠉⠑⠢⠀⠀⠈⠀⡀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠂⠂
⠀⠀⠘⢧⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⡀⠀⠀⢀⠇⠀⠀⠀⠐⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⢀⣀
⠀⠀⠀⠀⠙⠆⠀⠀⠈⠀⠀⠄⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠊⠀⠀⠀⠀⡀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⡤⢀⡄⠁⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠑⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡇⢠⠇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠄⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠌⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣒⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠄⠒⠀⠀⠀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠈⠤⠤⠤⠬⠥⠁⠠⠤⠤⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⢀⠈⠐⢀⠀⠀⡄⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠤⠄⠈⠀⠐⠂⠄⠆⠐⠀⠀⠀⠀⠀⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠀⠀⠰⠀⠀⠀⠀⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣄⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⢀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢈⡔⠘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠜⠀⠁⠉⠉⠉⠜⠀⠉⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⢤⠀⠀⢠⠤⠄⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⢀⠀⠀⢀⡀⠀⢀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠈⡁⠁⠀⠀⡆⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢘⠀⠀⢐⡃⠀⢘⡀⠀⢘⡃⠀⣚⠀⠀⢘⡂⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⠀⠠⠀⡀⠀⠁⠀⠀⠀⠁⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠈⠇⠀⢸⠀⠀⠀⡇⠀⢹⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠁⠙⠀⠀⠀⢤⠀⠀⠙⠀⠀⠀⠀⡇⢸⠀⠀⢀⠀⠀⢈⢈⠁⠀⡆⠀⢸⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⢀⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⡀⠂⠁⠀⠀⡇⢸⠀⠒⠐⡄⠀⢸⠀⠀⠀⡇⠀⢸⠀⠒⢂⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠈⠉⠉⠉⠀⠐⠀⠀⠀⠀⡄⠀⠚⠀⠀⠀⡇⠈⠧⠤⠤⠁⠀⠀⠦⠤⠔⠁⠀⠀⠦⠤⠜⠀⢸
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢂⡀⠀⠀⠀⠀⢀⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠁⠤⠤⠠⠄⠤⠤⠤⠤⠤⠠⠄⠄⠀⠤⠀⠀⠀⠈⠤

⣾⣿⣷⣾⣶⡷⢿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣷⣶⢷⡶⠶⣶⠶⠶⣶⡿⠶⠷⢾⠷⠶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣀⣀⣀⣀⡬⠀⠀⠀⠀⠀⠀⢏⣉⣉⣉⣉⣉⡕⠀⠀⠀⠀⠀⠀⠸⣉⣉⣉⣉⣉⣹
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠠⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠧⠤⠤⠤⠤⢧⢤⠤⠤⠤⠤⢤⡀
⠀⠀⠘⠂⠀⡖⠆⠀⠐⠒⠐⡖⠒⠒⠒⠐⡖⠒⠒⠒⠒⠒⠒⢲⠒⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⠒⡖⠒⠒⠒⡖⠖⠒⠁
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠇
⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡇⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⡇⠀⠀⠀⢷
⠀⠀⠀⠀⠸⡇⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⢸⣥⣤⣤⣤⢤⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢰⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢸⢸
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⠇⠀⠀⢸⢸
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠢⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠸⢸
⠀⠀⢠⠀⠘⠚⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡄
⠀⠀⠀⠁⠈⠈⠁⠀⠈⠈⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡞⠛⠛⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠀⠀⠈⠀⠀⠀⠈⠉⠙⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣴⠟⠉⠙⢳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠻⣤⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠛⠁
⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠛⠛⠳⣄⠀⠀⠀⠀⠉⠙⢷⠟⠉⠁⠀⠀⠀⠀⠉⠈⠙⠻⣦⣀⣠⡾⠷⠶⠞⠁
⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⠋
⠀⠀⠀⠀⠈⠛⠛⠛⠶⢤⣤⠀⠀⠀⢀⣄⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⠟⠉



⣶⠛⠶⣄⣀⠀⢀⣠⡴⠞⠛⠛⠳⢦⣀⡀⠀⣀⣤⠶⠛⠛⠛⠶⣄⣀⠀⢀⣠⡴⠞⠛⠛⠳⢦⣄⡀⠀⣀⣤⠶⠛⠛⠛⠶⣤⣀⠀⢀⣀⣴⠞⢻
⣿⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣀⣀⣠⣤⣤⣤⣤⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⢛⣷⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠛⠛⠛⢻
⣿⠀⠀⠀⠀⠀⠚⠛⠛⠳⠿⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⣤⡤⠦⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠈⠻⣦⣠⡾⠀⠀⠘⠳⠖⠛⠛⠛⠛⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⠀⣀⣀⡀⠀⠀⠀⠀⠉⠛⢶⡶⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠐⠛⠛⠓⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⢸
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠠⠤⠤⠤⠤⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠒⠒⠒⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⡆⠀⠲⡆⠀⢰⡀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⢓⠀⠛⠇⠈⠙⡇⠛⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠴⠀⢀⣀⠀⡀⣀⣓⡇⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⡇⠀⠀⠋⠀⠀⠨⢽⠛⠛⠩⠉⠍⠃⠒
⠀⠀⠀⠀⠀⠀⠀⢤⠈⢭⠀⣨⢽⣈⡳⢗⣐⠆⣎⡋⣧⡇⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⢫⣈⣉⠉⠈⠉⠉⠸⠭⢆⡋⣱⣬⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠈⠁⠈⠀⠁⠈⠉⠀⠀⠀⠉⠄⢀⣃⠀⠀⠀⠀⠀⠄⠀⠐⠀⠸⢄⠉⠉⠉⠉⢩⣀⣀⣀⣀⣨⣀⡀⠀⣀⣀⢠⣄⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣠⣤⣤⣶⣒⣒⡒⠒⠒⢀⣐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠸⣇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠛⠿⠿⠻⠛⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠃⠋⠉⠉⠉⠉⠉⠉⣉⣉⡉⣁⠤⣧⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣥⡀
⠀⠀⠀⠀⠘⠀⠀⠄⠤⠠⠤⠤⠇⠰⠒⠛⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⢄⣄⣀⢀⡀⢀⢀⠀⠀⠀⠀⠀⢀⣠⣠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠊⠐⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⢦⣤⡄⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⡦⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣎⣁⢠⠘⠳⠲⠙⠚⠉⠁⠀⠀⠀⢰⠀⠀⢈⠆⠀⠄
⠀⠀⢀⡰⠀⠘⣀⣀⡜⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠑⢦⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠑⠐⠣⠀⠀⡘
⠀⠀⠀⠑⠤⢴⠁⠀⠱⠀⢠⢠⡄⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠈⢸⡇⠈⠁⣏⠀⠀⠁⣿⠀⠈⢸⠁⠀⠀⠐⠉⠉⠣
⠀⠀⠀⠀⠀⠀⠑⠐⠃⠀⢸⢸⡇⠀⠀⠨⠁⠉⠒⠊⠀⠈⠠⠀⠀⢸⡇⠀⠀⣿⠀⠀⡇⣿⠀⠀⢸⠀⠀⠀⠘⠄⠠⠂
⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⠇⠀⠀⡿⠀⠀⠇⡿⠀⠀⢸
⠀⠀⠀⠀⠑⡃⣠⠴⡄⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⢀⡄⠀⠀⡄⠀⠀⡄⣤⠀⠀⢠⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠘⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢰⢸⡇⠀⠀⣧⠀⠀⡇⣿⠀⠀⢸⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⣶⢦⣀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣸⡇⠀⣀⣿⠀⠀⣇⣿⠀⢀⣸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠠⠀⣴⣿⣿⣿⣿⣿⣧⠈⢦⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢹⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡂⠰⣿⣿⣿⠿⣿⣿⣿⣆⠈⡇⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢺⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⡖⣿⡆⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⣿⢀⣀⣀⣀⣿⣇⣀⣽⣏⣀⣀⣸⣿⣀⣨⣿⣀⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄⠀⠀⢀⠆⣾⠟⠀⠀⠀⠹⡿⠿⡿⠿⠭⣿⣿⣭⣿⣿⣿⣯⣿⣿⡯⠽⠿⡿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠽⠭⠅⠚⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠇⠰⠘⢻⠰⠘⠃⠋⠀⠈⠈⠃⠃⠀⢀⠠⣦⣠⠄⠀⠀⣀⠳⣄⠀⡜⠁⠀⠀⠀⢠⠀⢠⣜
⠀⠀⢣⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠸⠧⠎⠀⠀⠀⠀⠁⠀⠈⠀⢧⠀⣀⣀⡤⠎⠀⡇⡟⠀⠀⢀⡀
⠀⢠⡈⢧⠃⠀⠀⡰⠢⣎⠀⠀⠈⠓⠀⠢⣀⣀⣀⠀⢀⠀⣠⠀⠀⢠⠞⠉⠉⠈⠉⠻⡉⠀⠀⠀⠀⢇⠣⠤⠚⣻⠄
⠀⠀⠘⢼⠄⠀⢛⡀⠀⢹⠀⠀⢀⡀⡠⠊⠁⠀⠀⠈⠙⠁⠉⠀⠀⠸⡀⠀⠀⠀⠀⠀⡇⢀⣀⡀⠀⠈⠓⠒⠊⠁
⠀⠀⠀⠈⡇⠀⠁⠘⠛⡍⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⣠⡼⢀⡤⠤⠤⠵⢤⠐⠤⠒⠋⡰⡟⠉⢠⢪⢂⢆⠀⣀
⠀⠀⢀⡀⢡⠟⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠚⠋⠀⢯⠀⠀⠀⠀⠀⢳⠀⠀⠀⢧⣉⠷⣇⡈⢁⠆⠸⣿⡇
⠀⠀⠀⠱⣾⠀⢰⠉⠉⡱⠚⢄⣸⠀⠄⠤⠶⠋⠉⠉⠓⠒⠶⢷⠤⠤⠴⠖⠚⠀⠀⠀⠀⠀⠈⠁⠘⢪⡤⡄
⠀⠀⠀⠀⠹⠀⠀⠳⣔⠁⠲⢀⡟⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⣤⡄⠠⣤⡄⢠⣤⠀⠀⢸⡂⡄
⠀⠀⠀⠀⠘⠀⠈⢀⡃⢑⡲⠮⠀⠴⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⢰⡆⠀⣶⠀⢸⢷⠀⠀⢸⠄⡇
⠀⠀⠀⠀⢀⣀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⢀⢢⠀⠐⠆⠀⠀⡔⡲⡀⠀⢸⡇⠀⣿⠀⢸⣾⠀⠀⢸⠀⡇
⠀⠀⠀⠠⠘⡍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢓⠃⢀⠀⠀⡯⣹⡁⠀⠘⠃⠀⠛⠀⠘⠛⠀⠀⠸⠤⠇⠆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡟⠻⡃⠀⠀⣐⠃⢱⡆⢐⠆⠣⠔⠁⠀⠀⡸⠁⠀⠀⢀⣄⠀⢀⠴⢆
⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⡰⠀⠀⢡⠀⠀⠉⠀⠀⠀⠀⣀⡀⢀⠀⠀⠘⠓⠠⣶⠊⠀⠀⠱⠃⢀⠀⠙⠆
⠀⠀⠀⢠⡪⢌⡆⠀⠀⠀⠀⢠⠃⣠⡀⡀⣆⠀⠀⠛⢸⠂⠀⠀⡄⠸⠄⠀⠀⠀⢀⡀⠳
⠀⠀⠀⡏⠀⠉⢹⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⡀⠀⠀⣷⠀⣦⠀⣺⠀⠀⠑⠂⠀⡤⠤⢴⠀⠀⠴
⠀⠀⠀⠙⠲⠶⠊⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡷⠀⠀⢸⠀⠀⠀⣿⠀⠻⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢰⣄⣀⡘⣧⣀⠬⢿⡬⠤⠄⠀⠀⠀⠠⢸⠄⠀⠀⣟⠀⠀⠀⢿⠀⢸⠁⠀⠀⣿⠀⢸⡇⠀⢴⡄
⠀⠀⠀⣶⠖⠈⠉⠟⠀⠘⠋⠀⠘⠁⠀⠀⠀⠐⡆⠒⠺⠓⠒⠚⡛⠚⠛⠒⠛⠒⠚⠞⠒⠲⠛⠃⠛⠃⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⢀⣀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠳⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠾⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡉⠀⠀⠀⠀⠀⠀⠀⠈⡉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠈⠑⠒⠒⠀⢀⠀⠀⣀⣤⠂⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⡔⢶⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠑⠒⠈⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠐⠉⠻⣦
⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠂⠤⠤⠀⠒⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠄⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⣼⠏⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⡠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⣸⡏⠀⠀⠀⠀⠀⠸⠀⠤⠀⠀⠀⠀⠠⠀⢀⣨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⢸
⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⢀⣠⣿
⠀⠈⠻⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠈⠙⢿⣿⣿⣶⣦⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⠿⠟⠛⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠹⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⡤⣄⢴⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠀⠀⠀⣸⠀⠀⣙⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠴⠋⠀⣸⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠛⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠞⠋⠉⠀⠀⠀⡋⠥⠤⢤⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⢉⣤⠀⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠈⠳⡽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠰⠋⠁⣠⠴⠒⠢⠄⡌⠀⠀⠀⠀⠀⠀⠀⠀⢷⢹⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣿⠀⠀⠀⠀⠀⠀⢰⣇⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢳⣤⣶⡄⠀⠀⠀⠀⣾⠀⣷⠀⠀⠀⠀⡖⠂⠒⡆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⢹⢃⡾⠃⠀⠀⠀⠀⠀⠈⠀⠈⠉⠛⢦⣄⠀⠸⡇⠀⠀⠀⠀⠀⢀⣻⡛⠁⠀⠀⠀⣰⠃⠀⣿⠀⠀⢀⣀⡇⢠⡶⠓⢀
⠀⠀⠀⠀⢻⠀⠀⠀⠀⡾⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠙⣄⠀⠀⠀⠀⠈⠙⡧⠤⡤⠴⣚⣡⡀⠀⣿⠀⠀⣏⢈⡇⢀⠽⠆⡼
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠙⢮⡛⠚⠛⠛⠛⠉⠉⠉⠙⠛⠒⠦⣄⠀⠈⢣⠀⠀⠀⠉⠑⠢⣀⠼⠁⠀⠀⠸⠁⠀⡸⢠⡇⠀⠀⠸⣼⡇⣸⠆⠀⡼
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠉⠳⢤⣀⡀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠉⠉⠀⠀⠀⣀⣀⡀⠉⠀⠀⠀⠀⠀⠀⠈⢀⡞⠀⠀⠀⠀⣽⡇⠈⢱⣀⠄
⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⢾⠀⠐⠃⠐⠋⢀⠀⠈⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢀⡞⢁⠀⠀⢸⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠦⣀⣀⡤⠖⠙⡄⠀⣧⠜⢤⡀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⡠⠋⠀⢸⠀⠀⠘⠢⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢳⠞⠁⠈⠀⠀⠀⣀⠔⠁⠀⠀⡠⠞⠁⠀⠀⠘⠦⠤⢂⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⣠⠖⠁⠀⢀⡤⠊⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⣷⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣼⠃⠀⠀⣠⠴⠋⠁⣀⡤⠞⠉⠀⠀⠀⢀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢘⠘⠒⠒⠋⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠷⠠⢤⡏⢱⠖⠒⠋⠁⠀⠀⢀⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠤⠤⠤⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠧⡀⢸⠀⢰⠁⣼⠇⠠⠤⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠈⠇⠀⢀⠟⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢘⠛⠛⣓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠈⡄⠀⠀⠀⠀⠀⠓⠀⠈⢡⠞⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆⠀⢠⡄⠀⣸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⣸⡄⠀⠸⡇⣰⣇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⣰⠃⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠛⠓⠒⠚⠛⠛⠓⠛⠛⠛⠛⡿⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠀⠀⠀⠀⠄⠄⠀⠆⠄⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠂⢀⠀⠆⠄⠀⠂⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⣿⡦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠲⠛⠈⠛⠋⠘⠷⣠⣶⣄⠞⠃⠙⠛⠁⠛⠆⣴⣶⡄⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⠀⠀⠀⠀⠀⠀⠀⣴⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠲⡦⣠⣤⣄⢠⠶⠙⠛⠋⠶⡄⣠⣤⡄⣤⠖⠃⠀⠀⢠⣦⣦⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠙⠛⠋⠶⣄⣠⣤⣄⠀⢀⣀⣀⣀⣀⣀⣀⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠀⠈⣩⣭⣭⣭⣭⣭⡍⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣷⣶⣬⣭⡅⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣶⣶⣿⣿⣿⣷⣶⣬⣍⣛⣛⣛⣛⣛⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠩⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⠫⣤⢹⣿⣿⣿⣿⡿⢛⠻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣷⣶⣿⣿⣿⣿⣿⣇⠿⠗⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠋⠀⠙⠛⠻⠿⠿⠿⠿⠿⠛⠋⠁