Poll Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠒⡒⠒⠒⠒⠒⠒⡄⠀⡇⠗⣀⣇⠰⠿⡁⠀⠇⢠⣼⠟⠲⠲⠺⡇⠀⡆⣶⠀⣴⡄⢰⡖⡆⠀⢰⣶⣤⡤⣠⣤⡄
⠀⠀⠀⠀⢀⣇⣅⣀⣠⣀⡀⠃⠀⡇⢸⠁⢩⠃⠀⡇⠀⣃⢤⠛⠾⣦⡁⠀⡇⠀⡇⠀⣄⣀⣤⠀⠀⡇⠐⡫⠹⠯⠛⡟⠉⠹
⠀⠀⠀⠐⣒⣓⠗⠒⠚⣚⣒⠇⠀⡇⠘⣖⡞⠀⠀⡄⠀⣝⠀⠦⠀⡛⡏⠠⡇⠀⡇⠀⠉⠈⠈⠀⡄⡁⠀⠿⠹⠦⢶⢵⠦⢶
⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠣⠤⠤⠧⠤⠤⠃⠀⠫⠧⠔⠐⠓⠫⠛⠇⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠁⠈⠉⠀⠉
⠀⠀⠀⠀⠀⠐⢒⠒⣒⠂⠀⠀⠀⢰⢒⣒⢒⣒⢲⠀⠀⠀⢀⣀⣠⣆⣀⡀⠀⠀⠀⠐⢲⡒⢒⣲⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡞⡶⡄⠀⠀⠀⢸⢀⠤⢀⠤⢸⠀⠀⠀⢸⡀⠐⠂⠀⡅⠀⠀⠀⠸⠭⠕⣚⠘⠀⠀⢀⠒⠚⠛⠫⠉⠁⡄
⠀⠀⠀⠀⠀⠀⢋⠀⠁⠇⠀⠀⠀⢸⣀⢀⢀⢀⢸⠀⠀⠀⢸⠣⢒⠶⣇⡇⠀⠀⠀⠐⠴⣋⣯⠀⠀⠀⢸⠘⠘⠂⠂⣞⠴⠒
⠀⠀⠀⠀⠀⠈⢹⣓⣻⠁⠀⠀⠀⢸⣁⣉⣘⣘⣈⠀⠀⠀⢸⣐⣄⣁⣁⡇⠀⠀⠀⠘⣒⣾⣖⡚⠀⠀⠘⠚⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⢀⣀⡀⢀⣀⣀⡀⣀⢀⣀⠀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠃⣿⠉⣿⠌⢹⡏⠁⣿⢸⡿⣶⣿⢸⡇⢬⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠛⠉⠀⠈⠁⠀⠉⠈⠁⠈⠉⠈⠙⠉⠁

⠀⠀⠀⠀⠐⠶⣄⣀⣠⠶⠂⠀⢠⠒⣰⡒⠒⣢⣢⢢⠀⡯⠭⣋⠭⠭⣹⠭⢭⠀⠀⡇⠨⠏⠫⠀⡇⠀⠀⠀⠈⠈⣉⠁⡁
⠀⠀⠀⠀⠟⠉⢜⠓⡃⠉⠻⠀⢸⡔⢡⠑⡄⣹⠀⠜⠀⡏⠉⢹⠉⠉⠉⠉⢹⠀⠀⣧⠟⣆⡞⢃⡇⠀⠀⠀⢀⣀⠀⣀⡆
⠀⠀⠀⠀⢀⠀⠬⠥⡧⠀⠐⠀⢸⡒⣒⠖⠒⠲⣒⣺⠀⡇⠀⢸⠺⠘⠓⠀⠸⠀⠀⡇⠤⠀⢠⠀⡇⠀⠀⠀⠀⢉⣄⡉⠃
⠀⠀⠀⠀⠘⠛⠚⠀⠓⠛⠋⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠣⠤⠼⠶⠶⠦⠤⠄⠀⠀⠧⠬⠤⠬⠤⠆⠀⠀⠀⠠⠬⠭⠤⠃
⠀⠀⠀⠀⠀⢀⡀⠴⣤⢤⠂⠀⠀⠀⠖⣒⡒⣒⡒⠆⠀⠀⠐⠒⠒⠲⠒⠂⠀⠀⠀⠀⢀⣦⢤⡄⠂⠀⠀⠐⢒⡲⣒⠒⣄⡀
⠀⠀⠀⠀⠘⡠⡆⡥⢁⡘⡮⠄⠀⠖⢓⡀⠐⣄⡀⠀⠀⠀⢀⡍⣈⢄⢤⡀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠰⠶⠀⠄⠀⠈
⠀⠀⠀⠀⠀⣑⡙⠹⠻⡄⠀⠀⠀⠀⠻⠿⠀⠏⠁⠄⠀⠀⢐⡗⠡⡠⢀⠀⠀⠀⡠⠾⠢⡤⠤⠄⠠⡄⠀⢘⣛⠀⡉
⠀⠀⠀⠀⡁⠀⣯⡄⢠⡇⠀⠀⠀⡀⠠⠄⢀⡀⠄⡀⠀⠀⢀⡇⣠⢇⣂⡃⠀⠀⠧⣀⠠⠇⠀⡀⢀⡇⠀⠐⢿⣒⣒⣒⣰⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣿⣿⣙⣿⣿⣛⣿⣿⣿⣿⣛⣿⣿⣋⣿⣿⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⠟⣽⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣹⣿⣞⣹⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣽⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⢿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣫⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣽⣿⣼⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣯⣯⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠂⠀⡂⢄⣠⡶⢲⡦⢶⢲⠰⢒⠒⠐⠂⠀⢰⡆⠀⠀⣆⣲⡖⣄⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⢻⣤⠆⣠⣟⣏⣴⣿⣟⡻⣨⣒⣾⣷⣀⣆⢀⣿⣧⠀⠀⣯⢻⠴⡂⡁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣦⢴⡏⠀⠀⠛⣾⡽⣫⡿⣦⢿⣽⣿⣿⣿⠾⣿⡇⠀⡄⣯⣾⡄⣻⣖⠄⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣵⣾⣶⡆⢠⣤⣼⣿⡿⣿⣷⣿⣝⢁⣻⡿⢯⣖⣫⣹⣩⣿⣿⢻⡿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡽⢿⣻⣿⣿⠯⡿⠟⡋⣀⣬⣷⣾⠛⣠⣾⡞⢛⣿⣿⣸⣿⣿⣇⢘⠷⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⠩⠈⢃⣈⣣⣠⣛⡶⠶⢼⣿⡏⠋⠠⢣⣟⠃⣿⣅⠀⣾⣿⣿⣿⣴⣶⣼⣿⣿⢺⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠂⠨⣾⣿⡏⣬⢿⣟⣷⡀⠉⠀⣄⣼⣿⡏⣛⡙⡯⠟⣿⣿⣏⣅⢘⢻⠗⡛⣿⣈⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⡇⠁⣿⡿⢿⠟⠏⣥⠀⣀⢰⣿⣗⢻⠿⣲⠢⣀⠘⢻⣧⣿⣿⣠⡶⣸⣷⠿⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠙⡗⡶⠉⢽⢿⡄⢠⡥⢖⣦⣸⣯⣿⣏⢨⢹⡏⢹⣺⠉⣿⣿⣿⡟⠻⣿⡿⣿⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⣧⣀⡀⣀⣀⣘⢳⡁⣀⣁⣁⠉⠁⠀⠲⠖⠚⠛⠛⠋⠉⠉⠁⠸⡿⢣⡇⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⡿⢀⠀⢀⣠⣝⣏⣈⣭⡭⠬⢭⠤⠀⠀⠀⢠⠤⠖⠁⠀⠀⠀⠀⠀⢳⣧⠐⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠈⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢻⡀⣼⣧⠀⡼⠀⣠⢤⡄⣤⠀⢀⡄⣠⠴⡆⠀⠀⠠⣾⡤⠀⣠⢤⡀⠀⠀⠀⣧⠀⣼⠁⣠⣤⡄⠠⢼⡦⠄⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠈⣷⠇⢸⣼⠃⠰⣇⢀⡇⠈⢧⡟⠀⡙⢳⡆⠀⠀⠀⢸⠀⠰⣇⢀⡇⠀⠀⠀⠸⣴⠃⠀⣏⢀⡿⠀⢸⡇⠀⣷⠚⣁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠉⠉⠉⠀⡞⠀⠀⠉⠉⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠈⠀⠀⠈⠉⠁



⠀⠀⠀⢠⣾⣳⠶⢖⡀
⠀⠀⠀⠘⠿⠿⠿⣿⠇⣠⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣻⣯⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⡟⠛⢻⠿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠛⠉⠃⠙⡆
⠀⠀⠀⠀⢠⣾⡿⠷⠽⠟⠿⢾⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⠀⣿⣧⣤⣤⣼⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠿⣷⢶⠶⣶⡶⡾⢿⠇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣿⣿⣿⣿⣻⣻⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣃⣴⣦
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠶⣿⠿⠿⠿⠿⠿⠿⣶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⡿⠇
⠀⠀⠀⠀⢸⣿⣏⣩⣏⣻⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣾⣿⣷⣶⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⡿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣻⣻⣿⡇
⠀⠀⠀⠀⢸⠟⠛⠛⠛⠛⠛⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠦⠄⠀⠄⠆⠐⠢⠤⠆⠄⠀⠀⠀⠀⠀⠀⠠⠠⡄⠤⡄⠀⡄⣤⠀⢠⡄
⠀⠀⠀⠁⠁⠈⠁⠀⠁⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠤⠀⢰⠀⣤⢀⠠⠀⡤⠄⠀⡂⢠⢠⠀⠀⠀⠀⠀⠀⢠⡄⠠⢄⡄⠄⡀⣀⡄



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

⠀⠀⠀⠀⠀⣶⡷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷
⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣾⡿⠿⠓⠀⣰⡿⠀⠀⠀⣿⠟⠿⠿⠿⠿⢿⣷⠀⢸⡿⠻⠟⠀⢠⣿⠁⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⡇⠀⢶⣶⡿⠁⣤⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠾⣶⣿⠃⣤⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⣇⣀⣀⣈⣁⣀⣿⠀⠀⣿⣀⣀⣀⣀⣀⣸⣿⠀⢸⣇⣀⣀⣈⣁⣀⣿⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⠀⢀⣤⡾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠿⣦⡀⠀⠀⠀⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣴⡾⠟⠁

⠀⠀⠀⠀⠀⣠⣶⠶⠶⠶⠀⠀⠀⠀⠶⠶⠶⢶⣦⣄
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⣰⡆⠀⠀⠀⠀⠀⣶⡆⠀⠀⣿⡇
⠀⠀⠀⠀⢿⣧⣤⣤⣾⣷⣤⣤⣤⣤⣤⣿⣧⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄

⠀⠀⠀⠀⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠆

⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⣾⣿⡁⠈⣿⣿⡄⠀⣼⣿⣿⡙⣿⣿⡆⠀⢰⣿⣯⡝⣭⣿⣷⠀⢀⣾⠿⠿⢿⣿⣿⡄⠀⣼⣿⠋⠍⢻⣿⡆
⠀⣿⡟⠋⠉⠻⣿⠇⠀⢻⡟⢿⣿⡟⢿⡏⠀⢸⣷⡞⢥⠝⣶⡿⠀⠘⣿⣀⣤⣼⣶⣿⠃⠀⢿⣿⡆⢀⣸⣿⡇
⠀⠈⠳⠶⠶⠞⠋⠀⠀⠀⠛⠿⠿⠿⠋⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠈⠛⠿⠿⠿⠋
⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⣼⣟⣏⣙⡻⣷⡄⠀⣰⣿⣿⡏⢹⣿⡆⠀⢠⣾⣯⡿⣯⣿⣦⠀⠀⣾⣿⣿⣿⣿⣷⡀⠀⣴⡟⠛⠛⠛⣿⡄
⠀⣿⣟⣷⣿⢧⣿⡇⠀⢿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣼⣿⣶⣿⣿⠀⠸⣿⠯⣿⣿⠽⣿⠇⠀⣿⣿⣨⣈⣹⣿⡇
⠀⠙⠷⣿⣷⠾⠋⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⠻⢿⣿⣿⠿⠃⠀⠀⠙⠶⣾⣷⠶⠋⠀⠀⠈⠻⢿⣿⡿⠟
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⣴⣟⣿⣿⡟⣦⡀⠀⢠⣾⠛⠛⠛⣷⡄⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⣠⣾⠿⣛⢿⣷⡄
⠀⣿⡷⠿⣿⡇⣿⡇⠀⣿⣯⣄⠀⣠⣿⡷⠀⢸⣿⢃⣭⣅⢻⣿⠀⠰⣿⣿⣭⣷⣿⣿⡇⠀⣿⣿⡜⢿⣹⣿⡇
⠀⠹⢯⣿⣯⣤⠟⠀⠀⠘⢿⣿⣿⣿⡿⠃⠀⠈⠻⣼⣿⣿⡼⠋⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠘⢿⣶⣿⣿⠟⠁
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⣠⡾⠿⣿⣷⣦⠀⠀⢠⣶⡿⠿⣿⣦⡀⠀⢀⣴⡾⢿⢿⣶⡄⠀⠀⣠⣾⣿⢿⣷⣆⠀⠀⢠⣶⣿⣿⠿⣦⡀
⠀⣿⣷⣖⡴⣦⣿⡇⠀⣿⡏⠀⢠⠈⣻⣷⠀⢸⣿⡠⣾⡎⢹⣿⠀⢰⣿⡟⠋⠈⢹⣿⡆⠀⣿⣟⠉⢽⣶⣿⡇
⠀⠻⣿⣿⣿⣿⡿⠁⠀⠹⣿⣦⣤⣿⡿⠃⠀⠘⢿⣦⣼⣤⣾⠟⠀⠀⠻⣧⣦⣤⣼⡿⠁⠀⠹⣿⣿⣷⣤⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⡙⣡⡀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡿⢳⡴⢿⡏⠉⠉⠉⠉⠉⠉⠉⢩⣿⣿⣿
⠀⠀⠀⠉⢉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣷⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿
⠀⠀⠀⢀⠻⣃⠀⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢧⣠⢿⡟⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠛⣳⡛⠃⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡬⢥⣼⣧⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿
⠀⠀⠀⣀⡻⣃⡀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢧⡴⢿⡏⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠛⠛⠃⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⣼⣧⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣦
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣿⣿⣿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣿⡇
⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⣿⣿⣿⠀⠀⢸⣿⠀⠀⣸⡇⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⣿⣿⡇⢸⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⣿⣿⣿⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⠉⠉⢹⣿⣿⡇⠀⠀⣿⠀⠀⣿⣿⡇⢸⣿⣿⠀⣿⣿⡇⠀⣿⣿⡆⠀⣿⡇
⠀⣿⣿⣿⣶⣶⣾⣿⣶⣶⣾⣷⣶⣶⣿⣶⣶⣾⣿⣿⡇⠀⠀⣿⠀⠀⠉⠉⠁⠈⠉⠉⠀⠉⠉⠁⠀⠉⠉⠁⠀⣿⡇
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠋






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣧⣼⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣦⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⣴⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⡴⠂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠙⠶⠋⠀⠘⠻⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣀⠀⣠⠞⢁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠛⠁⢀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢤⣀⡴⠋⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠉⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠲⣤⠞⠁⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⡴⠂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠙⠶⠋⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠀⡀⠀⣀⣀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⢛⡛⡛⠛⠛⢛⢛⠛⠛⠛⣷⠀⢰⡏⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⢀⠙⠛⢛⠁⡘⠛⢛⢃⢀⢀⣿⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠈⠻⠶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⣴⠶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⣩⣩⣉⡉⢉⣉⣩⡉⠉⠉⣿⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣀⣈⣉⣉⣀⣈⣉⣉⣁⣀⣠⡿⠀⠸⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠙⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢀⡶⠛⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢤⣤⣤⠄⢠⣤⣤⡄⠀⠀⣿⠀⢸⡇⢀⣤⣤⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠀⠸⡇⢸⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⢿⡇⠈⠙⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠁




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

⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⠀⠀⠀⠀⢰⠶⣒⡲⢖⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⢤⠀⠀⠀⠀⠀⣴⣶⣶⣄
⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠍⠉⡀⠀⠀⠀⠀⠀⢠⡎⠀⣻⡃⠈⣦⠀⠀⠀⠀⣾⣿⣿⢿⠀⣀⡼⠀⠀⠀⠀⠘⣿⣷⣿⣿⡄
⠀⠀⢠⣶⣿⣿⣿⣦⠀⠀⠀⠀⠀⣾⢂⣀⣀⡀⣶⠀⠀⠀⠀⠀⢱⣜⣛⣛⣴⠁⠀⠀⠀⠀⢻⠟⠛⠛⠀⠈⠁⠀⠀⠀⠀⠀⠘⣿⣿⠿
⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⢠⠄⢤⠠⢤⣤⠀⠀⠀⠀⠀⠀⣠⠖⡒⠒⢦⡀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⢸⠇⣭⠉⠁⢸⠀⠀⠀⠀⠘⠒⠚⠀⡟⣟⡀⠀⠀⠀⠀⠀⣿⠁⠓⣟⣻⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣛⢛⣛⣛⠻
⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⠉⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠲⠒⠒⠒⠺⡄⠀⠀⠀⠀⠀⡿⢈⣙⢛
⠀⠀⠘⠧⠤⠤⠼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠔⠁⠀⠀⠀⠀⠻⠤⠬⠤⠼⠁⠀⠀⠀⠀⠿⠿⠧⠤⠼⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⠀⠠⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⠁⢈⡆⠀⠀⠀⠀⠀⢰⠉⠉⠳⣀⡀⠀⠀⠀⠀⠀⠀⣴⠫⠝⢳⡀
⠀⠀⢸⢈⣬⠀⠀⢸⡇⠀⠀⠀⠀⠞⢿⣿⢿⡿⢛⠀⠀⠀⠀⠀⡾⠽⠷⠿⠽⡆⠀⠀⠀⠀⢸⠈⢫⡋⠀⠘⡆⠀⠀⠀⠀⠀⢳⣄⣀⡼⠃
⠀⠀⢸⣈⣉⣀⣰⣾⡇⠀⠀⠀⠀⡇⠀⠙⠋⠀⢘⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⡇⠀⠀⠀⠀⠸⠄⡈⠖⠀⠒⠁⠀⠀⠀⠀⠀⣾⠋⠉
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⡆⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠘⠉⢉⢭⡉⠙⠃⠀⠀⠀⠀⣾⣿⣿⢫⣼⣷⠀⠀⠀⠀⠠⠿⠋⠀⠀⢾⡇⠀⠀⠀⠀⢀⣤⣴⣿⣷⣶⠀⠀⠀⠀⠀⣴⣦⡖⠋⠉⠉
⠀⠀⢀⠀⠳⠤⠃⠀⡀⠀⠀⠀⠀⠻⣿⣿⣷⡝⠟⠀⠀⠀⠀⠀⢾⣦⡄⣴⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠙⠳⣶⣶
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⡆⡆⣶⢰⢰⣶⡆⡇⠀⢀⡁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠃⠛⠘⠘⠛⠃⠃⠐⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡆⡆⣴⢰⢰⣤⡆⡆⣦⣴⢰⢰⢠⡆⡆⡦⣴⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠇⠇⠿⠸⠸⠿⠇⠇⠟⠿⠸⠸⠸⠇⠇⠇⠿⠀⠾⠿⠾⠿⠆
⠀⠀⠀⠀⠀⠀⠀⡄⡄⣤⢠⢠⣤⡄⡄⣄⣤⢠⢠⢠⡄⡄⣄⣤⢠⢠⣠⡄⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⡇⡇⢿⢸⢸⠿⡇⡇⡯⢿⢸⢸⢸⡇⡇⡧⠿⢸⢸⠽⡇⢠⣶⣦⣶⣦
⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⢀⢀⣀⡀⡀⡀⣀⢀⢀⢀⡀⠀⠀⠞⠣
⠀⠀⠀⠀⠀⠀⠀⡇⡇⣿⢸⢸⣿⡇⡇⣯⣿⢸⢸⢸⡇⢀⣴⣦⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⠶
⠀⠀⠀⠀⠀⠀⠀⡇⡇⣿⢸⢸⣿⡇⡇⡟⣿⢸⢸⢸⡇⡇⣿⠀⣀⣤⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠈⠈⠉⠁⠁⠉⠉⠈⠈⠈⠁⠁⠉⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⣀⠀⣀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠐⠃⠃⠃⠀⠋⠙⠜⠻⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡡⠖⠒⠛⠓⠒⠦⣙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡴⠃⣠⢤⡀⠀⠀⠀⠀⠈⠳⡜⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⠀⠀⡇⠀⡇⣀⡀⡖⠒⡆⠀⢸⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠷⠴⣿⠧⢿⡷⠴⡇⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢹⡀⠀⡟⠙⣿⠋⢻⡟⠙⡇⠀⣸⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡱⣄⠉⠉⠉⠉⠉⠉⠉⢁⡴⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣑⠲⠤⠤⠤⠴⢒⡩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠋⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⡠⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿

⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⢽⣿⣿⣿⠋⠁⠉⢿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣯⢹⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡿⢉⣛⣋⢙⣿⣿⣿⠀⠀⠀⣿⣿⡿⣛⠋⠉⢛⡻⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣄⣀⣠⣾⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⢧⣦⢿⣿⣿⣿⡇⠀⠀⢸⣿⡟⢱⣿⣀⣼⣷⠙⣿⣿⠀⠀⠀⣿⣿⡇⣏⣉⣿⣿⡇⣿⣿⡇
⠀⠀⢸⣿⣿⠟⠉⠉⠉⠙⢿⣿⣿⠀⠀⢀⣿⣿⣏⣱⣿⣿⣷⣉⣿⣿⡇⠀⠀⢸⣿⣿⣞⢧⣤⡤⢏⣾⣿⣿⠀⠀⠀⣿⣿⡇⣿⣿⣿⡿⠇⣿⣿⡇
⠀⠀⢹⣿⣿⣤⣤⣤⣤⣤⣼⣿⣿⠀⠀⠠⣿⣿⣤⣼⣿⣿⣧⣤⣼⣿⡇⠀⠀⢸⣿⣿⣿⣶⣾⣶⣾⣿⣿⣿⠀⠀⠀⣿⣿⣷⣭⣭⣭⣥⣴⣿⣿⡇
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡇⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⢹⣿⡿⠿⢸⡗⠐⢸⣿⣿⣿⠀⠀⠀⣿⣿⠁⣿⣿⣿⣿⡇⣿⣿⡇⠀⠀⢸⣿⣿⡇⠠⠤⠤⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⡿⣳⣷⠎⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡿⣿⣭⣛⣿⣿⠀⠀⠀⣿⣿⠀⡿⠿⣿⣿⡇⣿⣿⡇⠀⠀⢸⣿⣿⣄⠀⠀⠀⢀⣽⣿⣿⠀⠀⠀⣿⣿⡟⢐⣂⣉⣐⠘⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣯⢿⣿⣟⣿⣿⠀⠀⠀⣿⣿⡀⣧⣴⣿⡟⠁⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⡇⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡇⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⡿⠿⠿⠶⢶⣶⣶⣿⣿⠀⠀⠀⣿⣿⣿⡇⣿⣿⡇⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⣿⣿⣇⣀⣀⢀⣀⣀⣿⣿⡇
⠀⠀⢸⣿⡀⠂⠂⠂⣸⣭⣱⣿⣿⠀⠀⠀⣿⣿⣿⡇⠿⠿⠇⣿⣿⣿⡇⠀⠀⢸⣿⣿⣙⣻⣛⣻⣋⣸⣿⣿⠀⠀⠀⣿⣿⣿⣿⡱⡶⣹⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣧⣀⣃⣠⣿⣿⣿⡇⠀⠀⢸⣿⣇⣉⣙⣿⣟⣉⣁⣿⣿⠀⠀⠀⣿⣿⣯⣿⣿⣼⣿⣯⣿⣿⡇
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⢸⣿⣿⣿⠿⠻⠿⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⠿⠛⡟⠻⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⡇
⠀⠀⢸⣿⡿⠁⠀⢀⡴⠋⢹⣿⣿⠀⠀⢀⣿⣿⡇⢠⣶⣷⣞⠉⣿⣿⡇⠀⠀⢸⣿⣿⠛⠛⣋⣵⣤⣾⣿⣿⠀⠀⠀⣿⣿⡿⠿⠟⠀⠛⠛⣿⣿⡇
⠀⠀⢿⣿⣷⡀⠀⠈⠳⣄⣸⣿⣿⠀⠀⠀⣿⣿⣏⠉⠻⡿⠋⢀⣿⣿⡇⠀⠀⢸⣿⣿⣦⣶⣍⡛⠛⠻⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣷⣶⣷⣶⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⠀⠀⠀⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⡇
⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠉⢹⣿⣟⣿⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣿⣿⣿⣿⣿⣯⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠉⠉⠉⠃⣸⡋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣀⠒⣲⠒⢂⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣽⣿⣏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⢹⡇⣻⠀⡿⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠀⠁⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣺⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠛⣛⣛⣛⣛⣛⣛⣃⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⣿⡿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⣿⣟⣛⣛⣛⣛⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡷⠶⠶⠶⠶⢾⣿⡇⠀⠀⠀⠀⠈⣿⣿⠀⣿⣯⣭⣭⣭⣭⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⣟⠛⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠸⣿⠀⣿⣯⣤⣤⣿⣿⢛⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⣿⣟⣛⣛⣻⣿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢻⡀⢿⣿⣿⣿⣿⡿⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⡇⢸⠟⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡏⣩⠭⠭⠭⠭⠭⠭⠭⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⣀⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢰⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⢠⠖⠚⠁⡇⢈⣉⣉⣉⣉⣁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠐⠒⠒⠒⠒⠂⠀⡇⠀⠀⠀⠀⠈⣇⠀⠀⡇⠠⠤⠤⠤⠤⠤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠒⠒⠒⠒⠒⠒⠀⡇⠀⠀⠀⠀⠀⠸⡄⠀⡇⠀⠖⠒⠀⠀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠘⠙⠛⠋⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⢳⠀⣇⣀⣀⣀⣀⣀⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠃⣿⢸⠀⠉⠉⠉⠁⠀⡖⣶⠇⠀⠀⠀⠀⠀⠀⠈⣇⠀⣀⣀⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠋⠉⠀⠀⣉⠉⠉⠉⠛⠛⠛⠛⠛⠳⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⣉⣩⡽⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠛⠋⠉⠀⠀⠀⠀⠠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⣡⡴⠶⠻⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣶⣶⠿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣴⣿⣷⠾⠿⠷⠶⠶⢶⡶⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣻⣿⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⣀⣴⣿⣿⣵⣶⣶⣶⣶⣾⡿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠀⢀⠈⠙⠻⣿⣶⣤⣀⠀⢄⡀⠀⠉⠙⠋⠁⠀⠀⠀⣠⣶⣿⣾⣥⠶⢍⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢶⣿⠛⠛⠛⢛⣛⣛⣛⣛⣛⡛⢿⣟⡶⣌⡁⠀⠀⠀⠀⢀⣠⣾⣿⠿⠛⢉⡰⢜⠃⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠾⣧⣤⣬⣤⣀⣀⣀⠒⠀⠒⠤⠻⠶⣹⣦⣄⢀⣴⡟⠯⢑⢠⡔⠊⠁⠈⠀⣀⡭⣷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣠⣤⣤⠠⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡭⣿⣁⣤⡶⠿⠃⣀⣤⡶⢾⡛⠉⠀⠼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⠻⡶⢦⣌⣉⠛⠛⠛⠶⢶⣦⣤⣤⣀⣒⣰⣟⣉⣤⣴⣶⣟⡋⠁⠀⢀⣹⣶⣤⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠶⣿⡀⠈⠄⠈⠉⠛⠷⢦⣤⣀⠀⠈⠙⠛⠿⠿⣿⡿⣿⡿⠛⢉⣠⣴⠾⠟⣛⣫⡭⠴⠶⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢷⡈⢿⡀⠀⠀⠀⠀⠀⣀⣀⠉⠛⠶⣤⣀⠀⠀⠀⢉⣠⣴⣾⣿⣿⣷⠾⣿⣿⣶⣦⣀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⢳⡈⠛⠛⠛⠛⠛⠙⠛⠛⠛⠷⠶⣤⣍⣛⣶⠶⢛⣽⠟⠋⣉⣤⣤⡒⠀⣀⣤⢼⡏⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣀⣀⡐⠒⠤⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣼⣿⣿⡿⠷⠛⢋⠛⢉⣠⣴⠾⢻⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⣉⣉⠉⠛⠓⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣴⠏⠉⢹⡇⡄⠀⢀⣴⠶⠟⠋⠉⠀⠀⠘⡿⠛⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡈⠉⠙⠓⠶⢶⣭⣍⣛⠶⣦⣄⣴⣿⣧⣴⣶⣾⡧⠶⠚⠉⠀⠒⠒⠒⠀⠀⠀⢠⣷⠀⠀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⠛⠛⠛⠓⠶⠦⣤⣌⣉⡛⠛⠿⢷⣿⡃⢄⡸⢦⣑⣲⠿⠿⡛⠛⠛⠛⠻⠶⣦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣷⣶⣤⣀⠀⠀⠀⠙⣯⠙⠛⠳⣶⣟⡉⠛⠿⢿⣶⡵⠞⡩⠐⢉⠄⠂⠀⢀⣠⠿⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⡍⠛⣷⣤⣄⠘⠓⠛⠛⠉⠉⠙⠷⣦⣤⣀⠀⠀⠀⠊⣁⣤⣶⣿⣯⣉⣠⠬⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣠⠟⠙⠛⢷⣤⣾⣿⣿⣶⣤⡀⠀⢀⣤⠾⠿⠬⣿⣷⣶⣶⣿⡿⠟⣋⡩⢬⡐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⣀⣀⠉⠛⠻⠿⣷⣾⣿⢾⣏⣁⣀⣠⣴⡾⠜⠛⣉⠵⢞⣋⣽⣾⣃⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉⣻⣿⢿⣶⡶⢦⣤⣤⡴⠟⠉⠛⣻⣗⠤⠔⠒⢊⣽⠿⢋⣵⣯⣼⠻⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢸⠁⠈⣷⡈⠙⠛⠷⢦⣤⣀⢖⣚⣿⡍⠑⣲⢛⣻⠶⣞⣛⣭⠴⠾⠃⠘⣆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠉⠲⢤⣀⢻⣤⣴⠶⠋⠉⠉⠹⠷⢮⣿⣿⣭⠷⠞⠛⣉⣥⠄⠂⠀⠀⠀⢹⡆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠐⠦⢄⡈⠛⠿⣤⣀⠀⠀⣰⡷⠲⣤⡉⠉⠀⠀⠐⠉⠁⢀⣀⣤⡶⢾⣛⣛⣡⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣤⣀⠀⠉⠓⠀⠀⠉⠛⣶⣟⠀⠀⡏⢻⣄⠀⣀⣤⡶⠿⠟⠛⠋⠉⠉⠉⢨⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣌⡝⠳⢦⣄⡀⠀⠀⢰⣿⠙⣿⠸⣧⠀⠹⠿⠋⠁⠀⣠⡀⠀⠀⠀⠀⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⠱⠤⠉⠛⠷⣦⣼⣿⣿⠃⠀⣹⠆⠀⠀⠈⣉⣡⣤⠴⠖⢊⡽⠋⠁⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣿⠃⠀⠀⠀⠀⠀⣀⡀⠀⢁⡀⠀⠀⣿⡤⠴⣚⠋⠉⠁⢀⣠⠞⠋⠠⣦⡴⠴⣿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣤⣍⣛⠶⢦⣤⣀⠀⠀⠈⠉⠙⢶⢧⣤⣾⢁⠤⠄⠉⠓⠾⠋⠁⠀⠉⠉⢀⣰⣤⣿⠃
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⡿⠿⠿⣶⣮⣭⣻⣶⣄⣀⠀⠉⠑⣺⣏⢀⡀⠒⣀⡠⠤⠔⣒⣠⠶⢟⡛⠉⠁⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣷⣦⣄⡀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣬⣿⣠⣄⣠⣤⠴⠖⠋⠡⠔⠊⣁⡤⠐⠀⠀⢀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣶⣤⣀⠀⠀⠀⠈⣙⢛⣿⠛⠛⣯⠳⣄⠀⠀⠀⠀⠊⠁⢀⣠⠤⠚⠋⠁
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣈⣉⢻⣀⣤⣬⢷⡌⣳⡀⠀⠀⠔⠚⠉
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠾⠛⠋⠐⠉⠉⠁⠛⠊
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⡉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠈⠁⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡤⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠻⡅⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⢀⣀⡤⠶⠚⠳⢤⠼⠃⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⡄⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⢻⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠤⠤⠾⠀⠀⠘⠧⠤⠤⠤⠴⠇⠀⠀⠿⠤⠤⠤⠤⠟⠀⠀⠸⠦⠤⠤⠤⠾



⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠙⠻⡿⠿⣿⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⡄⠨⠄⠀⠿⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠛⠿⠿⣿⡿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠿⠀⢀⠀⠀⠀⠀⠂⠘⠿⠿⣿⣿⢿⠶⠎⠙⠿⠿⠹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠭⠉⠋⠽⣶⡖⠒⠞⠠⡈⠀⠀⠀⠀⣀⡀⣀⣀⠐⠤⠤⠴⠲⠤⠄⠀⠠⠼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠔⠒⠒⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠚⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡈⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⡎⢹⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠏⢣⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣘⣒⣒⣒⣒⣒⣒⣒⣒⣚⣀⣀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠈⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣱⠀⠤⢀⣀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠐⠒⠒⠒⡖⡴⣊⠅⠒⠉⠉⠁⠒⠭⡢⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠟⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢎⢆⠇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⡼⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡊⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⡄⠀⣤⣤⣤⣤⣤⣤⣤⣦⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠃⠀⠛⠛⠛⠛⠛⠛⠛⠛⡜⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡏⠉⣾⣿⣿⣿⣿⣿⣿⣷⠘⠪⡓⢄⡀⠀⠀⠀⣀⠤⡪⣞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⡉⢉⣉⣉⣉⣉⣉⣉⣉⡁⠀⠀⠈⠑⠂⠥⠥⠌⠒⠫⣀⠜⢆⡃
⠀⠀⠀⠀⠀⠀⠀⢸⢨⠀⣇⣀⢿⣿⣿⣿⣿⣿⣿⡯⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⡽⢄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡖⠒⣴⣶⣶⣶⣶⣶⣶⣦⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠆⢘⢣⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⠓⠐⠛⠛⠛⠛⠛⠛⠛⠓⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⡳⡄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣀⠇⠈⢆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠾⠟⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢠⣤⣤⣤⡄⠀⣾⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⣏⡴⠀⠀⠀⠉⠉⠀⠀⠀⠀⠊⠁⠀⠐⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠚⠳⣄⠙⢧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⣏⠈⢳⡄⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣌⣧⠙⠳⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂




⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⣉⣉⣉⠉⠉⠉⠉⠉⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣉⣉⡉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡃⣤⠖⡄⣘⣛⣛⣛⡁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢄⠔⡀⢘⣛⣛⣛⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠒⠒⠒⠃⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠓⠒⠒⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠏⣉⡭⡀⠒⠒⠒⠒⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⣈⡅⠀⠐⠒⠒⠒⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠦⠭⠴⠇⠈⠉⠉⠉⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠧⠬⠤⠇⠈⠉⠉⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠋⢉⡩⠀⠴⠶⠶⠶⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠁⡁⠀⠠⠤⠤⠤⠄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣤⣭⣤⠇⠘⠛⠛⠛⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢇⣉⣠⠇⠈⠛⠛⠛⠃⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⢀⣠⠔⠙⠓⠛⠀⠀⠀⠰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠄⠀⠀⠀⣀⣴⣋⠀⠈
⠀⠀⠀⠀⠀⠂⠈⠀⠀⠈⠁⠒⠀⢤⢀⢀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠈⠀⠀⡠⢔⡾⠋⠁⡠⢔⡫
⠀⠀⠀⠀⠀⠀⡀⠄⠒⠤⢄⣀⠀⠀⠀⠀⠁⠉⠉⠋⡉⠀⠠⠀⠀⠀⣀⠠⠂⣩⠔⢁⡠⢒⠍⢚⠵⣺
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣄⢀⢀⠉⠒⠠⠤⣀⡀⠀⠀⠀⠀⣀⡠⠔⢊⡠⠖⡁⢔⡪⠔⠉⢀⡤⠖⠋⠩
⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠀⠈⠈⠙⠒⠒⠤⠤⣍⣑⣒⡋⢁⡠⢖⡡⣐⠥⠒⠁⣀⠴⠚⠉⠀⣠⡠⡇
⠀⠀⠀⠀⠀⠀⠗⠀⠒⠂⠈⠭⠅⣒⣀⠤⠄⣀⠀⠀⠈⢈⠤⣒⠅⣊⢄⣨⠵⠊⣁⣠⢤⡺⠟⣋⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠈⠩⠀⠚⠩⠭⠕⠊⣔⡯⠑⠉⢥⢴⣛⡽⠚⡹⢖⡯⠗⣻⠃
⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁⢀⣨⠴⢞⠍⠀⣀⠥⠚⡡⠔⡋⢸⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢺⣷⡾⣫⠕⠋⠄⠒⠉⣠⡔⠬⠐⠁⢠⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠐⠆⣭⠝⠊⠁⠂⢀⣠⠶⠛⠁⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠦⣄⣀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠄⠂⠀⢀⠤⠚⠉⡀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠢⠄⣀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠠⣐⡼⠞⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠄⣰⠿⠭⠄⠀⠀⠀⠀⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠵⠋⠀⠒⠒⠒⠒⠦⠄⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠒⠊⠉⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⡦⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡴⢲⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠘⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠴⠊⠢⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢀⠔⠉⠀⠀⢠⣤⣴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⣠⠖⠁⠀⠀⠀⠙⢆⠀⠳⡄⠀⠀⠀⠀⠀⢀⠔⠁⢀⠔⠁⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⢦⡀⠀⣠⠔⠁⣠⠖⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠈⢦⡀⠑⠞⠁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠑⢄⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡤⢤⠤⠤⠤⠤⣿⣿⡿⣿⢤⠤⠤⢿⣿⣿⣿⠤⠤⠤⢼⣿⣿⡿⠤⠤⡤⢼⣿⣿⣿⠤⠤⠤⢼⣿⣿⣿⠥⠤⡤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄
⠀⠀⠀⠀⠰⣖⣤⢄⡀⠀⠈⠓⠶⠚⠁⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⣙⣻⣷⣎⣾⠆⠀⢀⡄⢠⣄⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠀⢩⣭⠿⣿⡵⢎⢾⣌⣷⣾⡿⢭⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠷⣿⡿⣯⣙⠗⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠰⡷⡄
⠀⠀⠀⣀⢶⣘⣷⢰⡀⠀⠀⠙⠳⠆⠉⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠒⠒⠀⠛⠛⠃
⠀⠀⠈⠫⡿⢻⡿⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠁⠉⠀⠀⠀⢺⣷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡮⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢉⡀⢉⣉⣁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠞⡶⢄⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡤⢏⠉⣠⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢠⡤⢉⠤⡴⡩⠏⣀⠠⡾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡕⢠⠤⠤⢠⠄⢐⣈⠃⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣄⡈⠋⠴⠨⠛⠐⠒⠏⠠⣄⢎⡟⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡶⣧⡤⠃⡒⢠⠔⢲⣠⡦⠜⠒⡒⠲⠎⠁⠀⠀⠀⠀⠀⠀⠀⡖⠤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⠁⠀⠀⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣠⠚⣿⣷⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⡠⠊⠁⢸⣧⠀⠙⠛⡿⠀⠀⠀⠀⠀⣿⣿⢀⣀⠠⡤⠀⠀⠀⠀⠀⠀⠀⢠⣭⠀⠀⢀⣀⢸⣿⡇
⠀⠀⠅⠀⢀⣸⣿⣿⣶⠋⠀⠀⠀⠀⠀⠀⣿⣿⠈⣿⠰⣶⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⢸⣿⢸⣿⡇
⠀⠀⢳⣾⣿⣯⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣿⢸⣿⢰⣶⠀⠀⠀⠀⠀⢸⣿⢸⡏⢸⣿⢸⣿⡇
⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠈⠉⠈⠉⠀⠀⠀⠀⠀⠈⠉⠈⠉⠈⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⡄⢀⣤⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣆⢠⣄
⠀⠀⢐⡀⠀⠂⠂⡎⠒⠀⠂⠀⠀⠀⠀⠀⡇⢀⣄⠙⢡⠾⠁⠀⠀⠀⠀⠀⠀⢸⠀⡟⢇⢋⡞⠹⡦
⠀⠀⢸⡇⣤⡀⠀⡇⡂⠉⠀⠀⠀⠀⠀⠀⣇⡞⡋⠳⠃⢐⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠘⠿
⠀⠀⠸⠃⣿⡧⣶⠙⠁⠀⠀⠀⠀⠀⠀⠀⠿⠴⠶⠤⠤⠶⠦⠆⠀⠀⠀⠀⠀⠸⠧⠦⠤⠤⠤⠤⠴
⠀⠀⠀⠀⠈⠠⠧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠅⣼⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⣷⡡⠃
⠀⢠⡾⣆⠀⠂⣼⠏⢴⠆⠀⠀⠀⠀⠀⠀⠀⣠⡀⣸⣷⣴⣿⠀⠀⠀⠀⠀⠀⠸⠀⠙⢂⠜⣫⣾⣷
⠀⠀⠀⠹⡆⢠⡏⠈⣠⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⠀⠀⠀⠀⠀⠀⠘⢲⣾⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠹⡟⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠤⠿⠿⠿⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⠁

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

⠀⠀⠀⣀⣤⠖⠒⠒⠒⢒⣶⣶⣶⣶⣶⣶⣶⣶⠒⡒⠒⠒⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠿⠿⠷⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⣤
⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⢀⣠⠤⠖⠖⠶⠲⣶⣷⣤⣀⣀⣴⣿⣶⠒⠖⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣾⣻⣿⣿⣵⣶⣶⣶⣶⣶⣤⣀
⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠘⢳⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡞⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠕⠜⠉⠁⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠊⠁⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠱⡄⠀⠀⠀⢀⣔⣉⠀⠁⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠂⠁⠀⠀⠀⣈⣧⣤⣴⣿⡿⠿⠛⠃⠀⠀⠀⠈⠉⠐⣢⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠶⣉⡁⠀⠀⠀⠠⣤⣶⣾⣿⡿⠿⠛⠉⠁⠀⠀⠀⢀⣀⠤⠤⠒⠒⠉⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠉⠑⠢⢄⡀⠀⠉⠉⠀⠀⠀⣀⠤⠤⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠙⠒⡄⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡤⠂⣀⠀⠉⡏⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠰⡀⠀⡇⡌⠀⢳⠀⢡⠀⢸⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢃⠀⠃⡇⠀⠸⠀⢸⠀⠨⡤⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠀⠱⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⠀⠀⠀⣀⡠⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣙⣋⣍⣹⣝⣽⣻⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢠⠶⡄⠀⠀⠀⣀⠤⠖⠒⣶⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⠾⡅⠀⣠⠊⠁⠀⠀⠀⡯⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⠶⡇⡰⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⣶⣣⡇⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠸⠤⠻⡏⠉⠉⠉⠉⡩⢽⠏⠉⠉⠉⠉⠉⠉⢹⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢣⠀⣀⠔⠋⠀⡞⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠒⠒⠻⣦⡀⠀⠀⠀⣀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠀⡼⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢸⠀⠀⢀⡇⠙⣦⠔⠋⡥⠤⠤⡌⠢⣄⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠥⣀⣀⣀⠤⠔⠊⠁⠀⠀⠀⢸⠉⠉⠉⠗⠋⠉⠙⡆⣇⣀⣀⡇⠀⠈⠛⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠚⠚⡇⠓⠒⠚⡇⣇⣀⣀⡇⣎⣉⣉⣹⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠴⡏⡋⠉⠉⡇⣇⣀⣀⡇⡯⠤⠤⢽⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡎⣉⣉⣉⡇⣇⣀⣀⡇⣟⣐⣂⣻⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠀⠉⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠒⠒⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠝⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢤⠒⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡤⡞⢃⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⢀⡗⠁⠉⠻⣤⡼⠃⠀⠀⠀⠀⠌⠐⠀⠀⠀⠄⡊⠔⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡃⠀⡞⠔⠉⢩⠀⠀⠀⠀⠀⠀⠐⢈⡀⠀⡀⠀⠐⡪⢀⠁⢂⠅⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢼⠀⠀⠲⠳⠲⠀⠤⠤⠶⠒⠫⡀⠀⠢⠍⢡⡀⠐⠡⠊⠄⠅⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣁⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣀⣑⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣈⣀⣀⡀⣀⡀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠉⠉⠓⠋⠁⠀⠀⠀⠐⠒⠒⠒⠚⠛⠛⠛⠛⠃⠙⠛⠋⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡋⡀⠀⢀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⢦⠀⠀⠲⠆⠀⠀⠀⠀⠠⠄⡀⠀⢠⢠⠄⠀⠀⠀⠀⠔⠤⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠜⠁⠀⠀⠂⣂⢔⠐⠌⠢⢠⠘⠋⠀⢀⣤⡀⠀⢠⡞⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡘⠁⠈⠀⠠⠀⠂⠀⠁⠀⠙⠀⠀⡀⠸⠦⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠈⠐⠀⡇⠕⢣⣀⡀⠐⠈⠀⠀⠀⠀⠁⠑⠀⠀⡃⠂⣬⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢐⠁⠄⠀⠰⡀⠁⠨⢀⢀⠀⠐⠀⠀⠁⠀⣔⠀⠀⡁⠈⠶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠈⠂⠤⠤⠄⠊⠀⠀⠀⣄⣐⣂⠀⠀⣆⣈⣘⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠥⠤⠅⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡠⠄⡠⠀⡀⠀⠠⢀⠄⡀⠔⡠⠀⠠⠂⠠⢀⠄⡀⠄⠠⢀⠄⠠⠔⠠⠀⠠⠀⠔⡠⣀⢄⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠠⠐⠁⠔⠀⠂⠠⠂⠐⠈⠐⠁⠒⠁⠂⠁⠊⠀⠊⠐⠁⠊⠀⠈⠀⠁⠒⠁⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢤⣤⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠋⠊⠁⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧
⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿⠎⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠸⣿⣿⣷
⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢩⣶⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠮⠍⠐⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⣿⣿⡿
⠀⠀⠸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠹⡋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣿





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡃⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣯⡁⠛⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣀⣿⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣀⣿⣿⣿⣿⣿⣿⠿⠿⠿⠒⠚⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⣶⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⠿⠿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣟⣻⣿⣿⣯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣝⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠁⣴⠋⠹⠇⠈⠛⢮⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡏⠉⠉⠀⠉⠉⢱⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⡇⠘⠶⠴⠶⠶⠶⠶⠶⠶⠶⠴⠴⠃⡗⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠁⠢⡄⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⡀⠤⠇⠓⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢏⠉⠉⠉⡏⠉⠉⠉⣇⡤⠒⢞⡄⠀⠀⠀⠀⠀⠀⠀⡼⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⣀⡾⠁⠀⠀⠈⢞⠄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⣀⣄⣀⣀⣀⣀⣍⢉⠉⡔⠀⠀⠀⠀⠀⠈⢾⣄⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢸⠸⠤⠿⠿⠿⠿⠿⠿⠿⠸⢴⠁⠀⠀⠀⠀⠀⠀⠈⢋⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠉⣿⣿⣿⣿⣿⣿⣿⣸⡽⠞⠒⠳⢶⣄⠀⠀⠀⠀⢻⡻⡄⣀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⢸⢠⠄⣤⣤⣤⣤⣤⣤⣼⠋⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠉⠑⠋⠥⢖⡤⠄⣀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⢘⣒⣛⣛⣛⣛⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡪
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣀⠿⣿⠿⣿⢿⡿⡟⣷⣄⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⢨⡘⠒⠒⠒⠒⣒⠖⢊⡤⠚⠓⠛⠛⠛⠛⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⢀⣀⣀⡀⠠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣤⡶⠶⠶⢦⣤⣤⣤⣼⣟⠉⠁⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⠛⠷⠶⣿⣥⣀⡀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠉⠙⢻⡆⠀⠀⠀⠀⠀⢿⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⡶⠶⠶⠟⠛⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠸⣯⡀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡴⠞⠛⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠈⠙⠉⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⢀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣰⣏⣀⣀⡀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⣀⣀⡀⢀⣀⣀⣀⡀⢀⣀⣀⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⣇⠀⣾⢁⣾⠉⠉⢿⡌⠉⣿⠉⠁⢸⡏⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣴⠇⠘⣿⠀⠀⣼⠇⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠙⠓⠚⠋⠀⠀⠛⠀⠀⠘⠛⠛⠓⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⣷⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢳⣰⡏⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠋⠄⠸⠿⠿⠿⠛⠙⠛⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⡃⢸⣿⣿⣿⠿⠿⠿⠿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡈⠈⣤⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣴⡇⢠⣤⣤⣤⣤⣤⣤⣤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠂⠁⢂⠘⠛⠛⠛⠉⠉⠉⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢧⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢜⣛⠅⢸⣻⣿⣿⠛⠛⠛⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⠛⠛⠛⣻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⠯⠭⡿⠃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠋
⢿⣿⣿⣿⣿⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢀⠀⠀⢀⡤⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠓⡄
⠀⠙⠿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢀⠀⠈⠣⡆⣼⡑⠌⠢⡀⠀⠀⡀⠀⠀⢈⡂⠉⠩⠍⠉⠙⠒⠋⠀⠙⠲⣿⣶⠊
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⣦⡀⠀⠠⠂⠁⠌⠀⠀⠑⢄⠀⠉⢇⠁⠀⠀⠈⡦⠐⠀⠠⢐⡨⠄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣟⠟⠁⠀⠀⡐⠀⠀⠀⠀⠀⡑⠄⠈⢆⠀⠀⡰⠁⠆⠚⠠⢉⠔⠂⣀⠀⡀⠠⠤⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠘⡄⡰⠁⠘⠀⠞⢶⡁⠀⣐⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣦⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠂⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⡿⠟⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⣠⣀⢀⣀⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠸⣧⡽⠀⣿⠀⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣑⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠋⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⠉⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠀⣀⡀⠀⠐⠁⣀⣈⡠⡤⠶⠥⠚⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢉⣹⣋⣀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰⡏⡠⠊⠉⠀⡼⠁⠀⢈⠆⡷⠖⠒⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠧⡠⢤⠞⠙⠓⠒⠛⠚⠧⢤⡀⠒⠠⠀⠒⠠⡁⢶⣀⡂⠤⠬⠍⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠒⢏⣀⡀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⣄⢀⠀⠀⡀⠍⠀⠀⡀⠀⠀⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⠀⠀⠀⠰⠁⠀⠀⠀⠃⠀⠉⣹⡖⢒⡒⢠⣖⢚⡛⠲⠦⢄⣂⢭⣀⣒⠢⠭⡅⠹⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⢰⠦⠬⠃⠀⢀⠇⠀⠀⠀⠈⠀⢀⡴⢉⠁⣿⡏⠹⠼⠁⢹⣶⣤⣤⡈⠁⠀⠀⠀⠀⠈⡦⠤⠄⣀⡀
⢹⡿⣭⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⢢⣤⠀⠀⠈⠒⠒⠣⢄⡜⠀⠀⠀⠀⡷⠖⠉⠀⢸⠀⣿⡇⠀⠀⠀⣸⣿⠀⢸⡿⠀⠀⠀⣤⣀⠀⠇⠀⠀⠀⠙⡇
⠈⣿⠷⣿⣷⣦⢄⡀⣀⣀⣠⢤⣴⠶⢮⣛⠯⣻⠆⠀⠀⠀⠀⠀⢰⠁⢀⠠⡴⠶⠁⠀⠀⢠⡎⢸⣿⣧⣤⣶⣶⣿⣿⠀⣼⠇⡄⢸⠙⠛⠈⣠⠀⠀⠀⠀⢀⠁
⠀⢻⢔⠽⣿⣝⠑⢫⣉⡢⠼⠓⠂⠉⠀⠀⠀⢘⡇⠀⠀⠀⠀⢀⠃⠀⢀⠜⠁⠀⠀⠀⠶⣡⠁⣾⣿⣿⣿⣿⣿⣿⣿⢀⣿⢠⢠⠺⡄⠀⠀⢙⣤⠀⠀⠀⡜
⠀⢸⡗⠦⣦⢝⣳⠜⢢⡄⠀⠠⢄⡀⠠⠤⣀⣻⠀⠀⠀⢀⣀⡎⢀⠔⠁⠀⠀⠀⠀⠀⣠⡞⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠆⡌⠀⠓⢒⣋⡅⠀⠀⠀⢠⠃
⠀⠀⣇⡈⠀⠝⢿⣛⡴⢶⠛⠻⢟⣛⠛⠋⢭⡿⠀⠀⠀⢹⠟⠴⣿⠄⠀⠀⠀⠀⠘⠊⣰⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⠀⣇⣀⣀⣈⡅⢿⠀⠀⣠⠃
⠀⠀⣯⢻⢮⣔⣽⣯⠑⠒⠛⠉⠀⠀⠈⠉⢻⡇⠀⢀⠤⢿⠂⢴⠁⢀⡀⠀⠀⠀⠀⢠⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠁
⠀⠀⣟⠡⡳⢊⣛⡲⠤⠄⠒⠢⠤⠬⣤⣀⣿⡇⢀⡟⡧⡇⠀⣿⠀⡸⠀⠉⠒⠂⠤⠧⢈⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⣉⣉⠁⠈⠁
⠀⢈⡷⣮⣮⢥⡿⠿⠛⠛⡛⠛⠒⠒⠶⠴⣿⠀⡸⣤⡽⣤⡤⢭⡶⠥⠤⠤⠔⠊⠙⢦⣀⠙⠶⢦⣬⣉⣙⡛⣿⡿⠿⠾⢄⠀⡠⠊⠀⠀⠉⠒⠠⠄⣀
⠀⢸⢟⠗⡻⣷⠮⣍⣓⠒⠤⠉⠉⣉⣈⠀⣿⡀⠁⣈⣵⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣲⣆⡀⢋⠉⠁⠀⠀⠀⠀⠀⠫⡀⠀⢀⡀⣀⠔⠓⠂⠬⣀
⠀⣿⡭⡫⢺⠶⢿⡶⠾⠟⠛⠓⣂⣀⠌⠭⢿⣇⠼⠿⣦⣞⠳⣄⣀⣀⣀⡤⠤⠴⠂⣺⡋⠠⢚⡅⠈⠉⢣⡀⠀⠀⠀⠀⠀⠀⠙⢶⡁⠀⠉⠑⠒⠦⠤⣄⣉⠒⢠
⠀⠀⠀⠀⠻⠛⠲⠜⠛⢛⣛⡉⠥⠖⠓⠚⠋⠉⠑⢤⡀⠙⠯⣢⣄⣉⣑⣦⣤⣀⡴⣍⠂⠉⠉⠉⢁⠟⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⣠⡚⠕⠊
⠀⠀⠀⠀⠀⠀⠈⢫⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠉⠁⠀⠀⠀⠀⠀⣉⡵⢂⡴⠁⠀⠀⠰⠀⠑⢦⣀⣀⣠⡤⠀⠂⠂⠉⡀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣀⠀⠀⠀⣀⣀⡠⠴⢒⠩⠒⠙⠢⣀⣠⣤⠴⢖⡒⠉⢀⠔⠉⠉⠙⠲⣖⠃⠀⠀⢀⡨⣖⠉⠈⠁⠐⠛⠁⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠙⢏⠉⢉⡡⠐⠛⠀⡀⠀⠀⠀⠈⢁⡠⠔⠚⠉⠛⠁⠀⠀⠀⣠⠊⠈⠲⡤⠚⠁⠀⠀⠈⣁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠳⣤⠔⠊⠉⠁⠒⠒⠤⠄⣀⠀⠀⡔⠣⡀⠀⠀⢀⣀⠠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠙⠂⠈⠁

⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣧⡬⢉⣥⣤⣍⡩⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄
⡿⠱⣿⣿⣿⣿⡷⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣾⣿⠛⠙⣿⡆
⣷⣶⣿⣿⣿⣿⣷⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⢿⣇⠀⢹⣷
⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⣄⣴⡿⠋⣀⣠⣴⡄⠀⠀⠀⠀⠸⣿⡀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠉⠙⢻⣿⠿⠟⠋⣁⣤⣤⣤⣤⣴⡾⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠹⣷⣶⣶⠿⠛⠉⢻⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣄⠀⠀⠀⣤⡀⠀⣠⣤⣤⡀⠀⣠⣤⣤⣤⣤⠀⣤⣤⣤⣤⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⣆⠀⣼⡿⠀⣼⡟⠉⠙⣿⡄⠈⠉⣿⡏⠉⠀⣿⣏⣉⠉⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣿⣶⣿⠁⠀⢿⣇⠀⢀⣿⡇⠀⠀⣿⡇⠀⠀⣿⠟⠛⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠿⠃⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠿⠇⠀⠀⠿⠿⠿⠷⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣤⣶⣤⣶⣧⣴⣦⣴⣦⣤⣶⣤⣶⣤⣵⣦⣴⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⢻⣿⣾⣿⣾⣿⣶⣿⣷⡟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⣀⣀
⠀⠀⠀⢸⣿⢻⢻⣿⣿⡆⠀⢰⣾⣟⣻⣾⠀⠀⣿⡍⠰⢿⣿⡆⠀⢸⣬⣿⣬⣿⣼⡀⠀⣸⠀⡀⢠⣲
⠀⠀⠀⢸⣼⣼⣼⣼⣼⠃⠀⢸⣟⡾⣾⣿⠀⠀⢱⣿⣿⢀⣹⣇⠀⢸⣿⣼⣴⣼⣿⡇⠀⣻⠻⠟⠽⠿⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠸⠿⠞⠿⠿⠀⠀⠚⠛⠛⠉⠙⠉⠀⠈⠉⠉⠉⠉⠈⠁⠀⠙⠙⠋⠉⠛
⠀⠀⠀⠀⢠⣶⣤⣶⠀⠀⠀⢠⣶⣤⣴⡆⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⢠⠦⣶⣶⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⢸⡏⣐⡝⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣻⣿⣿⣟⠀⠀⠀⢰⣿⢿⣿⠀⠀⢠⣶⢾⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠟⠿⠳⠀⠀⠀⢘⣿⣛⣿⡇⠀⠀⠀⣾⣿⣶⣿⠀⠀⠀⠸⠾⠭⠽⠀⠀⠸⠿⠾⠿⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⣀⠀⣀⣀⡀⡀⢀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠇⣿⢉⡇⢹⡏⠁⡇⣿⢷⣿⢸⡉⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠈⠁⠀⠁⠈⠀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⣤⢆⣀⣦⣤⠀⢸⣟⣭⢿⡏⣭⡇⠀⣴⣮⣴⣦⣵⣦⠀⠀⢸⣿⣿⣻⡇⠀⠀⠀⣿⣯⣿
⠀⠀⠀⠀⣿⣯⠿⣿⣿⠀⠐⣿⣾⣿⣷⣿⡇⠀⣿⣿⣯⣯⣽⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠻⠟⠛⠛⠻⠟⠀⠀⠸⠿⠿⠾⠃⠀⠀⠀⠻⠶⠿
⠀⠀⠀⠀⢀⣀⣄⣤⢤⠀⠀⠀⣶⣆⣀⣶⠀⠀⠀⣶⡦⣴⣶⠀⠀⠀⠀⣀⣠⡀⠂⠀⠀⣴⣦⣰⣦⣀
⠀⠀⠀⠀⠹⠽⠿⠛⠊⠀⠀⣾⠛⢿⢙⣻⠀⠀⠀⣟⣟⣿⣽⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⢺⢿⠿⣿⣿
⠀⠀⠀⢠⣤⣸⣿⡇⠀⠀⠀⣿⣶⣾⣾⣿⠀⠀⠀⠯⣿⢻⣿⠀⠀⢰⣾⣷⣿⣷⣶⡆⠀⠽⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠉⠁⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⡠⠀⠂⠉⠐⠐⠀⠉⠛⠻⠿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠂⢐⣶⡶⠶⣤⣤⣄⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢬⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⢀⡖⠈⠉⣍⢳⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣄⢁⡀⠤⡀
⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠺⠶⠶⠀⠠⠘⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⣄⡀⠀⠀⠀⠠⠆⠠⠀⠄⣀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣓⢾⡧⠀⠀⠀⠀⣺⣿⣿⣿⡿⢿⣿⣿⡀⠀⠀⠀⡀⣀⣀⣭⣭⡏
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⢠⠤⠤⠤⡇⠀⠀⠀⠁⠭⢾⣷⠶⠷⠛⣛⢹⠶⠠⠀⠉⠉⢛⣭⡶⠯⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢔⣀⡀⠈⠧⠀⠢⠤⠀⠀⠀⠤⢄⣀⣶⣶⣿⡃⠀⢈⠀⠤⠒⣒⠚⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠍⠉⢅⣲⣇⠀⠀⠀⠀⠩⣔⣟⣻⣯⠽⣾⣿⣤⠀⣪⢠⡀⢠⠄⠬⠔⠓⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠛⠛⠛⠛⠛⠚⠰⠆⢰⣿⣿⣿⣛⠛⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠂⠈⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⠀⡒⠁⠀⠀⠀⠀⠀⠀⠊⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢀⣀⣀⣀⠜⠀⠀⠀⠀⠀⢀⠐⢁⠌⠁⠁⠐⠒⠒⠠⠤⠄⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⣤⣂⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠂⡤⠄⠀⠠⣦⠄⠀⠀⠀⠀⠀⠀⠀⠁⠒⣂⠴⢛
⠀⠀⠀⠀⠀⠀⣋⠂⠀⠀⠡⠄⢀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⢏⠊⢠⡭
⠀⠀⠀⠀⠀⠀⢳⠶⠴⢤⢄⣀⣀⡀⠀⠀⠀⠉⠀⠒⠂⠀⠀⠤⠄⠀⢀⣀⣀⢀⠀⠀⠀⣠⠴⠋⠁⢄⣤⢚⢃⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠉⠉⠉⠒⠒⠶⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢩⠃⣠⢴⠞⠣⢣⢂⢢⠇
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠐⠒⠲⠲⠦⠬⣼⢟⢍⡢⠡⠣⢃⢢⢺
⠀⠀⠀⠀⠀⠀⠘⠀⠈⡟⡇⠀⢠⠏⡗⡠⢒⠢⣄⠀⣄⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣏⢈⡶⡱⡡⠡⠣⠱⢹
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢷⢻⠀⣞⡼⡸⣠⠛⠻⡈⣆⠓⢳⡳⡷⢬⡇⡯⣃⣒⢶⡀⠀⣯⠞⡑⡔⡰⡱⡱⠡⢩
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⡚⡾⢱⠃⡇⡇⠀⠀⣗⠸⠀⢸⣒⡇⠀⠀⡯⣇⠈⠉⠁⠀⡟⠌⡜⡔⡐⡱⠡⠡⠫
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣧⢱⠇⠀⣇⢧⠀⢠⡂⡽⠀⢸⣊⡇⠀⢠⢗⡭⢼⠆⠀⠀⡏⠌⠌⡔⡐⡡⠡⠡⡃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠚⠀⠀⠘⢮⣛⣉⡞⠁⠀⢸⡢⡇⠀⢸⣵⡇⠀⠀⠀⠀⢎⠚⠘⠘⡴⡱⡱⡳⡇
⠀⠀⠀⠀⠀⠀⠀⢇⠐⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠾⠃⠀⠸⣬⣩⡻⣿⡄⠠⡏⠌⠜⠜⡜⡽⡱⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠠⢏⠎⠎⢌⣬⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠒⠒⠒⠤⠤⢀⣀⣀⣀⠀⠀⠀⠠⠀⣇⣮⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢂⠡⢊⠄⡊⠔⡡⢂⢔⠅⢈⠔⡁⢂⠔⡠⠂⢔⠔⡡⢀⢄⠄⡠⢀⠄⡠⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠊⠀⠠⠒⠕⡡⠀⡡⡪⢂⠕⢀⠔⡁⡠⢊⠔⡡⢂⠌⡔⡡⢊⠔⡡⡪⢊⢔⠕⡡⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠁⠊⠐⠁⠊⠐⠑⠁⠈⠀⠁⠁⠈

⠀⠀⠀⠀⠀⢀⣤⡶⣒⣒⣒⣒⣒⣲⣶⣒⣒⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⠒⠒⠒⢶⣦
⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣹
⠀⠀⠀⠀⠀⡏⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⢲⡆⣿⢸⡆
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠃
⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣼
⠀⠀⠀⠀⢘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⣿
⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿
⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⣿
⠀⠀⠀⠀⠀⢹⠀⠻⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⣧⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠤⠤⠤⠤⠄⡆⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣚⣛⣛⣛⣉⣉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣤⣀
⠀⠄⠀⠀⠀⠀⠀⠀⣰⢏⡭⣭⢭⡭⢭⡭⢭⠭⢭⠭⢭⡭⢭⡭⢭⡉⢩⠭⣽⠿⡿⢿⠻⣆⠀⠀⠀⣾⣡⠜⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⡾⢭⠯⢽⠭⢽⠭⢽⠭⢽⡭⢽⡯⠭⡯⠭⡇⠘⣟⣻⣛⣻⣚⣧⠹⣆⠀⠀⠹⣅⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣰⣏⣈⣓⣛⣒⣛⣒⣚⣒⣚⣒⣒⣓⣚⣓⣒⣓⣒⣛⣀⣱⣤⣷⣤⣷⣤⣇⣹⣆⠀⠀⠘⢧⡀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠻⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠛⠶⠴⠞⠋

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣤⣀⣠⡄⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣾⣟⢋⣉⣁⣀⠠⠤⠀⠋⠁⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣭⣤⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣼⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢪⣤⡀⣿⡇⣿⣦⢤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⣰⣿⣿⣿⣿
⠀⠀⣠⢔⣉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣯⣤⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠉⠒⠦⢄⣀⣠⠎⠀⣠⣼⣿⣿⣿⣿⣿
⠀⠈⡁⢸⣿⢀⣤⡕⢄⠀⠀⠀⠀⣀⠔⣥⡄⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⣜⣁⣠⣶⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣤⣕⢤⡠⢾⣥⡄⣿⡇⣿⣿⢸⣿⡆⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢰⣶⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿
⠀⣿⣿⢸⣿⢸⣿⡇⣿⣿⢸⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿⡇⣿⡇⣿⣿⢸⣿
⠀⠻⠟⠸⠿⠘⠿⠇⠿⠟⠸⠿⠘⠿⠃⠿⠇⠻⠿⠸⠿⠃⠿⠇⠿⠟⠸⠿
⣿⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠉⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠉⠁⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⠞⡟⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠎⡘⠀⠀⠀⠀⠀⢆⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠛⢌⡉⢰⠁⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢻⣄⠀⣀⣤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠶⠶⠚⠛⠋⠉⠉⠉⠙⠛⠛⠲⠶⠶⢦⣽⣿⣉⣀⣀
⠀⢀⣀⣤⡴⠶⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠠⣾⣋⣉⠓⠒⠦⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠓⠶⠶⢦⣤⣄⡀
⡼⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⣄⣈⣉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⠉⠉⠀⠀⢀⣀⣀⠤⠤⠔⠒⠚⠛⠉⠁⠀⠀⠀⣿
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠸⢦⣴⠦⠤⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣼
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠦⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠖⠒⠋⠉⠉⠉⠀⠀⠈⣿⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠤⣤⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⢲⠲⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⡇⠀⠈⢃⠘⠃⡹⡶⢋⡑⢾⠉⠒⠦⣄⣠⣄⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠈⢧⠞⠀⢱⣈⣋⡸⡇⢠⠖⢾⠃⠠⡼⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠀⠸⣄⣉⡷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠙⠲⠶⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣤⣄⣀⠀⠀⠀⠀⣀⣤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠇⠀⠀⢾⠋⠉⠉⠉⠙⡇⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⢻⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⡐⠀⢸⠀⠀⠀⡇⠀⢂⢀⠀⢸⡇⠀⠀⣿⠀⢀⠀⠂⠀⡇⠀⠀⢸⠀⠐⡀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⡄⠈⠁⠂⢹⠀⠀⠀⡇⠀⠈⠁⢠⢸⡇⠀⠀⣿⢠⠀⠉⠐⠀⡇⠀⠀⢸⠀⡂⠉⠀⡄⣿





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡴⠒⠛⠋⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⠀⢹⡆
⠀⠀⠀⣼⣧⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡇
⠀⠀⠀⢻⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⡏⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠘⡇⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⡇⢿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⢿⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡧⠀⡇
⠀⠀⠀⠀⢸⡘⡇⠈⣧⣤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⣀⣀⣇
⠀⠀⠀⠀⠈⣇⣧⣤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⢉⣉⣁⣂⣀⣀⡀⠘⣆
⠀⠀⠀⠀⠀⢿⠹⡄⠀⢀⡤⠤⡤⠄⢤⠒⡖⠒⡖⠘⣏⢹⣉⣙⣍⣉⣅⠨⣦⠽⡤⢧⠼⡄⠘⣆
⠀⠀⠀⠀⠀⠘⣆⢳⡀⠈⢧⠄⠺⡖⠚⡖⠙⡉⢹⣉⣹⣄⣵⡤⠴⡤⠼⡆⠲⡒⢳⠚⡗⠙⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠈⢷⠒⠳⠀⠙⡉⠹⡀⢣⠀⢱⣀⣣⡀⢵⡤⠼⡤⢷⠐⢳⠚⡂⠹⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⠘⣷⠒⢳⠒⠹⡉⢻⣉⣇⣀⣣⣀⣧⣠⣥⠤⠵⠴⠷⠚⠓⠛⠋⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠘⠒⠚⠛⠋⠉⠉⠉⠉⣭⡤⠤⠤⠴⠶⠒⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣀⡀⢀⣀⣀⡀⠠⢷⡀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠒⠒⢚⣒⣂⣈⣉⣤⡤⢤⣴⣶⣶⣾⠯⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡷⠤⣤⣤⣶⣖⣒⣚⣻⣿⣭⠭⠭⠽⠿⠓⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⢐⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⠒⠒⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠯⡇⢩⢳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂
⠀⠨⠒⠒⠒⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⡯⠭⣭⡇⢨⣼⠍⣷⠀⠀⠀⠀⠀⠀⠒⢤⠊⡚⠢⡀⡨⢴⠀⠀⠀⠀⠀⠀⣀⠀⢲⠈⠀⢸
⠀⢠⠤⠤⠤⠤⢄⠁⠀⠀⠀⠀⠀⠀⠀⠹⣫⢵⣢⣳⢮⣓⠏⠀⠀⠀⠀⠀⠈⠑⠃⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⢸
⠀⠘⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠦⠀⠼⠀⠄⠘⠄



⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣚⣭⣝⣶⣄⠀⠀⠀⠀⠀⠀⢠⢄⢔⢢⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠚⠚⠐⠒⠃⠒⠒
⠀⠀⠖⣀⡄⢸⠤⡀⠀⠀⠀⠀⠀⠀⢠⢫⢻⠃⠀⠈⢳⠹⡄⠀⠀⠀⠀⠀⢈⢮⡙⠵⣫⠚⢊⡲⡕⠁⠀⠀⠀⠀⢙⢙⢈⠋⡃⠀⢀⢀
⠀⠤⠇⡇⠀⢸⠀⠉⠃⠀⠀⠀⠀⠀⠘⣜⢾⣄⡀⠀⠈⢀⠅⠀⠀⠀⠀⠀⠘⠂⠈⠉⠡⡢⢪⠛⠺⠀⠀⠀⠀⠀⠉⢉⢈⠉⡅⠉⠉⠈⠉
⠀⣀⣅⢧⣀⣸⣀⣄⠆⠀⠀⠀⠀⠀⠀⠈⠣⠯⡂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠀⠀⠀⠀⣴⣰⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣹⢉⠽⣁⣙⠉⡃
⠣⠒⠂⢤⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⡇⠀⠀⢈⡄⠀⠀⠀⠀⠀⡇⡇⢠⡀⠀⠈⡐⠢⠀⠀⠀⠀⠀⠀⠀⡁⣸⣈⣶⢫⠼⣤⡇
⠐⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⡠⠊⡇⠀⠀⡠⠁⠀⠀⠀⠀⠀⢇⠇⠀⠀⢠⡄⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡸⢏⣸⣈⣲⣁⡅
⠠⠀⠀⠀⠧⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠉⠢⠤⡧⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢋⣱⣁⣸⣀⣨⡀⠅



⠀⢀⡀⡆⠄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢁⠸⣈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡒⢆⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠉⠢⡀
⠀⡇⠀⡇⠀⠤⢸⢸⠀⠀⠀⠀⠀⠀⢰⠁⡜⠁⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⢰⠉⠉⡆⠀⢀⠀⡇⢨⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡄⠀⠀⢘
⠀⡇⠀⡆⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠸⣀⠧⡀⠀⠀⡠⢄⡄⠀⠀⠀⠀⠀⢸⠀⠀⠈⠒⠉⠀⡇⢰⠀⠀⠀⠀⠀⠀⠰⣀⢔⢥⠃⠀⠀⡔
⠀⠣⠆⠧⠀⠤⠘⠼⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠉⠉⡠⠎⠁⠀⠀⠀⠀⠀⠸⠠⠄⠀⠠⠀⠀⠇⠨⠀⠀⠀⠀⠀⠀⠀⢴⠁⠸⠥⠔⠊
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⠉⢧
⠀⠀⠀⠀⠀⠀⠀⢠⣶⢖⣶⡶⠶⠶⠋⠰⢯⠗⠈⠳⢶⣶⣖⣶⣲⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣹⣿⡽⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣫⣿⠁⣠⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢯⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢟⣽⠀⣿⢼⣹⠆⡏⠀⠀⠉⠉⠉⠛⠀⠀⢸⡵⡳⡧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⡟⠊⠉⠓⡇⠀⠘⠋⠉⠉⠁⠀⠀⢸⣞⢟⡷
⠀⠀⠀⠀⠀⠀⠀⣼⣣⣿⠀⠙⠛⠋⠉⢁⡀⠐⠛⠛⠛⠓⠀⠀⢸⣫⢞⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡫⣿⠀⣰⢶⣲⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢟⡿⠀⠙⠚⠋⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⢸⢟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⡷⣫⡇⠀⢸⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢟⡇⠀⠈⠉⢩⠄⠛⠛⠛⠛⠛⠒⠂⠀⠀⣿⣿⢿⠇⠀⠀⠀⠀⢀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⢰⡿⢻⡇⠀⣾⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⠀⠀⠀⣀⡴⠋⣠⡏
⠀⠀⠀⠀⠀⠀⢸⡷⣻⠁⠀⢉⡉⠁⢉⡿⠟⠓⠒⠒⠒⠒⠀⠀⣿⡫⢿⢀⣠⠞⢁⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⢸⡟⣽⠀⠀⣿⣽⡶⣋⣀⣀⣀⣀⣀⣀⠀⠀⠀⣟⣴⡿⠋⣠⠶⠟⠋
⠀⠀⠀⠀⠀⠀⢸⣟⢾⣤⣤⣬⣭⣤⣄⣉⣉⣉⣉⣉⣉⣀⣀⣠⠟⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢸⣧⣫⢎⣞⡵⣱⣫⣫⢞⡝⡵⣱⣣⣫⣯⠞⣁⣴⣋⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡧⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠿⠛⠉⣿⡏⠙⠻⢷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣠⡀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⣿⡇⠀⣴⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠀⡀⠀⣿⡇⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡗⠀⣿⡇⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⢸⣿⠀⠀⠘⠃⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⣿⡏⠛⠻⢿⣿⠀⣶⡆⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⣴⡿⠛⠿⠷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠁⠀⣀⣿⡇⠀⠀⢸⣿⠀⠻⠃⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⠟⠀⢀⡀⠀⢀⣀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣴⣶⠿⠛⣿⡃⠀⠀⢸⣿⠀⠀⢠⡄⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⢸⡇⠀⠸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⢀⣀⣿⡇⠀⠀⢸⣿⠀⠀⢸⡗⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⡶⠿⠛⠛⣿⡇⠀⠀⢸⣿⠀⣀⡀⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⢸⡇⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣀⣠⣿⡇⠀⠀⠈⠁⠀⣿⡇⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⠈⠁⠀⠈⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⠛⠛⠉⣿⣷⠿⠻⢷⣦⠀⠉⠀⠀⠀⣿⡇⠀⠈⠁⠀⢸⣿⠀⠀⢰⡆⠀⢰⣶⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣠⣴⣿⠃⠀⠀⠀⢻⣧⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠿⠀⠀⠘⠃⠀⠘⠛⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠋⠉⢻⡇⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⠀⢠⡄⠀⢀⣾⠟⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⣿⣦⣀⣀⣠⣾⠏⠉⢉⣉⣉⣉⣉⣉⡉⠉⠉⠉⠉⣿⠀⠸⠇⠀⠸⣿⡀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡏⢻⣿⠋⠁⠀⠀⢸⡟⠛⠛⠛⢻⣿⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠙⠻⣷⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣿⠇⢸⣿⠀⣤⣤⣤⣼⣷⣤⣤⣤⣼⣿⣤⣤⣤⣴⣿⣤⣤⣤⣤⣤⣤⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠶⠤⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣲⠶⠛⠉⠉⣉⣈⠭⢟⣠⡀
⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡤⠖⠋⠀⢰⡇⢀⡠⠔⠊⣡⠴⢒⣉⣼⠃
⠀⠀⠀⠖⢻⡷⠶⠶⠶⠶⠶⠶⠶⢶⣶⣦⣤⡤⠤⣤⣤⣤⣤⣤⣤⣤⠴⠯⠬⠷⣖⣋⣻⡏⠉⣉⣉⠽⠋⡡⠴⠚⣉⠴⠊⠁⡗⡏
⠀⠀⢰⠶⠟⠒⠒⠒⠚⠋⠉⠍⠍⠀⠀⠀⣀⠀⣐⣂⣀⣀⣀⡀⢒⣀⣀⣀⣐⣒⡒⠒⠀⠤⠭⠭⠭⣶⡡⠔⢂⠩⢛⠨⢙⡩⠷⢇⣀⣀
⠀⠀⢸⡄⠀⢀⢀⣀⣀⠤⠿⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠬⠭⠭⢉⣉⣙⣻⡂⢩⣰⠼⠒⠊⠁⠀⠀⠀⠀⠈⣉⣳⡦
⠀⠀⣼⠀⣀⣉⣁⣤⠤⠤⠤⠶⠶⠒⠒⠒⢒⣛⠛⣛⣛⣒⣒⠒⠒⠒⠒⠒⠲⠦⠤⠤⠤⠤⠤⠴⠶⠾⠗⠉⠀⠀⠀⠀⣀⣤⣤⠒⠚⠉⠁
⠀⠀⠉⡠⠋⠁⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠬⠥⠭⠭⠽⢶⣶⣶⣦⣤⢤⣄⣀⣀⡀⠀⠀⠀⠀⣀⠤⠖⠩⢑⡒⠊⣻
⠀⠀⠚⠉⠉⣩⠉⠀⠀⠀⠀⠀⠐⠐⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠒⠒⠒⠛⠂⠀⠀⠀⠉⠉⢹⡲⠭⢀⣒⠤⠍⠀⣒⣀⡽
⠀⠀⠀⠒⠂⡇⠀⣀⣀⣀⣀⠀⠈⠉⠉⠉⠉⣉⣁⣒⣛⣛⣒⣒⣒⣒⣒⣒⣶⠦⣤⣤⣤⣭⠍⢉⡀⣼⠐⣊⡡⠴⠒⠊⡉⠉⠀⠠
⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠭⠛⠉⠉⠉⠩⠭⠍⠁⠈⢨⠔⠂⠀⣈⠠⠀⠀⠀⡇
⠀⡰⠁⠀⠤⠤⠐⠒⠒⠒⢈⣉⡉⣉⣉⣉⣉⠉⠉⠭⠷⠶⠶⠶⠶⠶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡀⠄⠒⠀⠠⢀⣰⣇
⠀⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⢸⠀⠀⠐⠚⠣⠤⠤⣼⣄⣱⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠒⢺⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠙⠛⠛⠋⠙⠋⠡⠤⠤⠤⠤⢀⣀⣀⡀⠀⠈⠉⠉⠉⡝⠀⢸⠀⢈⡀⠄⠀⠂⢁⡏
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠬⠭⢍⣉⣙⣛⣁⣀⣼⢈⠠⠀⠀⠈⢁⣰⠇
⠀⠀⠀⠀⠄⠚⠑⠠⡤⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣈⣙⣛⣛⣁⣒⠒⠒⠒⠢⠤⠤⠄⣇⢀⣀⠤⣔⡻⠿⡟
⠀⠀⠀⠀⠀⠀⢹⢎⡀⢤⡤⠄⠒⠒⠒⣊⣩⠭⠽⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⡉⢉⣙⠛⣋⠥⢒⣋⡁⠄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠉⠓⠀⠈⠁⠀⠀⠀⠀⠀⢠⣤⡤⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⣈⡥⠤⠖⣂⢹⠇
⠀⠀⠀⠀⠀⠀⠀⡸⠤⠤⠤⠤⠄⠰⠶⠶⠶⠶⠶⠒⠐⠲⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠁⠀⠐⠘⠋⠉⡟⠀⠀⠀⡀
⠀⠀⠀⢀⡠⠀⠀⢳⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⣶⡶⠶⠶⢶⣶⣤⣤⣄⣀⡀⠀⣽⠀⣀⣡⣤⣤⣶⡧⢀⡴⠊⡀
⠀⠀⣉⠤⢒⣩⠔⢊⡹⠗⢒⡲⠖⠒⣶⠶⠶⢶⡶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠦⠶⠤⣤⠤⢤⡿⠭⢗⠟⢉⠵⣪⠔⣉⠴⢊
⠀⢀⠥⠒⣉⠤⠚⢡⠔⠊⠉⠀⠔⠉⠀⠔⠊⠁⠀⠂⠁⠠⠐⠉⠀⠠⠐⠁⠠⠒⠁⣀⠔⠊⣠⠖⠉⠠⠊⠡⠚⠵⠊⡤⠊⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠶⢶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠁⠀⠂⠀⠐⠒⠒⠒⠀⠠⠬⠭⠭⠭⠉⠉⠛⢉⣙⣛⠛⠛⠒⠺⠶⢶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⢀⠀⠀⠀⠀⡇⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠉⣐⣒⡒⠠⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡉⠰⢠⠀⠖⢀⠄⠉⡁⠁⣒⠒⠠⠭⠅⢁⣒⠒⠠⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠤⠐⡠⠄⡀⠐⢂⠀⠙⠁⡄⠈⣠⠒⢁⡠⢄⠒⠤⠀⢉⡉⠁⣒⠂⠤⠭⠭⢁⣒⠂⠴⠃⡇⡀
⠀⠀⠀⠀⢀⡠⠒⠉⡀⠀⢁⠀⠈⠠⠄⠀⠡⠀⠈⠀⠒⠈⠔⠂⠄⠈⢡⠐⠈⠀⠠⠆⠀⠤⠈⢀⠀⠀⠆⠀⠤⠀⣀⢥⣟⣉⠇
⠀⠀⠀⠀⢨⣉⠉⠒⠒⠂⠥⠴⢅⣀⡐⠀⠄⠁⠀⠀⠀⠈⠁⠀⠀⠆⢀⡄⠀⣁⠀⠊⠀⠔⠂⡡⠀⠈⣁⠤⠒⣉⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠠⠤⠤⣀⣉⠉⠑⠒⠂⠤⠤⢄⣀⡠⠂⠐⠆⢀⠀⠈⡀⠀⠃⣀⠤⠒⣉⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⠤⣀⣉⡉⠐⠒⠒⠤⡤⠒⢉⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠓⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠓⠆⠈⠰⠀⠀⠀⠨⠄⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⢐⠀⠆⠀⠀⠀⠒⢌⢩⠉⠀⡇⡃⣀⣈⣈⣈⠀⠄⠀⠀⠀⢤⠀⠁⠀⠀⣀⠀⣱⢆⠀⠀⠀⠠⠀⠠
⠀⠀⠁⠐⡀⢨⢄⡅⠀⠈⠎⠉⠬⠵⠉⡇⡂⡀⢐⠨⡆⠀⠀⠀⠀⠘⠀⠀⠀⠘⠈⡈⠃⠨⠀⣀⣀⡰⠀⠀⠀⠀⠈⠁
⠐⠀⠐⠄⠌⠍⠠⠴⠶⠶⠖⠁⠀⠀⠀⡇⣀⣀⣀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠗⢋⠈⠂⢸⠉⠀⠀⡀⠈⠂
⠃⠀⠀⠐⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢒⢠⠉⠉⠘⣀⠀⠀⠁⠀⠀⠀⠐⣦
⠀⠀⠀⠀⠠⡄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠏⠸⠀⠀⠀⠘⡄⣀⠀⠀⡒⠛⢳⠒
⠀⠀⠀⡇⠐⠒⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠰⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠴⠆⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡴⠶⠶⠛⠳⠶⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡰⠀⡐⠁⠖⠀⠀⢀⣾⡶⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡘⠁⠀⠀⠀⣠⣾⣿⡀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠐⠀⠀⣠⡿⠿⠟⠁⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣯⡱⠶⠒⠛⠁⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⣸⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⠃⠐⠛⠛⠻⡏⠙⢧⠀⠀⠀⣠⠏⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣾⣧⡤⠞⠁⠀⠀⢀⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⣷⡀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⢠⣿⣿⣷⡶⠞⠛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⢀⣿⣿⣿⣿⣟⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⣶⣶⡶⠶⠶⠶⢶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡋⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣠⠴⠚⠉⠀⠀⠉⠛⠿⣶⣄⡀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠐⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣯⣤⣶⣭⣥⣤⣤⣤⣤⣤⣤⡀⠀⠀⠉⠻⢷⣦⣄⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠙⠻⢷⣦⣄⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⣿⡏⠙⢿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣿⡇⠀⠀⠈⠻⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢶⣶⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠿⠿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢤⠀⡠⢠⠆⢦⡀⢶⠆⢰⠆⠀⠀⡆⢤⢀⡴⠰⣄⢤⣠⠄⠀⠀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣤⠁⢣⣀⣠⠃⢸⠀⢸⣁⠀⠀⡇⣹⠘⣄⢀⠟⣠⠻⡀⠀⠀⣀⣀⣤⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠒⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⡇
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣟⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⡤⠀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔⠒⠒⠐⢛⣀⣉⣉⡈⠉⠉⠉⠉⠉⠁⠀⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠖⠞⠚⠒⠒⠒⠒⠚⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠒⠀⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠠⢁⠠⠡⡡⡁⡐⢀⢣⣧⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⢁⢁⠂⡄⡑⠡⠀⢀⢂⡴⡷⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠂⠂⠌⠘⠡⠡⢁⠂⣼⠂⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠀⠌⠈⢌⠄⠀⠁⠁⠆⣼⠃⠅⢿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠆⠜⡌⢀⠂⠆⢀⠀⠌⣰⢃⡮⠠⢻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡐⡀⠘⢠⠋⠄⡄⠀⠌⣰⠇⡾⢁⠐⢻⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡐⡰⡐⠠⡇⠀⡐⠌⡘⢠⠏⡸⢁⢂⠂⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⠐⡐⠁⡝⡐⠐⠀⡐⢠⠏⡰⢃⢂⠂⠲⢻⣗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⡐⢀⠛⠀⠁⠐⠔⢠⡎⣰⢃⢊⢎⢮⠡⢹⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⠡⠁⠠⠣⠡⠠⠁⠀⢠⡟⣰⢣⢂⠂⢢⢁⠃⢩⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢡⠣⠁⠡⢠⠁⠠⡰⠁⡙⡠⠃⠃⢌⢢⢁⢂⢒⠉⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠃⠃⠴⠡⢃⠶⠠⠁⠰⠴⠕⠌⢎⠊⠆⠀⠂⠞⠄⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⡂⣰⣆⣀⢀⣉⣉⣉⣉⣀⢀⣀⣀⣀⣒⣒⣒⣒⣒⣒⣒⣞⣒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣾⣾⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⠀⢈⠉⠪⣟⠝⢹⣬⡝⣭⣫⣝⣭⣋⢫⣝⣭⣪⡝⣭⢫⣭⣝⣫⣠⡉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠂⠀⠀⠀⢀⣀⣀⣀⣀⣜⣀⣠⣁⣀⣈⣀⣀⣉⣀⣒⣃⣰⣄⣱⡐⣒⣘⣆⣐⣒⣐⣄⣀⣀⣀⠐⢒⡢⠵⣆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠃⠀⣠⡀⣀⣀⣀⣀⣀⣇⠀⠀⣢⢐⠒⡀⢣⡐⣤⠚⢢⡀⠀⠀⠀⠀⠈⢆⠲⠈⢧⡙⡄
⠀⠀⠀⠀⠀⠀⢀⠎⡔⠀⠀⠀⠀⠀⠈⣹⠋⣉⣉⣃⡈⣨⡁⠈⠀⣉⠀⠀⢈⡀⠀⠈⡁⠀⠀⠈⣊⠢⡙⠀⡙⣬⣥⠛⣍⠀⠜⢏⠨⡠⡂⠀⢳⣜⡄
⠀⠀⠀⠀⠀⢀⠎⣈⠀⠀⠀⠀⠀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⣬⡍⠋⠁⠿⡹⢀⡀⢉⣈⡛⣁⣁⣹⡻⡀
⠀⠀⠀⠀⢀⠎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠥⠍⢭⠁⠉⠈⠀⠀⠙⠬⡑⠭⣋⠉⡋⠧⣉⡋⠙⠛⠦⣳⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠈⠑⠢⣉⠢⢕⠨⣙⠣⡢⢅⠂⢳
⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢍⡢⢅⠂⢅⠨⡀⢃
⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠀⠚⠢⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠬⣀⣌⡂⢸
⠀⠀⠘⠯⣀⣒⣖⣶⢖⣖⡰⣤⢤⢤⠤⠤⠤⢤⠤⠤⠤⠤⠤⢤⠤⠤⢤⢤⠤⡤⡤⡤⠤⠤⢤⠤⠤⠤⠤⡤⠤⠤⢤⠤⡤⠤⠤⢤⠤⢤⢤⠤⡤⣤⣤⣤⡼⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠘⠁⠙⠋⠊⠘⠉⠛⠋⠋⠚⠋⠛⠋⠚⠙⠛⠛⠉⠛⠛⠛⠛⠛⠘⠑⠃⠛⠉⠚⠑⠛⠛⠋⠋⠉⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣀⢀⠠⢀⠄⠠⠀⠒⠀⢐⠀⠀⠀⠠⠄⠀⠀⠀⡰⠄⡐⠀⠀⠐⢂⡒⠀⡐⠠⢀⠂⠠⣐⠀⠀⠂⠀⠐⠀⠀⠂⠀⠂⠠⢂⠁⢐⡀⠀⡢⠔⡀⢀
⠀⠀⠀⠀⠀⠁⠀⠁⠐⠈⠀⠀⠉⠐⠀⠀⠊⠀⠂⠀⠒⠁⠒⠁⠐⠊⠀⠊⠁⠂⠁⠐⠈⠀⠂⠁⠐⠈⠀⠂⠀⠔⠊⠀⠊⠁⠐⠀⠔⠊⠁⠂⠁⠐⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⢨⣷⣄⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣠⡴⠟⠛⠳⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣤⣀⣤⡶⠞⠋⠉⠙⠛⠋⠀⠈⠉⠙⠛⠳⢶⣿⠉⠉⣷⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⢿⡅⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢹⡏⠙⠁⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⠀⢸⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢰⡶⠲⠶⠶⠶⢲⡆⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣿⠀⠀⠀⠀⠀⣿⠀⢸⣇⠀⠀⠀⠀⢀⣿⠀⢸⡇⠀⠀⠀⠀⢸⣇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠉⠑⠒⠂⠀⠤⠤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠓⠒⠒⠶⠤⠤⢤⣄⣀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠤⢈⠑⠂⢅⠂⠄⡉⠲⡍⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⠘⠀⠈⠐⢨⠐⡢⠄⣉⠖⢩⠙⠲⢎⡑⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⡀⠀⠀⠀⠀⠎
⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠐⠢⠄⣉⠤⢈⡑⠂⡤⠂⠢⢤⣁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼
⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣁⠘⠠⠐⠋⠤⢈⡰⠆⠀⠀⠀⠠⠗⢀⠄⡀⠀⠀⠀⠀⠀⠀⠤⣀⠠⠇
⠀⠀⠀⠀⠈⠳⢄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢀⡠⠚⢠⡄⠈⠂⣄⡑⠄⠊⣉⡁⠤⣀⡨⠒⠎⠁⠀⠀⠀⠀⠀⢀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠠⠠⠂⠈⠀⠄⠒⠀⠄⠐⠒⠤⠀⠒⢠⠄⠂⠁⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⡀⠀⢤⠈⠉⠠⠀⠈⡁⠤⠒⠁⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠁⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⢀⣠⠴⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⢿⣿⣿⣿⠉⠛⠛⠛⠒⠶⠶⠦⢤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⡟⠉⢠⣴⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠁⢠⣿⣄⠀⢠⣶⣿⣿⠃⢸⠇⠀⠀⠀⠀⢀⣠⡴⠋⢸⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣧⣴⣦⡀⠹⣿⣦⠈⣁⣤⣿⣤⣿⠀⠀⢠⣟⠛⠋⠁⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⠋⣀⡈⠙⢿⣿⣦⣨⣿⣿⣿⣿⡏⢠⡏⠀⠀⢸⠏⠙⢛⡟⠒⢶⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⠃⣼⣿⣿⣶⠀⢹⣿⣿⣿⣿⣿⣿⣧⣾⣧⠦⣤⣾⠀⠀⢸⠇⠀⢸⣿⣿⣿⡟⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣾⠷⣤⣈⠉⠛⠋⢠⣿⣿⣿⡿⠟⠋⠉⠀⠸⣿⣶⣿⡟⠀⠀⣼⠀⠀⣼⠿⠿⠿⠃⠀⠹⠿⠿⣿⢿⡆
⠀⠀⠀⠀⢸⣿⣄⡈⠙⠿⣷⣶⣿⣿⣿⣷⡀⠀⠀⠀⠀⣀⣠⡍⠈⢷⡀⠀⣏⠀⢠⣿⣦⡀⠀⠀⠀⠀⠀⣀⣴⣿⣷
⠀⠀⠀⠀⣼⣿⡿⠿⠗⠀⠈⣻⣿⣿⣿⣿⣷⠀⠀⣶⢛⣉⡄⢹⡄⠈⢷⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣇⣀⣤⣴⣶⣾⣿⡿⠿⠿⠿⠿⢧⠀⠘⣇⠿⣛⣢⠷⠀⠈⣿⡉⠉⠛⠻⠿⣇⣠⣶⣿⣦⣀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⠻⢦⣄⡀⠀⠀⠀⢤⣼⡧⠤⠾⠿⠿⠖⠒⠛⠛⠛⠋⠉⠀⠀⠀⣀⣉⣙⣻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠈⠙⠷⣦⣄⣀⣀⣀⣠⣤⣤⣤⣤⣴⣶⠶⠶⣶⠛⠛⠛⠛⠋⠉⠉⠉⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠁⠀⠀⠀⢸⡇⢸⡳⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠐⣷⢟⢇⣟⢦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⣿⠿⠾⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡶⠶⠟⠛⠉⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⢸⡇⠀⠀⠀⠀⣀⣀⣤⣴⠶⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣴⡶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⢋⠉⠉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡏⠀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⡿⢛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣿⠠⣤⠖⡥⠀⣛⣛⣛⣛⣋⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⡇⣿⠻⢫⡟⣿⠶⠶⠶⠶⠶⣿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠿⠤⠥⠤⠗⠀⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣇⣛⣛⣛⣃⣿⣷⣶⣶⣶⣶⣿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣴⠒⠒⢒⡀⠀⣤⣤⣤⣤⣄⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⡏⣭⣭⣭⢽⣿⠿⠿⠛⠛⠻⣿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣿⠘⠶⠋⡖⠀⣤⣤⣤⣤⡤⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⡇⣿⣍⣵⡏⣿⡛⣛⠛⠛⢛⣿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠙⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣧⣭⣭⣭⣵⣿⣿⣿⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣿⢉⠉⣩⠀⠀⠶⠶⠶⠶⠶⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣛⠋⠚⠒⡶⣿⣿⣏⣙⣛⣛⣛⣿⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣿⣈⣛⣁⡷⠐⠒⠒⠒⠒⠒⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣟⡃⣻⣆⣚⠃⣛⣉⣩⣭⣭⣭⣿⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣧⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⡞⠙⠢⣀⠀⠀⢠⠏⠱⣤⠎⠈⠙⠲⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠑⢦⣸⠁⠀⠀⠈⠳⢤⡏⠀⠀⠈⠑⠦⣀⠀⠀⠀⠈⣉⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠂⠠⡉⠳⣄⡀⠀⠀⠀⠉⣻⠂⠀⠀⠀⣨⠗⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠢⣈⠳⢄⡉⢓⡆⠀⢰⡟⠀⠀⠀⢰⠃⠀⠀⡾⣄⣀⠀⢀⣀⣀⡤⠔⣄
⠀⠀⠀⠀⠀⢠⣖⠒⠒⠒⠳⣄⠐⢄⡑⠢⡀⢸⡶⠀⠈⣧⠀⠀⠀⣿⠀⠀⢸⣇⠀⠉⠉⠉⠁⠀⠀⠀⠈⠑⠢⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⣶⣄⣀⠈⠓⢤⣈⠓⠄⠸⣧⡅⠀⠈⠳⣄⠀⢻⡀⠀⠈⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢮⣟⣿⣿⣦⣄⠈⠙⠲⣄⡈⠳⣄⣀⠀⠙⢦⣀⠱⣄⡀⠘⢎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣯⣤⣶⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⢝⣿⡿⣿⣦⣄⠀⠉⠙⠚⠻⠿⣶⣤⣬⣟⣶⣿⣶⣦⣽⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⡶⠶⠾⢿⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡺⣷⣝⢷⣤⣖⠭⢭⣿⣿⣿⡿⣿⡿⢿⣿⣿⡿⠷⠶⢟⣛⡶⠾⠿⠛⠛⠚⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⢵⣛⣚⢩⣭⣦⡶⠮⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠋⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣷⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣶⠛⠓⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠶⠀⠀⠀⠀⠀⠀⠀⡠⣤⠶⠈⠓⣆⠀⠀⠀⠀⠀⢰⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣴⡄
⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢄⠀⠀⠘⣇⠀⠀⠀⢠⡭⠟⠀⠘⠗⣶⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢀⣴⠖
⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⢹⡄⠀⠀⠀⠙⢦⡀⢀⠜⠁⠠⠄⠒⢉⣀⣈⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⠀⢀⣀
⠲⢦⣄⠀⠈⠁⠀⠀⠠⢀⡜⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠸⣷⡀⠀⣇⠀⠀⠀⠀⠀⡉⠅⠊⣀⣤⠴⠚⠋⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋
⠀⠀⠈⠁⠀⠀⠀⠠⠱⡏⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⢀⠠⡏⣷⠀⠹⠅⡶⢀⠠⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠
⠻⠶⠤⠀⠀⠀⠀⠀⠀⠊⢕⠦⣄⠀⠀⠀⠀⠈⠳⣄⠈⠺⣏⡉⠀⠀⡰⠡⠂⢀⡼⠋⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠛⠀⠀⣀⡠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠊⠉
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡳⢦⡀⠀⠀⠀⠙⢦⡀⠀⠙⠲⣴⢡⠀⣰⠏⠀⠀⠀⠀⠀⣠⡶⠋⠀⡀⠀⢀⠀⣀⡴⠛⠁⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡝⢦⡀⠀⠀⠈⢷⡄⠀⣀⠀⠇⣸⠇⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢹⠿⠶⠞⠋⠀⣀⡤⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠎⢳⡀⠀⠀⠈⠛⠉⣿⠰⠀⡿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢐⢸⡀⠀⠀⢰⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⢸⠀⠈⠉⠲⣄⠄⣀⠣⣴⠖⠋⠀⢀⣀⡀⠀⠠⠧⠌⠀⣇⠀⠀⠀⠀⠀⠘⣏⣀⣀⣈⢀⡙⠧⠴⠶⠶⠿⠄⣠⡀
⠀⠀⠀⠀⠀⠀⠙⠓⠒⠦⠀⠈⠋⢹⠀⠈⠻⢦⣀⠈⠳⣤⠶⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⡼⠃⠀⠀⠀⠀⣴⠚⠁⠓⠛⠳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⣀⣀⣸⠆⠀⠀⠀⠙⠓⠦⣄⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠥⠤⢄⣀⠀⠀⠈⠑⠶⠟⠑⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣬⣥⠬⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠈⠉⠓⠲⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢈⣣⡀⠀⠀⠀⠀⠀⠀⢀⣾⣓⣚⣻⣿⣿⣟⣲⣤⠀⠀⠀⠉⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⣴⣒⣺⣯⣭⣭⣭⣽⣦⣄⣀⣀⠤⠴⠿⠿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢈⣙⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠓⠶⣤⣀⠀⠀⠀⠉⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠤⠤⠶⠖⠒⠛⠋⡩⠉⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣤⠴⠶⠶⠛⠒⢳⣽⠉⠉⠁⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⠀⠀⠠⣤⡶⠶⡄⡼⠋⠙⠳⣄⠇⠀⠀⠀⢀⡠⠾⡿⠀⠴⠒⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡟⠉⠉⠀⠀⢠⡇⠀⢀⣿⠀⢀⡀⠀⠹⠁⠀⢩⠀⠈⠀⠀⠓⡀⢷⣤⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢷⠀⠀⠘⡀⢸⠀⠀⢸⡇⠀⡎⢹⡀⠀⡇⠀⢸⡄⢸⠀⠀⢸⡇⢀⣸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢨⣇⠀⠀⢳⣸⠀⠀⣸⣥⠀⢣⡘⡇⠀⡇⠀⢸⡇⠀⠀⠀⢸⡇⢸⣄⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⡄⠀⠈⣿⠀⠀⡟⢻⡀⠈⠛⠀⠀⡇⠀⢸⡇⠀⡆⠀⢸⡇⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⡇⠀⢷⣄⠀⠀⡰⠁⠀⠰⣇⣠⠇⠀⠈⠓⠂⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣧⣀⣀⣀⣠⠃⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠈⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠴⠶⠒⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⢸⡀⢀⣀⣀⡤⠤⠴⠖⠒⠚⠉⠉⠉
⣶⣦⣤⣴⣦⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣬⣿⣿⣭⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢸⢶⠞⡋⡡⠀⠇⡇⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠐⠂⠉⠁
⠀⠀⠀⠈⣟⡩⠬⠐⠂⢁⠃⠀⠄⠀⠂
⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⡟⣿⢿⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠤⡥⠄⠄⠀⣿⣿⣷⡿⢳⣿⣮⡁⠀⠀⠀⠐⠚⠂⠀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⣷⢀⡀⠀⣿⢧⣦⣤⣮⠉⣍⠁⠀⠀⠀⣶⣀⣰⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠔⠲⠀⣿⣻⣏⣿⡟⣸⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣿⣿⡇⠀⢷⣿⣷⣠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢼⠆⠀⠀⡿⠿⠇⠀⠸⣿⣿⣄⣀⢀⠂⡄⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠐⠀⠀⠁⠀⠀⠄⠀⠄⠁⢀⣠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠋⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⡠⠒⠉⠙⠛⠉⠙⠒⢦⡀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠉⠐⠂⠶⠈⣀⠀⠀⠀⠀⠤⠐⠀⠀⠀⠀⠈⠉⠐⠲⠤⢀
⠀⠀⠀⡠⠚⠠⢐⣀⡞⠁⠀⡀⠤⠊⢡⡄⠀⠀⢹⡿⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⠴⢛⠢⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣴⣶⡂
⠀⢀⣼⠀⠀⢀⠚⠋⠐⠒⠀⠀⠀⡀⢉⣦⠀⠀⣸⣿⡧⢀⢀⠠⣂⣵⣶⣾⡄⣀⢔⠩⣶⠿⠟⣁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⣯⡥⠔
⠈⠀⢬⣑⠈⠁⠀⠀⠀⠙⠲⠒⠋⠁⠈⠉⠂⠰⢿⣿⣷⣏⣵⣾⣿⢿⣿⣿⢻⣫⠵⣾⣶⣿⣟⡍⢿⣦⣤⣴⣶⣿⣿⣿⣿⣟⣽⣿⣿⣿⠗⠊
⠀⠀⢚⠾⠷⣦⣄⣀⠀⠀⠀⠀⣀⣠⢴⣶⠶⠴⠒⣘⡿⣿⣿⢿⣧⠸⡿⠿⠾⣛⣲⣋⣫⣭⡖⠌⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠻⠟⠛⠛⠛
⠀⠀⠀⠠⣒⣲⣦⣉⠝⠒⠦⣝⡫⠶⡅⣴⣢⠾⢛⣋⣐⣼⣤⠬⠽⠿⠷⠖⠛⠛⠛⠛⠉
⠀⠀⠀⠸⠶⠾⢭⣟⣢⠤⣄⠀⠚⣳⢞⡩⢒⡽⠿⣿⠿⠛
⠀⠀⠠⣠⡍⢙⡛⠒⠚⠉⠓⠦⠄⠖⠁⢈⣡⡾⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣒⣒⣒⣒⣒⣒⣒⣒⣒⣶⣶⣶⣦⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠙⠛⠒⠒⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⡇⢀⣿⡇
⠀⠀⠀⠀⢀⣠⡴⠚⠋⠁⠀⣠⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⣿⠃
⠀⠀⠀⠀⢸⠉⠉⢉⣉⡙⡿⠁⣸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⢹
⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⡇⠀⢻⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣼⣿
⠀⠀⠀⠀⢸⠀⠀⢸⠀⠃⡇⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡿⡏
⠀⠀⠀⠀⢸⡇⠀⢸⢰⠀⡇⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣧⡇
⠀⠀⠀⠀⠀⡇⠀⠈⡉⠀⡇⠀⢸⡄⠸⣧⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⢸⢿⠇
⠀⠀⠀⠀⠀⣷⠀⠠⣦⠀⡇⠀⢸⣇⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠍⠩⠁⠀⢸⡾
⠀⠀⠀⠀⠀⢻⣀⣀⣀⣠⣧⠴⠋⢉⡏⠉⠉⠉⠉⢉⣉⡽⠭⠭⠭⠭⠭⡿⡟⠛⣟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣒⣢⣀⡀⣿⡭⠶⠶⠦⠤⢤⢼⠟⠁⠀⠈⠑⠒⠒⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⢽⡿⢛⣿⣻⣏⣩⣭⢭⠭⣭⠿⣿⠟⣒⢒⣒⣒⣶⠶⠦⣤⠀⠀⣀⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠰⠭⠿⢩⣞⣲⣓⣲⠫⣴⠯⡼⠭⡽⠏⡴⣡⣳⣛⡞⡳⢃⢎⡼⠀⣾⠥⣛⡒⠽⡆
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠭⠭⣭⣖⣒⣒⣒⡋⠭⠭⠭⠍⢑⣒⣙⣓⣣⡯⠝⡱⣣⠞⠀⠘⣧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⠒⠒⠲⠶⠶⠤⠽⠖⠁⠀⠀⠀⠻⢷⣶⡶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣄⣆⡀⠀⠀⠀⠀⠀⣀⠀⣤⡆⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⣟⣿⠣⠑⠄⠀⠀⠀⣠⣭⣭⣭⣥⡀⠀⠀⢸⠸⠾⠦⠤⠄⠀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠓⠓⠂⠀⠂⠀⠀⠈⠋⠉⠁⠉⠙⠉⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠏⠭⠭⠍⠀⠀⠀⠀⠘⣿⡿⢂⡀⠀⠀⠀⠀⠉⠉⠉⠉⠂⠀⠀⢰⠟⠋⠀⠀⠂
⠀⠀⠀⠀⠀⣖⠂⠀⠀⠀⠀⠀⠀⠀⠉⠣⣾⡧⠀⠀⠀⠀⢀⡖⡒⡂⠀⠀⠀⠈⢠⣤⠀⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢰⡆⠱⠀⠀⠀⠀⠀⡄⢰⡦⡇⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠈⢄⠰⠼⠅⣰⠀⠀⠀⢰⠇⡇⢾⠧⡇⠀⠀⠄⠠⠄⠤⠠⠄⠤
⠀⠀⠀⠀⠙⠛⠁⠛⠛⠀⠀⠀⠀⠀⠉⠀⠉⠸⣿⡦⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣾⠋⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠀⠀⣴⣶⣶⢶⣶⣦⡀⠀⠘⣧⣻⠿⣛⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿