Gale Emojis & Symbols

⠀⠀⠀⠀⣠⠤⠖⠒⢃⣀⣠⣤⠤⠤⣤⣄⡠⢄⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⣀⣠⡤⠤⠤⠤⣄⡀⢤⡀
⠀⠀⣰⠋⠀⡠⠖⢋⣭⠴⠶⠚⠛⣛⡶⠀⢹⠀⡧⣱⠀⠀⢠⠞⠁⣠⠔⠚⣉⠥⠖⠒⠚⠛⣲⠀⠈⡇⢹⢘⠆
⠀⠀⢇⠀⢸⡁⠀⠻⠶⠶⠶⠞⠛⢉⣠⡴⣿⠘⠁⠀⠀⠀⢸⡀⠰⡇⠀⠘⠳⠦⠶⠒⠚⠋⣁⡤⢾⠕⠋
⠀⠠⣌⠓⠦⢹⡟⠒⠒⠒⠒⠛⠛⢉⣡⠾⡅⠀⡄⠀⠀⠀⢤⠙⠢⠍⣗⠒⠒⠒⠒⠒⠚⠉⣁⡴⣏⠀⢄
⠀⠀⠈⠙⠒⠒⠙⢶⠶⠶⠶⠒⢛⣉⣡⡞⢡⠔⠃⠀⠀⠀⠀⠉⠒⠒⠊⢳⡖⠒⠒⠒⠚⢉⣁⡤⢋⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣛⣛⣋⣉⣉⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣉⣉⣉⣉⣉⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡮⠭⣭⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠬⢭⣉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠴⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀⡀⢀⢀⡀⠀⠀⠘⠶⢦⡀⠀⠀⠚⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⣦⣿⣾⡷⢟⣼⣇⣸⣧⣾⣷⣿⣿⣿⣿⣼⣧⣦⣤⣉⡛⢲⣄⠈⠛⣿⣿⣿⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣿⣧⣿⣿⣿⣿⣷⣟⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⣏⠁⠀⠀⠀⠠⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣏⣙⣿⢿⣿⣏⣻⣷⡌⢿⡓⠦⣀⣠⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠈⠙⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣿⠿⣯⣽⡿⢿⣷⠀⢛⣿⣿⢷⣿⣶⣤⣶⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣾⣽⡛⠛
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠁⠀⠈⠉⡁⣠⣿⣷⣦⣁⣀⣈⣿⣿⣍⡛⠛⠛⣿⣿⣾⣧⠟⣻⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣼⣾⠛⢿⣿⣿⣟⣿⣧⣼⡿⣛⣻⣿⣿⣿⣇⠀⠘⣿⣉⡿⣿⡿⣥⠻⣶⣄⡿⠿⣿⣿⣷⣄⠙⠛⠿⠿⠂
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⠁⠀⠀⠈⣿⣍⣿⣿⣉⣒⣿⣿⣥⣿⣿⣿⡀⠀⠹⣿⠟⣿⣇⣙⣿⣌⣹⡆⠀⠀⠈⠙⠛⠓
⠀⠀⠀⠀⠀⢿⣿⣿⣽⣿⣿⠛⣿⣿⣿⣧⣴⣿⣆⠐⠃⣠⣿⠟⠉⠛⢿⣿⣿⡿⠿⠿⠿⢹⡇⠀⠀⣿⣠⣿⢥⣭⡼⣩⢧⣽⣿⣄
⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣻⣿⣶⣿⣧⣾⣽⣏⣿⣆⠸⣿⣿⣦⣆⠀⢀⠉⣹⡇⣄⠀⠀⢩⣻⣦⣌⣿⣹⣿⣞⣱⣞⠀⣩⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣽⣿⣟⣛⣿⠿⣿⣷⣾⣾⣿⣿⢳⡟⢳⢾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⢻⣿⣿⣽⣷⣴⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢶⡿⠿⠿⠁⣿⣿⣟⣙⠧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⠉⠙⠉⠃⠘⢻⣿⡏⠈⣿⣻⣿⡿⠟⠿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶⣴⣶⣷⣾⣿⣿⣽⣯⣿⣿⣿⣿⢷⠿⢋⣽⣿⣿⣟⣷⣤⣄⣠⣀⡀⠀⢠⣿⣿⣧⠀⣿⣿⣧⣤⣤⣤⡀⠀⠙⡄
⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉⢤⣴⣿⣿⣿⣿⣿⠋⣩⢽⣿⣿⣿⣷⣤⣿⣿⣿⣆⣤⣀⣀⣤⣀⣿⣿⠀⢰⠃
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⠿⣿⣿⣿⠿⠃⣾⣿⣿⣿⣿⡿⢛⣿⣍⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣿⣷⣆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣿⠛⣩⣤⣴⣶⣄⣀⣀⣴⣦⣿⣿⣿⠯⠋⣴⠿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠿⣿⣛⣿⣼⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠈⢉⡸⣿⣿⣻⣿⣝⣿⣿⣿⣿⣿⣿⣯⢹⣿⣷⣿⣿⡭⠠⠄⠀⢀⠀⠀⣄⠉⠛⠙⠛⠛⠛⠋⠁⠀⢰⣿⣯⡷⢨⡷⢟⣿⣯⣿⠇⠀⡄
⠀⠀⠀⠀⠀⡀⢻⣆⣿⢻⡟⣾⡿⣿⣿⡏⣟⢻⢿⣟⢺⣿⣽⣿⣏⠀⣀⣤⣦⣠⣄⣀⣸⣀⣀⣠⣦⣄⡀⠀⠀⠀⣰⣿⢸⣴⢾⡇⣬⣿⡉⢿⠂⢰⣷
⠀⠀⠀⠀⢸⡇⢸⡇⣿⣾⣿⣿⣹⣯⣹⡿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⣻⠋⠿⠿⣾⣯⣭⣤⡀⠘⡿⣯⠚⡟⣶⡆⣿⣟⡀⠀⠉⢘⣿⢸⡇
⠀⠀⠀⠀⢸⡇⢸⣿⣾⣿⠿⣿⣧⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⡀⠀⠀⠈⢻⣿⡿⠇⠀⣸⣧⣶⣆⢿⡇⣿⣿⣿⣿⢶⣌⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡙⣍⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣤⣿⡇⠀⠀⣰⣿⣫⣥⣴⣶⣟⣿⢘⣿⣶⣉⣿⣿⣿⣿⢰⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣟⡛⣋⣴⣶⣿⣿⡿⠃⠀⠀⣸⣿⣿⣿⣿⣿⣷⣟⣴⠿⣿⣿⡇⣿⢿⣿⣿⢻⣿⣿⡏
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠘⣷⣾⣿⣧⣿⣿⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡿⢟⣷⣾⠿⣿⣿⣿⡟⠋⢀⣤⣴⣾⣿⡿⠿⠿⠿⠟⠛⠛⠻⢇⠀⠐⠀⢸⡿⣧⣿⣿⢿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢶⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣟⣡⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣶⣾⣿⣾⣅⢢⣼⠶⣾⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠹⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠟⠋⠉⠉⠉⠙⡟⠋⠹⢿⣾⣿⣿⣼⣿⡯
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⢀⣀⣤⣴⣶⣿⠿⠛⠛⠛⠉⠀⠀⢀⣀⣤⣤⣶⣦⣶⠶⠛⠆⠻⣀⢧⣽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠛⠉⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⠤⢔⠤⣠⣄⣀⠀⠈⠵⠲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠷⣤⣦⣬⡍⠉⢛⣩⣍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠿⡶⠶⣤⣭⣭⡭⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⣷⡾⡿⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣟⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡀⠀⠀⣀⣀⢀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠀⠀⠈⠀⠈⠉⠉⠉⠀⠉⠉⠁⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⣴⣶⠾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣄⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡄⠀⠀⠀⠾⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⠀⠀⢀⣶⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⢀⣸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠙⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠛⢿⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣇⣤⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⣶⣦⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⣴⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣶⣶⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠛⠉⠀⠀⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠙⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠉⠻⣿⣿⣶⡄⠀⠀⠀⠀⠈⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⢰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠻⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣶⡄⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⣠⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢰⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢺⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢙⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠉⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⡤⢤⡀⠀⠀⠀⢀⡴⠋⠀⢹
⠀⠀⠀⣠⠴⠊⠉⠉⠉⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠉⠁⠀⠀⢻⠀⠀⠀⠸⣄⡴⣄⣠
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠲⡄⠀⠀⣸⠆⠠⢤⠀⠀⠀⠀⠘⠲⢦⡀
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡄⠀⠀⠀⠀⠀⢹⠀⠀⢧⡀⠀⡸⠀⠀⠀⠀⠀⠀⣼⠁
⠀⢣⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⠤⠴⠃⠀⢀⣠⡀⠀⣸⣀⣀⣤⠽⠛⠁⠀⠀⠀⠀⠀⠀⠙⡆
⠀⣸⠃⠀⠀⠀⠀⠀⠐⠙⠦⣄⣠⠼⠁⠀⠀⠀⠀⠀⠀⠴⠖⠒⠒⠋⡴⠚⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣀⡴⠚⢲
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣤⣤⡤⠤⠤⠤⠶⠶⡖⠲⠀⠀⢸⠀⠀⢸⠁⢀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠒⠒⠦⡤⠇⠀⡇⠀⠀⠀⠀⠀⠀⠳⢤⣤⠴⠋⠀⠀⠈⠛⠉⠓⠋
⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠑⢦⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠁
⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⢀⣠⡴⣄⡀⠀⠀⠀⠀⢀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣿⣿⣿⣾⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣧⠄⢠⣿⠋⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⣿⣿⣿⣿⡿⠂⣼⣿⠃⠀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠺⢻⣯⣤⠶⠁⠈⠉⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣿⣿⣿⠿⠿⠟⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠸⣦
⠀⠀⠀⠀⠀⠀⢸⣧⡀⢀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⣥⣀⣹⣆
⠀⠀⠀⠀⠀⠀⣾⣿⣷⣾⣦⣿⣷⡀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡆⠀⠢⣠⠀⠙⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⣿⣿⣿⠗⠀⠀⠁⠀⠀⠀⠀⢠⣿⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠈⠛⠙⢦⡀⠀⠀⠀⠀⠀⠀⢨⣿⣷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⢹⣶⣦⣶⣶⣤⣀⠈⠓⣦⣤⣤⣀⣴⣾⣿⣾⣿⣿⡗
⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⠿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣾⣿⣿⣿⣿⡿⡀
⠀⠀⠀⠀⠀⢠⣾⣾⡁⠀⠀⠀⠐⠻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠘⢻⣿⣭⣽⣿⣶⣶⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⡧

⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⡀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⣤⣾⣿⣿⡿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣀⣴⢚⠻⢷⠄⢴⢿⣿⣿⡿⠿⠛⢻⣶⣿⡿⣟⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⠈⠀⠀⠀⠀⠀⣼⣽⣭⣽⣷⣶⣟⣰⣿⢿⡉⠥⠸⣈⣿⣿⡷⣟⣷⣟⡿
⠀⠀⠀⠀⢸⣿⣿⡏⠙⢿⡇⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠱⣤⢠⣿⣿⣰⣷⣟⣶⡟⣾
⠀⠀⠀⢰⣿⣿⣿⡟⢀⠀⠀⠀⠉⠓⠂⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠚⠓⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠠⣾⣿⣿⣿⣡⠞⣷⣤⡀⠀⠀⠄⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⡀⠀⠈⠀⣸⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⢸⣿⣿⣿⡟⠳⢬⢻⣵⣅⣀⣼⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡶⠿⡏⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠀⢸⣿⣿⠯⢭⡷⣤⠭⠿⣿⣿⣿⣿⣆⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣂⣀⣤⠞⠁⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⠀⣟⢡⡃⠑⢶⠻⣿⣿⣿⣿⣆⠀⠀⠀⢨⢿⠻⠛⠋⢭⣿⣿⡭⡰⠀⡀⠁⢻⠋⠋⠉⠉⠉⠉
⠀⠀⢸⣿⣧⠀⠸⢀⡁⠶⢠⡄⡈⢻⣿⣿⣿⡄⠀⢀⢺⣶⠀⠄⣴⣿⣿⣯⢀⠐⣿⠿⢧⠀⠃⣶⣄⡀
⠀⠀⢸⣿⣿⡇⠀⠘⣄⠶⠀⠻⣿⡮⢮⣿⣿⣿⣄⡀⠠⢿⡆⠀⠙⢿⣿⣽⡿⣿⡿⠂⢤⣀⠀⣽⣿⣍⠉⠄⠄
⠀⠀⢸⣿⣿⣯⣤⠶⢮⠳⠀⠦⣹⢷⣿⣿⣧⣻⣿⣿⣷⣾⣿⣤⣦⡀⠛⠿⣇⡽⣟⣠⡿⣿⢶⣿⠋⠛⢀⡔⢀
⠀⠀⢸⣿⣿⣿⣿⣤⣀⠠⠘⣆⣱⣿⣿⣧⣜⠉⣿⣟⣪⡹⠟⡿⣗⢀⠀⡥⢸⣷⡟⣽⣩⣿⣡⡏⠀⠀⠀⠂⢀
⠀⠀⠰⣿⣿⣿⣿⣿⡗⠣⣾⠟⡹⠿⠏⠿⠺⠿⠿⣎⠩⣻⣾⣀⠀⠀⠐⠛⠃⠙⢷⣿⣿⠛⠉⠁
⠀⠀⠀⠀⠉⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⣿⢳⣻⣿⣻⣟⣿⣿⣏⣿⣙⣝



⠀⠀⠀⢀⠔⠉⠉⠉⠑⠢⡀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⢀⡀⠤⠤⣀
⠀⠀⠀⠘⡄⡖⠉⢲⠐⠀⠘⡄⠀⡞⠁⠀⣀⣘⠳⣄⠀⢰⠃⠀⠀⠀⡈⣢⣀⠔⠊⠉⠉⠉⠑⢦
⠀⠀⠀⠀⠈⠛⠁⡸⠒⠋⠉⠙⢺⠁⡰⢉⣠⢤⡑⠈⢣⣸⡀⡔⠉⠑⡌⠁⠃⢀⠭⠥⡀⠀⠀⠀⢱⢀⡀⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠖⣉⡑⠀⠸⡄⢃⠸⠉⢱⡱⠀⡄⠣⠑⠺⠄⢀⠇⠀⠓⠁⠀⠂⢹⠀⠀⠀⢠⡗⡆⢱
⠀⠀⠀⢰⠉⠹⡇⠀⡎⠀⠘⠂⠀⠱⢄⣁⣠⠿⡁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡠⠔⠁⠀⠀⢀⡼⠀⠀⡘
⠀⠀⠀⢸⡀⢀⣑⣤⣑⡤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡔⠀⠀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠣⡳⠄⠈⠢⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠉⠉⠉⠁⠀⠀⠉⠁⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣛⣋⡩⠍⠍⣙⡛⠶⡄⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⣑⣚⠒⠂⠀⠀⠀⠀⠈⠁⠈⢂⡹⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠗⢊⡁⠀⠐⠒⠪⠢⣦⡀⠀⠪⢴⠐⡼⠈⠈⢙⡏⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠴⠊⠉⠉⠉⠒⠦⢀⠀⡈⠇⠀⠀⠀⢀⣠⠆⠀⠪⠸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠊⠐⠀⠀⠀⠀⠄⣀⠀⠁⢱⡀⢆⠀⡠⠈⠀⠀⡠⢂⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡔⠀⠀⠄⢀⠀⠤⠬⣕⡂⠀⠈⠁⠀⠀⠀⢀⢀⣀⠐⠁⠀⠀⠰⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⡸⢘⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡜⠠⠊⠀⠀⠀⠀⡀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠌⠀⠆⠀⡊⠄⡠⠃⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⠇⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠁⠀⠀⡤⠒⠶⠶⠀⣦⢁⢈⠀⠀⠂⠀⠐⠄⠠⠥⠠⠚⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠂⣀⡁⣀⠛⢈⠆⠌⠎⠀⠀⠢⠄⠀⠀⠀⡤⠂⠁⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠇⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⢠⠀⠀⠂⠄⡀⠀⠀⡁⢠⡟⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠀⠀⠀⠔⠀⠁⢠⣾⠇⡄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⠁⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⢈⡎⠈⢜⣕⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢇⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡁⠠⡀⠀⠘⡔⠁⡐⢙⢾⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣦⡑⣤⡐⠌⢦⡀⠠⡙⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⡫⠂⢤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡳⢍⡻⢷⣿⣷⣧⣽⣶⣤⣁⡂⠈⠉⠙⠛⣛⣛⣛⢟⣿⡿⠟⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⣓⡤⡩⢛⡻⠿⠿⣿⣿⣿⣿⣿⠿⠿⠝⠋⠉⠁⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀⣀⣀⠀⠀⠐⣀⣁⣀⣤⡤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠀⠈⠈⠁⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠀⠀⠐⠈⠁⠒⠒⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡿⡿⢻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀
⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢛⡿⢻⠛⠄⡁⢀⣼⠂⠀⠀⠤⠤⠔⡒⣦⣤⢄⡀⠀⠡
⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⢠⣧⠿⠉⡀⣌⣸⣤⡿⠚⠁⡀⠀⢀⠀⠀⢠⠀⠄⠡⠰⣧⠀⠀⠂
⠀⠐⠈⠈⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆⡀⠿⠓⠲⠿⠷⠛⠉⠁⣠⡴⠞⢿⣤⣾⣤⣽⣦⣼⣾⣷⣿⠟⠀⠀⢀
⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠂⠀⠀⢀⣠⠞⠉⠀⠀⠀⠈⠙⠿⠿⠻⠟⠛⠛⠁⠀⠀⠀⠰
⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡⣿⣀⣀⣠⢦⠶⠶⠿⣥⣀⡀⢀⡀⣀⠠⠐⢐⡖⢦⣄⡀⠀⠀⢀
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠘⠉⠛⠁⠀⠀⡀⠀⠀⠀⠁⠛⠛⡙⣯⣿⣄⣈⠀⠃⡀⢹⠳⠄⠁
⠀⠀⠸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠖⣡⠀⣀⡀⠀⠀⠠⡀⡀⡀⠀⠀⠂⠀⠙⢿⣿⣭⣴⢠⠖⡠⠊⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣼⣯⠛⠋⠈⠳⣄⡀⠀⠀⠈⠙⣿⡗⡴⣭⡁⣘⠲⣄⡀⠀⠀⠉⠛⠿⣿⣿⣷⡶⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠐⠠⠄⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⣀⣀⣀⠀⠀⠉⠡⢼⣀⠀⢉⡇⠰⠛⣤⠁⠀⡾⠉⠶⣀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢁⣤⡠⣭⡵⠖⠲⣶⣉⠉⢻⡖⢆⡀⠀⠀⠉⠛⢮⣣⣞⠻⢋⡠⠄⢠⠷⠀⠉⣠⠇⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⢶⠿⢛⣇⡋⠀⠀⠀⠀⠋⠁⣿⠀⠀⠙⣤⡀⠀⠰⣆⡙⠂⠛⠋⢹⣴⠦⣤⠖⣀⣙⢶⡄⡀
⠀⠀⠀⠀⡀⠔⠀⠀⢀⡞⠁⢠⣴⡿⠋⠀⠀⠀⢀⠀⠀⠈⡍⣄⠀⠀⠸⢷⣐⡀⠈⠛⠿⣇⡀⢆⠈⠙⠓⠴⠥⠎⠁⢾⠁
⠀⠀⠀⡠⠃⠀⠈⣴⢧⡄⣸⠟⠁⠀⠀⠐⠀⠀⠀⠀⠢⢀⣉⠈⢣⣀⢐⡼⢇⢴⢤⠀⠲⣌⡙⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⣿⠙⠊⠀⠈⠁⠂⠄⣀⠀⠀⠵⠀⠐⢮⠗⠉⡀⢳⠂⠓⠝⠢⠴⠓⠲⠿⢙⣋⣋⣠⡀⠀⣖⠦⠀⠀⠄
⠀⢀⠈⠀⠀⣠⡿⠟⠀⠂⠄⣠⠀⠀⠈⠁⠀⢀⠀⠀⠈⢩⡵⠄⣤⠀⠠⠠⢆⡐⢂⡦⠀⠀⠀⠀⠀⠉⠻⣷⡉⠒⠴⣂
⠀⠀⠅⠀⠨⠉⠁⠀⠖⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢘⢱⠞⢲⠀⠖⠚⣻⡄⢯⣶⠃⠀⠀⠀⠀⠀⠀⠈⠙⢧⡄⠈⢀
⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⡿⢇⣋⢘⣀⡆⣀⣯⡆⢗⣤⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢟⠥⣼⣠⠂⡀⣾⡡⣦⢨⠁⠀⠀⠈⡀⢀⡀⠀⠀⠀⠳⣌⡙⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⡔⠋⠚⣹⡗⠂⠤⢸⡿⠁⡙⠆⠐⠄⠐⠘⠫⠟⠀⠀⠀⠀⠀⠙⠛⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⣉⠁⠀⠈⠁⠈⠀⠀⣗⠼⠃⠤⢖⠀⢁⢜⣳⣂⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠉⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⡀⢙⣙⢦⢀⣎⢑⠲⣼⣿⠤⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡒⡛⣀⠻⡘⣍⠃⠘⣷⠀⠑⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠀⠀⠀⢻⡉⢹⠑⠇⣶⣼⠃⢮⣑⠖⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⡿⣃⠀⠚⠁⠀⡄⠠⣯⢘⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡠⢴⢄⡈⣳⡄⣮⠀⠠⠰⣄⠀⠀⠀⠘⠀⢀⠞⣿⣟⢻⠀⠀⠆⠀⢠⠙⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠌⠀⡂⣠⣻⠃
⣀⠀⠀⠀⣧⡌⠡⠀⠈⠃⠈⠀⠀⣴⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡆⡄⠘⢰⣷⡶⠏
⢧⡀⠀⠀⢻⡗⢀⠀⠀⠈⢧⠀⠸⠃⠤⠀⠀⠀⠀⠀⠉⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡄⠀⠓⠀⠨⠏
⠈⠓⣤⡀⠀⠓⠻⣜⣄⠀⠀⢐⡦⠀⠀⢀⡞⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠉
⠀⠀⠈⠛⣦⠤⠀⠒⠿⠂⠚⢉⠀⠠⠖⠉⠁
⠀⠀⠀⠀⠶⠒⠤⢀⡀⠀⡀⠉
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠒⠂⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠀⢀⣈⣡⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠄⠋⠀⠀⠀⠒⠀⠉⠀⠢⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠤⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠹⢀⠄⠀⠀⠀⠀⠀⠂⠠⠒⠂⠐⣛⠒⢒⠄⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠂⠤⡀⠒⠀⠀⠀⠀⠉⠀⠉⠀⢸⡀⠀⡀⠀⠀⢀⠀⣀⣠⣤⣤⣈⣀⡀⠈⣄⣁
⠀⠀⠤⣀⠀⠀⠀⠲⣤⡀⠀⠀⠀⠠⠀⠀⠧⠀⢀⡇⠀⠀⢂⠀⠈⡁⠀⠄⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠠⠙⣿⡿⣿⢿⣿⣿⣿⣆⠀⡿⠀⠀⠃
⠈⠒⠦⠤⠤⠌⠠⠤⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄⠀⠄⡀⠈⠐⠀⠴⠀⠀⢢⠀⠢⡀⠀⠀⠀⠩⠀⡉⠂⢉⡃⠙⡙⠛⡂⠉⠙⣭⣿⡿⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠐⠢⢅⣘⠆⠀⢀⠐⢄⠀⠈⠁⠻⠂⠉⡻⣆⢈⠢⣍⡙⠊⠉⢉⡴⠛⠒⠅⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⣸⠑⢄⠀⠂⠀⠄⠙⣿⡏⠀⠐⠉⠙⠋⠁⠀⠀⠀⠀⠀⠙⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠨⣠⠤⠤⠀⠀⠉⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠈⠉⠙⠢⡄
⠀⠀⣸⠀⣀⠙⡆⠀⠀⢰⠋⠙⠲⠖⠒⢤⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠒⠒⠲⠤⣀
⠀⠀⠈⠉⠉⠉⠀⢀⡤⠏⠀⠀⠀⢀⡀⢸⡇⠀⠀⡤⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢡⠀⢀⡗⠀⢸⠀⠀⠀⠀⢠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠙⠻⠤⢤⣽⣀⡠⠤⡀⠀⠳⠐⠁⠀⠀⠀⠀⢀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣠
⠀⠀⢸⠉⠙⠦⡀⠘⣦⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⡇⠈⠈⠉⠁⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠳⢤⠶⠤⠃⠀⠹⣄⡈⣀⠟⠉⠉⠉⠉⠙⡟⢲⡊⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣀⣀⣀⣤⠞⠒⠤⣄⣀⣀⡤⠞⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠉⠉⠉⠉⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⣄⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣠⡴⠒⢦⡀⠀⠀⠀⡴⠋⠀⠀⣇
⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠳⠴⠳⠤⠟
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣆⠀⠀⡼⠀⠐⢢⠀⠀⠀⠀⠉⠙⣦
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠆⠀⠀⠀⠀⠀⢹⠀⠈⠳⣄⣠⠎⠀⠀⠀⠀⠀⠰⣇
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣲⡄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠓⠚⠁⢀⡠⠖⣄⣀⣼⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⣀
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣶⠋⠀⣴⠶⠋⠈⡇
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡖⢲⡖⠒⠒⠛⠋⠉⠹⡅⠉⠀⢀⡏⠀⠀⢧⣠⣄⣀⡿
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠑⠚⠁⢠⡇⠀⠀⠀⠀⠀⠀⠙⠓⠚⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠉⠓⠒⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣤⡤⠖⠋⠙⠦⣄⣀⣀⣤⠤⠖⠋






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⣶⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⠛⠻⠿⠛⠛⠛⠛⠛⠛⠛⠻⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢾⣻⣿⣿⣿⣶⠀⠀⣠⣶⣿⣿⣿⣿⡛⠀⠀⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠘⠉⠉⠉⠈⠁⠀⠀⣿⡏⠁⠉⠉⠛⠋⠀⠀⢻⢿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄⠀⠀⠀⠀⢀⠀⠀⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠼⠿⠾⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣤⠞⢋⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢟⡉⣉⣠⣴⣿⣿⣿⣿⣿⣆⠀⠀⠘⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢳⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⡔⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠿⣿⣿⣿⡿⡋⠀⣸⣿⣿⣿⣿⣿⣶⣶⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⠇⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢟⠹⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⡾⠟⠿⣹⣿⣿⣿⡏⠀⠀⠰⢀⢰⡄⠀⠀⠀⠀⢀⠐⠋⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠁⠀⣠⣾⣿⣿⣯⡿⠁⠀⠀⠐⠀⠀⢠⡀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣤⠞⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⠁⠀⠀⠀⠀⠐⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠋⠁⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡴⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡞⠁⠀⠀⣾⠟⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠉⠀⠼⠁⠀⣼⡇⠀⠀⠀⠀⣸⡟⢰⡿⣀⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠐⠀⠀⠀⣿⢃⣿⣹⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢠⣼⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠿⠟⠿⠻⠿⠿⠿⠿⠷⠶⠶⠿⠟⠛⠻⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣰⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⠄⢨⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣶⣶⣶⡄⡿⢿⣙⣇⢻⣿⣆⣿⣿⣿⣿⠇⠀⢀
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣠⠄⠀⠀⠀⠈⣿⣿⣿⢿⡟⣟⠁⡧⢻⠘⣿⠈⣿⣯⣹⡟⢻⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠐⠉⠛⠀⢀⠠⠀⠀⠠⠟⠃⢸⡁⣾⠁⠁⢰⢈⡿⢀⠸⣿⣿⢄⡠⢿⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠀⠉⠙⠈⢠⡆⠄⠀⠀⠀⠈⠀⠀⠀⢸⡗⠃⠀⡀⢼⣸⢃⣿⡆⢹⣿⣾⡅⣾⠅⡀⠈⢷⠤⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠄⠀⠀⡀⠀⠀⠀⠉⡇⠀⠆⡃⣸⠇⢸⣿⡇⢸⣿⣏⣠⣿⣢⡇⠀⢰⣀⠀⡄⢉⡹⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠃⢄⠂⣼⡿⡇⠀⣤⣤⢉⢙⠛⠲⠧⠈⡡⠃⠀⠠⠘⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠁⠈⣰⣿⣷⢧⠸⣯⣿⣡⣾⣿⡟⢀⣟⡁⠀⠀⠐⠘⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⣸⢠⣿⠏⣿⡺⢞⡇⣌⠛⢣⠀⠀⠂⢀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠰⠀⣿⡿⣿⠠⡃⠸⢁⡴⠟⠀⣿⠇⠀⠘⠀⣧⡴⠂
⠀⠀⢳⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⣏⡿⠃⠁⠘⡡⢖⠀⠄⢩⠆⠀⠀⠀⠇
⠀⠀⠘⠀⡖⠉⠀⠀⣠⢦⣤⣀⣀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡠⡠⢀⡰⠛⡯⢅⠀⠀⠀⠑⠀⠀⠀⠈
⠀⠀⠀⠀⡇⠀⠀⠈⠀⢸⣿⣿⣿⣿⡯⣷⣿⡿⢓⠲⣖⠢⠀⠀⠀⡩⠈⣤⠟⠀⠀⠡⠀⣠⠂⣄⣂⢀⠀⣦⠀⠐⠀⠀⠀⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣏⢿⣿⢩⢸⣿⣻⢠⠘⠀⣠⠀⣀⡤⠊⡠⠊⠀⠀⠀⠀⠒⠸⠀⣴⣌⠙⢾⡧⢹⣤⡄⠤⠀⠔⠁
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠈⠵⢾⠿⣁⢸⣿⢿⢀⠔⠋⠹⠃⡀⠀⢘⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣹⣿⣆⠙⣯⡇⣡⣷⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠜⣿⠴⠫⢸⠀⠀⠁⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡹⢟⠹⣧⠈⡟⢿⡇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⢸⠀⠀⠰⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠐⠀⠐⠁⠀⠀⠀⣿⣧⠘⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢪⠹⡄⢻⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢱⢨⡀⠀⠀⠀⠉⠂⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⡀⠁⠀⠀⣠⠔⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠰⡄⢀⡴⠞⠙⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⡀⠀⠀⠀⠀⡠⠔⠂⠒⠈⠉⠀⠀⣀⢀⡀⣀⢀⣀⢀⡀
⠂⠀⠀⠀⠀⣀⠄⠄⠂⢀⣀⣨⠤⠥⡄⣐⣂⠂⢐⡐⠀⠄⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠠⡎⠀⠀⢠⠚⠉⢠⠔⣪⣥⣤⣤⠄⠀⠀⠀⠀⢢⠀⠈⠀⢀⣠⠲⠃⢈⠀⠨
⠀⠀⢠⡄⠘⠒⠀⠈⠀⠀⠈⠈⠁⠋⠲⠲⠌⠂⠀⠀⠐⠁⢐⠀⠈⠁⠀⢀⠔⠀⡸⠃
⠀⠀⠀⠉⠂⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⠂⠂⠁⢠⡢⠀⠁
⠀⠀⢀⠄⡌⠀⠈⠁⠉⠉⠉⠁⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⢀⠀⠠⠐⢁⡜
⠀⠀⠀⠂⠡⡀⠈⠠⠄⣀⣀⢀⡀⢀⣀⡀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠄⣈⠃
⠀⠀⠀⠀⠀⠈⠂⠕⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⣀⠤⠀⠀⣀⡎⡁⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣄⠀⠀⠀⠀⠀⠀⢀⣀⡠⠦⢴⠞⢋⣴⡁⠀⠀⡀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠑⢶⣤⡄⡠⠤⠤⠄⢒⡁⠀⡠⢆⢅⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣶⣚⣛⣓⠊⠴⢋⠆⡅⠀⠀⠀⡄⠂⣀⣤⠀⠀⠦⠀⠀⠖⠋⠐⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠁⠈⠹⣷⠶⠖⣷⠷⠊⡁⠀⠀⠀⠁⠀⢀⠘⠃⢀⡀⢷⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢰⣄⠀⠀⠉⠀⠐⠟⠒⢂⡺⢽⣻⡁⢀⠀⣀⠁⠀⠐⡃⠀⠠⡠⢀⠀⠂⠀⠀⠁⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠉⠀⠀⠂⢘⣾⣷⣶⡦⠝⣫⠁⠋⠐⢃⠤⢰⠁⢀⡄⢠⣦⣦⠈⠂⣈⠁⡀⠤⠀⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠀⠄⠀⡀⡀⢠⣾⣴⣶⠮⢿⣴⠃⠀⠀⡠⠁⠄⠌⠀⠈⠠⠄⠋⠉⡠⠔⠄⠀⢳⠠⠄⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠠⠀⠃⠀⠀⢳⡭⠤⠄⢰⣺⡵⠁⠁⠀⡁⠠⠁⠈⠴⠐⠁⣰⠃⠀⠁⠒⠤⠀⠀⣇⠠⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠒⢠⡀⠆⣭⠸⣿⡿⣛⣭⡿⣃⡶⠌⢀⠀⠈⠀⢁⠂⠀⠉⠉⠲⢄⠀⣀⣀⣈⠀⢀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣄⣰⣛⣿⠟⠀⠗⠉⢶⣿⣋⠙⠀⠚⠃⠀⠀⢠⠀⠰⣤⠢⠴⠦⠔⠊⠁⠀⢀⠀⠔⣸⠃⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠺⡷⠢⣤⣀⠀⠈⣩⡥⠤⠌⡀⠀⠠⡤⠤⠦⡤⠐⣬⣄⠀⢀⢄⣀⡀⣤⠠⢞⣃⠬⠒⠓⡡⠶⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠀⠀⠀⠐⠒⠒⠒⠂⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄⠤⠤⠀⠀⠀⠀⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣈⡉
⠀⠀⣀⠀⢀⡀⢀⠀⡀⠀⠀⠀⠀⣈⢡⢎⣼⣿⣾⣷⣤⡒⢤⡀⠀⠀⠰⠶⠒⠀⢀⣐⠂⣠⡀⠀⣀⠀⠄⢀⡀
⣀⣨⣤⣤⣀⠒⣀⣈⣉⣀⡈⢉⣿⡏⠐⣩⢟⣵⣿⢟⣿⣯⠀⠤⣄⡀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡷⢶⠛⡿⡲⠮
⠈⠙⠉⢉⣁⣈⡉⠛⠙⠛⠋⢉⡭⢒⡠⣸⡽⣛⣵⣿⣿⣿⣦⠀⠈⢝⣆⢠⣰⣠⣀⣀⣈⠉⠛⠛⠛⠻⠿⠾⠞⠛⠒
⠤⢆⣚⠫⠭⣛⣫⣭⠤⣒⡨⠔⣪⠵⣚⣽⣾⣿⡿⣿⣿⣿⡟⣄⡀⠐⠛⠍⡿⢤⡈⣍⣬⠍⣉⠈⠙⢃⡴⣧⡦⢴⡃
⡉⠄⣐⣚⣭⠵⠶⢚⣩⠵⣒⡭⣒⣯⣟⠽⣛⣥⢞⣿⡿⣿⣿⣯⣋⣐⢀⠀⡉⠐⠉⠁⠉⠩⣘⠉⠈⠉⠀⠀⠩⠟⠉
⣫⣭⣽⡥⢴⣒⣫⢽⣾⡿⣗⡿⢟⣫⠥⣐⢭⡾⢟⣽⡾⢟⣽⢿⣿⣿⣶⣤⣬⣥⣤⣬⣴⣮⣵⣍⣱⣀⣒⣵⡶⣾⢣
⠩⠽⠖⣛⠭⢑⣚⢭⣶⠏⣁⡡⢈⣠⡴⠞⣉⡤⠞⣩⠾⣫⣵⠟⣫⣴⠟⣋⢜⣹⠟⡯⢚⣿⣿⣻⣿⣿⠿⣫⠾⢃⣟
⣶⣶⠶⠶⠿⣷⣶⣿⣶⣿⣷⣿⣿⣷⣶⣿⣷⣶⣾⣶⣾⣿⣶⣾⣿⣷⣾⣷⣿⣶⣿⣶⣿⣿⣿⣿⣿⣷⣶⣷⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡗⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠠⠲⣤⣤⣀⢀⠀⠀⠀⣀⣠⣦⢠⣶⣀⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡗⡏⣯⣼⡆⢿⣿⡼⣿⣿⣿⠃⠀⠄
⠀⠀⠀⠠⠀⢸⠆⠀⠀⠀⠀⠀⣠⢔⡤⣶⠀⠀⠀⠠⠚⢻⡟⢹⠏⢿⠅⡇⠏⢸⡇⠘⣿⣿⢟⠙⣻⣇
⠀⠀⠀⠈⠀⠀⠀⠀⣀⠠⠄⠀⢸⢠⢈⢁⠀⠀⠁⠀⠀⠉⠁⢨⣶⡿⠀⠆⣧⣾⢣⣆⠹⣧⣯⠔⣹⡇⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢼⠀⠀⠀⠀⠀⠀⠈⠸⡇⠀⡀⡆⣵⡇⣾⣿⠀⣿⡿⠃⣼⡅⡤⡀⢙⠉⢀⠠⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠁⠂⢄⠀⠀⠃⠀⠀⠇⣋⠆⣿⣿⠂⢉⡙⠒⠿⠿⢥⡆⠸⢺⢼⠀⠋⣿⢿⣶
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠐⣰⣿⣽⠀⣸⣯⣟⢰⣶⡦⠈⣶⡃⠠⢀⢀⢨⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⣴⣯⡿⢹⠃⣹⡿⣷⣿⣿⣇⡸⠯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠸⣿⢿⡟⢨⠁⠋⣰⠿⠃⣿⡇⠈⠃⢸⣆⡤
⠀⠈⢦⣀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡻⢋⡟⠈⠀⣀⠥⠀⠠⢙⠄⠀⠀⠸⠋
⠀⠀⠘⢸⡹⠁⠈⢀⢰⣤⣀⡀⣀⣀⣄⡀⠀⠀⠀⠀⠀⢀⠀⠄⣀⠤⢀⡴⢹⡉⠂⠀⠀⠏⠀⠀⠀⠈
⠀⠀⠀⢸⡇⠀⠀⠁⠈⣿⣿⣿⣿⡾⣿⣿⠟⡂⢶⡶⠂⠀⠀⣨⠈⡤⠋⠀⠈⠀⢀⡖⠠⣆⠁⡀⣔⠀⠐
⠀⠀⠀⢨⡀⠀⠀⠂⠀⣿⣿⣻⡩⢺⣿⡏⡆⠑⣈⡄⢀⠰⠊⠠⠊⠀⠀⠀⠈⠂⠍⣠⣦⡉⠻⣧⣸⡤⣶⠇⠠⠊
⠀⠀⠀⠘⡇⠀⠀⠀⠈⠰⠻⠟⣥⢸⡿⣧⠰⠚⡅⡏⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣧⣿⣷⡈⢯⣇⣽⡿⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠊⡿⡚⠁⢉⠀⠀⠄⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢟⡇⠻⠉⣷⡄⢽⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⣰⠀⠀⠃⠀⠀⣀⠈⡄⠀⠀⠀⠀⠀⠁⠀⠁⠃⢀⠀⡝⣿⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡊⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠘⡇⠸⠃⠁⠁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡂⢸⠐⠂⠀⠀⠀⣠⠀⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠰⣊⡠⠬⠓⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⡤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣁⣁⣀⡀⠠⠄⠐⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⠖⠒⢉⠙⠛⠉⠥⢤⣤⣤⣤⣤⣤⡤⠤⣤⣤⣤⣀⣉⣙⣛⡛⠛⠛⠓⠲⠦⠤⢄⣀
⠀⠀⢀⠤⠒⠀⡀⠄⠂⠠⠐⠊⡉⢍⣐⣀⣠⡤⠤⠶⠖⢂⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣉⣉⣙⣓⠒⠢⠤⢄⣀⠀⠉⠑
⢀⠔⠁⣠⠖⠁⠀⢈⣀⣤⣶⠿⢛⣋⣭⡤⠶⠶⠞⠛⠛⣉⣉⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠙⠓⠒⠈⠉⠓⠶⢄
⡌⠀⠌⠁⢀⣤⢞⡿⣹⣥⡶⠟⢋⣡⣴⣶⠶⢖⣛⣫⠭⠉⠉⠀⠀⠀⠀⠀⠙⠛⠓⠒⠒⠒⠲⠶⠤⠤⣤⣀⣀
⠀⠀⢀⡶⢋⣸⠿⢋⠕⣫⡴⠞⠛⠉⢁⣀⠤⠔⣒⠒⠒⠒⠒⠒⠐⠀⠉⣉⣉⣉⣉⡉⠉⠉⠋⠐⠒⠀⠀⠀⠉⠙⠓⠦⣄
⠀⢀⣟⣴⠟⢁⢔⡵⢟⡩⡲⠼⢒⠩⠁⣀⣈⣤⣤⣤⣤⠤⠶⠶⠶⠶⠦⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⠤⢄⡀⠀⠀⠀⠀⠈
⠀⠈⡟⠁⣰⣻⢏⠔⠁⢀⣤⣶⡷⠞⢋⡉⠭⠴⠒⠒⠀⠀⠀⣉⣁⡀⠀⠀⠀⠀⠀⣤⣤⣄⣀⡀⠀⠀⠀⠉⠲⡄
⠀⠈⠃⠀⢱⠇⠀⢀⣴⣿⠟⢉⡤⣐⠄⡂⠤⠀⣀⣀⣀⣤⡤⠤⣤⡤⢤⡤⠤⠤⠤⠀⡀⠀⠀⠈⠑⠢⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢠⡟⡐⣡⢖⡡⢊⣠⣤⡶⠞⢛⣩⡥⠴⠶⠒⠒⠒⠒⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠜⠁⣡⢶⠟⢋⣡⠴⠚⣛⠿⠟⠋⠉⢉⣁⣉⣉⣉⣀⣀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⠁⡠⠊⠁⣢⡼⣆⣠⣤⣶⡾⣿⣿⠿⠿⠶⠦⠤⣄⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣡⠶⣫⡵⢟⣫⣭⣽⣶⣤⣤⣤⡤⠤⠤⠤⠄⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠞⣵⡿⣫⢖⣵⣿⡿⢟⣯⠏⣩⠤⠶⢒⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡾⡏⢐⡵⢋⡕⡡⡺⠋⣠⣮⠴⠚⣛⣩⣵⣞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠡⠀⡾⠁⠎⠈⡐⣡⡞⡫⢂⣴⣾⡿⠋⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⡇⠀⠀⢠⡾⢻⢊⡴⡫⠋⣩⣴⠿⣫⠭⠉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠸⡇⢆⡟⡐⣱⡿⢋⠕⣪⣴⣾⣯⡭⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠠⢊⣾⢟⢟⢫⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡾⠐⢋⣴⠿⣋⣤⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣧⠞⡻⢛⣩⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⢸⣶⢟⣥⣶⡛⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⡿⢫⣥⡶⠚⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⣭⣤⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣥⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣵⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡤⠶⠒⠚⠛⠛⠒⠒⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠈⢉⣀⡤⠤⠴⠤⣤⣄⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡾⠉⠀⣀⡤⠤⠤⢄⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⠈⠉⠉⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠉⢠⠔⠋⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣄⣤⣄⣀⡀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⣀⡤⠖⠛⠋⠛⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠈⠉⠳⢦⡀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⠇⠀⡠⠊⠁⠀⠀⠀⠀⠀⠢⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡈⠉⠛⢦⡄
⠀⠀⠀⣠⠖⠋⢀⣼⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠈⠉⠳⣄⠀⠹⣆
⠀⢀⡞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⢰⡄⠀⠀⠀⠀⠈⣧⠀⢹⡄
⠀⡾⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⡆⠘⡇
⠰⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠲⠾⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⡇
⠀⠻⣄⠀⠳⣄⡀⠀⠀⣀⡴⠞⠉⣭⠗⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⡞
⠀⠀⠈⠻⠦⣬⠉⠉⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠛⠉⣹⠶⠤⠤⠴⠚⠉⣠⠟
⠀⠀⠀⠀⠀⣿⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡋⠀⠀⣀⣠⠴⠋⠀⠀⠐⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠈⠛⠓⠶⠶⠤⣤⣤⠤⠶⠞⠛⠦⣤⣀⣀⣀⣀⣤⣴⠖⠋⠁⢀⡬⣭⣭⣅⣤⣤⣤⠴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢤⣀⡀⢀⡟⠀⠀⠀⠀⡾⠒⣒⣚⣭⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠚⠋⠁⠀⠀⠈⠉⡿⠁⠀⠀⠀⠘⠛⠋⠉⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⠴⠖⠛⠋⡇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠒⠶⢤⣀⣀⣀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢸⠁⠀⢀⡴⠋⠉⠀⠉⠙⢾⡁⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢇⣀⠤⠐⡲⠊⠁⠀⢠⠖⠛⠛⠒⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡄
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢇⠀⠀⠀⠀⡠⠤⡀⠀⠀⢀⠀⠈⢢⢀⠖⠀⠀⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠘⢦⣀⡔⠁⠀⠀⠀⢸⠁⢄⠀⠀⠀⠀⠹⣄⠀⠀⠀⣴⠁⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠂⡐⠀⠀⠀⣐⠠⣉⡇⢠⢋⡁⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠳⢤⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⢹⡍⠁⠀⠫⡈⠀⠐⢤⠜⠁
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⢀⣀⠇⠀⠀⠀⡀⠈⠓⢧⡀⠀⠀⠀⠀⠀⣠⠞⠀⡴⣻⢟⡳⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠓⢤⣀⠈⠑⡏⠁⠀⠀⢇⠉⠆⠙⠲⣉
⠀⠀⠀⠀⠀⢀⢆⠀⢀⣠⠇⠀⠀⠀⠀⠀⠈⢉⠉⠙⢦⡀⠀⠀⢹⠈⡀⠀⠠⡉⠢⢄⡀⠀⠀⣀⠤⣄⡀
⠀⠀⠀⠀⠀⠁⣿⠀⠀⠁⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠁⡇⠠⠀⠀⠈⠲⣄⢈⣑⠊⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢄⣀⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠐⠚⠀⠀⡀⠀⠀⢈⠀⢳⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠃⠀⠀⢀⠀⠀⠀⠀⠀⠠⠜⢀⡼⠆⠀⣀⡤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠈⢆
⠀⠚⠁⠀⠘⠀⠀⠠⠁⡸⠀⠀⠀⠀⠀⢠⢪⠀⠀⠤⠆⠀⠀⢀⡏⠀⠀⠀⠁⠀⠀⠀⠀⠈⠳⠀⠀⠒⢄⠀⡜
⠀⠀⠀⠈⢠⠂⠀⢤⣠⠇⠀⠀⠀⠀⢀⠁⢸⠀⠀⠀⠀⠜⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁
⠀⠀⠀⠢⠞⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠲⠊⠀⠀⠀⠀⢸⠀⠀⠻⢦⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠲⠊⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⢀⡴⠓⠒⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣒⣛⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⢹⠆⠀⠀⢀⣤⣠⡏⠉⠃⠀⠈⡗⠒⠀⠐
⠀⠀⠀⠒⠶⡶⢼⡶⣶⣦⡀⠀⠀⠀⠂⠀⠀⠀⠂⢤⠤⣘⣤⡟⠲⡁⠠⠒⠱⠋⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅
⠀⠀⠀⠀⠀⠀⠀⡇⠈⡌⢿⠄⠈⠀⠀⠀⠀⠀⠀⡸⢐⠛⠀⠀⢀⣄⠀⠀⣤⠐⠉⠁⠀⢠⣾⢻⣻⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣣⠀⡇⢸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠈⠀⠀⠀⠏⠈⠲⠒⠊⠉⠙⠋⠁⠀⠀⠀⠈
⠈⠺⡄⠀⠀⠀⠜⡿⠾⡟⠦⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠄⠀⠀⠉
⠀⠀⠈⢩⠖⠀⠀⡅⠀⢀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠀⠀⠀⠋⠀⠀⣠
⠀⠀⠀⠈⠀⠀⠤⢰⡤⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠶⠀⠃⠆⠀⠇⠶⠘⠰⠆⠖⠒⠓⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⢠⡴⠶⣶⠉⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠄
⢀⠔⠚⠂⠀⠈⠀⠀⠜⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣤⣾⡛⠛⠯⣃⢄
⢸⠄⠀⠀⠀⢀⣤⣤⡄⠀⠐⠋⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣀⣴⡜⠛⠦⠀⠙⠦⡈⢳⣃
⢀⠀⠀⢀⣨⣿⣭⣷⠁⠀⠀⠠⢄⠀⠀⠂⠐⢀⡀⠉⠀⠀⠀⠀⠆⠠⢤⣿⣤⣔⠉⠀⠇⠀⠀⠑⠄⠀⠱⠄⣿
⠈⠟⠉⠁⠀⠈⠉⠀⠀⠀⢀⣀⠈⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢡⠈⢇⠀⠀⠀⠀⠀⠈⠆⠀⣧⡟
⠀⠀⡀⢀⠄⠀⠀⢠⠀⠀⠀⠁⣠⠎⠀⠀⠀⠠⠀⠀⠀⠀⠢⣀⡀⢈⠀⠀⡰⢀⣿⠄⠀⠐⠀⠀⠀⢸⡴⠋
⣤⣤⣥⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣥⣤⣽⣶⣶⣿⣬⣥⣤⣵⣤⣤⣵⣦⣥⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣉⣁⣀⣀⡀⠀⠐⢶⣶⣶⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠛⣁⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠠⢬⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠉⡋⠀⠈⠑⠓⠄⠀⠀⠀⠀⠰
⠈⠋⠉⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡔⠀⡄⠀⠀⠀⠣⠄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠳⠀⠀⠐⡄⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠐⣆⠀⠀⠀⠀⢀⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡁⠠⠤⠄⠐⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄⡀⠀⠀⠀⠂⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣵⣤⢄⠐⠂
⠀⠀⠁⠀⠀⢐⣀⣠⡀⠐⠲⠶⣶⡶⠀⠈⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠋⠁⢠⠆⠀⠀⠀⢘⠀⠀⠀⢠⣤⣴
⠀⠐⠁⠀⠀⠀⠉⠁⣀⠀⠀⡄⠁⢐⡆⠀⠀⠀⠀⠀⠐⠈⠆⠶⣄⣀⡀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠈⠉⠑⠦⠀⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠀⠀⠀⣴⢀⠀⢰⠁
⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠧⠠⠀⠃⠀⠀⠠⢄⡂⠐⢦⠘⠱⡀⢧⠇⢀⡠⠤
⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⡓⠦⢙⣦⣁⣆⢀⡡⠀⡨
⠀⠀⠈⢀⠀⠠⡄⠁⠈⠀⠀⠀⠀⠀⡘⠖⠉⣠⡼⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠚⠛⠙⠻⢿⣿⣿⡶⠒⠛⠁
⠀⠀⠀⠈⠷⠢⣤⠀⡀⠀⠀⠀⢀⠐⢁⣦⠟⡉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⢰⣥⠴⡺⠋⢁⠞⡿⢹⢟⢣⣀⠀⠁⠁⠀⠐
⠀⠀⠀⠀⠤⠒⠄⠁⠀⠀⠀⠀⠀⣴⠏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠄⠤⠂⠁⠐⠀⣰⠈⣥⠜⢆⠉⠁
⢄⠀⡄⠂⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⠀⠀⢠⡄
⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⠎⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢐⡀⠀⠀⠀⠴⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠄⠀⠃⠀⠈⠸⠀⣴⡃⡀⠄⡠⠀⠀⠀⠀⠐⣡⠾⠀⠀⠀⠀⠐⠀⠂⡨⠿⢧⡀
⠠⠄⠀⠀⡀⠀⡐⠀⠀⠃⠀⠈⠁⠀⠊⠀⠀⠀⠀⢸⣿⣁⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⢸⠀⠀⠁⠈⠁⠀⠀⡠⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⠀⠀⠀⠀⠀⢀
⠅⠊⡀⠌⠀⠀⠠⠄⢀⠠⣈⣉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠁⣀⣒⠒⠂⠢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣈⠄⡤⠈⠓⠒⠂⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠊⠉⠄⠘⠉⠉⠐⡄⠀⠀⠀⠤⢀⡐⠠⠀⠠⣄⠉⣠
⠀⠀⠒⠀⠂⠉⠀⠀⠀⣀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠰⢄⡙⣳⣄⡀⠁⠘⠛⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠱⣤⠉⠛⠶⣶⣸⠀⠋⠃⡀
⣉⣯⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠐⢒⡀⠙⢛⢃⣀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠛⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠿⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣬⠉⠉⠉⠛⠛⠿⢿⣆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣋⣵⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡞⢭⣄⣀⣲⡶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢡⣾⣿⣫⣾⣿⣿⣿⣯⣭⣴⣿⣿⡿⠿⢿⣭⣭⣛⣛⠻⠟⢛⣻⠿⣽⡻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣋⢰⣿⣿⣾⣿⣿⣿⣟⣋⠍⠛⣉⠉⣍⠁⠀⠹⠽⣟⣓⠉⢉⣒⠀⠉⠛⠶⣭⡂⠈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣾⣿⠿⠫⣿⣵⣶⣶⣶⣾⣤⣀⠓⣈⠙⢂⡀⠈⣉⠓⠌⠻⣦⣙⢶⣌⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⣧⣿⣿⣿⣄⡶⢰⣿⡽⠶⣶⣾⣿⡻⣿⣷⡈⡁⢠⠙⢆⠈⠳⡄⠀⠀⠙⠷⣍⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⡿⠁⢺⣹⣟⠻⠎⣿⣿⢿⡼⣿⣷⢳⠈⡇⢠⡀⠳⡄⠀⡀⠀⠀⠈⢳⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠃⢻⣿⡜⣇⢦⠘⢿⣍⢿⣷⢿⣿⣾⢳⣿⣿⠘⠂⢀⠀⢳⠀⡉⠀⠹⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣦⠻⣿⣾⣖⠦⠀⠬⢁⣰⣿⢟⣣⣿⣿⠃⡀⠀⠀⢳⡀⠀⠸⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⢷⣝⠻⣿⣿⣶⣶⣾⣿⣶⣿⣿⠟⠁⠀⠁⠘⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣍⠡⠤⣬⠉⠙⠛⠛⣉⡥⠀⠠⠞⣠⠎⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠉⢀⣈⡥⠂⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣴⣶⠶⣒⣀⣬⣭⣴⣶⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣬⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⢲⣦⣤⣄⣀⣒⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢗⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢵⢟⣩⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⠏⣿⣳⣿⣿⣯⣞⣯⣿⢿⡿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠜⠁⠙⠙⠉⣾⠋⡩⣻⣟⢕⠿⣻⢻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠺⠿⣷⠿⣻⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢊⣠⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣿⣠⡔⠁⠀⠀⠀⠀⠀⠀⣠⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣻⣿⣾⣿⣟⠕⠀⠀⠀⠀⢀⠀⡺⣫⣽⢟⢽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢛⣿⠟⠁⠀⠀⠀⠀⠁⠀⠈⢫⣴⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢇⣬⢠⡔⠀⠀⠀⠀⠀⠀⢀⣤⠚⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⣾⣿⣿⣻⣷⠛⠔⠀⠀⠀⠠⢛⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⠺⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣴⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⢽⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢝⢦⡀⠀⢀⡤⡺⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⣽⣿⡵⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢔⡫⠚⠛⢮⡢⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⣾⠗⠉⠀⠀⠀⠀⠙⢮⣢⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣯⣄⡉⠁⠀⠀⠀⠀⠀⠀⠙⠪⡲⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠈⠪⡳⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⠪⡳⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⡿⡟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣤⣶⣵⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠚
⠀⠀⠀⠀⠀⠀⢰⡿⣻⣯⣵⣿⣿⡏⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣃⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠻⣿⡿⣿⣿⢃⣾⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡾⠿⠿⠿⠊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠃⠈⠓⢢⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢁⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡟⡗⠚⡇⠀⠀⠀⠀⣠⡶⠶⠂⠐⣧⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣏⠸⣇⣇⣀⣙⣲⡄⢸⣾⢟⣶⣄⣀⠚⢉⣀⣀⣀⣀⣀
⠀⣁⡀⠤⠒⠚⠃⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠀⠸⠖⠠⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⠀⠀⣠⠞⠁⣠⣶⣶⣲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣦⣾⡔⢀⠀⠀⠀⢆⠉⠀⣶⡇⢸⣷⣯⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⣿⣷⢮⣤⢄⣄⢻⡿⠘⢿⣼⣿⡏⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠠⠛⢿⠾⢀⣉⣺⡇⣸⢃⡨⠷⡴⢊⠡⢒⡨⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⠶⢮⡓⢿⡺⢃⡔⣻⣿⢁⡿⢛⣴⣨⠌⠱⡾⠷⠦
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡾⢺⣥⣤⣘⣯⣑⣅⣌⠲⣿⣿⣼⡗⣫⠞⠚⠋⠉⢀⣀⣀⣀
⠀⠀⠀⠀⡀⣦⡙⣿⣷⠃⡀⠈⠱⣿⣿⠟⢿⣼⡎⣹⡻⢯⡿⢋⣠⢴⣲⡾⠛⠀⠀⠉⠁
⠀⠀⠀⣀⣼⠻⣧⣽⣿⣿⣯⢉⡉⠩⣶⢍⠢⢿⣷⣿⣽⣦⣳⣾⢗⢏⠍
⠀⠀⣴⡿⢿⣟⣿⡿⣿⣿⣝⡉⣀⣀⢹⡃⠕⣬⢻⢿⣿⡏⣤⠝⣲⣍⠤
⠀⠀⠁⠀⣙⣿⣿⣿⣿⡋⣀⡈⠅⠛⠉⠻⣾⣿⡟⢶⡛⢿⣿⡼⠞⣙⣳⡴⠊
⠀⠀⠀⠀⡉⣿⣙⣻⢿⡟⢡⠀⠀⠀⠀⢤⡟⡍⢻⣾⣿⡜⡬⣴⣾⡏⡈
⠀⠀⠀⢠⣸⠛⣿⣳⣿⣿⣏⡤⠀⠀⠀⢼⠐⢿⣼⡿⠻⣷⣷⣞⣿⢿⠴⣯⠒
⠀⠀⠀⢈⣿⣶⣿⣿⣿⣏⣧⠄⠀⠀⠀⣀⣽⣼⣿⣷⡀⣹⣟⣿⣷⣿⡄⠃
⠀⠀⢀⣾⣟⣿⣻⣾⣿⣿⣿⣆⠶⠦⣾⣃⣱⣟⢻⣿⣎⣿⣿⢏⣇⣾⡟⠂
⠀⠴⠟⠛⠿⠿⠿⢿⣿⠿⠿⠫⠍⠀⠄⢈⣭⣿⣾⣿⣿⡸⢿⠚⠋⣼⣇⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡠⢾⣼⣧⣿⣿⠿⣿⣇⠈⣸⣅⢹⣿⢰⢲⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣞⣿⢱⣿⠇⣟⣶⢸⢸⡇⢃⣿⣌⣿⡆⣠⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣻⢻⡿⡀⢻⢿⢔⢿⣇⣜⡹⣿⣿⣷⣿⣧⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢽⣾⢿⣾⣿⣿⣩⢀⣿⣾⡏⡟⡟⣿⣿⡿⣯⢹⡟⣿
⠀⠀⠀⠀⣄⣀⣀⠠⠤⢤⣤⣰⢾⣿⣷⣿⡟⣰⢹⣧⣸⣿⣷⣿⣧⣻⣿⢹⣿⣾⣷⣿⠆
⠀⠀⠀⠘⠼⡗⡡⣠⠖⣃⠿⠾⣿⡟⢿⢿⣷⡿⢸⣿⣿⢸⡿⣿⢿⣿⣟⣿⣿⣿⣿⣿⠾⠖⢒⠶⢶⡴⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⣿⣷⠁⡾⠁⠀⢀⡇⢹⠀⡇⣻⡏⠻⢿⣿⣴⣾⣽⡸⣿⣻⡻⣿⣿⡟⠁⣠⠊⢠⠂⠁⢹⡟⣏⠹⡙⣦⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠁
⠀⠀⠀⢠⠄⠉⡻⢺⡇⠀⣤⣾⠧⠿⠾⣷⠋⢭⣄⣸⡏⡿⣿⣿⣿⣿⣿⣽⣿⣿⣇⢀⡇⢰⡇⠀⠀⣼⠁⣼⢰⣇⣿⠄⣼⣅⣤⣶⡉⢱⣌⡢
⠀⠀⠀⠐⡆⣐⣬⣿⣷⣼⣫⡼⠶⢅⠐⠠⡀⣠⣿⠿⣿⡿⢿⢌⠻⣿⣿⡟⡿⢙⣻⠾⠟⣙⣧⣶⡞⢻⣾⣿⣾⣿⡟⠺⠯⠿⣍⡱⣶⢀⡈
⠀⠀⠀⠀⠈⠿⣹⡞⢡⢠⡏⢀⠀⠈⢳⡄⣻⣶⣺⡦⣌⣥⡀⣄⢠⣙⠛⠡⣶⣯⠈⠑⠚⠪⣿⢑⢔⢿⣛⡿⢻⣻⣾⣿⡿⠤⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣇⢸⣀⣧⡘⣧⣄⣸⣿⣉⢻⣷⣅⣤⣴⣯⣥⡿⠏⠛⠒⠋⠁⠀⠀⠀⠀⠈⠉⠉⠈⢉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢽⣿⠿⠶⠞⠄⠼⣟⣽⠋⠹⠁⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂

⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣾⣿⡶
⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠈⠉⠉⣿⡿⠃⠀⠀⠀⠀⠘⠛⣿⣿⠟⠀⠀⠀⠀⠀⠸⢿⣿⠋
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⣀⡀⠀⠀⠀⢰⣿⣿⡗⠀⠀⠀⠀⢀⣾⣿⣇⣀⣀
⠀⠀⠀⠀⠛⢿⣿⣷⣦⠀⠀⠀⠀⠘⣻⣿⣿⣟⡀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠘⢻⣿⣿⠟⠁⠀⠀⠀⠈⢹⡟⠁⠀⠀⠀⠉⠁⣸⣿⡟⠁
⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠶⠶⠶⠶⠶⢤⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⣛⣋⣩⣍⣉⣙⣛⠻⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷⣿⣿⣴⣶⣶⣶⣤⣤⡤⢟⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠈⠉⠒⠛⠋⠉⠛⠛⠛⢛⣡⠄⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⢿⣛⣛⣛⣛⣛⣛⣛⣛⣉⣭⣴⣶⣿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⠤⠤⠤⠤⣤⣤⠤⠤⠶⠖⠛⠉⠁⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠢⠬⢭⣭⣭⣭⣭⣭⣭⣿⠿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⣛⣓⠒⠶⠶⠒⠒⠒⠂⣀⡀⠐⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢑⠒⠶⠶⠶⠶⠶⠶⠾⢟⣻⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣉⣉⣉⣉⣉⣉⣡⠤⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢭⣛⣛⣛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠦⠭⠭⠭⠥⠄⠀⠀⠀⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣒⠒⠂⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡑⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣓⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠀⠀⠀⣠⠤⡾⠿⠻⠿⠧⠤⠤⢤⣀⣀⡖
⢱⣖⠋⠻⣅⠐⠈⠀⢠⠔⠒⠒⠒⠦⠤⠤⣉⣉⠓⠲⠤⢬⣑⡒⠢⠄⣀
⢸⡉⠳⣄⡈⠓⠦⣄⣀⠀⠀⠀⠈⠉⠒⠒⠒⠬⠍⣛⠒⠦⣤⣉⡙⠒⠦⣄⣀⠀⠀⠀⡀
⠀⠑⢦⣀⠉⠓⠦⣄⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠈⠉⠓⠲⠤⣌⡙⠒⠤⣀⠁
⠠⣀⠎⠉⠙⠒⠤⢔⡉⠛⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣉⠒⢤⡀⠀⠀⠠
⠀⠈⢷⠒⠤⢀⡀⠀⠀⠉⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣈⡐⠒⠥⠤⣒⣂⡈⠙⣦⠈⠁⠸⡄
⠀⠀⢈⠑⢦⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠦⠤⠭⠭⠥⠤⠤⠤⢳⡀
⠀⠀⡀⢠⡆⠉⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠠⠤⢄⣀⣀⡀⠀⠀⠀⢂⣠⠾⠁
⠀⠀⠱⣧⣀⠁⠂⠄⠀⠀⠀⠉⠉⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⢤⣄⣀⣉⣉⣙⣏⣉⠄
⠀⠀⠀⠘⢯⡙⠓⠶⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⣀⣀⣀⣀⣀⣉⣭⠽⠋
⠀⠀⢈⠀⠀⠉⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⣀⣀⣀⣈⣉⣉⣩⡧⠄
⠀⠀⢺⡀⠘⣦⡃⠀⠈⠉⠛⠓⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣭⢽⡋
⠀⠀⠀⠙⢦⣘⡝⠓⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠉⠉⢉⣡⠴⣺⣃⠀⣴
⠀⠀⠀⠀⠀⠈⠙⢷⡶⠤⢤⣄⣀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠚⢛⣉⡤⠞⠁⢈⣽⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠴⠋⣠⣯⣀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⡈⠉⠉⠉⠉⠉⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠈⢁⡴⠚⠁⠀⣨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣛⣛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢚⡥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠛⢻⡛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠊⢀⡀⠂⠠⠴⢛⡷⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⢾⣧⣄⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠁⠀⠰⠚⢉⣠⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⠛⠉⠉⠀⠀⠀⠠⠄⠲⠶⠛⠋⢁⣴⣣⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣄⣀⣀⡰⠄⠀⠀⠀⢀⡀⣠⡭⠛⠁⣈⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢉⡹⠋⠉⠁⠀⠀⠀⠀⠉⢀⠠⠖⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣉⡉⠁⠀⠀⡀⠀⠀⠠⠖⠛⢧⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠉⠁⠀⠀⠀⠁⠀⠀⢀⣠⣤⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠛⠒⡀⠀⠀⠀⢈⣩⣥⠖⠋⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣹⣗⡒⠠⠁⠀⠀⠀⢀⣠⣬⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢭⣈⠉⠀⠀⠐⣀⣐⠣⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠀⠈⠙⠓⠲⠦⢼⡧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⠷⠀⠀⠀⠀⣀⢳⡖⠚⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠄⠈⠁⠀⠐⣶⠂⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠭⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣑⠶⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢣⠀⠀⢀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠸⠀⢈⠀⡔⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠈⢢⠀⠀⣀⡄⠻⡲⣕⣾⡰⠧⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢸⢠⠋⠀⠀⠀⠈⡩⠯⡖⠉⠁⠉⠑⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠐⠒⢤⠙⣤⣀⠴⢁⡤⠊⣀⠠⠬⢕⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠠⠷⣜⠀⡰⠻⠛⠭⡑⠄⠀⢀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠐⠳⠀⣸⠊⠀⠸⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⣆⠀⢀⡐⢤⣴⠗⠉⢉⠇⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠢⣗⡔⠁⡄⡸⠋⢰⣌⢡⠊⠉⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⢄⣀⣱⠾⣧⡀⠃⠃⢠⣜⢼⡶⠤⡴⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣍⣿⡟⠰⠀⡇⠘⣄⡥⢎⠀⢀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠉⡟⠲⡄⠡⣀⡿⢀⠄⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⢒⡩⠤⣀⢱⠀⣹⡗⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣀⡀⠴⣋⠻⡀⠈⠉⡩⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠆⠙⢄⠀⡁⢘⣠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠉⠑⠪⣢⡄⣴⣭⣛⣺⣾⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣒⣂⣉⠤⣏⠁⠀⡠⠤⠤⢬⠖⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡠⠁⠰⠁⠱⠀⣇⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠊⠀⠀⠈⢢⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣻⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣿⣆⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣿⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⢿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⣿⡇⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⡤⠞⠁⠀⠀⢀⣴⡿⠟⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⣿⠁⢀⠜⠀⢀⣤⣤⣾⡟⠉⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀
⠀⠀⠀⣠⣤⣤⣤⣴⣶⡾⠟⠋⠁⠀⢸⡟⠀⠁⠀⠚⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣧⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣇⣀⣀⡀
⠀⠀⢰⠟⠀⠀⠀⣠⡟⠁⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⢠⣾⣿⡟⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠈⠁⠈⢳
⠀⠀⡻⠀⠀⠀⣾⠁⠀⠀⠀⠀⣠⡤⡼⣷⣴⣶⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠀⢻⣷⣴⣶⣶⠶⠶⠶⣦⣀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠃
⡠⠞⠁⠀⠀⠀⢻⠀⠀⠀⢀⠞⠁⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⠃⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠛⣷⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠘⠀⠀⠎⠀⠀⢀⣀⣠⣤⡶⠶⠟⣿⡟⠋⠉⠙⣿⣿⣆⠀⠀⠀⠀⠙⠷⠶⣤⡀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣴⠏⠀⠀⢠⣿⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠘⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣤⣤⠤⠤⢄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⣿⣇⠈⠉⠛⠿⢿⣧⣄⠀⠀⠓⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⣰⠚⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣇⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠈⠿⢶⣶⣤⣤⣤⣄⣀⣤⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⢠⡞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠀⢻⣆⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⣫⣿⣿⠟⠀⠀⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠋⠁⣠⣾⣿⣿⠁⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣾⣿⠟⠁⢻⣇⠀⠀⠀⠈⠻⣆⣠⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠁⠀⠀⣴⣿⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠉⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣸⣿⠇⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⣆⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠙⠦⣀⠀⠀⠀⠀⠻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠞⢻⣿⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠷⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣤⡀
⠀⠀⢠⡆⠀⢠⡶⠀⠀⠀⣦⡄⠀⢀⣀⠀⠀⠸⡄⡏⡄⠀⠀⠀⠿⣿⣤⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠘⢷⡀⣼⠟
⠀⢀⠘⡇⡴⢋⡀⠀⣀⡀⠹⠿⣾⡿⠟⠀⢀⠳⣜⢊⡴⢀⠀⢀⣠⡛⠁⠀⠀⠀⣴⣿⣷⣦⡄⠀⠀⠀⠀⠈⠳⢁⣴
⠀⠘⣧⡈⢠⠾⠁⡄⠻⣷⣄⠀⣠⣴⡦⠀⣬⢷⡈⡡⠴⢋⡄⠘⠿⣿⣿⠀⢠⣷⣹⣯⣿⣾⣽⣷⡄⠀⠀⣈⢣⡿⠋
⠀⠀⢨⠛⠆⣠⡾⠃⠀⣨⡍⡀⠛⢋⠀⢀⡙⢇⢹⣠⠶⢟⠀⠀⢀⠀⡀⠀⣼⡛⣿⣿⣿⣿⣿⣻⣥⠀⠈⣿⢿⠁
⠀⠀⠸⣦⡀⠋⠀⠀⠀⠘⠿⣧⣾⡿⠃⠀⡳⡌⣯⣶⠞⠁⠀⢾⡿⣿⣿⠀⣻⠿⣯⣿⣿⣿⣿⢿⣿⢃⣀⠈⢿⢋⣤⡄
⠀⠀⠀⠈⠳⣠⡄⠀⠀⢿⣦⣏⣴⣶⠀⠀⠳⣘⣡⠆⠀⠀⠀⢀⣀⠀⣀⠀⢿⣋⣿⡟⢻⣏⣹⣿⣦⠈⠿⡀⡸⠚⡋
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙⠟⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⢿⠿⠿⠂⠀⠉⠉⠀⠀⠻⠿⠿⠃⠀⠀⠰⢡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢏
⠀⠀⠀⣶⣄⠀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣆⢀⣤⣤⠀⠀⠀⠀⠀⠀⣿⢀⡄⠀⢸⣧⠀⠀⡀
⠀⠀⠀⢻⣿⠆⠃⣀⡀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠈⢁⣈⠛⠋⠀⠀⠀⠺⣦⡀⢹⡿⠃⠀⠀⠙⠀⣾⠇
⠀⠀⠀⠀⠉⣥⣾⣿⠇⠀⠀⠀⠘⢿⡇⣠⣴⣶⠀⣤⡄⠸⡟⢷⡶⠀⢀⣤⣤⡬⢥⡄⣀⣀⣄⠻⣶⡤⡄⠀⣰⣿⠂
⠀⠀⣀⣤⣀⠘⠋⠀⣀⣀⠀⢀⣄⠀⡀⣿⡿⠃⠀⢙⣥⡄⠀⢰⣶⡆⠈⠛⣡⡀⠀⠱⠀⢉⡋⠀⠀⠀⠬⣀⠛⢁
⠀⠀⠻⣿⡿⠿⡇⣾⣿⡿⠂⠸⣿⣷⠸⡇⢀⣴⡆⠈⠻⠓⢤⠘⠛⡁⠀⠘⢿⣧⠀⢆⣾⡿⠃⠀⢀⣤⣀⢸⠐⠁
⠀⠀⠀⠀⠀⣀⣸⠈⠉⠀⠀⠀⠉⠻⡄⣶⣿⣿⠟⠀⠀⠐⣄⢀⠞⠀⠀⠀⠀⠉⢠⠋⠁⠀⠀⠀⠀⠙⢻⠘
⠀⠀⠀⠰⣾⣿⡿⣠⣴⣆⢸⣿⣷⡄⢸⠛⠋⠁⠀⠀⠀⠀⠈⡟⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⡄⡞
⠀⠀⠀⠀⠈⠁⠀⠟⠛⠁⠀⠙⠛⠓⠷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢀⠏⠀⠀⠀⠀⠀⢠⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⠶⠶⠶⢶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⣉⣁⣤⣤⣤⣶⣶⣶⣦⣤⣤⣈⡉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⣉⣤⣴⡾⠿⠛⣋⣉⣭⣥⣤⣤⣤⣤⣬⣍⡙⢿⣦⡀⠈⠃
⠀⠀⠀⠀⠀⠀⣠⡿⢋⣴⣾⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣧⠀⢀
⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣴⣿⢿⡏⢀⡾
⠀⠀⠀⠀⠀⠈⠀⠀⠻⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⣛⣋⣭⣵⣶⣿⡿⢛⣵⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠈⠻⣯⣝⣛⠻⠿⠿⠿⣿⣿⠿⠿⠿⠛⢋⣡⣴⣿⣿⠟⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣸⣻⣿⣿⣷⣶⣶⣤⣤⣤⣶⣶⣾⣿⣿⣿⡟⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡤⢤⣿⣿⣿⣿⣿⠥⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠟⠓⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⡶⠒⠛⠛⠙⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢣⠏
⠀⠀⠀⢠⡇⡰⣊⠩⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡟
⠀⠀⠀⢸⡇⠁⠁⠀⠙⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠘⠒⠒⠲⢦⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⠛⠛⠉⠉⣹⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⣹⠇⠀⠀⠀⡜⠀⣀⣀⡀⠀⠀⠀⣴⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠛⢉⡟⠀⠀⠀⠀⣼⠃⠀⠀⢀⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣯⣤⣄⠀⠀⠀⢲⡏⠀⠀⠀⠀⠉⠉⢉⡽⠃⢀⢀⡼⠁
⠀⠀⠀⣿⣀⣠⣤⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⣏⣴⡶⠟⢋⡏⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣢⡟
⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⣁⣀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠟⢛⡿
⠀⠀⠀⠀⠀⠰⡇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⢠⠏⠀⠀⠀⠀⠿⠛⠒⡖⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡧⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡤⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠟
⠀⠀⠀⠀⠀⢸⡇⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠋
⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠠⡇⢷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡴⠶⠖⠒⡒⣲⠆
⠀⠀⠀⠀⠀⢰⣶⠛⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⢀⡇⠈⢣⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⡶⠛⠁⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⢠⡶⠞⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡞⠉⠀⠀⢀⢂⡴⠛⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣰⢏⠀⠀⠀⠀⢠⣿⣃⣀⣀⡀
⠀⠀⠀⠀⣿⡄⠀⠀⠀⣠⣭⠇⠀⠀⠀⠘⣇⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠐⠉⠉⠉⣩⠞⠁
⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣧⣤⡄⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠠⣞⠃⠀⠀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠓⢲⠀⠀⠀⠀⠀⣸⠀⢀⡀⠀⢠⠞⠀⠀⠀⢰⡯⠶⠖⠚⣿⠀⠀⠁⣴⠋
⠀⠀⠀⠀⠹⣆⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡜⡀⢸⠀⠀⠀⠀⠀⠙⠋⢹⡟⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡄⠀⠀⠀⠀⠀⢀⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠹⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡇⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠓⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠄⠠⢀⠀⠀⠀⢀⠀⢑⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄
⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠁⠀⠀⠁⠀⠠⠀⠀⠀⡐⠀⠐⠦⣀⣀⡂⠀⠀⠈⠣⢄⠤⣁⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠁⠀⠀⠠⠀⠀⣀⠀⠐⠀⣀⡠⠒⠊⡑⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⡤⡤⠤⣤⣤⠊⠙⠀⢘⡆⠀⠀⠌⢳⢠⣍⠲⢠⠀⠬⠄⠐⠚⠒
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠈⠁⠀⠀⠀⠈⠁⠀⠀⠋⠩⡀⠐⠓⠀⢀⣀⠀⠀⠤
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠤⠀⠀⠀⠒⠄⠀⠀⠈⠭⠀⠦⠤⠄⠤⠅⠃⠈⠈⢄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⢀⠤⠂⠀⠀⢀
⠀⠀⠀⠀⠠⠌⠀⠐⠀⠀⠠⠀⠀⡀⣀⣀⡠⠀⠀⠈⠒⠂⢦⠀⠀⡠⠐⠈⠢⠐⠒⠀⡁⠀⡠⠀⠁⠈⠀⠁⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⢀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠔⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣶⣿⠯⠷⠦⠤⢤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⠻⣶⣮⣽⣿⣷⣶⣶⣭⠍⠛⠛⠿⠷⣶⣤⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡈⠉⠛⠿⠿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣈⣉⣿⡦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡀⠀⠀⠀⠀⠲⢤⣤⣬⣉⣉⣉⣉⣉⣉⣻⣿⣿⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣄⡀⠀⠀⠀⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⠀⠀⠀⣻⣿⣿⣷⣾⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣉⠁⠀⠀⣹⣿⣿⣿⣿⡷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣉⡁⠀⠀⢤⣭⣭⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡍⠀⠀⣈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣗⣂⠀⠈⠭⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣥⣤⡄⠘⢻⣿⣿⣤⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠙⠿⢿⣥⠀⠸⢿⣿⣿⣆⡈⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣿⣗⡘⢿⣿⣿⣿⣤⡁⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠀⢰⣦⠈⠀⠀⠸⠿⣦⠉⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠐⠀⠀⠙⠋⠀⡀⠀⢴⡄⠉⣧⢸⣿⣿⣿⠁⠸⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠈⠀⠀⠀⢈⡀⢀⡙⠀⣿⣾⣿⣿⠏⠁⠀⠀⡀⠈⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢀⠀⠀⢤⣤⣿⣿⣷⢰⣿⣿⡿⣣⣀⠀⠀⠀⠡⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠄⠁

⠀⠀⡀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢦⠤⡄⠀⠀⠀⠀⠀⠀⠄⠰⢠⠄⠀⡀⣀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⡋⢰⡄⠀⠄⠀⠄⠀⠀⠀⣀⠰⢊⡋⠀⠀⠀⡀⠐⠄⢀⠀⢐⠒⠳⠀⠸⠾⠶⢧⠀⠠⡀⠀⢂⠐⡄⠀⠃⠘⢃⣀⠈⡃⣀⠀⢰⢽⣳⠀⠇⠀⠀⣎⣷⠄
⠀⠀⠀⠉⣡⠟⠀⠠⣤⠲⠐⠀⠈⠤⠜⠂⣡⠤⠒⠒⠓⠒⠢⠄⠀⠀⢢⠄⠀⠐⠀⡐⠀⠀⠀⣈⠉⠋⠉⠀⡰⠄⠉⠀⢈⠀⠀⠉⣤⠈⠋⠀⢀⠀⠂⠐⠁
⠀⠀⠀⠀⠈⣀⢤⣅⠈⠏⢀⡀⠀⠀⡀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣺⠧⠄⠀⠀⠘⠙⠃⢀⣠⠈⠢⠠⠥⡂⠀⠵⠋⣀⠀⠆⢄⡀⠐⠂⠐⠂⠁
⠀⠀⠈⠑⢤⣿⡬⠅⠀⠀⠨⠁⠀⣴⠤⣤⣄⣀⣠⣤⡀⣀⣀⡀⡀⢤⣴⠀⠀⢈⣍⠓⢀⣀⢰⣥⣤⡆⠈⡤⠀⠀⠀⢠⠀⠤⠀⠀⠈⠀⡄⠀⢃⠀⠀⠀⢀⠆
⠀⠀⢀⠀⠀⠁⠀⢀⡒⠆⠀⠀⠀⠀⢠⣦⢀⠉⠀⢨⡇⠈⠁⠀⡀⠀⠀⣄⠱⠦⠀⠀⠀⠄⠀⠉⢉⢀⡀⠀⣎⡁⠰⣿⠀⠀⠀⢀⠤⡈⠀⢀⠊⠀⠦⠆⠈⢦
⠀⠀⢸⣦⣴⠆⢀⠠⠤⣤⣄⠀⡠⠀⠀⠉⠆⡀⠀⢸⡇⠀⠰⠀⠓⠀⠀⡈⡄⠀⢰⢻⢢⠀⠀⡠⢄⢈⡀⠀⡯⠅⢩⣿⠀⠀⢠⠃⠀⠀⢢⠏⡀⠀⠀⠀⠀⠈⢃
⠀⠠⠽⠯⠇⠠⠁⠀⢰⠁⠀⠀⣤⠄⠀⠈⢈⡴⠀⠘⡇⠀⠃⣔⠒⠲⠀⠰⣧⠀⠘⠨⡃⠄⠀⢧⣀⠬⠁⠀⢳⢨⠼⠿⣝⡆⢀⠀⠀⠀⡌⠀⢁⡀⠀⢠⠀⠁⠈
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢒⢚⡇⠙⠂⠐⠢⠂⠀⠀⡇⠀⠒⠃⠄⢀⠀⠐⡂⠣⠀⠡⣷⠃⠀⣀⠀⠀⡀⠀⠘⠂⠸⠴⠛⠁⠈⠀⠀⠀⠃⠀⠈⠀⠊⠃⠘
⠀⢀⣀⡠⢃⠔⠂⠐⠀⠀⠪⢹⡍⠑⠀⠀⠚⠁⠸⠥⠁⠀⡀⡄⠀⠈⠀⠀⠀⢄⢃⠀⢿⠀⠀⡏⠄⠀⣙⠄⠀⣀⠀⠈⠃⠀⠀⠈⠠⣀⢀⠀⠀⠀⢹⠀⡀
⠀⠈⠁⠰⠑⠖⠃⡀⠀⠀⢈⡐⠃⡄⠀⠀⢸⠂⠀⢠⡎⡃⢈⠤⠁⢰⠀⠀⠀⠀⠲⠀⠈⠀⠀⢃⠀⢊⠜⠀⠀⠙⠀⠀⢂⠀⡆⠀⣴⡀⠉⠃⠀⣁⡙⣨⠁⠈
⠀⠀⠑⠀⢤⠄⠈⠀⠀⣀⡀⠀⠀⠀⢀⠰⠋⠀⠀⠈⠓⠀⠀⡀⠐⠈⠀⠀⠀⢀⢶⡄⢀⡀⠀⠈⠉⠉⠀⠀⠀⠦⠁⠀⠾⢯⠃⢀⠆⠀⡀⠐⠆⠀⠀⠀⠀⣤⣠
⠀⠰⡀⠀⠀⠁⠀⢴⡀⠀⠧⠀⡄⠀⠉⠁⢢⠀⠀⠒⠉⠁⠈⠉⠎⠋⠀⠀⠀⡽⢀⡁⣀⠁⡔⠄⠀⢀⠔⠀⠢⠀⠐⠠⢦⠀⠀⠸⡀⢀⠄⢈⣀⣦⣀⡀⠀⢠⡀
⠀⠀⠷⠀⠖⠀⠀⠀⠉⠉⠁⠀⠐⠀⠀⠠⠎⠠⡂⣀⠊⠀⠈⠠⠿⡆⠀⣀⡌⠁⢮⡷⠶⠒⢂⣤⣞⢣⡄⢀⡄⠀⠀⡄⠀⢹⠀⠀⠀⡠⢖⠥⠎⡐⣁⣌⡴⡄
⠀⠀⣸⠀⢀⠀⡀⢠⠖⠀⠀⣤⣶⣴⡶⡆⠀⢰⡟⡄⠀⣀⡨⠤⠤⠤⠀⠀⠀⠈⠀⠠⠤⠐⠚⢼⠉⠹⡀⠀⠃⠀⠀⢀⠀⡀⠀⠀⣜⠽⡳⡴⢟⠴⡹⣾⣪⡵⡆
⠀⠀⠀⠀⣀⠀⣿⡶⠀⠀⠈⠉⠉⠀⠇⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣧⡄⡀⠀⠘⠑⠃⢼⡀⢠⠶⢮⠒⢺⡿⢿⣾⠖⢻⠟⢿
⠀⠀⢠⢀⠿⠶⡀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠻⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡽⡷⣜⡖⠸⡀⠀⠀⠀⠸⠀⠀⠀⠀⠃⠘⠸⠀⠀⠀⠘
⠐⣂⠸⠿⢤⠀⠄⠀⠀⠀⠀⡀⠀⠀⠉⠠⡭⠀⠀⢰⡴⢦⡔⠁⠁⢀⠈⠀⠀⠀⠀⠉⠳⣄⣤⣠⠀⠘⢓⠈⠉⠀⠈⡏⢁⠀⠀⣤⣤⣤
⠀⠙⡧⠤⠀⠄⣄⡰⢞⠀⢀⢉⡁⠉⠀⠈⠀⣀⣀⠐⠂⠑⢁⢀⠀⠦⡤⡤⠤⣤⠐⠁⠀⠛⠀⠊⠀⢀⡉⣄⠀⠀⣠⠴⠋⡀⠀⢳⠒⠻⠀⠀⠀⠀⠀⠷⣤⡤⣼
⠀⠀⠀⠀⢠⣀⠙⠓⠀⢰⡓⣭⠘⣦⠀⠀⠎⠀⠂⡀⠀⠤⡀⡀⠀⠀⠉⠙⠛⠉⠀⠤⡀⠀⠀⠐⠂⡀⠀⠀⠅⠀⡱⠚⠋⠀⠀⣈⠀⣰⣄⠀⣆⡄⠀⠀⠐⠂
⠀⠀⠰⢷⡐⠻⠂⠀⠀⠀⠛⠰⠼⠋⡀⣀⠐⠒⢎⢀⠀⠀⠠⣤⣠⣖⣆⠀⣠⠀⣈⡀⠅⠠⠈⠁⠀⡐⢀⠐⠀⠂⠈⠄⠀⣀⠀⢤⣀⣈⣹⠆⠀⡀⠀⠐⠦⠴⠃
⠀⠀⠀⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⠀⢠⣟⠂⠀⣾⡟⠁⠀⠀⠟⣿⠾⢿⡇⠀⠀⠠⡄⠀⠃⠀⢠⠐⠠⠞⠀⢢⠂⠀⠂⠀⠨⠁⠀⠀⡀⠀⠀⣿⣿⡆⠀⣠⡀⡀
⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠉⡀⠀⠈⠷⠆⢀⢀⢀⣷⠀⢰⠃⠀⠀⢠⡄⠀⢡⣤⡈⠀⠀⠀⠀⠈⠀⠀⣈⠀⠀⢠⣀⣁⣀⢠⣀⠈⠉⠀⠘⡳⢮⠜
⠀⠀⠀⠈⠀⠁⠸⠀⠲⠁⠇⠐⠈⠄⠀⠁⠀⠀⠀⠀⡀⢳⣟⠛⠁⡀⢸⡄⠀⠀⠘⠃⠀⡟⢀⣷⡃⠀⠄⠀⠀⠀⢉⠁⠀⠄⠈⠀⠁⠈⠀⠁⠀⠀⢔⣠⢉⡠
⠀⠚⠀⠒⠀⠐⠂⠙⠒⠀⡐⠊⠰⠚⠀⠀⠁⣘⢳⡖⠱⣀⠈⠉⠉⠉⢋⢅⠀⠀⣀⠀⠀⣓⢬⡧⠀⠀⠀⠀⣀⡀⢀⣥⡄⠈⠃⠤⢄⣠⠤⠘⠁⠀⠀⠁⢠⡛
⠀⠀⢠⡆⢰⡄⠀⠸⠀⢠⢇⠀⠫⣞⠃⠘⠁⠼⢏⠀⣀⠣⠄⠀⠀⢮⠸⠐⠀⠀⢧⡠⠀⡅⠘⠟⢂⡄⠐⠀⠋⠇⡀⠀⠀⠀⠀⠀⣠⠔⢂⠲⠄⠀⣄⠀⡀
⠀⢴⣆⡬⢤⠆⠀⠘⠀⠈⠀⠀⠀⠓⠀⠀⠀⠛⠈⣶⠚⢀⠃⠄⠀⠀⠈⢲⠐⡳⠀⠙⢀⠀⠈⠈⠀⢠⣶⠀⠀⠸⠀⣈⡄⠀⠀⡰⡁⠁⠈⠂⠀⠄⠀⠀⠀⠈⠙
⠀⣀⣈⢡⡄⠀⠀⢸⠀⠀⠀⠀⠁⡁⡀⠀⠀⠘⠦⣀⠀⠂⠀⠀⡖⠀⠂⠈⠀⠀⠃⢀⡏⡁⠀⠀⠀⣾⡇⠀⢀⡀⠛⢾⡇⠀⢰⢀⠀⠀⠀⠇⠘⠀⠀⣀⣱⠂
⠀⢈⢡⡤⠇⠀⠇⡸⠀⠀⣄⣀⡀⠈⠀⢸⠂⠘⢦⣡⠄⠂⠀⠈⠁⠀⣰⠀⠀⠀⠾⣘⡗⠁⠀⠀⠀⢱⠂⠀⠣⣠⢺⣦⢵⠀⠸⡆⠀⠀⢀⢇⠸⠀⠀⡠⡝⡀
⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠈⠉⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠈⢳⣤⣤⣤⣤⣤⣄⣴⠾⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠙⢿⣄⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠖⠂⠀⣠⣴⠶⠿⠿⠿⠿⠿⠿⢿⣷⣦⣄⣀⣼⠟⠋⠉⠀⠀⠈⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣧⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠉⢻⡟⢁⣤⠶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⣾⣡⣦⠘⡧⠀⠀⢀⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⡼⠿⣿⣇⠀⠀⣸⣵⣶⡀⣸⠃⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⡶⠶⠶⠶⠶⣤⣿⠿⠛⠳⠶⠤⠀⠉⠙⠻⢏⣰⠿⠋⠉⠉⠀⠈⠉⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⡽⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡶⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠶⠦⠤⠤⠤⣤⣤⡿⢱⣶⣶⡄⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⣆⠀⠀⠀⢀⣀⡤⠶⣻⡽⣿⣿⡇⣴⡶⠶⠿⠛⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡬⠿⣷⣖⣋⣩⡴⠖⣋⣡⢟⡶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠒⠒⠒⠚⠛⠋⠉⠉⢿⣆⠀⣀⡬⠟⠛⠛⠟⣻⠟⢁⣞⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠿⣧⣤⣀⣀⢀⣼⣁⣀⡼⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠙⡻⠋⠛⠉⢹⠇⠀⠀⠘⢿⢿⣷⣶⣤⣤⣤⣶⠟⠋⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠈⢣⡈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠐⠫⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⣀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠆⠀⠀⠀⠀⣰⣿⣿⡶⠛⠀⠀⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠼⢿⣿⡿⠋⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⢈⡽⠟⠁⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢀⠐⡁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠁⠌⠁⠀⠀⠀⠀⠀⠐⠁⢠⢌⠁⠀⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⢃⠔⠃
⠀⠀⠀⠀⢀⢀⡘⠂⡰⠀⠀⠀⠀⠈⠀⡪⠐⠁⠀⠀⠀⠀⠈⣱⠃⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢴⡩⠇⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠠⣸⣁⣀⡀⠀⠀⠀⡠⠋⢠⣎⠁⠀⠀⠀⣠⠊⢘⡂⠖⠀⠀⠀⠀⢠⠎⣠⠄
⠀⠀⠀⠀⠰⢓⢖⡴⠋⠀⠀⠀⠈⣩⡿⠒⠁⠀⠀⠀⠀⣩⠏⠋⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠰⠎
⠀⠀⠀⠈

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⠒⠁⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠈⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⠀⢈⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡠⠰⢀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⣠⠂⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢌⠀⢠⠃⢜⡂⠀⠀⠠⠀⣠⠖⠊⠀⠂⠀⡀⠈⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐
⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⠷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠿⠿⢷⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠛⠻⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣄⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠉⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠛⠿⣶⡄⢀⣤⣶⣶⠶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⠀⠙⠻⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⠶⠶⣶⣤⣄⠸⣿
⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠻⣿⣿
⠀⣠⣾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷
⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠙⠻⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⢀⣠⣾⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟
⠀⢠⣾⡿⠋⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁
⠀⣿⡏⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷
⠀⢿⣧⣀⣀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡀⠀⠀⠀⠀⣀⣴⣾⠟⠉⣿⡆
⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⣿⡇⠀⠀⢀⣾⡿⠋⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡆⠀⠈⢿⣦⣄⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢹⣿⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆⠀⢸⣿⡁⠀⠀⠀⣸⣿⠁⠀⠀⠈⠻⣷⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⣿⡇⠀⠀⠻⣷⣦⣤⣾⠿⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⡾⠟⠁

⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠃⠀⢀⡤⠴⣶⡶⢾⣿⣿⣿⣿⣿⠿⠛⠛⠛⢻⣟⣻⡿⢶⣶⣦⣄⠀⠀⠼⠃⣀⣀⣀
⠀⠀⠀⢀⣀⡀⠆⠀⢸⣷⣶⣤⣤⣭⣭⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣶⡿⠿⢿⣛⣫⡆⠀⠀⠀⠤⣤⡉⠛⡆
⠀⠀⠀⡏⢡⣾⡛⠀⠈⢻⣿⠿⠿⠿⠿⠭⠭⢭⣭⣭⣭⣭⣠⣤⣿⣋⣉⣉⣩⣭⣵⡶⢀⣀⣤⠤⠾⡛⠀⢤⡀
⠀⠒⡀⠀⠀⢈⡉⠛⠛⠛⢶⣶⣶⣒⣒⡒⠀⠀⠀⠀⠠⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⡷⠀⠀⠀⠀⢀⣹⡇⠐⠟
⠀⠀⠀⠀⣀⠻⢤⣤⣤⣀⣘⣿⣿⣿⠿⠍⠉⠩⠤⠤⠤⠤⠬⠿⢿⣿⣿⣿⣿⣿⡿⠤⠤⠶⠖⠚⠋⠉
⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠻⣿⣯⣭⣍⣉⣑⠶⠶⠤⠤⠤⠤⣬⣭⣵⣶⣾⡟⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠒⣤⡀⠛⣻⣿⣷⠒⢒⣲⣶⣤⣤⣤⣤⣶⣶⣿⣿⣻⣯⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣁⡉⠀⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣭⣉⣛⣿⣿⣿⣷⣖⠛⠛⠋⠀⠈⠡⠶⠿⠞⣛⡋⠀⢀⣉⣷⠘⡦⠀⠀⠀⠀⠰⡿⠉⠞⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠿⣿⣿⣯⢭⠝⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣏⡉⣐⣶⠀⠀⠀⠀⠀⠀⠙⠂⣴⠏⣠⠄⢈⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡗⠂⠈⠿⠿⢿⣿⣒⣶⣶⣖⠛⠛⡻⠿⠿⣟⣛⣿⣿⡁⠀⢀⣀⡀⠀⠀⠀⠘⠁⡴⠋⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠂⠀⠀⠬⠿⠿⠿⠷⣶⣶⣶⣾⣋⣙⡛⠟⠯⡿⣀⠒⠚⠋⢠⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⣤⣀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠉⢙⠛⠻⠿⢷⣶⣾⣿⣭⠿⣦⣤⠶⠟⠀⠀⠀⠉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠘⠉⠀⣹⣆⣤⠤⠴⠖⠀⠀⠀⠀⢤⡘⠦⠤⢤⠀⠰⠾⠿⢿⡿⢿⡧⠀⠀⠪⠅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⣤⠀⠀⠉⣥⠶⠖⠛⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣷⣿⡏⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⡀⠀⢀⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⠀⠰⠰⣇⣼⣿⣋⠀⠸⠀⠀⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⡀⢀⣴⠂⠀⠀⡀⠒⠀⠀⠀⠠⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⠟⠄⣹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠰⠆⠸⡍⠀⠀⠀⠀⠠⠀⠀⠀⠉⠉⠛⠳⢜⣀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠠⡔⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠈⠓⠀⢰⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⠶⣶⣿⡆⠀⠀⠐⠂⠀⠀⠀⠀⢀⠀⠀⠠⡀⠂⠀⠀⣨⠀⢀⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⣿⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⡧⠀⠠⡄⣿⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⣏⣽⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣩⣱⡿⠳⠀⠀⠘⣿⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡻⡿⢀⠀⠀⠀⠀⠀⠀⠁⢾⡣⠄⠘⠀⠸⡿⡇⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠤⠀⠻⠿⠁⠙⠿⡇⠐⠃⠀⠀⠀⠀⢁⡀⢀⡺⡇⠀⠀⠀⠀⢱⢻⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣶⠀⢰⡆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⡇⠀⠀⠀⠀⠈⢿⠱⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠋⠀⠟⠁⠘⡆⠀⠀⠀⠈⠂⠀⠀⠀⢈⠄⠀⢀⣂⣀⣂⣀⣲⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⣶⡆⠀⠀⠀⠀⠀⠀⠀⢷⡀⡀⠛⠀⠈⠈⠂⠠⠜⠞⠰⣿⢿⡏⠉⠛⢯⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⡶⡇⠀⠀⠀⢀⠀⠀⠀⡸⡴⠰⢦⡤⠶⠆⠄⡤⠀⠀⠸⠘⢸⡇⠀⡆⡸⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠆⠃⠀⠀⠀⡐⠀⠑⢲⠀⢿⡆⢸⡃⠀⠀⠀⠀⠀⣀⡰⠀⠸⡧⡄⡇⠐⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠤⠀⠀⠈⢇⣀⣀⡜⢀⣸⠀⠼⠧⠐⠒⠒⠈⠉⠉⠁⠀⢠⠄⠀⠤⣶⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡶⣶⠦⠀⠾⣷⡾⣿⣿⣯⠁⠀⠀⠀⠀⠀⠐⠛⠃⡀⠀⡄⠠⠶⡆⡀⢀⡯⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⡧⠀⠄⠀⠻⣿⣿⣾⣿⠀⢀⠀⢴⡀⠉⣀⠀⡀⠁⠐⣇⠈⠐⠻⡇⠉⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣳⡤⠀⠀⠊⠻⣿⡟⢛⣡⡎⠉⠉⠁⠰⣌⠠⠀⠀⠊⡀⠀⠃⠀⠀⡀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠆⠸⢹⣿⡀⠀⠀⠸⣿⣶⠿⣿⠀⠀⠀⢤⡄⣈⠓⠢⣤⣄⣁⣀⠀⠀⠀⢀⣧⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣁⠀⠈⠉⠀⠀⠀⠼⠟⠀⠛⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⡀⣴⣤⣾⣿⡿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣎⠁
⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀
⠀⠀⠀⠀⠐⠛⠛⠋⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣶⡶⠾⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣀⣀⣀⡄⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠋⠉⠉⢀⡇⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿
⠀⡀⠀⣠⡶⢻⡄⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠙⣦⠄⠀⠀⠀⠀⠀⠘⣿⠙⠛⠳⠿
⠀⠱⠞⠁⠀⠈⢷⣀⣴⠟⠉⣷⠀⠀⣠⠔⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⢠⣤⣄⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠘⣧⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡇
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡀⠀⠀⣼⠀⠀⠀⣀⡠⠒⠉⠀⠉⢳⡀⡀⢀⡀⠤⢄
⠀⠀⣀⠠⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠⠠⠄⠀⠀⠀⠉⠀⠀⢀⡨⠀⢠⠊⠀⡡⠤⠤⡀⠀⠈⠁⠈⡇⠀⠀⠀⠗⠢⡀
⠀⠉⠉⠁⠀⠀⠂⠀⠌⡑⠂⠤⢀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠪⣀⠠⠔⠀⠀⢸⠀⠀⠓⠀⠀⠹⠀⠀⠀⠴⠃⢀⣀⠀⠀⠀⠚⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢀⠈⠒⠠⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣣⣀⠠⠄⠐⠁⠀⠀⠀⢀⠀⠈⡸⠀⠀⠐⠀⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣐⠀⠀⠄⢀⡀⠉⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠉⠉⠀⣠⣤⡤⠶⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠒⠤⠤⢌⣁⡀⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣒⡒⠒⠬⣔⠄⡀⠀⠀⢠⠀⠀⠀⠈⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠢⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⡶⠶⠦⠤⣄⡀⢀⣸⡄⠀⠈⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢤⡀⢀⠤⠶⠦⣄⡀⠀⠀⢱⠄⠀⠖⠀⡹⠁⠈⠳⠤⠖⠁⠠⣄⠀⠀⠈⡧⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠤⠉⠒⠀⠀⠒⢿⡀⠀⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⢀⡴⠄⠊⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠦⠀⠤⠞⠉⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣤⣴⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣡⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣲⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣯⢛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣌⡙⠒⠤⣙⠛⠿⠯⣭⣟⣛⣛⠿⠿⢯⣭⣭⣭⠭⠉⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠠⣉⡓⠶⣤⣀⠀⢤⣔⣂⡀⠌⠉⠉⣓⣒⣒⣒⡀⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠝⡻⠓⠶⠤⢭⣭⣅⣒⣂⠠⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣌⣉⣙⡒⠤⠄⢀⣀⣈⣉⣙⣛⣒⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠭⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡽⠋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⣭⣿⣶⣭⣍⣙⣛⣛⣯⣭⣔⠒⢚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠽⠟⣉⣽⡿⢿⣿⠿⣟⣻⣯⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣤⢉⣁⣀⣉⣙⣿⣺⣟⠻⡯⢼⣒⣩⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢿⡄⠷⣜⠿⣷⣶⣶⣶⣶⣿⣯⣥⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢉⡓⢌⡒⠤⢍⠂⠉⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⠈⠁⠙⠩⠒⠒⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⢂⣀⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣦⣌⣀⣈⡍⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣾⢷⣶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠹⡛⠿⢛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣜⠟⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⡠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢋⠀⠀⠀⠀⠀⠠⠤⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⣽⠃⢴⠛⣃⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠖⢉⡀
⠀⠀⠀⠌⠉⠄⠠⠄⠄⠀⢲⣇⠀⠀⠠⢀⠀⠐⠉⠡⠘⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠄⡍
⠀⠐⡋⠀⠀⠠⢀⡀⡄⠀⠘⡉⢀⠀⠀⠈⠃⠀⡀⠀⠀⠀⠉⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⢠⡈⠂⠀⠀⠀⣠⠚
⠀⠀⠁⠀⠀⠁⠤⠀⢀⡀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⡩⠀⠀⡄⠀⠀⠀⣐⡖⠡⡄
⠀⠀⣿⡀⠘⠂⠁⠀⣼⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⡇⠁⠀⢀⠃⠀⠀⢀⠞⢉⠀⠀⠁
⠀⠀⢸⠀⠀⠀⠂⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⠀⠀⠀⡞
⠀⠀⢠⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⡅⠀⣸⠀⠀⠀⣀⡀
⠀⠀⠀⣧⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠧⠶⠛⠛⠛⠛⠛⠋
⠉⣩⡍⣹⣾⠻⠓⣾⡗⠙⠒⠉⠙⠛⠛⠿⠿⠻⠋⠉⠉⠛⠛⡉⣛⠛⠉⠉⠛⠛⠋⠙⢋⡁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂
⣴⢱⣿⣿⣷⣇⣶⣾⡗⢠⡀⡀⠐⠒⠀⡀⢀⠀⢀⣀⣀⡀⠈⠉⣤⣾⣯⣠⡀⢀⣠⣄⣠⣴⡤⠤⠐
⣷⣿⣿⣿⣿⣿⣿⣿⡇⣸⣟⣴⣦⡀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠺⠟⠻⠻⠿⠿⢦⠝⠋⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆
⣿⣿⣿⣿⡿⠿⢿⣿⠂⠿⠿⠿⠋
⠉⠀⠉⠉⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⢅⣀
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠊
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣦⣶⣴⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠋⠀⢀⣤⡴⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠈⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⣼⣿⠁⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠉⢻⣷⡄⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣌⡙⠻⠿⣿⣷⣶⣿⣦⣀⣠⣶⣶⣶⣶⣶⡦⢀⣀⣾⣧⣴⣾⠿⠿⢛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣦⣤⣬⣉⣉⣉⣙⣛⣛⣛⣛⣛⡋⠉⠉⠉⠉⣀⣠⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⣤⣤⣭⣉⣉⣉⣀⣀⣤⣤⣤⣴⠶⠾⢛⣋⣭⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⣭⣍⣉⣉⣉⣉⣭⣤⣤⣶⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⣭⣭⣭⣭⣥⣤⣶⠶⣟⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⢟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠾⠿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠋
⠀⠉⠉⠈⠈⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⢀⣾⣿⣿⠟⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢿⣏⣠⡤⠶⢿⣿⣟⣒⠒⠶⣶⣿⣿⣿⣿⡿⠷⠂⠀⠀⠀⠀⠀⠀⠄⠈⠙⠺⠧⠀⠀⠈
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡼⣿⣥⣤⡀⠈⠻⣿⣿⣷⡀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⢐⠀⠀⠀⢹⡀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠘⢿⣆⢀⡎⠀⠈⠻⣿⣿⡄⠀⠀⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠵⢀⠀⠀⠀⠀⠂
⠀⠀⠀⢶⣤⣄⠀⠙⡿⠀⠀⣀⣤⣤⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠬⠥⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢻⣿⣇⠀⢡⢠⡼⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠐⢛⡁⠀⠀⠀⠀⢁⠄
⠀⠀⠀⠀⠀⠉⠉⠙⠻⡏⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠥⠀⠀⠀⠘⠡⠂
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡘⢄⠀⢀⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣾⣷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⡀⢢⠸⡅⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⣻⠀⠀⠀⠀⠀⠀⠐⠪⠅⠳⡗⡨⠊⡁⠀⠀⠀⠀⠀⠹⣦⡄⠸⡇⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣦⠀⠀⡼⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢅⠀⠶⡗⢐⠄⠄⠀⠀⠀⠀⠀⠀⠘⢿⡄⣇⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⡆⢀⣇⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠑⢤⡷⠥⡐⢥⠀⠀⠀⠀⠀⢀⠀⠀⠻⣿⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡇⡼⠋⢀⣀⣀⣀⡀⠀⠀⠀⠀⢈⢢⡀⢐⣠⡧⠊⠀⢐⡀⠀⠀⠀⠀⠘⣿⡆⠀⢹⡇⠀⣠⣴
⠀⠀⠀⠀⠀⠀⡾⢆⠀⠀⠹⣾⣞⡥⠚⠉⠀⣠⠜⠁⠀⠀⠀⠀⠈⠂⠀⠠⣄⣧⣐⠈⠄⠊⠀⠀⠀⠀⠀⠈⠻⣄⢸⡇⢠⡿⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠳⡄⢠⠟⠛⠓⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⣬⣧⠐⠀⠈⠩⠀⠀⠀⠀⠀⡀⠀⠘⢾⣇⡞⠁
⠀⠀⠀⠀⠀⠀⢧⡀⠀⣷⠏⣀⡠⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠉⠐⠠⣀⡇⠄⠀⠀⠰⠀⠀⠀⠀⠀⣿⣦⠀⢸⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣤⣿⣚⣁⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠊⠉⠠⣁⠈⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⣦⢸⠃⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡐⠙⡟⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣾⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⡟⠃⠠⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⡠⠔⠒⠒⠒⠒⠊⠉⠉⠑⠒⠤⣄⡀⠀⠀⠀⠀⠀⠘⡄
⢠⠿⠗⠈⠭⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⣀⣁⠰⡀⠀⠀⢀⠤⠷⢄
⠈⢻⠀⠀⡄⠀⡀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⠁⠀⠀⠑⠷⠀⠀⠁⠀⡀⠈⢳⠀⢀⠄⠀⠰
⠀⢸⠀⠀⠀⠂⡀⠀⢀⠼⠄⣲⢾⢗⠒⠒⠒⠒⠒⠚⠌⠋⡢⣀⢈⣿⡄⢡⠀⣀⠐⠀⠀⠀⠳⣆⡀⠐⢀
⠀⠀⢣⠀⠀⠀⢠⢞⣧⢖⠏⣯⢃⠾⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠄⡀⠂⠀⢢⠈⠀⠀⠀⠀⠐⢏⠁⠀⠁
⠀⠀⠀⠑⠤⠔⡫⠎⡇⢸⢠⣇⠎⣸⠂⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠉⠂⠄⠱⣄⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢠⡇⠇⡇⢸⡜⣡⢞⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⡠⠔⠀⠁⢄⠀⠀⡈⣆
⠀⠀⠀⠀⠀⢸⣇⠸⡰⡼⢋⢔⠥⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡠⣴⢄⠀⠀⠀⠑⣠⠋⠈⢱⣀⣀
⠀⠀⢀⡠⢄⡀⢾⣷⣳⡯⢓⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣩⡤⡴⢸⠗⣟⢋⠀⠀⢫⠀⠀⠀⠀⠀⢸⡀
⠀⠠⠋⠀⢀⡨⠛⠳⡷⡬⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⠤⣔⢫⠔⠁⠀⠑⠄⠫⠬⡤⠚⠠⠄⠠⢍
⢀⠃⠀⠆⠀⠀⡇⢀⠇⠀⠀⢠⠤⠤⠔⢢⡀⢀⡤⠳⢄
⠀⡆⠀⠀⠀⠀⣠⠎⠀⠀⠀⢸⠀⠠⡀⠀⠑⠋⠀⠂⠘⢄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀
⠀⠸⠤⠒⠂⠈⠁⠀⠀⠀⠀⠰⡄⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠀⡀⢐⠁⠀⣠⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⡼⢉⠉⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⢀⠄⠀⡼⠯⢭⠤⠤⠤⠤⠤⢀⣀⡀⠀⠀⢀⠇⢄⠡⠢⡆⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠊⢏⣀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⡄⠄⣄⠡⣠⠈⢸
⠀⢀⣀⠀⢀⡤⠊⢉⡀⠀⠀⠀⠀⠈⠁⠀⡼⠹⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢄⡈⠉⣽⡖⠁
⠀⢻⠸⡶⠷⠒⣋⣉⡽⠀⠀⠀⠀⠀⠀⡰⢃⡄⢧⠀⠀⠀⠀⠀⢸⢹⠆⠀⠀⠀⠀⢀⣀⠈⠉⠈⠿⢦⡀
⢯⣉⡀⡶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢰⠃⡞⢧⢸⡀⠀⠀⠀⠀⡞⢸⠀⠀⠀⠀⠀⢸⠘⡇⠀⠀⠀⠀⠙⣄
⠞⠀⡇⢧⠀⠀⠀⣀⡤⠴⣄⠀⠀⢀⠟⡸⠁⢸⡀⡇⠀⠀⠀⢀⡇⣿⠀⠀⠀⠀⠀⠘⡇⣇⠀⠀⠀⠀⠀⠈⢆
⠀⠀⣻⠸⠶⢚⣉⣥⠴⠒⠃⢴⠒⠚⠐⠧⠤⣬⡇⢿⠀⠀⠀⢸⢡⡇⠀⠀⠀⠀⠀⠀⣇⢹⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢳⡀⡞⠉⠀⠀⠀⠀⠀⠀⡹⢁⠟⠓⠒⠒⢦⠀⣽⠆⠀⡞⢸⠁⠀⠀⠀⠀⠀⠀⢾⢸
⠀⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⢰⢃⡞⠀⠀⠀⠀⢸⡆⡇⠀⠰⡇⡏⠀⠀⠀⠀⠀⠀⠀⢸⢸⠆
⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⡟⢸⠁⠀⠀⠀⠀⠈⡇⢻⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇
⠀⠀⠀⢸⢸⡄⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⣤⣌⣉⣉⣉⣛⣛⡆⠀⠀⣧⠹⠤⡤⠴⠖⠚⣙
⠀⠀⠀⠘⢦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠻⠴⠒⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢉⣉⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠁⠐⢊⡄⠊⡁⠐⣤⠘⣀⠀⣞⣢⣭⡲⠄⠀⠀⠀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢨⣴⣦⠀⠀⣀⡢⢜⡣⣄⣬⡶⠶⠶⠶⣮⣬⡢⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣠⢿⣆⠉⣶⣷⣶⣸⣿⢊⣴⠟⠁⠀⡔⠀⠐⠀⠈⠻⣮⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣦⢘⠡⣿⣿⣰⣿⣿⣿⣿⣇⣾⡗⠠⠂⠘⡀⠋⠠⡁⠀⠐⣿⣷⢡⢠⣧⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠒⠂⢀
⠀⠀⠀⠀⠀⠀⡐⢢⢼⣤⣟⡫⢿⣿⣿⣿⣿⠿⣏⡿⢸⡟⢠⡁⢀⣔⡅⠀⡤⣸⣤⡀⣾⣿⣾⠏⣹⣶⡿⠶⠶⠔⠢⠀⣀⣠⣤⡤⠲⣄⠸⠄
⠀⠀⠀⠀⠀⠐⠶⢾⣎⠙⢦⣙⣹⣿⣿⣿⣿⢣⣌⣿⢻⣿⡾⣴⠁⠈⠀⠀⠀⠁⢈⣟⣿⣿⣇⢠⣿⣶⣧⣤⣤⠀⠆⣾⠛⠀⠀⢠⣆⣀⣀⡁⣀⣀⡀
⠀⠀⠀⠀⠔⠲⠶⠟⠃⠂⠀⠉⣿⠘⢻⡟⡿⢦⣿⣉⣿⢻⡾⣹⠀⠀⢀⣈⡀⠀⢸⢩⣿⣿⡘⠟⣻⠿⡍⠘⠛⡈⠷⣹⣦⠀⢀⠀⣀⠉⡍⢀⣩⠍⠁
⠀⠀⠂⠤⡀⢠⡤⠒⠒⢆⣀⡠⢻⡼⣯⡿⢢⠘⣷⡟⣿⣼⣇⢼⠣⢀⡀⠛⢀⡠⠾⣾⣟⣿⡗⠾⣿⣿⠟⣛⠀⠴⣃⣾⠉⠛⠋⠀⠈⣠⠡⠋
⠀⠀⠀⠀⠈⠑⣦⣤⣁⡀⠉⠁⢸⡟⢷⣐⡽⣻⣿⢰⣿⢛⣷⣷⠢⢀⣀⣍⣀⣀⡶⣹⣽⣿⣧⣤⡿⣱⠾⣯⣰⠟⢋⡏⠀⢠⠲⣞⠋⠁
⠀⠀⠀⠀⠀⠀⡓⡬⢭⡿⣣⠄⢀⡁⣼⠯⢳⡏⠉⠋⣹⣿⡿⠋⡇⠀⠀⠀⠀⠀⡇⢉⠿⡀⠃⠀⠙⢦⢤⡝⢛⣷⡀⠉⣁⠈⠀⣿⢤⠁
⠀⠀⠀⠀⠀⠀⡿⡶⡸⣧⣠⡔⠀⠀⠈⠳⣼⣇⠘⠐⢃⡝⡇⢸⠃⠐⠀⠐⠈⠐⢻⣤⢡⡝⢢⡀⢠⠔⠂⣿⡾⠁⠈⠉⠁⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢸⠁⡶⣤⣸⣧⠀⠀⠀⠀⢰⡆⠀⠁⣦⡿⢷⡌⠀⠲⠀⠔⠃⠀⠈⢳⡇⢹⠀⢳⡄⠘⠁⠙⠀⠀⠀⠀⣰⡟⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣄⢷⣏⣿⣹⣧⠀⠀⠀⠘⣧⠀⠀⡹⣧⠀⠈⢆⡀⠀⠀⠀⠀⣀⠎⠁⡘⠀⠸⣄⠀⠀⠆⠀⠀⠀⣴⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠈⣻⣿⣥⡘⢳⣄⣀⣐⣥⢴⣾⢃⠿⠀⠄⠘⡇⠀⠀⠀⡰⠁⢀⠎⠀⠀⣲⡏⡿⣶⣤⣀⣠⡾⡻⣠⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠿⣿⠿⣧⠾⣻⡝⣠⡿⠁⠁⠀⠀⠀⢀⠇⠀⠀⢰⠇⠀⡌⠠⣁⣾⠋⢿⡉⢺⡬⣭⡑⠂⡙⣌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣉⣦⢦⣝⣩⠞⢩⠀⡠⡆⠀⠀⠀⠉⠀⠀⠀⢿⠀⠀⢀⡀⠛⡇⠀⠸⣇⠀⠀⠙⠃⠄⠀⣀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣋⡵⠛⠁⠀⠘⠄⠈⠃⠠⡈⢦⠀⠀⠀⠀⠈⠳⣅⡀⠉⠁⢤⠠⡀⢻⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠸⠠⠆⣹⠘⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠷⠈⢧⠀⢑⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⢠⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢳⡼⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⠠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠈⣳⣄⣀⠀⠀⠀⠀⢀⣀⣀⣴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣝⡛⣿⠋⣻⡏⣉⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢿⡁⣿⠀⣿⠐⠉⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣱⡇⠀⣿⢸⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣸⡇⠀⣿⠸⢸⡏⠀⠀⠀⠀⠀⠀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⣿⣿⣿⠀⠀⠀⢀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⣿⣿⣿⡀⠀⠀⠘⠑⠈⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠋⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣿⣿⣿⡿

⢀⣀⡠⠀⠠⠢⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠤⡀⠀⠀⠀⠀⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠈⣢⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢁⠔⠂⠐⠖⠉⠢⠤⢄⣀⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠠⢀⡀⠄⡀⢀⣀⡀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠄⠀⠀⠀⠈⠀⠀⠈⠁⠀⠈⢦⠀⠉⠉⠉⠢⠄⠤⠒⠉⠁⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠉⠀⠐⠢⣄⠀⢀⡀⠀⠀⢀⣀⠤⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠪⡀⠀⢀⡀⠀⠀⠀⢹⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⡀⠀⠀⡀⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣨⣴⡀
⢀⣀⠤⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠁⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⣀⣀⣀
⠤⡀⠀⠀⢀⠄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠤⠂⠉⢉⡋⠀⠀⠀⠁⠀⢹⡀⠀⠘⠷⠦⠀⢀
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠻⡀⠀⡀⠉⠳⣀⡴⠉⠑⢤⡠⠃⠀⠀⠀⠀⠰⡉
⠀⠀⠀⠀⠀⠀⡔⠉⠙⠁⠀⠀⣑⡢⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠃⠀⠙⠢⠄⠤⠚⠀⠛⠋
⠀⠀⠀⠀⠀⠰⢅⠀⠀⡀⠀⠙⠹⠗⡅⠀⠀⡔⠲⣥⣄⠉⠉⢢⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠁⠈⠁⠉⠀⠀⠀⢎⠈⠉⠁⠀⠀⢈⠆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠛⠻⡄⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⠀⠀⠀⠁⠒⠈⠀⠀⠀⠀⠀⠀⢀⡠⠔⠢⡀⠀⠀⠀⠀⠀⠀⢘⠷⠂
⠀⠀⠀⠀⠀⠸⠛⠷⠀⠀⠀⠀⠀⠀⠋⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⣀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠶
⠀⠀⠀⠀⢠⣤⠄⠀⠀⠋⠙⠀⠀⠀⠀⠠⣤⡀
⠀⣄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⢰⣿⣶⡆
⠀⠀⠀⠀⠀⠀⣼⣿⡟⢁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⣼⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⣼⣏⣀⡀⠀⠀⠀⢰⣿⣯⣤⡄
⠀⠀⠀⠀⠰⢫⣿⣿⢏⣠⠄⠀⠀⠀⠀⢰⡿⢀⣠⡄⠀⠀⠀⢰⣿⣿⣶⠀⠀⠀⠐⠛⣻⡟⠀⠀⠀⠀⠞⢻⣿⡿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⢛⣿⣿⠃⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠚⢁⣿⣿⠃⠀⠀⠀⠀⠀⠟⠋⣽⡟⠀⠀⠀⠀⠀⣸⣿⣿⡶⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣸⣿⣃⡀
⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠟⣿⡿⠁⠀⠀⠀⠼⠿⣿⠁⠀⠀⠀⢀⣿⣿⡟⠁
⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣹⡟
⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣶⣶⣶⡶
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⣼⣿⣿⣿⣶⡦
⠀⠀⠀⠀⣰⣿⡟⠀⠀⣀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⣀⠀⠀⠘⠛⢻⣿⣿⣿⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣤⡄⠀⠀⢀⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠛⣿⣿⡟⠀⠀⠀⠀⣾⣿⣿⣷⣶⠂
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠾⢿⣿⡿⠁⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⣼⣿⣿⡶⠀⠀⠀⠀⠉⢹⣿⣿⠃
⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡩⠥⠬⠛⠋⠉⠉⠉⠑⠒⠤⠤⢤⣄⣀⣀⣒⡒⠀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣊⡭⠛⠂⢠⣤⣤⣤⣶⣦⣤⣶⣶⣶⣤⣤⣤⡽⠟⠧⢌⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠯⣤⣤⣶⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣊⣭⡥⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢫⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣔⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⡟⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣯⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤



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

⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣀⡀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⢀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⢠⣦⣅⠀⠀⠄⠀⡩⠒⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠎⠠⠄⠀⠦
⠀⠀⠀⠐⠮⠂⠜⠷⠀⡀⠒⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠂⡄⠀⡁⢅⠀⠂⠄⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⡠⠄⠢⠀⠀⡂⡸
⢤⡾⡤⡄⠀⢀⡀⠰⠀⠐⠀⠔⠼⠀⠀⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠀⠀⠴⠂⠀⠀⠂⠁⠀⠀⠀⠀⠀⠟
⠃⠂⠱⣭⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⣠⠄⡄⠀⠣⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠄⠀⠀⠀⡄⠀⠤⠀⠀⠀⠃⠀⠀⠘⠃⠀⠈
⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠈⠠⠀⠸⠀⡀⠀⠄
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡠⠀⠰⣄⡀⠠⠀⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⠈⠁⣀⠀⠐⠀⠀⣁
⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠂⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⢈
⠠⠠⠀⠘⠋⠀⠀⠀⠀⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠐⠀⠦⠠⡄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⡅⠀⠀⣀⢀⠀⠆
⠀⠦⡠⢀⢄⠂⠀⢀⠀⠑⠀⠀⠘⠁⠀⠀⠤⠀⠀⠰⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠚⠁⠀⠀⡀
⠀⠀⠹⠀⠀⠑⠄⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⠀⠀⠀⠈⠈⢣⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⣄⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⢡⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⡲⢿⠒⠐⠉⠠⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠪
⠀⠀⠀⠀⠀⢦⣬⡻⠂⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⢐⠁⠈⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠓⠐⠀⠀⠀⠀⠲
⠀⠀⠺⢭⢻⡄⣗⡽⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣈⣁⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠉⠁
⡀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⠛⠋
⣇⠀⠀⠀⠠⠾⢿⢛⣩⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠋
⣿⣿⠖⠀⠀⠀⠸⡿⢻⡟⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⣿⣿⠟⠁
⣷⣦⣤⣀⡀⠀⠘⠒⢻⣧⣿⣤⡞⠋⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⡿⠋
⣿⣿⣿⣿⣿⣏⣀⣀⡙⣿⣿⣿⣄⣤⣿⣥⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡟⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣼⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣴⣤⣤⣤⣤⣤⣤⣄⣤⣾⣦⣤⣄⣀⣴⣿⣿⣿⣤⣀⣿⣀⣀⣰⣾⣿⣂⣀⣀⣤⣶⣾⣦⣰⣾⣇⣸⡃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⠿⠋⠙⠿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠻⣿⣿⣿⣄
⠀⠀⠀⣼⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆
⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢸⣿⣿⣿⡀
⠀⠀⣿⣿⣿⣿⡟⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣬⣉⣹⣿⣿⣿⣿⣧
⠀⠀⣿⣿⣿⣿⡇⠀⠀⣡⣴⣶⣿⣿⣿⡦⠀⠀⠀⠀⠘⠛⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣆
⠀⢠⣿⣿⣿⣿⡇⠀⠈⠛⠛⠀⠈⠉⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟
⠀⢾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠈⢿⣿⡿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⣿⣿⣻⣿⡏
⠀⠀⠸⣿⣷⣹⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⠁⠀⠀⡀⢠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣴⡾⠟⠛⠛⠛⢻⣿⡶⠂⠀⠈⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠉⠹⣭⣤⣤⣤⣾⠟⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⣴⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⢻⣿⠏⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢻⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⠀⠀⢰⡶⣤⠀⣀⡀⠀⢀⣶⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠈⠙⣿⣿⣿⣿⣿⣿⡶⠀⠀⠶⣬⣧⠹⠸⣧⣀⣤⣤⣿⠳⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⠀⠀⢀⣤⣀⣿⣿⣿⣿⣿⣏⠉⠿⢶⣤⣾⣯⠀⠀⠙⡿⢟⣩⣏⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⢘⣿⣧⠀⠈⠉⣽⠟⠁⠀⣠⣴⣿⣿⣿⠙⠁⠀⠀⠀⠈⣿⣭⣟⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠁⠀⢀⣠⣶⣿⣁⠉⠋⠉⢀⣴⣶⣾⢷⣦⡀⠈⠉
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠠⣶⡿⣿⡿⡿⠛⠁⠀⠀⠚⢉⣿⣿⣿⣿⡇⠀⠀⠀⣶⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢋⣼⡿⣢⠄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠙⣧
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⡿⠛⠚⠁⠀⠀⠀⠀⣠⣼⣿⣿⢿⡿⣿⣿⣷⣄⠀⣆⢹⠁
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢨⡟⠀⣾⡟⠀⠀⢠⣾⣿⣿⡟⠀⠀⠀⠀⣸⣿⡇⠀⣯⣤
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣾⠟⠁⠀⠐⣮⣿⡇⠀⠙⠳⠶⠿⠟⠋⠙⠇⠀⠙⠏
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⣠⡾⠳⢀⡞⠋⠉⠀⠀⢀⣤⣤⣤⣄⡀⣤⠈⠁⠉
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠉⢀⠐⠋⠁⠀⠀⠴⣶⣿⣿⢿⣿⡿⣷⣿⣧⠉
⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠱⠟⠁⠰⠿⠋
⠀⠀⠀⢦⣈⣿⣿⡿⠁⠈⠛⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠀⠀⠀⠉⠳⣶⠟⠉⠛⢶⡶⠛⠉⠉⠉⠉⠙⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡶⠾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠛⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡟⠀⠀⠀⠀⢀⣱⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠠⣴⠿⠛⠀⠀⠀⠀⣀⢀⡝⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠶⢦⣄⡀
⠀⠀⠀⣀⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠬⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢦⢠⣀⣔⣬⡺⠛⠁⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠈⢻⡄
⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠢⡄⠮⠓⠋⠁⠀⠀⠀⠀⣠⡌⠁⠀⠀⠀⠀⠀⠀⢈⣀⢀⣠⡴⠎⠀⠀⠀⠀⣀⠠⣴⡍⠀⠀⠀⠀⠀⢀⣿⣀
⠀⢸⡇⠀⠀⠀⠠⡀⢀⡠⢀⠎⠀⠀⠀⠀⠀⠀⡠⠀⠈⠒⠙⠞⠓⢉⠀⠀⢠⣠⡀⣄⣦⣴⣿⣿⡿⠋⠀⠀⠀⠘⠎⠳⠿⠿⠊⠀⠀⠀⠀⠀⢸⠀⠉⠉⠛⢷⡄
⠀⠀⠻⢦⣤⣤⡄⠉⠀⠉⠀⢄⠀⠀⡀⢄⠠⡔⢊⡄⡀⢀⡦⣄⡴⠄⢠⡐⢤⣙⢻⣿⠿⣫⣾⠛⡱⣄⢦⡠⣰⣵⣦⢂⡄⠀⠀⠀⠀⠀⠠⡰⡅⠀⠀⠀⣠⠼⣿
⠀⠀⠀⠀⠀⢸⣇⠀⡀⠀⠀⠀⠀⠁⢈⣤⣤⢾⢯⡳⣝⡲⣝⡲⠭⢃⣼⢯⣷⡿⢋⣡⣾⡿⣷⣷⠙⠮⠻⠟⠛⢉⣥⣿⢭⡇⡄⢐⢦⡪⣕⠿⠃⢀⡠⠖⢉⣼⠏
⠀⠀⠀⠀⠀⠀⠻⣦⣍⡂⠓⠌⣓⣩⡀⣀⡀⣀⠓⠝⠳⠝⢒⣉⣴⣿⣿⠟⠉⠀⠋⠉⣠⡾⣟⢮⣿⡿⢲⣖⠿⡛⣝⣾⡟⣰⣏⣪⣕⠎⢚⢁⣬⣤⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠙⣷⣀⣑⣈⡙⡿⢿⢿⢿⡻⣽⣿⣥⣶⠖⠀⣠⡾⣏⣽⣮⡿⢏⣴⣟⢮⣓⣽⡾⢟⣴⡿⣤⣥⣶⡾⢋⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⢷⣌⠈⢯⣷⣻⣾⡿⠋⠀⣠⣾⢻⣶⣤⣤⣴⣾⣿⣏⣈⣓⣉⣥⡶⣿⢯⡷⠾⠛⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡽⠋⠀⣠⣾⣿⣿⣿⡿⠿⠟⢛⣿⣿⠿⠟⣽⠿⠷⠶⠶⠶⠶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣤⢦⣄⢀⡰⠋⠀⠀⠔⠛⠛⠉⠁⠀⠀⢀⣴⣟⣋⣤⠴⠋⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠛⠋⠀⠀⣹⠟⠀⢀⣀⣤⣴⣶⣾⠏⠀⢀⣴⠿⠋⠥⠅⠂⠀⢀⣤⣾⣥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠿⠃⠀⠀⠀⠀⣠⣼⣧⣶⡿⢟⣿⠁⣲⠞⠁⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⡐⢶⡄⢄⡐⠻⡿⠯⢠⡿⠋⢁⡼⠋⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢶⣤⣥⣴⠞⠛⠛⠋⠀⣠⠟⢁⣴⡿⡫⠂⠀⠀⠀⠀⠀⠀⠀⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡾⢋⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣼⠟⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠉⢤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡾⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣠⡟⠁
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠟
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⡤⠤⠦⢤⣀⠀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠈⢹⠃⠀⠀⢠⠶⠛⠉⠉⠙⠓⢶⡞⠉⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠔⠒⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡴⠋⠉⢠⠋⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⣆⠀⠀⠀⠀⣠⠖⠤⠀⠀⠀⠀⣀⠀⠈⠣⢰⠋⠁⢀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡰⠋⠀⠀⠀⠀⢰⠃⠀⣀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣰⠁⠀⠀⠀⠚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠦⠢⠄⠀⠀⠀⠀⠠⢈⠛⠀⢰⠛⢁⡀⠀⠀⠅⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡈⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠈⠳⣞⠛⠀⠈⢞⠑⠒⠀⢄⣀⡴⠁
⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⢱⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⢤⣀⡰⠓⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠈⠳⣄⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⣄⣠⣄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠡⠴⠃⠀⠀⠀⠰⠀⠀⠈⠙⣆⠀⠀⠀⣀⣀⣀⡴⠋⠀⢰⠃⢾⠓⢎⡀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢸⠀⠀⠀⠀⠈⠓⠦⣄⣀⣈⣇⠀⠀⠀⠘⢆⡃⢃⠀⠉⠣⣄⡀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⢨⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠣⣄⡀⠀⠀⢹⠈⠆⠀⠐⢄⡉⠒⠤⡀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠰⠁⡆⠀⠀⠀⠀⠀⠀⠠⢫⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠉⠉⠈⡇⠘⠀⠀⠀⠙⠦⣀⠀⠁⠒⠁⠀⠀⠀⠘⣦
⠀⠀⢀⠀⠀⡁⢀⡇⠀⠀⠀⠀⠀⠠⠁⢸⠀⠀⠀⠠⠜⠀⠀⢀⠀⠀⠀⢈⠀⢷⠀⠀⠀⠀⠀⠀⠚⠉⠉⠙⠂⠀⠀⠀⠀⡼
⠀⠀⢁⠀⠀⠉⠉⠀⠀⠀⣠⡄⠀⠠⣠⠞⠀⠀⠀⠀⠀⠀⠠⢸⠀⠀⡀⡸⠀⢸⣀⡀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠊⠑⢆
⠀⢀⠌⠀⠀⣠⠀⠀⠀⡰⠁⡇⠀⠀⠀⠀⠀⠀⢠⢖⠀⠀⠀⣸⠀⠀⠈⢀⡜⠁⠀⠀⠔⠋⠉⠉⠉⠙⠓⢦⡀⠀⠀⣀⠀⠀⢈
⠀⠀⠀⠀⡐⢠⠀⠀⠐⠀⢸⠃⠀⠀⣀⠀⠀⠠⠁⣸⠀⠀⠉⠁⢀⡄⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⣄⠞
⠀⠀⠀⢀⠀⢸⠀⠀⠘⠦⠞⠀⠀⠀⢠⠀⠀⡁⠀⡟⠀⠀⠀⢀⠊⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠠⣀⡼⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⡇⠀⠀⠙⠷⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⢀⣠⠞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⡨⠋⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢠⠂⠀⠀⢀⠃⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠙⢢⣠⠖⠩⢠⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠦⠤⠖⠒⠒⠢⡀⠰⡇⣀⡴⠤⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⣴⡟⠁⢥⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢆⠀⠀⢀⣀⣸⣿⣿⣤⣮⣀⡀⢀⡤⠤⠤⠤⠖⠛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠉⠰⠒⠚⡶⠞⠋⠉⠙⣿⣿⡏⠉⠙⠛⠛⠒⠒⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⢀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣿⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣄⣄⣤⣤⣤⣀⣀⣀⣦⣠⣀⣀⣀
⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠴⠶⠒⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⢳⢏⣠⡄⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠞⣉⠁⠀⢀⣤⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⢀⡎⠈⡇⢠⡏⠀⠘⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⠖⠒⠚⢿⣆⡇⢸⣦⣄⡠⠟⠋⠉⠁⠀⠈⠉⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠉⠀⣇⠀⠀⠀⠀⠩⠤⢬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢦⣀⡀⠠⢤⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠒⠀⠲⠤⠤⠀⠄⠀⡀⣾⣀⡀⠀⠀⣰⠛⠓⠚⠛⠃⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⣉⣡⡽⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠤⠤⢶⡀⠈⠁⠈⠳⠶⠛⠦⣤⣀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⢤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⠀⠀⣀⡤⠟⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠶⠛⠁⠀⠈⠙⠲⠶⠒⠛⠉⠀⠀⠈⠓⠶⠤⠤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠠⠒⣐⣢⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⡀⠤⠀⠀⠠⢄⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠉⠢⠀⣄⠀⡤⠊⠀⢄⠀⠀⠀⠀⠑⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⠺⠀⠀⠀⠀⠀⠀⠶⠄⢁⠀⠄⠀⠀⠀⠀⠀⠁⠀⠱⢤⡀
⠀⠀⠀⠀⡤⠠⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢀⠀⠀⠐⠀⠀⠀⢻⡀⠦⢄⡀⠀⠀⠀⠀⠆⣀
⢠⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠦⡀⠀⠈⠀⠀⠀⠈⠒⠒⣄⣀
⠈⠂⠖⠢⠤⠄⢤⠔⠢⠀⠠⠄⠒⠤⠀⢀⡠⠄⣀⠀⠀⡀⠠⠀⣠⠠⡔⣤⠤⠔⠁⡀⢄⣀⢀⡀⣤⣀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠈⠒⠓⠠⠤⠀⠒⢲⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠁⠀⠀⠀⠀⠀⡈⢉⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠑⠒⠒⢒⠿⠖⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠲⣌
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠘⠅⣀⡀⢀⣀⣀⠠⠄⣀⣀⢀⡀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠒⠒⠉⠁⠒⠒⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠄⠄⠐⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠒⠒⠂⠴⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⡀⢤
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣷⣶⣦⣀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀⣀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⣀⣀
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣥⣾⣿⣿⣿⣿⣿⣿⣦⣀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⣷⣀
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠻⢿⡛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠘⠷⣽⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠈⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠗⠶⢦⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⢛⣤⣀⡀⠀⠀⠀⣴⣿⣿⣥⣶⣶⢶⣶⠤⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠉⠀⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⠀⠈⠘⢿⣯⡬⠿⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⠀⠀⢺⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣉⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠿⠋⠉⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣴⣾⣿⠟⠛⠋⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣶⣿⡄⠀⠀⠀⠀⠀⢀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⢀⣴⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠶⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠙⣿⣿⣷⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⡆⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠷⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⣿⣿⡀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠿⠋⠀⠙⣿⣧⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⢸⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠶⢀⡠⢠⠤⠐⠀⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠘⠇⠀⠈⠛⠒⠚⠡⠶⠤⠤⠤⠄⠀⢈⠀⠀⠀⠱⠆⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⠀⠀⠀⠁⠀⠀⠀⡀⠒⠒⢒⣒⡉⠉⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠣⡂⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣐⡤⣴⣾⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣆⡀⠀⠀⠀⠀⠀⠈⠭⣐⣰⣽⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⢹⠿⡛⠋⠋⢀⠀⢀⣭⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⣶⣴⡤⠀⠀⢸⣻⠅⠀⠀⠠⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣏⣉⣄⠤⢠⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢀⢀⠀⠀⠠⣉⣙⣛⣏⣄⠘⡛⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠁⠐⠀⢱⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠀⠀⠀⠀⠈⠄⠈⠂⠀⡁⠇⢌⣻⣛⣷⠌⢀⣊⠧⡀⠈⠂⠠⡀⠀⠀⢀⣀⡠⢀⡀⠀⠀⠀⠀⣈⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠒⠰⠁⠀⡀⠏⠈⠈⠀⢈⣛⠡⠉⠀⣘⣿⣿⠀⠈⠀⡠⠀⡅⡁⠀⠀⠀⠂⠀⡱⠀⠄⢀⡄⢂⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠂⠀⠦⡂⢁⠀⠄⡀⠀⠀⡷⠦⣀⣀⢬⣿⡯⠡⡀⠀⠁⠀⡈⠣⠀⠀⠀⠠⠄⠁⠀⠀⣘⠈⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠂⢀⠐⠕⡂⠠⠀⢠⡀⠂⠀⡀⠁⣻⡏⠁⣜⢫⢾⣀⢥⢊⡀⢪⠁⠀⠀⠀⠀⠈⠢⠀⠀⢤⢋⣈⣁⠔⠏⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡄⠀⣠⠋⠠⠀⠘⠱⠂⠜⡴⢦⣍⡍⠹⣿⣒⡰⢝⡊⠄⣸⣄⠀⠁⠀⢁⡂⠠⠀⠀⠣⡀⠀⡞⠨⠦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠉⠀⠐⠀⠰⠀⠀⠀⠀⢁⠀⠙⠿⣣⠈⠻⣶⣟⣦⣥⠀⡅⢠⠀⠀⠀⠈⠀⠀⢐⣄⠃⡆⠘⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠘⠻⠇⠀⠀⠀⢀⠀⡆⢀⠄⠁⠈⠁⠠⠭⠿⣿⠏⡉⡴⠃⠀⠀⠀⠄⠊⠙⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠀⠀⠑⠈⠀⠁⠂⠀⠀⠀⠀⢀⠀⠀⠈

⢲⣤⣤⣤⣤⠠⠄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢠⣀⣄⡀⠀⠀⠐⡶⣶⣶⡶⠶⣶⣶⣤
⠀⢹⣧⡉⠙⠷⣷⣷⡘⡦⣇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣈⠩⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⡟⠛⡄⠀⠀⢘⣶⢮⡱⣞⢀⠓⠀⠁⢂
⠀⠀⠍⠿⢗⣀⠀⠈⠉⠃⠁⣿⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠴⢒⠓⠲⠸⠄⠀⠀⢿⣿⢿⣦⣈⢣⠀⠀⣼
⠀⠀⠘⢿⣭⡽⠞⣣⡄⡀⠀⠙⠃⠀⠀⢀⣔⠀⢸⡁⣤⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⠀⢀⠀⠀⠀⢀⠀⠀⠂⡄⠀⠀⠨⠿⣷⣎⡈⠁
⠀⠀⠀⠀⠑⠂⠁⠀⠈⠺⠷⠦⣀⠠⠀⠀⠙⠛⠎⠓⠚⠓⠲⠇⠀⠀⠀⠀⠀⠉⠀⣤⣀⢻⣄⣄⣬⡿⣿⣷⠿⠆⠀⠀⠀⠻⠯⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣅⢹⣶⣽⣟⣿⠿⠛⠀⢀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠔⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠰⠀⠀⠀⡄⡏⠉⠠⠀⠢⡀
⠀⡀⠀⠖⠀⠀⠀⠀⠀⠨⢻⣤⠀⠀⠀⠀⠀⡐⢿⠇⢰⢃⠀⠀⢀⢠⢤⡄⢀⢀⠀⣤⡤⢀⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢩⠀⠸⠀⠀⠰⠆
⢸⣷⠋⢰⠆⠀⠰⡄⠀⠀⠂⢽⡇⠀⠀⠀⠀⡝⣿⠀⢨⡟⠀⣰⢻⠿⣼⢣⡤⠾⠋⣁⢴⠖⠀⠀⠀⠀⠀⠀⠈⣑⠦⠀⠠⡀⠀⠀⠀⠀⠀⠀⣆
⣸⣿⡿⠁⠀⠀⣤⠘⣆⠈⣻⣲⡇⠀⠈⡐⢀⣤⡌⠀⣼⡧⣠⣿⠂⠴⠋⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠄⣀⠐⠀⠀⠀⠀⠀⠀⠀⣸
⢹⣛⡄⠂⠄⠀⠀⢀⠀⢠⣜⡟⠀⠀⠀⠁⠀⠙⠁⠀⠻⠁⠛⠁⠀⠠⠤⣼⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡤⡄
⠀⢿⠡⢎⡄⠀⠀⣄⢁⡰⡎⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡐⠷⠶⢶⣔⡶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠴⣬⣌⡀
⠀⠘⢷⣎⠄⠀⢦⢰⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠿⠳⠦⢤⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⢻⡆⢼⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⢛⠑⠀⠀⠀⢀⡀⡐⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⣧⣄⣀
⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢳⠘⡅⠘⢧⢉⠀⠙⠟⠟⠈⠉⠀⠀⠀⠀⠀⢫⣄⢀⠀⠘⠀⠀⠸⣯⠿⡿⣷⣶⡆⢠⣄
⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣀⣤⣤⡶⣾⣾⣷⣦⣄⠀⠀⠙⢿⡄⡀⠀⢠⣶⣿⣄⠸⠇⢸⡟⢿⡿⣠
⠀⠀⠀⠀⠈⠳⣯⣟⣷⢮⡙⠆⠀⢀⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⠋⠻⣮⠻⣿⠝⣻⠆⠀⠀⠀⠙⠁⠀⢘⡻⢿⡖⠂⠀⠘⠀⠞⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠟⠂⢤⣄⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣽⡷⠖⣚⣶⣦⡆⠁⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣯⣙⡀⠀⠈⠐⠋
⠀⠀⣴⣶⡶⠶⢷⠖⠂⠀⠀⠘⠳⣶⣿⣛⠲⡄⠀⠀⠀⢀⠴⢿⣿⢯⣭⠄⠀⠈⠙⠉⠄⠀⢀⠀⠀⠀⠀⠁⠀⠈⠻⢿⣿⣽⣥⡄⠀⠰⡀
⠀⠀⢈⣙⣛⣛⣛⡛⠃⠀⢤⡙⢷⣤⡀⠉⠙⠉⠀⠀⣠⣾⡛⣛⣫⣤⠴⢀⣀⣴⣤⡾⠂⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⣾⣿⣿⣯⣿⣯⡅⢀⡐⢦⡙⠳⣽⣻⣆⠀⠀⠀⠀⠉⠛⠟⠿⣿⣿⣿⠟⣻⣿⡏⡅⣴⣟⣰⣿⠇⠀⠀⠀⠀⠀⠀⢦⣶⡆⠀⠀⣠⡄
⠀⣈⣭⣭⣽⡶⠄⠠⣌⣳⣦⡛⢷⡌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⢳⣿⣿⠿⠏⠀⠀⠀⠀⠀⢰⠤⢹⣙⣤⡞⣽⢟⡏
⢼⣯⣿⣿⡿⠟⠀⠸⣿⡿⣿⣟⠿⠿⠀⠀⠀⢀⣠⣤⠤⢦⠀⠠⣤⣤⣀⠀⠈⠉⠁⠁⠁⠀⠀⠀⠀⢠⣶⠀⢀⡶⣢⣾⡿⠋⣻⣿⣆⣋⡜⢳⡀
⠀⣤⡶⠟⢛⡯⢠⣷⣬⡻⢿⣿⡆⠀⢀⣤⡾⠿⡛⠣⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢸⣤⡌⠀⠛⠙⠳⢚⢉⣭⡸⣟⣿⡯
⠘⠻⠟⠛⣩⣧⡌⢻⣿⢿⡄⠀⠀⢠⡾⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠁⠀⠀⠀⠀⣀⠄⠤⠚⠀⠉⢀⡈⢠⣥⣈⡘⠻⣷⣾⣥⣿⣧
⠀⠀⠀⢸⣿⣿⡇⠀⠙⠻⠃⠀⣠⣏⠀⠀⠀⠀⠀⠀⢠⢂⢂⣾⣶⡾⠁⠀⠀⠀⠀⠀⠊⠁⣀⣠⠴⢤⣯⠉⠙⠿⣿⣿⡛⢦⣬⣽⣣⡌⠉
⠀⠀⠀⠘⢿⠟⠁⠀⠀⠀⠀⠈⠉⠉⠙⠓⠀⢠⣤⣤⣷⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠂⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠔⢘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⠠⢀⣀⣀⡠⢞⢅⠀⠀⠀⠀⡎⠁⠀⣺⠋⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⢄⠀⠀⠀⡇⠸⠀⠀⠀⢠⢇⣀⣰⡗⠂⠀⠈⠛⢒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠃⠀⠀⣤⠀⢰⠉⠀⡐⠀⢀⡾⡠⢒⠟⠂⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡔⠤⡀⠹⣧⠸⢘⡐⠠⢠⣻⡉⠤⢊⡆⠀⠠⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⠓⣟⣇⠀⠡⢔⠊⣏⠡⢲⡞⠉⠐⠲⠀⠀⠇⠀⠤⠤⢤⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠂⢁⢻⡖⢃⠲⣄⣁⠴⣉⡏⠨⠀⠈⠀⠂⠀⠀⠀⢠⠞
⠀⠀⠀⢄⣀⣀⡤⠞⢛⢾⡻⢁⡆⠃⠼⠋⠽⣦⠖⢅⡈⢢⡏⢠⠄⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠈⢷⢯⣒⢌⢆⠱⣎⢩⣷⣭⣭⡭⡼⢿⡂⠨⣤⡆⠁⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠘⢤⠈⠛⣻⣦⣈⠢⣝⢦⡀⠀⠀⠀⠙⢶⣄⣥⠁⠀⠀⠀⠀⠀⠧⣀
⠀⠀⠀⠀⠀⠀⢠⠇⢈⠃⠀⠉⠛⠾⢿⣿⣶⣶⣶⢶⡶⣿⣵⡷⠖⠐⠐⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠙⡄⠻⠄⠐⠈⢉⡽⢫⠟⠉⠝⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣁⡐⠊⠀⠀⢠⠀⢘⠔⡀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⡤⠥⠄⠐⢢⡠⠔⠊⠁⠛⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣛⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⣿⣿⣶⣿⣷⣿⣿⣿⣿⣶⣶⣯⣭⣽⣻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡟⢻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣟⣿⣿⣿⡿⠟⠿⣿⡿⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣯⡿⠿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⠷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣏⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁


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