Syrupy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠏⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣟⣉⣉⣈⣹⣝⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢩⣽⣿⣿⣿⣿⣶⣦⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣿⣿⣿⣅⣀⣿⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣿⣿⣿⠉⢹⣿⣿⡿⣿⣥⣔⣂⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣿⣿⠀⣸⣿⣿⣧⡬⠻⠿⣟⠿⢆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⣿⣿⠀⣿⣿⠋⡏⠀⠀⠀⠈⢣⠘⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⡀⣿⣿⢸⡇⠀⠀⠀⠀⠸⡆⣿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⡇⣿⣿⡘⡇⠀⠀⠀⠀⢰⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⣿⣿⣿⣷⢸⣿⡇⣧⠀⠀⠀⠀⣸⢠⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⣿⣿⣿⣿⢸⣿⣿⢸⡀⠀⠀⢀⡇⡼⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣇⢧⠀⠀⡸⠀⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠿⣿⣿⣼⣿⣿⣾⣆⣰⡇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢟⣵⣶⣶⣶⣍⢻⣿⣿⣿⣿⢹⡁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢣⣿⣿⣿⠿⣿⣿⣷⡹⣿⣿⣿⣎⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣱⡿⠻⢿⠍⢘⠈⡿⡿⣷⡘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢡⣿⣷⡈⣄⠠⡅⠈⠴⢙⣿⣧⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⣿⣿⣿⣯⣉⢦⠄⠋⣰⣾⣿⣿⡆⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⢸⠟⢻⡿⢷⢶⠿⠿⠿⡿⢿⠿⠿⢿⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⣼⣗⣀⠱⠀⣾⣀⣷⣆⣁⣸⠀⣃⣼⡆⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣶⣿⡿⠀⢸⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⡄⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠛⠛⠉⠀⠀⣸⣿⠃⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢹⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠻⣿⣷⣭⣝⣛⡓⠒⠶⠶⠶⠶⠶⠒⢚⣛⣛⣯⣽⣿⡿⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⠿⠛⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠁⠀⠉⠩⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠤⠤⠤⠤⢤⡟⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠘⣏⣉⠤⢄⣈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣀⣀⣀⣿⡀⠀⠀⠈⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠰⠄⢠⠀⠀⠀⠀⢳⠀⠀⠀⡏⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠂⠀⠀⠀⠀⠀⠀⠀⢾⡀⣠⠞⢡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡀⠈⠀⠀⣠⠤⠤⠤⢤⣀⠀⠀⠙⢆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡴⠋⠀⠀⠀⢀⡀⠈⠳⣄⠀⠀⠳⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢠⠎⠀⡆⠁⠰⠄⠦⠄⢁⡆⠈⢣⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢠⠋⠀⠀⠈⣉⠀⡀⠉⣉⠥⠀⠀⠀⢳⡀⠄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢠⠏⢀⣀⡀⠤⠀⢀⠀⣠⠀⠀⠀⢠⣄⣀⢷⡘⡄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠃⡞⠀⠠⠥⠒⠚⠀⢹⠛⠉⢦⠀⢀⡜⠀⠀⠀⢧⢡⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡐⠌⢸⠁⠢⣄⠀⠀⠀⠠⡴⠀⠀⠀⠙⠋⠀⢠⠀⡆⠘⡆⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠃⠀⡎⠈⠢⣀⠑⠲⠤⠤⠸⡄⠀⢣⠀⣴⠀⠈⢳⠁⠀⢣⠀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⢠⡇⠀⠀⠀⠉⠐⠂⠀⣰⠇⠀⡀⢷⠁⠀⢀⡎⡠⠀⢸⠀⠈⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠄⠀⠀⠀⠀⠈⢻⠉⠀⠀⠀⠙⠺⡀⠀⡼⠋⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠈⡆⠐⠤⡀⠀⠐⠋⠠⠤⢄⣉⠑⢦⡇⡰⢁⣰⡀⠀⢸⠀⠀⠀⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢣⠀⠀⠈⠓⠢⠤⣀⣀⣀⣈⣳⠀⠛⠉⠉⠀⠀⠀⡞⠀⠀⠈⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢎⡴⠚⠉⠛⠲⣤⠞⠁⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠒⠚⠒⠒⠚⠉⠉⠀⢀⡀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠉⠉⠉⠙⢃⣀⣀⢀⠀⠀⠀⠀⠀⠤⠤⠤⠒⠒⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣄⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⡤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⠀⣀⣀⣠⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡀⠀⠀⠀⠀⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⣇⣁⣐⡀⠬⢿⣁⣠⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠸⡝⢁⣀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠶⠟⠒⠲⠶⢶⣦⡿⠋⠀⠙⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠉⠉⠉⠉⠁⠀⠻⡇⠀⠀⢀⠇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢲⣴⠋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⣀⠐⠈⣁⣀⣀⣀⣀⠀⠀⠀⠉⠳⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠖⢁⠔⣿⣥⢖⠠⠓⢌⡙⠶⡀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⡀⠀⣰⠏⣸⠟⠀⣹⠲⣄⡴⠉⢆⠹⣆⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢠⠃⡐⠡⣠⡼⠏⠀⠘⢿⠀⢈⣂⣹⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠁⣀⢸⣿⡏⠙⠛⠧⠀⠎⠀⠟⠋⠉⣿⣿⡿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⡙⣧⠀⠶⠀⠀⠖⠀⡠⠀⣠⣿⣧⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⣿⣯⢤⡀⠀⠀⡦⠀⠀⢤⣿⠏⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠀⢾⣥⣤⣾⣿⣶⠦⠼⣋⣤⣽⡷⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠰⣄⠋⠐⠐⠋⡛⣐⣀⣀⢈⣷⡏⠀⠀⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢀⠀⢻⡁⣉⣉⣉⡉⣉⠉⡤⢌⡼⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠁⢀⠛⠾⠧⢽⣶⠤⠾⠶⠟⠁⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠈⠳⠤⠤⠀⠠⠤⠄⠤⠤⠤⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢶⣶⢾⣴⡋⣀⡀⠁⣘⣿⣷⣶⡦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠈⠌⠉⠉⠉⠩⠑⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⠒⠚⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠒⠒⠉⠉⡏⠁⠀⠀⠀⠀⣨⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⢀⠀⠐⠈⠛⢧⠀⠀⠀⣣⡤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⡜⢀⠴⡞⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⣀⡝⠀⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⣀⣠⠴⠦⢄⣀⣉⣠⠤⢴⠖⠶⠖⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⠴⠅⠀⠀⠈⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢇⠀⠀⠀⠘⠓⠲⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠤⠤⣄⣀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠙⣆
⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢤⠲⠶⠃⠈⠓⢒⠶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠈⢧
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠈⣧
⠀⣸⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠧⠒⠲⠤⠏⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠸⡆
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⠤⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⡄⠀⠀⠀⣿
⢸⠀⠀⠀⢀⠀⠀⠀⡤⠄⢰⠋⢂⠀⢘⠁⣀⠑⠈⠀⣤⠀⢳⠁⡇⠘⠁⢉⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢸⡆
⡟⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⢀⠀⠠⣸⠀⢋⠀⠰⡄⢠⠶⢛⠀⠇⣤⠀⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⡏⠡⣀⡽⠄⠎⠑⠒⠳⠚⠀⠘⠂⠠⠿⠤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇
⣇⠀⠀⠐⠀⠀⠀⠘⠒⠃⠙⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⡇
⢹⠀⠀⠀⠀⠀⠀⠀⡠⠆⠶⡄⢤⠒⠠⠊⢱⠜⠁⡀⠈⢁⡖⢂⡰⠤⡴⠒⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠇
⢸⡄⠀⠀⡀⠀⠀⠸⣄⠀⠘⠳⡄⠑⡄⢀⠃⠀⠀⠉⠰⢏⡇⠀⠁⢰⡇⢀⣉⣀⡼⠀⠀⠀⠄⠀⠀⠀⡸⠀⠀⠀⣾
⠀⢧⠀⠀⠀⠀⠀⠀⡤⠜⠃⢀⡄⡘⠁⡜⠀⡇⠀⡛⢆⣀⠷⣄⣀⠼⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇
⠀⠘⣇⠀⠀⠀⠀⠀⠒⠒⠒⠉⠀⠙⠚⠀⠀⠉⠒⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡜
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡔⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣼⠁
⠀⠀⠀⠘⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⢀⡞⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠖⢆⠀⢀⡰⠁⠀⠀⠀⢀⠎⠀⠀⠀⣸⠦⠤⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⠆⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠶⠴⠤⠤⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡯⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠲⢦⣀⠀⠀⣀⣀⡤⠤⠶⠒⠒⠛⠓⠒⠦⠤⣤⣤⣤⣤⣤⠤⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠁⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⣁⣀⣀⣀⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⢫⣀⠀⠀⠀⢨⠏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠉⣳⠀⢀⠼⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠁⡎⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⣀⣀⡀⠈⠊⣁⣸⡤⢄⠤⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠉⠁⠀⠀⡭⡉⠁⠀⠃⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠑⠁⠀⠀⢸⠀⠀⠀⠙⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠤⢄⡀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⣀⣤⠜⠐⠄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢑⠄⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠲⠤⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠤⡀⢠⠄⠠⢀⡄⡀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠘⠁⡜⠀⠀⠀⡇⢦⠈⠀⠘⠂⣸⠃⡃⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠀⠔⠀⠐⣇⡰⠄⠸⣆⠏⠀⢀⣉⣇⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠦⠎⠑⠃⠈⠉⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠄⠒⡄⢔⠢⠔⢱⠞⢁⠈⠁⡔⠂⡤⢤⠖⠒⠒⡄⠀⠀⠀⠀⠀⠀⡈⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠣⢄⡈⠁⡄⢱⠀⡀⡀⠈⡀⠚⢥⠀⠀⢸⠀⡍⠠⠖⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢰⣊⣁⠴⠲⣅⡜⠀⠧⢠⠍⠦⠚⠂⠀⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠸⠄⠀⠀⠀⠀⠀⡰⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠠⡀⢀⡠⠂⠀⠀⠀⡔⠁⠀⠀⡷⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠓⠒⠒⠲⠶⡶⠶⠀⠠⠤⠤⠼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠶⠶⠤⢤⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⣀
⠀⠠⠤⠤⠤⠤⢀⡀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⡀⠉⢓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⣤⣤⣤⣤⣾⣾⠏⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠙⠋⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢐⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⢀⡠⢀⠙⠊
⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣝⢌⠎⣆⡆
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠙⡜⣾⣼⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠠⠀⢠⣄⠉⠁⠀⠮⢼⠀⠤⠶⠆⢠⣤⢂⠀⢀⡀⢀⠀⠀⠠
⠀⠀⠀⠀⠁⠐⠂⠀⠀⠙⠉⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣠⣴⣶⣦⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧
⠀⠀⠈⢿⣿⣿⣿⡿⢋⠙⣦⠦
⠀⠀⠀⠀⠙⡿⠣⢡⠀⣫⣶⡀⡀
⠀⠀⠀⠀⠀⠀⠬⠲⣿⣿⣿⣇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢘⢽⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣖⣠⣦⠶⣿⣥⣽⢶⡈⣷⠟⠧⠴⠠⠤⢲⠄⠂⠀⣙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⡟⣿⠿⢧⣤⣶⣋⢳⢸⠁⡝⠉⢧⡀⢀⣠⢤⡔⢧⣀⣒⡡⠎⠙⡖⢤⠰⡤⡀⠀⠀⠐⣶
⠀⠀⠀⠀⠀⠀⠀⣰⠦⠥⡾⢩⠇⢸⡄⠀⠹⣦⣍⣹⡿⢿⣷⣤⣤⣟⡻⠴⠒⡲⠶⣆⣛⣀⣀⣄⡹⣆⠢⣠⠀⠀⠀⠘⠿⡍
⠀⠀⠀⠀⠀⠀⠈⢡⠞⠛⠀⠛⠀⢸⠷⣄⣴⣻⡞⠋⠻⢬⣙⡿⣮⣙⠛⠛⠛⠲⢤⣤⠀⠬⢉⢬⠡⠁⠁⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⡛⢇⠀⠃⣔⢰⢀⡟⠗⠝⣷⣠⢿⣀⣂⣀⣸⣉⢿⠿⣻⣄⢥⣂⣀⠜⠷⠖⢀⣀⠁
⠀⠀⠀⠀⠀⠘⠀⠸⡚⡩⠃⣤⠎⠙⠀⠀⣼⢸⢻⡿⣅⣐⢋⡀⠀⢷⠀⠊⢢⢈⠁⠀⡸⢄⠀⠈⠳⣤⠤
⠀⠀⠀⠀⠀⠈⠀⠤⢁⢆⢟⡎⠀⠀⠢⣍⠀⡱⡇⢳⡄⠙⣝⠲⠶⠶⢤⣀⡀⠀⠀⢀⠁⠬⡄⡀⠀⣰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠴⠖⠒⠚⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠉⠙⢁⡉⠀⢈⠏⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠕⡠⠞⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠐⠔⠀⣃⣀⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠡⡈⠉⢙⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠈⡀⢀⠈⠉⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⣀⡠⠋⠄⢠⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠘⢆
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢱⢀⠀⣐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⡆
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⢀⡊⠀⠀⢡⡄⠈⠰⢈⠏⠃⠉⣵⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢤⠐⢂⣰⠤⠂⠸⠬⠁⠄⠽⠄⠹⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢁⡉⢀⡀⡠⣀⠤⠤⠀⡀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⢈⠀⠘⠈⢀⠀⠁⠐⠠⢞⠈⠁⡇⣐⢀⠆⠀⠀⠀⠀⡄⠀⢰⠃
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠰⣈⠠⠒⢅⠂⠠⠨⠑⠜⠂⠐⠃⠂⠀⠀⠀⠀⠀⠀⠂⠀⡜
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⡀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⡀⠈⠁⠀⠀⠀⡐⠁⠀⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⢊⡠⠄⠊⠀⠀⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⣖⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠒⠤⣀⡠⠤⠐⠒⠒⠒⠒⠲⠤⠤⠤⠔⠒⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠉⠛
⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠁⠛⢿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠙⢻⣿⣿⣿⡿⠛⠛⠉⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠋⠉⢻⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡟⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⢻⡏⠀⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠈⣿⡄⠀⠀⢠⡄⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⣶⣿⣷⣶⡿⠿⠿⠿⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣶⠶⠂⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⣶⣶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠈⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣀⠈⠙⣿⣷⣄
⠀⠀⣀⣤⣾⡿⠛⠁⠀⠀⠈⠙⢻⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⠀⠀⣀⣤⡾⠟⠋⠁⠈⠙⠻⠿⣿⡿⠛⣿⣷⣄
⣴⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣶⣶⡶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣽⣿⡿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⣿⣿
⢸⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⣷⡄⠀⣠⣾⠟⠛⠛⠻⢷⣄⠀⣾⡟⢻⣧⢠⣶⢶⣦⣀⣴⣶⣾⣷⡄⠀⢠⣶⡾⠿⣿⣠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⣸⡟⠁⠀⣰⡿⠁⢠⣿⠁⠀⣠⣴⣶⡄⠙⣿⣿⡇⠀⠿⠟⠃⠀⣿⣿⠁⣾⣿⣿⣷⣤⣼⡿⠀⠀⠙⠟⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠻⢷⣶⣼⣿⠀⠀⢿⣥⡿⠁⠀⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⠀⠻⠟⠃⣀⣼⣿⡇⠀⣴⣶⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠙⣿⣤⣀⣀⠀⣻⡇⢿⣷⣄⡀⠉⢀⣠⣾⠏⠈⠿⣶⣶⣶⣾⠿⠋⢻⣦⣤⣿⣆⣿⡏⢹⣧⣠⣿⢻⣧⣤⣼⠟⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠟⠁⠀⠉⠛⠛⠻⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠁⠀⠈⠉⠉⠁⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣿⡇⠀⠀⠀⠀⠀⣀⣴⣶⠶⣶⣦⡀⠀⣀⣴⣶⣄⠀⠀⢀⣀⡀⠀⢀⣠⣤⣤⣤⣀⠀⠀⢀⣠⣄⠀⢀⣀⣀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⣿⡇⠀⠀⠀⠀⢰⣿⠁⠀⠀⣤⣿⠃⣾⣟⠉⠉⠙⣷⡾⠛⣿⡇⣶⣿⡏⠉⠉⠉⠻⣷⠀⣿⠟⢿⣦⣿⠏⢹⣇⠀⣿⠟⠉⠉⠉⠙⠻⣶⡀⠀⠀⠀⠀⠀⣿⡇
⠀⢻⣧⠀⠀⠀⠀⠈⠿⣶⣦⣄⠉⠛⢷⡌⠻⣶⣄⠀⠀⣀⣤⣿⠁⣹⡇⢴⡦⠀⢰⣾⣟⠀⣿⡇⠀⠉⠉⠀⠈⣯⠀⣿⠀⠸⠿⠟⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⢀⣤⣿⠏⠀⠀⢸⡿⠀⠈⣿⡇⢀⣿⠉⠀⠀⢸⡇⠀⢀⡀⠀⠈⠻⣶⣿⡇⠀⠀⠀⠀⢸⡿⠀⣿⠀⢠⣤⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⣴⠟⠋⠁⠀⠀⢀⣿⠇⠀⢰⣿⠀⢸⣿⠀⠀⠀⢸⡇⠀⣿⢿⣆⣀⣼⠟⢻⣇⠀⠀⠀⣠⣾⠃⢰⣿⠀⢸⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⢿⣶⣶⣶⣶⠾⠿⠋⠀⠀⠘⠻⠿⠿⠏⠀⠀⠀⢸⣧⣴⣿⠀⠙⠿⠋⠀⠀⠙⠷⠶⠾⠟⠁⠀⢸⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣼⣿
⠀⣾⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⢿⣿⣆
⠈⢿⣧⡀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿
⢠⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿
⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿
⠀⠙⢿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣿⣿⠟
⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣹⣿⠟⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⡇
⠀⠀⣸⣿⠁⡇⠀⠀⠀⢸⣤⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⢠⣶⣴⠄⠂⠆⠀⠸⣆⣠⠃
⠀⠀⢻⣿⡗⠁⠀⠀⠀⠸⢂⠖⠀⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸⣿⣾⡀⠀⣃⣀⠀⢸⠁
⠀⠀⣿⣿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⡀⡏⠀⠀⢸
⠀⠀⠙⠉⠉⠈⠁⠀⠀⠀⠳⡶⢄⢀⠀⠘⠆⠠⡆⢀⢏⠀⠀⠀⢀⠄⠀⠀⠀⠀⢸⣿⡇⠀⠃⠁⠀⠀⢘
⠀⠀⠀⠀⠐⢆⡀⠀⠀⠀⠁⠠⡄⠀⠁⠉⠀⠀⠀⡘⠀⡄⠠⠆⠠⠀⠀⠀⠀⠀⢸⣿⡷⠖⠀⣇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⢄⠀⠀⠀⠠⠀⠀⢀⠉⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉⠈⠈⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⠈⠁⠀⠐⠂⣀⠀⠀⡐⠁⠀⠀⠈⠁⠀⢀⠔⠒⠊⠉⠓⢦⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡎⠀⠀⠀⠀⢀⡀⢀⠂⠀⠀⠀⠀⠀⠈⢳⣄⠀⠐⢶⣿⠷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡠⠀⢠⠀⠚⠁⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠶⢿⠦
⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡄
⠀⠀⢠⡟⠀⠀⢸⣇⠀⠀⠝⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣿⡇
⠀⠀⠘⠋⠀⠀⢸⡁⠀⠠⠀⢷⠀⠀⢰⠀⠀⠀⠀⢀⠀⠀⠀⠈⠐⠆⠀⠒⠛⢻⡟⠿⠿⠿⠿⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠁⠊⠀⠀⠄⠀⠆⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢠⣤⣄⣀⡐⠀⠒⠀⠀⠀⢁⡀
⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣟⣒⡮⠭⡒⠦⠖⠈⠩⢉⡁
⠀⠀⠀⠠⡀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠇⠀⠀⠂⠘⢿⡫⠭⠑⣒⡒⠒⢒⣋⡩⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠁⠂⠍⢉⡁⠚⠛⠉⠉⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠶⠒⠶⠶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⢁⣀⣀⣠⣒⣀⣠⡴⠶⠶⠶⠷⠦⢤⣄⡀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠜⣁⠤⠶⡶⠛⠍⠁⠀⢠⣭⣽⣳⣶⣶⣤⣈⣩⣷⡶⣦⣍⠙⠲⣤⡀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⣬⢪⠅⠁⠀⣠⣤⣶⣶⣾⠟⠛⠿⠽⠻⠯⠿⠿⢝⠿⣄⠠⢿⣷⣷⣄⠹⣦⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢠⠞⣱⣁⠀⣠⢊⡄⠈⠁⠐⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠷⣜⣷⡌⠉⠉⢻⣷⠈⢳⣄⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠃⠈⣭⣿⣿⣱⠇⠀⠀⠀⠀⢀⠐⠄⠲⠠⠿⠌⠀⠀⣄⠀⢿⣿⣿⣶⣶⡿⠛⠀⠀⠹⡆⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⡹⠀⠀⠃⢀⠀⠀⠉⣀⠀⢀⣘⠨⠀⠋⢠⠁⢰⣦⣹⡿⢿⣷⡀⠀⠀⠀⣹⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠃⠀⠀⠀⣾⡇⠀⡀⠀⠀⠁⠀⢀⠀⠀⠁⠊⠀⠤⢠⢀⢦⣼⡿⠟⠁⢸⣿⣗⠁⢁⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠙⢿⣿⣦⠶⠶⢦⣄⠀⠐⠆⠸⠀⠋⠲⠆⣰⡿⠋⠁⢈⠀⣼⠿⢃⣠⠞⣼⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⡿⣆⡂⠀⠀⠀⠈⠉⠉⠉⠛⢿⣛⣗⣚⣿⣿⣿⣟⢭⣿⡇⠀⣠⢈⣾⣯⠴⠛⣱⣿⣿⣷⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⢿⣿⡂⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⢷⡹⣷⣨⠕⣿⣿⣥⣶⣿⣻⣶⡿⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⠻⣝⡶⣤⣑⣶⠦⣤⣤⠴⠤⣤⣤⣤⣄⣀⣤⣿⣷⣻⣿⣾⣿⢿⣿⡿⣻⠵⠋⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣉⣉⣛⠛⠛⠛⠓⠒⠒⠒⢛⣋⣥⣿⣽⣿⣿⣿⢾⣟⣶⠁⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⢿⡻⣟⡻⠿⠿⢿⡿⢿⣿⠿⠿⡛⠲⣶⣾⣿⠿⣊⡵⠃⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⡀⠀⠀⠉⠛⠻⠿⣶⣶⣶⣶⣶⣶⣶⣾⣯⡽⠷⠚⠋⠁⠀⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢥⣄⣀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣢⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠲⠶⠶⠒⠒⠒⠋⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠉⠉⠉⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠻⣿⣀⣀⣀⣀⣿⡍⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠀⡼⠀⠀⠀⠀⢀⡿⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠰⡀⣀⠀⢠⡏⠀⢰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⡆⢸⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠈⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢁⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠀⡇⠀⠀⠛⠶⠶⠶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢧⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈⢧⠀⠀⣀⣠⣤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠓⢄⠀⠀⠈⣇
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣄⡏⠀⠀⠀⠀⠸⡶⠋⠁⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢹
⠀⠀⠀⠈⡇⠀⠀⠈⠀⠀⠀⠹⡀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠄⠀⠀⠀⠱⣄⠀⠀⠀⠀⠂⠀⠈⠀⠔⠀⠀⠰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣰⠃⠀⠀⢸
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⡴⠃⠀⠀⢀⡟
⠀⠀⠀⠀⠈⣇⠀⠀⠀⡀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣀⣀⣈⡽⠂⠀⠀⠀⡀⠀⠀⠀⠉⠓⠒⠋⠉⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠻⠴⠒⠒⠒⠚⠉⣇⣀⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠓⠶⠤⣤⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣰⠧⢀⣤⣀⣀⠀⠀⠀⠀⠀⢀⣶⣶⢄⡀⡀⢰⠆⡀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢸⠀⠀⠸⣬⢻⣽⣥⣦
⠀⠀⠀⠀⢀⡜⠻⠶⣾⣿⣉⣀⠀⠀⠀⠀⠀⠈⣩⡝⠛⠻⡟⠙⠚⠋⠀⠀⢠⡔⢿⣯⣏⠀⠀⣦⣤⣼⣤⣴⡄⠙⢯⣘⣿⢿⣴⡆
⠀⠀⠀⠀⢺⣷⡶⢤⡀⢈⠿⣹⡏⠂⠀⠀⠀⠀⣭⠷⣶⠶⠶⢶⡷⠂⠀⠀⠀⠙⣿⣿⣿⡀⠀⠱⢄⣀⣨⠟⠀⠀⠀⠻⡤⠞⠋⠁
⠀⠀⠀⢰⣟⣈⣶⣀⣘⠿⠗⠈⠘⡀⡀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠀⠀⠀⢠⣾⠟⠛⠻⣷⣆⠀⠀⠀⠀⢀⡀⣀⣠⣤⣄
⠀⠀⠀⣸⠋⢉⣀⣸⡿⣷⠊⢻⠿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⣠⣶⡄⠈⣿⠀⠀⢸⣹⣿⡿⢿⢿⣿⣟⣣
⠀⠀⠀⢸⠿⣿⣇⣀⣉⣿⣿⣳⢶⣄⠀⠀⠀⠀⣞⣾⣿⣯⣙⣲⣦⡀⠀⠀⣧⢦⠘⠛⢃⣄⣿⠀⠀⠀⠻⣾⠃⢽⣷⣦⣿⣏⣰⡄
⠀⠀⠀⠹⣦⣬⠀⣿⣧⣤⣭⡿⣷⣤⠁⠀⠀⣨⣿⣿⣿⠿⢽⣿⣫⠀⠀⠀⢿⣿⣗⣒⣾⣿⡿⠀⠀⠀⠀⠈⠛⠻⢤⣼⣛⣫⡟
⠀⠀⠀⠀⠀⠉⠉⠉⢻⠋⠈⠉⠉⠀⠀⠀⠀⢹⣿⣯⣿⣿⣿⡿⢋⠀⠀⠀⠘⠿⠿⠿⠿⠿⠃⠀⠀⢀⣠⣴⣤⣤⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠛⠛⢾⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⣦⣿⣿⣿⣿⣛⣿⣿⣛⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⣈⣳⡶⣤⣈⢙⣯⣿⡯⡀⠒
⠀⠀⠀⠀⠀⠲⡤⡄⠀⣠⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⣿⣿⡿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⢀⡘⢞⠋⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠁⠀⡁⢘⣿⠿⠂⡴⠄⢻⣀⠀⠀⢀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⡴⣶⣳⣿⣥⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠠⠴⢴⢻⣿⣄⢆⠀⠂⣴⣥⠛⠀⠠⡟⠈⢚⡅⢠⠂⠀⠀⠀⠀⡜⢲⠡⠀⠰⢤⣿⣿⣏⡂⠀⠀⢀⡀⣀⣦⣿⡨⡔⢠⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠠⢧⣔⠀⠀⠀⠀⠶⢗⣦⣟⣄⣁⠵⠆⠀⠀⠀⢳⡈⠛⠲⠾⢿⣥⣾⣿⠆⠀⠀⣩⣷⣿⢿⣿⡟⢷⣝⢤⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠁⠙⠫⢙⡈⠁⠀⠀⠀⠀⠀⠉⠿⣿⣾⣿⡿⠟⠁⠀⠀⠀⠴⢿⣹⣟⣹⣿⣿⡕⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⠖⠒⠛⠛⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⢁⣀⡀⣀⣀⣀⣀⣀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠈⠉⠩⡁⠀⠀⠀⠀⢀⡍⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠄⠀⠐⠈⠙⡆⠀⠀⣸⠤⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⣴⢃⠴⢺⠁⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣦⡸⠀⠠⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣀⠴⠒⠒⠄⣤⡤⠶⠲⡟⠑⠚⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⢢⠀⠀⠀⢱⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠀⠀⠀⠀⠉⠉⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠤⢄⡀⠀⠀⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢤⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠫⡉⠉⠀⠀⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢳⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣀⡀⢀⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢷
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⣇
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⡤⠔⠲⠀⠖⠉⠑⢢⡴⢠⠀⡤⠤⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⢁⠀⠀⠀⢻
⣼⠀⠀⠀⠀⠀⠀⠸⠉⢢⡇⠀⠡⠀⠀⠰⠆⠀⠀⠘⢃⣰⠇⢸⠀⡀⠐⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠶⡀⠐⣆⢀⠦⣀⣹⡄⡏⠀⡄⢈⣹⡃⠈⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⢠⡠⠙⢤⠆⠑⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⢸⡇
⢷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡠⢄⣀⠦⣄⠖⠚⠓⢦⢀⠤⠀⣀⣀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸⡇
⢸⠀⠀⠀⠀⠀⠀⢰⠉⠀⠤⢄⠑⢄⠀⠀⠋⠀⠺⠀⣠⢻⠀⠛⠀⣿⠀⠴⠀⢱⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡜
⠀⣆⠀⠀⠁⠀⠀⠈⣑⡲⠀⠈⡄⡼⠁⡌⠀⠀⢰⢆⠈⢹⡄⠀⢀⣟⠐⡗⠒⠋⠀⠀⠀⠁⠀⠀⠀⡜⠀⠀⢠⡇
⠀⢹⡄⠀⠀⠀⠀⠀⣅⠠⠤⠊⠘⠤⠞⠀⠘⠢⠼⠀⠙⠁⠈⠁⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡾
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣰⠃
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠁⠀⠀⠴⢄⠀⠀⠀⠀⠀⠀⡰⠁⠀⢰⠏
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⢤⠀⠀⢀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢹⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⣠⠎⠀⣀⡠⠔⠉⠀⠀⠀⠀⠈⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⠥⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⣀⣈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠶⣤⣀⠀⠀⢀⣀⣠⡤⠴⠶⠖⠒⠶⠶⢤⣤⣀⣀⣀⣠⣤⣤⠴
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢶⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠶⠤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⣴⠂⠐⠃⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡀⠀⠇⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢇⡈⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢭⠈⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡖⠩⣟⣣⣶⣾⣇⣄⡐⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢩⣺⣰⣶⣿⣤⣠⠂⠀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠀⠻⢤⡆⢠⣿⣿⣿⡷⣾⡿⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢦⣴⠆⣽⣿⣿⣿⣾⣿⠟⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣹⠇⠛⠈⢻⡟⢻⣿⣿⡏⢉⣳⣶⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⢷⠙⣿⡛⢿⣿⣿⠁⣩⣿⣮⣝⢠
⠀⠀⠀⠀⠀⠀⠀⠰⡟⢹⡄⠀⠀⡾⠓⠒⢿⣿⣿⡿⠛⠿⠿⢷⣤⠆⠀⠀⠀⠀⢠⡿⢻⡄⠀⠠⡿⠛⠛⢿⣿⣿⠟⠙⠛⠛⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡿⠁⠀⠀⠀⢹⣿⠁⠀⠀⠐⢸⡏⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡟⠁⠀⠀⠀⣿⣿⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡇⠸⣇⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠸⢷⡄⠸⡇⠀⠀⠀⣠⣿⣻⣄⠀⢀⡠⠍⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⣶⣶⠶⣿⣿⣿⡈⢲⡤⡃⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡔⠿⣶⣶⣿⣟⣿⣿⣾⣿⣻⢋⣬⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡥⣥⣾⣟⣴⣾⣿⣿⣾⣷⣿⣿⣟⣹⣷⠄⠀⠀⠀⠀⠀⠀⣿⣿⣯⠀⢻⣿⣦⣼⣯⠋⢿⣿⣿⡟⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⢿⣷⣾⠶⠛⠁⠀⠀⠀⠉⠛⣿⣿⣏⡆⠀⠀⠀⠀⠀⠀⢻⡏⢿⣄⠀⠹⠿⣿⣷⣶⣶⡿⠋⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠛⢉⠀⠀⢀⣴⣶⣦⢤⣶⣜⣿⣏⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⢤⣀⠀⠉⠉⠁⠀⣀⣴⣿⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⣸⣷⣦⣿⣿⣿⣟⡛⠻⢿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⢀⣴⣿⣿⣿⣿⣿⣿⢻⣿⣿⡯⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢏⣵⣻⠅⠘⢊⣈⠥⣶⣶⡖⣴⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣹⣿⣿⠗⣀⣤⡻⣾⣿⣷⣦⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣷⡄⢘⣟⣷⡾⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣴⣿⣿⣿⢿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⣿⢛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡿⠄⠠⣿⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢚⣫⣭⣍⣉⣽⡓⠶⣄
⠀⠀⠀⠀⠀⠀⡾⢫⠖⠃⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢰⣦⣼⣦⣹⣿⣿⣿⣧⣾⣦⣶⠀⠀⠀⠀⠀⣿⢠⣿⠿⠿⣿⣿⠿⠟⣀⣿
⠀⠀⠀⠀⠀⢸⡇⢀⣄⣾⣿⣠⡀⠸⡇⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄⠀⠀⠀⠀⣿⣌⣿⠚⠲⠶⠖⠚⠛⣉⣿
⠀⠀⠀⠀⠀⢸⡇⠘⢿⣿⣿⡿⠃⢰⡇⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⣿⡄⣿⠘⠒⠒⠒⠛⢋⣡⣿
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⣄⡟⠘⠛⠒⠚⠛⢋⣡⡞
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⣤⢤⢠⣤⡄⡤⣤⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⡴⣽⣬⡿⣤⣧⣧⣿⣾⣴⣣⢤⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⡔⢫⣽⣿⣿⠋⣭⡿⢏⣹⣿⣉⡆
⠀⠀⠀⠈⢹⡿⠾⠿⠷⠿⠷⠿⠿⠷⢷⡞⠁⠀⠀⠀⢠⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠳⣞⣛⣒⣒⡻⠭⠭⠥⠴⠶⠛
⠀⠀⠀⠀⠸⡇⣧⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⣇⣾⠀⠀⠀⠀⣠⢤⣴⣶⣶⡖⣒⣲⡶⣒⣲⢦
⠀⠀⠀⠀⠀⢿⠸⠄⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⠏⠀⠀⠀⠸⣅⣿⠿⣽⡭⠼⠿⢿⣒⣛⣓⡾
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠛⠚⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢀⣤⣤⣤⣤⠤⠤⠤⢤⣤⣄⡀⠀⠀⠀⡄⡄⡄⢀⣴⡞⠉⠀⠀⠉⢳⣄⠀⢀⣤⠀⠀⠀⠀⠀⢀⡤⢀
⠀⠀⠀⣴⠋⣤⣶⡶⠲⣶⣷⢶⣶⣤⣌⠙⣦⠀⠀⣧⣧⡇⣾⣿⡇⠀⣤⡄⠀⢸⣿⣧⢸⣿⠀⠀⠀⢀⣾⢫⣾⣿⣿⣶⣤⣀
⠀⠀⢸⡇⢸⡟⠙⣷⣶⡿⣧⣤⣿⡋⢙⡇⢸⡇⠀⠙⡏⠀⣿⣿⡇⠻⣿⣿⠟⢸⣿⣿⠘⣿⠀⠀⠐⣿⢱⡿⠿⠿⢿⢿⣿⣿⣿⣦⡄
⠀⠀⠘⢧⡈⠛⠿⣿⣿⣤⣼⣿⣿⠿⠟⣡⡾⠁⠀⠀⡇⠀⠹⣿⡇⠀⠀⠀⠀⢸⣿⠇⠀⣿⠀⠀⠀⡗⠁⠤⠘⡄⠌⡚⣬⢈⠨⢸⡇
⠀⠀⠀⠀⠙⠓⠶⠶⠤⠤⣭⣤⠴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⡤⠤⠚⠁⠀⠀⠁⠀⠀⠀⢿⣭⣥⣮⣥⣬⣤⣤⣧⣬⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠴⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⢀⡠⠤⠤⠤⠤⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠣⢤⠀⠀⠠⡇⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⢠⢇⣤⠞⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢇⠸⠀⡇⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠤⠒⠐⠂⠠⠔⠒⢻⠙⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠐⠵⠀⠀⠐⠀⠀⠀⠳⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⣀⡀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⣀⡌⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠢⠴⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⠤⠀⡄⠀⠀⡠⢄⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⡬⠀⠀⠀⠀⠆⠀⡀⢃⣠⠇⣀⡀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⡄⠚⢀⣸⠦⠗⠀⠿⠼⠀⠀⠤⠯⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠂⢉⡆⠪⡑⠁⡙⠁⠄⢀⡰⠁⠖⢰⡋⢠⠉⠂⠀⠀⠀⠀⠀⢠⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢈⡂⠄⢀⣀⠜⡰⠀⠀⢰⡈⢙⡄⠀⣸⠀⠒⠐⠃⠀⠀⠀⠀⠀⠸⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠘⠤⠐⠊⠘⠒⠁⠀⠓⠊⠈⠁⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠖⠀⢠⠀⠀⠀⠀⠀⠀⡰⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⠀⠈⠁⠀⠀⠀⠀⡴⠁⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠉⠀⠀⢀⠎⢀⣀⠤⠊⠉⠉⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠶⠶⠤⠤⠤⠤⣤⡤⠄⠀⣀⣀⡸⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠶⠒⠊⠉⠁⠀⠀⠀⠉⠉⠛⠓⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡴⠶⠶⠶⠒⠛⠛⠛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⣇⣀⣀⣠⣄⣠⣥⣤⣿⣟⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠈⠀⢠⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⢠⠖⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⠤⡾⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣄⡀⠀⠀⠀⠀⢀⡴⠶⢦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠙⣦⠀⠀⠀⣸⠁⠀⠀⠷⡴⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠶⠞⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠲⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠖⠒⠊⠛⠛⠛⠛⠛⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⠠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⢸
⢰⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣼⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⡤⠤⣀⣀⡀⠀⠀⠉⠀⠀⠀⠙⣆⢀⣠⣤⢶⠶⢶⣤⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠉⠙⠳⠶⠇⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠸⠏⠀⠀⠀⠀⢸⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢠⡀⠀⠀⢰⡇⠀⠀⣠⠄⠀⢀⡴⠃⠀⠀⠀⠀⠀⠸⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⣯⡀⠀⠀⠀⠀⠀⠀⢀⡇
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠞⠁⠀⠀⠀⠳⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢀⣀⣤⣆⠀⢠⣄⣀⣀⣀⣀⣠⣽⠄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠛⠒⠛⠛⠋⠉⠀⣿⠀⢸⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⡤⠈⠂⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠋⠁⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢺⣍⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢢⡤⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢣⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠲⠎⠉⢳⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⡼⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢰⠲⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠁⠀⠀⠀⠀⠀⠀⠸⢠⣠⣤⠀⠀⠈⣇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡉⠉⢳⣎⣠⣃⣀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠃⠀⡁⠘⣉⠉⠳⡒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢋⠭⠒⠒⣏⠀⢀⡀⠀⡁⠀⣰⠯⠭⣍⣓⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢯⠀⠀⠀⠘⠢⠤⠤⠤⠖⠋⠀⢰⣂⣐⠆⣹⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⠙⠒⠶⠶⠶⠶⠶⠦⠤⠤⠤⣤⠀⡰⠊⢁⣀⡠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣟⠑⠭⢭⣓⣒⣒⣒⡒⠲⠖⢒⣒⣒⣒⣒⣾⠁⢻⡭⠭⠴⢚⣿⠉⢻⢲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⢶⣤⣤⣀⣀⣀⣀⣀⣀⣈⣬⣭⣭⣤⣤⣬⡇⡿⠴⠖⢒⣩⡟⠀⡇⡼⢠⢷
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⣄⡉⠓⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠗⠒⠋⠉⣠⡗⠺⣥⠇⣞⣼
⠀⠀⠀⠀⠀⠀⠀⠹⣗⢾⣉⠻⢭⣟⣒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠖⢒⣒⣩⣭⣛⣀⣠⣼⠾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠺⣭⣭⣛⣛⣛⣛⣓⣒⣒⣒⣒⣒⣒⣒⣻⣿⠭⠭⠭⠿⠛⠛⠋⠉⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣹⡈⣇⢣⢹⠉⣏⠉⠉⡏⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠾⠶⠶⠾⠾⠦⠶⠶⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⡤⠀⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠛⠛⠀⠀⠀⠀⢀⣀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⡄⠀⠀⠈⠙⠋⠈⠁⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⡇⡆⠀⠀⠀⠀⠀⡄⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢭⠧⣈⣀⣨⠤⠥⠤⢟⣀⣠⣤⠄⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠩⠋⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⣀⣤⣄⣀⣤⣤⣄⣠⡀⣳⡶⠞⠛⠲⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣉⣏⢀⣀⢀⣉⡈⠭⣿⣿⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⢀⡞⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⠞⠀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⡠⠂⠁⣀⡤⠔⠲⠖⠶⠢⡤⣄⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠅⠀⠀⠀⠊⠀⡴⢛⣧⡶⠞⢃⠐⠉⠃⠄⡈⠓⣄⠀⠀⠀⠀⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢀⠈⠀⢠⡞⠁⢪⡟⠀⠐⣵⠶⡄⢀⠄⠁⡀⠘⢷⡄⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠁⠀⢠⡟⢀⡴⢃⠀⠀⣰⠃⠀⠉⣠⡄⠀⠀⢂⠈⣶⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⣴⢠⡄⣀⢀⣾⡗⠋⠀⢀⠀⢈⠷⣈⣀⣤⣔⣸⡇⠀⠀⠀⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⣿⣿⡇⠀⠉⠙⠷⠀⠀⠋⠀⠘⠛⠉⠀⣸⡟⣭⢿⠃⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠈⠀⡿⠉⣿⠀⠀⣤⠀⠀⠰⡄⠀⠀⡄⠀⠀⣼⠃⣥⡾⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠠⠀⠀⠀⢿⠠⣾⠃⠀⠀⠀⠀⢀⡄⠀⠈⠁⠀⣺⡿⠊⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠸⣀⠋⠛⢻⠄⠀⠀⣸⡇⠀⠀⠀⠻⣿⠁⢠⣻⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠧⠀⠰⣧⣴⡴⠶⢿⣿⠶⠶⠲⠞⡋⢀⣀⣾⠟⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠸⣄⠐⠈⠠⠀⠤⠘⠿⠀⠀⠂⠀⠀⣿⢡⡟⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⠀⠀⠀⠀⢻⠤⢚⠤⠴⠶⠶⠶⠒⠒⠛⠋⠠⠟⡿⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠰⡀⠈⢧⣤⠐⠶⠢⡄⠀⠐⠄⡠⠘⢡⣼⠃⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣶⠀⠀⠁⢀⠈⠛⠲⠷⠦⢿⣼⠧⠴⠷⠾⠞⠁⠀⠀⢀⣤⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠳⣦⣄⢀⡀⠀⠀⠀⢀⠀⡀⠀⣠⠄⢀⣾⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⣀⣀⣤⡤⠀⡤⠄⠀⠀⠤⠀⢠⣤⣤⣀⣉⡛⠛⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢏⣛⠛⠛⠻⠷⢤⣠⣤⣄⣀⣠⣮⣵⡿⠿⠟⣋⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉



⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⠿⠿⠿⠿⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠛⠛⠛⠛⠛⠛⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠸⣿⣶⠶⢶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⣤⣀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⠉⠙⠻⣷⡀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⣻⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣿⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣀⣴⣶⠿⠿⠟⠻⠿⢷⣶⣤⡀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⠁⠀⠀⠀⠀⣠⣾⠟⠻⣷⣄⠀⠀⠀⠀⠘⣿⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡏⠀⢀⣤⣤⣤⣿⡇⠀⠀⣼⣿⣤⣤⣤⡀⠀⣿⡇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢸⣷⠀⢸⣿⠉⠙⣿⣿⠀⢀⣿⡿⠋⠉⣿⡇⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠘⠻⣷⣄⠈⠉⠀⠀⠉⢀⣤⣾⠟⠁⠀⣿⠃⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⠀⠀⠀⠈⣿⣧⣶⣿⣷⣦⣼⣿⠀⠀⠀⠀⠉⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣿⡇⠀⠀⠀⠛⠋⠁⣿⡇⠈⠙⠋⠀⠀⠀⢠⣄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣾⡇⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠈⠻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⢀⠂⠀⣀⠀⠀⠀⢠⣢⡘⠋⢈⠷⣣⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢃⡸
⠀⠀⠀⠀⠀⢤⣈⠂⠀⠀⠎⠁⣏⠹⢗⣋⢁⠙⠧⢔⡄⠀⠀⠠⠺⠇⢀⠀⠀⠀⠀⠈⠁⣠⣞⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠙⠁⠛⢣⣄⣓⢬⠄⣬⡒⠇⠀⠠⠀⠀⡀⠀⢉⠖⠀⠀⠾⢿⡟⠉⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢶⢄⣤⣼⢀⠁⠾⠧⠕⡀⡄⠀⠀⡀⣠⢴⠛⢋⠠⢂⠀⠤⣀⡄⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠡⠐⠂⢌⠉⠰⠆⠁⠀⠈⠉⢿⡁⠊⠁⠀⠀⠀⠀⠈⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠂⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢃⡄⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠀⠀⠈⠑⠂⠤⡀⠀⠀⠁⠈⠼⠀⠀⠀⠈⠙⠷⠤⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠢⣄⠉⠒⠄⣐⠂⠀⣆⡀⠀⢀⠀⣀⠂⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⠀⠄⡠⠀⠀⠤⠀⠀⠀⠠⢠⠐
⠀⠀⠀⠀⠉⠀⣠⣿⡟⠓⠚⢷⣾⡚⠛⠋⠙⠲⠀⠀⠀⠠⡀⣠⡆⠀⠈⢹⣦⡇⣀⠀⠂⠠⠠
⠀⠀⠠⠊⢀⠾⠛⣿⠷⠒⠂⠀⠀⠘⠯⠶⠲⢾⠆⠀⠐⣎⠂⠀⠀⠀⠀⠈⠨⠛⡧⢶⣠⣇⣰
⠀⠀⠀⠈⠁⠜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠁⢀
⠀⠀⠀⠀⠘⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠖⠦⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢡⣶⡀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⣦⠀⠉⠀⣐⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠉⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢐⠒⠒⠒⢦⡀⡴⠁⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣆⡈⢦⢹⡇⢸⠀⠘⣦⠀⠀⣸⠀⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠁⠀⠈⠉⠙⠻⢧⣸⡄⠀⣽⣤⠾⢃⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣴⠀⠀⠀⠾⠁⠀⠀⠀⠈⠑⣄⠻⠓⠚⠳⠶⠦⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠀⠀⣠⡤⠀⠀⠛⠃⠀⠀⠈⠡⠊⠀⣰⢂⠀⢻⡟⠛⠻⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠈⣧⠈⠦⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣴⠀⠀⢠⠆⡀⠀⠒⣃⠄⠀⠀⢚⡡⠂⠀⣀⠀⡇⢻⢿⣦⢄⣀⢙⡂⠄⠀⠀⠀⠀⠀⠀⠀⢀⡞⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡾⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠫⠂⠀⣿⠄⢹⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣤⣀⢸⠀⣿⡜⢉
⠀⠀⠀⠀⠀⠀⢸⠛⠃⠀⢞⠄⠀⠰⢆⡄⠀⠀⠻⠷⠂⠀⠴⢖⡠⠀⢀⠄⣼⠃⠀⠂⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠈⠻⣆⠘⡇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⠀⠁⢠⠶⣀⠀⠀⠀⢀⣈⣡⠴⣊⣵⠞⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠐⢦⣀⠀⠀⠀⠘⣧⣁⣾⣀⡀
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠘⠁⠂⠀⠀⠀⠨⢴⣒⣺⣭⠭⠶⠞⠛⠉⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢄⣠⠽⠛⠉⠀⠀⢀⠀⠀⠈⠻⡶⢬⡉⠉⠒⠒⣂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠶⠖⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡶⠊⠁⣀⠄⠀⣠⠶⠋⢀⠀⠀⢀⠹⡆⠈⡓⢏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⠝⢋⡠⠖⢋⣤⠶⠋⣁⣤⠖⠹⠀⠀⣸⣸⡿⠦⠤⠤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠁⢠⣶⣭⠶⢛⣩⠴⠒⢻⠁⠈⠁⠀⠀⢀⣵⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⡠⠀⡤⠈⠀⠞⠀⠀⣀⣀⣠⣤⠴⠖⠋⠉⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣤⣤⡤⠴⠶⠶⠶⠚⠛⠛⠏⠛⠒⠈⠉⣀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠴⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡿⠯⠥⢤⣱⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣸⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣀⡀⣀⠀⡀⠀⣶⠀⠲⠀⠐⢁⡠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⢲⠀⢷⠀⢸⠀⠀⡆⠀⠀⢨⢸⡁⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠒⠈⠚⠈⠊⠈⠊⠀⠉⠀⠈⠋⠑⠁⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⢤⣖⣒⣛⣛⣛⣛⣛⣛⣛⣒⣶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡇⠀⠀⢸⣤⣒⣒⠉⠛⠛⠛⠉⠑⣒⣒⣦⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠠⠭⠍⠀⡇⠀⣠⠖⠉⠉⠲⢤⣤⡤⠖⠋⠁⠀⠀⡇⠀⠀⢸⡀⠀⠉⠉⠉⠉⠉⠉⠉⢉⡁⠀⡾
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠥⠖⠀⡷⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣒⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣧⣄⣀⣀⣀⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠧⠤⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠈⠛⠒⠲⠶⠶⠖⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣭⣭⠿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⣿⠁⠀⠈⠹⠏⢹⣧⣤⣄⣤⡉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠡⣿⣶⣶⣤⡴⣣⡜⣿⠋⠀⠈⢡⣾⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⣶⣤⣤⣤⣤⣤⣾⣧⠘⣄⣀⣠⣟⣡⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⣾⣿⣿⣶⠶⠶⠴⢿⣿⣆⠘⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣽⣿⣿⣛⣒⡛⢛⣟⣛⣥⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠜⢷⣿⠿⢿⣿⣿⡿⠿⠿⢞⡓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⣴⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⠿⣿⣿⣿⣿⣿⣦⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⣼⣿⣿⣿⢿⣿⠿⠶⠛⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⡿⣿⣤⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⣾⣿⣿⣟⣽⠟⠁⠀⣀⠀⠀⠤⠴⠦⢤⣀⡀⠀⠙⢿⡟⠂⢻⣇⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⣿⡿⠁⠀⡀⠊⠁⠀⠀⢠⡀⠀⠀⠀⠙⠲⣄⠀⠻⡇⠀⣿⡆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣿⣿⡿⠁⠀⡜⠁⠀⠀⠀⠤⠈⠛⣶⠀⠀⠀⠀⠸⡂⠀⢻⠀⢹⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡇⠀⢠⠁⠀⢰⣦⠒⢾⠀⢰⠺⠾⢂⣶⠀⠀⠛⠀⠈⡇⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡇⠀⢸⠀⠀⠠⣭⡲⣤⡀⣿⢣⠔⢨⡿⠀⠀⢈⠀⠀⡇⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣷⠀⢸⡀⠀⠀⣀⡛⢯⣿⣾⣿⠇⣿⣄⠀⠀⣼⠀⢀⡇⢸⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸⣿⣿⣿⡀⠈⣇⠀⠀⠉⠁⠀⠈⠁⠈⠛⠁⠀⠀⢀⡟⠀⢸⠃⣸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣇⠀⢿⣀⡀⣀⣀⠀⠀⠀⠀⣀⣀⢀⣀⣾⠃⢠⡟⠀⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⣿⣿⣾⣿⡄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣾⠁⢰⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢹⣿⣿⣛⡿⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠃⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⣿⣿⣿⣟⣿⣿⠓⠒⠂⠀⠀⠀⠀⠀⢀⠈⠉⠹⠿⢷⣼⣿⡴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣿⣛⣛⣒⣒⣐⠒⠀⠀⠐⢂⣛⡒⣒⣒⣚⣫⣭⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆⣿⣿⠿⡍⣿⠿⠿⠿⡋⢹⡿⠿⠿⣏⢉⠿⠻⣿⢰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⠼⣿⣶⣦⣿⣄⠠⠤⡗⢾⠠⠤⠂⣿⣼⢆⣼⠫⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣛⣿⣉⣿⣿⠿⣃⠘⣷⠾⢛⣁⣻⣿⣧⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⢹⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡍⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠤⠤⣄⡀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠙⠲⢤⣙⠀⡧⠒⠉⠑⢶⣴⠞⢳⡀⠀⠀⣠⠔⠋⠉⠉⠉⣒⣦⣤⠉⠺⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡤⠛⢛⡿⠛⢀⡠⠀⠈⠀⢀⡀⠀⠠⠴⢿⣆⠀⢳⢀⡾⠁⠀⠀⢀⣴⣯⣥⣤⣤⣤⠀⠀⠘⠙⠓⠶⢤⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠰⣯⠴⢚⠏⢀⣦⠀⢱⣄⠉⢳⣶⣤⣄⣹⡆⢸⡾⠀⣀⠀⠀⠧⠀⠀⠀⢀⣞⡡⠚⡀⢸⡀⠑⢦⡄⣈⡒⢄
⠀⠀⠀⠀⢠⡏⠀⣟⡆⠀⠀⢸⣠⠏⢸⡀⢸⣨⠟⠉⠀⠉⠙⠫⣽⢸⠗⠉⠀⠉⠓⠂⠤⡀⠀⠟⢋⠀⠀⡇⢸⠈⢦⡀⠹⡌⢿⡉⠃
⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠸⠇⡴⢄⣳⠞⠉⡽⠋⢉⡠⠄⡀⠀⡀⠤⣀⠐⠒⠒⢖⡚⠻⡄⠀⢸⡄⠀⢸⣸⠀⠀⠙⢦⣿⠘⣇
⠀⠀⠀⠀⢸⡇⠀⠀⣠⢤⠀⠀⠀⠙⡼⠁⣀⠸⠷⠚⢹⠁⡰⣇⠀⠹⢦⡈⠻⡓⠶⢤⣵⡄⠙⣆⠀⠀⠀⠈⠿⠀⠀⠀⡶⠟⠀⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠘⠎⠀⠀⠀⢸⠃⠀⣯⠇⠀⠀⣾⡼⠁⢹⡀⢸⠀⠻⣦⡹⡄⠀⠀⠣⠄⢹⡆⠀⡌⠀⠀⠀⠀⠀⠘⠁⠀⡿
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡤⣾⠀⠀⠀⠀⠀⠀⠘⠁⡞⡆⠳⣼⠀⠀⠈⠙⠷⢠⠄⠀⠀⢀⡇⠀⠀⠀⢰⢳⠄⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⢹⡆⠀⠀⣶⢦⠀⠀⠀⠉⠀⠀⠉⠀⣖⢲⠀⠀⠈⠒⠃⠀⢸⡇⠀⠀⠀⠈⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⡵⠀⠀⠀⢿⡀⠀⠈⠃⠀⢰⢦⠀⠀⠀⠀⠀⠈⠁⠀⢀⣀⠀⠀⠀⡾⠰⣉⠀⠀⠀⠀⢰⢄⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⣀⠀⠈⣷⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⠃⠀⣼⠃⠀⠁⢀⠀⠀⠀⢈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠓⠀⠀⠸⣧⠀⢀⡀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⣰⣏⠀⠀⠀⢨⠇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠦⣄⣘⣧⠸⠜⠀⠀⠀⠀⠀⠈⠁⠀⠀⢾⡷⠀⣰⠏⢯⠃⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⣧⡀⠀⢄⠀⠀⢀⣀⠀⠀⠀⠈⠀⣴⠷⠤⠤⠖⠡⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠉⡀⠀⠘⠜⠀⢀⣄⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠾⠲⢤⣄⣴⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣟⢳⣄⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢷⣝⢦⡀⠀⠀⠀⠀⢀⡾⣏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠙⣯⣻⣆⠀⠀⠀⣼⠁⣿⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠘⢧⡈⢿⣽⣧⡀⣸⠁⠀⢸⠀⠈⡇⢀⡤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢀⡀⠓⢦⣿⣿⣷⡏⠀⡀⣾⣷⢰⣿⠋⡴⠁⠈⠉⣹⣭⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⣧⡁⡠⣌⢿⣿⣇⣷⣷⣿⣿⣿⠇⣼⠇⠀⣠⣾⠟⠋⠙⠳⠶⣦⠴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣶⣿⣿⣾⣿⣧⣿⣿⢟⣫⠁⣀⣈⠛⣯⣶⣖⣯⣶⣿⠗⠂⠀⣠⡔⠛⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣯⣷⣾⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣾⣿⠶⠿⠿⢿⣦⡀
⠀⠀⠀⠀⢀⣤⣶⣶⡟⠀⢰⠋⠁⣴⣥⡾⣻⣿⡟⣻⠏⣝⣿⡳⡝⢿⣿⣙⠉⠀⠂⠈⠉⢢⡀⠀⠈⠁
⠀⠀⠀⣴⣿⢝⣽⣿⡇⠀⢼⣀⠀⣿⣿⣿⢯⢿⣼⣿⣄⠈⠙⢳⡀⠀⢿⣿⣦⣴⣶⣿⣿⣶⣿⣦⣀
⠀⢀⣴⣿⡷⣋⣿⣿⣷⣴⡘⣜⡀⣿⣿⣿⠀⣾⠏⢠⠟⢦⡀⠂⢧⠀⣸⡇⠻⣶⠈⠡⠞⢷⣿⡭⢻⣷⣄
⠀⢾⣯⠵⢚⣿⣿⣙⡟⠻⠿⣮⠇⣿⣿⡸⢸⡏⠄⡀⠈⠀⠛⣆⠘⠀⡟⠀⠊⠁⠰⠆⠳⠌⠠⢞⣹⡿⠹⣆
⠀⠀⠀⠀⣾⠓⠮⠐⢇⠁⡒⠝⢷⠟⢻⣧⣸⠃⠁⣿⡆⠀⠀⢹⡄⢸⡗⠂⣿⣷⠀⢸⡇⠴⠾⣿⢯⡩⠽⠻⣆
⠀⠀⠀⣸⣿⠵⠠⠆⣾⠀⣩⡄⠙⠻⠋⠿⠏⣿⡆⠘⠁⠀⠀⠀⡙⠻⠀⢀⠁⠁⠈⢀⡀⠀⢶⣯⡀⢂⣴⡀⣿
⠀⠀⠀⡇⠐⠊⠈⢷⠄⡌⢧⣿⠀⠦⠀⠀⠀⠉⠀⠀⠐⣶⡄⠀⣁⣀⠀⠿⠇⠀⠀⠀⣄⠀⢬⡷⠢⣗⡻⠂⢻
⠀⠀⠀⣿⣾⣖⣛⠃⢨⣀⠀⠀⣠⣶⡄⠀⠀⣶⢶⠀⠈⣷⠇⠀⠘⠋⡀⠀⠀⢂⡀⠀⠛⠀⠈⠔⠃⢖⣴⡄⢻
⠀⠀⠀⢸⠀⠲⡜⢳⢸⡅⠀⠀⢿⣿⠀⠀⠀⠹⠾⠀⠀⠀⠀⢰⡀⠀⢸⣧⠀⠿⠇⠀⡄⠀⣦⡭⣀⡐⠉⠩⢼
⠀⠀⠀⠘⣧⠀⠾⠤⠀⢡⠃⣰⡄⠀⠀⣴⡄⠀⠀⢘⣷⠀⠈⡾⠃⠀⠀⠀⠀⢀⠀⠀⠠⡏⠕⢂⣛⢠⣶⢠⡗
⠀⠀⠀⠀⢿⣀⡄⢉⢲⢶⣃⣿⡷⠀⠈⢷⣿⠀⠀⠸⠿⠁⠀⠀⠀⣿⠁⠀⠘⢿⠃⠀⠀⠀⠀⣸⢂⡄⣁⢰⡇
⠀⠀⠀⠀⠈⢿⡿⠀⠘⠾⣖⡀⡀⢴⣦⠀⠀⠀⣂⠀⠀⣸⣧⠀⠀⠀⢠⣄⠀⠈⠀⠀⣰⡄⢀⣁⠈⣱⠇⣾
⠀⠀⠀⠀⠀⠈⢷⡀⣶⡔⠈⣴⡧⠘⠿⣠⠀⢰⣟⡇⠀⠻⠼⠀⠀⠀⠴⠋⢀⣰⡆⠈⠉⢡⠎⠙⠀⣈⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠻⣎⣡⣖⠙⡿⠃⠀⣽⣷⠈⠙⠁⣴⡄⠀⠀⣼⡆⠀⠀⠈⠙⠃⣀⣠⡕⡐⡁⣸⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣏⢱⡄⢰⠔⢈⣩⣦⠀⠀⠿⠟⠀⠀⢉⠁⠀⢠⡦⠀⡀⣛⡶⠰⠷⣄⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡓⢎⠀⣤⠚⠟⠀⠀⠀⣦⠀⠀⢸⣧⠀⠛⠃⢀⡃⣠⣀⣦⣌⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⡄⠻⣅⢀⣽⡆⠀⠙⠁⠀⠀⠀⠀⠀⠠⠙⡁⠻⢁⣛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣣⡂⠻⠯⠀⣸⣷⠀⣸⣧⠀⣷⡴⠛⠀⠿⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣶⣦⠀⢈⡋⠀⢨⠅⡜⠉⠤⠰⣨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⡀⠠⠀⢠⣾⠀⡀⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠦⢾⣥⠶⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⢄⠒⠂⠉⣉⣉⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣥⢀⣀⣀⡀⠤⡇⣰⠉⠀⠀⠀⠉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠚⠂⠦⠤⢤⡚⠠⣏⠀⠀⠁⠀⠐⠀⠀⠈⠢⡤⠤⢀⣀⡀
⠀⠀⠀⠀⣀⡠⠔⠒⠉⠉⠁⠀⡰⠁⠀⠀⠠⠀⠄⠀⠈⠙⣌⠢⡀⠀⠁⠄⠀⠠⠄⠀⢸⠀⠀⠀⠀⠉⠉⠒⠢⣀
⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢂⠄⢀⠀⠁⠀⢀⡿⠢⣈⠑⠒⠂⠠⠤⠤⠔⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⠀⠀⠀⠤⢔⡪⠕⠁⠀⠀⢉⠗⠒⠀⠠⡔⠚⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⢣⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣔⠤⢑⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢆
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⡇
⠀⠀⡌⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡆⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣀⠔⠊⠱
⠀⠀⢰⡀⠀⠀⠈⠉⠓⠠⠭⢍⣈⡁⠉⠉⠑⠒⠣⠬⠬⢰⣁⣀⣀⣀⣀⡀⠠⠤⠤⠔⠒⠉⠁⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⡰
⠀⠀⡎⠈⠓⠤⢄⣀⣀⢀⠀⠀⠀⠀⠉⠉⠁⠚⠒⠐⠒⠒⠒⠂⠀⠤⠤⠤⠤⠤⠀⠤⠤⠤⠔⠒⠊⠉⠀⠀⠀⠀⡠⠒⠉⡇
⠀⠀⡱⣄⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠔⠊⠀⠀⣴⠁
⠀⠀⢇⠈⠑⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⠒⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⣀⠤⢊⡎
⠀⠀⠀⣱⢤⡀⠀⠈⠑⠢⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠤⠤⠤⠤⠤⠤⠀⠒⠒⠈⠉⢀⡴⣋⠆
⠀⠀⡎⠸⡀⠙⢕⠢⢀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡶⠁⠀⠈⠆
⠀⠀⠱⢄⡙⠢⢄⠉⠒⠬⠕⣲⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠒⠊⣁⠔⠁⠀⠀⠀⣤
⠀⠀⢠⠀⠈⠑⠂⠬⣑⡂⠤⢀⣈⠉⠉⠛⠃⠭⠄⠈⣀⣀⣀⣈⣀⣈⣉⣉⣉⠁⠤⠄⠐⠒⠒⢂⡠⠋⠀⠀⢀⡀⠔⠊⠀⡇
⠀⠀⠀⢦⠀⠀⠀⠀⠀⠈⠁⠒⠢⠬⣉⡁⠒⠒⠒⠐⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⢀⠔⠁⠀⢀⡠⠞⡃
⠀⠀⠀⠈⠳⡢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠠⠤⠤⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡤⠔⣁⠤⡴⠚⠁⠀⡠⠁
⠀⠀⠀⠀⠀⠹⣑⠒⠩⢄⣒⡒⠠⠤⠤⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠒⠂⣁⠔⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠉⠉⠁⠒⠒⠠⠤⠭⠬⠩⣭⡍⠉⢉⣉⣉⣉⣉⣉⣉⣁⣀⠠⠤⠒⠉⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠐⠒⠂⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠭⠭⠭⠭⡽⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢫⠥⠴⠤⡉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⡒⠒⠶⠶⠚⢆⠀⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠈⠀⠀⠀⢸⠀⣠⠞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⡠⠔⠒⠒⠢⣄⠀⠈⢌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⠊⠀⠀⡀⣀⠀⠈⠳⡄⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡰⠁⠀⠒⠀⠀⠐⣒⠀⠀⠘⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣸⠁⣀⠀⠄⢀⡀⣀⠀⠀⢠⣄⡸⡄⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢡⠇⠀⠒⠀⠁⠈⡍⠉⢦⣀⠎⠀⠀⠹⡐⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠰⠀⡸⢀⠑⠠⣀⠀⠻⡀⠀⡀⢀⠀⠰⡸⠀⢇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡆⠀⡇⠀⠈⠐⠀⠈⢡⡇⢀⠰⡌⠀⢀⠇⠀⢸⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠄⠀⠀⠀⠈⡋⠁⠀⣀⠓⠇⠀⡜⠉⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠈⠢⣀⠈⠁⠒⠂⠤⡁⢦⡜⠤⠆⠀⢸⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠸⣄⠀⠀⠈⠀⠒⠒⠒⣇⡠⠤⠤⣀⣠⠇⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠒⠒⠶⠶⠤⠤⠎⠴⠖⠒⠊⠁⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡶⠶⢶⣶⡶⠶⠾⠿⠿⢿⢿⡿⠿⠿⠿⣷⢶⠶⠶⠶⠶⢶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⣿⢸⡀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡏⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣀⣀⣧⣷⣀⣀⣀⣀⣸⣠⣇⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣿⣸⣀⣀⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠉⠉⠉⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡿⣤⣀⡀⣀⣀⣀⡤⡞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠳⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣬⡉⠉⠉⠉⣁⣠⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠹⣄⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠙⣦⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠈⢧⠈⣿
⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠈⢳⣽⡇
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⡤⠖⠛⠓⢦⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⢠⣿⡀⠀⢀⡞⠁⠀⠀⠀⣸⠃⠀⠀⠀⡟⠀⠀⠀⠹⣄⠀⠀⠀⣴⠖⠒⢦⡀⠀⠀⠀⣿⠷⠦⣄⠀⠀⢠⠶⠒⢦⡀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⡟⠙⠛⠉⠀⠀⠀⠀⠀⠻⣄⠀⠀⣠⠇⠀⠀⠀⠀⠙⠲⠤⠞⠃⠀⠀⠀⣇⠀⠀⢸⡏⠀⢀⡟⠀⠀⢸⠀⠀⠀⡇⢀⣼⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⣻⡇⠀⢸⣇⠀⣀⡼⠀⠀⠀⠉⠉⣿⠃
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⣿⢿⣆⣴⣾⡆⠀⣀⠀⠀⣿⠇⠀⣿⠀⠀⣀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠘⣿⠈⠛⠁⢸⣇⠀⣩⡀⠀⣿⠀⠀⣿⣴⣿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠹⠇⠀⠀⠸⣿⠀⢿⠇⠠⡿⠀⠀⣿⠉⠻⣦⡀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣿⣀⣤⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠶⠞⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠞⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⢸⡆⠀⡇⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣾⠀⠀⢸⠀⠀⣿⠀⢸⡇⠀⢸⣀⣀⣧⣤⣧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣸⣀⣠⣽⣤⠶⠷⠛⠛⠉⠉⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⡖⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠹⣄⠀⠀⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢀⡼⠃⢀⣀⣤⡤⠴⠶⠚⠛⠉⠁⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢠⡿⠷⢿⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠿⣦
⣿⠁⠀⠀⠀⠈⠉⠛⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠛⠋⠉⠁⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⢤⣀⣘⠆⣀⣀⣤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⢸⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣴⡄⠰⡟⣷⠞⣹⠆⠸⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⢸⠀⠀⢰⠲⡆⠀⣾⣆⠈⣏⢿⢻⡘⡇⠀⣷⠀⢠⣏⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⡆⠀⢸⠀⠙⢶⠇⠹⡆⢻⢸⣼⡇⣿⣀⣽⠀⣶⣝⢦⡀⢿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢀⠀⢀⡀⢹⣼⡎⣿⣧⣈⣠⣿⣧⡽⠘⠾⠁⢸
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠀⡟⣧⢘⣿⣦⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠚⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠠⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⣿⠀⠀⣒⠦⣭⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⢠⣭⣟⡒⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠈⡇⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⢹⡀⠀⢇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⣧⡀⠈⠳⢦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞
⠀⠀⠀⠀⠀⠉⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⣤⡴⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣄⣀⠀⣸⣤⡶⠞⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠙⣏⠐⠀⠃⠁⠀⠀⠀⠄⠸⠧⣄⣀⣿⡵⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠶⠝⠛⠀⢠⠈⠑⠒⠂⠁⠀⠠⡽
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠷⠤⠄⠈⢀⣄⠃⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣯⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢀⣵⠟
⠀⠘⠀⠀⠂⡀⠀⢀⠀⠀⠀⠀⠠⠂⠁⠂⠀⠀⠀⠀⠉⠚⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⢀⡼⠁⠐⣀⠤⠒⠋
⠀⠀⠀⠠⣶⢋⣀⣠⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠁
⣾⡆⠀⢠⡕⣎⢉⡀⠀⠀⡠⠀⠄⠀⠀⠁⠀⠀⠀⠀⠰⠿⠀⠀⠂⠉⠘⠛⠀⠁⠂⠀⡆⠀⠀⠀⠀⠀⠀⠀⢰
⠚⣁⢀⣐⣾⠛⢷⣶⠾⣎⣀⠀⠐⠄⡀⠾⠷⠒⠂⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⢀⡄⠀⠑
⣄⡨⢫⠏⠸⣆⣼⠁⠀⣘⢽⣇⠀⠀⠀⠀⠠⠀⠀⡡⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⢠⡽⠂⠀⠀⢠
⠉⢁⣿⠟⡓⠀⠉⠀⠀⠅⣼⠇⠀⠤⠄⣴⠀⣰⠏⠀⠀⠸⠀⢀⡀⠀⠂⠠⠔⠀⠀⠀⢀⡀⠀⠁⠀⠀⠁
⢠⠋⠀⠀⠀⠀⠀⠡⠤⠞⠁⠀⠀⠀⠰⢷⢸⣏⣠⢖⣹⡷⠚⠭⡛⢷⡄⠀⠀⠀⠈⠂⡀⠈⢠⡄⠀⠀⠆⠀⠸⡶
⠁⠀⠀⣠⣴⠉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠶⢸⣿⠙⠻⡆⠀⠀⢆⠸⡇⠀⠀⠇⠀⠀⠘⣤⠀⠠⠎⠀⢀⠀⠀⠀⢀
⠀⠀⠙⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⢷⣦⠀⠀⣭⣏⠳⠶⠝⠂⠄⡀⢈⠇⠀⠀⠘⢦⡀⣀⣈⣿⡟
⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢐⠊⠀⠙⣛⢻⣤⡞⠂⠀⠀⠀⠁⠀⠀⠀⠂⠈⠈⠁⠉⠙⠛⠒⠊⠀⠀⠀⠙⠃
⠀⢸⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠉⠁⠀⠀⣠⠾⠀⢹⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠀⠂
⡕⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠂⠀⠀⠀⠀⠀⡠⠀⢸⡀⠀⢷⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⣤⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠪⣟⢳⣦⠀⡇⠀⠀⠣⡀⠀⠀⢀⣼⡃⡀⠀⠀⠀⠈⠁⠀⠐⠀⠀⢠⡴⠞
⢀⡴⠂⡀⠀⠀⠠⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠈⠥⠁⠀⡀⠀⠀⠀⠢⠄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠁⠀⠀⡇⠀⡀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡤⡀⠀⠀⠐⢀⠄⢀⠠⠤⣴⠀⢠
⡤⢂⣾⠖⠺⢛⠳⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠎⠀⠂⠀⠀⠈⠁⠀⠀⠀⠲⣾⣸⣏
⡟⠛⢿⠀⠀⠸⠁⢿⠀⠀⠀⠀⠄⠀⠠⡀⠀⠘⠛⠀⠀⠀⠀⠋⠀⠀⠀⡴⠁⠀⠀⠀⠀⢠⣤⠀⠀⠘⢀⠀⠀⡾⢹
⠳⠦⠬⠶⠠⡀⠀⣹⠀⠀⠆⠀⠐⢦⠀⢠⠄⠠⠀⡄⠀⠡⠁⠀⠀⠀⣿⠃⠀⠠⠈⠀⠀⠀⠉⠀⠀⠀⢷⣭⠄⠀⣉
⠇⠴⠲⠀⠀⠀⠈⠁⠀⠀⠆⠀⠀⠸⠆⠨⠠⠂⠀⠂⠀⠀⠀⠀⠀⠘⠋⠈⠀⠀⠐⠀⠠⠀⠀⠠⠀⠀⠡⠸⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠶⠶⠒⠚⠛⠛⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣂⣀⣀⡀⠤⠤⠤⠤⢼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠁⠀⠀⠸⣄⣀⠀⠀⠀⡔⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠂⠀⠁⠀⢸⠆⢀⡠⠛⠉⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡋⠸⠉⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⢀⣠⣀⠀⠓⢁⣀⣡⡤⣤⠤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠉⠁⠀⠀⠨⠉⠉⠀⠘⡀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠑⠊⠀⠀⠀⡇⠀⠀⠀⠛⠲⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠤⣄⡀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⢄⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠋⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⡀
⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠪⡉⠉⠀⠀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠹⡄
⠀⢠⠏⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⣃⣨⡀⡠⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢃⠀⠀⠀⢻⡀
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠈⣧
⢸⠁⠀⠀⡀⠀⠀⣀⡀⢠⠲⠀⠀⠖⠉⠑⠐⠍⣉⠁⣶⠓⠂⠒⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢹
⡟⠀⠀⠀⠁⠀⠀⠇⠀⠆⢀⠀⠄⠀⠘⡃⠀⡀⢩⠤⠟⠸⢄⡄⢨⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⢲⡀⡏⢀⣀⡧⠴⠑⠆⠻⠞⠀⠐⠀⠴⠧⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⣇⠀⠀⠂⠀⠀⠀⠓⠃⠉⠁⠀⠀⠀⡀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⢻⠀⠀⠀⠀⠀⠀⡤⠒⢒⡆⢮⡁⠚⢉⠎⠁⣄⠀⣡⡎⢦⠖⢲⠋⢉⠑⢂⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠺⠃
⠸⡄⠀⠀⠀⠀⠀⠢⢄⡈⠑⠀⢹⠆⡌⠀⠀⢠⠘⠣⡇⠀⠀⣼⠀⡬⠤⠞⠀⠀⠐⠂⠀⠀⢰⠂⠀⠀⡶
⠀⢷⠀⠀⠀⠀⠀⢘⢉⣀⠴⠣⣅⡼⠁⠸⣀⡄⠑⠖⠉⠒⠒⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠃
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢠⠏
⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⢸⠄⠀⠀⠀⠀⠀⢀⡎⠀⢀⡞
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⡤⢤⠀⠀⢀⠀⠀⠀⠀⢠⠎⠀⠀⠘⣧⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠳⣦⠄⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⢀⡴⠁⣀⡤⠔⠊⠀⠀⠀⠀⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠶⠤⠤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣰⡯⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⣀⣀⠤⠴⠒⠚⠛⠉⠉⠙⠓⠲⠶⠦⠤⠴⠶⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⡒⢲⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣉⣉⣉⣉⣁⣉⣉⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠛⠻⠿⠿⠿⠿⠿⠷⠾⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠿⠿⠿⠿⠿⠿⠿⠷⠾⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠦⠀⠀⠐⠶⠶⠶⠶⠶⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣧⡀⠀⣀⣤⡄⠀⣤⣼⢀⣴⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⢹⣿⡇⠀⣿⣿⠸⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠇⢰⠀⢸⣿⡇⠀⣿⣿⡄⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⣿⠀⢸⠀⣿⣿⡇⠀⢸⣿⣿⢨⡉⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢛⣾⣿⠀⡄⢀⣿⣿⡇⠀⢸⣿⣿⡼⣇⠀⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣼⣿⣿⠀⠇⢸⣿⣿⡇⠀⢸⣿⣿⡇⢿⡄⠀⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣰⣿⣿⠇⡆⠀⠈⠛⢿⡇⠀⠸⣿⣿⣿⡘⣿⣄⠰⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠃⢀⠃⠀⠀⠀⢸⡇⠀⠀⠙⢿⣿⡇⠻⣿⡷⠆⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣭⣤⣤⣤⣿⣤⣀⣀⣀⣤⣶⣄⣀⣀⣀⣀⣿⣦⣤⠠⡶⠿⠿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣛⣛⣛⣋⣩⣿⣿⣿⡿⠯⣍⣹⡛⠉⠩⣭⣽⠟⢿⣿⣿⣿⡿⠟⠓⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⠟⠋⣉⣿⣿⣋⣴⣤⣴⡶⠊⠀⠙⣿⠗⠈⠳⣦⣬⣿⣿⢿⣿⣿⠿⠛⠺⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠟⠛⣹⡿⢁⣴⡟⢁⣈⠉⠁⢾⣿⣋⣀⡹⠷⠂⠙⣅⡀⢱⣦⣸⣿⣿⣿⠶⠾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⢛⣿⢿⠟⢉⡽⠃⣹⣿⣾⡿⠿⠛⠛⠛⠿⢿⣷⣦⡙⣷⡞⣭⡉⢉⡹⣿⣿⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⣩⡟⣥⣿⡷⠻⣷⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⢿⡿⢺⣷⠾⣿⣶⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⢛⡟⠁⢻⣾⣀⣾⡟⠁⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣿⣧⡜⢿⣦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⣿⣸⣷⠚⠙⢿⠉⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⣴⣿⣿⡷⠀⠀⠀⠀⢽⣿⣿⠿⠟⢿⣿⣧⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⡟⠁⣼⡄⠹⠀⠀⠀⠀⠀⣀⡀⢀⣻⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠻⣿⡔⢷⣼⠙⣿⡉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⡟⣧⡰⠿⠻⡆⠀⠀⠀⠀⣼⣿⡇⠘⢿⣿⣿⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠙⣷⢠⡟⢳⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⡇⠙⡇⠸⣇⠀⠀⠀⠐⣶⣿⣿⣿⡖⠈⢻⡿⣿⡿⢿⣿⠟⠋⠁⠀⠀⠀⠀⠸⢻⣿⢧⣦⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⢇⠀⣿⡆⠉⠀⠀⠀⠀⠙⣿⣿⡿⠇⠀⡸⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣀⢻⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢾⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⢿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣷⡈⢡⣄⢻⠀⠀⣤⣤⣤⣄⣤⣄⠀⣤⣤⡄⠀⢠⣤⣴⣤⠰⣿⠆⣠⣤⣤⡀⠀⢸⣿⡇⣿⣇⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣿⠃⢼⠋⢸⠀⠀⣿⡏⢸⡿⢸⣿⢠⣶⣿⣷⣤⢸⣯⣸⣿⠀⣿⠆⣿⡷⢻⡇⠀⢸⣿⣿⠏⣿⣿⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣇⣼⠉⠳⠀⠀⠈⠁⠈⠃⠈⠋⠀⠙⠋⠙⠉⠸⠿⠉⠀⠀⠿⠆⠙⠛⠛⠁⠀⣾⣿⠉⣼⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⢻⡿⠁⢸⡆⣿⠀⢰⣿⣛⠂⠺⣷⣀⡾⢰⣿⠾⣷⢴⣿⠀⣾⡆⢰⣿⠿⣷⡄⠀⣿⣿⡾⣛⣿⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠻⣿⣀⡇⠠⠀⡆⢠⣍⣿⡿⠀⢻⣿⠃⢸⣿⡇⠁⠘⣿⣦⣿⡇⢸⣿⣤⣿⠃⣸⣿⣅⣴⢿⣿⠛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡷⣯⣝⠿⣤⣤⠁⡄⠀⠀⠐⣷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⢰⣿⣿⢿⡏⣼⡿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣣⣼⣿⣄⣽⡉⣆⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣿⡁⣼⣷⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⢿⣏⢻⡷⢿⣦⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣩⡿⠿⠏⢡⣿⠋⣡⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣝⣛⣿⣿⡿⣶⡤⢶⣌⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⣴⡟⣠⣾⣷⣿⠷⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣯⡉⠻⣷⣈⣣⣈⣽⠂⣨⡐⠂⣀⠀⠀⠀⠀⠀⢠⡔⢒⣩⡶⠿⣿⠗⣭⣽⡿⣡⣴⠾⣫⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣍⣀⣈⣿⣿⡟⠻⣷⣬⣿⡿⠒⢶⣦⣬⡵⠶⣶⣾⣻⣥⣤⣾⡿⢾⣿⠏⠘⠛⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡛⠻⠍⠉⠛⢿⣄⣹⣏⡈⠻⣷⡦⣉⣭⣿⣿⣿⡿⣫⣽⣷⣿⣶⣟⣥⣤⣶⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣶⣶⠀⠀⢈⣿⣿⣿⣷⣉⣲⣿⣿⣷⡛⢛⣾⣿⣿⠟⠋⣀⣩⣽⡿⠿⠛⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡜⠛⢻⣶⣾⣿⣟⣙⣻⣿⣿⣿⣿⣿⣿⠿⣛⣻⣿⣿⣏⣉⣠⣤⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣾⣯⣍⠉⠯⠛⠛⠛⠛⠛⠛⠛⠛⠛⢯⣭⣭⣟⣛⣿⣯⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠶⠲⠶⠖⠲⠶⠖⠲⠒⠚⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⡴⠋⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⣷⣿
⠀⢸⡿⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⢟⣭⣴⢿⣿⢿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⠿⣻⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢽⣶⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣏⣽⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⢀⣄⣀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⢀⣾⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⡇⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠻⣷⡀⣼⡟⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⣷⣾⣿⡋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⡄⢻⣶⣿⠁⣿⣿⠀⠀⢿⣿⡇⠀⢸⣿⣇⠀⠀⠀⠀⢸⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠇⠈⠻⠃⠀⣿⣿⠀⠀⢼⣿⡇⠀⢸⣿⣿⣶⣶⣶⡆⢸⣿⣇⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠈⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣧⣧⣿⣿⣿⣿⣿⣿
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠙⠉⢉⣭⣍⡙⠿⢶⣶⣶⣶⠶⠋⠉⠻⢿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠯⡭⠤⠤⣤⣀⡘⠻⣿⣇⠀⠈⠛⠛⠋⠠⣿⣆⢠⡀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⢡⣿⠇⢠⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⣄⠀⣿⣀⠀⠀⣀⣠⣤⣤⣀⡀⠈⢆⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢻⣿⣿⣿⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡄⣿⠀⠀⠉⠁⠈⠛⠛⠛⠙⣦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⢻⣻⣿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢡⠟⡀⠀⠀⠀⠀⠀⠀⡧⠞⠀⡿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣾⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣤⣤⣤⣿⣧⣼⣿⣥⣤⣤⣤⣤⣤⣼⣿⣥⣤⣷⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣵⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠹⣧⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⠀⠀⢠⣾⣿⣷⣶⣶⣶⣶⣿⣿⣿⣷⣶⣴⣶⣾⣿⣿⣿⣦⣤⣴⣾⣿⣿⣿⣶⣤⣤⣤⣶⣿⣷⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⢸⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢸⣿⣿⣿⣿⡿⠋
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢦⣤⣤⣀⠀⠀⢀⣴⠞⢻⣇⠀⠀⠀⠀⢀⣠⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⠉⠛⢦⣾⡃⠀⠀⢻⡆⠀⣀⣴⠟⠋⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠹⣷⡀⠀⠀⠀⠠⣈⢧⡀⠀⠀⣧⢞⣩⡴⠞⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠤⡄⠈⠛⢦⣄⣀⣀⣀⣑⣷⣴⣴⡷⠿⠷⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠳⠾⠃⠀⠀⣠⣾⠿⠋⠉⠭⢿⣿⣿⠿⣄⠉⠀⠀⠉⠻⢷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡔⢲⡄⠀⠀⢠⣤⣴⡿⠋⠁⠀⠀⢀⣠⡾⣿⡀⠑⢀⠑⢦⣤⣤⣤⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠘⠗⠋⠀⠀⠀⣀⡈⠛⠿⠷⣶⣶⠿⠟⠋⠀⠹⣷⣄⠀⠁⠈⢿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣞⢁⠇⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠘⠻⢿⣶⣤⣬⣧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠒⡄⠀⠀⠀⠀⠙⠉⠀⠀⠀⣼⠁⣽⠀⠀⠀⢰⡏⣹⠀⠀⠀⠀⢩⡤⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠻⠾⠃⠀⠀⠀⢀⡔⢲⡀⠀⠀⠘⠛⠁⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⢸⣇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⠀⠀⠸⣷⡾⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⣠⠒⡄⠀⠈⠉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣾⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣠⠇⠀⠀⠀⠀⣷⡴⠃⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠛⠁⠀⠀⠀⣠⠤⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⣀⣄⠀⠀⠀⠀⠀⣯⣴⠇⠀⠀⣠⠴⢦⠀⠀⣰⠊⣩⠂⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⡃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⠞⠀⠀⠙⣋⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡉⠉⠀⢀⠔⠆⠀⠀⢀⡴⢲⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⢿⡴⠃⠀⠀⠸⠶⠞⣀⣤⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣤⣤⣶⣶⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣏⣩⣬⣩⣿⣹⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡤⣄⠀⠀⠀⡠⢤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠃⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣤⠀⣀⠈⢧⠀⠐⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⣞⠀⠀⠀⠀⠀⠱⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠍⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣠⡍⠠⠇⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠤⠀⠀⠐⠉⠀⠀⢀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⢘⠂⡢⠀⠀⠀⣇⣸⠁⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⠀⠀⠔⠀⠀⠀⠀⠀⠈⠱⠀⠀⠄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⠀⠀⢠⣀⠀⠁⠀⡀⠀⠀⠀⢀⣀⠀⠀⠄⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⠀⠀⠀⠚⠋⠀⠀⠄⠁⠀⠀⠀⡀⠀⠀⠠⠀⢠⠀⠀⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⠀⠀⠎⢀⡀⢀⠀⠆⠀⣀⣀⠇⠀⠀⠄⠂⠀⠒⠐⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⢀⠔⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠶⠒⠒⠚⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠶⠖⠒⠚⠛⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠦⠔⠒⣖⠀⠀⠀⠀⡛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠈⠙⡆⠀⢀⠧⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢐⢡⠚⡇⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢀⠤⣀⠘⢊⣀⣣⠤⠤⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠬⠁⠀⠈⡀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠉⠀⠀⠀⣇⡀⠀⠈⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⡀⠀⠀⠉⠲⢄
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠱⡄
⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠲⡙⠁⠈⢩⠙⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢄⠀⠀⠘⣆
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢰⡡⢤⣀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢂⠀⠀⠸⡄
⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠤⢀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢷
⡜⠀⠀⠀⠀⠀⠖⢀⠋⠁⠀⠁⢤⠀⠀⠲⢀⡇⡈⠈⠀⡼⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⡄⠰⢆⠐⣆⡰⢄⠸⣬⠍⢄⢈⣿⣀⣱⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠖⠁⠚⠀⠉⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⢧⠀⠀⠀⠀⠀⠔⠒⣢⠲⡁⠊⠱⠋⢠⠀⣡⠊⠴⢲⡎⢉⠑⡄⠀⠀⠀⠀⠀⠸⠀⠀⢸⠃
⠸⡀⠀⠀⠀⠀⢓⡤⠈⠂⡸⢂⠀⡀⠠⡌⠑⡆⠀⣸⠃⡖⠐⠃⠀⠀⠁⠀⠀⡏⠀⠀⡏
⠀⢳⠀⠀⠀⠀⠠⠐⠒⠁⠓⠊⠀⠓⠒⠈⠉⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣸⠁
⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⢠⠀⠀⠀⠀⠀⠀⡌⠀⢠⠃
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⣀⡀⠈⢁⠀⠀⠀⠀⠜⠀⠀⢯⣀⣀
⠀⠀⠀⠀⠀⠙⠲⡤⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⡠⠊⣀⡠⠔⠉⠀⠉⠉⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠶⠦⠤⠤⣤⠤⠤⠤⠤⠤⠼⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠓⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠄⠀⠠⢄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠛⠃⠀⢰⡁⠀⠀⠀⠀⠱⡄⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠋⠹⠲⡆⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢉⠏⠱⡀⠑⠒⠦⢄⡀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠴⠃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣤⡎⠀⠀⡈⡐⠠⢀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⠵⢮⣤⠈⡀⠈⠀⠀⠁⠀⠨⠱⠄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡴⠃⠀⠀⠀⠉⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣋⠀⠀⠸⠀⠠⠐⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠠⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⢀⡞
⠀⡠⠖⠉⠉⠙⠲⡀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢰⠒⠊⠀⠀⠀⠀⢠⠎
⡞⠀⠀⠀⠰⠉⢀⣇⣀⣠⠴⠚⢅⡠⣆⠐⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠧⠄⠠⠔⢢⠀⢀⠇⠀⠀⠀⠀⠀⣠⠏
⣇⠀⠀⠀⠀⠀⠀⠁⠀⣀⢴⡴⠋⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⢠⠏
⠈⢦⣀⣀⣀⣀⠤⠔⢊⡵⠋⢠⠞⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠏
⠀⠀⡠⠋⠁⠀⠀⢠⠟⠀⠀⣏⠘⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠤⠤⠤⠤⠤⠖⠚⠉⠙⠁
⠀⢰⠇⠀⠀⠐⠀⠙⢆⠀⠀⠈⠉⠉
⠀⠘⡇⠀⠀⠈⠀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠃
⠀⠀⠀⠀⠀⢀⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠒⠀⠀⠀⠀⠳⣀⣠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠺⠀⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠙⠂⠠⠤⣤⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠀⠀⠀⠈⢿⠛⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠿⠉⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠂
⠀⠀⠀⠁⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠼⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡠
⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠢⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡼⠁
⠀⠀⠀⠀⠀⠀⣒⠀⠉⠀⠠⠤⠤⠤⠶⣤⣿⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠰⠂⠀⠠⠀⠐⠋⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢀⠀⠀⠀⠠⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠈⠀⠆⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠈
⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⢄
⠀⠀⠀⠀⠀⠈⠉⢳⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣈⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⡞⠉
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠖⠢⠀⠐⠊⣿⠃⢠
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠴⠄⠈⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠂⣀⡠
⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠃⠄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠳⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠀⠰⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡲⠀⣀⠀⠀⠀⠈⠀⠱⠀⠀⠀⠠⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⠒⢻⠷⣄⡈⠀⢐⠻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⡄⠀⠤
⠀⠀⠀⢠⢦⡤⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠐⡶
⠀⠀⠀⠀⠗⢦⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠈⠓⠠⡄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⢠⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣦⢀⣀⡀
⠀⠀⠀⠀⠀⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠐⠠⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠻⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣀⡤⠀⠘⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠴⣠⣿⠖⢲⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣤⣄⣟⠃⠀⠀⠀⠀⠀⠀⢤⣴⣄⣁⡀
⠀⠀⢠⣍⠀⠀⠀⠀⠂⢿⠀⠀⠀⠀⠀⠀⠐⠰⠁⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠇
⠀⠀⠘⡇⠀⠀⠀⠀⠈⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠣⡄⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠉⠉
⠀⠀⠀⠀⠀⠠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⠟⠁⢀
⠀⠀⠀⢤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡸⣿⣫⣉⣉⣛⠂
⠀⠀⠤⠞⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡴⠿⡅⠻⢿⣗⣳⣂⣁⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠑⠂⠀⠀⠀⠐⠀⠀⠀⡃⠀⠀⠈⠛⠶⢌⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠓⠐⠒⠲⠤⠖⠒⠲⢤⠴⠒⠒⠒⠊⠉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠀⠀⠐⠲⠶⠶⠶⠶⠒⠒⠒⠒⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠀⠀⠀⣤⠒⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠉⢧⡀⠀⠀⠀⢨⠁⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠈⢶⣄⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠳⢦⣀⡰⠎⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠹⣆
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⡙⢇⠀⢠⠻⡆⠀⠀⢸⠋⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡉⡇⠀⣀⡤⢶⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠸⠃⠈⢷⡎⠀⠻⡀⠀⠘⠀⠇⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡆⠰⠞⠁⣠⠏
⡇⠀⠀⠀⠀⠀⠀⠇⣠⢦⡀⢀⣤⡀⢣⡀⠀⡆⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠠⣞⠁
⡇⠀⠀⠀⠀⠀⢰⢀⡉⠀⠳⠎⠀⢳⠈⢧⠀⣇⢸⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⡇⢸⣧⠘⠦
⣿⠀⠀⠀⠀⢀⡇⡜⠁⠀⠀⠀⠀⠀⢧⡈⡇⣿⢈⠀⠀⣇⠀⠛⠛⠛⠂⠒⡄⠃⠸⠈⢧⠘⢣
⡇⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠳⠇⠈⠛⠀⠀⠘⠓⠒⠲⠖⠒⠋⠀⠙⠃⠀⠈⠳⠞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠐⠒⠒⠲⠶⠶⠶⠶⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠉⠉⠉⠉⠁
⡇
⢣
⢸
⢸⡆
⢸⡇
⢸⡇
⢸⡇
⢸⡇
⢸⡇
⠘⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢻⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠟
⠀⠀⠻⣷⣶⣤⣤⢤⢤⣤⢤⣤⣤⢤⣤⣤⣤⣤⣤⡤⣤⡤⠤⣤⡤⠤⠤⢤⡤⠲⠲⠖⢶⠖⠒⣩⡟⣻⠟⠁
⠀⠀⠀⠈⢿⣀⣳⣀⠀⠳⠀⠈⠃⠀⠘⠃⠀⠉⠀⠀⠈⠀⠀⠘⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⣞⣡⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⢀⣠⣆⣬⣿⣿⣿⣦⠀⠀⠀⢀⢀⠀⠤⣤⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣓⣶⣤⣤⣖⣲⡄⠀⠈⠮⢻⠿⠟⠋⠉⠁⠀⠀⠀⠘⣜⢷⣾⣿⣿⣿⡄⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠
⠅⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⢀⢷⣾⡿⠟⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⡠⠀⠀⠐⠒⠒⠢⢤⠀⠙⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠘⣦⣄⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸
⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⢀⣢⣾⠋⠀⠀⠀⠀⡐⢶⡄⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢠⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠣⡀⠀⠀⠀⠀⠇⠀⠀⡌
⢠⠀⠀⠀⣈⠀⠀⠠⠀⠀⢀⢸⡏⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⢰⣻⣭⣭⣑⠒⠶⠤⠴⣷⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⣄⠀⠀⢰⠀⠀⠀⠇
⠸⣌⠀⢀⠜⠀⢠⠁⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⡇⢿⣦⣀⠈⠳⢤⣿⣿⡿⢷⣶⣶⣿⣿⠀⠀⠀⠀⠄⣸⣷⣦⠀⠀⠀⠀⠀⣸⢿⡇⠀⠘⡄⠀⢸⠀⠀⠸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢈⣿⣇⠀⠀⠀⠀⠀⠙⢦⣽⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⢪⣿⡀⠀⢀⣴⣿⣿⡿⠏⠀⠀⠀⠀⣴⡿⣸⠁⠀⠀⢱
⠀⠀⡇⢁⠀⠀⡇⠀⠀⢀⣼⣿⣿⣷⣵⣄⠀⠀⠀⠀⠀⠈⠙⢮⠻⢿⣾⣿⣶⣄⣀⣀⣼⣿⣿⣾⣿⠟⠉⠁⠀⠀⠀⢀⡠⣺⡟⣡⡟⠀⠀⠀⢸⠀⠀⠀⢠⡇
⠀⠀⡇⢸⠀⠀⡇⠀⠀⢸⣿⣿⡟⢿⣿⠻⣕⡤⣀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠿⣭⡽⠟⠛⠳⣽⡇⠀⠀⠀⠀⣠⠔⣫⠾⢋⣴⣿⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⡇⠀⡆⠀⢳⠀⠀⠘⣦⣿⣿⡀⠹⣗⢄⡙⠮⣝⡒⠤⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣷⣤⠤⠒⣋⡴⠞⣡⢔⣵⣿⠇⠀⠀⠀⠀⡞⠀⠃⠀⢸
⠀⠀⡇⠀⢡⠀⠈⢧⠀⠀⠈⠳⢽⣾⣄⠈⠳⢭⣒⠤⣉⠛⠶⢤⣤⣈⣉⣙⠒⠒⠒⢒⣈⣩⣭⣭⣿⣷⠖⣋⡥⢖⣩⢶⢟⣽⠋⠀⠀⠀⠀⣼⠁⢸⠀⠀⡏
⠀⠀⠀⠀⢸⡀⠀⠈⠳⡄⠀⠀⠀⠹⣿⣷⣖⢤⣉⠛⠲⢯⣕⡒⠦⠤⢍⣉⣉⣉⣉⣉⣉⣉⠿⡿⣻⣿⣯⡴⢚⡫⢞⣡⠞⠁⠀⠀⠀⢀⡾⠁⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠢⡀⠀⠀⠈⠪⠿⠷⢮⣙⠢⢤⣀⡉⠛⠓⠲⠶⠶⠒⠒⠒⠒⠛⢻⣿⣷⣿⡯⠖⣋⣴⡿⠃⠀⠀⠀⢀⣴⠏⠀⠀⢠⠇⠀⢰⡇
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣍⣑⣒⣒⣒⣒⣉⣉⣩⣭⣭⣵⣹⣷⣟⡿⠋⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⡸⠀⠀⢸
⠀⠀⢸⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠑⠈⢓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠗⠚⠛⠉⠁⣀⣠⢴⡺⠝⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⡀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠭⢒⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠤⢖⡒⠍⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣠⣄⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢉⣉⣐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣧
⠀⠀⠀⢼⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⡿⠀⢿⡄⠀⠀⢀⣠⡴⠶⠶⠲⠶⠶⣤⡀
⠀⠀⠀⢸⡈⠛⠶⣤⣄⡀⠀⠀⣠⡴⠖⠛⠛⠳⢦⣼⠛⠦⣄⣼⣁⠀⠘⣧⢀⣴⠟⠉⠀⠀⠀⠀⡀⠀⠀⢙⣧
⠀⠀⠀⠘⣧⠀⠀⠀⠈⠉⣹⣞⡁⠀⠀⠀⠒⠢⣄⣿⠀⠀⠀⠈⠉⠙⠓⢿⣿⠃⠀⠀⠀⢀⡴⠋⠀⣠⣶⡋⠁
⠀⠀⠀⠀⠘⢷⣄⡀⠀⠐⠋⠉⠙⠳⣦⡀⠀⠀⠀⠙⢦⡀⠀⠑⢦⡀⠀⠀⠙⢧⣀⣠⠴⠋⠀⣠⡾⣯⣀⠙⠲⠦⠤⠴⠶⣶⠟
⠰⣶⡶⠶⠶⠶⠾⠛⠛⠛⢋⣉⣡⡤⠾⠿⣷⣤⣄⣀⣀⣹⣆⠀⠀⠙⣦⠀⠀⠈⢿⣁⣠⡴⠞⠻⠶⠶⢦⣤⣤⣤⠴⠶⠛⠁
⠀⠈⠛⠶⢦⣤⣤⣀⣀⣈⣀⣉⣠⣤⠶⠛⠉⠀⠉⠉⠉⠉⠻⣦⣄⣀⣹⣆⣤⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⢀⡶⠋⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⡄
⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⢹⣄
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠢⠀⠀⠀⠀⠀⠀⢿
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠤⢀⠆⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠸
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⣿⠀⠀⡠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠎⠂⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⢀⠀⡓⠀⣸
⢸⡇⠀⠐⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢰⢒⡄⠀⠀⣠⡀⠀⠐⠀⠀⡀⠤⠀⠀⠀⠀⠆⠀⢀⠐⠀⠈⠀⠃⢠⡿
⠀⢿⡄⠄⠀⠀⠀⠇⠀⢐⡂⠀⠀⠀⠣⡼⠀⠀⠀⠀⠘⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠀⡜⠐⢁⡤⠀⢀⣿⠃
⠀⠈⢷⡈⠀⠀⠄⠀⠠⢈⠁⠀⠀⡀⠀⠀⠀⠀⠐⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠚⠀⠀⣾⠃
⠀⠀⠈⢻⣄⠈⠂⠀⠀⢫⡇⠀⠀⢰⠳⡀⠀⠀⠀⣧⣹⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⢠⡌⠄⠀⠔⠃⠀⠈⠂⠀⣼⠇
⠀⠀⠀⠀⠹⣦⠀⠀⠀⡄⠀⠀⠀⠀⠓⠃⠀⠠⢄⡈⠁⠀⠀⠀⠁⠈⠠⠀⠘⠲⠃⢀⡀⠀⠀⡀⠀⠠⠄⠀⠠⢄⣽⠃
⠀⠀⠀⠀⠀⠈⢷⡄⠀⠘⠀⠀⠀⠰⡄⠀⠀⠈⠰⠇⠀⠠⠤⣄⠀⠀⠀⣠⠔⠂⠘⠁⠀⠀⠀⠀⠀⠃⣡⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⢀⠀⠀⠑⠁⠀⠀⠒⢤⠀⠀⠀⠐⠈⠁⠀⠘⠀⠃⠀⡴⠓⠀⠀⢀⠒⠀⠈⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠀⠀⠀⢄⠀⠠⡀⠘⠀⠀⠀⠒⡄⠀⠔⡒⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢢⡈⠀⠀⢁⠀⠀⠒⣀⠈⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠁⢀⠤⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡁⠀⠀⠄⠀⡀⢀⠁⠀⠀⠈⠉⠑⠀⢀⣀⠀⠀⠀⠀⠃⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⡄⠀⠀⠁⠀⠉⠀⠀⠀⡀⠑⠀⠀⠋⠠⠄⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠙⡀⠀⠀⠠⠂⠈⠀⠀⠒⠂⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠠⣄⠀⠐⠒⢄⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣤⡶⠾⠛⠁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⡯⢚⣭⠶⠞⣛⣉⣉⣩⣭⣄⢠⣿⣿⣿⣿⡿⠇⠀⠸⠀⠀⢠⠃⠀⠀⢀⣼⡿
⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⢟⣩⢞⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠉⠀⠀⠀⠃⠀⠀⠁⠀⠀⣠⡾⠟⠁
⣠⡶⠋⠁⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⡿⢟⡵⣋⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠁⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣫⡶⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⠞⣡⣾⣿⡿⠟⠉⢁⣾⡟⣹⣿⣿⣿⠟⠁⠀⣠⠤⠶⠚⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⠞⣡⣾⣿⡿⠋⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⣫⡶⢟⣵⣾⣿⠟⠉⠀⠀⠀⢀⣼⣿⣿⡿⠋⡠⠒⠋⣁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⣿⣿⣿⡿⣛⣵⠿⣫⣶⣿⡿⠛⠁⠀⠀⠀⠀⣠⣾⣿⣿⢋⠔⠈⠐⠉⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿
⣿⢟⣭⡾⢟⣵⣾⣿⠿⠋⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⣑⣥⣴⣤⡻⠄⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠿⣫⣶⣿⠿⠛⠁⠀⠀⠀⠀⢀⣴⣿⣿⢿⣯⡾⠛⠛⠛⠛⠛⠛⠳⠷⢦⣤⣈⠛⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣵⣿⠿⠋⠁⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡷⣋⠥⠒⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠛⠳⠦⣬⡓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠟⡃⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣭⣓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣓⠦⢄⣀
⠀⢁⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠭⣕⣒⡠⠤⢀⣀⣀⣀
⣴⣿⣿⣿⡿⠋⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶
⣿⣿⠿⣋⠤⠂⠁⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡀
⢟⡷⠊⣁⡤⠴⠚⠋⠉⠁⠉⠙⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠉⠁⠀⠠⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⠿⣿⡿⠟⠉⠉⠉⠉⠙⢛⣿⠿⠶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠀⠀⠀⠀⣿⣅⣀⣀⣀⣀⣀⣴⡿⠋⠀⠀⠀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡾⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣄⣀⢹⣧⠀⠀⢠⣶⠿⠿⢷⣦⡀⠀⠀⣰⡿⠛⠉⠉⣀⣠⣴⣿⣁
⠀⠀⠀⠀⠀⠀⣠⣶⠿⠻⣧⡈⠉⠛⢻⣿⠀⠀⣿⣧⣤⣤⣤⣽⣷⣤⣴⣿⣷⠶⠾⠟⠛⠉⣁⣼⠟⠿⣶⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⢸⣟⠛⠷⣶⣬⣿⣄⣸⣟⠀⠀⠉⠉⠉⠉⠈⠁⣀⣀⣀⣤⣤⣶⠾⠛⣻⡆⠀⠀⠙⢷⡄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣶⣤⣀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⣀⣤⣶⠟⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠉⠛⠛⠿⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠛⠉⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠷⠶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢠⡐⠉⠙⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠀⠀⢺⠀⠀⠀⠘⠀⠐⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠘⠒⠊⡁⠀⠈⢹⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣭⣍⣛⣛⡻⠿⠷⠶⠶⠶⠒⠒⠀⠀⠐⠉⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣀⣈⡉⠉⠛⠛⠿⠿⠷⠒⠚⠛⠃⠀⠀⠀⢴⠿⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣮⣉⡙⠛⠛⠿⠷⠶⠆⠀⢒⠀⠀⠀⠀⠐⠈⠀⡴⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢤⣉⠛⠛⠻⠷⠶⠶⢶⣤⣤⡤⠀⠀⠀⠒⠀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠙⠻⠶⣶⣶⣤⣤⣤⣤⡤⠤⠄⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠙⠀⢀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠠⡇⠀⠂⠀⠀⠀⠈⠃⠔⠀⠠⡔⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⢠⠀⠀⠀⠀⠀⠐⡀⠀⢠⠔⠁⠀⢈⠜⠏⠀⢀
⠀⠀⠀⠀⠀⠐⠢⡀⠁⠀⠈⠀⡀⠈⠠⠀⠀⠐⠄⠸⢀⠄⠈⠀⠘⠀⡀⠿⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡁⠈⠃⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⡡⢁⠊⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡁⠀⠀⠀⠈⢀⣂⠔⠁⠀⠓⠁⠀⠀⣀⠀⠀⡐⠢⢤⡤⠄
⠀⠀⠀⠀⠀⠰⠄⣀⣀⣀⠀⠈⠀⠱⡄⠀⠀⠂⠘⠁⠀⠀⠀⠀⢀⣀⠀⠁⢀⠠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠐⠢⠀⡱⣐⠀⠄⠿⢀⠔⢀⡤⠴⠉⠀⠂⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡠⠳⢌⢆⠐⠎⣠⠔⣋⣀⣤⡄⡀⠠⠦⣄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢱⣶⡾⠃⠉⠁⠀⠂⠀⡀⢀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⢂⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡗⢺⢣⠏⣠⡀⣀⢠⣠⢏⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠀⠀⠈⠈⠀⢡⠏⠉⠈⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠶⠖⠒⠛⠛⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⣈⣠⢤⡤⠤⠤⠤⠤⠴⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⢀⠣⢤⡀⠀⠀⠜⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠈⠀⠀⠀⢀⠇⣀⡴⠋⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡌⢸⠁⢰⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢄⣀⠤⠦⢄⣈⣁⠤⢼⡖⠶⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠢⠀⠀⠀⡄⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⢁⠀⠀⠀⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠤⢀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⢀⡤⠤⠇⠐⠤⢠⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠱⡄
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢣⠀⠀⠀⢹⡀
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠲⠴⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢷
⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡠⠤⠀⡤⠖⠠⣀⡤⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢸⡆
⡼⠀⠀⠀⠂⠀⠀⠚⢠⡋⠀⠀⠈⠠⡦⠀⡀⠰⠀⣸⢀⡁⠀⠠⡾⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣧
⡇⠀⠀⠀⠀⠀⠀⠀⡄⢠⠦⠈⢢⣀⡶⣀⣹⣀⡏⢠⣈⢉⡇⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠠⠤⠋⠼⠀⠑⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⢸⡀⠀⠀⣿
⢷⠀⠀⠀⠀⠀⠀⣠⠤⢄⢠⡴⢤⠔⢦⠒⢉⠉⠂⡤⢀⡤⣤⠴⠒⠦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡿
⢸⡀⠀⠀⠀⠀⠸⣁⠀⠚⢄⠑⣄⢀⠃⠀⠘⠀⢞⡇⠈⠁⢸⠀⣘⣀⡼⠀⠀⠠⠀⠀⠀⡴⠀⠀⢰⠆
⠈⣇⠀⠀⠀⠀⠀⠴⠚⢀⡴⣐⢁⡎⠀⡀⢈⠳⣠⠟⠤⠤⠻⢤⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡾
⠀⠘⣆⠀⠀⠀⠀⠁⠉⠁⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠃
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠀⢰⡀⠀⠀⠀⠀⠀⢠⠇⠀⢠⠇
⠀⠀⠀⠈⠳⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⠀⠀⢀⠀⠀⠀⠀⢠⠏⠀⠀⢺⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠓⢦⣤⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠑⠐⠉⠀⠀⢀⡴⠁⢀⡠⠔⠋⠉⠉⠉⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠶⠶⠶⠦⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣰⡯⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣀⣀⣀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣠⠤⠶⠒⠚⠋⠉⠉⠓⠒⠶⠦⠤⠴⠶⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠚⢻⠛⠉⢹⠉⠉⠏⠉⢹⠀⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣤⠼⠦⠶⠾⠖⠒⠛⣛⣋⣉⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⠽⢦⡀⠀⠀⠀⠀⠀⣾⡏⠀⠙⡆⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡯⠚⠁⠀⠈⠳⣄⠀⠀⠀⠀⠻⣿⣦⠾⠁⠀⠀⢀⣈⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣀⣠⡤⠶⠚⠛⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠛⠛⠛⠛⠒⠶⠶⠿⣶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠤⠶⠶⠖⠒⢹⡇
⠀⠀⠀⢠⣟⣛⠓⠒⢶⣄⠀⠀⠀⠀⠀⠀⡷⠤⠤⠤⠤⠤⠖⠒⠒⠢⠤⣀⣀⣇⣀⡠⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⢧⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡜⣷⠿⠿⠿⠿⠿⢿⣿⣶⣶⣶⣤⣄⠀⠀⠀⠀⢀⡇⠀⣄⣀⣀⠠⠄⣶⢸⣿⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⣿⢿⡶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣯⢾⡇⠀⠀⠀⢸⡇⢸⡏⠉⢹⡜⠇⠙⠈⠁⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⡌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣧⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠙⠒⠒⠒⠊⠉⠉⠉⡏⣿⠀⠀⠀⠀⢸⡇⠀⠀⢠⠖⠳⣄⡰⠃⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠉⢆⠀⡴⢺⡏⢦⣀⠜⠀⠀⠈⠀⣀⠀⢆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⡇⠀⠀⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⢸⠀⠀⢀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⡟⠉⠓⠶⢤⣄⣸⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣭⣒⣒⣒⣤⡤⢤⣔⣚⣥⠞⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠁⠀⠀⠀⠀⠀⠉⠻⣿⣄⣀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠋⠁⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣴⣾⠟⠛⠉⠉⠉⠙⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣀⣀
⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠛⠛⠻⢿⣶⣄
⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣷
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠈⠙⠻⠿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠈⠛⠁⠀⠀⠀⠛⠃⠀⠀⠀⠘⠋⠀⠀⠀⠘⠛⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣤⡄⠀⠀⠀⣠⣄⠀⠀⠀⢠⣤⠀⠀⠀⢀⣤⡀⠀⠀⠀⣤⡄
⠀⠀⠀⢀⣾⡟⠀⠀⠀⣼⡿⠁⠀⠀⣰⣿⠃⠀⠀⢠⣿⠏⠀⠀⢀⣾⡟⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠘⠋⠀⠀⠀⠘⠛⠁⠀⠀⠀⠛⠃⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛⠀⠀⠀⠘⠛⠁

⢠⣤⠀⠀⠀⢀⣤⡀⠀⠀⠀⣠⡄⠀⠀⠀⣠⣄⠀⠀⠀⢀⣤⠀⠀⠀⢀⣤⡀
⣿⣏⣀⣀⣀⣸⣟⣀⣀⣀⣸⣿⣁⣀⣀⣀⣿⣃⣀⣀⣀⣿⣏⣀⣀⣀⣸⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⡀⠀⠀⠀⠀⠀⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⠀⡘⠀⠰⠀⠀⠀⠀⠆⠀⢀⠀⢀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠉⠁⠀⠀⢣⠀⠀⡜⠀⠀⠈⠉⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⡀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⢀⠄⠀⠃
⠀⠀⠀⢀⡤⢀⡀⠀⠀⠀⠀⢀⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡀⠀⠀⠀⠀⢀⡀⢤⡀
⠀⠀⠀⠀⠆⠀⢉⠙⠲⠀⢤⠞⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠳⡄⠄⠂⠋⡉⠀⡰
⠀⠀⠀⠀⠘⡄⠈⠑⠦⡀⠀⠀⠀⠰⡀⠀⠑⠢⢄⡀⡠⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣄⠀⡠⠔⠋⠀⠀⠂⠀⠀⠀⢀⠴⠊⠁⢠⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠳⢄⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡠⠞⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠱⢄⠘⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠁⣠⠞⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠤⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠠⡄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢠⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⡤
⠀⠀⠀⠀⠀⡀⠀⠂⠀⣀⡀⠀⠀⠀⠀⠈⢆⠀⠀⢰⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡆⠀⠀⡰⠃⠀⠀⠀⠀⠀⣀⠠⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠁⠒⠂⠤⠄⣀⣱⡀⠈⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠁⢀⣎⣀⠠⠤⠐⠒⠉⠁⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠇⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠁⢀⡀⠊⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⠀⠐⠒⠠⠤⠤⠄⢀⣀⣀⣱⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣏⣀⣀⡀⠠⠤⠤⠄⠒⠂⠀⣴⡁
⠀⠀⠀⠀⣀⡠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⣀
⣤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠉⠑⠒
⠀⠑⠢⣀⠀⠀⠀⠀⢀⣀⣀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣀⣀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⢉⣩⠟⠃⠀⠒⠒⠒⠒⠒⠂⠀⠀⠠⠄⢠⡷⢾⣤⠤⠄⠀⠀⠀⠐⠒⠒⠒⠒⠂⠘⠻⢍⡉⠁⠀⠀⠀⠀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⣰⣆⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠎⠁⡌⠳⣄⠀⠑⢄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⡠⠊⢀⠴⠊⠀⠀⠀⡇⠀⠀⠑⠦⡀⠑⢄⡀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠔⣋⠤⠚⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠓⠤⣙⠢⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢡
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣟⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣄⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⢹⡏⠉⠁⠀⢈⡽⡅⠀⠀⢿⣯⠉⠉⠉⢉⡟⡏⠛⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⣿⠀⠀⠀⠀⡇⡇⠀⠀⢸⣿⠀⠀⠀⢸⡇⡇⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣼⣿⣤⣤⣤⣴⣷⣧⣤⣤⣼⣷⣤⣤⣤⣼⣷⣧⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣿⢦⣄⣀⣀⣤⠴⡎⣿⠀⠀⠀⠀⠀⠀⠀⣼⢳⡀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣤⣤⣴⠶⠛⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠱⡄⠀⣷
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠙⣆⢹⡆
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⢮⣷
⠀⠀⠀⠀⢠⡏⠀⠀⣠⠖⠋⠙⣆⠀⠀⠀⡤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⢸⡧⢤⠴⠁⠀⠀⠀⡇⠀⠀⢸⠃⠀⠀⠹⣄⣀⣠⠋⠉⠑⡄⠀⢀⡟⠉⢳⠀⠀⡞⠉⢱⡄⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠓⠤⢼⡇⠀⣏⠀⢀⡇⠀⠈⠓⢺⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⠉⠉⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣴⣦⠀⣠⡄⠀⠀⠀⣶⠀⢠⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣿⠘⠟⢹⡇⠐⣃⠀⣿⠀⢸⣤⡾⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠙⠃⠀⠀⠿⠀⠟⠀⠟⠀⢸⠏⠻⣆⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡴⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⠶⠟⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠒⠶⠶⠶⠶⠶⠾⠛⠋⠉
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⡀⢂⠐⠀⠀⢦⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠒⠀⠀⠐⠀⠈⡇⠀⠀⠀⠸⠀⠁⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠀⠀⠈⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢢
⠀⣬⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡠⠿⡟⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢱⡐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠀⠀⠀⢈⠀⠀⠄⠀⠀⠀⠄⠀⠙⠀⢡⡀
⠀⠀⠀⢠⡈⠰⠄⠀⠀⠀⠀⠀⠀⠀⣀⣈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⡠⠛⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢈⡀⠀⠈⠃⠀⠀⠀⠀⠃⡀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠹⢿
⠀⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡔⢉⡈⠉⠀⠐⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢷⣧⡀
⠀⡘⠈⠀⠀⠀⠈⡄⠁⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⡀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⠀⡀
⠀⣕⠢⠀⠀⠀⠘⠁⠀⣀⣀⠀⢀⠀⠄⠀⠀⠀⠀⡀⠀⢀⣀⢑⡂⡙⠃⡄⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⡀⠀⠀⠃⠀⠈⣦⠁
⠀⠐⠦⠀⠀⠀⡠⠀⠀⠀⠁⠀⠀⡀⡆⠀⠀⢀⠲⠤⡄⠀⠀⢀⡄⣀⠀⢡⠀⠀⢀⢀⣀⠈⠃⠀⠀⠀⠀⠀⠀⠟⣶⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⣀⡈⢴⡄⠀⠈⠛⠁⣴⠏⠀⣀⡭⠃⣈⣿⠆⠀⠀⠀⠀⠀⠀⠠⠻⡧
⠀⠀⠀⠀⠂⣤⠀⠀⠀⠀⢰⣷⠰⡂⢠⠀⠀⠀⠿⣿⣆⣀⠀⠀⠉⠉⠁⡠⠀⡈⣑⢙⠝⡡⡃⠀⡀⠀⠄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣔⣤⣉⠄⠀⠀⠀⠈⢎⠻⣶⠎⢐⡠⣺⠁⡔⠃⠚⢹⠀⡇⠁⣴⡁
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠹⠁⠁⠀⠀⢀⠠⡀⢚⠓⡄⠑⠦⢀⡡⣾⣖⢀⡄⠈⠓⠾⣷⣄⣤⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡗⢤⣀⣀⠀⠀⠈⠂⢐⠀⠀⠘⢦⡀⠀⠀⠀⠬⠕⢦⣀⡀⠀⣀⡙⠣
⠀⠀⠀⠀⠀⠀⠀⠐⠂⣶⠀⠀⢴⠈⠿⢴⣦⠃⠀⠀⠐⢄⣀⣀⡙⠢⣀⠀⠀⠀⠀⠈⠣⠀⠈⠙⠳⡀⠘⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠆⠀⠀⠀⠀⠀⠘⠃⠀⠑⠤⠀⠀⠉⠀⢠⠀⠄⠋⡀⢨⡷⢀⡗⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠲⠶⠷⠐⠁⠘⣧⣤⡥⠜⢁⠂⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁⠐⠈⠁⠀⠀⠆⠙⠛⠉⠁⠀⠁⠀⠰⠾⢷⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣻⠏⠀⠀⣀⣤⡴⠖⠛⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡆⠀⠀⠀⣠⠞⠋⣰⢏⣠⡶⠛⠉⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠃⠀⢀⡾⠁⠀⢰⡿⠟⠁⠀⠀⠀⣠⣴⠟⠛⠓⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⠀⣿⡸⡇⠀⡼⠁⠀⣰⠟⣠⠖⢁⣠⠶⢛⣉⣀⣀⣀⠀⠀⠀⠀⣈⡟⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡙⠻⢿⣇⢳⡼⠃⠀⡼⢁⡼⠁⠰⠿⠛⠛⠉⠉⠉⠉⠉⠉⠙⢋⣩⡟⠱⣦⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣠⣿⣎⢷⡄⢸⢃⣾⠁⣠⡶⠚⠉⠉⣀⣀⣠⣤⣴⡶⠾⠛⠁⠀⠀⠉⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⢉⣁⣀⣿⣆⠹⣆⢸⣷⠟⠁⣠⠴⠖⠛⠋⠉⠁⢠⣄⠀⠀⠀⣦⡀⠀⠀⢠⣄⠀⢸⣇
⠀⠀⠀⠀⣠⠞⠁⢠⠴⠛⠉⠉⣉⡥⢹⡿⠛⠘⠁⠀⠀⣿⠀⢀⡀⠀⠀⠀⠈⠻⠂⠀⠀⠙⠁⠀⠀⠘⠿⠀⠀⣿⡄
⠀⠀⠀⢰⠏⠀⢀⣠⣴⠶⣺⡿⢋⡴⠋⠀⡖⠀⠀⠀⠀⣿⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢠⢏⣴⠾⠋⡿⠁⣰⠏⢠⠞⠀⢀⡾⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠀⠀⠀⡀⠀⠀⢸⣆⠀⣿⡇
⠀⣠⣿⠟⠁⠀⣼⠃⢰⡏⠀⠘⠀⣠⠏⠀⢿⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠘⣿⡄⠀⠀⠋⠀⢻⣧
⠴⠋⠀⠀⠀⢠⡟⠀⢼⡇⠀⠀⣴⠃⠀⠀⠈⠻⣦⣀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⣸⠏⠀⠀⠸⣷⠀⠈⠙⣷⡈⡇⠀⠀⢸⣷⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⣄⠀⠸⣿⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⢷⡀⡿⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣿⠀⠀⠀⠉⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠘⠿⠀⠀⢻⣇
⠀⠀⠀⠀⠀⢿⡇⢷⡄⠘⣿⡇⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠸⣿⡈⠟⠀⢻⠀⠀⣀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠘⠀⠀⠹⢷⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠰⣷⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠻⣶⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⢼⣦⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣦⣤⣀⡀⠀⠈⠛⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣦⣀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠸⣷⡄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⣠⠞⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⢀⣤⠞⢉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⢀⣠⠴⠞⣋⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⡴⢞⣫⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⢀⣤⣶⣿⡷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣞⣫⡥⠶⠶⣦⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣚⣭⠶⠛⠉⠀⠀⠀⠀⢸⡇⠀⢠⡾⠁
⠀⠀⠀⠀⠀⣀⡴⣛⡵⠞⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣰⡟⠁
⠀⠀⠀⣠⢞⣭⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⢾⡟⠁
⠀⣠⢞⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣴⠞⣡⠟
⣼⢫⡟⠁⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⣁⡴⠋⣡⡾⠃
⣿⠈⠷⣤⣤⣤⣤⠶⠾⠛⠉⣀⡴⠞⠉⣠⡾⠋
⠘⢷⣤⣄⣀⣠⣤⣤⠶⠞⠛⢉⣀⣴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠱⢃⡴⠖⠶⢤⣀⠀⠀⠀⠀⠀⢠⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡀⠀⠀⢌⠳⢄⡀⠀⢀⣼⣜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠏⠉⠉⣩⠍⣓⠳⢤⣵⡒⣮⠻⠟⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣆⣀⣈⡴⠲⢤⣵⢤⠤⠃⣼⡷⠖⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⡙⠷⣤⡗⣿⠁⠈⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⡄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣛⡷⡟⢸⠀⠀⠀⠀⠀⠀⣴⣋⠭⠖⠒⠂⠈⠁⠀⠉⠀⡨⢿⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠟⠀⠛⠀⠀⠀⠀⠀⠀⢸⡷⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⢦⡀⠀⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣥⣤⣄⣁⣀⣀⣀⣀⣀⣀⣸⣇⠀⠠⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡉⠙⠯⠍⠉⠙⠛⠛⠛⠉⠉⠩⠽⣶⣬⢹⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡓⠆⠀⠀⠀⠀⣀⠤⠤⠶⠤⣄⣀⡀⠀⠸⣧⠿⢽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣦⣤⣀⣀⣠⠞⠁⢀⣠⣤⡀⠀⠙⣦⠀⠀⢰⢾⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⣴⣿⣿⣿⣿⣷⡄⠘⣿⣿⣿⣛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⡿⢭⣿⣧⠀⢿⣿⣿⣿⣷⣿⠁⢠⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣒⡢⠔⠛⣦⣬⠉⠛⠿⠋⠡⣪⠟⢩⠤⢌⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣗⡀⠀⠀⠙⠳⠶⠶⠖⠛⠁⠀⠠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠐⠩⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣄⣀⣀⣤⣤⣄⣀⣀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠶⠶⣤⣌⡒⠦⣄⡀⠀⠀⠀⠀⠀⠸⣯⣀⣉⣛⠙⠛⠛⠛⠉⢙⣻⣿⠇⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⠬⠙⠓⠶⣭⣓⠦⢄⡀⠀⠀⠀⠈⠉⠙⢛⣛⠛⢛⣛⣋⣉⣁⠀⠀⣀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢙⡻⢶⣮⣀⠤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠢⡈⠙⠽⠳⢶⣄⡀⠀⠀⠈⠑⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡈⢦⣄⣐⣐⠉⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣄⣭⣤⡁⢂⣡⡼⣁⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⠛⠋⠥⠤⠤⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣷⣶⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣶⠛⠉⠉⠉⠈⣻⣟⣻⣿⣿⠿⠛⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣟⠾⡅⠓⣶⡿⡯⠽⣏⣹⣟⣥⣬⣶⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠲⢭⣽⣼⣉⣉⣉⣩⣽⣿⣶⠿⠛⣩⢜⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢮⣅⣺⣾⡇⠤⠤⠤⠽⠾⠿⣾⣻⣽⣿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠷⣌⠹⣿⣿⠭⠭⠭⠿⠿⠟⠋⣋⣊⡭⣛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠤⣈⢹⢿⡷⠬⣭⣭⣙⣭⣬⣭⣷⣶⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠳⢮⣍⣿⣧⣤⣤⣬⣵⣶⡿⠷⣿⣿⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⠤⢤⡟⡟⠉⠩⠉⠉⢭⣒⡫⢵⣾⣿⣧⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠐⠁⠀⢀⠴⢾⣿⣔⠝⠒⡿⣯⣭⣭⣉⣭⣥⣴⡾⣛⣭⣿⢿⡋⠒⢄⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣰⢼⡄⠙⠿⣿⣾⡧⠤⠤⠬⢭⣯⣭⣵⣾⣿⣿⣿⣗⡰⣎
⠀⠀⠀⠀⢀⠀⠀⠀⠲⣷⣿⣿⣶⣤⡾⣹⣏⣉⣛⣛⣛⣭⠽⢶⣻⣽⣿⣿⣿⡶⠋⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠉⢻⢿⣟⠨⠿⢿⣿⣶⣶⢶⣾⡿⢿⢿⡿⠟⠋⠉⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠉⠓⠛⠛⠛⠛⠛⠒⠒⠚⠉⠉⠁⠀⠀⠀⣀⡠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠂⡠⠤⠀⠀⠀⠄⠀⠀⠑⡄⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⣄⣀⡄⠠⠀⠀⠀⠀⠇⠀⠐⠁⠀⠠⠰⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⢀⠈⡁⠈⠛⠓⠛⠛⠁⠀⠓⠛⠛⠇⠈⠛⠁⠀⠀⣢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠐⠀⠀⠈⠛⠋⠤⢄⣀⡀⠀⣀⣀⣐⣂⣀⡀⠊⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠁⠒⠢⠤⣄⣀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠐⣒⣀⠔⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠅⣀⣀⡀⠀⠀⠀⠈⠉⠙⠀⠉⠉⠉⠉⠉⠉⠉⠀⣂⡴⠞⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠈⠉⠉⠛⠚⠒⠒⠆⠀⠲⠶⠶⠒⠚⠚⠉⠉⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣤⣀⣀⡇⠀⠀⣀⣀⣠⣤⣤⠤⠴⠒⠋⢁⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⣀⡀⠀⠀⠈⠀⡀⢀⣉⣉⣁⣤⣤⡤⠶⠖⠛⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠲⠤⣄⣀⠈⠉⠉⠉⠉⠉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠃⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠤⣤⣀⣀⣀⠀⠈⠉⠛⠒⠒⠶⠶⠶⠶⠶⠶⠒⠒⠚⠛⠉⠉⣀⣀⣀⡤⠄⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠩⠭⢉⣛⡓⡒⢒⣒⣒⣒⡒⠒⣒⣻⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠏⢯⣭⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠿⠛⢉⣇⣴⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⢀⣀⣾⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⣉⣠⣤⣤⠶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣥⣶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣡⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠉⠉⠻⣿⠉⢠⡟⠁
⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠃
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠋
⠹⢴⣤⣤⣤⣤⣶⣶⣞⣻⠭⠭⠚⠋⠁

⠄⢣⣶⢶⡄⠀⠀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠻⠆⢣⣾⠻⣆⠀⠀⣰⡿⣷⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣎
⠀⠈⠻⡾⠃⠀⠀⠙⣻⠟⠀⠀⠀⠀⠀⠀⣾⣿⣧⣛⣚⠃⠀⠀⠈⣛⠛⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏
⠀⠀⠟⠉⠀⣀⣀⢾⡁⢰⠄⠀⠀⠀⠀⠀⠙⠛⠃⠛⠁⠂⣤⣄⢺⣁⢰⠆⣀⣀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠈⠺⣉⣹⠆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⠃⠉⠁⢺⣏⢹⠂
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⡄⣄⣼⣗⠶⠖⠒⠒⠒⠒⠒⢦⠒⠒⠚⠓⠖
⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣤⣇⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⢀⡆⠈⠁⠀⠀⠈⠉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⢷⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈
⠀⠀⠀⠀⠀⢈⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠀⠀⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⡠⢄⡀⣤⣶
⠀⠀⠀⠀⠀⢿⠶⡄⠀⢘⡃⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⣀⠉⠀⠈⠛⣀⡄⠀⠀⠀⠀⠲⣿⡵⠂⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠰⠋⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⢁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠾⡇⠀⢀⠀⠀⠠⠐⠊⠁
⠀⠀⠀⢠⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠈⣄⠀⠀⠸⣿⡂⠀⣀⠀⠀⠀⢰⣾⠚⢗⠀⠀⠀⠀⠀⠀⠀⣴⣂⣽⠒⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⣷⣰
⠀⠀⠀⠀⡅⠀⠀⠀⠘⠻⠃⢻⠀⠀⠈⠉⠿⠶⠞⠁⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠰⣤⢧⠆⠀⠀⠀⡉⢟⣿⣻⢏⠛
⠀⠀⠀⠀⠛⠀⡀⠀⠀⠀⠀⡘⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠛⠞⠀⠀⠀⠀⠀⣼⠈⢑⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⣿⢧⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⠀⣇⢀⡾⡆⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢄⣀⣠⡗⠈⠀⢣⡾⢀⡽⠛⢁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢆⠈⠀⠉⢄⠀⠈⠋⠞⠁⠐⣈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠣⡀⠀⠀⠑⠘⡞⠀⣠⠎⡠⡎⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⢦⡀⣀⠀⠀⣰⢦⡀⠈⢻⣦⡉⠐⠠⡀⠀⠗⠊⠁⠀⢀⢋⡿⠀⠀⣠⡾⡇⠀⠀⡀⠀⣀⢠⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⣆⠈⠈⡙⠶⠋⠀⠻⣄⠀⣿⠁⡀⠀⠈⠒⠀⠀⣀⣴⠋⢸⡇⢠⡞⡹⠠⠉⠑⠉⡈⢉⡵⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⢹⣆⢀⡇⠀⠀⠀⡀⠙⠛⢃⠀⠈⠠⡀⠀⠀⣷⠚⠁⠀⠈⠙⠋⢀⡇⠀⠀⠀⠀⣦⡽⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠓⠌⢿⣄⠀⠀⠀⢠⠀⠀⣄⠀⠀⠀⠀⠙⠺⡇⠢⡰⠀⡄⠌⠀⠘⠀⠀⠀⠰⣒⠼⠰⣯⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢯⡐⠦⠤⢬⠶⣄⡀⢸⠀⠀⢹⠀⠀⠑⢄⠀⢸⣿⡘⡐⠂⠐⠀⠀⢃⠀⠷⠛⠶⠀⠐⠋⠷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡸⠇⠠⢤⣀⢤⠬⠟⠾⣀⠀⢸⠀⠀⡆⠀⠙⠺⣿⢰⠇⠀⠀⡦⠀⣨⠃⠄⠀⠠⠤⠀⠐⢋⢉⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣐⠬⣅⣀⢀⡀⣀⣀⣴⠾⠻⣾⡀⠀⣧⠀⠀⢀⣿⠞⠠⠀⠀⣆⡜⠁⠠⠀⠤⠤⠶⠴⢶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⡭⢁⠀⢀⠐⣀⣨⣷⣤⡏⠀⠀⢸⣿⡆⢀⠆⣰⠏⡀⠀⠀⠀⠀⣠⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠄⢬⠹⠀⠈⢰⣼⢿⡤⡀⢸⣷⡆⢊⡶⣻⣾⣄⣤⡴⠽⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠛⠓⠲⢤⡀⠀⢆⠘⠙⠿⣯⣿⣏⣴⠟⣱⡧⠚⠁⠀⠀⠀⠐⢛⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⢢⣤⡤⠴⢲⠟⠛⠉⣩⠞⠛⣿⣿⣿⣿⠿⠿⣤⣄⣠⢚⣁⢀⣔⣉⣈⣭⣛⣳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣿⣖⢚⡟⠋⠀⢀⠐⠁⠀⠀⠔⢁⣠⠞⣻⠿⣴⠁⠉⠓⢦⡄⠀⠉⠙⠯⣉⠉⢉⡭⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⠶⠖⠛⠛⡶⠾⠞⠋⠁⢠⡿⠀⠈⠙⢦⡠⢤⣙⢦⣀⣴⠒⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟

⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⡀
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⣦⣄⣀⡀⠠
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠐⠦⠄⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡠⠔⠒⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠘⠢⡀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠙⠦⣤⠶⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⣀⠀⡠
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⡰⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⡞
⢀⡴⠀⠀⠀⠈⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣷⣾⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣦⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢻⣿⣿⡦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣴⣿⣿⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⣟⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⡿⢡⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣷⣿⣿⣿⢉⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣯⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⡀⣐⣶⡾⠿⠿⠿⠿⠿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣀⣤⣤⣽⣷⣍⢿⣯⣥⣤⣤⣤⣤⣾⣭⢭⣥⣶⣶⠶⢶⣶⡿⢉⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣈⣤⣞⢿⣿⣟⣙⣿⣿⣆⢻⣿⣿⣯⣯⣽⣫⣵⣯⣥⣾⣧⣴⣾⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠶⣿⡿⣿⡟⠛⠛⠛⠛⠛⠛⠻⣦⢹⣟⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⢇⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢠⡿⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢻⣆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣼⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢀⡿⢡⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢻⡄⠀⠀⠀⠀⠈⣿⣾⣿⣿⣷⠘⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⣼⢃⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢷⡀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣧⠹⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡘⣧⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢹⡆⠀⠀⠀⣹⣿⣿⣿⡇⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣿⡀⠀⠀⣿⣿⣿⡏⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢹⡇⠀⠰⣿⣿⣿⡃⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣧⠀⠀⢻⣿⣿⣧⢹⣿⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⡄⠀⣼⣿⣿⣿⡜⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣏⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠸⣇⢸⡇⠀⢻⣿⣿⣿⣿⣿⣿⡟⣿
⠀⣀⣀⣤⣴⣶⠶⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠶⠾⣿⡈⣿⣤⣼⣿⣿⣿⣿⣿⣿⣦⣿
⣾⣟⠉⠁⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⣽⣧⢸⣷⡄⠉⠉⠛⢻⣿⣏⢿⣿
⠉⢻⣿⠿⣷⣶⣶⠶⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⠉⠆⢿⣷⣿⡿⣿⣿⣿⣿⣾⣿
⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣈⣙⢻⣿⣶⣴⣿⣭⠄⣏⣿⣿⣿⣿⡿⣿
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⣭⣽⣻⣿⣿⣿⣿⣥⣴⢄⣻⣿⣿⣿⣿⣗⣿
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣤⣽⣯⣭⣿⣿⣿⣿⣯⠅⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⣾⣯⣿⣿⣿⡿⠶⠶⠂⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⣶⣿⣿⣿⣋⣉⡁⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢉⣿⣿⣿⣿⠿⠏⠉⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣽⣿⣿⢿⡶⠦⠀⣸⣿⣿⢿⣿⣿⣿⣿⠉⣿
⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⣶⢾⣿⣿⣿⣿⣶⠶⢀⣿⣿⣿⣾⣿⣿⣿⡏⢸⣿
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⡿⣿⣿⣷⣏⣿
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣧⣿⡿⢡⣿⠿⣿⣿⣿⣿⣿⣿⡿⣿⣾
⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠻⣶⣿⣿⣿⣟⣛⡀⣼⡿⠀⣿⣿⣏⠹⣯⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣛⣻⣻⣿⣿⣿⣿⠯⢠⣿⠃⠀⢻⣿⠙⣷⣿⣿⣷⢻⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⢴⣶⣿⣿⣿⣾⣛⠁⣾⡏⠀⢀⣼⣿⣷⡘⣿⣿⣿⡞⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣶⣿⣿⣿⠛⠁⣰⣿⠁⠀⢸⣿⣿⢏⣿⣿⣿⣯⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣻⣿⣿⣶⣶⣟⢀⣿⡏⠀⠀⠀⢻⣿⢾⣿⣿⣿⣻⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣭⣭⣽⣿⣯⣭⣤⣶⣼⣿⠁⠀⠀⠀⣿⣿⣾⣿⣿⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣛⣉⣉⣉⣉⣉⣭⣭⣭⣭⣍⣉⣩⣭⣭⣭⣥⣥⣼⣿⠇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢿

⢳⣄⠀⠀⠀⢰⣄⠀⠀⠀⠀⡄⠀⠣⠾⠃⠀⢸⡧⡄⠀⠈⣿⣼⣷⠀⠀⢻⡈⣳⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⢶⣄⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀
⠈⢿⣳⡄⠀⠀⠻⠄⠀⣀⠀⠙⠀⠀⠀⠀⠀⠈⢧⣿⡆⠀⠈⠉⠁⠀⠀⠈⠻⣯⡿⠀⠀⠀⠀⠀⠘⣟⣆⠀⠸⣟⢧⠀⠀⠀⠸⡟⣦⠀⠀⣦⡈⠀⠰⣦⡀
⣄⠈⠉⠁⠀⢄⠀⠀⠀⣿⢦⡀⠀⠀⢸⡷⣄⠀⠈⠛⠃⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠙⠋⠀⠀⢿⣾⠇⠀⠀⠀⢿⣼⠁⠀⠘⣿⡆⠀⢡⢝
⢾⣆⠀⠀⠀⠈⠁⠀⠀⢿⡨⢻⡄⠀⠈⣷⢹⣦⡀⠀⠀⠀⠀⠀⢹⡷⣄⠀⠀⠀⣄⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢧
⠛⠃⠀⠀⠀⠘⣷⣄⠀⠈⠿⣾⠇⠀⠀⠹⣮⣿⠇⠀⠰⣄⠀⠀⠈⣇⣿⣧⠀⠀⢸⡷⣄⡀⠀⠀⢿⡳⣄⠀⠀⠀⠀⠀⠀⣟⣦⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈
⠀⠀⠀⣦⡀⠀⢻⢻⣆⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⡟⣧⡀⠀⠙⠿⠟⠀⠀⠈⣿⣼⣷⠀⠀⠘⢧⣿⠆⠀⡷⣄⠀⠀⢧⣸⡇⠀⢀⡀⠀⠙
⠀⠀⠀⠘⠛⠀⠈⠛⠃⠀⠀⠀⠀⠀⢿⡦⠀⠀⠀⠀⠀⠳⣿⠇⠀⢤⠀⠀⠀⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⠸⣷⠀⠀⠉⠀⠀⠈⣿⢦⡀⠀⠀⢷⣤
⣱⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⢲⣄⠀⠀⠀⠀⢰⣄⠀⠀⠀⡀⠀⠀⠈⠳⠀⠀⢿⣦⡀⠀⠀⢦⡀⠀⠀⠀⢠⡀⠈⢷⠿⠀⠀⢸⢦⡀⠀⠸⣧⣿⡄⠀⠀⣷
⣿⣿⠀⠀⠀⠀⠀⠀⣟⢦⠀⠈⣿⢳⡀⠀⠀⠈⣟⢧⡀⠀⢸⣦⡀⠀⢠⣄⠀⠘⣿⣻⣆⠀⠘⢿⠀⢠⣄⠀⠓⠀⠀⠀⠀⠀⠸⣏⣷⡀⠀⠙⠛⠃⠀⠀⠘
⠀⠀⠀⠘⠦⠀⢤⡀⠈⠛⠀⠀⢹⣷⡇⠀⠀⠀⢹⣮⠇⠀⠀⢿⣻⡆⠘⡏⠳⣄⠈⠙⠋⠀⢀⠀⠀⠘⣯⠳⣄⠀⠀⢸⢦⡀⠀⠙⠛⠁⠀⠀⠀⡀
⢠⡀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⢹⣀⢘⣆⠀⠀⠀⠀⠁⠀⠀⢻⡄⢻⡆⠀⢸⡇⢿⣦⠀⠀⠀⠀⠀⠀⣿⢦⡀
⠀⣿⣦⡀⠀⠀⠀⣷⡙⢻⡄⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠻⠿⠏⠀⠀⠀⢸⣦⡀⠀⠈⠻⣼⡗⠀⠀⠻⣼⣿⠀⠀⢰⡀⠀⠀⠸⣇⣿⣆
⠀⢿⣈⢻⣆⠀⠀⠸⣿⣶⡿⠀⢿⢦⡀⠀⢿⣩⣧⠀⠀⢤⡀⠘⠂⠀⠀⠀⠀⠀⠀⠰⣆⡀⠀⣏⣿⡄⠀⠀⠀⠀⢠⡀⠀⠈⠁⠀⠀⢸⠹⣆⠀⠀⠻⢿⠟
⠀⠈⠿⡿⠇⠀⠀⠀⠉⠉⠁⠀⢸⡌⢷⡀⠀⠉⠁⠀⠀⢸⡟⣦⡀⠀⠀⣦⡀⠀⠀⠀⠛⠇⠀⠘⠟⠁⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠘⢷⣿⠂
⢦⣄⠀⠀⠀⢰⡀⠀⠀⠀⠠⡀⠀⢣⣿⠇⠀⢰⣦⡀⠀⠘⣷⣼⣷⠀⠀⢇⠙⣦⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣦⡄⠀⠈⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠈⣟⣷⡀⠀⠈⢿⡄⠀⡀⠀⠙⠀⠀⠀⠀⠀⠈⣧⢻⡄⠀⠈⠛⠁⠀⠀⠘⣿⣿⡇⠀⢦⡀⠀⠀⠈⡿⣄⠀⢹⣟⢦⠀⠀⠀⢸⡿⣆⠀⠀⣄⠀⠀⠠⣄⡀
⡀⠈⠛⠁⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⢸⣦⡀⠀⠘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⢿⠄⠀⠀⠙⠛⠀⠀⢿⣾⡇⠀⠀⠀⢿⣽⠂⠀⠸⣷⡄⠀⢣⠙
⣾⡆⠀⠀⠀⠀⠀⠀⠀⣿⣈⢷⡄⠀⠘⣷⢻⣦⠀⠀⠀⠀⠀⠀⢻⣦⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠈⠉⠀⢠⡀⠀⠀⠁⠀⠀⠀⠉⠃⠀⠈⢯
⠿⠇⠀⠀⠀⠐⣦⡀⠀⠘⢿⣼⡇⠀⠀⠹⣌⣿⡇⠀⢠⡄⠀⠀⠈⣇⣿⣧⠀⠀⢸⡶⣄⠀⠀⠀⣿⡻⣦⠀⠀⠀⠀⠀⠈⡿⣄⠀⠀⠀⠀⠰⡄⠀⠀⠀⠈
⠀⠀⠐⣦⡀⠀⢿⢻⣆⠀⠀⠉⠀⠀⡀⠀⠈⠁⠀⠀⢀⡟⣦⠀⠀⠹⣿⡿⠀⠀⠈⣷⣨⣷⠀⠀⠘⣦⣿⠇⠀⡷⣄⠀⠀⣿⣸⡇⠀⠀⣄⠀⠹
⠀⠀⠀⠘⠿⠀⠘⠿⠃⠀⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⠈⢷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠉⠀⠀⢻⡙⣧⠀⠈⠉⠁⠀⠀⢹⡳⣄⠀⠀⢶⢤
⢳⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⣄⠀⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣦⡀⠀⠀⠀⣦⡀⠀⠀⠀⢀⠀⠘⢶⡿⠀⠀⢰⣄⠀⠀⢸⣧⣽⡇⠀⠈⣧
⣿⣿⠀⠀⠀⠀⠀⠀⣿⣆⠀⠘⣟⢧⡀⠀⠀⠘⡿⣦⠀⠀⢤⡀⠀⠀⣤⡀⠀⠸⣿⣦⠀⠀⠘⣷⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸⡎⣷⠀⠀⠙⠛⠁⠀⠀⠘
⠈⠁⠀⠱⣄⠀⠀⠀⠙⠚⠀⠀⢹⣯⡇⠀⠀⠀⢻⣜⡇⠀⠘⣟⣆⠀⢸⡙⢦⡀⠙⠛⠀⠀⠀⠀⠀⢠⡿⣄⠀⠀⠀⢰⣤⡀⠀⠙⠿⠃
⠀⢀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢀⠀⠉⠁⢀⡀⠀⠀⠉⠀⠀⠀⠈⠋⠀⠀⢧⠆⣽⡄⠀⠀⠀⡀⠀⠀⠸⣧⠙⣧⠀⠀⢸⡏⢷⣄⠀⠀⠀⠀⠀⠀⣷⣤⡀
⠀⠸⣷⣄⠀⠀⠀⢸⡷⣄⠀⠈⠁⠀⠀⠀⣿⢦⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈⠳⠿⠃⠀⠀⠀⢿⣦⡀⠀⠹⣧⣹⠇⠀⠀⢷⣸⣿⠄⠀⢠⡀⠀⠀⢸⡎⣷⣄
⠀⠀⢿⣎⣷⡄⠀⠈⣧⣿⡇⠀⣷⣄⠀⠀⢿⣘⣇⠀⠀⣀⡀⠘⠆⠀⠀⠀⠀⠀⠀⠰⣄⠀⠘⣷⣽⣆⠀⠈⠉⠀⢀⠀⠀⠉⠁⠀⠀⢸⠻⣄⠀⠀⠻⣿⡿
⠀⠀⠈⠛⠛⠁⠀⠀⠈⠉⠁⠀⢸⠈⣷⡀⠈⠙⠁⠀⠀⢸⡿⣦⠀⠀⢲⣤⡀⠀⠀⠀⠻⠗⠀⠈⠛⠛⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠘⣦⣿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠤⠼⠤⢠⣄⣠⣰⡀⡇⣰⣶⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠿⠛⠁⠉⠢⡀⠀⠀⠀⠀⠀⠀⠐⠔⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠗⠋⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠈⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠈⠳⣄⣀⠠⠤⠖⠒⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣠⡠⠤⠔⠒⠈⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠀⠀⠈⠉⠉⢉⡍⢹⠀⢀⡀⠀⡄⠀⡟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣶⣀⡀⢀⠀⢀⠀⢸⠀⢸⡇⣼⠁⠀⠇⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠸⠁⠃⢸⠀⣿⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠈⠀⠛⠀⢸⠋⢷⡄⢀⡀⠀⣠⡠⠴⠚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠄⠒⠒⠒⠒⠒⠒⠒⠀⠁⠀⠀⠬⠭⢼⠗⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠓⠒⠒⠒⠒⠶⠤⠤⠤⠤⠤⠤⠼⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠒⠒⠂⠀⠐⠒⠠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢹⠆
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⢤⡀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠰⡄⠆⠀⠀⠀⠰⡆⠀⠀⢀⣀⢉⢲⣄⣄⢀⣀⣠⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢀⠇⠨⠧⠦⠆⠦⠽⠶⠤⠤⠰⠒⢳⣳⣋⡭⠶⠷⢿⠛⠿⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡜
⠀⠀⠀⠀⢰⣅⡰⢀⢠⣀⠴⢬⣀⢀⣠⢀⡴⠖⠉⠈⣃⠀⡀⡄⠈⡆⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠱⠚⠓⠊⠉⠓⢳⢹⣔⣃⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡼
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢰⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠛⠶⢦⣀⡋
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡴⠛⠂⠉⠒⠀⠤⡀⠈⠑⢆
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣸⡅⠤⣠⣤⡀⠀⠀⠀⠐⠢⣈⡇
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣼⣁⡀⠀⠀⠀⠀⠉⠙⠢⣀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠑⠒⠂⢤⡀⠀⠀⠹⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠶⠴⠶⠒⠐⠋⠁⠀⠀⠀⠀⠀⠀⢸⣀⣀⣤⣤⣀⠀⠀⠀⠈⠒⢴⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠀⠀⠀⠉⠒⠤⡂⠄⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠈⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠠⠄⣀⣤⣈⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠳⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠐⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢤⠂⠁⠠⠀⠐⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢑⠆⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠑⢦⠀⠀⡀⠀⠈⢀⠀⠰⡆⠁⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠤⠚⠃⠀⠀⠁⠀⡃⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡀⠀⡄⠒⠂⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣈⡥⠆⠀⠀⢄⣀⣀⣠⡀⠈⠀⢤⡄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠄⠀⡀⢀⠄⠚⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⢵⠀⢠⣄⣀⣹⠀⠀⣠⣄⣒⡂⠤⠭⠍⠐⠒⠊⠉⢢⢀⡠⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⢀⣀⣀⡀⡉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠒⢟⡈⢀⡨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⢀⣀⠁⠀⠀⠀⠈⢱⡀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⠀⠑⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢄⣉⠀⠀⠐⠒⠂⠶⡇⠠⠤⠤⠒⠒⠒⠋⠉⠉⠁⢀⣀⡗⠐⠈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣭⠈⠒⠒⠒⠒⠶⣇⠠⠤⠤⠤⠔⠒⠒⠊⠉⠉⠀⢀⡠⠔⠊⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⡓⠤⣀⡈⠑⠒⠒⠒⠂⠆⠀⠠⠤⠤⠤⠤⠔⠒⠚⠉⠽⣧⡀⠀⢀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⢃⠢⢄⡉⢁⠒⠶⠦⠄⣴⠀⣀⣀⣠⣤⠤⠤⠤⠴⡶⠋⠀⠁⠀⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠙⠲⢤⡄⠀⠋⠁⠀⠙⠶⣼⠁⠀⠀⠰⢤⡤⠤⠤⠤⠤⠤⠤⠴⠶⣶⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢄⣀⠤⣉⠀⠀⣀⠠⢄⣀⣀⠠⠝⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⡑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠱⡀⠙⠦⠚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠼⠀⠀⠀⢱⡀⠀⠀⠹⠶⡦⠀⠀⠀⢀⠀⠀⢀⣴⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠐⢤⡀⠀⠀⢣⠀⡇⠀⠀⣧⠀⠀⢠⠏⠓⠖⠉⡼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠈⠳⣄⠈⢷⡇⠀⠀⠘⢦⡴⠋⠀⡄⠀⣸⠁⠸⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠈⠓⢾⣇⠀⠀⠀⠀⠀⠀⠀⡇⢠⠇⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⡤⣄⡀⢸⡄⠀⠀⠀⠀⠹⡄⠀⠀⢠⠀⠀⠀⣧⠏⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠸⡿⢍⡉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢳⠀⢠⠏⠀⠀⣰⢋⡤⠖⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠓⠤⣀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⢢⡀⠈⣧⠏⠀⠀⣰⠟⠉⠀⠀⢠⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠈⠉⠒⠦⢤⣀⣈⢣⡀⠀⠀⠀⠀⠙⢄⣸⡄⠀⣰⠃⠀⠀⠀⠀⠘⠒⠒⠋⠉⠩⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⢉⡹⠿⠦⢄⣀⡀⠀⠀⠙⣇⣰⠃⠀⠀⢀⣠⣤⣒⣁⣀⠤⠤⠤⠜⣻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢦⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠈⠉⠙⠲⢤⣼⣧⡤⠔⠚⠛⠉⠉⠉⠀⢀⣀⡀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡴⢺⠟⣿⣍⠒⢤⣀⠀⠀⡴⠉⠁⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠙⣆⠀⠀⠀⠀⡠⠋⠀⡞⠀⠟⣇⠱⡄⢈⡑⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⣠⠞⠁⠀⢸⠃⢸⠀⢿⡀⠘⠏⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⠀⠀⣠⠞⠁⡴⢦⣄⠘⣠⠞⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡠⠊⠀⢀⠞⠀⠀⠸⣤⠃⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠤⠖⠒⠒⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠖⠒⠒⠒⠲⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣀⣀⠀⠀⢀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢠⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡅
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢻⣿⣿⣷⣦⡀⠀⠀⠀⢰⣿⣿⡇⠀⢀⣀⡤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠸⠿⣿⡇⢰⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⠟⠋⠁⠀⠀⢀⣤⡀⠀⠘⡇⢸⠁⠒⠀⠤⠤⠤⠤⠤⠀⠚⠉⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡦⠀⠈⠁⠀⠀⠀⠀⠀⣼⣿⣟⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⢀⣴⣿⣾⣿⣿⣿⣆⠀⠇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⢀⣤⣄⡈⠛⠿⠻⠿⣿⡿⠃⠀⡆⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⠀⢀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⡇⣠⣤⣾⣿⣿⣿⣿⡇⠀⠀⠀⠸⢿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠙⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⢛⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⢀⠜⠀⠀⢸⣰⢀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠢⠤⠤⠼⠿⠿⠟⠛⠒⠁⠀⠀⠀⠀⠉⠓⠒⠤⠤⠤⠖⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣠⠆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠈⠛⠤⣤⣤⣤⣤⣀⠀⠀⣠⣤⣶⠏⠁⢀⡤⠞⠁⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⢿⣧⣧⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠂⠈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⡀⠉⠛⠷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠄⠀⢀⠀⠀⠀⠀⠐⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣤⣦⣃⠀⠀⠀⢀⠐⠀⠀⢤⣄⣀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣴⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣼⢄⡀⠉⠉⠛⠶⢤⣿⡀⠀⠀⡦⠭⠉⣝⣓⣦⣄⣀⡦⣆⣀⣄⣀⣠⣤⡶⣞⣻⣯⣵⠶⣫⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠛⠷⣧⣈⠀⠀⠀⠀⠀⠀⠋⠃⠘⠛⠒⠿⠿⣿⣿⣿⡗⣿⠿⠿⠿⠟⠛⠛⠉⢉⣁⡤⡾⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⡀⠈⠉⠛⠻⠷⣶⣀⣾⣤⣤⣀⠀⠀⠤⠀⠀⠀⢠⠀⡀⠀⠀⢀⣀⣠⣤⣼⣿⣽⠷⠾⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡯⣙⠳⢤⣤⣄⣠⡀⣎⡴⢈⠉⠉⠛⠳⠶⠶⠶⠴⠞⢠⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⢀⣰⣶⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠻⠾⣷⡀⠀⠀⠈⡉⢉⡟⠛⠲⠶⠤⢄⣑⣀⡀⠀⢐⠊⡀⣀⣀⣀⣠⣦⡴⠴⠶⠚⠋⠉⣹⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠉⠑⠶⡆⢰⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠀⣀⣀⣀⣤⡤⠶⠛⠉⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⠿⢦⣀⡒⢀⣀⢃⠀⢡⠈⠉⠉⠛⠲⠲⠦⠤⠤⠄⠒⠶⠖⠒⠒⠋⠉⠁⠀⢀⡀⠀⢀⣄⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣤⡉⠙⠻⠿⠾⢧⣼⣠⣈⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀⣉⣀⣠⣶⠶⢿⣛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣤⣀⣈⡙⠛⢿⣟⢿⣾⢶⣶⣴⣤⣤⢤⣶⠶⠶⣾⡿⠿⢋⣡⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠉⠉⠛⠛⠿⠷⠶⢦⣭⣭⣭⣤⣬⣥⣭⣭⠤⠖⠓⠛⠉
⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠁
⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠾
⠀⠀⠀⠂⠘⠛⠋⣿⣿⣿⣿⣿⣿⡿⠿⠿⠤
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣴⢯⣉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⠛⣦⡼⠟⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠳⠶⢿⠁⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠳⠦⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⣴⠎⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣱⣾⡄⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢿⣯⡼⠛⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠘⣃⡀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡿⣶⡛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⡾⣿⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠈⢧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢉⣿⣾⠟⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠟⠷⠟⠈⢇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⣠⠴⠖⠛⣛⣛⣛⣋⣉⣉⣉⣉⣍⠩⠭⠭⠽⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⠤⠤⠤⠴⠶⠶⠶⠶⠶⢶⣟⣛⡳⢶⣤⡀
⢠⡞⢻⡷⠖⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡿⠉⠙⡆
⢻⡄⠈⠳⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣤⣤⡤⠴⠶⠞⠋⠁⠀⠀⠀⢹
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⣄⠀⠀⢀⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢠⡇⠀⠀⠀⠀⣠⠛⢳⡀⠀⣸⠀⠀⢧⣀⣠⠟⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⡄⠀⠀⠀⢰⠋⠉⢻
⢸⡇⠀⠀⠀⠀⣿⠆⠈⠳⠞⠃⠀⠀⠈⠉⠁⠀⠀⠀⢨⡇⠀⠀⠀⣠⠞⠋⠛⠳⣄⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣿⠀⠀⠀⢀⡄⢀⡼
⣾⠀⠀⠀⠀⢸⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢠⡇⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢸⠇⠀⠀⣀⣹⣦⣤⡴⠾⠛⢻
⡿⠀⠀⠀⠀⣿⠿⠦⠤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠖⠂⣾⠀⠀⠀⠘⡷⠒⠒⠲⠖⠛⠷⢶⡄⠀⠀⣿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡆
⠱⣄⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⣿⡿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⣀⣤⠴⡇
⠀⢹⠶⣮⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⠤⠤⠤⠤⢾⡀⠀⠀⠀⣸⠓⠚⠛⠛⠉⠉⠁⠀⠀⢸
⢀⡟⣁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠁⠈⠁⣈⡉⠉⠉⡁⠀⢠⡀⢀⣀⡀⢀⡀⠛⣷⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⣿⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⡆⠀⡟⠉⢹⠀⠘⡛⠳⣼⣇⢸⡦⠀⠀⠙⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡶⠁
⠀⠈⢳⣾⣥⣀⣀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠛⠀⠙⠶⠞⠀⠀⠁⠀⠈⠋⠈⢉⣉⣀⡀⣥⣤⣤⣤⣤⠤⠄⠶⠶⠒⠉⠉⠀⠀⡇
⠀⠀⡿⠂⠀⠈⠉⠉⠉⠛⠋⠙⠛⠛⠒⠋⠘⠀⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠰⣟⡡⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠈⠙⣿⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠄⠒⠛⠋⠉⠹⡆
⠀⠀⠀⠸⣧⠀⠈⠉⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠈⠉⠉⠉⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠸⣧⠔⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠃
⠀⠀⠀⠀⠀⠈⢹⡷⢦⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠶⠶⠶⠖⠊⠉⠉⠸⠁
⠀⠀⠀⠀⠀⠀⠘⣶⡠⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣋⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠚⠛⠛⠙⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⢂⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⡟⣩⣉⣐⣀⡜⣁⠀⣠⣀⡜⣠⠆⣠⠆⣤⣤⣼⣷⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⢳⣄⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠛⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⡀
⠀⠀⠀⠀⣠⠶⠞⣋⣥⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⡀
⠀⠀⠀⠀⣷⣄⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣀⣤⡿
⠀⠀⠀⠀⠻⣬⡉⠛⠶⢤⣤⣀⣀⣀⣠⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣈⣉⣉⣩⣥⡴⠞⠋⣩⣼⡃
⠀⠀⠀⠀⠀⢰⡟⠛⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⠉⠛⠛⠛⠛⠉⢉⣉⣉⣀⣀⣀⣀⣀⣀⣠⣤⡴⠖⠋⠁⠉⠙⢳⡄
⠀⠀⠀⠀⠀⣸⡷⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⣿⠃
⠀⠀⠀⢀⣞⠉⠙⠶⣤⣉⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠚⠋⣁⣤⣾⠋
⠀⠀⠀⠀⣿⠳⠦⣤⣀⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⢖⣻⡿⠿⢻⡏
⠀⠀⠀⠀⢈⣿⢦⣄⣉⠛⠛⠒⠒⠒⠒⠒⠒⠲⠶⠶⠤⣤⣤⣤⣤⡤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠚⢋⣁⣤⠾⠿⣤⡀
⠀⣀⡤⠶⢻⣧⣀⡀⠉⠛⠛⠒⠒⠒⠒⠒⠒⠲⠶⠶⢤⣤⣤⣤⣤⣤⠴⠶⠶⠒⠒⠒⠒⠒⠒⠚⠛⠋⠁⠀⢀⣠⣼⡷⠦⣄⡀
⣾⠉⠀⠀⠈⢷⣌⡉⠛⠶⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⠶⠛⢉⣥⠟⠀⠀⠈⢹
⠙⠷⣄⣀⠀⠀⠈⠙⠳⠶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣉⣉⠉⠙⠛⠛⠋⠉⣉⣉⣀⣀⣠⣀⣀⣀⣠⣤⣤⠶⠚⠋⠁⠀⢀⣠⡴⠟
⠀⠀⠀⠉⠛⠓⠶⠦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠖⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣽⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣭⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠋⠉⠁⠀⠈⠉⠙⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣤⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠉⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢿⣿⣿⣆
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆
⠀⠀⠀⣼⣿⣿⠟⠋⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠐⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⢻⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⣷⡆⠀⢀⣤⣤⠀⠀⣠⣤⡀⠀⢀⣀⡀⠀⢀⣤⡀⠀⢀⣤⣄⠀⠀⣿⠁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣼⣿⡿⠁⠀⣾⣿⡿⠀⢰⣿⣿⠃⢀⣿⣿⡇⢠⣿⣿⠇⢀⣿⣿⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠻⠟⠁⠀⠻⠿⠋⠀⠘⠿⠟⠀⠈⠛⠛⠀⠈⠛⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⢠⣶⣄⠀⠀⣠⣤⡀⢀⣴⣦⠀⠀⣴⣦⡀⠀⣴⣶⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⣿⣿⡟⠀⢰⣿⣿⠃⣼⣿⡿⠀⣼⣿⣿⠃⠠⣿⣿⠃⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠻⠟⠁⠀⠹⠿⠋⠀⠙⠟⠁⠀⠻⠿⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣆⠀⢀⣤⡄⠀⢀⣠⡀⠀⢀⣠⡀⠀⠀⣠⣄⠀⢀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⣾⣿⡟⢀⣾⣿⡏⠀⣾⣿⡗⠀⢰⣿⣿⠂⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠁⠸⣿⣿⠁⢸⣿⣿⠁⠰⣿⣿⠃⠀⢼⣿⡟⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⢠⡶⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢂⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⣤⣤⠋⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣹⡷⠴⣠⣄⠱⣿⣶⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣚⠋⠶⣦⣝⣿⣷⠘⣿⠇⠸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠞⠉⠐⢿⣦⣈⣿⣷⣻⣧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠏⡀⠀⠐⠄⠙⡿⣮⣿⠛⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠦⡀⣊⣢⠟⡫⠃⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢈⠭⠁⠜⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠤⠄⠠⡇⠀⣤⠀⠤⠤⠤⠤⠤⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣂⣉⣀⣲⡤⢶⠤⠃⢀⠰⣶⠦⠴⠄⠀⣀⣀⡀⠀⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠴⢏⣙⣇⣸⡠⡧⠼⣦⠀⠈⠐⢦⣤⠤⠤⠤⢤⣄⠀⠙⠢⡀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣶⢮⣉⣉⠭⠄⣀⠀⠠⠀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⢑⣤⠃⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⣽⣂⠈⣉⠃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠀⡄⠐⣁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⣝⡻⣟⠀⣴⡂⠀⠀⠀⢠⣶⡄⢰⡯⢦⡀⢂⡀⣇⢈⣀⡥⡞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢈⢀⡇⠁⠈⠋⢮⡇⠀⠀⠂⢨⠟⡻⢿⠺⠾⠟⠛⠋⠙⠁⠃⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠜⠻⠀⠀⠀⠀⢀⠀⠀⢀⠓⢸⡃⢌⢂⢃⠀⡀⠠⠠⠀⠀⡘⢰⠀⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠃⠀⠀⠀⠀⣠⣆⠀⠤⠅⣼⡷⠶⠶⢄⠀⠀⠁⠆⡀⠀⠃⡈⠀⠁⢈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⣠⡾⠃⠈⠳⠶⠞⣦⣀⠀⡀⢋⣆⠀⠀⠠⠀⡘⢀⠃⠃⠠⢸⢠⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠂⠀⠀⣔⠝⢁⡀⠠⠠⠤⠤⣴⣿⣤⡟⢀⠄⠘⣧⡀⠀⠈⠁⠸⢰⡖⠓⠮⡈⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠉⠠⢄⠀⣠⣶⣯⢁⣿⠃⠀⠀⣀⠘⢟⡄⠀⠀⠃⣼⠀⠀⠀⠡⡆⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠆⠀⠨⡥⣀⡠⢶⣦⣙⡿⣿⠀⢀⡄⠀⠀⡜⡌⠀⠀⢀⠻⠀⣄⠀⠀⠀⠠⢄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⣁⠀⠔⢸⠷⣶⣝⣿⢷⠻⡃⠈⣠⠂⡘⡘⠀⠀⢨⡘⢀⠶⣈⡁⠾⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠴⠐⠀⠀⠀⠀⠰⠘⢆⠶⠻⠷⠼⢋⢾⡈⠤⠁⠐⠁⣲⠌⠀⠀⠀⠀⠃⡈⡀⢁⠁⠆⠀⢠⡸⠃
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠱⠈⠖⢎⢦⡀⠋⠀⠀⠀⠖⢠⡳⠁⠀⠠⢀⠂⠈⠰⢡⠁⠄⡘⠰⠀⠂⡉
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠚⠭⠴⠶⠖⣂⣀⣀⣰⠏⠀⠀⢀⠂⠌⡀⢀⢃⢃⠆⡠⢁⢃⠂⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠐⡐⡀⢀⠎⠎⡰⢁⠎⣀⠎
⠀⠀⠀⠀⠀⠀⠐⠢⠀⠂⠀⠊⠑⣠⣉⠂⠑⢊⠔⡠⠂⠂⠐⠀⠁⡠⢁⠌⠔⠜⠀⠀⠎⠈⠀⣤⠞⣣⠀⡠⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⢂⠠⠀⠄⢉⡰⠛⡰⠦⣤⢀⣀⣀⢀⢀⡀⣀⡀⢀⡤⠀⠄⣐⢞⠕⢁⠈⡡⢊⠐⢁⠔⡡⢂⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠡⠒⢁⠔⢉⠔⢊⠤⠂⠠⠀⠀⠀⡁⠀⡠⠂⠔⠠⠂⡠⢁⠄⡡⠊⡄⠁⠔⢁⠘⠁⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠊⠐⠀⠐⠀⠊⠀⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠒⠛⠛⠉⠉⠉⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠀⠀⠀⢀⣀⡤⠤⠒⠒⠲⡄⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢠⠤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣳⣹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠜⢯⣷⣷⣿⣿⡗⠛⠛⠳⢦⣀⠀⠀⠀⡇⢀⣀⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠠⠃⢠⢁⠎⠀⠜⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠛⠉⠁
⠀⠀⠀⠀⢀⣀⣌⣀⣘
⠀⠀⢰⡯⣉⣩⣉⣉⣍⡭⢿⠂⠀⠀⣀⣤⣤⣤⣤⣤⡄
⠀⠀⢸⡗⠲⢚⠲⢼⣄⣀⣾⠀⠀⠀⣿⠉⠛⠛⠉⠉⡿
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⣄⣠⣤⣀⣴⠇
⠀⠀⠀⢹⣤⣀⣄⣀⣀⣴⡇⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⠐⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠄⠀⠀⠐⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠁⠀⠡⡀⢀⠌⠀⠈⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢄⠀⠀⠀⠘⠃⠀⠀⠀⡠⠀⡇
⠀⠀⠀⠀⠀⠀⠒⠒⠠⢀⡀⠀⡀⠑⢄⠀⠀⠀⠀⢸⠀⠀⢢⠀⠀⠀⠀⠀⠀⡜⠀⠀⠃⠀⠀⠀⠀⡠⠊⢀⠀⢀⡀⠄⠒⡂
⠀⠀⠀⠀⠀⠀⠈⠄⠐⠠⡀⠀⠀⠀⡀⠑⠢⢄⡀⡸⠀⠀⠀⠡⡀⠀⠀⢀⠌⠀⠀⠀⢆⢀⡠⠔⠈⢀⠀⠀⠉⢀⠄⠂⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⠐⢄⠀⢡⠀⠀⠀⠀⠀⢀⠀⠀⠀⠡⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠊⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠑⢼⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠘⠃⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡥⠊⠀⠀⠀⠠⡁
⠀⠀⠀⠀⠀⠀⠣⠀⠠⠀⠀⠀⠀⠀⠈⠢⡀⠀⢃⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡘⠀⢀⠔⠀⠀⠀⠀⠀⠀⠄⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠉⠐⠂⠤⠀⣐⣄⠸⠀⠀⠀⠈⠆⠀⠀⠀⠀⠰⠁⠀⠀⠀⠃⣠⣂⠀⠤⠐⠂⠉⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠈⢲⠀⠀⠀⠀⠈⢆⠀⠀⡰⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠱⠎⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠀⠉⠀⠀⠀⠉⠉⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠉⠁⠀⠀⠀⠙⠀⠀⠄⣀
⠀⠀⠀⢠⠐⠂⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠉⠐⠂⡄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠠⠤⠌⢦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠔⠠⠤⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠠⠔⠈⠀⠀⠀⠀⠀⠉⠉⠉⢉⠝⢛⡛⠫⡉⠉⠉⠉⠀⠀⠀⠀⠀⠁⠢⠄⡀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡠⠋⡗⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠔⡁⠔⠉⠀⠀⡇⠀⠑⠢⣈⠢⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠊⠔⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑⠢⠵⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠳⢤⡼⠀⡀⠹⣤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢟⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠅⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⠓⢦⣄⣀⢀⣴⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⢸⡇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠠⠈⠉⠉⠀⢆⠛⠏⠉⠀⠀⠀⠈⠄⠀⠈⠛⠋⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠈⠢⠀⠀⠈⢧⡀⠈⠢⡀⠀⠀⠈⡄⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠭⢥⣀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡷⠀⠀⠀⠐⠀⠀⢸⠀⠀⢠⢆⠀⢸⠷⢤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠊⠀⠉⠁⢀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⣧⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⡦⠀⠀⠀⠀⠀⠀⠀⡠⠂⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠁⠐⠒⠚⠋⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠃⠀⠀⠀⣀⠤⠖⠉⠀⠀⠓⣶⡆
⠀⠀⠀⠀⠀⢦⡴⠶⠖⠚⠁⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⣀⠂⢄⢠⢀⠄⠂⠉⠀⠠⠄⠀⠀⡠⠞⠉
⣀⣀⣤⣤⠴⠞⠁⠀⠀⢀⠠⠤⠤⠶⠂⠀⢀⣠⣤⣤⡀⠀⠀⠀⠈⠁⠸⠛⢳⡶⣶⣦⡤⣤⣤⣬⣵⡦⠄
⠈⠉⠉⠉⠉⠁⠲⠀⠀⠀⠀⠀⠘⠁⢠⠞⠉⠀⠀⢈⣿⠀⠀⠀⠴⠀⠀⠀⠀⢳⠈⠙⢯⡲⣄⡀
⠀⠀⠀⠀⠀⠀⡀⢀⠤⠒⠛⢷⢀⠖⠁⠀⠀⢀⣠⠾⠋⠀⠀⢀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠱⡄⠻⣆
⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠘⠋⠀⠀⠀⠐⠿⣧⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠹⡀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠀⠀⢠⠀⣠⠔⠁⠀⠀⠀⠀⠀⢧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣀⣠⡀⠈⠀⢀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⣧⠀⣰⠋⠳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠁⠀⠃⠴⡟⡁⠀⠀⠐⠒⠢⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣾⣷⡀⠀⠀⡴⡠⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠂⠉⠁⠀⠀⠒⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢄⡀⠈⠙⠓⠲⠶⠶⠤⠤⣤⡤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠒⠚⠋⠁⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠂⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠙⠒⠶⠤⠤⣬⣭⣉⣁⣀⣀⣠⡤⠤⠤⠖⠒⠊⠉⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠓⠤⣀⣀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀⢀⠀⠄⢀⣀⣀⣤⠤⠴⠒⠉⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠁⡀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠉⠓⠲⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠤⠶⠖⠒⠛⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠙⠦⣄⡀⠈⠉⠛⠓⠷⠶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠖⠛⠊⠉⠀⠀⢀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠛⠓⠶⢶⣤⣤⣤⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠂⠉
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠀⠉⠉⠁⠀⢀⣀⡤⠶⠛⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣀⡀⠀⠀⠀⠀⠉⠛⠒⠶⢤⣤⣤⣤⣤⣄⣀⣤⣀⢤⣀⣀⣀⣤⣤⠴⠖⠒⠚⠉⠉⠀⠀⠀⠀⠀⣀⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣠⡤⠤⢶⣖⡖⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠚⠦⠤⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠤⠴⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉
⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠊⠀⠂⠠⢶⠂
⠀⠀⠀⠀⠀⢀⠀⠀⢖⣿⡀⠀⠀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠠⠀⠐⠚⠉⠁⠀⠂⠀⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈