Typhoon Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠶⠦⢤⣀⣀⡀⠀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⡝⠁⠀⢀⡴⠋⠉⠉⠉⠙⠶⣏⠁⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢸⠀⣀⡤⠤⡤⠒⠉⠁⢀⡤⠖⠒⠦⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠈⡟⠁⠀⢸⠁⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢢⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⢇⠀⢀⡤⠓⠀⠀⠀⣰⠉⠀⠁⠀⠀⠀⠲⡄⠀⠁⠁⠀⡤⠂⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡎⠀⠀⠀⠀⠀⢡⣀⣉⠀⠀⠀⠀⠀⠙⠦⠀⢠⠾⠁⠀⠀⠐⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡐⠀⠀⢀⡄⠀⠀⠀⠀⠀⠐⢦⡽⠾⠀⠡⡖⠥⠀⢀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⢧⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⣀⣀⡴⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢦⣘⡆⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣀⣀⣠⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠢⠊⠀⠀⠀⠰⠀⠀⠈⠳⣄⠀⠀⠀⣀⣀⠴⠋⠀⢸⢡⠞⠲⣁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢸⠀⠀⠀⠀⠈⠓⠦⢤⣄⣧⡀⠀⠀⠘⢄⡘⡄⠈⠑⠦⣀
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠤⠜⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠑⠤⣀⣀⡀⣅⠰⡀⠀⠢⣄⠑⠂⢄⡀⢀⠴⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠁⠀⠀⠀⠑⢦⡤⢬⡁⠀⠀⠀⠀⢹
⠀⠀⠄⠀⠰⣤⠞⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠈⠊⠀⠀⢠⠀⠀⠀⡄⠈⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⢼⡀
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠜⡇⠀⠘⠚⠁⠀⠀⢠⠀⠀⠀⢸⠀⠀⠐⠃⡠⠓⠂⢀⡤⠴⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠹
⠀⠉⠀⠀⡐⠁⠀⠀⠌⢀⡇⠀⠀⠀⠀⠀⠠⢁⠀⠀⠢⠞⠀⠀⠀⣼⠁⠀⠀⠈⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠢⡀⣰
⠀⠀⠀⠀⠀⡇⠀⠰⣄⡼⠁⠀⠀⠃⠀⠀⠁⢸⠀⠀⠀⠀⠐⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠳⠴⠃⠀⠀⠀⠀⠀⢀⠀⡄⠀⠘⠲⠊⠀⠀⠀⠈⠀⡇⠀⠘⠷⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⠞⠁⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⣠⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠊⠡⠂⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⣀⣤⠄⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠠⠒⡁⠀⡀⡈⠠⣼⠁⠈⠀⠁⠈⠕⢒⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⢑⠂⠉
⠀⠐⠂⠀⠠⢀⠀⡀⠀⠛⠀⠀⠀⠙⠀⢀⠀⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢠⠐⠀⠀⠀⣀⠒
⠀⢈⡀⠀⠀⢲⠀⢠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡩⠀⠀⡄⠀⠀⢠⣶⠁⢄
⠀⠀⡷⠀⠁⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⠀⠀⢠⠁⠀⠀⡜⠡
⠀⠀⠃⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸⠀⠀⡸
⠀⠀⢸⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⣇⣰⢷⡤⠴⠶⠆
⠒⣲⠚⣧⠲⠦⣾⡏⠦⠄⠒⠴⠶⣶⡶⠶⠒⠒⠶⠔⠒⠶⠒⠒⠶⠶⠖⠖⢒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⡴⣱⣿⣿⣎⣤⣽⡇⣄⡀⠀⠒⠀⡀⠀⠀⣀⣀⠀⠈⢉⣵⣶⣀⡀⢀⣤⣀⣤⣤⠤
⣷⣿⣿⣿⣿⣿⣿⠀⣿⣥⣶⡄⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠿⠿⠮⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒
⣿⠿⣿⡿⠛⢻⡿⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣭⣴⣭⣿⣽⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⣀
⠀⠀⠀⠀⣠⣴⣶⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⡀
⠀⣠⣶⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣤⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿
⣰⣿⣿⣿⡿⠁⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⣰⣿⣿⡿⠋⠁⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠈⢿⣿⣿⡆⠀⢀⣾⣿⣿⣿
⠘⢿⣿⣿⣿⣧⡀⢰⣿⣿⣿⠀⠀⠀⠀⣠⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⢀⣼⣿⣿⣿⣠⣿⣿⣿⣿⡿
⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⣿⣿⡁⠀⠀⠀⢤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⣸⣿⣿⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠟⣫
⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣤⣀⠀⠈⠛⠿⠿⣿⣿⣿⣿⠿⠿⠿⠛⠋⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣩⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣍⡙⠛⠛⠿⠿⠿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⣁⣠⣴⣾⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⢿⣿⣷⣶⣤⣀⡀⠀⣀⣠⣤⣤⣀⣀⣀⣉⠉⠁⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⣿⣿⠿⠿⠛⠉⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣯⣛⠿⢿⣿⣿⣶⣦⣬⣉⣉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣀⣤⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣌⡉⠛⠛⠿⢿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣶⣦⣤⣀⡈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣀⣈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣦⣤⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⡈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣶⣤⣤⣤⣤⣍⣉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠈⠛⠿⠿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣶⣶⣶⣶⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢀⡀⣠⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⡖⠝⢋⠠⢄⣒⠾⠗⠂⠀⠀⠀⠀⢔⣶⠖⠀⠀⠀⠠⠚⠐⣀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢊⡠⠴⢖⡛⢉⣀⠀⠀⠀⠀⠀⠀⣠⣆⠌⠀⠀⡠⡢⠀⠀⠼⠿⠋⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠧⠶⠶⠟⠛⢛⠉⣁⡀⠀⠀⠀⠤⠼⠿⠋⠀⠀⠚⠒⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣠⣴⣿⠷⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡄⠀⢀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢀⠞⠛⠋⢂⣲⢾⡣⠄⠀⠀⠀⢠⠤⠊⠀⠈⠉⢉⠀⠀⠀⠀⠀⢠⣾⠵⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠋⢀⣤⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠠⠚⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡰⠁⠀⠀⠰⠁⠀⠋⠁⠀⠐⢀⠌⠀⠀⠀⣀⣴⣾⣿⣿⣿⡄⡰⢁⣴
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡐⠁⠐⠀⠀⠁⠀⠀⠠⠈⠀⢀⠊⢀⡷⠻⠿⠛⠛⠛⢩⣍⠛⠃⠁⠈⠡⠄
⠀⠀⠀⠂⠀⠀⣼⠁⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢴⡿⠟⠀⠛⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠠⠤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣒⠚⠋⠉⠀⠀⢀⠀⠀⠀⠈⠉⠛⠓⠲⠶⣦⣤⠾⢥⣥⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⡴⠆⠉⠁⣐⣶⣶⣶⣿⣿⣶⣾⣿⣿⣿⣷⣶⣶⡶⠛⠉⢛⣧⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠶⣶⣶⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⡾⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢺⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢘⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣽⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⢠⡟⣻⠄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⣀⣀⣠⡤⠤⠤⠶⠶⠶⠶⠶⠶⠤⠭⢭⣄⣀⡈⠉⠉⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣀⣐⠟⠉⠀⠀⣠⡤⠴⠶⠶⠶⠶⠶⠶⠤⢤⡄⠀⠀⠉⠳⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠉⠉⠙⠛⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⣛⡋⠉⠁⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠤⣤⣤⡄⢰⠒⣦⠀⣄⣤⣤⣤⣤⠤⠤⠶⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣀⣀⠀⠀⠀⠀⠈⠛⠁⠀⠤⠄⠀⠀⠀⣠⢤⡄⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢉⣉⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠀⠹⠴⠋⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠙⠲⠦⢤⣤⠀⣴⠲⡄⠀⣀⣀⣀⣀⣀⣠⣤⠤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⣶⡴⠚⠁⠙⢦⡀⠀⠀⠀⢠⣀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠋⠀⢠⠶⣄⠀⠈⠳⣄⠀⠀⠈⠉⠛⠓⠒⠒⠒⠒⠒⠒⠒⢚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠘⠆⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠇⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠿⠆⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⡄⠀⡴⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡤⢶⠶⣄⠀⢀⡴⣶⢦⡀⠀⡇⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠴⠏⢀⡼⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣥⣼⣤⣼⠀⣿⣤⣿⣤⣷⠀⡇⠀⠀⢀⡴⠚⠁⠀⠠⠤⠄⠀⠘⢁⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣀⣸⣀⣸⠀⣿⣀⣿⣀⣿⠀⡇⠀⢰⡏⢀⣠⠶⠶⠶⠶⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠁⠀⡇⠀⠈⢧⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣇⣛⣛⣛⣛⣛⣛⣛⣀⣛⣛⣛⣀⣇⣀⣀⣈⣻⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣅⡹⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢏⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀⠀⠀⢀⡀⢀⡀⠀⢀⠀⡀⠀⣀⡀⠀⢀⡀⢀⣀⢀⣀⠀⢀⢀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡃⡇⠈⠀⡬⢥⢸⠒⡇⡇⠀⡷⠀⠀⠑⢤⠀⡇⢸⠤⡁⡁⠀⡇⡷⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠀⠁⠀⠀⠀⠈⠀⠉⠁⠉⠁⠉⠁⠀⠈⠁⠀⠁⠈⠀⠀⠈⠉⠀⠁⠈⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⢀⡀⠤⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠷⢛⡫⠵⢒⡊⠭⠥⠄⠒⠀⠒⠊⠓⠒⠛⠒⢲⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠞⢁⣠⠔⣒⣠⣤⠤⢖⠲⠒⠂⠀⠀⠀⠀⠰⠚⠠⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢫⢦⠞⣡⠴⣾⣣⣤⠾⠉⣁⠤⠚⠁⠀⢀⡀⠴⠴⠶⠄⠤⡐⠒⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡋⢁⡙⠉⠀⠠⠖⠉⠀⠀⠀⠀⠈⠀⠀⢀⣄⣀⣀⣀⣁⠚⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠅⣂⡤⠐⢂⡀⠤⠔⠐⠂⠁⠀⠒⠒⠉⠉⢉⡩⠂⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡴⢖⠉⠁⡀⢊⡡⠄⢀⠄⢐⢀⡠⠄⠀⣀⠀⣈⣁⡒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⣀⠁⡠⠊⠀⠉⠀⠈⢀⣀⡬⣄⠆⠀⠀⠀⢈⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢑⡴⢊⠤⠞⡫⠕⢒⠉⠀⢐⡀⠈⠀⠀⢀⣀⣼⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣉⠶⣫⠔⢊⡡⣖⡮⠑⠂⠉⠀⠐⠀⠤⠤⢤⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⣁⡠⠟⠋⠁⠀⠒⡂⠉⠉⢉⠉⡦⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢙⡴⠒⢀⠄⢂⡤⠀⠠⠴⠚⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠾⠥⠄⡴⠋⡔⣉⡤⠖⠂⠀⡐⠛⢓⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⣋⣁⡤⣺⠝⠉⠐⠒⢛⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⣘⣣⠖⠖⡒⠉⠁⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠳⠖⢡⣊⣁⠄⠹⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠍⠠⠉⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣶⡟⢞⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠁⠀⠀⠀⠏⠀⠀⢄⠀⠀⠀⠤⠀⠀⠀⠐⠂⠀⠀⠈⡂⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠀⠀⠀⠱⢀⠈⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⢄⠂⠀⠀⠈⠀⠀⠀⠀⣀⠤⠜⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠘⠄⠀⠉⠒⠒⠒⠈⠀⠀⠜⠋⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠐⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣠⣴⠶⠶⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢸⣇⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠙⣧⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⣠⡾⠛⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠛⢶⣄⣠⡴⠶⠶⢶⣤⡀⠀⠀⠀⠀⠉⢻⣆
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠈⠻⣦⣀⣀⣀⣀⠀⠹⣇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠙⠻⣶⣿
⢀⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄
⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠙⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣀⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⣠⡾⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁
⢰⣿⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⢀⣤⡶⠟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣿⣇
⠀⠛⠷⠶⠿⠋⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢿⡇⠀⠀⢠⡾⠟⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠈⢿⣄⣀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⢸⡇⠀⠀⣿⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢀⣴⠾⠛⢹⡇⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⡾⣿⠀⠘⣿⡀⠀⢀⣿⠃⠀⠀⠙⠷⣶⠾⠋
⠀⠀⠀⠀⢠⡿⠁⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢀⣶⠟⠉⠀⣿⠃⠀⠘⠛⠛⠛⠁
⠀⠀⠀⠀⠈⢿⣤⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣿⣿⣿⣿⣋⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣶⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣽⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⠳⢍⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠻⢿⣿⣿⣦⡈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓⠀⠈⠈⠙⢿⣆⠈⠢
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠳⡄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠛⢿⣧⠘⠆⠀⠀⠀⠙⡆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣶⡶⢬⠻⣿⣿⣿⠀⠀⠘⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡟⢻⡏⣵⠛⠃⠀⠁⠀⠈⠻
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⢿⣿⠀⠛⠃
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣧⡈⠳⢄⡀
⠀⠀⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣷⡙⢷⡙⢯⣖⠤⣀⠁⠀⠀⠄⠀⣠⡄
⠀⠀⢻⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢻⣿⣿⣿⣿⣿⣗⠄⠈⠓⠌⠀⢀⣴⣆⣀⣤⠾⠋
⠀⠀⠈⢿⣿⡿⣿⡆⣼⣿⣿⡏⢿⣿⣧⠹⣿⣿⣿⣷⡄⠹⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⡀⠀⠀⣀⣀⠠⠀⠀⢀⡔
⠀⠀⠀⠀⠻⣿⣿⣆⠘⢿⣿⣷⣜⣿⣿⣆⠙⣿⣿⣿⣿⣦⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⠍⠈⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠈⠉⠻⣆⠀⠻⣿⣿⢿⣿⣿⣆⠈⣿⡋⠻⣿⡿⢮⣝⡟⡿⣯⣟⣻⠿⠿⣿⣿⣥⠤⠶⠖⠛⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠹⣿⢿⣧⡈⠛⠲⢄⠙⠶⣍⡻⢶⣦⣍⣙⣛⡛⠒⠒⠒⠒⣒⣒⣈⣡⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠈⠁⠉⠳⡀⠀⠀⠀⠀⠈⠙⠳⠦⣭⣍⣛⣛⣛⣛⣛⣛⣋⣭⠥⣖⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣒⣛⣛⡛⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣛⣽⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣾⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠈⠉⠙⠊⠉⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠉⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠛⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⠟⠉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣰⣾⣿⣿⡟⠋⣡⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣿⢁⣾⣿⣿⡇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣤⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

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


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

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

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

⠀⠀⠀⣀⠰⣀⣤⣀⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢦⣰⣤⣀⡆⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣄⢠⣀⣠⡄⢀
⠀⠀⣄⣹⣷⣿⣿⣿⣿⣶⣿⠄⠀⠀⠀⠀⠀⢠⣌⣿⣾⣿⣿⣿⣷⣾⡥⢀⠀⠀⠀⠀⢀⣘⣿⣾⣿⣿⣿⣷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣶⣠⡄
⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣃⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⡷⠂
⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⣺⣿⣿⣿⡿⠛⠉⠉⠀⠈⢑⠀⠀⠀⠠⣿⣿⣿⣿⡿⠟⠉⠉⠁⠩⠳⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⠟⠁
⠐⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠈⢩⢿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠉⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠺⠿⣿⣿⣿⣿⠻⡍⠁⠀⠀⠀⠀⠀⠀⠐⠻⢻⡷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⡦⣤⣴⠂⡄
⠀⠀⠀⠀⠀⠉⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠄⠠⠡⠀⠀⠀⠀⠄

⠀⢀⣀⠀⢀⠠⠐⠒⠠⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢀
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠠⠁⠀⠌⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢠⣶⣟⣿⢿⣶⡠⡀⠀⠀⠀⠀⠀⠄⠒⠒⠂⢀
⠀⠀⠀⡠⣢⣾⣻⡿⣶⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⣿⡻⣶⡀⡀⠀⠀⠀⠀⢀⣸⣶⡿⢱⣿⡟⢸⣿⡆⢿⣿⡜⡄⠀⠀⡐⠁⠀⠀⢀⠔⡴⠁⢄⣀
⠀⠀⡜⣼⣿⠃⣿⣧⢹⣿⡌⡆⠀⠀⠀⠀⠀⢀⣂⣾⣿⢰⣿⡇⢻⣿⣘⡄⠀⠀⠤⣿⣿⣿⣿⣿⣿⣇⣸⡿⠇⣸⠟⠛⠛⠀⠀⢡⠀⠐⠈⠡⠈⠀⠀⠈⠀⠀⡄
⠀⠀⠉⠙⠻⠔⠛⠛⠸⠟⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠻⠘⠛⠓⠼⠋⠉⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠁⠐⠂⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠙⠛⠆⠁

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

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠒⠒⠤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠈⢢⣀⣀⣀⠈⠢⠤⣄⡠⠤⢳⠤⢀⣀⣀⡠⠤⡚⠤⢄⣤⠤⠔⠁⣀⣀⣀⡔⠁
⠀⠀⠀⠀⠀⢠⣃⠀⣞⠧⣀⣀⡠⡴⠀⠀⣮⡀⠀⠀⠀⠀⢀⣵⡀⠀⢦⢄⣀⣀⠼⣣⠀⣘⡄
⠀⠀⠀⠀⠀⢠⠊⡠⠊⠀⢀⠀⢠⠁⠀⡼⣤⠀⠉⠉⠉⠉⠀⣤⢧⠀⠘⡄⢀⡀⠀⠑⢄⠑⡄
⠀⠀⠀⠀⣰⡡⠊⠀⠀⠀⡎⣣⠉⡝⠀⡰⢡⡄⠀⠀⠀⠀⢠⡈⢆⠀⢫⠉⡎⢡⠀⠀⠀⠑⢜⣆
⠀⠀⠀⠰⠋⠀⠀⠀⡴⡀⠈⠁⢰⢁⠞⠀⢇⡰⠀⠀⠀⠀⢇⡕⠀⠳⡈⡆⠈⠁⢠⢆⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠥⠇⣀⣠⡷⠧⠒⠒⠒⠒⠒⠀⠀⠒⠒⠒⠒⠒⠼⢾⣄⣀⠸⠼
⠀⠀⠀⠀⠀⠀⠀⢀⣔⠋⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢀⣀⣤⠔⠒⠒⠋⠉⠉⠉⠉⠉⠉⠓⠒⠒⠤⣤⣀⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡊⠀⢈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙⡄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠⠋⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

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

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




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠉⠉⠉⠉⠑⠢⣄⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⠉⠀⠀⠀⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠒⠒⢤⡀
⠀⠀⠀⠀⠀⢀⡤⠖⠚⠒⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢹⠴⠶⠤⣄⡀
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣀⠼⠗⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⣄⡀
⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠴⠒⠲⢤⡀⠀⠀⠀⠀⠀⠙
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠙⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⠞⠁
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠓⠒⠚⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠦⣀⣀⣀⣤⡀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠲⢤⣀⣀⣀⣠⢤⡀⠀⠀⢀⡴⠓⠒⠚⣉⠀⠀⠀⠀⠀⠀⢠⡆⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⢠⢿⠀⠀⠀⠀⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⣏⡼⠀⠀⢸⢧⡀⠘⣇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣶⠀⠀⠀⠀⢠⡏⢣⠀⠀⠀⠈⠀⠀⠀⠸⣄⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡾⡀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⠀⠈⠳⠞⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡞⠀⡇⠀⠀⠀⠀⡀⠀⠀⠈⠶⠃⠀⠀⣄⠀⠀⠀⠀⠀⢰⠉⡆⠀⠀⠀⠀⠀⣰⢻⡀
⠀⠀⠀⠀⠀⠀⠙⠞⠁⠀⠀⢀⡴⢻⠀⠀⠀⠀⠀⠀⢠⠋⡄⠀⠀⠀⠀⠘⠚⠁⠀⠀⠀⠀⠀⢇⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡼⠁⠀⠀⠀⠀⠀⠘⠖⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠾⡄⠀⠀⠀⠀⠀⠀⢠⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡞⢱⠀⠀⠀⠀⠀⠸⣤⠇⠀⠀⠀⣴⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⡜⠹⡄⠀⠀⠀⠀⠀⠈⢧⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⡄⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⣄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠙⠴⠃⠀⠀⠀⠀⠀⢸⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⢰⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣆⠀⠀⠀⠀⠀⠸⣀⡼⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡄⠀⠀⠀⠈⣇⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⡝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⢦⡤⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠓⠀⠀⠀⠀⠀⠀⠀⢿⠖⠋⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠤⡄
⠀⠀⠀⠀⣀⡤⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡤⠶⠶⠦⣄
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢳⡀
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀
⠀⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢳⡀
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢦⡀
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⢀⡿
⠀⠈⠳⣤⣀⣀⡶⠒⠛⣲⣞⣂⣀⣀⣀⣀⣀⣀⣈⣻⡉⠉⠉⠙⣅⣁⣀⣀⣀⣀⣀⣀⣀⣀⣲⡶⠓⠒⣲⣋⣀⣀⣈⣛⣏⠉⠉⢧⣀⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⢠⡟⠀⠀⢰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠿⢤⡄⠀⠀⠀⢈⡗⢀⡴⠋⠀⠈
⠀⠀⠀⢠⣯⣤⡤⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢴⡶⠚⠃⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠉⡿⠀⢀⡴⠋⠀⠀⠀⠼⢥⡄⢈⡽⠂
⠀⠀⠀⠀⢠⠞⢀⡴⠋⠀⠀⠀⢀⡞⣇⠀⠀⠀⠀⠙⢢⡀⠀⢶⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⡼⣁⡴⠋⠀⠀⠀⠀⠀⣠⣏⡴⠋
⠀⠀⠀⣠⢏⡴⠋⠀⠀⠀⠀⢠⣯⣄⠈⣹⠇⠀⠀⠀⠀⠙⢦⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⠀⢀⣞⠓⡆⠀⠘⠛⠁
⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠉⢷⡇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠂⠀⣄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢀⡠⠠⠕⠊⠁⠈⠲⠂⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠤⠂⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣢⣄⠀⠐⠈⢀⡀
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠉⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⢀⢐⠀⠀⠀⠄⠀⠀⠒⠮⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠤⠄⢀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⡉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

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

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⢀⣤⣤⣤⣤⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣤⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣤⣤⣤⣬⣤⣤
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢋⣀⣹⣿⣿⠿⠛⠛⠛⣛⣉⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠛⠛⠉⠁⠀⠀⠀⠸⣿⣿⡿⠿⠟⣛⣁⣀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⡶⠚⠓⠀⢀⣀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠙⠿⣿⣿⡿⠋⠁
⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⣿⣿⣧⣿⣻⣿⣾⣿⣿⣿⣿⣷⣾⣆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⠀⢀⣀⣀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

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

⣀⠀⠀⠠⡄⠀⠀⠀⠀⠀⡀⠰⡆⠀⠀⠀⣤⠀⢰⠄⠀⢀⠀⢰⡄⠀⠀⢀⠀⠀⠀⠀⠦⠀⠀⠀⢀⠀⠀⠀⣀
⠼⢀⡄⠈⠃⠀⠘⡇⠀⠀⣧⠀⠀⠀⡄⠀⠛⠀⠀⠀⠀⠿⠀⠈⠁⢀⠀⢽⠀⠀⠀⠀⠚⠀⢀⡀⠸⠆⠀⠀⣽
⠀⠘⠗⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠨⠆⠀⠀⠀⢶⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⢠⡄
⢀⠀⠀⠀⠀⠛⠀⠀⠀⡧⠀⠀⢀⠀⠀⣠⠀⠈⠆⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⣆⠀⢘⡄⠀⠘⠖
⢸⠀⠀⢼⡄⠀⢀⠀⠀⢀⠀⠀⢨⡆⠀⠸⠀⠀⠀⠀⢸⡇⠀⠋⠀⠀⢸⡅⠀⠀⠀⢀⠀⠀⠸⣥⠀⠁⠀⠈⠁⠀⠀⣆
⠀⠀⠀⠈⠀⠀⢻⠆⠀⠘⠇⠀⠀⣀⣀⡀⠤⢀⠲⠒⠺⢧⠤⠄⣀⣀⣀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⢀⡄⠀⢠⠀⠉
⠀⠀⣷⠀⢾⠀⠀⠀⣀⠔⢂⡩⡾⢝⢉⡒⢕⣊⠅⠀⠀⢨⡆⢅⠂⢀⠈⠉⠛⠳⢖⡒⠠⠤⣀⡀⠀⠀⠸⣾⠀⢪⠄
⢀⠀⠙⠀⠀⣠⡐⠙⠀⡠⢺⠭⡢⢋⠥⡂⠅⠂⠀⠀⠀⠀⠘⡀⢇⠂⠈⢀⠀⠀⠀⠈⠳⣕⠆⠈⢉⠢⢄⡀⠀⠀⣤
⠸⠆⢀⡔⡫⠔⠁⠀⡰⠁⢑⠥⡪⢔⠨⢒⠁⠀⠀⠀⠀⠀⠀⠃⢑⠄⠀⠀⠢⠀⠀⠀⠀⠈⢧⡢⢀⠠⠀⠈⠢⡀⠙⠁
⠀⡠⢧⣀⡄⠀⠀⢰⣁⢔⢕⣪⡲⠑⠍⡀⠄⠀⠀⠀⠀⠀⠀⢸⠴⠊⡀⠠⠀⠁⠀⠀⠀⠀⠂⢳⣕⡠⢁⡀⠀⢜⢆
⣼⠖⠓⠓⠒⠧⢀⡨⢔⢅⣕⠵⠮⠖⠒⠖⠤⠦⣤⣠⣀⠀⠀⢐⠨⡐⠠⣘⣪⣥⣀⣂⠤⠀⣀⠄⢳⣊⣔⠤⠙⠒⠺⡆
⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⠦⣘⡵⠟⠋⠁⠀⠀⠀⠀⠀⠉⠈⠉⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⡇⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣇⠀⢀⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣉⣉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠐⠒⠄⠢⠀⠄⡪⠐⡡⢂⠅⡪⢀⠔⡠⡂⢔⠔⡠⢂⢔⠔⡠⡢⢀⠀⡢⢀⢄⠀⡀⡀⡀⢀⠄⡀⡀⢀⢀⠀⠀⡀

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⠁⠀⠈⠙⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⠋⠉⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠽⠿⠿⠛⠿⠟⠿⠿⣿⠿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠄
⠘⠛⠛⠋⠘⠿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠁
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⣿⠂⣰⣶⣏
⣀⠀⠀⠀⠀⠀⠋⠙⠉⠈⢿⣿⠤⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣠⣤⣶⣶⣤⣿⣿⣷⣿
⣯⡙⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⡤⠀⡟⠉⠉⠙⢉⡛⠉⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠇⢠⣬⣤⣥⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣉⣿⣛⡛⢀⡩⠀⠀⢀⠀⠀⠀⠀⠀⠄⣠⣠⣦⣤⣆⠀⠀⠀⣴⣾⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣻⣤⣄⣶⣆⣂⣴⣿⣿⣿⣿⣧⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⢔⡗
⠀⠀⠠⢂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡠⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠰⣿⣿⣯⣦⡀⢀⣀⣤⣤⠄⠀⠀⠀⠀⠀⣀⣴⣆⣤⣜⣿⣏⠁⠆⠀⠀⠀⠀⣸⣦⣤⣀⣴⣴⡶
⠀⠀⠀⠈⠛⠋⠁⢨⣓⠿⠿⠿⢛⢦⣴⣶⣶⣿⣿⣿⣿⡿⢘⣿⣿⣶⣿⣦⣤⣤⣼⣿⠾⠿⠋⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣿⣿⣟⡿⣿⣿⣿⡿⠛⠁⣿⠿⠿⠿⠋⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠂⠀⠈⢧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠉⠀⠢⣷⣴⣄⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠉⠁⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⣤⣴⠤⠴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣶⣤⣦⣤

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠚⠛⠓⢲⠞⠉⠀⠀⠀⠀⠀⠉⠳⣴⠒⠛⠛⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⡴⠋⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠳⡄
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠈⢳⣤⣀⣀⠀⣀⠀⠈⠓⠦⠤⠤⣤⠤⠖⠚⢹⠗⠦⣄⣀⣀⣀⣀⡤⠖⢿⠙⠒⠦⢤⡤⠤⠤⠖⠋⠀⢀⡀⢀⣀⣠⣴⠋
⠀⠀⠀⠀⠀⢠⠎⠀⠈⣹⢿⡉⠀⠀⠀⢠⡀⢠⠏⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⢧⠀⣠⠀⠀⠀⠈⣹⢯⡁⠀⠈⢧
⠀⠀⠀⠀⠀⠿⢶⠆⠘⢓⡖⠙⠲⠦⠤⠶⢋⡏⠀⠀⠀⡾⠚⠧⢤⣄⣀⣀⣀⣀⣤⠤⠟⠺⣆⠀⠀⠘⣏⠳⠦⠤⠶⠚⠑⣖⠛⠀⢶⠾⠇
⠀⠀⠀⠀⠀⡴⠃⢀⡔⠋⠀⠀⠀⣀⠀⠀⡼⠀⠀⠀⣼⡥⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣽⡄⠀⠀⠸⡄⠀⢀⠀⠀⠀⠈⠳⣄⠀⠳⡀
⠀⠀⠀⢀⡼⢁⡴⠋⠀⠀⠀⠀⣰⠻⡆⠸⠗⢺⠂⠀⠁⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠁⠀⢺⠒⠿⢀⡞⢷⠀⠀⠀⠀⠈⠳⣄⠹⣄
⠀⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⢿⣀⡇⠀⢀⡏⠀⢀⡞⠁⡼⢧⠀⠀⠀⠀⠀⠀⠀⣰⢳⡀⠙⣆⠀⠈⣇⠀⠸⣄⣸⠂⠀⠀⠀⠀⠀⠈⠳⣜⣆
⠀⠠⠿⠋⠀⠀⠀⠀⠀⣰⡄⠀⠀⠉⠀⠀⣼⠀⣰⠋⠀⠰⡇⢈⡇⠀⠀⠀⠀⠀⠀⡇⢈⡇⠀⠈⢷⡀⢸⡄⠀⠈⠁⠀⢀⣦⠀⠀⠀⠀⠀⠈⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢹⡀⠀⠀⠀⢰⢃⡜⠁⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠹⣄⢷⠀⠀⠀⠀⡾⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠞⠀⠀⠀⢀⣿⠏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⡇⠀⠀⠀⠹⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡾⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⡄⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠛⠛⠲⠊⢁⣀⣀⣀⠀⠈⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠊⠁⣀⡤⠤⣀⡤⠚⠉⠉⠀⠀⠉⠙⠲⣄⠀⠘⠒⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠉⢁⡔⠉⠀⠀⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣦⣤⣤⣀⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⡞⠁⢀⠤⠒⠉⠉⠉⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡄⠉⠳⡄⠀⣧⣀
⠀⠀⠀⣀⠼⠀⣔⡁⠀⠀⣴⡿⣿⢿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠛⢦⠀⠈⠛⠀⣠⠤⠘⠲⢤⡀⠙⣆
⠀⢠⠞⠁⡠⠊⠁⠀⠀⠀⠈⢰⡇⠀⢸⣟⣷⣄⣠⢿⣤⣤⣾⡟⣿⡆⠀⠈⡇⠀⠀⡎⠠⡄⠀⠀⠀⠙⡆⠘⡇
⢀⡏⠀⣼⠁⠀⠀⠀⠀⠀⠀⠸⣧⣀⡈⢙⣫⡿⠁⠀⠉⢹⣍⣙⠋⠁⣀⡼⡇⠀⠀⠉⠙⠁⠀⠀⠀⠀⢹⠀⣿
⠸⡇⠀⣇⠀⠀⠀⠀⠀⣀⡤⣄⠀⠈⠉⢋⣁⣠⣀⡀⠀⠀⠙⠋⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡟
⠀⠙⢦⣈⠳⠤⣤⠴⠋⠁⠐⠋⠀⢠⠞⠋⠉⠉⠙⢿⣦⡀⠀⠠⣄⠀⠀⠀⡀⠀⠀⡤⢤⣀⡀⢀⣀⠴⢃⡼⠁
⠀⠀⠀⠈⣿⠀⠘⢆⡀⠀⠀⠀⠀⠀⢠⠤⣀⠀⠀⠀⠈⠛⠿⠿⠻⡆⢀⡰⠁⠀⠀⣀⡜⠈⠉⢭⣤⠴⠋
⠀⠀⠀⠀⠹⣄⠀⠀⠙⠒⠶⠤⠤⣤⡤⠤⠞⠓⢤⣀⣀⣀⣀⣤⡴⠚⠉⣨⢭⣍⣉⣁⣤⠤⠴⠋
⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⢀⣠⠖⠦⣤⣀⡼⠀⠀⠀⣸⣓⣺⡭⢴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⡼⠁⠀⠀⠀⠉⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⠴⠖⠚⡆⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⢑⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡠⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⣀⠀⠔⠒⠤⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠉⠀⠀⠀⠓⠐⠤⠤⠤⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔⡠⠂⢊⠄⡐⠡⠂⠄⠠⢂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠄⠀⢀⠔⠀⡀⠊⢔⠑⡡⠊⠀⠀⡡⢊⠔⠀⢀⠔⡀⡁⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠀⠀⠂⠀⠀⠀⠀⠊⠀⠁⠀⠈⠀⠁⠁⠈⠀⠀⠀⠈⠐⠁⠀⠐⠡⠊⠀⠠⠒⠆⢀
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡄⠀⢀⡄⠀⣠⠀⠀⡀⠀⠈⢀⠋⠉⣭⠶⢀⠔⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠼⠁⠬⠁⠈⣠⠀⢈⠀⣀⠁⠀⠜⣠⠀⡰⠃⠀⠐⡡⠂⠀⣀⣀⠈⠀⣀⠀⠀⠞⠐⢃⡔
⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⡄⠀⠁⠀⠁⠀⠁⢀⠀⠐⢁⠜⣀⣀⣀⣈⡉⠁⠀⢈⠒⡊⠀⢁⠀⠀⠞⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠊⠀⢠⠆⢠⠎⠀⠐⠋⠀⠀⠈⢀⡀⠀⠠⠊⠀⠀⠀⡨⠊⠀⠀⠁⠀⠀⠐⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠚⠀⠀⡘⠀⠀⣠⠀⠐⠃⠀⡄⠀⢠⠀⠈⢀⠔⠀⠀⠀⡠⠊⠀⠀⠀⢠⠆⠀⠀⠖⢀⡄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠆⠀⠁⡴⠀⡤⠀⠀⠀⠋⠀⠔⠁⠀⠀⠐⠀⠀⠞⠁⡄⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⢀⡀⠀⠀⠊⠀⡠⠂⠀⢀⠔⠁⠀⣠⠀⠀⠀⠀⡠⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠄⠀⠀⠚⠀⠀⠜⣀⠊⠀⢀⠀⠀⠀⠀⠀⠁⢀⠄⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⠤⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠁⢀⡨⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠁⠀⠀⠎⠀⠀⠀⠀⡔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠻⠶⣦⣀
⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣠⣴⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⢸⣇⣀⣀⢀⡾⠋⠀⠀⠀⠀⠀⠙⢷⡀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣠⡾⠛⠉⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠛⢷⣄⣠⣤⠶⠶⠶⣤⣄⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠀⠀⠀⠀⠀⠙⣷⡀⣀⣀⣀⠀⠈⣷⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠛⢷⣼⡇
⠀⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠿⣤
⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠹
⠀⠈⠙⠻⠶⠶⠶⠶⢶⣶⠶⠶⠶⢶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⣠⣿⣁⣘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾
⠀⠀⠀⠀⠀⠀⣰⣟⣁⣀⣀⣈⡉⠉⢉⣿⠋⠙⠿⠶⠶⠶⠶⠶⠶⠶⣶⡶⠶⠶⢶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⡿⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠠⠾⠿⠶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠷⠶⠶⠶⣶⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠐⠠⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄
⠀⠀⢀⡀⠀⠐⠀⠀⠀⠄⠀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠄⠀⠀⠈⠁⠠⠀⠀⠐⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠆⠐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠁⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⡈⠀⠀⡀⠀⠀⠄⠀⠀⠁
⠀⠀⠂⠀⠀⠀⢀⡀⠀⠀⠀⡀⢈⣠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣪⣴⣶⣾⣶⣿⡋⠁⠀⠀⠠
⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⢊⣼⣿⣿⣿⣿⡇⠀⠀⣀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⠿⠗⠁⠀⠀⠀⠈
⠀⠀⣀⠀⠀⠀⠀⠈⠡⠡⢞⣿⣿⣿⣿⣿⡗⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⢳⠶⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⠀⢏⠻⣁⠀⠐⣨⢋⣠⣤⣤⡀⢂⠀⠀⠀⠈⠀⠀⠂⠈
⠀⠀⠀⠈⠀⠀⠀⠁⢈⣴⣿⣿⣿⣿⣿⣿⣿⣄⢀⣑⣘⣟⣤⣦⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠈⠀⠀⠀⠀⠐⢻⣿⣯⣭⣽⣾⣿⣿⣿⣿⠧⠤⠴⠿⠛⠛⠋⠁⢻⣿⣿⣿⣿⠈⢄⠀⠀⠀⠄⠀⠠⠄
⠀⠐⠀⠀⠀⠀⠀⠈⢿⡿⢿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⣄⡀⠀⡀⠄⠀⠀⢀
⠀⠀⠀⠄⠀⠀⠀⠀⠈⢿⣦⣙⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠚
⠀⠈⠀⠠⠀⠀⠀⠂⠀⠄⠙⠿⣳⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣟⢏⢱⢃
⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠈⠛⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠼⣧⠀⠉⠸⠀⠀⠈⠡
⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁⠠⠀⠈⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠆⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡔⠁⠀⠀⠀⠀⠀⠀⢲⣆⣮⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⢮⣤⡝⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣟⣛⣟⣛⣻⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⢀⠀⢀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⣀⢈⣵⣾⣷⣾⡁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣧⠤⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣴⣦⣄
⠀⢈⢾⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⣀⠤⠴⠋⠉⠉⢪⣻⢿⡟⠒⠀⠀⠀⠀⠆⠀⠀⣼⣿⣇⡐⠂⠀⠀⠀⠀⢀⠤⠴⠋⠀⠈⠳⣙⡿⣿⣿⣿
⠀⠉⢘⡿⣿⣿⡿⡃⠉⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣱⠈⠀⠀⠀⠀⠠⣤⣴⣾⡝⠋⠀⠈⠒⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢸⠘⠛⠋
⠀⠀⠈⠁⠐⠆⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⢀⣶⡄⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⣴⠄⡀⠀⢠⣄⡄⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣶⣦⡀
⠀⢀⡞⣿⣿⡆⠐⠿⠏⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠛⢳⡞⡴⠛⠁⠘⠏⠃⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⢸⡘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠶⠀⠀⠴⠀⠀⠀⠀⠀⠀⠻⠏⣹⡉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠰⠂⠀⠆
⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐


⠀⢀⣠⣾⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⡐⢒⡒⠂⢠⢒⣒⠢
⠀⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠠⠀⠀⢠⣿⡄⠒⠀⠰⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⡅⢸⡇⠀⢨⢠⢿
⠀⠀⠀⠄⠤⢔⣒⠀⠀⠀⠀⠀⠀⠀⣠⣦⣰⣿⣿⣿⣆⣴⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣧⠀⠘⡸⣿⡀
⠀⠀⠀⠍⠭⠽⠒⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠽⠯⠄⠐⠼⠿⠴


⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣭⣍⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣠⣴⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡼⠀⡚⠋⠰⠶⠂
⠙⠛⡿⠛⠏⠁⠀⠙⡤⣀⠀⠐⡆⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⢀⡴⠋⠺⠟⠂
⠀⠈⢦⣀⣀⣀⣀⣀⣤⡼⢌⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⣠⣾⣀⣴⣏⣀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⢉⣾⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠉⠙⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⠲⠶⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⣠⠴⠖⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀
⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⡟
⢿
⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠙⠶⣄⣀⣀⣀⣀⣠⣤⣀⠀⠀⠀⠀⠀⣀⡴⠶⣄⡀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⢀⠀⠈⠉⠉⠁⠀⠀⠉⠛⠓⠒⢛⡋⠁⠀⢰⠏⠉⠛⠒⠚⠛⠉⣼⠉⠙⠲⠦⠤⠤⠶⠞⠋⠀⠉⠙⠛⠛⠋⠉
⠀⠀⢠⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣷⠀⢀⡟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣆
⠀⠀⠈⠛⠁⠀⠀⠀⠀⢀⡾⡆⠀⠀⠀⠙⠋⠀⣸⠃⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀⡀⠀⠀⣸⢻⡄⠀⠀⠀⠀⠀⠀⠈⠷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⠀⠀⠀⠀⠀⢠⣯⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢀⡞⢀⡀⠀⠀⣰⢧⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡿⠀⠀⠀⠀⠀⠀⣿⣤⣤⡀⠀⠀⠈⠉⣹⠏⠀⠀⠻⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡾⢛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣵⣤⣤⣀⡀⠀⠀⢶⣄⣴⣿⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣴⣄⣀
⠀⠀⠀⠀⠀⢀⣰⣯⠾⣻⣭⣴⠖⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⠿⠷⡦⣤⡀⠀⠀⢸⣿⣿⣿⣶⣮⣅⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠷⢦⣄
⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣻⣶⣤⣤⡄⠀⠀⠠⣿⣿⣿⠛⠿⣿⠛⠃⠀⠁⠀⣠⡾⡿⣿⡿⣿⣧⠈⠉⠀⠀⠀⠀⣼⣿⣿⣿⡻⣯⠛⠃⠀⠈
⠀⠀⠀⠀⠀⡸⣿⠻⣟⠿⣯⡉⠉⠁⠀⠀⠀⠀⣿⡘⣿⠳⠂⠘⠇⠀⠀⠀⠀⣿⣸⣿⡈⢷⡄⠻⡆⠀⠀⠀⠀⢀⡇⠙⠻⠉⠙⠛⠇
⠀⠀⠀⠀⢰⠃⢹⣖⠙⠧⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠙⢸⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢀⡏⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⢸⣷⣿⢿⣛⣁⡀⠀⠀⠀⢸⡇⠀⣰⠟⣋⣵⣤⠶⠀⠀⠀⠀⢀⡇⠀⠀⢠⣾⢋⣤⡶⠀⠀⣿⠀⢀⣀
⠀⠀⠀⠃⠀⢶⣿⣿⣶⣿⣍⡉⠉⠀⠀⠀⠈⣀⣰⣷⣾⣿⣽⣤⣤⣤⠀⠀⠀⠘⠃⣤⣰⣿⣴⣿⣧⣤⣄⣀⠀⠀⠀⠙⣧⢀⣴⣞⣛⣛⡉
⠀⠀⠀⠀⣠⣾⣿⡿⣿⣿⣯⣟⠛⠁⠀⠀⢠⣿⣿⣿⣿⡟⠿⣿⣉⠀⠀⠀⠀⠀⣠⣿⣿⣿⣟⡛⠻⣷⣾⠍⠀⠀⣠⣶⣿⣿⣿⣿⣿⣍⡋
⠀⠀⠀⠰⢏⣿⢹⢷⣄⠙⠷⠄⠀⠀⠀⠀⠀⢹⠻⣿⢿⣿⣀⡀⠉⠑⠀⠀⠀⠀⣿⢿⣿⣋⢛⣛⠛⠀⠁⠁⠀⠀⣿⡏⠘⣿⣿⣿⣿⣏⠛⠳⠄
⠀⠀⠀⠀⠈⠁⢸⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢻⠇⠀⠉⠁⠀⠀⠀⠀⠀⢠⠏⠉⢻⣽⣯⡉⠙⢦⡀⠀⢀⡾⠙⠟⠀⢹⣏⠻⠯⡙⠃
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠿⠿⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠿⠿⢷⣦⣴⡿⠉⢀⣠⣤⡀⠀⠀⠙⢿⣶⣶⡶⢶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⣀⣤⡀⠀⠈⠉⠀⢠⡿⠋⠉⠀⠀⠀⠀⠈⠁⠀⣀⣀⠀⠉⠻⣧⡀
⠀⠀⠀⢀⣠⣤⣼⡟⠀⢠⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣆⠀⠹⣷⣤⣄⡀
⠀⣠⣾⠟⠋⠉⠉⠁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠙⠻⣷⡀
⣰⡿⠁⢀⣴⡾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣦⠀⠘⣿
⣿⡇⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢹
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠈⢿⣦⡀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠄⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠙⠻⢷⣶⣶⣶⡾⠟⢻⣧⡀⠈⠛⠿⠷⠀⠀⠀⣠⣤⡀⠀⠀⠰⠾⠿⠋⠀⣰⡿⢿⣶⣶⣶⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⣀⣠⣤⣾⠟⠛⢿⣦⣄⣀⣀⣠⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠹⣷⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣼⡟⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⣹⡇⠀⢀⣼⡿⣷⡀⠀⠀⢸⣿⡀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠻⠷⠾⠟⠁⠀⣾⠏⠀⠘⣷⠀⠀⠀⠙⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣴⡿⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢸⣿⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⠉⠁⠀⠀⢀⣾⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⡾⠋⠀⠀⠀⠀⣴⡿⢻⣦⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⢹⡇⠀⠀⠀⠀⠀⠈⠻⠷⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⣶⠿⠃

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⡀⢀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⢤⡂⠀⠀⣄⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⡄⢤⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡤⡻⣄⠐⠈⢷⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣔⣀⣤⠬⠶⠒⠃⣁⠆⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡛⢶⣄⡀⡉⠙⠛⠓⣾⣾⠾⢲⠾⠿⠿⠿⠟⠋⠉⠉⠉⠉⠁⠁⠀⢀⡠⠟⠀⡈⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡖⢊⠛⣷⡦⡤⣄⡉⡀⡀⢀⡠⠀⠀⠀⢀⡀⢀⣀⣀⣤⢴⠴⠋⠁⠀⢰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣵⣀⡷⠀⠀⠀⠀⠀⠁⠀⠀⠐⠐⠛⠛⠉⠉⠉⢉⠀⢠⣀⣤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠈⠹⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⠑⠚⠉⠁⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣫⣄⠈⠙⠛⠓⠗⠒⠳⠒⢸⣀⣣⠴⠋⠁⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣈⡛⠞⠄⠉⠉⠐⠛⠙⠋⠉⠁⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⠒⠛⠉⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣴⣳⡔⣄⡠⢠⣩⣈⣤⣰⣏⣠⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⡛⠛⠛⠛⠛⡯⡼⢟⣉⠙⢉⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡁⠀⠀⠀⣼⣿⢏⠚⠛⣂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢋⠿⠵⢶⣶⣿⣿⡷⡷⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⣼⣿⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣇⠀⢿⣿⠀⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢟⡨⣻⠀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣄⢹⡆⠿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠁⢽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡌⢟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣼⣷⡤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣾⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠶⠚⠳⠶⢦⡴⠚⠉⠁⠀⠀⠀⠈⠙⢳⣦⡴⠶⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠶⠿⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⢶⣟⠋⠀⢀⣿⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣤⠿⠶⠖⠒⠒⠀⠀⠀⠀⢀⡤⠶⠒⠲⢾⣅⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⣿⣿⣿⣿⣿⡆⠀⠐⠉⠉⠳⠄⠀⠀⢀⣾⣿⣿⣿⣿⡿⠿⢿⡇⠀⠀⠀⠀⣀⣴⣏
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡾⠛⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠻⣆⠀⠈⠀⠀⠀⠀⢨⣁⠀⣹
⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠓⠀⠀⠀⠀⠀⠀⣾⠁⠀⢨⣿⣿⣿⣶⣾⠇⠀⠀⠀⣷⣶⣿⣿⣿⣇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠹⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⠻⠿⢟⡿⠋⠀⠀⠀⠀⠈⢿⡛⠷⠿⠋⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠛⠓⢶⣶⣺⠟⠁⠀⠀⠀⠀⠀⠀⠀⠹⢷⣲⡶⠖⠛⠋⠀⠀⠀⠀⠀⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠙⠳⠶⣤⣤⣤⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⣤⣴⣶⠟⠉
⠀⠀⠀⠀⠀⣠⠴⠻⣇⣀⠀⠀⠀⠀⣰⠇⠀⠙⠦⣤⣀⣀⡀⠀⠀⣀⣰⣦⣤⣄⣀⣀⣀⣀⣀⣠⣤⡶⠟⣧⠀⠀⠀⢸⣧⣤⠤⣦
⠀⠀⠀⠀⠛⣻⠟⠀⣀⠭⠛⠀⠀⣴⠃⠀⠀⠀⢴⣏⣉⡉⠛⠛⠛⠛⠉⢸⡆⠈⠉⠉⣿⠉⠉⠉⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢰⡇
⠀⠀⢀⡴⣚⡵⠖⠋⠁⠀⠀⢀⡾⠁⢀⣀⠀⠀⠀⠀⠉⣩⠗⠀⠀⠀⠀⢸⡇⠀⠀⢸⡷⠶⠛⣿⠀⠀⠀⠀⠸⣦⣤⢾⡆⠀⢸⡇
⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠛⠋⠉⢩⠋⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⠇⠀⠀⣄⠀⠀⠈⠀⠈⣧⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣼⣁⣤⠴⡶⠀⠀⡾⠀⠀⠀⡟⠶⠖⡶⠀⠀⢹⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣠⡴⠋⠀⠀⠀⠀⢀⡴⢿⣀⡀⠀⠈⠉⠀⠀⣧⠀⢸⠃⠀⠀⠠⠧⡆⡼⠁⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠞⠉⠀⠀⠀⠀⠀⢠⡾⢴⢀⡼⠁⠀⠀⠀⠀⠀⣿⢀⡏⠀⠀⠀⠀⢰⣷⠃⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⠎⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠓⢋⣀⣠⣤⣀⡀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠎⢁⡠⠤⠤⢤⡴⠋⠁⠀⠀⠀⠀⠉⠳⣄⠀⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⢠⠔⠋⠁⣀⣠⣾⣁⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣶⣶⡦⢤⣀⠀⠘⣆
⠀⠀⠀⠀⢰⠃⢀⠔⠊⠁⣀⣀⣤⣹⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⠿⣿⡷⠀⠈⢳⡀⠹⠦⢄
⠀⠀⣠⠔⠋⣠⠧⠀⠀⠸⠟⣻⠛⠻⣿⣆⠀⠀⣀⠀⠀⣠⣿⣏⠀⠙⢧⠀⠈⠁⡤⠂⠉⠉⠑⢦⡀⢳⡄
⠀⡼⠁⢠⠎⠀⠀⠀⠀⠀⢀⣿⠀⠘⢷⣿⣷⠾⠙⠷⣾⠿⣭⡿⠃⠀⣸⡆⠀⠸⣄⡶⠀⠀⠀⠀⢳⠀⣷
⢰⠇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠋⠓⠦⢤⠶⠋⠀⠀⠀⠘⢷⠦⢤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼
⠈⢧⡀⠣⣀⠀⢀⣠⠴⠒⣵⠆⠀⣠⡴⠶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢠⠇
⠀⠀⠙⠲⡌⠉⢏⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠻⣶⣤⣼⢷⡀⠀⣰⠀⠀⠘⢉⡗⠲⠖⠒⢉⡴⠋
⠀⠀⠀⠀⢧⠀⠀⠑⠤⣄⣀⣀⣀⣀⣁⣱⢦⣀⠀⠀⠀⠀⢀⣸⠥⠚⠣⠤⠤⠒⠋⢀⣀⡼⠋⠉
⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠉⠉⠉⣩⣄⣀⠈⣹⠋⠉⠉⣹⠤⢤⣚⣩⡍⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠖⠒⠋⠁⠀⠉⢹⠇⠀⠀⠀⠛⠋⠉⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣀⡤⠤⢶⡄⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⡇⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁

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

⠀⠀⠀⢀⡀⠀⠀⠹⠂⠀⠀⠓⠀⠀⠰
⠀⠀⠀⠀⠉⠁⣀⠤⠒⠈⠉⠉⠉⠑⠂⢄⡀⠁
⠀⢀⣀⠄⠀⡜⠁⠀⠀⢀⣄⠀⠀⢀⠀⠀⠙⡄
⠀⠀⠀⠁⠸⡇⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⡝⠀⠐⠂
⠀⠀⠴⠀⠀⠙⢄⡀⠀⠉⠒⠒⠀⠀⠀⣀⠜⠁⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠒⠉⠀⠀⡀
⠀⠀⠀⠀⠐⠋⠀⠀⣤⠀⠀⠀⡀⠀⢤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠒⠃⠀⠀⠉⠊⠁⠉⠒⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⣀⠀⠀⠀⣠⡶⠄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⣳⠀⠴⠱⠾⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣠⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠒⠖⠚⠋⠀⠀⢰⡏⠀⠑⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢀⡰⠦⣀⡀⠀⠀⠀⠀⢀⡀⠀⠠⢴⡳⠀⢈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠊⠁⠀⠀⠀⠀⠉⠉⠁⠀⠁⠀⠀⠀⠀⠉⠂⠌⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⠻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣠⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠉⠉⠛⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠙⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠛⠛⠛⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠛⠛⠛⠿⣶⣄⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡿⠋⠉⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣶⣶⣶⣦⣄⡀⢿⡆
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠙⢿⣿⡇
⠀⠀⠀⢠⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠈⠻⣧
⠀⠀⠀⠘⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⢛⣿⠟⠛⠛⠛⢛⣿⠿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⣰⣿⣃⣀⣻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠈⠛⠛⠛⢻⣿⠋⠛⠷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⢻⣿⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⢀⣾⣿⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣀⣀⣀⣀⣈⠉⠉⠉⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣻⡏⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢇⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟

⠀⠀⠀⠀⠀⣀⠀⢤⠤⡄⡐
⠀⠀⣴⣌⠃⠀⣀⠠⠤⠒⢿
⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⢨⠀⠀⢠⡄⢠⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⢳⠤⠀⠀⠀⠀⡸⠀⠀⡀⠀⠀⠀⠀⢘⡁⠀⠀⠈⠀⠰⠀⠀⠰⠀⠁⠀⠀⠀⠀⠐
⠀⠀⠀⢠⠅⠀⠀⠀⣤⠊⠀⠀⢺⣠⣔⡭⠀⠠⠀⣀⠀⢠⢌⠖⣂⣠⡄⡀⠀⡀⠈⡰⠁⢀⠘⠁
⠀⠀⢰⠇⠀⢀⠤⣶⠁⠀⠀⠀⠀⣿⠛⠀⠐⠀⠀⠀⠀⣿⣿⠿⠥⠀⠀⠁⠀⠐⢿⠇⠀⠈
⠀⠀⠈⢷⣚⡁⠚⠑⡀⠀⠠⠀⣼⡇⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⢀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠙⠚⠤⠼⠀⠄⠚⠻⠷⠀⢀⡀⠀⠀⢠⣿⣿⣿⣀⠀⠀⠀⠀⢀⣴⣇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⡠⠘⠃⡴⡀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⢱⠯⠀⠐⢿⠲⠀⠀⠀⠀⠀⠀⠀⢠⣼⠷⠇⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢰⠲⠤⠀⠀⣾⠀⠀⠀⠀⠘⠰⣤⠄⠀⠀⠀⠀⠀⢠⠀⠀⠋⢉⠈⣏⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣆⠀⠀⠀⠀⠀⠟⠮⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠉⠀⠈⠳⣤⣏⣠⡀
⠀⠀⠀⠀⠀⠀⣡⡞⠀⡀⢠⣄⡤⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⠀⠿⣿⢿
⠀⠀⠀⠀⠀⠴⠏⠙⠀⠻⣾⣿⡇⠈⠇⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⣂⣄⣠⣿⡀
⠀⠀⠀⠀⠃⠀⠀⠚⠦⠀⠈⠋⡇⠀⣀⠀⠁⠈⠀⠀⠀⠀⠒⢷⡀⠀⠃⠤⠀⠀⠀⠀⠀⢉⡉⡀⠀⠀⢀⡷⠉⠉⠸⣟⡄
⠀⠀⠀⠀⠆⠀⠀⠀⠠⠀⡀⣀⢷⡀⠸⣶⠠⡀⠀⠤⠀⠀⠘⣿⣧⣠⡤⣤⣤⠄⠀⠀⠀⣈⠻⣿⣶⣿⠉⠀⠀⠈⠀⠼⠃
⠀⠀⠀⠈⠲⣷⠗⠀⠀⠣⠁⠀⢸⣿⣦⡈⠋⣑⣀⣀⡀⣀⣠⡜⠋⠙⠿⣋⣥⠦⠀⠀⠀⠘⠛⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠈⢶⣔⣲⡄⠀⡠⢿⣿⡞⢷⡶⣿⠿⣬⣙⠛⢯⡁⠀⠠⠀⣤⢤⠀⠀⢀⠀⠀⡆⠠⣄⣄⢀⡃⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠈⢷⡀⢛⣆⡾⠃⢻⡀⠹⢿⡿⠛⢠⡀⢠⡐⠘⠈⠇⠀⠈⠂⠀⠸⠀⠙⠀⠈⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠛⣆⠀⠀⠈⠇⠀⠀⢆⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠄⠈⠁⠀⠄⠘⠀⠀⠘⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠦⢄⣠⢄⡀⠀⠀⠀⢀⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠈⢠⠞⠉⠉⢳⠤⣇⡀⠀⠈⠓⢦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⠓⠀⠀⠀⣰⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠁⣠⠖⠒⠲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡋⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⢇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠤⠤⠂⠀⠀⠂⠀⠀⡹⠋⠙⠤⠀⠀⠀⠀⢚⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⡀⠉⠒⠀⣄⠠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠋⠰⣄⠀⠀⠀⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠦⠶⠚⡉⠓⠾⠖⢶⣉⢀⡵⠲⢲⡀⠀⠀⠀⠀⠉⢲⡖⠚⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡖⣸⠡⡝⠀⢠⡆⠀⣈⢡⢀⣶⠀⠑⠦⡤⠤⠚⠉⠉⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣸⢡⠇⢱⠁⠀⡾⠁⢠⡇⡞⣼⠇⠀⣴⠀⢀⡆⡼⣸⡏⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠇⡞⠀⠆⠀⣰⡇⠀⡾⣸⢡⡟⠀⢀⡟⠀⡼⢰⢣⡛⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡞⣸⠠⡘⡐⢀⡟⠀⣸⢃⠇⣾⠁⠀⣾⠁⢰⠇⡎⡾⠏⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣸⢡⠃⢣⠁⠀⣼⠃⢠⡏⡜⣸⡏⠀⢰⡇⠀⡟⢸⠰⡽⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢀⢣⠇⡎⡄⠄⠀⢰⡏⠀⡼⢠⢡⣿⠀⢀⣿⠀⣸⠁⠇⢆⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⡆⡞⡘⠀⠀⠀⠀⡿⠀⢰⠃⡆⣾⠇⠀⣸⠃⢀⡏⡘⠀⡌⠀⠀⡟
⠀⠀⠀⠀⠀⠀⡰⢰⢁⠃⠢⠀⠀⢸⠃⢀⡏⡸⣰⡜⠀⢠⡟⠀⡼⢀⠁⢰⠁⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠋⠀⠸⠀⠃⠛⠁⠀⠘⠁⠀⠃⠈⠀⠃⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⠀⠀⢀⣀⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⣀⣿⣿⣿⣿⡿⠋⠁⠈⢮⠩⠁⠀⠃⠉⠙⠗⠒⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠀⠀⣀⠁⢀⣁⣀⣤⣤⠤⠤
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣯⣽⣠⣖⣀⠀⠀⠀⠀⠀⡄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠳⢄⢠⣄⡀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠉⠙⢷⡄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢰⢦⣄⢒⠟⠁⠀⡐⠋⢤⣭⡓⣦⡄⠀⠐⠀⠂⢀⣀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⣸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢯⠀⣤⣊⠀⢀⡴⠛⠛⢿⡆⡀⠀⠀⠀⠚⣿⠾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⣿⣿
⠀⠀⠀⠀⠀⠀⠠⠲⠻⠁⠀⠙⢵⣿⣧⣎⡀⠀⠠⠊⠀⠀⠀⠀⡀⠌⢱⡄⠀⠉⢓⠤⣀⠀⠀⠀⢀⣼⣿⣷⣿⡿⠁⣠⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢻⡼⣽⠛⠂⠀⠀⠀⠀⠀⠈⠂⣀⢹⣶⣦⣤⣽⣮⣽⣶⣦⡟⠛⣿⣿⠟⡑⣨⣻⠟
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠐⣀⣠⣴⡿⣿⣿⣛⡏⡁⠘⠁⠁⠉⠁⠁⠈⠐⠒⣠⣴⣿⡯⠊⣬
⠀⠠⡀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣿⣶⣿⣿⣿⣿⠿⡿⣶⠤⠂⠒⠒⠺⠟⠛⠛⢯⠵⠛⠠⡚⠁
⠀⣴⣄⣄⠢⠀⠀⠀⠈⠁⠀⠀⠉⠉⠸⠿⣿⣿⡿⠿⠊⢉⠀⠉⠹⡦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡦⠖⠁⡀⢀⡔⢠
⠀⢈⠻⣿⣶⠤⠀⠤⣀⠀⠀⠀⠀⠀⠐⠈⠉⠈⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠁⣠⠆⣪⡀⡊⠄
⠀⠀⠁⠉⣭⣷⣶⣄⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠛⠉⠀⢀⣠⣶⣦⠀⢿⡛⠀⠙⠒
⠀⠀⠀⠀⠨⠙⠛⠿⢿⣇⣄⣴⣏⣿⡇⠀⢈⣦⠀⣀⠀⠀⠀⠀⢒⡂⢠⣴⣾⣷⠶⣿⣷⡤⣿⣿⣿⣿⠗⠀⣝⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠥⠀⠀⠛⠁⢲⣿⡝⠻⢿⡖⠿⡿⠿⢿⣿⠿⠻⢿⣿⠿⠉⠉⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣦⣤⣀⣴⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠿⠿⠛⠻⠿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⣀
⠀⠀⠀⠀⢀⣠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⣠⣾⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣦
⠀⣴⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧
⣸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠘⠻⣿⣿⣿⣶⣦⣤⣤⣶⣾⣿⣿⣧⡀⠀⠀⢠⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣤⣀⣀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠛⢿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣤⣀⡀⠀⠀⠀⣤⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠙⠟⠁⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⢠⣿⣿⣆⠀⠀⠀⠀⠀⢀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⡿⠋⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣄⠀⠀⠀⠀⠀⢀⣼⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣧⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⢿⣿⣿⣿⠟⠀⠀⠀⠀⠹⣿⣿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⡀⠀⠀⠀⠀⠐⠂⠐⠂⠈⠡⡐⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠦⢌⡓⣆⣤⡤⣤⡀⢀⠒⠒⠛⠛⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣦⣭⣰⣭⣭⣍⡙⠛⠓⠭⣉⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⣺⣒⣀⣈⡉⡛⠛⠿⢒⡒⠚⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣭⣈⡹⢯⣭⡛⢿⠿⡗⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣙⣻⣿⡻⠿⠿⠿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⣿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠱⠄⠀⠀⣶⣶⢦⣤⠤⠤⡤⠤⠤⠴⠤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣿⣻⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣭⣿⣭⣿⣯⣯⣯⣿⣽⣭⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⠋⣿⠏⣿⠷⠚⣻⣿⣿⣿
⣇⡀⣿⣿⣿⣿⣿⣿⣿⣯⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡛⣵⠋⠀⣼⣿⣿⣿⣿
⠿⣿⣷⣤⣮⣿⢛⠛⢛⣽⣧⣀⡸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠳⣀⣠⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂
⠀⠈⠛⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠋⠅
⠀⠀⠀⠘⠗⠀⠀⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠟⠉⠎⠈
⠀⠀⠀⠀⠀⠙⠿⢷⣦⣀⡘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠐⠩⠤⠤
⠀⠀⠤⠐⠀⠠⠒⠀⠀⠀⢄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡗⠠⠔⠉⠉
⡁⢠⡀⠀⠀⠀⠀⠀⠠⠀⠄⠄⠀⠀⣾⣿⣿⡿⠩⠀⠀⠀⠀⢀⠀⠀⠀⠉
⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣷⡀⢠⣦⣶⣄⠀⣴⣧⣄⣤⣄
⠀⢀⣤⣄⣀⣀⣠⣤⣄⣤⣀⣤⣄⣀⣻⣿⣿⣷⣤⣶⣤⣶⣦⣤⣤⣴⣾⣟⣬⡻⠃⢻⣿⣿⣿⣿⣿⣿⣿⠛⠏
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠅⣤⣤⣀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣯⣯⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠐⠚⠛⠛⠒⠒⠢⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣉⣤⣴⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠻⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣠⣤⣶⣴⣶⣼⣾⣯⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⣷⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⢰⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣴⣶⡵⠂⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠻⣿⣿⣿⡧⣄⡠⣤⣷⣶⣃⡀⠀⠀⠀⠀⣠⣴⣿⣧⣾⡆⣿⣿⣆⢈⡁⠀⠀⠀⠀⣼⣷⣿⣶⣶⣿⣿⠇
⠀⠀⠀⠀⠈⠉⠁⠀⢳⣽⡛⠛⠛⢡⡳⣶⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⠟⠟⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⣿⡿⢿⣮⣛⣿⣿⣿⣿⡿⠋⠀⢹⡛⠿⠛⠛⠁⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠧⠀⠀⠻⡄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠉⠉⠁⢠⣷⣴⣄⠀⠹⡄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⣟⠛⠓⠀⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⡿⣛⣭⣥⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣔⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢢⣿⣿⣿⣿⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣙⢿⣿⣿⣿⣿⣿⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣭⡷⢮⣭⣛⡻⠿⢿⣿⣶⣶⣭⣭⣭⣭⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣥⢻⡆⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢮⣝⣛⠷⢭⣝⣛⡻⠿⠶⠶⣮⣭⣭⣝⣛⣛⣛⣛⣛⣛⣛⣛⣛⣫⣭⣴⠾⢛⣥⠾⠣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢫⣽⡷⣶⣦⣭⣭⣭⣛⣛⣛⣛⣛⠿⠿⠿⠿⠿⠿⠿⠿⢟⣛⣛⣛⣬⣭⡶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣮⣝⡳⠬⣛⣛⡛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠛⠛⢁⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠻⣶⣮⣭⣍⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣭⣭⣿⣷⡾⢟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣭⣭⣭⣝⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣭⡉⣵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⠾⣿⣝⣛⣛⠿⣛⣛⣛⣩⣽⣿⣿⠿⢟⡋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣶⢶⡶⠭⠭⢭⣭⡭⢭⣵⣶⣶⠟⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣭⣭⣛⣛⣒⣢⣤⣭⣭⣵⡶⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣥⣋⣉⡉⠉⠉⠹⣛⣫⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡋⠙⠛⠛⣛⡿⢟⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠻⠓⠚⠶⠞⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠛⠱⠟⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢙⣛⣫⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠿⣫⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⡆⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠛⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠓⠐⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡀⠄⠂⠐⠀⠂⠂⠐⠐⠀⠄⢀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠐⠀⠀⠀⠠⠀⠀⠀⠰⠐⠀⠁⠀⠀⣠⠾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠒⠶⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠴⠒⠉⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⡀⠉⠓⠒⠶⠤⠤⠤⠤⠤⠤⠖⠒⠊⠉⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠉⠒⠦⢄⣀⣀⡀⠀⠀⣀⣀⣀⠤⠔⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠲⠤⣄⣀⡀⠀⠉⠉⠉⢉⣀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠑⠦⢄⣀⣀⡀⠉⠉⢉⣉⣉⠥⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⠤⣀⣀⠀⠈⠉⢉⣉⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠓⠦⠤⠬⠭⢭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣑⡒⣒⣒⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣚⣻⠏




⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⣠⡶⠟⠛⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⣀⣿⣦⣤⣤⣾⠋⠀⠀⠀⠀⠀⠈⢻⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⢠⡾⠋⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⢻⣦⡶⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⠙⢷⡄
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⢻⣦⣤⠶⣤⣤⡈⣿
⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠻⣿
⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠙⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⢀⣴⠾⠋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣙⡳⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠛⠁
⢠⡟⠁⠀⠀⣿⠁⠀⠀⠀⠀⠀⢀⣠⣴⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠈⢿⣤⣤⡾⠋⠀⠀⠀⠀⠀⣰⡟⠉⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⠀⠀⠀⣠⡶⠟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢻⣆⡀⣀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠁⣿⠃⠀⢸⡏⠀⠀⢠⡿
⠀⠀⠀⠀⠀⢀⣤⡾⠛⣿⠀⠀⠉⠛⠋⠁⠀⠀⠀⣀⣼⣷⠀⢸⣏⠀⠀⢠⡿⠀⠀⠘⢷⣤⣴⠿⠁
⠀⠀⠀⠀⢰⡟⠁⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠾⠋⠁⣿⠀⠀⠙⠷⠾⠛⠁
⠀⠀⠀⠀⠘⣧⣄⣠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⢠⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⢒⣒⣂⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡶⠟⣋⣥⠶⠛⠋⠉⢀⠀⠀⣀⣜⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠦⠀⠀⠀⠀⠀⣀⣴⠶⠛⠛⠋⢁⣴⠞⠋⠁⠀⢀⣤⣾⡿⠟⠛⠋⠉⠉⠉⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣉⣤⡤⠄⢀⡴⠋⠀⠀⣀⣤⡞⠛⠉⢀⣀⣤⡤⠤⠀⠀⠀⠀⣠⠟⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⡴⠛⣉⣠⡤⠀⣀⣤⡶⠞⠛⠉⠀⢀⣤⡞⠛⠉⠀⣀⣤⡖⣀⣠⣞⡁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠰⠚⠛⠁⠀⠀⠀⠀⠀⢈⣴⢞⣡⣶⠟⠋⢁⣰⠾⠋⠁⠀⢀⣠⡶⠟⠋⢉⣠⣤⣶⠿⠛⣋⡿⠛⠉⠉⠉⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⣿⡿⢋⣠⠴⠛⠁⣀⡤⢀⠴⠛⠁⢀⡴⠞⠛⠋⠁⠀⣀⣈⣁⣀⣀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠾⢶⣯⡶⠟⣻⡟⠁⢀⣴⠟⠁⠀⣀⣤⠔⠂⠀⢀⣠⣴⠶⠛⠛⠉⠉⣁⣀⣠⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠋⠀⣰⣟⣥⠆⣠⠾⠋⠀⠀⠠⠞⠋⠁⠀⠀⠤⠶⠟⠛⠋⠉⢉⣩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⢙⣦⣶⣿⠟⠁⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⡀⠀⣀⣠⣤⣾⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣼⠋⠘⠿⣶⠿⠷⣿⠟⠋⠉⠀⠠⠤⠶⣾⣻⣭⣶⣿⣯⣥⣤⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠻⠶⡶⠶⠿⣦⣘⣿⣶⣤⣤⡶⠶⠒⢉⣁⣤⣤⡤⠀⠀⠀⢈⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣤⡶⠟⠛⢁⣤⣾⣿⣟⣩⡄⠀⠀⣤⣶⡿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⡟⠁⠠⢤⣶⣿⣯⡭⠍⠉⠀⢀⣤⣶⠟⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⠶⢛⣉⣀⣠⣤⡤⣶⡶⠟⠋⠀⢹⣾⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣀⣬⣿⣿⣿⠾⠿⢀⣀⣤⡶⠟⠁⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⢉⣽⠟⠉⠁⢀⣀⡚⠉⠩⠶⠞⠛⠛⢛⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⠤⠌⣉⡡⣤⡴⠶⠶⢟⣪⣶⡟⢁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠈⠠⠤⠾⠛⠉⠑⠚⠛⠛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠶⣶⡶⠶⠛⠃⠀⣀⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⣾⣉⠀⠀⠒⠛⠉⠉⠰⠒⣳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⡿⠛⠂⢀⣀⣀⣠⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⠋⣉⣉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡉⣉⠽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠍⠉⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡻⣥⠀⠀⣶⠄
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⠀⣤⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⣷⠄⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣍⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⣠⠟⠰⠞⠋⣩⠏⠀⠀⣤⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠀⠀⠀⢰⠟⠛⠇⠀⠐⠛⠁⡀⠠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⣤⠤⠄⠘⢦⡸⡄⢆⣠⠴⠚⠀⠀⠊⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⣁⠤⠤⠟⠀⣀⣀⣀⣀
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠒⠒⠁⠀⠉⠁
⠀⢀