Absorb Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣧⡀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡦⠀⠀⣼⣧⠀⠀⠀⢠⡿⠁⠘⢷⡀⠀⠀⢀⣾⣆⠀⠠⣾⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣼⠋⠙⣧⡀⢠⡟⠀⠀⠀⠈⢿⡄⢀⣾⠃⠻⣦⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⢷⡈⠀⠀⠀⠀⠀⠈⢁⡾⠁⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⠘⣧⠀⣶⠀⠀⠀⠀⣼⠃⣤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣧⡈⠓⠀⢀⣴⠏⠀⠈⠻⠀⠀⠀⠹⣆⡙⠓⠀⢀⣼⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠈⠙⠛⠛⠛⠁⠀⠛⠷⠶⠶⠾⠛⠀⠈⠛⠛⠛⠋⠁⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⣦⣤⣤⠶⠷⠶⣤⣤⣴⠶⠷⢶⣤⣤⡶⠾⠶⣦⣤⣤⠶⠾⠶⣤⣤⣴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣤⣴⠞⠛⠷⣦⣤⡴⠟⠛⠶⣤⣤⠶⠛⠻⢦⣤⣴⠾⠛⠳⣦⣤⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠶⣤⡶⠟⠛⠳⢦⣤⠶⠛⠛⠷⢦⡴⠾⠛⠛⠶⣤⡴⠞⠛⠻⢶⣤⠶⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠚⠛⠛⠛⠒⠒⠚⠛⠛⠛⠒⠒⠛⠛⠛⠛⠒⠒⠛⠛⠛⠒⠒⠓⠛⠃

⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣶⣦⣸⣿⣇⣴⣶⠀⠀⠀⠀⠀⠀⠀⣶⣶⣬⣿⣧⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⢴⣶⣄⣿⣿⣴⣶⡆
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⢿⣿⡷⠀⠀⠀⠀⠀⠰⣿⣿⠆⠀⠀⠺⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⠇
⣠⣤⣀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⢀⣠⣤
⠛⠻⢿⣷⣦⣤⣴⣿⡿⠛⠿⣿⣷⣤⣤⣾⣿⠿⠛⠿⣿⣦⣤⣴⣾⡿⠟⠻⢿⣿⣦⣤⣴⣿⡿⠛⠻⣿⣷⣤⣤⣶⣿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⣰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⢰⣿⣿⣿⣿⣿⣿⣿⡄⣤⠞⠋⢉⣉⣉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠙⢿⣿⣿⣿⠇⠀⠀⠸⣯⣭⣽⠗⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠆⠛⢿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⢛⡛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡀⠀⠸⣧⣤⡽⠃⠀⢀⣠⣾⣿⡿⠻⣿⣿⣿⣿⣿⣬⣥⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣦⣶⣶⣾⡿⢿⣿⣿⣄⣃⣼⣿⡏⠤⢙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠐⢀⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠡⠌⣿⣿⡟⡛⢿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣾⣿⣿⣧⣥⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣬⣿⣿⣴⣾⣷⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣾⣷⣤⣿⣿⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢾⣿⣿⠆⠀⠀⠀⠰⣿⣿⡇⠀⠀⠀⠀⠀⠀⢺⣿⣿⠆⠀⠀⠀⠰⣿⣿⡷⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣠⣤⣄⡀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠀⠀⢀⣠⣤
⠻⠿⢿⣿⣦⣤⣤⣴⣿⣿⠿⠿⢿⣿⣦⣤⣤⣴⣿⡿⠿⠿⢿⣿⣦⣤⣤⣴⣿⡿⠿⠿⢿⣿⣦⣤⣤⣴⣿⡿⠿⠿⣿⣿⣦⣤⣤⣴⣿⡿⠿
⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄
⠀⠀⢠⣾⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢈⣿⣿⠀⠀⠀⠀⠀⣼⣿⡿⠿⣿⣷⡀⠀⠀⠀⢰⣿⡿⠛⢻⣿⣧
⠀⠀⣿⣿⡇⠀⣹⣿⡇⠀⠀⠀⣠⣾⣿⣿⣷⣦⠀⠀⠀⠀⠻⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣄⣠⣿⣿⠇⠀⠀⠀⠸⣿⣷⣤⣾⣿⡟
⠀⠀⠘⢿⣿⣿⣿⠿⠁⠀⠀⠀⣿⣿⡁⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⣷⡀⠀⠀⠀⠀⠀⣼⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠘⣿⡀⠀⠀⢀⣼⠏⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⣸⡇⠀⢀⣾⠋⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣶⠟⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⡀⠀⠀⠀⠀⠀⣿⡄⠸⠟⠉⠀⣠⣤⣤⣤⣀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠸⣧⣀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠸⣯⣥⣤⣭⡿⠀⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠹⣷⣄⣈⠉⠀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⠛⠉⠉⠁⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠶⠾⠛⠛⠛⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⣤⣶⢶⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠉⠀⠀⠀⠀⠀⢠⡾⢻⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢷⣤⡀⠀⠀⠘⣿⣤⣠⣼⡿⠀⠀⠀⢀⣠⡾⠋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⠛⠷⣶⣤⣄⣉⣉⣁⣀⣤⣴⡾⠟⠉⠀⠀⢸⣏⣹⡇⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⢀⣴⢶⣄⠀⠀⠀⠉⠉⠀⠀⠀⠘⠷⠾⠃⠀⠀⢀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠷⡾⠏⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⡶⠶⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⢰⡟⢻⡆⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣠⡾⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣈⠛⠛⠁⠀⠀⠸⣧⣼⠇⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⣦⣤⣤⣤⣤⣤⣶⠾⠛⠁





⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣿⣿⣿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⢀⣤⡀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣾⡿⢁⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠈⣿⣷⣿⣟⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠟⠀⠀⠀⢀⡀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢻⣿⣴⣿⢏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠿⠃
⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀
⠛⠻⣿⣶⣤⣤⣾⡿⠟⠻⣿⣶⣤⣤⣾⡿⠛⠿⣿⣦⣤⣤⣾⡿⠛⠿⣿⣦⣤⣴⣾⡿⠛⠿⣿⣦⣤⣴⣾⡿⠛
⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⢈⣙⠛⠛⠉⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠈⠙⠛⠛⠉⢀⡀⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⣿⣆⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣤⣿⡟⠀⠀⠀⠀⣾⣿⠉⢹⣿⡆⠀⠀⠀⠀⠀⠸⣿⣦⣴⣿⠇
⠀⠀⢀⣤⣤⣄⡀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠀⠀⠀⠀⢀⠀⠀⠉⠛⠛⠉
⠀⢠⣿⡟⠙⣿⣷⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣷
⠀⠀⠻⣿⣾⡿⠏⠀⠀⠀⠀⠀⠀⢿⣷⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣤⣼⣿⠃⠀⠀⠀⠀⠀⢸⣿⣤⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⢠⣾⣷⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⣿⣿⣿⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⢿⣿⣿⡿⠃⠀⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣀⠉⠉⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⡿⠟⡛⠻⣿⣿⠿⢛⡛⠿⣿⣿⠟⢛⠻⢿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣴⣿⣿⣷⣦⣤⣶⣿⣿⣶⣤⣴⣾⣿⣿⣦⣤⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣻⣿⣟⣛⣛⣿⣿⣛⣛⣿⣿⣛⣛⣿⣿⣛⣛⣻⣿⣟⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⣿⠿⠿⣿⣿⠿⠿⣿⣿⠿⠿⣿⣿⡿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣿⣿⣶⣾⣿⣿⣶⣶⣿⣿⣷⣶⣿⣿⣷⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢹⡆⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡄⠀⠀⠀⣿⠀⠀⠀⢸⠇⠀⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡙⠃⢀⣴⠏⠀⠀⡀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⢀⡾⠻⣆⠀⠀⠐⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠹⣧⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠛⠓⢀⡾⠋⢻⡄⠠⠤⡤⠀⣾⠁⠀⠀⠀⢹⡆⠘⠛⠋⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣀⡾⠁⠀⠀⢻⡄⠀⠀⢀⣻⣄⣀⣀⣤⣼⠷⠄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣦⣤⣀⣀⠈⠙⠛⠓⠶⠶⠾⠷⠆⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⣠⣤⣶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠛⠛⠳⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠛⠛⠋⠉⠁⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠟⠶⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⣄⡀⠀⠀⠉⠙⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⡤⠶⠶⠶⠛⠛⠉⠁⠀⠀⣀⣤⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⠦⣤⣤⣤⣤⠴⠶⠶⠛⠋⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⡀⠀⠀⡰⠃⠀⠀⠳⡀⠀⠀⣠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠳⠀⡼⠁⠀⠀⠀⠀⠹⡄⠰⠃⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢰⠃⡔⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢲⡀⠀⠀⢿⠀⡁⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠦⢭⣥⠄⠘⢆⠙⠦⠄⠀⠀⢀⠞⠀⢤⣤⠤⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡉⢉⣍⡉⠉⠉⠉⠉⠉⣉⣉⠉⠉⠉⠉⠉⣉⣍⠉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠻⠤⠃⠀⡤⢤⠀⠘⠧⠼⠀⢠⠤⡄⠀⠣⠼⠃⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠟⠸⠾⠨⠆⠣⠇⠛⠙⠾⠨⠆⠀⠓⠃⠥⠛⠨⠆⠸⠘⠤⠸⠚⠸⠄


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠭⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠐⠒⠶⠚⠁⠀⠲⠒⠈⠐⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣦⡀⠤⠀⠀⠀⠎⠹⠃⠘⠛⢁⡴⣤⠀⠀⢶⡶⠈⢙⡒⠄⢠⣀⠀⠀⠀⠀⠀⠀⡞⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠓⠦⢀⡫⠗⠀⠀⣠⢄⠀⠀⠀⠠⠦⠀⠀⡀⠘⠋⠀⠀⡠⡉⠒⠤⣀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⣔⢂⠈⠉⠲⡖⣌⠉⠀⢴⠶⠀⡴⢦⠈⠛⣀⠤⡀⠀⠉⢠⠖⢲⠌⠓⠶⠤⢤⣀⡀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠃⠈⠁⠀⠀⡀⠀⠀⠉⠒⠤⣄⡀⠉⠁⠀⠀⠓⠚⡁⠀⢀⡈⡙⠉⢠⣶⠆⠀⠀⠈⠙⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⣏⠆⠀⠘⠬⠀⠀⠰⠵⠀⠀⣬⡑⠢⣤⠠⢄⡘⠊⠀⠈⠉⠀⠀⢀⢤⡀⠐⠛⢃⡴⢹
⠀⠀⠀⠀⠀⠀⣀⣠⢤⡄⢀⡤⡄⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⣈⣀⠀⠀⠉⠒⠓⠄⡀⠀⠀⣔⠄⠈⠛⠁⣠⠴⢋⡄⣼
⠀⠀⢀⢠⠖⠋⠁⠀⠀⢧⣀⠓⠃⠀⠀⢰⢢⠀⠘⠶⠚⠀⠀⠀⣇⠀⡇⠀⡀⠀⠀⡤⢬⡑⠤⣤⣠⡤⠔⠊⢁⣠⠀⡅⣸
⠀⠀⠏⣯⠀⠀⠀⠀⠀⠀⠈⠉⠒⢦⡀⠀⠁⢀⡀⠀⠀⠨⠇⠀⠈⠉⠁⠀⠉⠀⠀⠓⠶⠋⠀⠀⡔⠢⠀⠸⠆⠀⠈⠒⢻⡄⠀⠀⠀⣀⣀⠀⠀⡀
⠀⠀⠀⠈⠓⠶⠶⢤⡀⠀⠀⠀⠀⠀⠛⢄⡀⠘⠚⠀⢀⣀⠀⠀⠀⠀⢰⣓⠀⠀⠀⠛⠃⠀⠀⠀⡇⠰⠳⠀⠈⢁⠐⠂⠘⡟⠉⠉⠉⠉⠉⠙⢳⡀
⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣄⡘⠚⠀⠀⣤⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢠⢶⠀⠀⡀⠀⠣⠃⡠⠚⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⡤⡄⠀⠀⠉⢀⡀⠀⠈⠋⠀⠀⠽⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠴⠞⠃⢤⡀⠀⠘⠞⢠⡀⡄⠀⡀⠔⠉⠀⠀⠀⠀⠀⢀⡴⠚⠋⠁
⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠶⠶⠖⠒⠒⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠽⠵⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠤⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣊⣤⠄⠀⠀⠂⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠽⠀⠠⠀⠀⠀⠀⠠⠄⠂⠀⠠⠠⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠤⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠈⠁⠀⠀⠁⠰⠚⠋⠁⠀⠠⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠠⠂⠈⢁⡀⠀⠀⠠⠄⠀⠀⠐⠂⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠐⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠇
⠀⠀⠀⠀⢀⡤⠀⠀⠁⣤⡌⠁⠀⠀⠀⠁⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠒⠒⠀⠀⡄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠉⠀⠈⢦⠃⠉⠀⠀⢡⡎⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⢀⠀⣀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠀⠐⠀⠀⠀⠀⢂⠄⠀⠠⣀⡠
⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠒⢢⡔⡂⠀⠀⠈⠀⠀⠐⠀⠀⠀⡅⠀⠈⠠⠀⠀⠀⡠⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⢀⡀⠉⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠄⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⡀⡀⠃⠂⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⠄⠀⠀⣠⠄⠠⠄⠀⠰⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠔⠈⠀⠄⠀⠀⠂⢀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁
⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠠⠠⠂⠆⠀⢠⠎⠁⠀⠀⠁⠀⠀⠀⢀⠄⠀⠐
⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠠⠄⠢⠀⠤⢄⠀⠀⠀⠀⢀⠀⠘






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⣾⣿⡟⠻⣿⣿⡄⠀⠀⠀⠀⢀⣀⣀⣸⣉⣉⣹⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⠛⢻⣿⣶⡀
⠀⠀⠀⢸⡿⠟⠛⠚⠛⢿⡇⠀⠀⠀⢸⣿⣿⣿⡿⠛⡛⠻⣿⣿⣿⣿⠀⠀⠀⠀⣀⣀⣸⣿⣇⣀⣀⠀⠀⠀⢰⣿⠿⠿⠇⠀⠸⠿⠿⣿⡄
⠀⠀⠀⠈⠀⠀⣀⣀⡀⠀⠁⠀⠀⠀⢸⣿⣿⣿⢠⣴⣿⣤⢸⣿⣿⣿⠀⠀⠀⠀⠿⠿⢿⣿⡿⠿⠿⠀⠀⠀⢸⣿⣀⣀⡀⠀⢀⣀⣀⣿⡇
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⢛⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠈⠻⣿⣿⠿⠃⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⡿⠿⠟⠋


⠀⢰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣤⣾⣿⣷⣆
⠀⢸⡇⢰⣦⠀⢀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⡿
⠀⢸⡇⠸⡿⠀⠸⡯⠀⠸⡧⠀⣿⠀⠀⠀⠀⠀⠀⣹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣿⣿⣷⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠸⠗⠚⠛⠒⠚⠛⠒⠚⠓⠚⠿⠀⠀⠀⠀⠀⠀⢾⣻⠇⠀⠀⠀⠀⠀⠀⠀⠠⣟⣿⣿⣤⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⢠⣾⣿⣿⣷⣶⢰⣶⣿⣿⣿⣧⠀⠀⠀⠀⣴⣶⡟⢛⠒⣦⠀⠀⠀⠀⠀⠀⠎⠉⣠⡴⣭⡑
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⠀⠻⠿⣿⣿⡏⠀⣿⣿⡿⠿⠋⠀⠀⠀⠀⣿⣿⣌⠻⠋⣼⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡼⠃
⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⢀⡠
⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠠⠤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⠟⢷⡜⠀⠀⠀⠀⠀⠀⣿⣿⠋⣴⡌⢻
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⢱⠀⠀⠀⠀⠀⠀⣿⣿⣧⣍⣥⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⠻⠿⢿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠙⢻⡿⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⣿⠀⢶⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⣴⣿⣾⡂⠀⠀⠁⠀⠀⠀⠀⢸⡇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠏⢀⣬⡄⠹⢿⡟⠁⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢶⣄⣹⡏⠀⠈⠛⠛⢀⣤⡄⢇⣀⠀⠀⠀⠀⢴⣿⣿⣿⣦⠀⠀⠀⠀⡄⠈⠀⠀⠀⠈⠀⡿⢃⣤⡀⣀⠀⠀⠀⡴⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠚⣿⣭⣻⡷⠖⣰⢃⣼⣶⣷⡄⠀⢀⠀⢒⠂⢻⣿⣿⣿⡿⠃⠀⠀⠈⠀⢀⣄⠈⠶⠄⠈⠁⣘⠛⠙⠿⠁⠞⣿⠁⠀⢠⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠸⠈⣿⠿⠏⠁⢀⡉⢸⣿⣿⣿⡿⢀⡃⠀⣀⠀⠀⠙⠙⢿⡠⡆⠀⠐⠛⠐⣤⡈⠑⠀⢀⡢⢠⡟⢰⣇⣠⠄⠀⢤⣀⡙⢾⠀⠀⢙⡃
⠀⠀⠀⠀⣠⠟⢠⣶⣷⠀⠐⣶⡿⣧⡤⠉⠛⠉⣠⠘⢧⣀⠁⠰⠐⠆⠀⠸⠗⢀⣠⡄⠀⠀⠀⠉⠀⣀⠀⠀⠛⠠⢾⣯⣿⣾⣿⣷⡄⠀⣠⣦⡀⠀⣛⠄
⠀⠀⠀⠀⠿⠀⣿⣿⢏⡀⢿⣿⣿⣿⡄⠂⠀⢾⠃⠀⢀⠙⠀⣠⣤⡀⢀⣀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠙⠿⠲⠄⣰⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⠏⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣿⠏⠸⡇⢸⣿⣿⠋⠀⢄⠀⠀⠀⣀⠸⣇⠀⢨⣽⣷⣿⣿⣷⡀⠀⡙⠁⠀⠀⠀⠄⢠⢀⠀⢀⡘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡆⠄
⠀⠀⠀⢀⡀⠘⣃⠀⣤⣆⠀⠈⡀⠀⣀⢚⡄⠧⠻⠌⠀⠉⠀⠸⣿⣿⣿⣿⣿⠇⠀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠘⠁⢈⠻⣿⣿⣿⣿⠿⠃⢰⠀⠈⠀⠀⠹
⠀⠀⠀⠈⣇⠈⣿⣷⣈⢻⣿⣠⣤⡀⠈⣉⣁⡀⢀⠘⠓⠚⠛⠀⠙⠻⠛⠛⠁⢰⣤⣴⡆⠀⣬⡃⠀⠀⠀⠀⣰⣿⣿⠄⠈⠛⠋⢉⡴⠖⠘⢄⠀⠄⠀⠈⣤
⠀⠀⠀⠀⠙⠁⠛⠛⠛⠷⣿⣏⡉⢀⡾⢹⡉⢻⡝⠷⠀⢀⢈⠁⢀⣠⣬⠑⠒⢈⡀⠈⣠⣄⠈⠛⠶⠀⢀⡀⠈⠙⠏⠀⣰⣷⠀⠙⢁⣤⣤⣠⣤⡄⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣷⠀⣾⣿⣿⡿⠘⢧⣈⢁⣼⣯⠑⣀⣤⣾⣄⠘⣿⣿⠃⠀⠘⠿⢾⣿⠏⠀⠞⠀⢠⠞⣿⣄⠀⣠⣶⣿⣿⠀⠀⢻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⢻⠃⠀⢀⣉⣉⠀⠉⢸⡏⢸⡄⢻⡄⣈⠁⣠⣾⣿⣿⣿⣷⡆⢸⠀⠣⡈⠛⠛⠁⠀⠻⣿⣿⠟⠀⠀⠄⠙⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⣄⠀⠘⣿⣿⡟⠀⠀⠈⠳⢦⡤⠟⢠⡟⣼⣿⣿⣿⣿⣿⣿⠃⠰⡄⠀⠉⢠⣼⣿⡆⠀⠀⣠⠀⠞⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⠓⣄⡀⠀⠁⠘⠀⠀⠐⠀⢠⠀⠋⠀⠈⠻⣿⣿⡿⠿⠋⠀⣠⣴⣾⣆⠙⠿⠟⠁⠚⠋⢩⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢯⣽⡶⠖⠀⠀⠀⡀⣼⡼⠆⠀⠀⠀⠀⣠⣶⣾⡂⠈⢳⣿⣿⠏⠰⣾⣿⣿⠀⣾⣿⡄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢶⡌⢳⠀⠀⢰⣶⣶⡆⠀⢉⠛⠋⠃⠀⠀⢈⡀⠠⣄⠈⠋⠋⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⡄⠀⠀⠀⠀⠻⠏⠀⠀⠁⠈⢰⢄⠴⠀⢸⠀⠀⠀⢰⠇⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢠⣶⣥⣤⡀⠀⠀⢌⠡⠇⠀⠀⠀⠀⠈⠳⠄⠀⢠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⣿⣿⣶⣤⡀⠀⠀⣴⣿⣿⣿⣷⡆⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⣤⣤⡈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠷



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⠒⡶⠶⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⣤⣤⣀⡀
⠀⠀⠀⠀⢀⡴⡋⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⢬⠍⢩⠍⠉⠀⠀⠂⠀⠆⠀⡀⠆⠉⠓⢦⡀
⠀⠀⠀⢀⡞⡀⠋⠀⠀⠈⠀⠄⠀⠀⠋⠀⠀⠰⠀⠠⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⢀⡀⠀⠁⠀⠹⡄
⠀⠀⠀⢸⠃⠁⠀⠈⢀⠀⠀⣀⠀⡀⠀⠁⠀⠀⡀⠀⠀⠰⠆⠀⠐⠀⠸⠆⠀⠄⠀⠀⠠⠀⠁⡀⢻
⠀⠀⠀⢸⡀⠉⠀⠀⠈⠠⠀⠛⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠄⠀⠖⠀⠀⠀⠤⢠⡄⢸
⠀⠀⠀⢸⢳⡀⠀⠀⠐⠗⠀⠀⠀⠂⠀⠀⠠⠀⠴⠀⠀⡠⠆⠀⠀⡀⠀⢀⡀⠀⠀⠂⠀⠀⡀⠀⣾
⠀⠀⢀⣼⡀⡝⢦⣀⠈⠀⠈⠃⠀⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡒⠀⠰⠆⠀⠀⠀⠀⠤⠀⢀⣾⠇
⠀⠀⡏⠈⣧⠀⢀⠈⢹⡗⠒⠶⠶⠶⡒⠒⠟⠉⠉⡁⠐⠀⠀⢨⠉⠙⠓⠲⠤⠤⠤⠤⠴⠚⠁⡞
⠀⠀⢧⠀⠈⠳⣄⠠⠀⠤⠈⠀⠄⠀⠅⢀⡀⠀⢀⠠⡦⠀⠄⢀⡀⠀⠀⠁⠀⠂⡀⠰⠄⣀⡞⢁⣤⣄
⠀⠀⠈⠳⣄⡀⠈⠙⠲⠦⠤⣤⣤⣤⠤⠴⠖⠒⠚⠒⠒⠶⠶⠬⠥⠤⠥⠤⠭⠤⠤⠴⢚⡟⠀⠛⠒⠚
⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠙⠒⠦⢤⣀⠀⠀⠀⢀⣀⡤⠖⠒⠛⠛⠒⠒⠦⠤⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠰⣏⣉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠁⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⡜⢩⣦⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠰⢧⣤⣿⣇⠀⡎⢀⠜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⢠⠃⠈⢿⣿⣿⠀⠳⠧⠤⠊⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡎⠀⡠⠤⣉⠀⠀⠀⠀⢀⣴⠋⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡸⠀⢘⢀⠔⠉⡇⠀⠀⣼⣿⣿⣷⠯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣰⠃⠀⠈⠓⠒⠊⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠈⠑⠒⠁⠀⣸⠀⠈⠑⠒⠉⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠉⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠒⠒⠒⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣤⣤⣤⣶⣾⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⠤⠶⠒⠒⠛⠛⠛⠉⠉⠛⠛⠲⠦⡤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⢀⡖⠛⠋⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠦⢤⣀⣀⡀
⠀⠀⠀⠀⠈⡷⢤⣀⡀⠈⠉⠉⠉⠑⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⣆
⠀⠀⠀⠀⢀⡇⢀⣄⠉⠉⠒⠲⢤⣀⠀⠀⠀⠈⠙⠒⠤⠤⣀⠀⠀⢀⣠⣀⣠⠤⠤⠤⠒⠒⠒⠒⠉⠋⠉⠁⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠁⣠⡀⠀⢀⠀⢀⠉⠙⠒⠦⢤⣀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠖⠒⠚⠋⠉⢹⡆
⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠙⠃⠀⡦⠀⠀⠈⠙⠲⠤⣄⣠⣤⠤⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠴⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠺⠂⠰⠆⠀⠂⠀⠀⠀⠀⠀⠁⠰⠆⠀⡇⠀⠀⠀⠀⠀⠘⠛⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣀⣠⠤⢇⠀⠰⠆⠠⠐⠀⠀⠀⠀⡀⠀⠐⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠄⣠⡀⠰⠷⠀⠀⠔⠀⠀⠀⢸⣇⣀⣀⠀⠀⠀⠾⣭⠿
⡴⠊⠁⠀⠀⠀⠑⠤⣀⠀⠀⠛⠃⢀⡄⠀⠀⠀⠀⠀⠐⠂⠀⠀⣇⠀⠰⠆⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠓⠲⢄
⢷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡶⠀⠀⣠⡈⠀⠀⠀⠣⠃⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠉⠓⠶⠤⠤⠤⠤⢤⣄⣀⠀⠀⠈⠉⠓⠦⣄⣀⠀⠈⠀⠀⠀⠛⠀⢀⡀⠀⠀⠀⠀⠉⠀⢀⣀⡤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⣤⠖⠒⢦⡄⠀⠀⠀⠀⠉⣳⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠈⠁⣀⣠⠤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠒⠚⣁⣀⣀
⠀⠉⠛⠛⠋⠁⠀⣠⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠙⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⣥⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⠀⠀⠈⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⠤⠤⠴⠖⣖⡒⠶⠒⠲⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⣷
⠀⠀⠀⠀⠀⠠⣤⠴⠶⢚⡋⣉⡉⠛⢁⡀⢘⣋⠀⢰⡮⠀⠛⠀⣤⡀⢶⠄⠙⠓⠦⢤⣀⡀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⢀⡤⠞⠰⠶⠀⠀⠓⠚⠒⠮⠥⢈⡉⠀⠀⠀⠀⠀⡤⡄⠀⢠⠒⡄⠀⠰⠭⠆⠉⠙⠶⠦⣄
⠀⠀⠀⠀⢸⠃⢰⠒⡄⢠⠤⡀⠀⠀⢠⡚⡆⠀⠈⠻⣉⡹⠄⣈⡀⠶⠀⠉⠀⣖⡀⣤⣠⠔⠒⠂⠉⢹⡆
⠀⠀⠀⠀⢸⡄⠈⠉⠁⠘⠒⠁⣤⢄⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⠈⢱⠂⠈⠉⠀⠀⠀⠀⠀⡤⡀⠁⠀⣷
⠀⠀⠀⠀⢸⡇⠀⢠⢤⠀⠀⣀⢌⠁⢀⣀⠀⠛⠁⡤⡀⠀⣠⡀⠀⠸⠀⠐⠝⠀⡴⢆⠀⠈⠒⢁⣀⠀⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠉⠀⠀⠧⠴⠃⠘⠆⠃⠀⠀⠉⠁⠀⠓⠃⢀⣇⠀⢀⡀⠈⠓⠊⠀⠀⣀⠀⠛⢸⡇
⠀⠀⠀⠀⠀⠳⣤⣛⣀⢰⢒⡄⠀⢠⢤⠀⠀⢀⡀⠀⢴⠄⠀⠀⠰⠎⠀⠈⠁⠀⠀⡠⢄⠘⠿⠃⠀⣾⠥⢤⡀⠀⠀⣠⠶⠒⠒⠶⡄
⠀⠀⠀⡴⠊⠉⠀⠁⠘⠮⣍⡀⠀⠳⠔⠁⠀⠣⠜⠀⠀⢀⣀⠀⢠⡇⠀⢀⡞⢲⠀⠓⠊⠀⠠⢄⠀⢸⠀⠸⣄⠀⠀⠓⠶⠖⠒⠚⠁
⠀⠀⠀⠳⢤⣄⣀⣀⣀⠀⠀⠉⠓⠖⢢⡀⠀⠀⣠⡀⠀⠐⠚⠀⣸⠀⠿⠀⠉⠁⠀⠸⠧⠀⠈⠁⢀⣸⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣳⠄⠀⠀⠀⠈⠑⠲⠬⣠⢄⠀⠀⠈⠛⠀⠀⠐⠆⠀⣀⣀⠠⠤⠖⠒⠉⠀⠀⠀⠀⠀⠀⢀⡇⠀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⠦⠤⠔⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢋⣡⣤⣤⣌⡙⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⢠⣿⡟⢋⡙⢻⣷⡄⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣇⠸⣿⣇⠛⠛⣸⣿⠇⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⡙⠿⣿⣿⠿⢋⣰⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣶⣶⣶⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉







⡿⠿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⡿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠺⠀⠈⠉⠙⠛⠒⠶⠶⠶⠶⠶⠶⠒⠛⡛⠉⠉⠉⠉⠙⢛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⢠⡞⣡⡄⠀⠁⢀⡀⠀⠐⠂⠀⢀⡀⠀⠈⠁⠀⠀⠿⠂⠘⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠂⠉⠁⠀⡈⠙⢦⡀
⠀⠀⠀⠀⢠⡏⡀⠈⠁⠀⠀⠈⠁⠠⠀⠀⠀⠈⢁⠀⠀⠘⠃⠀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠀⢀⡄⠀⠀⠁⡀⠀⢷⡀
⠀⠀⠀⠀⢸⠇⠁⠀⠀⠁⢀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠠⠆⠀⠀⠐⠀⠰⠷⠀⠀⡀⠀⠀⠀⣀⠀⠋⠀⠈⣇
⠀⠀⠀⠀⣼⠀⠰⠂⠀⠀⠈⠁⡀⠐⠿⠀⠁⠀⠀⣤⠀⠀⠈⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⣄⠀⠀⠀⠀⣀⠀⠁⠀⣿
⠀⠀⠀⠀⣿⣇⠀⠀⠒⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠉⠘⠛⠀⣿
⠀⠀⠀⠀⣿⠙⢦⡀⠀⠀⠀⡉⠀⣀⡀⠀⠁⠀⠀⠀⠉⠀⠘⠃⠀⠀⠄⠋⠀⠀⠀⠄⠀⠀⠰⠄⠀⠀⠈⠀⠀⠀⠆⠀⣰⡇
⠀⠀⠀⣠⢿⡀⠶⠙⢦⣀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠐⢀⣀⣀⣤⣤⣤⣄⣀⡛⠀⠐⠿⠀⠀⠤⠀⠀⠰⠆⠀⢀⡼⡿
⠀⠀⢸⠃⠘⣧⠀⠀⣀⠉⢙⡓⠲⠶⠶⠶⠶⠶⠶⠒⡛⠋⠉⢉⠀⠰⠀⠀⠀⣈⠉⠛⠓⠶⠦⣤⣤⣤⣤⣤⠴⠚⠉⢰⠇
⠀⠀⢸⡀⠀⠘⢧⡀⠀⢀⠈⢁⠀⠒⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⢁⡀⠀⣀⠀⠁⠀⠆⠀⠚⠂⠀⠄⠀⠀⣀⠀⠉⣠⠟
⠀⠀⠈⢳⡄⠀⠀⠙⠶⣬⣀⠈⠁⠀⠀⠁⠀⠈⠀⠘⣃⣀⣀⣆⣈⣁⣀⡀⠀⠲⠂⠀⠆⠀⠀⡤⠀⠐⠀⢁⣠⣾⠁⢰⣋⣉⣳
⠀⠀⠀⠀⠙⠳⢤⣄⣀⣀⣉⣙⣛⣒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠀⠀⠉⠉⠉⢉⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⣉⡵⠋⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠙⠓⠦⣄⡀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠉⠉⠉⠛⠛⠲⠶⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠠⣞⣋⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠁⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣤⠖⢋⣉⣉⡙⢶⣤⣀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠟⠁⢀⣤⣾⣿⠃⣴⣿⣷⣦⡘⡄⢻⣿⣷⣦⣀⠈⠻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄⡀⠙⠻⣿⣿⡄⢻⣿⣿⣿⣷⠃⣸⣿⣿⠟⠋⠀⣠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠟⠉⣤⣉⣙⣉⣡⡼⠟⠋⠁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣀⡀⠉⠉⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⡿⠿⠿⠟⠛⠉

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠈⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡤⠄⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠤⠶⠤⠶⠋⠀⠶⠶⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣆⡀⢤⠄⠀⠀⠶⠭⠖⠐⠟⢃⡤⢤⠀⠀⢴⣺⠉⠓⡒⠄⢀⣀⠀⠀⠀⠀⠀⠀⢰⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠒⢤⣀⡶⠶⠀⠀⢀⣀⡈⠉⠁⢠⣤⠀⠀⠀⠘⠛⠀⠀⣀⡁⠲⢤⣀⠀⠀⠈⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠠⢄⠈⠑⠢⣤⢌⠉⠀⣤⣠⠀⡠⢤⠀⠛⠃⣀⡀⠀⠑⢁⡤⠒⡌⠓⠤⢤⣀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠫⠃⠈⠋⠀⠀⠀⠈⠈⠑⠒⠤⣀⠀⠉⠉⠀⠀⠻⠔⠃⠀⠀⢈⡓⠚⢁⣤⡦⠀⠀⠈⠙⣦⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡞⡆⠀⠀⠌⠂⠀⠠⢲⠀⠀⢉⡒⠠⣤⠤⢀⡀⠟⠀⠀⠓⠁⠀⠀⡉⡀⠀⠶⠆⢠⠿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡆⠀⣠⡀⠈⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢈⡁⠀⠈⠁⠲⠷⢄⠀⠀⠀⣠⣄⠀⠓⠃⠀⣠⠖⠁⣰⡇
⠀⠀⠀⢀⡤⠚⠉⠁⠀⢇⠀⠳⠃⠀⠀⢠⢤⠀⠐⠧⠼⠀⠀⠀⣎⠈⢳⠀⠀⠀⠀⢀⣌⡒⢤⣈⣀⣀⡠⠔⠋⡀⡀⢉⣸⡇
⠀⠀⡶⣿⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⡀⠈⠈⠀⠀⠀⠀⠰⡣⠀⠈⠓⠋⠀⠘⠀⠀⠹⢤⠽⠀⠈⢉⠍⠀⠠⣦⠉⠁⠣⠾⡇
⠀⠀⠀⠙⠶⢤⣤⣄⠀⠀⠀⠀⠀⠈⠳⢄⠀⠸⠵⠀⠀⡀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠸⠆⠀⠀⠀⢸⠀⣔⡄⠀⠒⠀⠤⠀⣷⠒⠒⠒⠛⠛⠛⠲⡌
⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⡀⠘⠽⠀⠀⡠⡀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠸⠵⢀⡴⠛⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠁⠀⢀⣄⠀⠀⠛⠁⠀⠀⠈⢲⠃⠀⢟⠄⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣠⣴⢎⣄⡀⠀⠀⠿⠆⡀⢸⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⣀⡤⠖⠚⠁
⠀⠀⠀⠀⠈⠓⠦⠤⣤⣤⣤⡤⠤⠤⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣘⣏⡼⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⠴⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠉⠙⠓⠒⠒⠲⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠚⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠦⠀⠀⢀⠀⠀⠀⠀⠀⠀⢾⠄⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠀⠰⠉⠀⠆⠁⣸⠆⠀⠀⠀⠰⢶⣶⣶⣦⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠁⢀⠀⣴
⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡄⠀⠈⢫⢿⣿
⠀⠀⠀⠀⠀⣶⣾⣿⣶⡆⠀⢀⠀⠀⠀⠀⢙⣿⣿⣿⠋⠀⢀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⢸⡀⠀⠀⠀⠀⢸⣾⠃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢀⡞⠀⠀⠀⠀⠻⠿⠛⠿⠋⠀⡼⠀⠀⠀⠀⠸⠛⠁⠀⠀⢀⣾⠃⠀⠀⠀⠀⡾⠙⢷⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣄⣦⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣄⣀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠦⡀⠀⠠⠀⣠⠏⠀⠀⣀⣤⣤⣤⡀⢹⣿⣿
⠀⠀⠀⠀⠀⢠⡞⢻⣏⠀⠀⢰⠀⠀⠀⢸⣿⣇⠆⣽⣿⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⢠⣧⢴⠀⢀⣶⣾⣿⣬⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠟⠃⠀⢀⡞⠀⠀⠀⠀⠉⠉⠉⠉⠉⣠⡿⠀⠀⠀⠀⠐⠤⠤⠤⠂⢀⣾⡟⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠤⠴⠞⠋⠀⠀⠀⠀⠐⠀⢀⣠⣤⡶⢿⡉⢸⣿⣿⣷⣦⣤⣤⣤⣶⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠚⣴⣾⣿⣿⠟⠉⢀⣀⣀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣾⢿⣷⡆⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠀⠠⡇⠀⠀⣶⣶⣶⡄⠀⠈⣿⣿⡏⠀⠀⢸⣿⡿⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠺⣷⣴⣿⠋⠀⢰⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⢰⡇⠀⡄⠀⠀⢹⡟⣿⠀⠀⢀⣿⣿⡇⠀⠀⣶⣿⣷⡄⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢆⡘⡔⠀⠀⠈⠀⠉⠀⣠⣾⣿⣿⣷⢄⠀⠿⠿⠿⠃⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠀⠏⠁⣤⠾⣷⢶⠶⡶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣿⡷⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⢰⣟⠋⠉⠉⠉⠉⠙⠛⠶⢤⡈⠉⠙⣳⡄⠀⠀⠀⠀⠀⠈⠙⢷⡄⠉⠙⠛⠶⠶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠷⣦⣄⡀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠶⠶⠾⠷⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠛⠳⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡴⠶⠛⠉⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⡿⢦
⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣄⡀⠀⠀⠀⠀⠀⣼⠅⣠
⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⣀⣤⡶⠾⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠉⠛⣛⣷⣶⣶⠿⠞⠋
⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠁⠒⠦⢬⡙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⣿⠻⣦⣀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⢦⣤⣤⣬⣶⣤⠴⠿⠛⠛⠛⠛⠳⠶⠶⠶⠶⢶⣶⣶⠞⠛⠛⠋⠉
⣿⠀⠀⠙⠛⠶⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠁⢸
⣿⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⠳⠶⠶⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠚⠋⠉
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠞⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡄
⠀⠀⠀⠀⠀⡆⠀⣧⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣴⠛⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡇⢀
⠀⠀⠀⠀⠀⠙⣦⠘⣷⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⣷⠀⠢⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡾⠀⣸
⠀⠀⠀⠀⠀⠀⢸⠀⣼⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⣾⣇⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⣰⠃
⠀⠀⠀⠀⠀⠀⠋⢰⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⣏⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢧⡀⠙⣦⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⣦⡢⠔⣣⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠳⠄⢰⡄
⠀⠀⠀⠀⠀⠙⢶⡄⢹⡆⢳⡄⠀⠀⠀⠀⠀⠀⠀⢼⠉⠉⢻⡻⣼⡧⣂⡽⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢀⠀⣟
⠀⠀⠀⠀⠀⠀⡼⢱⠞⡵⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠀⠃⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⠃⠈
⠀⠀⠀⠀⠀⠀⠉⠈⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄


⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠘⢦⡐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡀
⠀⠀⠀⠀⠀⢠⡀⢻⡆⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⠇
⠀⠀⠀⠀⠀⠀⢹⡄⡿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⡜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠸⠃⢃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⡀⠀⠀⠀⣷⣑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⣘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠒⡚⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠰⠄⣀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡁⠒⠁⠈⠓⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠀⠸⣶⠆⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⠐⡀⢾⠁⡀⠀⠀⢀⠃⠀⠄⢠⠀⣰⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⣲⠊⡠⠌⠀⠈⠀⠀⠀⠈⠐⠰⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⣀⡅⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡔⡈⠁⠀⠈⡀⣠⢐⡀⠤⢂⠀⣦⣸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣾⠄⠀⡈⠀⠆⢙⣦⣕⣁⣆⠀⠠⠈⠈⠈⠹⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠐⠛⠿⠿⡃⢐⣴⠃⡀⡀⠄⠋⠹⡟⠶⢆⠄⠀⢀⢠⠀⡞⠐⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠞⡡⢀⣠⠈⣴⣽⢺⠏⠀⠂⠐⡨⠄⠀⠈⠠⢂⠀⡀⢀⠋⠈⡃⠀⠀⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡇⢀⡴⣻⡷⠃⠀⠀⠀⠀⠀⡀⠐⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠉⠀⢓⠉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢬⠄⠨⠀⠀⠀⢠⣫⡎⠀⠁⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⢀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠃⡐⠁⠀⠀⠐⠀⡀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠁⡐⠀⡄⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠰⠀⠀⠰⠂
⢀⠀⠘
⣮⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣲⣯⣭⣩⣭⣍⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⡆⢸⠯⡀⠀⣙⡤⠀⠀⠀⠀⠠⣄⣀⣤⣀⣀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣇⣿⣿⣤⢸⣞⠆⠆⣹⡇⠀⠀⠀⠀⠀⠹⣟⡻⢾⣿⣷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢀⣴⣌⣿⣟⣛⣩⠟⣾⡤⠖⠒⠓⠯⣌⠀⢀⠀⣼⣣⣿⣽⣿⣿⠄⠀⢐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣧⣿⣼⡿⢿⣿⡻⣶⣶⡗⠚⠂⡆⢀⠲⢶⣿⣿⣿⢿⣽⣿⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠀⣠⠤⣶⣿⣿⣿⣆⣀⣽⠛⠉⠉⢯⠹⠁⠓⡲⠖⠏⢴⠞⢃⡼⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡶⠟⠲⢭⡿⠋⣹⣿⣿⠂⠀⢠⣶⣾⣶⣦⣤⣄⣨⡤⠤⠴⣿⡀⣱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡞⣿⣔⢙⣀⣨⣼⠛⠋⣹⣷⠖⠉⠁⠈⠹⣏⠻⣿⡇⠀⠀⠀⣀⡼⣿⣿⠟⠂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣃⢹⣍⣿⣭⣤⠟⠢⢀⣿⡅⠀⢸⡇⠀⠀⢹⣆⣯⣤⡴⠶⢾⡁⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣯⢘⣿⡇⠤⠄⠀⣀⠞⢿⡧⠹⣿⡇⠀⠀⢸⠟⠉⠀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠃⠀⠀⠀⢀⣾⢇⠘⣏⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣴⡿⣸⠄⠙⣤⣤⠀⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢁⡾⠀⢷⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠠⢈⠦⠝⠒⡏⢁⠂⠀⠀⠀⠀⠆⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠄⠁⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠁⠀⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⢀⠎⠀⠀⠘⠀⠀⠢⡀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠄
⠀⢀⠂⠀⠀⠀⠀⠡⣠⠄⠈⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠚⠂⠐⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠈⠁⠀⣧⣤⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⣰⡟⠡⢠⣤⡄
⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠈⠉⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠘⠃⠀⠀⢨⣞⠁⠀⠈⠂⠤⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⢀⠀⠠⢀⠄⡀⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠙⠁⠂⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠈⠠⠀⠧⠑⠀⡂⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠰⠂⠀⠀⠀⠀⠀⢀⣐⠀⢀⡒⢲⣬⠀⠀⢈⡁⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠈
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠁⡀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⡠⠂
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣉⡀⠠⠂⠀⠁⠀⠀⠀⢀⡠⠊
⠀⠀⠑⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠐⢷⠓⡶⠀⠁⠐⠄⠄⠂⠁
⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢠⡄⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⢐⠐⣋⠀⠈⠃⠀⠀⣠
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠀⠀⠀⠰⠀⠀⠀⢀⠤⠀⠉⠀⠀⠀⠀⡐⠐⡒⡀⢀⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠈⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢹⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⢨⡧⣀⣀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢹⡇⠀⣨⠻⠉⠁⠉⠰⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⢀⡴⣷⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⢀⡾⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⢀⣾⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⡾⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠁⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⣿⡁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣇⠀⣿⣅⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⣿⠀⠙⠋⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⢰⡿⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡆⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡿
⢸⡇⢀⡞⠀⠀⣴⣿⡿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣠⡾⠁
⢸⣧⠞⠀⠀⠀⠉⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠁⢀⣴⠋
⠀⠻⠿⠶⠶⠶⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⣀⣤⢄⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠼⠃⠘⣷⡈⠷⠀⠷⡤⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣺⠋⡝⢻⢶⡴⠟⠻⠿⠯⢡⡙⢆⡁⣼⢹⠲⣄⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠻⠔⠢⣽⣆⢿⠁⠛⠀⠸⢡⣄⢁⣈⡜⣂⠖⠀⣼⠸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠏⠠⢖⡠⠌⡳⠾⠧⣦⣘⣂⠓⠎⢘⠋⡳⠤⠠⠶⣿⡄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡙⠀⠀⠉⣠⠔⠀⣀⠀⠁⢀⡹⢳⣯⢱⠂⠓⢌⠀⠂⠘⣷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣬⡄⢀⠚⠃⢘⢄⠲⣭⠀⠀⢹⡇⠠⣋⠀⠀⠀⢈⠙⢀⡶⠉⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡷⠖⠊⡀⠘⠱⢢⡐⠈⠰⠲⠤⣴⡶⠿⠲⣆⠠⢤⠖⠞⠀⠀⢡⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣤⡀⠁⠘⠋⠀⢀⡀⢴⡀⠀⢰⣞⡀⢤⠹⣦⠀⡄⠰⣮⣤⣶⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣯⠠⠀⢰⡖⠀⡿⠏⢀⠠⣌⠳⡖⢿⣿⣶⠑⠁⠐⠸⢠⣿⣼⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣐⢀⠀⡼⠂⠆⠛⢁⡀⠠⣶⡍⢛⢷⡄⠀⣻⣌⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣉⠀⠸⠃⠅⠾⢀⣻⣿⣭⣀⡶⡬⠋⠛⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠯⠽⠧⠞⠓⠶⠋⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⢶⣤⣤⣀⡀
⠀⠀⢀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠁⠀⠀⠀⠀⠀⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣤⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⣿⡟⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⣿⡇
⣿⡇⠀⠉⠻⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣟⣋⣁⣀⣿⣇⣤⣤⣤⣤⣤⣀⣀⣀⣀
⣿⡇⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠷⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⢶
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠙⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⠶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡊⣂⡀⣔⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣘⡇⡇⢹⣇⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣏⣉⡅⢡⣾⢁⡙⣮⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⠛⠁⣙⣂⣠⣤⣤⣀⣀⡉⠙⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⣂⣵⠾⠟⠛⠉⠉⠉⠉⠉⠛⠛⠿⣶⣤⡀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣡⠞⠋⠀⡠⡀⠀⠀⠀⣾⡇⠀⠀⠀⢀⡄⠉⠻⣷⣄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢣⠞⠁⠀⠀⠀⠙⢶⠀⠀⠀⠙⠃⠀⠀⢠⠎⠀⠀⠀⠈⠻⣷⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢣⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠈⢿⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠇⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠘⢷⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠚⠀⠉⠀⠀⠀⠀⠀⠀⠀⠮⠗⠒⠂⠀⠐⠒⠂⠐⠛⠛⠛⠀⢸⡇⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⣴⠾⡓⠀⠀⠀⡀⠊⠀⠈⠀⠀⠀⠄⠀⠀⠰⣷⣦⣄⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠁⠊⠀⠀⠀⡀⠁⠀⠀⢠⡀⠀⠀⠘⠀⠀⠀⠀⠙⠃⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⢀⡞⠘⢀⠄⠀⠈⠀⠀⠀⡈⠑⢄⠀⠀⠀⠀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡽⠃⠈⠣⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⡽⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡴⠒⢚⣏⠉⠉⠩⠯⢉⡍⠉⠉⠉⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡼
⠀⠀⠀⢷⡛⠛⠫⠥⢤⡤⣄⠉⢴⡦⢨⣥⠀⠐⠿⠂⠈⠁⠰⠶⠀⠻⠃⠀⠈⠙⣓⡶⢤⣄⣀
⠀⠀⡖⠚⠁⠛⠀⠀⠀⠁⠉⠉⠓⠒⠤⠅⣀⠀⣀⣀⠀⠠⣶⠄⠀⢰⣋⡳⠀⠘⠧⠜⠀⠀⠙⠒⠳⣤
⠀⢸⡇⢠⡞⢲⠀⡴⢢⠀⠀⠀⠰⣏⡣⠀⠀⠘⠥⣤⠵⠤⣄⡀⠟⠀⠀⠀⣿⣃⡠⣖⡶⠒⠙⠉⠉⢸⡇
⠀⠘⡇⠀⠈⠁⠀⠓⠚⠀⡔⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠂⠉⠉⠀⡀⠀⠀⠀⢀⠤⠀⠀⠀⣿
⠀⠀⡇⠀⢀⡤⡀⠀⠀⣀⣈⠁⠀⣀⠀⠙⠃⢀⣄⠀⢀⣀⠀⠀⢸⠀⠀⠯⠂⣠⠦⡀⠀⠙⠚⢀⡀⠀⡏
⠀⠀⡇⠀⠀⠑⠁⠀⠰⣅⡼⠀⠸⠤⠇⠀⠀⠙⠊⠀⠘⠶⠃⢀⡇⠀⠀⡀⠀⠳⠔⠃⠀⠀⠀⠈⠛⢰⡇
⠀⠀⠷⣄⠰⠆⠀⡄⢤⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢫⠞⠀⠘⠋⠀⠀⠀⣀⠀⢺⡹⠀⠀⣼⣃⣀⠀⠀⠀⠀⣠⡤⠤⠤⢤
⢀⠴⠒⠋⠙⠉⢇⡓⠚⠀⠀⢏⡨⠂⠀⢾⣙⠆⠀⠉⠀⠀⠀⢸⠄⠀⠀⡠⢦⠀⠸⣤⠇⠀⢀⡀⠀⢻⠁⢸⡇⠀⠀⠸⣥⣤⣤⣤⠴
⢯⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡠⣄⠀⠀⠀⠁⠀⠀⠀⣎⡕⠀⢸⠀⢤⡀⠣⠖⠁⠀⡤⡀⠀⠫⠑⠀⠸⠀⠀⠙⠲⠤⣀
⠀⠉⠙⠒⠒⠒⠶⢤⣀⠀⠀⠀⠈⠲⠤⣀⠘⠇⠀⠀⠀⠀⠰⡿⠀⠉⢀⣀⠀⠀⠀⠉⠁⠀⣀⡠⠴⠂⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⣀⡤⠤⠟⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⢦⣀⠀⠀⡇⠀⠀⢀⣁⢠⠔⠲⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢿⣭
⠀⠀⠀⠀⣾⣁⣀⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⡀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢙⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠒⠚⠉
⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠉⠻⣿⠻⣿⠟⠛⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⣀⣄⡀⠀⠻⠿⠶⠠⢼⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠶⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢴⠀⠀⠀⢀⡴⠋⢠⡄⠘⠋⠀⠀⠀⠀⠀⠓⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠆⠀⣾⣿⡇⠀⠀⢰⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣴⣤⣄⣀⣤⡄⠀⠀⠀⠀⢿⠟⠁⠀⠀⢼⣿⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⣠⣿⣶⣄⠀⠻⠁⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠀⢀⡝⠛⠀⠈⠉⠙⢳⠀⠀⢀⣀⣰⠷
⠀⠀⠀⠀⢠⡄⠀⠀⣀⠈⠁⠀⢠⣿⡿⠶⠶⠶⢀⣠⡌⠀⠀⠀⠈⢀⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⣇⠀⠐⣿⣀⣀⡀⠀⣹⣷⣄⠀⠀⠀⣿⡀⢀⡰⠶⠀⠙⢻⣶⠀⢶⡆⣀⣀⠀⠀⠀⠀⠠⠴⠶⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⡷⠀⠀⠀⠈⠙⠁⠀⠀⠙⠛⠛⠻⠛⢿⣿⢷⣶⣾⣿⡿⠋⢀⣤⠄⠀⠄⠉⠀⣠⠀⠉⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠟⠛⠛⠁⠘⠋⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⠖⠀⠀⠀⠐⢿⡄⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⢿⣷⡄⠀⠰⠊⠁⠀⠀⠈⠙⠿⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷⠀⠠⣦⠀⠀⠀⠠⢤⣄⠙⠛⠷⣶⣶⣦⣀⣀⣀⣀⣀⣀⠤⣶⣶⠎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠀⢰⡀⢸⣿⠀⠁⣤⡄⠙⠀⠀⠀⠀⠀⠀⠘⠿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠸⣷⡄⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠃⠀⠈⠻⣆⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣧⡀⣀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⣿⣇⠛⠁⠙⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠙⢿⣷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣿
⠀⠀⠀⠀⢀⡀⠀⢠⡶⠀⠘⠛⢂⣀⡀⣾⡷⠀⣾⡏⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⡇⠀⣿
⠀⠀⠀⠀⠸⠇⠀⠀⠙⠷⠶⠶⠟⢻⣿⣿⣇⠀⢸⣿⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⢠⠈⢳⡆⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⠏⠹⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⠛⠀⢸⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠜⠃⠀⠀⠉⢸⣿⡷⠈⢡⡄⠀⠀⠀⠀⢀⠛⠀⢠⡌⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡏⠀⠀⣼⡇⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠹⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠀⠀⠀⠘⢇⠀⠀⣤⠁⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢿⡆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡾⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⢀⣀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣀⣀⣀⣀⣀⣀⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⣁⣀⣀⣀⣀⣀⣼⡗
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⢿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣦⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠛⠋⠉⠉⠉⠙⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⢀⡤⠤⣀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⢤⣄⡀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢷
⠀⠀⢀⡠⠒⠒⡄⠀⠀⡤⠖⠛⠉⠉⣳⠀⠀⢀⡴⠒⠛⠲⣄⠀⠀⡴⠒⢺⡄⠀⠀⠀⠙⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⡆
⠀⢠⣋⣀⠀⢸⠁⠀⠀⣷⠀⠀⠀⢰⣃⣀⣀⣸⠀⠀⠀⢰⡏⠀⣀⡇⠀⣼⢧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠉⠉⡷⠈⠙⠛⠉⠁⢀⣀⣀⣀⣈⣉⣁⣀⡀⠀⠀⠀⠉⠉⠉⠀⠸⡇⠀⠳⣄⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢰⠇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠉⠉⠉⠉⠉⢰⡇⢀⠀⠀⠙⠲⣤⣀⠀⢠⢾⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡴⢄⢀⣟⣦⣤⣀⠀⠘⠟⠁⠀⠀⠀⠀⢸⡇⠸⡇⠀⠀⠀⠀⠈⠉⠉⣾⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⠏
⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣟⠙⠛⠋⠁⣠⠀⠎⣧⠈⠁⠀⠀⠀⠀⠀⢠⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⡇⠀⠀⣠⠏
⠀⠀⣇⠀⠀⠀⠀⠀⢰⣃⡟⠀⣴⣷⣯⢛⣤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢤⣀⠄⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⠇⢀⠔⢻
⠀⠀⠸⡄⠀⠀⠀⠀⠈⢹⡤⠞⠀⠉⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⢤⢤⣤⠶⠚⠙⡏⠉⡇⠀⠀⠀⠀⢸⡟⠁⠀⣼
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠲⣴⠇⠀⣠⢶⠏⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣼⠹⡄⠀⡿
⠤⠤⠤⠤⠬⠷⢦⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠥⠤⢦⠀⠈⠻⣏⠉⣿⠀⠀⠀⢰⠃⢠⡇⠀⠀⠀⢠⠇⠀⢳⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠈⠳⡇⠀⠀⠀⣟⠀⢸⠃⡄⠀⠀⡞⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡇⠀⠀⢠⠟⣆⣼⡼⡇⠀⡼⠁⠀⢀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠁⠀⢀⡞⠀⢿⠣⣄⣧⡼⠷⠒⠋⠉⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⠞⠁⠀⢸⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠶⢿⣛⣻⢟⢦
⠀⠀⠀⠀⠀⠀⣠⡤⠶⠤⣤⡀⠀⢀⣴⣻⣟⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡻⣿⣏⣧
⠀⠀⠀⠀⢠⠞⢁⣴⣶⣶⣦⡙⢧⣼⣿⣿⣿⡎⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⠀⠀⣿⠀⣿⡏⣠⠖⠛⢿⣾⣿⣿⣟⣝⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃
⠀⠀⠀⠀⢻⣠⣿⣿⣿⣦⣤⡀⠉⠉⠉⠉⣿⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣿⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣿⣿⡞⣿⡟⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣀⣀⣠⣤⣤⠤⠤⠴⠶⠶⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠁⠀⠅⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⣀⠀⣤⠦⠐⣄⠤⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡕⠊⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠈⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠫⢀⠔⠢⠤⠁⢀⠀⠀⠀⠰⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⠣⡈⠢⢀⢠⠐⠁⠀⠀⠀⠤⡀⠐⠃⠀⠲⢿⠁⠂⠄⡌⠀⠠⠒⡂⠑⠃⠀⡘⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠉⠌⢊⠢⡀⠀⠀⠀⠑⢄⢡⡀⡀⢀⡞⢞⣳⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠁⠜⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⠀⠂⠀⠀⠺⠓⠂⠉⠀⠢⡠⠴⠀⠀⡃⢀⠃⡝⢑⠂⡰⠈⠀⠐⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⢀⠰⣀⡐⠁⠀⡠⢠⡟⡏⠅⠀⠀⠠⢈⡤⡁⠂⠄⠀⠰⠖⡐⠋⠀⡀⠔⣡⡮⠀⡑⠀⠀⠀⠠⠄⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡣⠄⠠⡄⠕⠐⣬⡂⡊⡤⠋⢈⠄⠨⡐⠪⡴⠛⣁⣤⣄⡀⢤⢲⠿⠤⠤⢤⣆⣁⣼⣀⢨⣥⠀⣉⠉⠠⠔⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢜⢯⡑⠞⠫⠒⠀⠐⢁⣭⡈⢀⡙⢹⡆⠀⠈⠓⢓⠈⡀⡀⠀⠠⢌⠀⢨⡼⠀⠈⣣⣰⢁⣾⡷⢎⣡⠁⡰⠀⢂
⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠑⠩⠑⢤⣀⠀⢠⡟⠄⡽⠿⣉⡉⠀⠀⣠⡤⠗⠛⠲⣤⡤⠋⠷⠔⡉⢳⠀⠉⠈⠡⠈⡋⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⣤⠀⣎⠆⠀⠆⠐⠂⠀⠰⡙⡖⠫⠉⢉⣤⠴⠃⠂⠐⠑⢀⡤⣐⡒⠖⠠⠤⠐⠒⠀⠂⠉⣀⠀⢄⠁⠉⠠⡩⠂⠀⠛⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣀⡆⢠⠒⠀⠀⠀⠀⠀⠀⠀⢰⡈⠀⠘⠚⠑⠲⠓⠉⠙⣻⣮⢔⣲⡭⠁⢠⠒⢒⡂⢁⠀⠂⠐⠀⠡⠈⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⠆⠀⠂⠀⠀⢀⡄⠰⢆⢬⡁⠉⠀⠠⢀⠀⢗⠀⡆⢀⢀⠀⠀⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠁⡄⢠⣅⡨⡗⠀⠦⡭⣤⠴⠻⠄⠀⠀⠀⣄⠴⢀⡾⠞⠛⠁⠒⠒⠂⠁⡀⠀⢇⡱⠈⠊⠀⣤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⠠⢄⡁⠀⠀⠀⣊⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠜⣆⠈⠀⠀⠑⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠠⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠑⠀⠀⡠⠚⠉⠋⠐⠀⡀⠀⠐⠂⠠⠤⠎⠬⠄⢒⠄
⠀⠀⢀⠔⠰⠶⡇⠀⠒⠀⠒⠂⠀⠂⠠⠆⠒⠨⠄⠒⠠⢣
⠀⠀⠈⠉⠀⠀⠐⠈⠀⠀⠘⠄⠐⠐⠠⠄⠉⢀⣘⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠓⠒⣄⠀⠀⠀⢀⡤⠖⠤⠒⠓⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⣨⠇⠀⠀⣾⠀⠀⠀⠒⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡤⢤⠼⠁⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢨⠆⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠉⠉⠙⠦⠤⢤⡀⠀⠀⠈⠣⢤⡀⠀⢀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠓⢆⣠⠖⠉⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣠⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⢀⠀⠀⣠⣀⢀⡠
⠀⠀⠀⠀⠀⠀⢇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣠⠞⠉⠲⠃⠀⠀⠙⠉⠉⠱⡄⠀⠀⠀⠀⠀⠈⠑⠒⠚⠇⠀⠸⣠⠎⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠠⠂⠀⣀⣠⣞⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⡴⠁⡠⠖⢤
⠀⠀⠀⠀⢀⣀⣤⠷⠖⠶⠒⠒⠚⠉⠉⠉⠉⠀⠉⠉⠙⠓⠒⠶⠤⠤⣤⣀⣀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡸⠐⡞⠁⣼⠴⠦⠴⠁
⠀⠀⠀⠐⣏⠀⠀⠀⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⣇⡀⠀⠀⠀⠀⠀⡴⠁⠘⠲⡄⠈
⠀⠀⠀⠀⡏⢉⠑⠒⠠⠄⡀⠀⠀⠉⠑⠤⢄⣀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠤⠤⠐⠐⠒⠀⠀⢿⠀⠀⠀⣠⠎⠀⠀⠉⡰⠃
⠀⠀⠀⠀⡇⠈⢀⡄⠀⠀⠀⡉⠑⠒⠤⣄⡀⠀⠉⠒⠊⠁⠀⠀⠀⠀⣀⣀⠤⠤⠔⠒⠒⠈⢹⠀⢀⡜⠁⠀⢀⠔⠊
⠀⠀⠀⠀⡇⠠⠀⢀⡀⠀⠀⠃⠀⠆⠀⠀⠈⠑⠢⠤⠤⠒⠒⠊⠉⣉⠀⢀⠀⠀⠀⠀⠂⠀⢸⢀⡞⠀⣠⠞⠁
⠀⠀⠀⢀⡇⠀⣀⠈⢁⠈⠀⠀⠀⠀⢀⠀⠀⠘⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⣠⠀⠀⠀⠀⠀⢸⡞⢀⠞⠁⢀⡤⢤⡀
⢀⠔⠊⠉⠑⢤⡀⠀⠀⠤⠀⣀⠀⠀⠀⠀⠀⠄⠀⢀⠀⢠⡄⠀⠀⠀⠚⠀⠈⠀⠀⠀⠀⠀⢸⠃⠙⠒⠂⠠⣍⠉
⢧⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⡉⠀⠀⠲⠀⠀⠀⠀⢨⠀⠀⠀⢰⠆⠀⣀⠀⠀⠀⠪⠂⢀⣀⠜⠀⠀⠀⠀⠀⠀⢱
⠀⠉⠒⠲⠄⠰⠦⠤⣄⡀⠀⠉⠑⠢⠤⣀⠈⠀⠀⠈⠀⢠⡀⠀⠀⠀⠁⣀⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⢾⣉⣹⠆⠀⠀⣀⣠⠗⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⡀⠤⠴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠉⢉⡥⠤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠘⠦⡤⠴⠋
⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣠⣤⣤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣶⣦⣄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠶⠴⠶⠦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⡆⠀⠀⠀⠀⢿⣇⣄⢸⣿⡆⣿⡏⢹⡿⣿⣷⣶⣶⣶⡆⠀⢀⣀⢸⡯⠭⢽⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢰⣿⣿⢿⣷⣀⣀
⢷⣾⣿⣶⠀⣈⣉⡉⢸⣿⡇⣿⡇⢸⣧⣿⣿⢿⣿⣿⡿⠀⠸⣿⢸⣟⣛⣻⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡿⣿⣼⣿⣿⣿
⣼⣿⣿⣿⠀⣿⡏⢹⣿⣿⡇⣿⡇⢸⡿⢿⣿⣼⣿⣿⡇⢸⣿⣿⢸⡷⠶⢾⣾⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣸⣧⡿⣿⣿⣰⣿
⣿⣿⢿⠋⠀⠉⠁⢺⣿⣿⡇⣿⡇⢸⣧⣼⣿⣿⠉⢉⣀⣈⣉⡛⢸⣯⣭⣽⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⣧⣿⣿⢿⣿
⣿⣿⣼⣤⣤⣤⣤⣼⣿⣿⣧⣿⣧⣼⣿⣿⣿⣿⣤⣬⣭⣭⣭⣵⣬⣷⡖⠚⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣼⣧⣿⣿⣿⣤⣤
⣿⣿⣿⠀⢸⠠⢾⣶⣷⣶⡇⣐⣁⣸⡀⢸⣿⣿⠀⡯⢽⠯⢽⠭⢍⠨⣧⠄⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⡿⢻⡿⢹⢹⣿⡿⣧⣿⣿⢸
⢿⣿⣿⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⠀⡯⢽⡯⢽⠭⠭⢘⣿⣳⡄⠀⠀⠀⠀⠙⠛⠉⠉⡟⠻⠇⢸⠀⢸⢸⣿⣧⡿⣿⣿⢸
⣼⣿⣿⠀⢸⠀⠸⠛⡿⠻⡟⠻⠟⢿⠗⢸⣿⣼⠀⠛⠙⠋⢹⠋⠋⣰⣷⣺⡇⠀⠄⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⢸⢸⣿⣿⣧⣿⣿⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠒⠛⠛⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠓⠋⠀⠀⠉⠀⢀⣀⣀⠘⠒⠒⠛⠓⠶⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡼⠋⣀⡀⢀⡴⠲⣄⠰⡍⠇⠰⣏⣀⣈⡷⠰⡲⠀⣀⣀⡀⠙⠶⠶⠚⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢊⡥⠤⣄⠀⠀⠉⣀⡈⠓⠒⠁⠀⣠⢦⠀⠈⠉⠀⠀⠀⠀⢸⡁⠀⣸⠆⠀⠀⠀⡴⢻
⠀⠀⠀⠀⠀⢀⣀⣀⣠⠟⠀⠀⢏⣀⣠⡟⠀⢰⡋⠀⣹⠇⣠⣄⠀⠈⠁⡴⠚⠛⣦⠀⢀⣀⠀⠉⠉⠁⣠⠶⠶⠾⢣⠟
⠀⠀⠀⠀⣴⠋⠉⠉⠁⠈⢁⡤⠤⣍⠁⢠⢤⠀⠉⠋⠁⠀⠓⠚⢀⣀⠀⠳⠤⠴⠋⠀⠓⠋⣠⠴⠶⡞⠁⠀⠀⢀⣘⠳⣆
⠀⠀⠀⠀⢹⣦⣤⢦⡄⠀⣇⣀⣠⡼⠃⠈⠉⣠⠖⠒⢦⠀⠀⡞⠉⠉⡧⠀⠀⢀⣠⣄⣀⣤⠿⢤⡴⠃⠀⠀⠀⣏⣨⠇⣿
⠀⠀⠀⠀⠘⣦⠀⠀⠻⢦⣀⣩⣤⣄⠀⠀⠀⢻⣤⣤⠞⠀⠀⠙⠒⠛⠁⠀⠀⣾⠀⠈⡟⠀⠀⠀⢀⡴⠶⣄⠀⠀⠀⢸⠁
⠀⠀⠀⠀⢸⠃⢠⠶⣄⠀⣿⠀⠀⢸⡇⠀⣶⠄⠀⠀⠠⣏⣹⠀⢀⡤⠶⣤⠞⠋⠓⠚⠁⣠⡄⠀⢸⡀⢀⡟⠀⢀⡤⡌⢳
⠀⠀⠀⠀⠘⣷⠘⠒⠁⠀⠙⠶⠴⠋⠙⠶⣤⠶⠛⠛⣦⠀⠀⠀⡏⠀⢀⠏⠀⠃⣠⣄⠀⠈⠀⠀⠀⠉⢉⣀⡀⠀⠋⠀⢸⠄
⠀⠀⠀⠀⠀⡇⠀⣀⡀⠀⠶⠀⢀⡀⠀⠀⢸⣀⠀⣀⡞⠳⡶⣚⠉⠉⠁⠀⠤⠘⢥⠼⠁⠀⢰⡱⠀⠀⠻⠴⠃⠀⠀⠀⡞
⠀⠀⠀⠀⠀⣷⠸⣍⣹⠂⠀⠰⡋⢹⡆⠀⠀⠉⠉⠩⡦⠀⡇⠀⠀⣰⠛⠻⡆⠀⡀⠀⢠⠴⢦⡀⠀⠀⠀⠀⠀⢀⣠⡴⠃
⠀⠀⠀⠀⠀⠙⡄⠀⢰⡦⠀⣀⡈⠉⢰⠆⣴⠋⢳⡄⠀⢀⡇⠀⠀⠘⠦⠞⠁⠀⠉⠀⠻⡤⠞⠁⠀⡴⠋⠙⠒⠋⠁
⠀⠀⠀⠀⠀⢰⠁⠀⡤⠤⡄⠒⢡⠖⢦⠀⠘⠲⠚⣀⠀⠘⡇⠀⠾⠃⠀⢀⡀⠰⠀⢀⡀⠀⠓⣀⠴⠃
⠀⠀⠀⠀⠀⢸⡆⠀⠳⠶⠃⠀⠘⠦⠞⠀⠀⣀⡀⠉⢠⠞⠛⢲⡄⠀⠀⢏⡽⠀⣴⠋⠉⠳⠊⠁
⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣫⠤⣄⠀⠆⠀⡎⠁⢹⠂⢿⡀⠀⢸⡇⢀⣀⡀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⢦⣀⠀⣙⣚⡁⢰⣢⠙⡟⢋⠀⡏⠀⠘⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⣷⠠⠅⠈⡇⢛⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠷⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⠛⠙⠉⢉⠁⠆⠖⡄⡀⠀⠀⠀⠀⣠⣟⣀⣀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠙⠨⠐⠈⠀⠀⢤⠀⠀⢀⠀⡀⠉⡐⠀⠀⠀⣹⣦⢻⣄⡗⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢑⠐⠀⠰⣝⢦⡀⠀⠜⠇⠀⠼⣿⠃⠀⢈⠀⡘⢌⢳⡼⢷⣽⣿⣥⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣡⠋⣴⣄⡀⠘⠁⠀⠀⠀⡆⠀⠀⠀⠀⢠⣿⢦⡘⡄⠪⢆⠀⢹⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡿⡍⠂⣘⣖⠿⠁⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠈⠛⠉⠀⠸⢈⠽⡃⠈⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡺⣿⣳⣄⠀⠀⠀⢸⡇⣣⠀⣉⣈⠀⠀⠀⠀⠀⠀⣫⠛⠀⠀⠀⠀⠀⢀⣄⣤⡄⣃⠩⣧⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢻⣿⣦⡳⣄⠀⢸⡇⣧⠸⠗⣛⠀⠀⠀⠀⠀⠀⠛⠛⠻⢅⡒⡀⠦⠀⠀⠀⠀⡏⠐⣿⠀⠀⢸⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢿⣿⣷⡈⠣⣘⣷⢿⡀⠊⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢦⣔⡶⢤⢀⠃⢸⡇⠀⠀⢸⣟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⢻⣿⣿⣄⠙⣮⣳⣷⡀⠆⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠉⢃⡞⡐⣾⠃⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠻⣿⣿⣦⠈⢯⣻⣿⣄⠋⠀⠀⣴⠄⠀⢀⣤⣤⠀⠸⣿⣿⠆⣠⠎⠌⣼⠃⠀⠠⣶⡟⠉⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⣿⣿⣷⡄⠹⣟⢎⡓⢄⡀⠘⠀⠀⠸⣧⠿⠀⢀⣈⡴⢚⠅⣠⠞⠁⠀⣰⣴⠟⠀⠀⠀⠀⠀⣠⠞⣿⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠈⠻⣎⢿⣶⠘⠟⠷⣴⣄⡘⡂⠒⠶⡖⢒⠁⢀⣤⣴⠾⠛⠁⠀⠀⢠⣽⡏⠀⠀⠀⠀⢰⡷⡟⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⠘⢻⣿⣽⣏⣄⠈⠉⠛⠛⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⢘⠻⡦⣀⠀⢀⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣷⢍⠻⣾⡻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢻⣽⡿⠷⠟⠃⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⢀⠤⡶⢶⣶⡖⣷⡆⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛
⠀⠀⣶⡵⠟⠳⠒⠉⠉⠁⠈⠙⠒⠶⣾⣧⢀⣄⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣺⣶⠟
⣀⣷⡙⠁⠀⠀⠀⢀⡖⠽⠦⠶⣤⡀⠀⠈⠉⠛⠛⠻⠯⢴⣾⠗⢶⣖⠶⣶⠀⣤⣤⣤⣤⢶⣤⠤⣤⣶⢦⣤⡒⣶⣮⡟⠷⠞⠊⠉
⠻⣿⣡⡀⠀⣠⣀⣸⡷⢤⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠙⠉⠉⠛⠉⠙⠋⠙⠛⠊⠛⠋⠉⠉⠁
⠀⠫⠽⣿⣽⣷⡿⣿⣿⣿⡃⢀⣠⣞⡗
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠙⠙⠛⠉






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⣠⣄⠀⣿⡀⠀⠀⠀⡠⢺⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠙⠋⢀⡘⠇⡴⠒⢲⠀⣿⣀⠤⢄⡀⠀⡠⠤⠤⡀⠰⡹⡄⡘⣷⡀⢠⣤⠀⢾⡷⢀⢃⢀⠅
⠀⠀⠀⠀⠀⠀⠈⠰⣔⠶⡁⠈⢲⠃⢀⡠⠾⠟⠀⠠⠀⢱⣼⠀⠀⠀⠘⠆⠼⢃⠀⣿⡇⣀⡄⠀⠀⠀⣸⣸⠏⠀⠄
⠀⠀⠀⠀⠘⠇⠀⠐⢸⠛⠛⠢⣬⣧⡎⠀⠀⡜⠂⠀⠀⣼⣿⣄⠀⠀⠚⡄⠀⠈⣇⣿⢏⡼⣥⠋⠉⣴⣵⠏⠤⠂
⠀⠀⠀⠀⠀⢀⠤⠊⠉⡕⠂⠉⡝⠁⡣⠀⢀⡅⠀⠠⠊⢸⣿⠈⠂⠀⠀⣄⠀⠐⠁⠙⣟⠀⠀⠀⠀⠋⠉⢱⠀⢠⣄
⠀⠀⠀⠀⠀⠈⠙⠻⢿⢷⡖⠈⠁⠀⠀⠀⠘⡀⠀⡀⠀⣿⣿⠇⠀⡀⢀⠄⠀⠀⠁⠐⠈⠑⣤⣄⠀⠀⠐⠣⡀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣊⣴⢿⠀⠀⠘⣤⣀⠀⢐⠁⠈⠁⠀⢸⣿⠀⠀⠀⠀⠱⠀⢀⠤⡈⠀⠀⡸⠋⠀⠠⣀⣠⣃⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠛⠉⠁⡜⠳⢄⣘⠀⢨⡀⠀⠐⠂⠀⠀⢾⣿⠆⠀⠄⢢⠀⠀⡇⠀⢘⡀⠔⠙⡄⢦⣀⣨⣭⠟⢩⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠄⡀⠘⠀⠐⠄⠀⢸⠀⠀⠀⣸⣿⡄⡀⠀⡼⠀⠀⠆⠀⠈⠀⡤⠐⡧⠚⠫⡒⠲⡀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢻⣿⠋⠀⠀⠘⢄⠀⠀⠀⠁⠈⠀⠀⣅⡰⢦⠍⣉⣀⡀⣠⠄⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⠀⢀⠀⢀⠎⠀⠒⠄⡴⣿⣿⡷⠤⠐⠂⠈⣆⠀⣀⡃⠀⠘⠑⠋⢷⣾⡿⠟⣻⣿⣋⣀⣀⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠐⠀⠉⠉⣄⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣀⡋⠉⠀⠝⠈⣤⣅⠤⠈⠁⠉⢫⡉⠉⠐⠶⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⠔⠉⠈⠑⢆⠀⠐⢻⣿⠓⠀⢠⠊⠀⠈⠱⢄⣀⡤⠻⠛⠢⢄⣸⠷⣦⣘⣄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⠁⠀⠀⠈⡄⠀⢸⣏⠀⠀⠆⠀⠀⠀⠀⢠⣉⠀⠀⣇⢀⣠⡦⣙⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⠀⠀⢀⡜⠘⣄⠀⠀⣠⠤⠄⠒⠁⠀⠉⠍⢩⡅⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠁
⠀⢀⣤⠶⠶⠶⠶⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⢠⡟⠉⠢⡀⠈⠠⡀⠀⠈⠙⠶⣄⣿⠛⠒⠒⠒⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⢧⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠘⣆⠀⠀⠀⠈⠳⣄⠙⢦⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠸⣆⠈⢢⡀⠀⠄⣀⠈⠳⢄⠙⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠁⠀⢀⡾⠁
⠀⠘⢦⡀⠙⢦⡀⠀⠀⠀⠀⠑⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⢦⣀⠈⠓⠶⢤⣤⣀⣤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⢉⣽⠟⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠾⠋⠁⠠⠄⠀⢀⡤⠞⠉⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⡠⠚⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠦⠤⢤⣄⣀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⣠⠖⠁⠀⣠⠖⠉⠀⠐⠀⠀⣠⠎
⠀⠀⠀⠀⣠⣴⡛⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢴⣟⡁⢀⣠⠞⠁⠀⣀⣀⡤⠖⠋⢁⡼
⠀⠀⠀⣰⠏⠈⠙⠲⢦⣀⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠉⠉⠉⠉⠉⢉⣀⣠⡤⠶⠋
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⢹⣏⠑⠦⣄⡀⠀⠂⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢦⣤⢿⡆
⠀⠀⠀⠀⠻⣆⠀⠀⠉⠓⠒⠤⠤⣀⣀⣀⣀⣬⠽⠶⠒⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⡴⠞⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠛⠙⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⠛⠛⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢠⡶⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣆⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⡟⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠻⣏⠀⠀⠀⠹⣆⣀⣀⣀⣀⣀⠀⣿⢸⡇⠀⢀⣀⡼⠁⠀⠀⠀⠹⣆⡀⢀⣠⣤⣄⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⠉⠉⠉⠉⠙⣷⣿⢸⡇⣼⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠉⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠸⠿⢸⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠒⠒⠒⡶⠢⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⢀⡴⣋⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⠍⠩⠍⠉⠀⠀⠂⠀⠆⠀⠀⠆⠉⠓⢦⡀
⠀⠀⠀⢀⡞⡀⠉⠀⠀⠈⠀⠄⠀⠀⠉⡀⠀⠰⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠁⠀⠹⡄
⠀⠀⠀⢸⠃⠀⠀⠈⢀⠀⠀⣀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠀⠀⠘⠂⠀⠄⠀⠀⠠⠀⠁⡀⢻
⠀⠀⠀⢸⡀⠉⢀⠀⠀⠠⠀⠉⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠖⠀⠀⠀⠤⢠⡄⢸
⠀⠀⠀⢸⠳⡀⠀⠀⠘⠓⠀⠀⠀⠂⠀⠀⠀⠀⠶⠀⠀⡀⠆⠀⠀⡀⠀⢀⡄⠀⠀⠂⠀⠀⡀⠀⣾
⠀⠀⢠⢾⡀⠜⠦⣀⠈⠀⠈⠁⠀⠂⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⣒⠀⠰⠆⠀⠀⠀⠀⠦⠀⣠⢾⠇
⠀⠀⡏⠈⢧⠀⠠⠈⢩⡟⠒⠒⠒⠒⡒⠚⠏⠉⠉⡀⠐⠀⠀⢠⠉⠙⠓⡲⠶⠤⠤⠤⠶⡚⠁⡞
⠀⠀⢧⠀⠈⠳⣄⠠⠀⠤⠈⠀⠄⠀⠅⢀⡄⠀⢀⠠⠦⠀⠂⢀⡀⠁⡀⠁⠀⠂⡀⠰⠀⣠⠞⣡⠤⣄
⠀⠀⠈⠳⢤⣀⠀⠙⠓⠦⠤⠤⠤⠤⠤⠶⠒⠒⠚⠓⠒⠒⠲⠶⠶⠤⠤⠤⠭⠤⠤⠶⣚⡟⠀⠙⠒⠚
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠓⠲⢤⣀⠀⠀⠀⢀⣠⠴⠖⠚⠉⠙⠓⠒⠲⠤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠰⣯⣭⠗

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠉⠀⠀⠈⠉⠙⠻⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣾⣿⢹⡇⣾⣿⣿⣿⣦⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢼⣿⣿⣾⣿⣿⣿⣿⣿⣿⠆⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⢸⣿⡅⠀⣽⣿⣿⡏⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⣼⣿⡇⠀⣿⣿⣿⢀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣇⣀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠴⣿⣭⣿⣿⡿⢆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢰⡾⣿⣷⡾⣶⣶⣿⣿⣿⣿⣶⣶⣷⣾⣿⣿⣿⣷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣯⣿⣷⣼⢷⣿⣿⠗⣾⡿⢿⡇⣈⢋⣅⠙⠛⠛⠉⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣽⣿⣿⡟⣿⣿⢾⣿⣿⡿⠟⠥⠜⢶⡿⠿⢿⣤⡤⠀⣥⣴⣾⣿⣭⣉⡉⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣇⣿⣿⢿⠿⣿⣿⣿⣔⠶⢢⣷⡶⣾⣾⣖⣷⣿⣯⡕⠋⠿⣿⡦⣼⣿⡿⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⡿⣿⣧⣾⠿⢇⡸⠟⠛⣻⣿⣍⠛⠀⠀⠉⢻⣿⣋⣴⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣻⣿⠙⣿⠿⠿⠿⡿⢻⣷⡤⠳⡖⠘⣿⣿⢿⡆⠀⠀⠀⠀⠸⠿⠏⠁⠈⠛⠀⣘⣿⣷⡄
⠀⠀⠀⣼⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣟⣡⣿⣣⣿⣿⣶⣼⣷⠀⠞⠻⣶⢾⣿⢦⡽⢿⡼⠷⢦⠄⠀⠀⠀⠀⣿⣄⠀⠀⠀⠃⠿⣿⣿⡄
⠀⠀⣴⣿⣄⠹⢿⣷⠮⠟⠛⢻⣭⣵⣿⣩⣭⣿⣋⣼⣭⣇⣈⣥⣿⣿⣿⠏⠺⡁⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠛⡁⣿⣿⡄
⠀⢰⣿⣯⢻⣷⠛⢿⣿⣿⣷⣬⣿⣿⣿⣿⠿⣟⡙⠿⣿⣿⢿⣿⠍⡋⠁⠀⠀⠀⠀⠀⠀⠐⠶⣤⡀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠉⢛⣿⣿
⠀⣷⠛⢉⠀⣹⡿⢠⣟⣻⡟⣿⣯⢶⣭⡿⠾⠿⠟⡁⠉⠉⠃⠘⠀⠈⠀⠈⠀⠀⠀⠀⢀⠀⠀⠈⠙⠻⣤⡀⠀⠀⠀⠘⣿⣿⠄⠀⠀⠈⢿⣿
⢰⣿⣽⣧⣾⣿⣷⣶⣏⣹⣿⠛⠁⠘⡁⠁⠂⠀⠀⠀⢀⣤⣶⣮⣿⡿⠇⠛⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠸⣿⣇⠀⠀⠰⠚⣿
⢸⡟⣽⣿⡿⠉⢐⢀⡙⠛⠁⠂⠀⠁⠀⠀⠀⠀⠀⢠⣾⣿⣿⡽⢅⣀⣸⡷⢾⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⢀⣿⣿
⢸⡇⠛⠀⠀⠀⣹⣿⠆⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠻⣏⡔⠀⢰⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢿⢃⢠⣄⠙⣱
⢸⣿⣶⡆⠀⠀⣼⣿⡇⠀⠀⠈⠛⠿⣷⣄⡀⠀⠀⣿⣿⣿⣿⣿⡀⠞⢻⣼⣥⣴⠿⠛⠃⠀⠀⠀⠀⠀⠀⢠⠀⠠⠂⢃⣀⣷⣶⣬⣿⣿⣷⣿
⢸⣿⠏⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠉⠻⠦⡀⠹⣿⣿⣿⣟⣿⣦⣐⣏⣀⣩⡖⠀⠀⠂⠀⢀⣀⠒⣤⣾⣤⣶⣄⣿⣿⣬⡿⣿⣛⡙⢻⣿
⠘⣿⡇⠀⠀⠀⠀⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⠟⣠⡀⣼⠶⢿⣬⢻⣿⣇⣽⣿⣿⢿⣛⣿⡿⣿⣿⣌⠻⣸⣿
⠀⢻⣧⠀⠀⠀⠀⠘⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⡉⢙⠛⢛⣭⡄⣶⣾⣿⣿⣚⣃⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣶⣿⣟⢹⣿⣫⣿
⠀⠘⣿⣇⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⡀⠐⣤⣼⣱⣾⣧⣶⣾⢃⣍⣿⡇⣦⢹⣏⠙⣉⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡿⣨⣿⣻⡟
⠀⠀⠹⣿⣄⠀⡄⠀⠀⠸⣿⣖⡀⢰⣶⢠⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⢿⣻⣏⣴⣯⣿⣿⣿⢏⣤⣶⣿⢦⡄⢸⡿⠟⣿⣿⣿⡿⠁
⠀⠀⠀⠙⣿⣠⣀⠓⠶⣶⣝⣿⣿⣾⣿⣶⣿⣿⣿⣿⣝⠿⣏⠻⡞⣿⢱⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣉⣧⡼⢦⡟⢸⣿⣦⡙⢳⡿⠁
⠀⠀⠀⠀⠙⢷⡻⢿⡟⠻⠿⣿⣯⣿⣿⢯⠻⢿⣷⣯⡛⠳⢌⣁⣬⣻⣦⢿⣿⣷⣤⣽⢿⣿⣻⠿⢟⣿⡏⠛⠼⠖⢚⣠⣾⣿⣿⢡⠟
⠀⠀⠀⠀⠀⠈⠳⣿⠛⣿⣻⡏⣿⡁⣴⣿⣶⣿⣽⣿⣥⣀⣄⣤⣽⣿⠛⣂⣽⣿⣿⢻⣾⠱⣿⣷⣾⣿⣿⣿⣶⣦⣿⠿⣿⢋⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⣿⣃⣽⣿⣿⣿⡿⢿⣿⣿⣿⣽⣍⢩⣾⣿⣿⣋⢡⣿⣿⡴⣿⣿⣟⠛⢿⣿⡿⠿⢣⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡃⢿⠽⢿⠃⣭⣤⣦⡅⠹⣿⣿⣟⢀⣾⣿⣁⣾⣾⣹⣿⣷⣼⣿⣿⡧⣴⣥⣤⣲⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⡛⠐⠂⠙⠀⠛⢻⣯⡌⢹⣿⣿⠟⡿⣫⣬⣙⣻⣿⣿⣷⡷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠨⠶⢦⣤⣄⢀⡀⠀⣐⣉⣁⣹⣞⣻⣿⣿⡿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⣉⣩⣤⣤⣤⣤⣤⣉⡙⠻⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠛⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⡛⠻⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢿⣿⣿⣿⣿⣷⠈⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⡆⢿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣁⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠻⢿⣿⣿⣷⣦⣌⣉⣙⠛⠛⠛⣋⣉⣤⣴⣾⣿⣿⡿⠋⣠⣶⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⣴⡾⠛⣉⣤⣤⡄⣤⣤⣌⡙⠻⣿⣿⣿⣿⣶⣤⣀
⠀⠀⢀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⢛⣉⣉⣡⣤⣶⣿⡿⢁⡀⠿⣿⣿⣿⣇⣿⣿⣿⣿⠇⣈⠻⣿⣿⣿⣿⣿⣿⣦⣄
⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⢠⣾⣷⣤⡽⠛⣉⣭⣌⡙⠻⣧⣾⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⡟⢠⣿⣿⣿⣿⣿⣦⠸⣿⣿⣿⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣤⣤⣼⡇⢸⣿⣿⣿⣿⣿⣿⢀⣿⣤⣤⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣄⠙⠿⣿⡿⠟⢁⣼⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣄⠻⠟⣡⣿⣷⣶⡶⣶⣾⣿⣦⡙⠿⢁⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣄⠛⠿⣿⣿⡇⣿⣿⡿⠟⢃⣴⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⣄⢰⡶⢶⡄⣴⠶⠀⣶⠶⠂⣴⠶⣆⢶⡄⢰⡆⢰⣦⠐⢶⡶⠆⣶⢀⡴⠶⣦⢠⣶⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⡿⢸⡟⣻⡄⣙⣻⡆⣿⠛⡁⣿⠻⣇⠈⣧⡿⢀⡿⢼⣇⢸⡇⠀⣿⠸⣇⢀⣿⢸⡏⢻⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣾⣿⣿⣶⣿⣿⣶⣿⣿⣷⣿⣶⣿⣶⣿⣷⣾⣷⣶⣿⣾⣷⣶⣿⣶⣾⣿⣷⣾⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠶⠒⠛⠚⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠲⠶⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠂⠠⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠙⠒⠒⠤⢄⣀⠀⠀⠀⠉⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢀⡠⠤⠤⠄⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠈⠃⠀⠀⠀⠈⠁⠒⠶⠤⣄⡀⠀⠈⠉⠑⠢⢤⠞⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣠⣤⠤⠤⠶⢿
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠀⠀⠁⠀⡦⠀⢀⠀⠈⠙⠲⢤⣀⡀⢀⣀⣀⣠⡤⠤⠴⠒⠊⠉⠉⠀⠀⠀⣀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⣶⠀⢠⠀⠀⠀⠈⠀⠀⠀⠀⢠⡌⠉⠋⠉⠀⠀⠀⠀⠶⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⣠⡇⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⡐⠀⠀⠀⢸⠀⠀⠀⠀⠀⢴⣖⣲⠆
⠀⠀⢠⠔⠋⠁⠀⠈⠢⣄⡀⠀⠐⠶⠀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⣴⠄⠀⠈⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠓⠒⠦⣄⡀
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⡈⠀⠀⠰⠖⠀⠀⠀⠀⠀⡁⠀⢀⠀⢰⡆⠀⢀⡀⠀⠀⠀⠺⠇⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠙⠲⠦⠤⠤⠤⢤⣤⣀⡀⠀⠈⠉⠓⠢⢄⣀⠀⠈⠀⠀⠀⠣⠀⣀⠀⠀⠀⠀⠈⠁⠀⢀⣠⠤⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⢠⠶⠲⢦⡄⠀⠀⠀⠀⢙⡆⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠉⢀⣀⣤⠤⠐⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⢠⡤⠤⠒⠒⢋⣀⣀⡀
⠀⠀⠀⠈⠙⠛⠋⠁⠀⣠⠖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢾⡉⠉⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁





⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠔⠒⠒⠈⠉⠉⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣒⡚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⣼⠋⠠⡌⢳⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⣿⠀⠈⠁⣸⡀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⡞⠉⠛⠵⣶⡼⣀⣀⠀⠀⠀⠀⡜⣁⣤⣶⠿⢧⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⣿⠀⠀⠳⣄⠀⠀⠀⠉⠑⠒⠭⢵⣒⠮⠝⠋⠉⠀⠀⠈⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢻⡀⢠⣤⢸⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠊⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⠳⠤⠴⠋⠀⠀⠀⠀⢫⠉⠙⠒⠒⠒⠛⢅⣀⣀⣠⠞⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⣀⣠⡴⠟⠊⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣤⣤⣤⠤⠤⢤⠀⠀⠀⠀⠀⣄⣀⣀⣀⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⢰⠃⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⡇⠀⢾⣉⣤⣀⣉⡷⠀⢸⠀⠀⠀⠀⢰⠃⠉⠉⠁⠀⠀⠉⠉⠈⢧
⠀⠀⠀⠀⣠⠤⠤⡀⢠⠇⢸⠀⠀⠀⠀⠀⢀⣀⣿⣀⡴⣠⣶⣠⣼⣤⡀⠀⠀⠀⠀⡇⠠⠒⠛⠛⠛⠉⠀⢤⢸⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠀⠀⠀⠀⡼
⠀⠀⠀⠋⡁⠀⢀⡙⠚⠁⢸⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠛⢈⡠⠋⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⢸⡗⠒⠒⠚⠒⠒⠒⢾⠁
⠀⠀⠀⢀⡉⠉⠉⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣄⣀⣸⠀⠀⠀⠀⠀⠀⣧⠀⢠⠏⠳⡄⠀⠼
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄
⠀⠀⢘⣙⣉⣋⣉⣋⣙⣋⡛⢳⠀⠀⠀⠀⣠⠤⠤⡄⠀⠀⠀⢠⡴⠲⣆⠀⠀⠀⠀⠀⠘⣶⢴⣦⣶⣴⣶⠟⠀⠀⠀⠀⠀⠀⢤⢸⡀⠀⠀⠀⢸⡡⡄
⠀⠀⢸⡏⡿⡏⡿⠿⢿⢹⢻⢸⠀⠀⠀⣼⢹⠀⠀⣷⠤⠤⠤⣼⠁⠀⡏⣷⡄⠀⠀⠀⠀⣷⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⣇
⠀⠀⢸⡯⡏⡇⡏⢹⢹⢸⢸⢸⠀⠀⠀⢹⣼⡄⠀⡟⠉⠉⠉⢹⡂⠀⣧⡏⠀⠀⠀⠀⠀⣿⠽⣏⢱⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡶⡀⠀⢸
⠀⠀⢸⣙⣓⣓⣓⣚⣒⣚⣚⣸⠀⠀⠀⠀⠈⠓⠚⠃⠀⠀⠀⠘⠓⠒⠋⠀⠀⠀⠀⠀⠀⢸⠖⠉⠉⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⣧⠀⢸
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠀⠘⠀⠉

⠀⠀⠀⢀⡔⠲⣖⣒⠒⡛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣤⣤⣀⣀
⠀⠀⢀⡞⣄⢘⠀⠉⠀⢀⠀⣳⠀⠀⠀⠀⠀⠀⣠⠴⠚⢋⣉⣥⠽⠷⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠶⠶⠶⠶⠶⣶⡞
⠀⠀⠀⢳⠮⠯⣥⠤⠭⡥⢾⣁⡀⠀⠀⠀⠀⠘⣧⠶⠯⣭⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣳⡟⢻⢿⣁⣄⡀⠀⠀⠀⠀⠀⣼⣡⣄⣀⣀⣀⠀⠀⢸⡆
⠀⠀⠀⢸⠀⡇⠀⡇⠀⡇⢸⠓⢸⠄⠀⠀⠀⠀⣉⣲⣦⡤⣤⣍⣙⡇⠀⠀⠀⠀⠀⣿⢹⡏⣿⣘⡿⣾⣿⡍⣿⡆⠀⠀⠀⠀⡇⢠⣤⡀⣀⡀⣀⣀⢸⡇
⠀⠀⠀⢸⠀⡇⠀⡇⠀⡇⢸⢀⢸⠀⠀⠀⠀⠀⠻⣤⡤⠶⠶⢾⣯⣅⠀⠀⠀⠀⠀⠻⠷⠶⠋⠈⠁⠘⠷⠶⠟⠁⠀⠀⠀⠀⣧⠈⠀⠉⠉⠁⠉⠉⢸⡇
⠀⠀⣴⠉⣷⣇⣀⣇⣀⣣⡿⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣤⣭⣍⣉⣉⣉⡽
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣏⣹⠆⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⣀⣀⣀⣀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢠⣠⣿⢷⠿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢨⢷⡀⠀⠀⠀⠀⠀⠀⠸⣟⡻⢡⢎⣕⣶⣖⡊⡳⡄⢸⢿⠀⠀⠀⠀⡤⠒⠋⣚⣗⡛⢛⠦⡀⠀⠀⠀⠀⠀⠀⡾⢯⠼⡇⢰⠯⡏⢷
⠀⠀⠀⠀⠀⠤⠿⣤⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⣾⠙⣿⠍⣻⣿⣿⣾⢸⠀⠀⠀⢸⡁⠀⠀⠀⠉⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⡇⣟⡇⡇⢸⢸⣿⢸
⠀⠀⠀⠀⠀⠀⣞⠸⠦⠤⡄⠀⠀⠀⠀⠀⢸⣿⠳⡙⢙⢶⡋⡡⣪⠏⢹⣿⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣇⣿⣷⡇⢸⣼⣿⡼
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠓⠚⠛⠋⠁⠀⠸⠿⠀⠀⠀⠀⠀⠙⠦⡤⠶⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠈⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠐⣯⠀⠀⠀⣀⣠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢰⡏⠁⠙⣧⠀⢻⠀⠀⠀⠉⠁⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢘⣇⠀⠀⠘⣧⣸⠇⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣴⠟⠁⠀⠀⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣭⠰⠞⠛⠷⠞⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣸⡏⠐⠞⠻⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠉⠛⠉⠀⢠⣤⠄⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣴⣶⣦⣤⣤⣤⣿⣿⣿⣷⣤⣴⡶⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢈⣉⣭⣭⣁⣉⣁⣌⣤⣀⡀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠋⠉⢉⣿⠛⢻⣿⡉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢿⡛⠿⠿⠿⠟⠛⠛⠻⢶⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣴⠾⠛⠛⢷⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡉⢻⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈⢻⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⣼⣶⣿⣡⡟⠀⠀⠀⠀⠀⠀⢀⣴⡾⠷⠾⠿⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⣿⣍⡙⠛⠛⠛⠛⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⡉⢿⣦⣀⢀⣀⣀⣀⣿⠟⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠙⠶⢭⣙⣛⣛⣛⣩⡥⠞⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣦⣤⣬⣭⣭⣥⣤⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣦⠈⠉⠉⠉⠉⠉⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⠀⠀⡄⠀⠀⠀⣿⡇⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡧⠀⠀⠀⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⢰⡇⠀⠀⠀⢸⣿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⠀⢸⡇⠀⢠⡄⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠃⠀⢸⠇⢸⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⠀⠀⢠⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡟⠀⣀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⡏⢸⠀⠀⠀⢸⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⠀⡇⠘⠀⠀⠀⢸⣇⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢸⡟⢠⠇⠀⢀⡀⠀⢸⣿⠀⠀⢰⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣀⡀⠀⢸⡇⠀⠀⠀⢸⠃⠀⢸⡇⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣴⡟⠛⠻⠇⠸⡿⠀⠀⠀⠘⠀⠀⠘⣧⠀⠘⠋⠉⣠⣄⠀⠀⠘⠋⠉⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠇⠀⢀⣤⡄⠀⠀⠀⠀⢀⡀⠀⠞⠁⠀⠲⣦⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠸⣤⠀⠀⢰⡟⠁⠈⠙⠃⠠⣇⠀⠀⠀⠀⠐⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣠⡄⠀⠈⠀⣀⠀⠀⣀⠀⠈⢉⣤⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠿⠛⢿⣤⣀⣠⡿⠃⠀⠻⣶⣶⠾⠛⠿⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢃⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢃⣾⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣶⣶⣾⣿⣿⣿⣼⣿⣿⣿⢃⣴⣶⣶⣶⣶⣶⣾⡟
⠀⠀⠀⠀⣴⣎⠳⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢃⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⡴⢋⣶⡆
⠀⠀⠀⠀⠸⣿⣧⠹⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡼⢡⣿⡿
⠀⠀⠀⠀⠀⢹⡇⣶⣦⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣠⣶⡆⣿⠁
⠀⠀⠀⠀⠀⠀⢿⣌⢻⣿⣦⣀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⣠⣾⣿⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣧⡙⢿⣿⣿⣷⣶⣤⣀⠀⠀⠈⠙⠿⠟⠋⠀⠀⢀⣠⣴⣶⣿⣿⣿⠟⣡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣎⠻⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⡿⢋⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣬⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠁⠀⠀⠙⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⢛⣛⣛⣛⣛⡛⠛⢛⣛⣛⣛⣛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣏⠀⢠⣿⣿⣿⠠⡏⠀⢠⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠈⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡔⠚⢛⣒⣳⣦⢀⡖⠛⢛⣚⣲⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠳⠦⠼⠿⠿⠛⠀⠳⠤⠾⠿⠿⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣠⠤⠤⠤⢤⣄⠀⣠⠤⠤⠤⢤⣄⢸
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣼⣤⣡⣀⣀⣛⣛⣛⠘⢇⣀⣸⣿⣿⡿⢸
⠀⠀⠀⠀⠀⢰⣿⡿⠿⢿⣿⣿⡿⠿⢿⣿⣿⡿⠿⢿⣿⣿⣿⠿⢿⣿⣇⣀⣀⣀⣀⣀⠀⢸
⠀⠀⠀⠀⠀⢸⣿⣠⠞⢁⣿⣿⣠⠔⠉⣿⣿⣁⠔⠋⣿⣿⣇⠴⠋⣿⣿⣿⣿⣭⣭⣽⣷⢸
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣶⣾⣿⣿⣿⣶⣾⣿⣿⣿⣶⣾⣿⢸⣿⣿⡟⠛⠛⠁⢸
⠀⠀⠀⠀⠀⢸⣿⠟⢉⡿⣿⣿⠟⢉⡿⣿⣿⠟⢉⡻⣿⣿⡟⠋⣹⡟⠘⠛⣛⣷⣶⣖⣦⣸⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣖⣉⣠⣿⣿⣶⣋⣠⣿⣿⣶⣋⣀⣿⣿⣷⣊⣠⡇⠀⡞⠉⠀⠀⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣷⣀⣀⣀⣈⣭⣭⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠋⠙⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠂⢀⡀⠀⠘⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⣠⣴⠟⠃⠀⠀⠦⠀⠀⠀⠈⠀⠀⠀⠀⣶⠀⠓⢄
⠀⠀⠀⢀⣴⠞⠋⡀⠀⢀⠀⠀⠀⣤⠀⠞⠀⡀⠀⠀⠀⠁⠀⠀⠈⠳⡀
⠀⢀⡰⢟⡋⠀⠀⠁⠀⠉⠀⠀⠀⢁⡀⠘⠃⠀⠀⠘⠀⠀⠀⠙⠀⠀⠙⢦⣀
⢀⡾⠃⠀⠀⠰⣦⠀⠀⠀⠀⠀⠐⣿⣿⠆⠂⠀⠀⠀⠀⠼⠃⠀⡴⣷⡄⠀⠈⠳⢦⣄⡀
⣼⠀⠀⠘⠁⠀⠉⠀⢀⡀⠰⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⡀⠀⠈⠻⡖⠤⢤⣀⣀
⣿⠀⠀⢤⡤⠀⢀⢠⡄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⠙⠲⢦⡀
⣾⣄⠀⠈⠀⠀⠈⢀⣠⣶⣴⣓⣴⠆⠀⢠⠀⠀⠀⡄⣠⠀⢀⠈⠉⠀⠈⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠞⠓⠀⡀⠨⢿⣆
⠸⣿⣷⣤⡀⠘⠃⠈⠙⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠰⣶⣶⡌⡀⠀⢰⡆⠠⡄⠀⠉⠀⢘⡿⠀⠀⠀⣠⠀⠀⠻⣦⢸⣿
⠀⣯⣭⠻⣿⠓⠦⣤⣅⣤⣄⡀⠒⢀⠀⠀⠄⠀⠘⢂⣾⠉⣼⠋⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⢀⣴⡄⠀⣸⣿⡆
⠀⣿⣟⣸⢇⣶⠟⠘⠁⠈⠙⠻⠛⠿⠆⢠⣤⣀⡀⠘⠃⠾⠂⣄⠈⠉⠐⢒⠀⢠⣄⠘⣋⡉⢁⡀⠀⠀⠀⠉⢴⣾⢫⡿⣷
⠀⢿⠦⢛⠋⠙⠇⣠⣦⠀⢀⡀⠀⠀⠀⠟⠉⠈⠛⣶⣟⣤⠀⠈⡀⣄⠀⣀⠀⠈⠀⠀⠛⠁⠀⠀⠀⣾⣦⠀⡶⠃⢉⡾⣿
⠀⢸⣿⢦⣤⡄⢠⠈⠁⠀⠘⠇⠸⣇⠀⣀⠀⠠⠟⠋⢻⣧⡀⢉⣵⡿⠀⠉⠀⠀⠀⠸⠿⠁⠀⠀⠰⠋⠁⠄⡀⠀⠉⡷⢸
⠀⠀⢻⣮⣀⠀⠀⣤⡆⠀⠠⠆⣸⣿⠘⠋⠀⠀⠀⠸⠈⠿⣇⠈⢉⡀⠀⠦⠀⠐⠀⢀⣄⠐⠂⢀⣠⡴⠁⠀⠀⠘⠀⠀⢸
⠀⠀⠀⠉⠻⢧⣤⡉⠀⠀⢠⠀⠀⠁⠀⠀⠲⠀⠀⡀⠀⠰⠙⣷⡟⢁⣀⠀⣰⠂⠀⠈⢁⣠⣴⣿⡟⢡⣀⠗⠀⠐⠀⠯⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢤⣄⣀⠀⠀⣦⡄⠀⠀⡀⢁⢨⡈⢋⠛⣿⣿⣄⣤⣾⣷⣾⣿⢿⠏⠙⠃⠀⠐⠀⣤⠀⠀⡦⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣤⡄⠘⢠⡄⡈⠠⠀⢨⡶⠙⢏⣿⣿⣷⡏⠽⠿⠡⠄⠷⢰⣦⠀⢙⣤⡀⠐⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣾⡆⠀⠁⢚⣤⡶⣇⢨⣧⣄⣹⢙⣛⣃⠘⠀⢀⠘⠋⠁⠸⠛⢃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡈⠀⠙⠟⠸⣷⣶⠙⢛⠻⣿⣏⣻⠂⠀⠀⠠⠀⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡁⠃⢀⡹⠿⠃⠶⢀⣀⠹⠅⠀⠐⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠘⠻⠆⣘⡀⠀⠍⠠⠆⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣯⣤⣢⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡦⢰⡄⠄⠀⠀⢠⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢃⣿⠇⡖⠂⣧⣿⡃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣼⡿⠀⠰⡔⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠚⠉⢀⣬⡿⠃⠀⠠⢻⡜⣿⢆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠒⠓⠒⢲⣶⡿⠋⣀⠀⡀⢘⢩⣴⣶⠀⠀⠰⠈⠉⢿⡀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢥⡀⠀⠀⠈⠿⣿⣯⡀⠀⠀⠄⢙⣿⣿⠀⠀⢰⠀⣧⠘⣷⣵⠁⠀⠤⠀⠠⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣭⣒⠰⢤⣄⠈⠻⢿⣿⣖⠋⢩⣿⣿⣄⠀⢰⠀⣨⠳⡉⣿⡍⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣷⣦⣈⣿⡄⠬⠟⢻⣿⣿⣩⣽⣿⣆⢠⡀⣌⠃⠱⣧⡀⠀⣴⣖⣚⣡⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣾⣿⣆⠀⣠⣿⣿⢉⣿⣿⡁⢈⢹⣿⡀⠀⠈⢧⣸⡏⠀⠈⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡻⢿⡈⠀⠀⠉⢻⣿⠟⠉⣸⣿⣿⠇⢠⠘⠙⢿⣄⠀⢤⣹⣇⠀⢀⡿⢃⡼⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢷⡈⠀⠐⣄⣿⡄⢠⣾⡿⣙⠁⠠⠀⠀⠀⠛⣷⣴⣍⢻⠂⠈⣤⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣞⠷⣄⣤⣸⣿⣿⣿⡟⠀⣹⠀⠆⠀⠀⠀⠀⠈⠙⠻⢮⡉⢛⣥⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⣠⣿⣿⡿⠍⠻⡅⢠⡈⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠋⠀⠀⠀⢻⣾⡇⠀⡄⠀⠀⠀⣈⡤⠓⠃⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠃⠀⠀⠀⢈⣲⣿⣇⠀⡀⠀⠐⠋⣱⠾⠃⠀⠌⠬⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⣿⡟⠉⠀⠀⠀⠔⢋⣴⠶⠂⠤⡀⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⣄⣄⣛⠏⠁⠀⠀⠃⠀⠾⠛⣿⣿⠶⠤⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣀⡾⣿⣏⠀⠀⠀⠘⡃⠀⠠⠛⠁⠤⠆⢀⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢿⣿⣿⠁⠀⠀⠀⠀⡁⠀⠀⠀⠀⢒⠀⡏⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣿⣿⣿⠇⠀⠀⠀⠀⢂⡁⠀⠀⠀⠀⠈⠀⠱⣤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣎⣼⡯⠿⠛⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣆⠄⠀⠸⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡤⠋⠀⠉⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠐⠀⠀⠀⡔⠁⡾⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠛⠇⠀⠀⠃⠀⠀⠟⠀⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠀⠀⠀⠀⠀⠚⣷⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠻⡍⠀⠀⠀⠀⠀⠠⠐⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠙⡇⠀⠀⠀⠀⠀⢶⡾⠀⠀⠀⠀⠀⢀⣺⠶⠆⠀⠀⠀⠀⠀⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠀⠈⠓⡄⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠐⢻⡿⠀⠀⠀⠀⠀⠢⠴⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠉⠊⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⢸⠗⠀⠀⠀⠀⠀⠀⢺⠇⠀⠀⠀⠀⠀⣤⡅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠐⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⡤⠶⣶⠛⠻⠷⠶⠦⠀⠈⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣆⠳⠚⡇⠙⠃⠀⠌⠛⠉⢁⣀⡉⠰⠉⠉⠀⠈⠁⣀⡛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠤⠟⠁⠀⠀⠀⠀⠀⢂⠐⠓⠛⠑⠆⠀⠙⠋⠀⣵⠆⠀⠙⢦⠑⢋⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡀⠀⢠⠖⠀⠀⠐⠚⠂⠀⠉⣠⠞⠃⠂⠀⠺⠿⣿⣤⠀⠋⠛⠠⡄⠀⠠⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠉⠃⠀⠙⣠⠖⢀⡆⢠⣦⠀⠀⣤⠀⠀⠀⢀⡀⠀⠀⠷⠈⠁⠲⣤⡈⠀⢦⡀⢶⠙⠛⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠙⣿⡇⠀⢀⣀⠀⠹⠟⠂⠉⠀⠢⠉⠀⠀⠀⠚⣥⣴⢶⡆⠀⠀⠈⠛⠃⠀⠘⠁⠀⠒⣶⠆⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠐⢣⠟⠀⠀⠨⣭⡷⠆⠐⣶⢶⣄⠀⠀⢲⣤⣾⣇⠈⠳⠀⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⣦⡈⠢⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⡴⢦⠀⠛⠁⠀⠸⡏⠀⠈⠁⠀⠈⣿⣼⠏⠀⠀⠀⢶⡐⠀⠸⣿⠁⠀⠰⣄⠀⠙⠧⠀⠀⠉⣻
⠀⠀⠀⠀⠀⠀⢰⡏⠇⠙⠠⠄⠀⣄⡀⣀⠀⠐⠀⢀⠘⠆⠀⠀⠀⠉⠉⠀⢀⣠⡦⠀⠀⠀⠀⠀⠀⠸⠆⠚⠁⠄⠒⠀⠾⡇⢉⡆
⠀⠀⠀⠀⠀⠀⠸⠃⠀⢴⠀⠂⢰⣯⣷⠋⠀⠀⠰⠈⠑⠀⠀⠀⣀⡀⠀⢠⣿⣿⠛⠃⠀⠀⠶⠀⠀⠀⠀⣴⣦⠀⠶⣀⠀⠐⠈⢱
⠀⠀⠀⠀⠀⠀⢠⡀⠘⠃⢤⠀⠈⠙⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠙⠛⠀⠈⠙⠿⠿⠃⢀⣠⠀⠀⡀⠂⠀⣹⡆⠐⠆⠹⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠚⠷⠀⢂⣠⣄⠀⠈⣿⡿⠃⠀⠀⠠⣀⣤⣀⠀⠀⠀⢠⣤⡌⠃⠀⠀⠉⠀⠀⠉⠋⠀⠀⠀⠀⣓⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠈⢿⣏⠀⠘⠉⠀⠰⣦⠀⠀⠹⠿⠟⠃⠀⠀⠈⠉⠀⠈⣀⡄⠀⠀⠒⠠⢿⣦⡀⠀⠠⣼⡄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⡀⣤⢠⡄⠸⠉⠁⠀⠀⠁⣀⣤⡀⠃⠀⠀⠀⠐⠒⠗⠀⣶⣶⣿⡏⠹⠄⠀⠀⠐⠛⠈⠀⠀⡀⢀⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⢄⡈⢧⡀⠖⢈⡛⠀⠀⠀⠛⣿⠿⠃⠀⡛⢿⡆⠀⠄⠀⠛⠿⠿⠁⠀⢠⣴⠟⢀⡀⠒⠀⣰⠄⠘⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡈⠁⠀⠄⠁⢸⣿⣄⠀⠀⠀⠀⠀⠠⠀⠀⢀⣤⡄⠀⠀⢠⠀⠰⠏⠁⢴⠩⠁⠀⠚⠉⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡘⢦⡄⠀⠲⣦⡈⠛⠿⠄⠀⣀⣴⣶⣤⣄⠀⠀⠀⠀⠀⠋⠀⠀⠀⣀⠀⢀⡀⠀⣀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣌⡇⠈⢳⣌⠁⠀⡧⠈⢃⠙⠟⠛⠋⣉⠀⠀⠚⠛⠀⠀⠁⠀⠀⠈⠀⠎⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⠏⣀⠠⠶⠀⢀⠁⠘⠁⠀⣤⣤⣤⠀⠀⠀⠾⣏⡉⠀⣀⠠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⡓⠦⠈⠛⠛⠓⠀⠀⠀⢀⣠⡤⠤⠶⠀⠴⠆⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠴⠊⠋⠋⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠉⠰⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠔⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠰⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠖⠂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠀⠈⠉⠒⠶⠆⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄⠦⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡔⠉⠉⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⢀⡄⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠋⠩⠀⡄
⠀⡠⣤⠒⠺⠏⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠆⠉⠉⠈⠁⠀⠀⠠⢄
⣾⢧⣤⣥⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠠⠀⠸
⢘⠷⣿⠷⡿⢿⣟⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠐⠂⠈⠁⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠚⠄⠐
⡼⠷⢛⡀⠓⢀⡟⠻⢊⠻⠰⣶⢂⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠠⠄⠀⡀⠸
⣹⣆⣭⣵⠧⠌⢀⣻⡌⢐⢻⡆⠹⠋⣼⣷⣤⣤⣄⠀⡀⠀⠀⠀⠀⠀⡀⠀⣼⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀
⣝⢣⡉⣥⡶⣶⡈⢻⣷⣌⢏⠀⠶⠚⣣⢤⡦⠄⠈⠀⢙⣦⣠⡀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠂⠀⠀⢀⣰⠒⠊⠈
⣹⠇⣃⣸⣠⠏⡁⠊⠠⢍⡾⠁⠰⠈⣇⠠⠴⣆⡀⠁⠄⠉⣿⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⠂⢀⣀⠲⠆⠋
⡿⠛⠹⠟⣃⡘⢡⠿⠐⠄⢁⣀⣠⠀⠸⠀⠆⣬⢻⠐⣆⡉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠀⠠⠀⢀⠀⠄⠀⢄⣀⡔⠂⠒⠋⠉
⠷⠶⠷⠚⡼⢋⠀⣶⣀⠰⠎⣛⢁⡀⠶⣔⠿⠠⠏⠙⡯⠅⣸⠆⢀⡀⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⠀⠁⠀⠀⠀⣈⠀⠴⠛⠈⠁
⠀⠀⠘⠻⠖⠛⡎⠁⣰⠃⠀⠉⢄⣁⠘⡄⢀⠀⣤⢻⣷⠄⣻⠄⠠⡄⢀⠈⠠⠀⠀⠀⠀⠀⠄⢰⠀⠠⢄⠦⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⣤⣦⣨⡍⢠⡄⠞⣀⣈⠛⣿⡶⢿⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠹⣤⣝⣠⠛⠭⣄⠳⣶⣠⠄⠀⠀⠈⠀⠀⢀⡤⠆⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢆⣤⡚⣙⣿⠀⠀⠀⣀⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⣁⠀⠋⠁
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠑⣄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⠀⢠⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠠⡆⡤⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠐⠂⠉⠀⠀⠀⠙⠢⣤⠤⢄⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⢄⣀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠐⢤⠘⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠁⠐⠒⠒⠀⠀⠀⠀⠀⡘
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠡⡀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⠀⢀⠔
⡇⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⢀⠁⠀⠀⠐⡄⠀⠀⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠚⠤⠤⠤⠄⠂⠉
⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈⡄⠀⠳⠄⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠙⠢⠄⢠⣄⠀⠄⠊⠈⢂⣀⡀⠀⠂⠁⠀⠀⠀⠀⠣⠀⠀⢄⣀⡀⠖⠊⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠀⠀⠀⠒⠒⠒⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡏⡀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠇⠄⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢣⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠐⠚⠀⠀⢀⡘⢒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣿⣷⣿⣿⡿⠿⠿⣿⣿⣿⣿⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⡿⠿⠇⣿⠀⠿⠿⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣇⠛⠛⣿⠟⠛⢂⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿⣇⠛⢀⣿⣿⣿⣿⡇⠀⠀⣿⣧⡴⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢃⣠⣶⠟⠉⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣀⣠⣤⣤⣄⣀⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠈⠉⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣄⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣤⣄⣀⣀⣀⡀⢀⣴⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣰⣿⣿⡟⠀⠀⠘⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⠀⠉⢉⣡⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⡇⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣨⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⠀⠀⠀⠀⠀⠀⠀⢼⣶⡆⠀⠀⠀⠀⠀⠀⢸⣯⡁⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⢠⣅⢀⣠⣮⢉⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣿⣦⡀⠀⠀⠀⢀⣴⣦⣤⣶⣆⠀⠀⠀⢀⣤⣴⣹⣷⡄⠀⠀⠀⠀⣀⡬⣿⠥⣤⡀
⠀⠀⠻⠿⠋⣿⣿⡟⣿⡄⠒⠀⠂⠀⠀⠀⣾⣿⣿⣿⢻⣧⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⢤⣨⣽⣿⣿⣿⠀⠀⠀⠀⣾⡟⠁⣾⠄⢙⣿⡆
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠷⠷⡄⠀⠀⠀⣿⣾⣿⣿⣿⣿⠀⠀⠀⣿⢿⣯⣿⣿⢹⡇⠀⠀⠀⠁⢸⣿⣿⡆⠀⠀⠀⠙⠿⢲⣿⣶⠙⠛
⠀⠀⠀⠀⠘⣿⣿⣿⠈⠁⣤⠠⡄⠀⠀⠀⣿⢸⣿⣿⡇⠈⠀⠀⠀⠷⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⣿⡿⣿⡆
⠀⠀⠀⠀⠀⢻⣿⡇⠀⢈⣉⣁⡁⠀⠀⠀⠛⠈⣿⣿⠀⠀⠀⠀⠸⠿⢿⣿⢻⣿⡄⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⢸⣿⠃⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⡇⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠶⠆⣒⣦⠠⠒⠒⠒⠒⡄
⠀⠀⢰⡞⠃⠀⠀⠀⠀⠀⣀⣘⣉⣃⡀⠀⠐⠺⢛⢻⡯⢓⣀⠀⡇
⠀⠐⣿⣿⣷⣶⣀⣠⠀⣾⡟⢿⣿⣿⣿⡆⠀⠀⠀⠈⠉⠀⠉⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⡀
⠀⠀⣿⣿⡿⠿⠿⢛⣄⡙⡷⠺⠟⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡅⠀⠀⠀⠀⠀⠀⠈⡇⠙⠁⠀⠀⠀⠀⠀⣀⠀⠀⣶⡏⠉⠉⢉
⠀⢸⣿⣿⡇⠀⠀⣻⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇⡀⠛⡏⠘⠃⠀⣤⠀⠀⠀⠀⠀⡇⠀⢀⣄⢀⣀⣀⣼⣿⣷⠀⣿⣧⣤⣤⣶⡀
⠀⠀⢿⣿⡇⠀⠀⠛⠛⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⢛⣿⡀⡇⠀⠀⠀⣿⠂⠈⠉⠉⠉⡏⢿⣶⠾⠿⣿⣿⣿⣋⣉⣹⡿⠀⠀⠀⠀⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⣿⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⠛⠛⠛⠛⠛⠛⣟⠀⠀⠀⠀⡇
⠀⠀⠘⠿⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠈⠁⠀⠀⠀⠻⠀⠀⠀⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶
⠀⠀⢀⣴⣶⣤⣶⣦⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⣽⣿⣤⡀⠀⠀⠀⠀⠀⠀⢠⣼⣯⠁⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣥⣄
⠀⠀⣾⡟⣿⣿⡏⣿⡇⠀⢴⣿⣷⠄⠀⠀⣿⣻⠋⠘⢛⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⣠⣽⡟⠁⠀⠀⠀⠀⠀⢀⣿⠟⢻⣿⢻⣧
⠀⠠⢿⡇⣿⣿⣇⢹⣧⣤⣾⣿⣿⣆⠀⠀⠈⢿⣀⣠⡮⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢿⣦⣤⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⣿⢰⣾⣿⠈⠁
⠀⠀⢸⢸⣿⢻⣿⡀⠀⠀⢸⣿⣿⠿⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠉⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣾⡿⣿⡇
⠀⠀⠈⣿⡿⠈⣿⡇⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⣿⡇⢰⠁⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⢠⣿⡇⢻⣿
⠀⠀⠆⠻⠇⠀⠿⠇⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣿⡇⠘⢄⡀⠀⣠⠞⣿⡿⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁