Yayy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡟⠙⢻⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⠃⢀⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠻⣿⣷⣤⡀⠉⠀⣼⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣿⡇⠀⣀⠈⠻⣿⣿⡆⠀⢻⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣟⠉⠛⠿⠇⠀⣿⣿⠁⠀⢁⣀⣄⠀⣻⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⢿⣿⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⠛⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠛⠋⠉⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠚⢲⣆⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⣠⣿⡟⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠉
⠀⠀⠀⠀⣴⣿⣿⠏⠀⠀⣴⣿⡿⠁⠀⣰⣿⣿⠏⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⣰⣿⠏
⠀⠀⠀⣼⣿⣿⠃⠀⢀⣾⣿⡟⠀⠀⢰⣿⣿⠏⠀⢀⣴⣿⡿⠋⠁⠹⣦⣤⣄⠀⠀⣸⣿⣿⠀⠀⣰⣿⡿⠁⢠⣿⠏
⠀⠀⢸⣿⣿⠃⠀⣠⣿⣿⠟⠀⠀⣠⣿⣿⡏⠀⣠⣿⣿⠟⠀⠀⢀⣼⣿⡟⠁⠀⣰⣿⣿⠃⠀⢰⣿⡿⠁⠀⣾⡟
⠀⠀⠈⢿⣿⠀⢰⣿⣿⠏⠀⣠⠞⣿⣿⣿⣠⠾⣿⣿⠏⠀⠀⢀⣼⣿⡟⠀⠀⣴⣿⡿⠁⠀⢠⣿⣿⠃⠀⠀⢿
⠀⠀⠀⠀⠈⠀⠸⣿⣿⡤⠞⠁⣼⣿⣿⠋⠁⢸⣿⡟⠀⢀⡴⢻⣿⡟⠀⣠⢾⣿⡿⠁⢀⡴⣿⣿⡏⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣿⣿⣿⠃⠀⠀⠀⠻⠷⠖⠋⠀⠈⠻⠷⠚⠁⠘⠿⡷⠶⠋⣼⣿⡿⠀⠀⠀⠿⡿⠃
⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠉⠉⠛⠲⣦⣄⣼⣿⡿⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣾⡇⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡃⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⣤⣤⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣤⣤⣴⠾⠟⠋⠀⠀⠉⠛⠛⠻⠛⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣩⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣩⣵⣾⠿⠛⠋⠁
⡿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣫⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣉⣤⣶⠿⠟⠋⠁
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⣉⠘⠛⠚⠛⢻⣿⣿⣿⡿⣿⣻⣽⣿⣿⣿⠿⠟⠋⢉⣠⣴⠾⠟⠋⠉
⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢛⠉⠻⠛⣋⣩⣤⣶⣾⣴⠿⣿⠛⠛⣿⠇⣼⣿⣯⣷⣿⡿⢿⡟⠋⣡⣀⣴⠶⠞⠋⠉
⣛⣭⣶⣿⣿⣿⡿⢋⣡⣶⢿⡿⠿⢿⣟⢟⣿⣾⣿⣿⣅⣿⠀⢰⡟⢰⣿⠿⠛⠉⠁⣠⠃⣧⣾⡟⠻⡤⠂
⣿⣿⣿⣿⣿⣿⡇⢹⡏⠀⢸⡟⢁⣾⠁⢸⡇⠘⣿⡏⠓⠃⢀⣿⢁⡿⢷⣶⣠⣴⡾⠁⢀⣿⣿⢷⡎⠉⠉⠈
⣿⣿⣿⣿⣿⣿⣷⠘⣷⠀⠈⠁⣸⡟⠀⢸⡇⠀⣿⣧⠀⠀⣼⠇⣼⠃⠀⠉⠉⠁⠀⢠⡏⢹⣿⢺⡃⠀⠀⢀⣠
⣿⣿⣿⣿⣿⣿⢿⡀⣿⡀⠀⢀⣿⡇⠀⠈⠀⠀⢹⡿⠀⠀⣿⢀⡏⠀⠀⠀⠀⠀⠀⣸⡁⠀⠙⠛⠓⠒⠛⢹⡇⣠⣀⣰⠃⠀⣇
⣿⠿⢟⣫⣭⡶⠿⠇⣿⡇⠀⢸⣿⠀⠀⣿⡇⣀⣸⣧⣤⡾⠟⢸⡇⠀⠀⠀⣴⣦⣽⣿⣿⣿⣶⡄⠀⠀⠀⣿⣯⢹⣯⠍⢦⡀⠈⠳⢤⣀
⠶⠟⠛⢉⣡⣤⣾⠀⣿⣠⣤⣿⣿⡶⠟⠿⠟⠋⣉⣩⣤⠶⠞⠛⠀⠀⣀⣴⣿⣿⣿⣿⣶⣾⣿⣿⣄⠀⠀⢿⣶⠘⡛⢦⣀⠙⠲⣤⣀⡈⠉⠒⠂
⣴⣶⣿⠿⠛⠉⠈⠀⣋⣩⣄⣨⣤⠶⢶⠶⠻⠛⠉⠁⠀⠄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⣯⣥⢤⠈⠙⠒⠠⠈⠉⠛⠒⠦⠤⢄⣀
⠋⠀⠀⠀⠀⠀⠘⣛⠭⠟⠋⠑⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠉⠛⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠉⣡⢶⠶⡶⢷⢶⣤⠀⠀⢀⠀⠐⠀⠀⡀⠀⠂⠨⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣤⣶⢶⠶⠶⠶⠶⠗⡖⢒⠤
⠀⠀⠀⠀⠈⠉⠛⠛⡛⠛⢯⡽⢩⡭⣷⠀⠄⢀⠁⠀⠠⠀⡈⠀⠂⠻⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠆⠶⠐⠂⠃⠛⠘⠁⠁⠉⠀⠂⠒⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⢀⡇⠀⢳⣧⣶⡛⣿⣦⢁⡂⠠⠀⠌⢀⠀⠛⢦⣿⡛⢿⣿⣿⣿⣿⣿⠿⠹⣞⡃⣋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢨⢭⠬⠇⠶⠺⣠⣴⢷⢶⡄⠄⣈⢀⣂⡨⢙⡛⠛⢛⣿⡟⡷⢶⢤⣿⡥⠰⠀⠀⠀⠀⠀⠀⢀⣠⡴⣾⣿⡟⠾⠋⢋⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠿⡍⣭⠨⠤⠿⡦⣾⡛⣋⢛⢦⡀⠉⣾⠠⢄⣦⣴⡐⡊⣛⢘⡀⠀⢀⣀⡴⢶⣯⠽⠛⠃⣤⠴⢒⣋⣴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⠀⠀⠉⢹⡉⡅⠥⠸⠄⠷⣠⡗⡂⣏⢙⣈⣁⠁⠈⡤⢐⡿⠼⠔⢋⣍⢰⠦⢛⣭⣰⡾⣼⣯⣟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠱⡈⡥⠧⠸⠁⢀⡍⠐⢂⣭⣴⡚⠟⣈⢥⣖⣿⢷⡯⣽⣻⡟⣾⢭⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣶⢖⣈⡍⣴⢦⠟⢛⢁⡤⣴⣹⡟⣾⢿⣯⣿⣳⠧⣿⣽⣟⡿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣴⣺⠿⠞⢉⣁⡴⢶⣯⣿⣻⠷⡾⣽⣟⡿⢶⢯⣟⣻⢳⡿⣽⣛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡶⢤⣏⠿⠒⠋⣥⣰⡒⡾⣼⣏⣟⣲⢧⣿⣹⣓⡿⢼⢍⣛⣺⢧⣯⣽⣳⠿⣼⣽
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣼⣶⣧⣿⣹⣟⣈⣤⣤⣿⣿⣧⣯⣽⣿⣿⣾⣭⣟⣿⣷⣏⣉⣡⣶⣾⣭⣏⣿⣶⣯⣿⣹⣷⣾
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⢦⡀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣾⡇⠀⠀⠀⣸⣿⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⣼⣿⠀⠀⠀⢠⣿⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠙⢿⣄⣀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⢰⣿⠃⣠⣶⣶⣤⣶⡆⠀⠀⣴⠂⠀⠀⣠⡄⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⢀⣨⣿⡧⢠⣿⠏⣼⡿⠋⠀⣿⣿⠁⠀⣼⠃⠀⢀⣼⣿⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠙⠉⣡⣿⣯⠞⣿⡇⢀⣴⣿⣯⢀⡴⣿⣀⣠⣾⣿⠃⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣤⠾⠟⠋⠀⠀⠈⠓⠛⠉⠈⠛⠋⠀⠙⠿⠟⢹⡿⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⢻⣿⠋⠉⠀⠀⠈⠙⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⠉⢹⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⢀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠛⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⢾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠀⢸⣿⣿⣿⣿⡿⠉⠀⠈⣿⣿⡿⠋⠉⠁⠀⠉⠙⠿⣿⣿⣿⠏⠉⢻⣿⣿⣿⡟⠉⠉⢿⣿⣿⣿⣿⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⡆⠀⢀⣴⣶⣦⡄⠀⠙⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⣶⣿⠿⠟⠛⠻⠆⠀⢹⣿⡀⠀⠸⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠉⠉⠀⠀⣰⠀⠀⣿⣿⣿⠏⠁⠀⢀⣀⣀⡀⠀⢸⣿⣷⡀⠀⠈⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣤⣄⣀⣤⠴⠞⠟⠀⢀⣿⣿⡟⠀⠀⢠⣿⣿⣿⡇⠀⠀⣿⣿⡿⠗⠒⠲⠶⡿⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣠⡀⠀⠀⠙⢿⣷⠀⠀⠈⠿⠿⠟⠀⠀⠀⣿⣿⠁⠀⠶⠦⠀⠀⢠⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⠀⠙⠉⠀⣼⣇⣀⣸⣿⣷⣄⣀⣀⣀⣠⡀⠀⢠⣿⣿⣷⣤⣀⣤⣤⡀⠈⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣥⣭⣭⣥⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠇⠁⠀⠉⠀⠀⢼⣷⣼⣥⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠴
⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠇⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⣶⡀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⢰⣿⡿⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⢠⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⢠⣿⣿⠇⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣰⠃⢸⣿⣿⡿⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡆⠀⢰⣿⣿⠁⠀⠀⣰⣿⣿⡿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⡼⠃⠀⣼⣿⣿⣇⣠⣤⣴⣿⣿⡿⠉⣼⣿⣿⠃⠀⣿⣿⡏⠀⢀⣼⠋⣿⣿⡇⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡴⠞⠁⠀⠀⣿⣿⣿⡏⢁⣿⣿⣿⠏⠀⣼⣿⣿⡏⠀⢰⣿⣿⠀⣠⡿⠃⢸⣿⣿⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣿⣿⣿⠀⣼⣿⣿⡏⢀⣾⣿⣿⣿⠀⢠⡟⠻⢿⠿⠋⠀⠀⣿⣿⡇⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢰⣿⣿⣿⠀⣿⣿⡿⢠⠟⣿⣿⣿⡇⢠⡿⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⣿⣿⡿⠀⠸⣿⣷⠏⠀⣿⣿⣿⢡⣿⠁⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡆
⢀⣴⣶⣶⣶⣶⣶⣶⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣶⣶⣶⣶⣶⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠙⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⣸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⣼⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣇
⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠈⠙⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡄
⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⢠⣿⡟⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⣠⣿⣿⡿⠋
⠀⠀⠀⠀⣼⣿⣿⠏⠀⣰⣿⡿⠁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢰⣿⣿⡟
⠀⠀⠀⢰⣿⣿⡏⠀⣰⣿⡿⠁⣸⣿⣿⢀⣴⣾⠟⣿⣿⠗⣰⣿⡿⠃⢠⣿⡇⠀⣾⣿⡇⠀⣿⣿⡟
⠀⠀⠀⠀⢿⣿⡇⠀⣿⣿⠁⣰⣿⣿⢇⣾⡿⢁⣼⣿⠏⢠⣿⡿⠁⢀⣾⣿⢃⣼⣿⠏⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠉⠁⠀⢿⣷⠾⣿⣿⡟⠸⣿⡴⠋⢿⣿⣤⠾⣿⣧⡶⢻⣿⣿⠟⠋⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣴⣿⣟⠀⠀⠀⠀⠀⢀⣅⠀⠀⣀⣴⠾⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⡟⠁⠀⠀⠈⣹⣿⠿⣿⣿⣿⣶⣶⣾⣿⡟⣠⡾⠋⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡷
⠀⠀⠀⠸⣧⣀⣀⣠⣾⡿⠋⠀⠀⠉⠙⠛⠛⠛⠉⠀⣿⡀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠁

⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⢀⣤⣤⣤⣀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⣄⣠⣤⣤⠀⠀⣤⣤⣤⣄⠀⠀⢠⣤⣤⡀⣤⣤⡄
⠀⠀⠀⠀⠈⣿⣿⡋⠙⣿⠹⣿⣿⢻⣿⡇⠀⠀⢻⣿⡟⠀⠀⠀⠀⠀⠀⠙⣿⣧⡾⠋⠀⠀⢀⣿⡧⣿⣿⣆⠀⠀⠙⣿⣶⡟⠉
⠀⠀⠀⠀⠀⣿⣿⠛⠇⠀⢠⣿⣿⣾⣿⠃⠀⠀⢸⣿⡇⠀⢀⣤⣤⣴⡆⠀⣸⣿⣷⡀⠀⣠⣾⣿⡄⣼⣿⣿⣦⠀⣰⣿⣿⡇
⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⢸⣿⣿⠙⣿⣷⡄⠀⣸⣿⣿⠀⠛⠛⠋⠉⠀⢠⣿⣿⠿⠃⠀⠉⠉⢉⣀⣈⡉⠉⠀⠀⠉⠛⠻⠿
⠀⠀⠀⠀⠘⠿⣿⣧⡀⠀⣼⣿⣿⣆⣿⣿⣧⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣄⢴⣶⠟⠀⠀⣿⠟⣿⣷⡀⠀⠙⢿⣷⠀⣤⣀
⠀⠀⠀⠀⠠⣦⣄⡉⠁⠀⠛⠿⠿⠛⠘⠿⠟⠋⠛⠉⠉⠀⠀⠀⠀⠀⠘⠻⣿⣧⡾⠁⠀⠀⢀⣿⠿⢿⣿⣇⠀⠀⠈⣿⣶⡿⠛⠃
⠀⠀⠀⠀⠀⣿⣿⡛⠿⣷⣤⣶⣤⣤⣤⡀⠀⠀⢾⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⢀⣼⣿⡀⣸⣿⣿⣆⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠛⠦⠉⢈⣿⣿⣸⣿⠇⠀⠀⢸⣿⣇⠀⣠⣶⣶⠿⠃⢀⣿⣿⣿⡆⠀⠾⠿⠟⠃⠙⠛⠛⠋⠀⠻⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⢸⣿⣿⠻⣿⣦⡀⠀⣸⣿⣿⡀⠉⠁⠀⠀⠀⠘⠛⠋⠉⣀⣀⠀⠀⣤⣶⣶⡄⠀⠀⣠⣤⣄⠈⠉
⠀⠀⠀⠀⢀⡉⠛⠷⠀⠀⣾⣿⣿⣆⢿⣿⣧⡠⣿⡿⠿⠃⠀⠀⠀⠀⢠⣴⣶⡗⢺⡿⠋⠀⠀⣿⠟⣿⣿⡀⠀⠈⢻⣿⠁⣿⣶⡄
⠀⠀⠀⠀⠈⣿⣷⡶⣤⣄⢈⣉⣉⣁⣈⠉⠉⠀⣠⣤⣶⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠈⣿⣿⡟⠁
⠀⠀⠀⠀⠀⣿⣿⠷⡦⠛⢙⣿⣿⣹⣿⠇⠀⠀⢸⣿⡇⠀⢀⣤⣤⣴⡆⠀⣸⣿⣷⠀⠀⠀⣸⣿⠀⢸⣿⣿⡀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⣸⣿⣿⡙⣿⣶⡄⠀⣼⣿⣿⡀⠙⠉⠉⠉⠀⢠⣿⣿⣿⡆⠀⣴⣿⣿⡆⣾⣿⣿⣷⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠛⠛⠛⠋⠘⠛⠛⠊⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠉⠙⠉⠁⠈⠙⠉⠁⠀⠈⠛⠛⠋






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣦⣴⣶⣿⣶⣦⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠛⢿⣿⣧⠀⣼⣿⣿⣿⣿⠿⣿⡿⠛⢿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣤⣄⠙⣿⡿⠀⣿⡿⠿⣿⣁⡀⠻⠇⢰⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣶⣌⠁⢰⣿⠁⣤⡙⢿⣿⣦⠀⣾⣿⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣼⣿⠀⢈⣤⠈⢙⡿⠀⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠻⣧⣤⣴⣿⣾⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠀⠛⠿⠟⠛⠋⠁⠀⠉⠻⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⠈⠛⠛⠛⠋⠀⠀⠈⠉⠉





⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣶⣤⡀⠀⠀⠀⠀⠈⠉⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⢀⣠⣴⣿⡿⠋⠸⣿⣆
⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⣀⠀⠀⢴⣶⣤⣀⣀⣀⣠⣤⡄⠀⠺⡿⠿⠿⠿⢿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⢰⣶⣿⣿⠿⠿⠋⠁⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⢠⣾⡿⠂⠀⠘⣿⡟⠻⠿⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣤⡀⠀⠀⠙⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀
⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⢠⣿⡇⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠈⢻⣿⣆⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿
⠀⠀⠀⣾⡿⠁⠀⠀⠀⣰⣿⠏⠀⠀⠀⢀⣾⣿⣷⣶⣶⣾⣿⣇⠀⢀⣴⣶⣶⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⢻⣿⡆⢀⣴⣿⠿⠋⠉
⠀⠀⣸⣿⠇⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠈⠙⠉⠁⠀⠈⠉⠻⠟⠀⠀⠉⠉⠀⠀⠀⠀⠉⠙⠻⣿⣦⡀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⣿⣿⣿⠟⠁⠀⠀⠐⣿⣆
⠀⠀⣿⡟⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠘⣿⣧⠀⠀⠻⡿⠃⠀⠀⠀⠀⠀⢻⣿
⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⢰⣷⡆⠀⠀⠀⢸⣿⡇
⠀⠸⠿⠃⠀⠀⠀⠸⠿⠃⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠈⠿⠇⠀⠀⠀⠘⠿⠇⠀⠀⠀⠈⠿⠇
⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⣰⣿⡿⠛⠛⠿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⠻⢿⣷⡀⠀⠀⠀⣴⣿⠟⠛⠛⠻⣿⣦⠀⠀⠀⢀⣾⡿⠛⠛⠻⣿⣧⡀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠛⢿⣿
⣿⣿⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⢈⣿⡇⠀⠀⣼⣿⠃⠀⠀⠀⠀⠘⣿⣧⠀⠀⢸⣿⡁⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⣿
⠹⣿⣦⠀⠀⠀⠀⠘⢿⣷⡄⠀⣰⣿⠟⠀⠀⠀⠀⢀⣾⡿⠃⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠘⢿⣷⡀⠀⠀⠀⠀⠻⣿⣆⠀⢀⣾⡿⠃⠀⠀⠀⠀⣰⣿
⠀⠘⣿⣷⡀⠀⠀⠀⠈⢻⣿⣾⣿⠏⠀⠀⠀⠀⣠⣿⡟⠁⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠈⢻⣿⣄⠀⠀⠀⠀⠙⣿⣷⣿⡟⠁⠀⠀⠀⢀⣼⣿⠋
⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⣾⣿⠁⠀⠀⠀⢀⡀⠀⠀⠀⠈⢿⣷⡀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⣼⣿⠃⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠘⠿⠃⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⢰⣿⠏⠀⠀⠀⢀⣿⡟⢻⣿⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⢀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⣼⣿⠁⠈⣿⣧⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣽⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠰⣿⣿⣶⣶⣿⣿⠆⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⢸⣿⡇⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⢠⣿⡟⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⢻⣿⡄⠀⢸⣿⡇⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣾⡿⠀⠀⠀⠀⢠⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⡄⠀⠀⠀⠈⣿⣷⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⢀⣼⣿⠁⠀⢿⣷⡀⠀⠀⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⢀⣿⡿⠀⠈⣿⣧⡀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠟⠁⠀⠀⠈⠻⢿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⠟⠁⠀⠀⠈⠻⢿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣫⡶⣮⢫⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⣿⣾⣿⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢰⢫⡷⣯⡻⣾⣶⡄⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢳⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣮⡻⠾⣷⠼⣿⣿⣄⠗⣵⢹⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⡇⣿⢸⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡅⣿⣸⢹⣿⣿⠀⣿⢸⣿⣿⣷⢀⣠⣤⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡞⣸⣧⠇⣿⡘⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⣿⢸⣸⣿⣿⠀⣿⢸⣸⣿⣿⢟⣽⢾⣿⢾⣽⣿⣷⡄⠀⣲⢰⡆⣄⣀⡀⣿⣣⡜⣹⣿⣦⠀⠀⡇⣿⣹⣸⡟⢇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⣤⣿⣿⣯⣻⣿⢸⣾⣯⢏⣾⣳⣿⣿⣿⡽⠺⣧⣠⣀⣿⣾⡇⣷⣿⣿⣃⢸⡇⢿⣿⣿⠀⢸⢳⣿⣏⣻⡇⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⡿⣟⣿⣳⡿⣿⣦⡇⡏⣾⢧⢻⣿⡿⢏⣾⡿⣹⣿⣿⣿⢿⡇⣿⣿⣇⢻⣸⡇⣸⣿⣿⠀⢸⢸⣿⡾⣿⢧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⢿⣜⣚⣵⢯⣴⡟⠟⣼⣏⢿⣼⠿⠽⢟⣼⡟⣽⣿⣿⣿⣼⣸⣇⣹⣿⢭⣬⣿⠏⣿⣿⣿⠀⢸⢸⣧⢇⣿⣸⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣝⡻⣗⣇⢻⡇⢸⣿⣿⣿⣝⣛⣛⣻⠭⡻⣮⣓⣚⣓⣸⢧⡻⢾⣷⠿⡟⣿⣠⣿⣿⡿⠀⢈⠈⡿⢾⣇⢧⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣸⢸⡷⣼⣿⣿⠀⠉⠲⢶⣿⣿⡾⢶⣭⣭⡭⣗⣬⣽⣷⣾⡄⡇⣿⢹⢿⣿⡇⠀⢸⢖⣷⣿⢩⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣡⣵⢷⣛⣻⡲⣤⣄⠀⠀⠀⠀⠈⣛⣿⣛⢿⠛⠋⠉⢱⡇⣿⢸⢸⣿⡇⠀⠈⣧⣟⢫⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢴⠶⣞⣓⣻⣭⣷⣾⡿⢟⣫⣶⣿⣿⡇⠀⢀⣴⢝⣭⣷⡶⣳⣿⣿⣆⢸⠃⣿⢸⣿⣿⡇⠀⠀⣀⡁⢾⣿⡿⠟
⠀⠀⠀⠀⠀⣠⡾⣛⣭⡶⢶⣟⣛⣛⣹⢭⢖⢾⣷⣴⣿⣿⣿⡿⠛⠀⢀⡾⣱⢟⣏⣭⣾⣷⣿⣿⠏⣼⢸⡟⣇⣿⣿⡇⣴⢟⣭⣬⡻⣶⣦⣄
⠀⠀⠀⠀⠐⡟⣾⠏⡖⣝⣳⣲⢭⡷⣢⣟⣼⣿⣿⡿⠉⠉⠁⠀⠀⠀⢸⠇⣿⢾⢿⣿⡿⠟⠋⡡⡾⣫⡿⣽⣾⣿⣿⠠⣉⢿⡱⢮⡿⣼⣿⣿⡄
⠀⠀⠀⠀⠘⢷⣝⣳⢾⣭⣭⣽⠶⡟⣿⣮⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⢷⣝⢷⣕⣛⣥⣘⣛⣭⡾⣫⣾⣿⣿⣿⠃⠀⠙⢮⣝⣫⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠉⠳⢽⣖⣒⣺⣧⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢼⣏⡋⣏⣻⣴⣿⣾⣿⣿⠟⠁⠀⠀⠀⠀⠐⠺⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠰⠛⠀⠀⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠘⠛⠄⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠐⠦⡀
⠀⠀⠀⡀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠆
⠀⠀⠀⠀⠀⠈⠋⠠⠍⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⢄⠉⠀⡠⠀⠀⠀⠀⢀⡀⠁⠀⠀⠚⠇
⠀⠈⠣⡀⠀⠀⠈⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢁⠐⠀⠄⠀⠄⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⠀⠀⠰⠂⠁
⠀⠀⠀⠀⠑⢆⠁⠂⠀⠂⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠄⠀⢆


⠀⠀⠀⠀⢠⠖⠒⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠃⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⢰⠾⠀⠀⠀⠀⠶⡄
⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⡐⠂⠀⠀⡀⠀⠀⠀⢀⡀⠀⠰⣿⣿⡷⠰⣿⡿⠂⠀⢀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠂⠀⠊⠀⠀⠀⠀⠈⡄⠀⠀⠈⠩⠔⠁⠂⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠠⣤⡀
⠀⠀⠐⠺⠶⢦⣤⣀⣀⣠⣤⠶⠞⢛⠀⠀⠀⠀⢠⠐⠋⣁⠀⡎⢉⡓⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠰⠀⠀⠁⡨⠉⢌⠉⠀⠠⠂⠀⠀⠀⠀⠘⠄⠀⠁⢀⠆⠈⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡖⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠁⠀⠀⠊
⠀⠀⠀⠀⡤⣊⣀⠸⠀⡠⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀
⠀⠀⠀⢸⢀⠀⠸⠀⠀⠂⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠈⠉⠙⠻⢶⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⣀⠀⠀⠀⠀⠀⠀⢻⡆⠀⡠⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣼⣿⡆⠀⠀⠀⠀⠀⢸⡇⣾⣄⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣨⣿⡿⠃⠀⠀⠀⠀⠀⣼⠇⠘⠧⣼⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠈⠻⢷⣤⣄⣀⠀⠀⠀⣀⣀⣤⡾⠟⠁
⠀⢠⣾⡿⠿⣿⡟⠛⠛⠻⢦⡀⠀⠀⠈⠙⠷⣦⣤⣀⣀⣀⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠉⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣰⠏⠉⠛⠻⠷⢶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⡶⠶⠿⠛⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠈⣿⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣿⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡛⠛⠛⣿⠛⠛⠋⠙⠋⠙⠙⣿⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣤⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠿⢷⣶⣤⣤⣤⣤⣴⡶⠿⠿⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢻⠿⠿⢿⠿⡹⠿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣛⣛⣛⣛⣿⣿⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠟⠛⠛⠛⠛⠛⠿⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⡀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠛⠋⠉⠉⠛⠛⠻⢶⣦⣀
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣤⡀⠀⠀⣿⡆⢀⠄⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⢤⣿⣿⠀⠀⢸⡷⣿⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢼⠿⠋⠀⠀⢸⡟⠈⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣄⣀⣀⣀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠻⠿⣷⣤⣀⡀⢀⣀⣤⡾⠟⠋⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⠉⠉⠛⢿⣷⣦⣄⣠⣤⣾⡿⠃⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠈⠛⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣽⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣾⣿⣷⣾⣶⣷⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⢤⡀⠀⠀⠀⡸⠀⠀⠀⠀⠈⣆⠀⠀⠀⢀⠤⠒⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠙⢦⡀⠀⡇⠀⠀⠀⠀⠀⢸⡀⢀⠔⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠱⣼⠁⠀⠀⠀⠀⠀⢸⣧⠋⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠑⠲⢌⣆⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣼⠧⠒⠋⠉⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣈⣧⠤⠤⠤⠼⣄⣀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠔⠋⣁⠤⢦⠀⢠⠦⠤⣈⠑⢶⡋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠀⠀⢸⠀⢸⠀⠀⠈⠑⢄⠙⢦⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⢀⡤⠖⠒⠉⠉⠉⠉⠉⠑⠲⠤⠤⡼⠀⡞⠸⡀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠈⢆⠈⣷⠒⠊⠉⠉⠉⠉⠉⠑⠒⠢⢄⡀
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⢳⡀⠀⡸⠀⠸⣄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠸⡀⠀⠀⡹⠊⢀⠀⢀⠈⠳⣄⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠈⠉⠒⠲⠶⢤⣤⠴⠖⠒⠊⠉⢳⠀⠳⡤⠊⢀⡴⢻⠀⢸⠷⢄⠈⠳⣶⠏⢠⠏⠉⠑⠒⠦⣤⠤⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢰⡋⠀⢸⠀⢸⠀⠀⣱⠄⠀⣠⠋⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠲⢄⡈⠙⠚⠀⠈⠒⠉⣁⡤⠊⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣤⠚⠁⠀⠀⠀⠀⢨⠋⠙⠒⠒⢻⠉⠁⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠑⠤⢄⣠⣤⠤⠶⠛⢻⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠹⡌⠑⠲⠤⠤⣀⡠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢀⣾⠻⡄⠀⠀⠀⠀⠀⢸⡏⢢⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢀⡴⠟⠁⠀⣇⠀⠀⠀⠀⠀⣼⠁⠀⠙⢦⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠉⠀⠀⠀⠀⠸⡄⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⣠⠶⠛⠉⠉⠉⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⠀⠀⠀⢹⢀⠠⣤⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢀⣀⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢗⣺⣿⠀⠀⠀⢸⠻⣴⡿⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⠠⠶⢶⢦⡄
⠀⡴⠛⣏⠉⠃⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠀⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠳⣤⣀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣤⣤⡤⠶⠋⠁⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠯⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣰⣿⠿⠿⠦⣤⣤⣴⣿⣿⣦⡀⠀⠀⣀⣴⣿⣇⡀⠀⠀⠀⢰⣿⣴⡆⣠⡆⣠⡴⠋
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢹⣶⣄⣀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⡿⠛⠉⠁⠀⠉⠙⠳⣄⠀⠉⣁⣼⣿⣿⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣇⣀⡾⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡯⠙⠻⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠐⠶⣤⣤⣴⣶⣿⠿⠛⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠙⢿⣿⠋
⠀⠀⠀⠀⠀⠈⢻⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢰⣿⡇⠀⣾⡇⠀⢀⣴⣾⢿⣶⡆⠀⣼⣿⠀⢰⣿⠂⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢰⠀⣿⠀⠀⠀⠀⠀⠀⣾⣿⠀⣼⣿⡇⢀⣿⡟⠁⣸⣿⠃⢰⣿⡇⢀⣿⣿⠀⢀⡀⠀⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡆⣿⣀⣤⠀⠀⠀⠀⣿⣿⣾⣿⣿⣴⣾⣿⣧⣠⣿⣿⣤⣾⣿⣷⣿⣿⣧⣶⠿⠃⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠈⠛⣡⣿⡟⠉⠀⠉⠛⠋⠀⠙⠋⠁⠙⢋⣵⣿⠏⠁⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠛⠁⠀⠻⣿⣦⣄⣀⠀⣰⡿⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣽⣿⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣟
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣀⠀⠀⠀⢀⣠⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⢀⣠⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣾⣿⣿⣿⣿⣿⡟⠂⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠁⠙⢿⣿⣿⣷⣦⣤⣄⣠⣤⣴⣾⣿⠟⠁⠈⠉⠋⠉⠛⢿⣿⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠉⠛⠿⠿⠿⠿⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣠⡴⠞⠋⠉⣿⠀⠀⠀⠀⣿⠉⠙⠳⢦⣄⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢻⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⡟⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⣇⣴⠞⠁⠀⠀⠀⠀⣠⡴⣿⠀⠀⠀⠀⣿⢦⣄⠀⠀⠀⠀⠈⠳⣦⣸⠇⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠞⠉⠀⣿⠀⠀⠀⠀⣿⠀⠉⠳⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣿⠀⠀⠀⠀⣿⣀⣠⡴⠞⠋⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠦⢤⣤⣤⡤⠴⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣿⠀⡰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡆⠀⡠⠞⡟⣡⠋⠈⠷⡄⠈⢧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡴⠁⠈⣧⠊⠀⠀⢯⠁⠀⠀⠀⠀⠀⢸⠃⠘⣦⠶⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣠⠏⠀⠈⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢦⣼⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⢀⡠⠶⠒⠒⠒⠒⠦⣄⠀⠀⣇⠀⠀⠀⢀⡠⠔⠒⠢⣄⠀⠀⠀⠀⠀⠀⢸⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⡀⡆⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢽⡶⠤⠒⠁⠀⠀⠀⠀⠈⢦⡀⠀⠀⢰⠀⠈⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣀⡘⡄⠀⢣⡙⢤⡎⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⢠⡀⠀⠈⠙⢲⣼⡇⠀⠀⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠙⣏⠁⠀⡱⢭⣻⡇⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣀⠀⠳⠄⠀⠀⠀⣟⡵⠀⡴⠓⣹
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠍⣲⣶⡇⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⣿⣶⣤⠀⠀⠀⡿⣊⠀⢀⣴⣃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢑⣒⡶⠋⠀⠀⠀⠀⣀⣤⣶⣿⣿⡇⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠙⢿⠇⠀⠀⡿⠁⠴⠋⢁⠎
⠀⠀⠀⠀⠀⢀⣀⣸⠄⠀⠀⠀⡸⠁⠠⠶⠾⠟⠛⠛⠛⠻⣟⠋⠁⠀⠀⠀⠀⣠⣤⣤⣤⡤⣤⡤⠤⢤⡤⠀⢰⡇⠀⡰⢠⠏
⠀⠀⠀⠀⠀⠀⠉⢢⣄⣆⠀⢀⡇⠀⠰⡗⠒⢶⣖⣲⡶⠒⢻⡟⠛⠀⠀⠀⠀⢸⡄⠀⠈⠛⠛⠁⢀⣾⠁⠀⣸⠁⡼⠓⠻⣄
⠀⠀⠀⠀⠀⠀⢰⠋⣁⢹⡄⠘⡇⠀⠀⠱⣄⣀⠉⠉⣁⣠⣾⠁⠀⠀⠀⠀⠀⠸⣯⡓⠦⣤⣤⠶⢻⠏⠀⢀⡏⢸⠃⡤⡄⢹
⠀⠀⠀⠀⠀⠀⣿⢰⠁⠀⣧⠀⢷⠀⠀⠀⠘⠛⠿⠛⢛⡽⠃⢀⣤⠤⠶⠶⣤⡀⠙⠍⠂⠀⠀⠀⠁⠀⠀⢸⠃⣾⠰⠛⡷⣸
⠀⠀⠀⠀⠀⠀⢸⣼⡩⠗⣾⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣟⠀⠀⠀⠀⠀⣹⡄⠀⠀⠀⠀⠀⣰⡄⠀⢸⠀⡿⡀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢳⡝⢦⠜⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠚⠋⠀⠀⠀⠀⣠⠞⠁⠀⠀⢿⠼⣳⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣻⠁⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠴⠖⢒⣋⣡⣴⣾⡟⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠳⣶⣿⡿⠛⠛⢋⣠⠞⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠈⠛⠶⠶⢒⣋⡥⢠⣄⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠿⣄⡀⠀⠀⠉⢉⣡⡾⠋⠉⠁
⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⠛⠋⠁
⣿⣏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣿⢻⣿⢻⣿⡄⠀⠀⣼⣿⣏⣻⣟⣻⠄⠸⠷⢾⣿⣥⣿⣿⣧⣿⣷⠾⠇⠀⣿⡛⣿⣛⣿⣇⠀⠀⢠⣿⢿⣿⣟⣹⡇
⠀⠀⠀⠀⠛⠻⢿⣿⠻⣷⡀⣼⣿⣹⣿⠛⠛⠛⠀⠀⠀⣸⣏⣽⡏⠀⢿⣭⣿⡀⠀⠀⠛⠛⠿⣿⡟⣿⣆⢰⣿⣿⣾⡟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⣿⡏⠀⠀⠀⠀⠀⠀⣿⢿⣿⠀⠀⢸⣿⢿⡇⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡟⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢰⣿⣾⡏⠀⠀⠀⣿⣾⣿⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢛⣿⠃⠀⠀⠀⠀⠀⠀⣸⡟⣿⣿⣿⣻⣿⣿⡟⣿⡆⠀⠀⠀⠀⠀⠀⠘⣿⣏⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⡟⠀⠀⠀⠀⠀⠀⢠⡿⣿⡟⠋⠛⠛⠛⠻⣿⢿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣾⠃⠀⠀⠀⠀⠀⠀⣼⣶⣿⠁⠀⠀⠀⠀⠀⢿⣶⣿⡀⢀⠀⠀⠀⠀⣸⣧⣾⡏
⠀⠀⠀⠀⠀⠀⢰⡟⣿⣟⣹⣿⣛⡿⠀⠀⠀⢘⣹⣿⣙⣿⣻⡆⠀⠀⢰⣏⣿⣏⣿⣿⣿⠀⠀⣏⣿⣟⣻⣿⣛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⡏⣴⣶⣶⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢇⣬⣭⣭⣭⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⠒⠒⠒⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠜⡻⠿⢿⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⢱⣾⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⣀⣀⡏⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠉⠉⠉⠉⠉⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠛⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠛⠛⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣼⠋⠀⣠⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃⠀⣼⣇⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢿⣭⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢉⣿⣿⡄⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⣿⣿⣿⠇⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⣀⣤⡄⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠋⢠⡿⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⣠⣾⣿⠛⠁⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣴⠟⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⣰⡟
⢀⡾⠋⠈⢿⣧⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⠀⠀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠜⠁⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠈⠛⠿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⡇⠀⠉⠉⠛⢻⡿⠿⠷⠶⠶⢶⣶⣶⣶⡶⠶⢶⡾⠿⠛⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡶⠶⢤⣧⣀⡀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠉⠉⠑⠛⠓⠒⠒⠂⠀⠼⠃⠀⠒⠒⠒⠚⠓⢹⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⢀⣠⡶⠾⠛⠛⠛⠛⠛⠻⠶⣦⣴⡟⠉⠀⠀⠀⠈⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢹⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠋

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣀⢀⠀⡀⠀⡀⡀⠀⠀⡠⠀⣀⠀⡀⠀⠀⡀⡀⠀⡀⢄⠀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡐⣥⡜⠀⠓⠐⠀⣏⠀⠈⠢⡅⠀⠀⠈⠁⠘⣄⣠⠇⢀⣠
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣀⢀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡧⣶⣼⠂⠀⡀⠀⠀⢀⠀⠀⡇⡿⢄⠀⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⠤⣀⢀⣀⠀⡀⡀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⢟⣿⣿⣿⣿⣿⡯⣿⣿⢣⣄⡇⡏⢂⡇⢎⠠⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢀⠀⢀⡉
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣚⣿⡇⣿⣿⣜⣧⣿⣸⣿⣦⡛⠜⢳⠰⡀⡿⠀⠨⢜
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡶⠛⠉⠀⠀⠈⠉⠀⠉⠁⠀⠈
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠿⣾⣟⠿⡏⠉⡖⠁⠀⠀⠱⠊⢰⠉⠈⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠞⠉⠉⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⠊⠀⠈⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠐⠧⡀⠈⡇⠀⡴⡄⠀⣏⠇⠀⠉⠀⡯⠀⡊⠑
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠂⠁⠀⠁⠈⠀⠈⠀⠁⠁⠀⠀⠀⠑⠂⠐⠈
⠀⠀⠀⠀⠀⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠉⠁

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠐⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠒⠛⠋⠀⠙⠒⠂⠒⠖⠀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣀⣀⣀⣤⣥⠤⠴⠷⠶⠶⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⡓⠻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⡟⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⡜⢧⣀⣀⣀⣤⣼⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡌⠀⢀⡼⠿⢯⢉⣿⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠏⠀⠀⢸⠀⠀⢸⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠀⠀⡈⠀⠀⠘⢢⣤⣤⣠⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠶⠶⠒⠒⠊⠉⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣸⣿⡟⠉⠛⠉⠉⠛⠓⠶⠶⠶⠶⠶⠦⠤⣤⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡋⠈⢡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠣⣤⣤⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡎⠉⠻⠛⠛⠻⢶⣦⣤⣤⣤⣤⡤⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢿⡶⠆⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢸⡟⢷⣦⣀⡀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠶⠶⢶⣶⣯⣴⣥⣤⡤⠾⠷⠞⠛⠛⠛⠛⠛⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣸⠃
⣿⣿⣿⣿⡟⢰⣶⣶⣶⣦⢉⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⡌⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⠋⠉⠉⠉⢹⡞⠁⠀⠀⠀⠙⠛⠛⠛⠛⠛⢳
⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⢻⣿⣿⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿
⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢋⣼⣿⣿⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴
⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢘⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠗
⣿⣿⣿⣿⣧⠸⠿⠿⠿⠟⣠⣙⠿⢿⣿⣿⣿⣿⣿⡿⢂⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣅⣀⣀⣀⣸⠟⢤⣄⣀⣀⣀⣀⣀⣀⣈⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠖⠉⠉⠉⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⢐⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⡀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢴⠂⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠵⠂⡀⠀⠀⢀⡀⠀⠠⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡭⠴⠒⠛⠁⠀⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣲⠀⠀⠀⠀⠀⠀⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣗⠶⠤⠀⠀⠀⠀⠀⠀⠚⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡦⠤⠤⠄⠀⠀⠀⠀⠀⣀⣘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⣠⣀⣤⡤⠀⠀⠀⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⣳⣶⣤⣀⠶⢚⡋⠟⣿⢹⡗⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⢠⠞⢁⡄⠚⠁⠀⠁⠀⢉⡟⠛⠾⣷⠚⢡⡌⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤⠴⡫⢻⣄
⠀⠀⢠⠃⣠⠈⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠈⠳⣾⣆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠎⡏⡷⡄
⠀⠀⢸⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠟⡄
⠀⠀⢸⢸⣿⡀⠀⠀⣀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢀⠛⡄
⠀⢠⠏⡸⠻⢷⣶⠿⠷⣮⣌⣐⡠⢀⠀⡀⡀⢸⡇⣼⣯⡑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢻⡀
⣰⠃⠀⠀⣰⠈⠀⠀⠀⠀⠈⠉⠙⠓⠾⢴⣦⣯⣼⣿⡧⣍⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⣇
⣇⢰⡆⣶⡇⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣷⣌⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢱⠿
⢹⣾⡄⣿⠣⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⡜⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣰⢏
⠀⡟⢷⣽⡀⠀⣼⣀⣄⣲⣁⣀⣀⠠⡠⢀⢈⡿⣾⣰⣿⣿⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡟⣼⢀
⠀⡇⠀⠉⠋⠙⠉⢩⠏⠉⠙⣝⠋⠛⠻⠛⠛⠻⠛⢿⣿⡌⢿⣶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡜⢠⠇⡼
⠰⡇⠀⠀⡀⠐⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⡻⢮⡓⠙⠷⣀⠭⢁⡀⠀⠀⠀⠀⠀⠀⢠⡞⢠⠏⢰⠁
⠀⢇⠀⠀⡇⢀⠀⠀⠀⡆⠀⣤⠀⠀⠀⠀⠀⢀⠸⣧⣿⡗⠽⡀⠀⠈⣻⣶⡤⠤⠤⠴⠄⠀⠘⠋⣰⡟⢀⡇
⠀⠘⡞⢦⡙⣌⣄⢠⡀⣿⢀⡏⠆⣀⣀⣠⣴⡼⣶⣿⣟⠀⠉⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠇⣼⠁
⠀⠀⡇⠀⣩⠿⠿⣷⡿⠿⠿⠛⠉⠉⠉⠀⠀⡀⠸⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣰⠇
⠀⠀⢷⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡏
⠀⠀⢸⡀⡀⡀⠀⢧⠀⡄⠀⢠⡄⠀⠀⠀⢀⣾⣹⡏⠓⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠙⢿⡜⣔⠠⠷⣼⣤⣀⡇⢆⣳⣠⣾⣿⢻⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡄⢀⠄⢀⡏
⠀⠀⠀⠀⠀⠙⠻⣿⣷⣶⣿⣿⣾⣿⢛⠝⣡⣠⢏⣀⣠⢀⣀⣰⡆⣦⣄⣤⣄⣴⣧⣮⣶⣧⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠉⠁
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣟⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⣀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠐⡀⠀⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠥⠀⠀⠀⢴⡈⠀⠀⠁⢁⣂⡴⠶⠤⠼⡂⠀⣀⠀⠀⠀⠀⠀⠀⣶⡆⠀⣤⠄
⠀⠠⠀⠀⠄⣀⠀⣶⠖⠀⡄⠀⠀⠀⠀⠈⢷⠊⠙⢿⠇⠸⡿⠀⠈⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⠒⠃
⠀⠀⠁⠀⠀⠙⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠠⠖

⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠁⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠘⢠⣴⠀⣤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣨⠀⠀⢸⠄⠠⠞⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠈⡍
⠀⢀⠀⠀⢠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⢀⡀⠰⠖⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⡠
⠀⠃⠀⠀⠈⠄⠀⠀⠠⠀⠀⠰⠄⠀⠀⠀⠀⠀⢨⠟⠀⠈⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⣀⠂⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⠒⡄⠀⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠂⠀⠀⠈⠃⠀⠈⠄
⠀⠀⠀⠀⡀⠀⠀⢅⢀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣶⡌⠁⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠴⠘
⠀⠀⠠⠂⠁⠸⢌⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⣀⠠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠒⠠⡀⠈⠀⢤⠀⠀⠀⡆⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠠⣤⠠⢀⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⢢⠀⠀⢀⡄⠐⣶⣶⣶⣶⣶⡆⠠⡀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠍⠒⠻⠘⠡⠨⠄⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⠀⢺⠀⣀⣀⠴⡐⢀⡄⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠂⢁⡀⠀⢁⡉⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠂⠀⠁⠀⠀⠀⠀⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠐⠏⠊⠑⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⢀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠔⠀⢀⣘⡆⠀⣃⡀⠀⠀⢐⠋⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢡⡆⠓⠋⠀⠀⠄⠀⠀⠀⠀⢃⠀⠀⠚⡇⠠⢆⠙⠘⠛⠚⠃⠓⠙⠓⠁
⠀⠀⠀⠀⠀⠄⠀⠤⠀⡨⡄⢂⡄⠀⠤⠍⠀⠂⠀⠀⠀⠀⠈⡄⠀⠐⡞⠀⠀⠀⣈⣁⢀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠶⠖⠤⠲⠠⢒⡢⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡠⠍⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠠⢴⠒⠲⠑⠂⠐⠫⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠆⡀⣆⡆⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠦⠄⠘⠀⢓⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢠⠱⡷⢸⡜⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢍⣡⡄⡅⡈⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠰⣿⡟⡇⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠂⠒⠂⠆⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠋⠉⠉⠛⠛⠶⣤⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣀⣀⣀⠀⠀⣀⣤⣤⣀⠀⠀⢻⡾⠁⢀⣀⣀⡀⠀⣠⣤⣄⡀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣾⠟⢹⣿⣿⣾⣿⣿⣿⣿⣷⡀⠀⢿⣰⡟⢉⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⠉⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣼⠀⠀⠀⢀⣤⠶⢶⣶⣄
⠀⢀⣴⢿⣟⠛⠳⠄⠀⠀⠀⢷⡀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣰⣯⡀⠀⠀⠙⠻⠿⠛⠉⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⣼⠀⠙⠃
⠀⠈⠀⠘⣿⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⢀⣴⠏⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⠀⠀⢀⣀⣤⡶⠛⠁⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⢦⣤⣤⣤⣤⣤⣤⠶⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⣀⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣦⣀⣀⡀⠀⠀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠛⠉⠙⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣨⡽⠿⣿⣿⢿⣧⣾⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠚⡿⢁⡀⠈⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⠇⠀⢀⣿⠗⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣿⣿⣧⣀⣠⣿⠏⡀⠀⢠⣿⣿⣿⢿⣿⣿⣿⢟⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠛⣃⠌⠀⠀⠰⣷⠀⣑⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣴⣶⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠃⠀⠀⠀⢠⣶⣿⣿⣿⣿⠿⠉⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠉⠉⠻⠿⡤⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣗⠚⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣷⣴⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⠀⠀⠀⣀⢀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠻⠿⣿⣿⣶⣄⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠒⠒⠒⠒⠢⢤⣀⠘⢿⡇⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠦⣤⡉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡁⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣇⣨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢎⢀⡀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠲⢤⣀⠀⠀⠀⣼⠃⠈⠧⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⠐⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⢰⡿⠿⣿⡛⠛⠓⢦⠀⠀⠀⠈⠙⠒⠦⠤⠤⠤⠴⠖⠋⠁⠀⠀⠀⠀⠈⠛⠲⠦⠤⠤⠴⠖⠛⠉⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠉⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⡼⠁⠉⠉⠛⠓⠶⣶⠶⠶⠦⠤⠶⠶⠶⣶⠖⠒⠛⠁⠁⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠉⠙⡟⠉⠉⠉⠉⠉⢻⠉⠉⠉⠉⠉⠉⢻⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣇⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⠧⣤⣤⣤⣤⣤⠴⠷⠞⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠁⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠋⠁⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⣠⣴⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠁
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠄
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠃
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠸⣿⣧⣤⣤⣤⣤⣤⣤⣾⡿⠿⠿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠉⠛⠿⣿⣶⣶⣤⣤⣤⣤⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡇
⠀⠀⠀⢸⠉⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⡆
⠀⠀⠀⠈⠓⠒⠒⠒⠓⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠚⠁
⠀⠠⠒⠀⣀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠻⠿⠛⠛⠛⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠿⠿⢿⣟⠷⢦⡄⠀⠀⠀⠀⠀⢸⠀⠀⣀⠀⠒⠄
⠀⠈⠉⠉⣀⣤⡶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠉⠉⠁
⠀⠀⢠⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠈⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠉⠛⠶⠦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠤⢤⣀⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣠⠴⠾⠷⢦⡀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⢀⡔⠉⠀⠀⠀⠀⠉⠳⠀⠀⠀⣠⡿⠛⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣤⣄⣀⣀⣀⣤⡤⠶⠖⠛⠉⠀⠀⠙⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠴⠶⠛⠋⠉⠁⠉⠉⠉⣉⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⢰⣧⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠈⠙⠛⠿⣷⣶⣤⣤⣤⣀⣀⣠⣤⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⢠⡄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠿⠟⠁⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠿⠾⣿⣶⣶⡶⠶⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣾⠟⠋⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁⠀⠀⠀⠠⣤⣴⡾⠛⠙⠛⠿⠿⣶⣶⡶⠶⠾⠿⠿⠟⠛⠛⠛⠋⠉⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣤⣠⣇⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣾⠿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣤⣤⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⣿⡇
⠀⠀⣀⣠⣤⣶⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⣤⣤⣤⣤⣌⣹⣷⣦⣾⣆⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡿⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡸⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡘⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣀⠄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠒⠴⣒⠶⠒⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠛⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣠⠶⠶⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⢠⡔⢂⡿⠶⠶⠶⠤⠤⣄⣀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⠉⠙⣛⠒⣲⠞⠁⠀⠀⠀⠀⠀⠀⢴⣶⠶⢶⠚⠋⣴⣤⣤⠀⠀⠀⢉⠉⠙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠁⠀⣼⣿⣿⣯⣀⣠⣾⣋⣄⡀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⠶⠾⢿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠈⠁⠀⠀⠀⠀⠀⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡾
⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⢳⡀
⠀⠀⠀⠀⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⠉⢿⠟⠳⢤⣶⣶⣿⣿⡷⢀⢀⠀⠀⠀⢀⣀⡀⠄⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠈⣷⡑⡀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠈⠙⠶⣭⣉⣥⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⢿⢿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⡿⠇⠀⠀⣀⣀⣐⣾⣿⡄⠀⠀⠀⠀⢀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⡵⠋
⠀⠀⠀⠀⢸⣸⣿⣿⣧⠀⣠⣶⠲⣠⢸⡿⣇⠀⠀⢴⣾⣿⣿⣿⡿⠛⠲⠤⣒⠾⣿⣾⢄⡀⡀⠀⢀⣀⡤⠤⠐⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⡿⣦⣿⡜⣷⣝⠀⢀⣷⣿⣿⣿⣿⣧⠀⠀⠀⠈⠙⠲⠯⠤⠥⠶⠚⠋⠁⠀⠀⠀⠀⠀⡀⣠⠏
⠀⠀⠀⠀⠘⢧⣍⣛⣛⣛⣻⡿⠿⠿⠇⢷⣼⠓⠦⣍⣻⠿⣿⣿⡟⠷⠦⢔⡾⣿⣿⡄⠀⡀⠀⠀⢀⣀⣀⠠⠀⠀⠈⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠛⠛⠷⣄⡀⠀⠈⣉⢛⠳⠤⠶⠖⠛⠉⠁⠀⠀⠀⣀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠿⣿⡿⠦⠀⠀⠀⣀⣠⡤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢸⡇
⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢸⣷⣄⣀⣀⣀⣤⣴⣶⠾⠿⠷⢶⣤⣀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣦⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠸⡇⠉⠛⠛⠉⠉⠀⠀⠀⠀⢠⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⣿⣦⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⡶⠛⠉⠉⠛⠶⣾⣤⠶⠒⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣶⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢙⣿⣶⣤⣀⠀⠀⠀⢠⡇⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣡⠀⠀⠙⠛⠷⠶⡾⠷⠿⠛⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⠋⠙⠳⢦⣤⣀⣀⣀⢀⣿⢀⣀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢿⣧⡇⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣸⠟⠿⣦⡀⠀⠀⠀⠀⢸⡄⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠘⢹⡿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠀⠈⠙⠻⠶⢦⡤⢾⣧⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠃⠈⠙⠛⠷⠶⢶⣶⣶⠶⠶⠖⠛⣿⣄⠀⠀⠀⠀⠀⣤⠀⣸
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣤⣿⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⡀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠁⠈⠙⣿⣿⣿⢠⡞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⣤⣤⣄⠀⠀⣀⣴⣿⣿⢣⣿⠁⢀⡏⠀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣡⡾⠟⠛⢿⣿⠀⠀⠉⠉⠉⠁⣾⡏⢀⡞⠀⠀⠿⠿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢦⠀⠘⠁⠀⠀⠀⠈⠀⠀⢀⣶⡏⠀⢠⣿⣥⠏⠀⠀⢀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣧⣤⣼⣿⠁⢠⣤⠀⢸⣷⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡋⢹⣿⠀⠀⣿⠀⢀⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⡟⣠⠏⠀⠀⠀⠀⠀⣀⠀⣠⣤⣾⣿⠀⣽⡇⢸⣿⡀⠘⣿⣄⡾⣿⣷⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⡏⠀⢠⣿⡿⠃⠀⠀⣠⡶⢶⠀⢹⣿⢹⣿⢸⣿⠀⣿⣧⡀⠻⠿⠇⠈⠉⣰⣿⡏⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣾⣿⡇⠀⠀⣼⡟⠀⣶⡆⢸⡇⢸⣿⠘⠿⠆⠘⠛⠁⠀⠀⠀⠀⣼⠋⣿⡇⠀⠾⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠉⢁⣼⣿⣧⣠⠖⣿⡇⢠⣿⣇⠸⠷⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠙⣆⠘⠿⠋⠈⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡤⠂⠀⡏⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡄⠀⢀⣀⣤⣤⡀⠀⢰⣿⣷⣤⣴⡾⠋⣠⠆⠀⣇⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⢸⠇⣴⣿⣿⡟⠋⠁⠀⠀⠙⠛⠋⢁⣠⠞⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⣿⣿⣿⣧⣀⠀⠀⣀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀⠀⢀⣠⣶⡾⠿⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢁⡴⠚⠛⠛⠛⠲⣝⠻⠃⠻⢋⡵⠞⠛⠛⠛⠲⣤⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡀⠀⠀⠈⢷⣠⣠⠏⠀⠀⠀⡀⠀⠀⠈⢳⡀
⠀⠀⢀⣀⣤⠤⠶⠶⠒⠚⠛⣿⠀⠀⠀⣿⣿⡇⠀⠀⢸⡏⣿⠀⠀⠀⣿⣿⡇⠀⠀⢈⡟⠛⠒⠲⠶⠦⢤⣄⣀
⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠉⠀⠀⠀⡼⠁⠸⣆⠀⠀⠈⠉⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⡤⠞⠁⠀⠀⠙⠶⣄⣀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⢀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣀⣀⣀⣀⣹⣶⣄⣀⣀⣀⣀⣴⣾⣁⣀⣀⣀⣠⣤⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠿⡿⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⢏⢹⠉⢹⠀⢸⠀⠀⠀⡇⠀⡇⠉⡏⢙⠟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠉⠒⠼⠤⣼⣀⣤⣀⣇⡠⠧⠔⠋⣁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⠀⢀⣀⣀⣤⠴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠈⠁⠀⣀⠎⠑⠀⠀⠈⠀⠀⠀⠀⠈⠶⠀⠀⠦⡞⣣⠄⠀⢠⣏⣹⠄⣿⣙⠀⣾⣿⡄⣴⠟⠀⣿⣙⠀⠀⢶⣦⣰⣿⠀⢀⡄⠀⠓⠃⠑⠂⠀⡆
⠀⢒⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⣠⣴⡄⠀⠀⠀⠀⠀⠋⠑⠀⠀⠈⠯⠁⠀⠿⠶⠀⠿⢿⠇⠻⠶⠂⠿⠶⠀⠠⠀⠙⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸
⠀⠈⠂⡀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠍⠁⠀⣠⠞⠉⡏⠙⠲⡄⠀⠦⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢀⣿⡀⠀⠀⠀⢠⡄⠀⠀⠀⠁⠀⠒⡀
⠀⠀⠀⠑⢄⡀⠀⠀⠂⠀⠀⠈⠁⠀⠀⢰⠇⠒⢀⣧⡒⠃⢹⠀⠀⠀⠀⠀⢀⢿⢑⣤⠀⠁⠀⠀⠀⠁⡀⠀⠈⡟⠁
⠀⢀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠶⠀⠈⢧⡴⠋⣿⠙⢲⡼⠀⠀⠀⠐⠉⠙⠈⠀⠁⠀⢀⡀⠀⠱⠇⠀⠀⠀⠂⠀⠀⠀⠵⠎⠀⠈
⠀⢒⡋⡈⠈⠄⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⡈⠓⠶⠿⠖⠋⠀⠀⠀⣀⣤⣴⣶⣦⣤⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠺⠇⠀⠀⠀⠀⠄⡀⢺⣿⣿⣷⣿⣿⡿⠀⡀⠀⠀⠤⠀⣠⣾⣿⣿⣿⣿⣿⣿⢿⣷⣄⠀⠘⢻⡟⠃⠀⠀⡀⠀⠀⠘⢟⡿
⠀⠀⠀⠀⠀⠚⠀⠀⠀⠈⠀⠙⢿⣿⡿⠋⠀⠈⠀⠀⠀⠀⢰⣿⣿⡏⠋⣯⠋⣍⠏⢸⣟⣿⣇⠀⠀⠀⠀⠀⠸⠿⠄⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⡀
⠀⠀⠢⠤⠄⠀⠀⠀⠃⡀⠀⠀⠀⠛⠀⠀⠀⢀⠘⠀⣤⡂⣿⣟⣿⣷⣿⣷⣷⣿⣶⣾⣿⣿⣟⠀⠋⠀⠉⠉⠉⠉⠉⠉⠃⢠⠀⠂⠀⠀⠀⠀⢐⠀⠀⠖
⠀⠀⢀⣠⡤⣤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⡔⠀⢻⣿⣿⣿⢦⡰⢃⡓⣇⣾⣿⣿⡿⢀⣤⣤⡀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⣠⠏⠁⠀⣼⠀⠀⠙⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠁⢸⡇⠈⠻⣦⠶⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⡏⠀⠀⢀⣿⡀⠀⠀⢹⠀⡼⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡿⣿⠿⡿⠟⠉⠀⠀⠀⢻⣄⣴⣿⣄⢀⡼⠃⠀⠀⠁⠀⠀⠀⠐⠂
⠀⢻⡤⠞⠋⣿⠙⠳⣄⡾⠀⠀⠀⠀⠀⠀⠂⠀⠘⠄⠀⠂⠀⠘⠂⢀⣤⣶⡖⠰⢦⡀⠀⠈⠢⡄⠀⠹⣾⣧⠿⠋⠀⠀⠖⠀⠀⠠⠐
⠀⠀⠙⠦⣤⣿⣠⡴⠟⠁⠀⠒⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⢠⣿⣿⢫⠁⠀⣽⡿⡆⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⢀⠀⠀⠀⠁⠠⣤⣴⡶⠶⠦⡔⠁
⠀⠀⠀⠀⡀⠀⠀⢀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠒⠠⠥⠀⠀⣤⠀⠈⣭⣥⡄⠀⣴⣾⣶⡿⠀⣤⠀⠀⠀⣀⡄⠀⠀⠈⠀⠀⠤⣤⣴⣻⣛⣯⡽⢯⡾⠁
⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠸⣧⠀⢮⢿⢟⠀⢻⣿⠏⠀⡴⠇⠀⠀⠉⠟⠣⠀⠀⠀⠀⠀⠈⠉⠚⠑⠋⠉⠀⠋⠀⠘⠁
⠀⠀⠀⠄⢀⢀⢀⠀⠈⠀⠀⠠⠄⠄⠀⠀⠀⠀⠀⢀⠀⠀⠁⣉⣴⢀⠀⠀⠐⠊⠁⠐⣦⡉⠁⠀⠀⠀⠐⠀⠐⣦⡶⣰⢦⠀⣲⡀⢰⡆⡰⣦⠀⣶⣦⡄
⠀⠀⠀⠀⠀⠈⠉⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠘⠛⠟⠘⠃⠀⠸⠟⠻⠃⠙⠛⠄⠻⠻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⣀⠀⢰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⣻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠉⠀⠀⠈⠉⠙⠛⠛⠉⢹⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⠻⣿⣿⣿⣷⡀
⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠘⣿⣿⣿⣿⡀
⠀⠀⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠘⣿⣿⣿⣿
⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢸⡿⠃⠀⠀⠀⠸⣿⣿⣿⡇
⠀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣀⣿⣿⣿⣷
⠀⢹⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠉⠹⢿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡄⠈⠻⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠈⢿⣿⣿⣧
⢀⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿
⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢴⣾⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⣿
⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣶⣶⣶⣦⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣙⠋⠉⠀⡀⠀⠀⠀⠀⣾⣿⣿
⠀⠙⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠻⠿⢿⣷⣦⣾⣿⣶⣄⠀⢠⣿⣿⣿
⠀⠀⠀⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⡿⠿⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢹⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⣿⣿⣿⣿⡟
⠀⠀⠘⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣤⣀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣶⣶⣄⣤⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠇⠀⢀⣠⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⠈⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣶⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠁
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡿⠋
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡿⠋⠉⠙⠛⠛⠿⢿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣶⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣤⣤⣄⢀⣀⣀⣀⣀⣀⣀⡀⣠⡤⢤⡀⠀⠀⠀⣴⠖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⡀⣿⠀⡿⢸⣿⠀⠀⠀⣀⣈⡇⣿⡆⠀⡇⡾⠙⣦⣿⠀⢸
⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⠀⠀⢿⣿⠀⡇⢸⡇⠀⢸⠋⠉⠉⠀⢻⡇⠀⣷⡇⠀⣿⡏⠀⡟
⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⠀⣀⠘⣿⠀⡇⣿⡇⠀⠈⠓⢲⡆⠀⢸⡇⠀⣿⢠⡄⢸⡇⢀⡇⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⡀⠀⠀⡇⣿⡇⠀⢰⠓⠚⠀⠀⢸⣷⠀⠁⢸⣇⠘⠃⢸⠃⠀⠒⠒⠂
⠀⠀⠀⠀⠀⣀⠀⠀⣿⡇⢸⡇⢧⠀⢠⡇⣿⠀⠀⠈⠓⠒⠚⡇⢸⣿⠀⠀⡾⣿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠧⠼⠀⠘⠶⠾⠃⠛⠓⠒⠒⠒⠒⠒⠃⠈⠿⠤⠴⠇⠻⠤⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀⠀⣠⠤⢤⠀⠀⣀⡤⢤⡄⠀⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡞⢓⡆⢹⢰⢃⡞⡆⡇⢰⠃⡴⠚⠁⠐⠷⠶⡄⣠⠤⠼⠃⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠛⣉⡴⠋⡟⢸⠀⡇⡇⢸⡀⠳⡀⠀⠀⠀⢀⡇⣿
⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⡇⢰⡏⠁⠀⠀⡇⢸⢰⠃⡇⠀⠑⢦⡈⣧⠀⠀⢰⡇⡗
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣷⢸⠁⠀⠀⠀⣇⠘⠞⢰⠇⢀⣀⣀⠇⡾⠀⠀⢸⡇⡇⠀⠀⠀⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠀⠀⠀⠀⠘⢦⡤⠎⠀⠸⠭⠤⠞⠁⠀⠀⠈⣧⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⢝⣃⡙⠉⠙⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣣⡽⠎⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⠤⢎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡣⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠙⢀⡄⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠂⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠀⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣋⡀⠀⠐⡛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢬⣭⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢸⠸⠀⠛⢶⣶⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠒⢰⠀⠀⠀⠀⠈⠙⠛⠂⠶⠖⠆⠶⠚⠂⢀⣛⠣⠿⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠟⢰⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠨⠐⠛⠘⢑⡌⠀⡆
⠀⠀⠀⠀⠀⠀⢀⣤⠀⢸⡁⠙⢠⠆⠃⠀⠑⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⢸⡇⡃⢉⠠⠀⢀⡀
⠀⠀⠀⠀⣀⠰⡫⣴⠀⢬⢋⣹⡆⠐⠘⢯⣢⣜⠚⡖⡀⠈⠀⠀⠀⠀⠀⠐⢿⡼⠁⡇⠨⢒⡠⠏⠁
⠀⠀⠀⢈⠥⠋⠁⠀⢠⡾⠀⠀⢡⠨⣨⠈⠉⠉⠀⠘⢟⣄⢨⡄⠀⠀⠀⠘⣿⠇⢠⠇⠀⡎⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠁⠘⠶⠋⠀⠃⠁⢰⣿⠘⠃⠐⠲⠖⠻⢿⠁⡈⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣰⠶⠀⠀⠀⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠀⠀⠀⠀⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠈⠀⠁⠢⡠⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡠⠐⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣠⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠁⢰⠀⠀⢰⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣬⠈⠁⠀⠀⡆
⠀⢀⣤⠀⠠⢀⠀⢀⠠⠓⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠔⠃⠄⠀⠤⠂⠀⠀⠀⠀⠀⢴⠀⠁⠀⠀⠈⠙⠂⠀⠀⠐
⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠀⠆⣄⡀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⣈⠓⠂⠄⠀⠠⠐⠒⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠈⠢⡒⠁⠒⠀⠒⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠉⠈⠁⠁⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠀⠀⠀⠚⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠃⠀⠀⠀⠀⠻⢦
⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠈⠢⡤⠂⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⢄⡀⠀⠂
⠀⠀⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠖⢸⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⡆⢷⣿⣿⠆⠀⠀⢀
⠀⠀⠀⠐⡄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⠠⠓⠄⣀⡠⠔⠀⠀⠀⠀⠀⢻⠀⠀⠀⠙⠛⢋⠔⠢⢉⡁⠔⠃⠀⠈
⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⢈⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⣠⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⡀⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⢠⣦⡀
⠀⠀⠀⠴⠶⢶⣤⣄⣀⢀⡀⣀⣤⣴⠶⠿⠒⠀⠀⠀⠀⢀⠤⠖⠋⠉⠐⡔⠉⠓⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⢀⠠⠐⠀⠁
⠀⠀⠀⠘⡀⠀⠉⠛⠉⡛⢋⠋⠋⠀⠀⡸⠀⠀⠀⠀⠀⠘⡀⠐⠘⠋⠀⡅⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⡄⠈⣤⡄⠀⠀⢡
⠀⠀⠀⠀⠈⠐⠀⠀⠊⠀⠀⠁⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠈⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠰⢄⡀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⣀⠀⢐⠂⠀⢐⠒⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⣠⠮⠧⢤⠘⠂⠠⠞⠒⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢏⡢⠤⠘⠒⠒⠘⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⢉⡛⠳⣄
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⠔⡃⢸⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠿⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡠⠜⡁⢸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣽⡇⣿⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠠⠴⠊⠁⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⣿⡁⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠖⠋⠀⡟
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣸⣿⣇⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠙⠛⠇⢸⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠲⣾⣿⣿⣏⢹⡄⠀⠀⠀⠀⠀⢰⡇⠀⠀⠒⠒⢋⠁⣼
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢾⣿⣿⢸⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢙⡁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⣟⡟⠘⠆⠀⠀⠀⠀⢠⠏⠀⠀⠀⢉⣀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣍⣛⠂⣦⠀⠀⠀⠀⢸⠀⠄⠀⣀⣠⡤⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣩⣭⣦⠟⡀⠀⠀⠀⢸⠀⠀⢀⣴⣶⣶⡗⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠛⠷⠟⢻⣽⢸⡇⠀⠀⠀⣮⠀⠀⠉⣡⣤⠴⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢠⣽⡎⢷⠀⠀⠀⣿⠀⠀⢀⣀⣠⡴⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢻⣽⠿⠀⠀⠀⡦⠀⠀⠀⢀⣀⣠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠤⢙⢦⡇⠀⠀⡇⠀⠀⠘⠛⢉⡁⠀⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣤⣴⣿⣶⡰⣀⢀⠇⠀⢀⣛⢉⣟⣻⡿⠿⠿⢿⣍⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠛⠿⠉⣉⡨⣿⣤⡾⠛⢋⡽⠯⡁⡄⠀⠀⠀⢻⡆⠀⢿⣦⡴⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠷⣾⠿⠛⠉⠀⠀⠀⢹⣆⠀⠀⡿⠀⠀⠈⢸⡇⠀⣾⣿⠁⠀⢀⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⣿⣿⠹⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⡤⠰⢂⣾⠁⢰⣿⡿⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⣿⠃⠈⠄⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢔⣶⡿⠏⠀⣾⣿⠃⠀⢰⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠃⠀⠀⠀⠀⠀⠢⡀⠀⠀⣻⣿⡿⣛⣵⠟⠋⠀⣰⢧⣼⠇⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠏⠀⠀⠀⠀⠀⢀⢠⣬⣥⣿⣿⣿⣾⡟⠀⠀⠀⢶⢿⣿⠃⠀⠀⣾⣿⣿⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⠀⢀⡐⣷⣿⣟⣹⣿⣿⣿⣿⡇⠀⠀⢤⣤⣸⣿⣠⣦⡾⢋⣿⠇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠐⠈⠙⢿⣿⣿⣿⣿⣿⣿⣯⡧⢀⣘⣻⣟⣿⣁⣡⣽⣷⣿⣿⡷⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠟⠋⣿⣿⣿⣿⣿⣥⣤⣿⣿⣿⣿⣿⠿⠟⠋⠉⠋⠠⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡿⢹⡟⣿⣿⣿⣿⣿⣿⣿⡛⠆⠀⠀⠀⠀⠀⢀⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠸⠃⠟⢻⣿⣏⠀⠀⡱⠀⠀⠀⣠⠖⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⢸⢠⣿⣆⠉⠀⠀⡀⣼⠯⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠲⡋⠿⢻⣄⣤⣸⠏⠀⢈⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠃⠀⣿⢹⠆⠈⠀⠀⠀⢸⣷⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠙⠘⠆⠀⠀⠠⣴⣿⣿⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣉⠀⠠⢘⣧⠞⣤⣶⣰⡆⢸⣿⣿⢏⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠠⣄⡀⠀⠀⠀⠀⠀⠀⡾⠛⠾⠃⣺⣿⢿⡛⣿⣿⣿⣿⣿⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠔⣁⡀⠀⠀⠀⠤⠤⢤⡽⠿⠟⠗⠉⠈⠛⠋⡹⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠺⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠴⠀⠀⠀⠀⡀
⠀⢠⠤⠤⠤⠤⠤⡄⢀⣀⣠⠃⠀⠀⠀⠀⡔⠒⠂⠀⣨
⠀⢸⠀⠀⠀⠀⢠⠀⢸⠉⠙⠦⠀⠀⠀⠠⣏⣈⡉⠠⢼⠀⡎⠑⡄
⠀⢸⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⡧⠤⠀⢠⠌⠀⠠⠀⠈⠢⡀
⠀⢸⠀⠀⠀⠀⡇⠀⢸⣀⣀⡀⠀⠀⠀⢀⣳⣤⡤⢊⣀⣀⣀⣑⡄⠀⠈⢆
⠀⢸⣀⣀⣀⡠⠓⠂⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⢘⠁⠐⠒⠠⡀⠀⠀⠀⠈⢦⡀⢀⣀⣀⣀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⣁⣀⣀⡧⠀⠀⠀⢀⠜⠉⢹⠀⢡⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⡠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣤⢤⠃⠀⠀⠀⠀⠀⣀⣸⠀⠀⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠀⠈⠓⠂⢇⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠄⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣠⣿⣧⡆⠀⣴⣿⣿⠇⠀⠀⣀⣤⣒⣴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣒⢦⣽⣿⣿⣿⣿⣧⣾⣿⣿⣏⣤⣶⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣟⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⡿⠟⣉⣀⡠⣄⣀⣀⣠⠴⠚⠉⠉⠉⠙⠲⣤⡙⢦⡈⠉⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⡼⣫⣾⣿⠟⡠⠊⠁⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⢆⠀⢠⡶⠛⠋⠙⠻⢶⡀
⠀⠀⠀⠀⣾⣽⢿⣿⡏⣸⠁⠀⠀⠀⠀⢀⣶⣶⠀⢹⣴⣶⡀⠀⠀⠀⠀⢸⡇⠀⠀⠳⠋⠀⣴⡟⠳⣦⡀⢻⡀
⠀⠀⠀⢰⡟⠁⣾⣿⠀⡇⠀⠀⠀⠀⠀⠈⠻⠟⠀⢸⠛⠻⠃⠀⠀⢀⣰⡿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠘⡇⢸⡇
⠀⠀⠀⠀⠀⠀⡟⠹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠙⢒⣲⣶⣾⡿⠯⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠘⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⡀⠀⠀⠀⠀⢀⣤⣾⠷⠒⢻⠉⠁⠀⠀⢳⠀⠀⠀⠸⠈⢳⡀⠀⠐⡖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠳⢦⣤⠴⢞⠟⢫⠀⠀⠀⢸⡀⠀⠀⠀⠈⣧⣠⣤⣾⣷⣾⣿⡄⠀⢹⡀
⠀⠀⠀⠀⢀⡴⠛⠛⠉⠀⠀⠀⠀⣴⠏⠀⠸⡀⠀⠀⢈⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⣇
⠀⠀⠀⠀⡾⠀⣰⠾⠂⠀⠀⠀⢸⠁⢇⠀⠀⣱⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿
⠀⠀⠀⠀⣧⠀⢿⡀⠀⠀⠀⠀⢸⣄⢈⣷⣾⣿⣿⣿⣿⡿⢻⠟⠉⠉⠉⠛⢿⣿⣿⡿⠛⣿⡿⠀⢸⡆
⠀⠀⠀⠀⠹⣦⠈⠛⠶⠀⢀⣀⠀⢻⣿⣿⣿⣿⣿⣿⡏⠀⠘⠀⠀⡠⠴⢄⡏⠀⠘⣿⡀⢸⠇⠀⣾⡇
⠀⠀⠀⠀⠀⠙⠻⠶⠶⠖⠋⠘⢆⠈⢿⣿⣿⣿⣿⠿⢧⡶⠛⠙⢦⣇⠀⠈⢿⡀⠀⠸⣧⠎⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢻⣿⠟⠻⡄⠀⢣⠀⠀⠈⢿⡄⠀⠀⢷⡠⠔⣁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠙⢷⣄⠙⢦⠀⠣⡀⠀⣀⡷⠤⠒⣉⡤⠞⣡⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠙⠚⠛⠊⠉⠉⠀⢀⣈⣩⣵⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠶⠶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠋⠉⢹⣿⡏⠉⠙⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⣠⣾⣿⡿⢻⣿⡟⢿⣿⣷⣄⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⣠⣾⣿⡿⠋⠀⢸⣿⡇⠀⠙⢿⣿⣷⣄⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣾⣿⡿⠋⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⢿⣿⣷⣤⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣏⡀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⣹⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣦⣤⣀⣼⣿⣇⣀⣤⣴⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠉⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠒⠶⢤⣌⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡴⢋⣁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣴⠏⣼⢃⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠳⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣹⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣿⣿⡿⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⣠⣴⡾⠿⠗⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⡼⠋⢹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⠙⠳⠦⢤⣤⣤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠙⠻⠷⢶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠶⣾⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢀⣀⣤⣄⣀⣀⣀⣀⡀⠀⢀⣠⠴⠛⠛⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡶⠛⠉⠀⠀⠀⠀⠉⠉⠙⠛⠯⣀⠀⠀⠀⠀⠀⠈⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄⡀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣄⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⡀⢀⢠⣀⡄
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠀⢀⠀⡤⢀⠐⠀⠦⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠂⠘⠀⠁⠆
⠀⣥⣀⡀⢶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠆⠠⠄⠀⠤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⣈⣀⣀⣀⣤⣥⣤⣤⣬⣯⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡗
⠀⠘⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⠿⠿⠿⠿⠛⠛⢻⣿⡟⠛⠿⠟⢛⣛⠛⠛⠁
⠀⠀⠐⠀⠉⢻⡿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢿⣿⡟⢛⠛⠋⡉⠉⢽⣯⠀⠰⡄⠠⠀⠄⠀⠀⢿⣿⠀⠀⠀⠀⠀⠃⠸⡿⢇⣀⣀⣀⡘⠁⠤
⠀⠀⠘⠶⠆⢠⡙⣂⠰⠧⠄⠀⠐⠀⠆⠈⠀⠸⠿⠇⠀⠀⣊⡁⠀⠈⠋⠀⠀⣄⠀⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠄⠻⢷⣶⣾⣾⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠈⠁⣀⣀⣼⣿⣽⡏⠉⣉⠉⡀⢩⣭⣭⣭⣭⣭⠉⢤⢀⢀⠀⣀⢠⣀
⠀⠀⠀⠀⣀⣈⣀⡀⠘⣿⣿⣿⡛⣛⢻⠛⠛⠛⠉⣭⣵⣶⣶⣿⣧⠀⠇⠀⠀⠿⠿⠿⢿⣛⣛⣀⣍⣀⣁⣙⣛⣛⣟⣿⣇⣀⣉⣈⣀⣄⣀⣬⣴⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠇⠋⠛⠀⠀⠀⣀⣹⣛⣛⣻⣯⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⡿⢿⠿⠛⠛⠛⣛⠛⢛⡛
⠀⠀⠀⠀⠀⠰⠄⠉⠭⠙⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣋⠹⣿⣿⣿⣷⠀⡀⠈⠠⠀⣶⣶⣶⣶⠀⠤⠀⠀⠴⠀⠀⠀⠀⠛⠘⠈⠛
⠀⠀⠀⠀⣴⣶⣶⣶⣦⣤⣤⣤⣄⣻⣿⣿⣿⠉⣡⠉⠨⠀⠰⠆⠰⣀⣃⣀⣿⣿⣿⣿⣦⣧⣤⣤⣤⣿⣿⣿⣿⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠛⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣤⣤⣴⣶⣶⣶⣾⣿⣿⠿⠿⢿⠟⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⢟⠛⡋⠁
⠀⠀⠀⠀⠇⠘⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠛⢩⡅⡆⠸⠚⠃⠀⠸⠡⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠨⡈⠁
⠀⠀⠀⠀⠀⣀⠠⠀⢨⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿⣇⠈⡄⢩⣄⠀⠐⣿⣿⣶⠀⣬⡁⢤⡴⡶⠀⠀⡶⠀⠀⢶⣿⣿⣿⠃⠰⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠁⠀⠀⠀⠸⠿⠿⠿⠿⠿⠂⠠⠎⡈⣿⣿⣿⣿⣿⠀⣀⣈⣀⣡⣤⣽⣭⣽⣥⣄⣁⣀⣀⣉⣈⣁⣁⣀⣤⣬⣭⣥⠀⢀⠀⠀⡂⠀⠂
⠀⠀⠀⠸⠆⠹⠐⡇⠀⠰⠔⡃⠀⠶⢨⣴⠂⣠⠙⢋⣭⣥⣶⣿⣿⣿⣿⣿⡿⠿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣛⣀⣀⣀⣀⣤⣤⣤⣤⣤
⠀⠀⠀⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣁⠁⠆⠘⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿
⠀⠀⠀⢿⢿⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠛⢿⣟⣿⠛⢻⣹⣟⠛⠛⣿⣿⣿⣿⣿⣿⣿⡏⠉⣯⠅⠀⠀⠀⠤⠂
⠀⠀⠀⠐⠐⠄⠞⠀⠀⢀⡀⠠⠀⣠⠀⠠⢯⡋⠉⣿⣿⣿⣿⣿⣿⡿⢃⠀⠀⣆⢉⣁⢁⡈⢉⣉⣉⣀⠙⡿⢿⠟⢛⣛⣛⣃⣤⠀⢀⠀⡀⠠⠀⠂
⠀⠀⠀⠀⡀⠄⢠⠀⡄⠀⢠⣤⣤⣤⣶⣆⠀⢈⣁⠀⢸⣿⣿⣿⣿⠂⠟⠒⣀⡏⢊⡹⠏⠁⠼⣿⣿⣿⡆⠻⠄⠀⢹⣿⣿⣿⢿⡆⠈⠁⡀⠀⣀
⠀⠀⠀⠀⠉⠐⠉⠀⣁⡀⢸⡿⢿⠿⠿⡟⠀⠈⠋⠀⣨⡉⢉⢲⣴⠀⠗⠀⢿⣿⠘⡉⠋⣹⣂⠉⢾⠁⣿⡄⡆⢶⡂⡀⢹⣷⠀⠀⠀⢸⡇⠀⠛⠃
⠀⠀⠀⠀⠠⡄⠷⠀⣿⡧⠠⠀⠀⠦⡆⠀⠀⢀⡇⠰⣿⡇⢀⣈⠁⠡⡄⢰⡀⠰⠆⠇⢸⣿⣿⣿⣧⡤⢛⡇⢀⠀⢠⠀⢸⣷⣶⣷⠀⠀⠷⠀⠸⠷⢤⠆
⠀⠀⠀⢰⠀⠆⢠⠀⣾⣶⣶⣶⡆⣖⠀⣤⠀⠄⣴⢀⣿⣿⣿⡿⠀⠛⠛⢂⣬⣵⣶⣦⠠⢈⡠⠉⠽⠠⠘⠒⠁⢃⠘⢀⣘⣛⣛⣛⠀⢈⠀⠀⠀⠠
⠀⠀⠀⢈⡀⠃⢈⣀⠛⠛⠛⠛⣁⣉⣀⣅⠀⣀⢀⣤⣹⠉⡿⠴⡀⠾⠀⣿⣿⠿⢿⣛⣀⠈⢄⠑⢢⡄⣀⠙⠷⢌⡇⢸⣿⣿⣿⣿⠀⠀⠘⠁⠀⠀⡀⡀
⠀⠀⠀⠘⠛⡄⠸⠄⠀⠒⠀⣿⣿⣿⣿⡿⠀⣉⣡⠉⣩⣤⣶⣶⡆⠰⣦⡿⠀⢾⠤⠙⣛⣀⣨⣤⣴⣮⣭⣄⣘⠄⣰⣿⣿⣿⣿⠀⠷⠆⠱⠶⠐⠂⠙⠉
⠀⠀⠀⠿⢠⡇⣴⡰⣦⠀⣸⣿⣿⣿⣿⣥⣷⣿⣷⣤⣿⣿⣿⢛⢧⡤⢀⠌⠀⠆⣰⡿⢟⣟⣛⣛⠛⠛⠻⠿⢿⣿⡏⢉⣭⡉⠡⣦⢠⡄⠖⠴⠆⠐⠀⠛
⠀⠀⠀⠘⡀⢉⣏⢦⡛⣨⣿⣿⣿⠹⠼⠇⠄⠈⠈⠟⢩⣿⣷⣶⡴⠷⠆⠘⠈⠀⠀⠛⢸⣿⣿⣿⠉⠘⠃⠛⠀⣩⠀⣌⢨⣤⣤⣬⡀⢀⡄⠠⠀⠖
⠀⠀⠀⢀⡇⣼⣠⣌⢁⠁⣾⡉⢉⠀⢠⣄⡘⠀⣶⣶⡆⠼⠰⡇⠀⣃⢸⠇⣾⣿⣿⡇⠀⢁⡁⠠⢀⡁⢐⡆⢠⠀⠀⢁⠸⡿⢿⣟⡁⠈⡀⠀⣀⠀⠀⠄
⠀⠀⠀⢠⣄⢀⡈⢃⢘⠓⠟⠃⣈⣀⣨⡉⠉⢈⡛⢛⢡⡄⠀⢠⣶⣶⡄⣠⠉⠉⣭⡅⢠⠀⠄⠀⣤⣴⣦⠀⠀⠀⠤⣈⠙⠁⣿⣿⣟⠀⠉⣉⠀⠀⡀
⠀⠀⠀⢨⣍⣛⣃⣘⡉⠤⠷⠀⣿⣿⣿⠇⢈⠈⣑⣉⠈⡁⠀⠘⢻⠟⡀⠀⠈⢁⣀⣀⣈⣀⣀⣀⣻⠿⡟⠀⠀⠀⠀⠈⠁⠓⢸⣿⣿⡇⠁⠙
⠀⠀⠀⢿⠿⣿⣿⣿⣿⣿⣷⣦⣿⣶⣶⣾⣿⣿⣿⣿⠐⠀⠋⠂⠸⠀⠓⠀⠀⢸⡿⠿⠿⠿⣿⣿⣿⣿⠇⠸⠃⠃⠀⠀⠐⠓⠨⣭⣅⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠆⠀⢩⣭⢨⠍⠟⣹⢏⡻⣿⠛⣋⣩⡅⠀⠀⢸⡰⠶⠈⠴⠀⠾⡆⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠐⠲⠃⠀⠆⠀⠰⢦⠀⠀⠀⠀⡄
⠀⠀⢠⣶⣶⣦⣤⣤⣤⣦⣌⣈⣙⣂⣲⣉⠁⣠⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⠀⠀⢀⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠛⣛⠛⠿⠿⠿⠿⠿⢿⠿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣁⣀⣀⣀⡀⠀⠀⠀⠃
⠀⠀⠈⠀⠉⠡⠄⠂⠦⠈⠄⠂⠐⣮⢤⣌⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⠛⠀⠀⠀⠀⢰⣿⠀⡆⠶⠀⢰⡆⠉⢁⡀⢸⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠘⠷⠀⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⢩⡍⠉⠉
⠀⠀⠠⣦⣄⠀⣶⣦⠈⠉⢀⢠⣌⣄⡘⠃⠀⠘⠓⠀⠉⠄⠀⠀⢸⣿⠄⠀⡀⢠⣶⠀⣴⠀⠀⠠⠀⣿⡏⠀⠈⢩⠉⣿⡇⠀⡀⢈⢀⡀⡈⠀⠉
⠀⠀⡀⣄⠀⢤⣙⡇⢠⣤⣬⣤⣍⣈⣀⡀⠂⠘⠁⢸⣧⠀⢄⡀⣼⣿⠀⠂⢀⠄⠉⢠⠄⡀⢠⣤⠀⠭⠡⠀⠄⠀⠀⠛⠃⠀⠁⠈⠀⠋⠘⠀⠸⠃
⠀⠀⠀⠉⠀⠈⠉⠁⠻⠻⠿⠿⠿⠿⢿⣿⣿⣿⣶⣾⣷⣤⣈⣁⣙⣋⣤⣤⣤⣀⣀⡈⠀⠁⢛⡛⠀⠀⠀⠀⠀⠀⠉⠎⠀⠆⢸⣿⡇⠀⢠⠰⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣰⢰⣤⠈⠉⠀⠀⠆⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⣿⣏⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋


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

⠀⠀⠀⢀⡴⠒⠋⡟⠙⠲⢄⠀⠀⠀⠀⠀⢀⣤⠖⠛⣿⠛⠳⢦⡀⠀⠀⠀⠀⠀⣠⠶⠛⢻⡟⠛⢶⣄
⠀⠀⢠⠏⠀⠀⠀⡇⠀⠀⠀⢳⡀⠀⠀⢀⡞⠁⠀⠀⣿⠀⠀⠀⠹⡆⠀⠀⠀⣾⠁⠀⠀⢸⡇⠀⠀⠙⣧
⠀⠀⢾⠀⠀⠀⢀⣧⡀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⢀⣿⣄⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢀⣼⣇⠀⠀⠀⣻
⠀⠀⠸⡄⢀⠔⠁⡇⠙⢦⡀⣰⠃⠀⠀⠘⣇⢀⡴⠋⣿⠙⢷⡀⢠⠏⠀⠀⠈⢷⣀⣴⠟⢹⡏⠳⣄⢠⡿
⠀⠀⠀⠘⠣⣄⣀⣇⣀⣠⠟⠁⠀⠀⠀⠀⠈⠻⣤⣀⣿⣀⣠⠽⠋⠀⠀⠀⠀⠈⠻⣧⣄⣸⣇⣠⡼⠟
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣀
⠀⠀⢀⣴⠟⠉⠀⣿⠀⠉⠻⣦⠀⠀⠀⠀⣴⡿⠋⠁⣿⡇⠉⠻⣷⡀⠀⠀⠀⣴⡿⠛⠉⣿⡏⠉⠻⣷⣄
⠀⠀⣾⠃⠀⠀⠀⣿⠀⠀⠀⢹⡇⠀⠀⣸⡏⠀⠀⠀⣿⡇⠀⠀⠸⣷⠀⠀⢸⣿⠁⠀⠀⣿⡇⠀⠀⠘⣿⡄
⠀⠀⣿⡀⠀⣠⣾⣿⣦⡀⠀⢸⡟⠀⠀⢻⡇⠀⢀⣴⣿⣷⣄⠀⢠⣿⠀⠀⢸⣿⠀⢀⣴⣿⣿⣦⡀⢀⣿⡇
⠀⠀⠘⢷⣾⠋⠀⣿⠈⠻⣦⡿⠁⠀⠀⠈⢿⣶⡟⠁⣿⡏⠻⣷⣾⠏⠀⠀⠈⢿⣶⡿⠋⣿⡏⠙⣿⣾⠟
⠀⠀⠀⠈⠙⠷⠶⠿⠶⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⠶⣿⠷⠿⠛⠁⠀⠀⠀⠀⠀⠙⠻⠷⣿⣷⠿⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣦⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠐⢶⣴⣿⣿⣿⣿⣿⡯⠉⠀⠘⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⢠⣴⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⢁⡴⠋⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡗⢀⠀⠀⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣀⡀⣠⡶⠋⢿⣿⣿⣿⣿⣿⣿⠇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⡄⠉⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠈⠻⢿⣿⣿⡿⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⡀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣦⠀⠀⠀⠀⠀⠼⠿⠷⠄⣀⣀⣸⣆⠀⠀⠀⠀⠀⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠁⠀⠀⠀⣿⠟⠀⠀⣠⠾⣿⣿⣿⣿⣇⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡋⠀⢀⣠⣾⣥⣀⣀⠚⠁⠀⠈⣻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⡏⠻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠀⢰⣄⠀⣿⡇⢠⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⢙⡿⠻⣿⣿⣿⡿⠋⠀⠀⠈⠻⣷⣿⡟⣀⣿⣿⡿⢋⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⢠⣉⣀⡀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣷⣡⡾⠏⠀⠛⡫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠋⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⠀⠀⢀⡾⢛⢏⠢⡀⠀⠀⣸⣿⠋⠻⣧⠀⠀⠈⠀⠀⣠⠆⠀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⢸⣅⣒⣲⣦⡐⣢⠖⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⢼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⠀⠀⢠⡄
⠀⠀⠀⠀⢀⡶⠏⠁⣀⡥⠥⡌⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠁⡀⣿⠀⣠⠋⢻⡀
⠀⠀⠀⡖⠃⢂⡶⠊⠁⠀⢠⠇⢐⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣄⠀⣿⠐⠀⢸⠇⡴⠃⠂⠂⣳⠄
⠀⠀⢀⣷⠄⢽⠀⠀⢀⡴⠋⠠⣸⠃⢀⠔⣚⣛⠳⢄⠀⠀⠀⣾⠿⢉⣀⠀⢸⢇⡿⠀⠀⣿⡜⠅⠅⣥⣾⠃
⠀⠀⠀⣿⡇⠸⢀⡴⡛⡁⠀⣼⠃⣴⠏⡞⠁⠀⢻⠨⢧⠀⣾⠏⣥⠋⠘⠛⠋⢸⡗⠀⢰⡏⡀⣠⡿⠟⠁
⠀⠀⠈⣿⡇⡚⡋⠀⠀⣂⡾⠁⣼⡇⣼⠀⠀⠀⠘⣇⢸⡄⡏⠀⡏⠀⠀⠀⠀⣾⡇⠀⠀⢠⡞⠉
⠀⠀⠀⣿⡇⠄⡄⠄⢶⡋⠀⢠⣿⡃⡇⠀⠀⠀⠀⣗⢸⣿⡇⣺⠁⠀⠀⠀⠀⣿⠇⠀⣤⡅⠍⠳⣦⡀
⠀⠀⠀⣿⣿⠁⣿⢇⠁⠓⡄⠸⣿⠆⡇⠀⠀⠀⢸⠡⢸⢿⠅⢼⡀⠀⠀⡶⢦⢹⠀⢀⡟⠻⣧⡀⡈⠛⣦⡀
⠀⠀⠀⢸⣿⡆⢺⠈⢦⡂⠘⢦⢹⡇⢿⡀⠀⢀⠟⢨⣾⠸⡇⡀⣧⣤⡜⢀⡇⣸⠐⢸⡇⠀⠈⠱⣆⠂⠊⣿⠇
⠀⠀⠀⠈⣿⡇⠄⡇⠈⢷⡠⠨⠳⡘⢆⡓⠒⢛⣰⡾⠁⠀⢷⣆⣀⣀⣴⡿⠁⡿⠴⠾⠁⠀⠀⠀⠈⢳⡴⠁
⠀⠀⠀⠀⠚⠟⠓⠗⠀⠀⠻⣄⣈⣹⣦⠉⠙⠉⠉⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⢿⣶⣤⣤⣴⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⡿⠟⣛⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡅⢟⣭⣶⣿⣿⣿⣿⣿⡿⢛⣵⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟⣴⣿⣿⣿⣿⠿⢟⣫⣥⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢢⣭⣭⣶⣶⣿⣿⣿⣿⣿⠿⢛⣩⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⣡⣦⡙⠻⠿⢟⣛⣛⣭⣥⣶⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣴⣶⣶⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⠿⠿⣛⣩⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢨⣭⣭⣶⣶⣾⣿⠿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣭⣭⣭⣵⣶⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⡶⠀⠀⣀⣠⡤⠶⠖⠚⠛⠛⠛⢛⣛⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢁⣴⣫⡴⠞⠋⠉⠀⠀⠀⠀⣀⣤⣖⡛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣩⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢛⡽⢖⣠⡤⣤⣄⡈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢯⣥⠀⠀⠈⡼⠋⠀⠀⠀⠀⠙⠳⣄⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⢀⣹⠀⠀⠀⠀⠀⢴⠚⠉⠀⠀⢀⠞⠛⠀⠀⠸⡇⠀⢀⣤⡀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠙⢦⡀⠀⡤⠤⢬⡀⠀⢌⠉
⠀⠀⠀⠀⢰⢋⣽⠀⠀⠀⠀⠰⣶⠓⠀⠀⠀⠏⠀⠀⣤⠖⠛⠛⠲⣥⣿⡿⠀⠀⠀⠀⣸⠃⠀⢀⣀⣀⡀⠙⣎⠣⣤⡞⠉⠙⣦⡤⣄
⠀⠀⠀⠀⠸⢣⡏⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⡆⠀⠀⠀⣀⡴⠋⠀⠐⠉⠀⠠⡟⠉⠻⣦⠏⠉⠉⠜⢁⣀⣸⠃⡴
⠀⠀⠀⠀⠀⢸⡇⣴⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⣧⡀⠀⠀⠈⣹⡟⠒⠚⠛⠉⠀⠀⠀⠀⠀⢀⣠⠟⠙⢲⡈⠀⠀⣠⠞⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠻⠙⣆⠀⠀⠀⣀⣀⡀⢹⠀⠀⠀⠀⠈⠙⠒⠒⢊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡡⠤⠀⣸⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠋⢉⣉⣉⣉⠉⠃⠀⠚⢯⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢾⡉⠲⢤⡴⢣⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠛⠷⠶⠋⢁⣴⣄⠀⠀⠈⠙⠒⠦⠤⣤⣤⣤⡤⠴⠖⠛⠉⠙⣆⠀⠈⠣⡀⠙⢧⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⠴⠞⠋⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⢩⣷⡀⠀⠀⠀⢀⣧⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⡋⠀⠀⠀⠀⠀⢠⠖⠀⠀⠈⠉⠙⠓⠒⠒⢚⡿⠯⠭⢤⣤⣀⣸⠶⠿⠦⠤⠶⠋
⠀⠀⠀⠀⠀⢀⣰⡟⢻⡀⣠⣿⣶⣄⣀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢯⡀⠙⠜⠛⠁⣨⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⣾⡋⠑⢂⣠⠼⠉⠓⠶⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⠴⠞⠁
⠀⠀⠀⠀⡼⠁⠉⠉⠉⠀⠀⠀⠀⢀⡴⢋⡴⠋⠉⠉⢀⡤⠞⠁
⢀⣠⠤⠾⣧⡀⠀⠀⠀⠀⠤⠤⣴⣏⣴⣏⡀⠀⣠⡴⠋
⣟⠀⢠⠞⠧⠽⠛⠲⢤⣤⣄⣴⣋⣤⡤⠴⣶⣫⢿⠉⢳⡄
⢿⣓⣛⣓⣚⣛⣛⣭⠭⠵⠚⠯⢽⣓⣒⠒⠾⠶⠛⣒⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣣⣧⡌⡤⣀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠁⠃⠛⠓⠋⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣄⢀⣠⣀⢰⡦⢸⡧⣄⢨⠁⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢷⣿⡸⣄⣸⠘⡇⢸⠀⡟⣾⢸⡇⣼⢁⣏⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠙⠁⠉⢁⡄⢀⣤⠀⣤⠤⣨⠀⠉⠛⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣲⣦⣦⣴⠀⠀⣞⡶⠘⡸⢰⡋⠘⡳⠀⠰⣤⣤⣤⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⡀⢀⣀⠉⠁⣈⣀⣈⠀⢈⣁⠀⣀⡀⢀⣀⣀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣿⡇⣼⡟⢠⣾⠟⠛⠛⠀⣿⣏⣰⡿⠁⢛⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣏⢠⣿⠃⢻⣿⣤⣼⠏⢸⡿⢛⣿⠇⠀⣸⡟⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠈⣁⣀⠀⠈⠉⢁⡀⢨⡁⡀⠥⠤⢤⣩⠥⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣦⢆⢸⡆⠀⠈⢠⣴⣾⠇⠁⠰⢶⡾⠟⢳⠀⡶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣼⢿⡏⣽⠃⠸⠸⠧⢄⢸⠠⠶⠿⣄⠀⣶⠾⠇⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡿⠨⠛⠃⠀⠈⠶⠶⠾⠃⠶⠶⠶⠟⠳⠿⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠐⠲⠶⢤⣤⡀⠈⢀⣤⡤⠤⠀⠐⠶⠶⠤⠀⠙⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠤⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠬⣌⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠲⣤⣬⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⠻⠷⠿⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⢩⡍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠬⢍⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⢸⣿⣿⡿⢻⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⠁⠀⣀⢲⣿⣿⣿⣿⣁⡌⣉⣉⣀⠀⣄⡀
⠀⠀⠀⢀⡤⢤⣤⠤⡤⣤⣤⣤⣄⣀⣠⡤⡤⣄⣤⡄⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⢰⣾⡿⢟⣷⡿⠛⠉⠁⠀⠀⠀⠠⠉⠐⢤⡀
⠀⠀⠀⠈⠉⠉⠃⠘⠋⠉⠁⠉⡉⠉⠁⢷⠟⠉⢉⡭⠷⠦⠄⠐⠃⠀⠀⠀⡀⢦⣤⣴⡿⠋⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠉⠢
⠀⢀⠄⠀⠀⢀⡀⢀⡈⣁⡀⢀⡀⣈⣡⡟⠀⢠⡾⠂⠀⠀⠀⢸⡄⠀⠀⠀⠀⡜⣻⣿⣇⣆⣴⣧⢀⡆⠀⠀⣀⢰⢰⣷⣇⣬⣢⠀⠈
⣰⣾⣶⣰⣿⡿⣿⣿⣿⠿⡟⡿⣿⣿⣿⠁⠀⢸⠀⠀⠀⠀⠀⡸⣧⢃⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣼⣰⣸⢛⡎⣼⣿⣸⣿⠿⣟⣾
⣿⠟⣿⡆⣿⠗⣿⣿⣿⡆⣹⡇⣾⣿⢹⠀⠀⢸⡴⡴⣴⢸⣀⢁⣿⣿⣿⣾⣶⣿⣿⢿⣿⣿⣷⣶⡾⢿⣧⣬⣿⣶⣥⣤⣤⣤⣴⣬⠁
⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣷⣿⣿⣾⠀⠀⢸⡇⠇⠇⢷⠟⠀⢿⣿⣿⣿⣯⣿⣃⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠉⠉⠈⠉⠉⠉⠉⠣
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣇⠀⠀⠀⠀⡇⠈⠉⠛⠻⡿⢿⣿⣤⣦⣇⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⢠⣴⡄⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣧⣶⣴⣘⣷⣶⢀⠀⣄⣱⡾⠻⣮⣛⠿⠟⠈⡆⡀⢰⡇⢠⣿⣿⣼⣿⣿⡿⠿⡡
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡀⢹⡭⣍⢹⣿⣿⣿⣦⠁⠙⣿⣄⡈⠹⠿⣿⣷⣤⣡⣫⣤⣬⣭⢿⠷⠶⠤⡖
⠀⢹⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⠿⣛⡻⢯⣿⡴⠀⢸⡝⠿⠞⠁⠙⢿⣿⣷⡼⢿⣿⣷⡄⠀⠘⠈⠉⠉⠉⠉⠁⠀⠀⡀⠀⢀⠁
⠀⠈⣿⣿⣿⣿⣿⡇⠠⠁⡄⣿⠀⠺⠈⢨⠿⣧⣴⣾⠀⠀⠀⠀⠀⠀⠹⢿⣿⣄⢿⣝⢷⣦⣤⡢⣤⣄⣠⣤⣤⣦⣶⣿⢗⠞
⠀⠀⢻⣿⣿⣿⢿⣿⣶⣦⡶⠿⠷⣶⠶⠟⠀⣻⡏⡙⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⠝⠿⣿⣿⣿⣛⣿⣿⣿⣭⡿⠟⡋
⠀⠀⠈⠉⣀⣤⣤⣬⣤⣤⣴⣶⣶⣶⣶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⣿⣷⡎⠀⠈⠉⠉⠉⠉⢀⣀⢠⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡾⠿⠎⠈⠈⠊⠁
⣿⣿⠟⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣶⣾⣿⣿⡿⠿⠿⠿⠿⠆⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠶⠾⠟⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣤⣴⣶⠾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠹⣷⣄
⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠘⢿⣦
⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠘⣿⣆
⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⣤⣄⠀⢸⣷
⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣾⡟
⠀⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠘⢷⡀
⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⣿
⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢹
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⢸
⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⡿⠿⠿⠿⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢲⣄⠀⠀⠀⢠⣿
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⢹⣷⣄⣠⣿⡿
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⢿⣿⣿⡟⠁
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢸⣿⣏
⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢸⣿⡿
⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠈⠻⣿⣇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠂⠀⣀⣠⣤⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣷⣶⣿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣀⣀⣀⣹⡆⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠉⠉⠉⠉⣷⠀⠀⠀⠀⠀⠀⣿⠿⠒⠀⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠛⠒⠦⠤⢴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣿⡄⠀⠀⣸⠇⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢀⣠⡟⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣤⡶⠾⠛⠛⠛⠙⠛⠶⠦⣤⣀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣱⣿⠟⠿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⡀⠀⢸⣿⠛⠳⠶⣤⣀⣀⣀⡠⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⡀⠙⠛⠛⠁⠀⠀⢠⡿⢿⣯⡁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠘⢿⣤⡀⠀⠀⠀⠀⠘⠁⠀⠈⠻⣶⣤⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⣿⡿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢷⣄⠀⠀⢀⣼⣇⠀⠉⣻⣷⣤⡀⠀⠀⠀⠀⠀⠙⢿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠺⢿⣦⠀⠈⠉⣡⣄⣰⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢷⡀⣴⣿⣿⣿⣿⣿⠛⢿⣿⡇⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠙⠿⢿⡟⢉⣿⠏⠀⠀⠙⠇⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⡿⠿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠛⠟⠛⠻⣿⠀⠉⠁⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⠀⢸⣇⡗⠀⠀⢀⣀⣴⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠋⠉⠉⠉⠉⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⣴⠟⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠏⠀⣴⡏⠉⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⠓⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠹⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⢿⣶⣄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣠⣼⣿⣿⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⣿⣿⣿⡟⢀⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⣠⣴⣾⡿⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠿⠋⢠⡾⠃⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⢠⣾⠿⣿⡄⠀⠀⠀⠀⠀⠀⠹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⠻⢦⣄⣀⣀⣀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⢰⠟⠁⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣤⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠁⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠈⠉⢻⠿⢷⣶⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢸⠀⠀⠀⠈⠉⣻⠛⠛⠛⠛⠛⢿⠿⠿⠿⠿⠟⢻⠟⠛⠉⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠒⠢⠿⠤⣀⣀⡀⠀⣿⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⢸⣇⣀⣀⣠⣤⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠈⠉⠋⠁⠙⠋⠉⠉⠉⠉⢉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⣠⣴⠶⠛⠛⠛⠛⠛⠛⠻⠶⣦⣶⡛⠁⠀⠀⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠘⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠁