Mold Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢶⣄
⠀⠀⠀⢈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡼⠟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠡⠈⢹⡆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⢲⠄⠀⠀⠀⠄⠇
⠈⠒⠄⠰⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⡁⣰⠆⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠨⠀⠀⡇⠀⠃⡠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⣠⠀⠀⠀⠀⠐⠀⠀⠀⠀⡐⠠⢾⢁⡰⠀⠀⠤⢄⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣉⡐⠀⠀⠀⠀⠀⠀⠈⠉⡀⣀⡰⢠⠃⠀⠀⠈⢀⡀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠙⠒⠓⠶⠦⢠⡤⠤⠯⢴⠒⠒⠒⠖⠂⠀⠁⠙⢢⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠉⠁⠀⠊⠀⠀⣒⣉⠀⠀⡠⠺⠋⠉⢂⠀⠀⠣⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣁⠖⠁⠙⠛⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠂

⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣘⠃⠰⢈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠟⢀⡉⠐⠂⠀⣥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠘⠃⠈⣁⣡⣈⠀⢤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠃⢸⣿⣿⣿⣷⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠠⠌⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠐⢨⢠⣿⣿⣧⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣤⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠛⠀⢀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠁⠀⣰⠂⠠⠀⠀⠀⡇⠀⠀⠉⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⠸⠷⠀⠀⠀⢸⣿⠁⠘⠗
⠀⠀⢰⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠁⠰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠈⢘⣿⡆
⣤⣤⣬⣥⣤⣤⣤⣤⣤⣥⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠚⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⠐⣰⣯⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⢬⣥⣾⡽⣿⣿⣿⣿⣿⣿⣟⣯⣿⣿⣛⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣿⣆⢀⠀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣎⣳⣛⣿⣿⣿⣿⢿⢿⣿⣿⣿⠿⡟⠛⠻⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⢺⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿⣎⣿⡿⠿⠿⣿⣿⣿⡘⣿⣿⢋⣵⣌⠻⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⡗⢽⣿⣿⣿⣿⡟⣧⣀⣴⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⡟⣿⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢶⣾⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⡿⣾⣷⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣩⣿⡿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣟⣿⡾⣿⣟⣿⣿⣿⢿⣿⣿⣿⣿⡿⠉⠀⢽⡀⢿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡿⣷⢿⣯⡿⣯⣽⡿⣿⢿⡿⣿⣿⣿⣿⡿⣸⣤⠛⠗⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡻⢿⠿⠿⡿⣿⡿⠋⣹⣿⢸⣿⡏⠙⠛⠚⠗⣿⣾⣿⣿⣾⣷⣿⣿⣿⣟⣿⣯⠿⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⡘⣿⣿⣿⠟⢀⢰⣿⣿⢸⣿⣿⣦⣀⠀⠀⢹⣿⣿⣿⣿⢇⣿⣿⣿⣯⣿⣿⣷⣄⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢈⣿⡏⠀⢸⣿⣿⣿⡇⢿⣿⣿⣿⣷⣆⣾⣿⣿⣿⣡⣾⣿⣿⣿⣿⣽⡿⣿⣿⣎⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠇⢫⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡽⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣉⣷⡿⣿⣿⣿⣽⣿⣿⣷⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣃⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠴⣿⣹⣿⣷⣾⣿⣾⠿⣶⣿⡇⡈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⡝⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣡⣿⣿⣿⣿⣿⣷⣷⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⡇⣾⣿⣛⣿⣞⢿⣿⡾⢳⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢉⣿⣿⠋⣽⣿⣿⡟⣿⢁⣾⣿⠟⣻⣿⣿⣿⣯⢼⢿⡿⣧⡎⢸⣿⣿⣿⢻⣿⣿⡟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣈⣿⣿⣸⣿⣿⣿⣅⣻⣷⡿⣿⣘⣿⣿⡿⣿⣷⣤⡿⡇⢸⣷⣿⢻⣿⣿⣿⣿⣿⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣹⡿⢿⣛⢛⣿⣿⣟⡛⢉⢀⣌⣿⣿⣿⣥⣿⣿⡿⢋⣁⣼⣿⣿⣿⣿⢿⡛⠛⢟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⠉⠀⠀⠀⠀⠈⠉⠉⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡴⠒⠋⢉⡳⠀⠀⠀⠀⠀⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⠃⠀⠀⠀⠉⠧⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⢄⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⢰⢮⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠤⠤⠃⠀⠰⣁⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⢉⡏⠛⠶⠦⠴⠶⢛⡿⠁
⠀⣿⠀⠉⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠖⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⢀⣠⠶⠚⠛⠦⠤⠶⠖⠛⠉
⠀⠹⢦⣀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠘⣶⠀⠀⠀⠀⠈⣹⠆⠀⠀⠀⠀⣀⡤⠖⠋⣁⣠⠴⠖⠋⠉
⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠙⠓⠒⠋⠙⠉⠁⣀⡤⠶⠚⠋⣁⣤⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣤⣀⡀⠀⠀⠉⠛⠦⣤⡠⠴⠖⠋⠁⣀⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣄⣀⠀⡇⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢻⠇⡻⠀⡀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣆⣹⡐⣆⡄⠀⡄⣼⣿⠻⢟⢡⡖⣇⡞⣷⣼⢫⢌⡟⣱⠃⠀⣤⢆
⠀⠀⠀⠀⠀⠀⠓⠲⣔⣤⣝⡮⣇⣈⣹⣬⣿⡜⢺⢿⣷⣻⣶⡿⣸⠯⠞⡅⡿⡊⣜⡡⢟⣫⣄
⠀⠀⠀⠀⠀⡄⣄⢲⡨⣽⡛⣻⣺⡍⢧⣿⣿⣏⢿⣷⣫⢿⢼⢡⢧⣞⢊⢜⡹⢷⠍⣆⡯⠛
⠀⠀⠀⠠⠤⡷⢮⣋⡿⣇⣇⡇⣍⣟⢲⣍⣟⢸⢰⣻⡧⣿⣫⣟⡯⣤⠞⡜⣸⡭⠜⣱⡼⣙⢁⡄
⠀⠀⠀⠀⠈⢒⡶⡖⣝⠾⣍⢷⣞⣮⡹⣵⡬⡟⣧⡳⣷⢻⢻⣻⡏⡴⡶⡵⣫⢟⣽⢵⡦⢗⣋⡭⠅
⠀⠀⠀⠀⠉⠽⠋⣛⣺⣉⠿⡍⡳⣷⣟⠤⢷⡛⢿⣮⣳⣟⣻⣫⡇⣟⣱⠟⣩⣷⠶⠯⢖⣉⠞⡷⡊⢛
⠀⠀⠀⡽⢛⣲⢭⣙⢻⡬⢿⠓⡿⡆⣺⣻⣷⣛⣦⢷⣿⣿⡾⣻⢳⠯⡖⣫⢿⣷⣍⡴⢓⣛⣿⣻⠷⣯⠄
⠀⠀⠀⠐⠲⠬⣕⣦⣟⡿⣛⣻⣗⡷⣼⠿⣿⠶⣟⡿⢾⢹⣟⡾⣭⣭⣭⣽⣻⣣⡹⠭⣙⣒⣒⡥⢞⡦⢆
⠀⠀⠈⠹⠖⣩⠥⠔⣛⣶⢓⣏⣡⣿⣯⢷⣿⡽⣟⣿⣾⣿⣯⣽⢯⣿⣷⣞⣫⢿⢭⡿⢍⣒⡢⢅⣠⠼⡥⠆
⠀⠀⠀⢶⠤⢤⣬⣹⣿⣶⣛⣛⣯⠭⢿⢽⡿⣺⢗⡯⣾⣼⣭⡺⣕⣒⢲⡮⡷⣯⢽⡒⣺⠴⣬⣝⣲⢯⡃
⠀⠀⠀⢀⣻⠯⠼⣿⡿⣻⢖⣻⡮⢭⡯⢝⣫⡵⣫⢿⢾⡝⣿⡝⣜⠧⣷⡽⣝⢮⣳⣽⣿⠳⠬⣷⡯⠅⠋⠁
⠀⠀⠀⠈⠴⡫⠟⠵⠏⠟⢯⡿⣷⣫⢮⣟⣜⡽⣱⢫⢏⣟⣟⢦⣯⣽⡸⡿⣾⢷⡦⣥⣘⢯⣯⠗⣀⣄
⠀⠀⠀⠀⣐⣒⣿⣩⠭⣿⣏⡞⣳⣵⢣⢞⢻⣾⢧⢮⣾⣾⢽⢻⡜⣦⣿⣯⣏⣿⣶⣿⣮⠽⣼⣿⡙⠂
⠀⠀⠀⠀⠘⣁⡶⢚⡩⣞⡭⣪⣵⢻⢺⣾⡿⣏⣾⣌⡮⡼⡴⢳⢳⡍⡏⡿⣭⢷⠧⡟⡾⣝⣏⠳
⠀⠀⠀⠀⠀⠀⠃⢺⡥⢫⠞⡵⣫⢬⡞⡷⣎⣟⣼⡹⣿⣧⡟⡿⣼⡜⣟⠻⣹⡟⠿⣭⠲⠜⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⠄⠑⠇⡿⠼⡗⣣⣾⢋⣇⢛⣇⡗⣟⢮⢳⣫⣗⡎⡿⣇⣛⡻⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⣿⣽⢢⢛⣇⡿⣸⣼⣏⢳⡝⠸⡍⢼⢿⠦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠹⠹⢹⠁⠏⠁⡯⠇⠁

⠀⠀⠀⠀⠀⠨⠈⠀⠈⠉⠉⠉⠁⠈
⠀⠀⠀⠀⢀⢶⡀
⠀⠀⠀⠀⢈⣼⣧⡀
⠀⠀⠀⠀⠈⢺⠟⣅⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠈
⠀⠀⠀⠀⠈⠋⡸⣿⡀⡀⠄⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠃
⠀⠀⠀⠀⠀⢸⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣳⣆⠄
⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⣀⡀⢀⠀⡀⢀⡀⠀⠀⠀⠸⣟⣧⡰⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢰⠄⣷⣤⣠⡀⠁⢄⠀⠀⠀⢨⡻⣿⣴⣷⣶⣶⣶⣴⣬⣩⣷⣿⣿⠩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠺⣦⠎⠈⣟⠒⢰⣼⣷⣇⢐⣆⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠝⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠓⡝⠒⠻⢾⡿⠿⢿⢿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠠⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠻⠇⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠄⠀⠀⠀⠈⠠⠀⡀⣤⣔
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠻⢽
⠀⠀⠀⠀⠀⠈⡀⠀⠠⠀⡀⢀⠀⠀⠀⠀⠀⠀⡀⠨⢦⠀⠀⠀⠀⠀⠠⠄⠐⠁⠀⠀⢀⠀⠀⠐⠄⠒
⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠘
⠀⣀⣀⡀⢀⠀⠀⠀⠁⢄⠀⡀⠂⠀⠁⠀⠀⠀⠀⠨⠘⠀⠇⣀⠈⠀⠀⠀⠀⠈⠂⠀⠈⠇⠐⠀⠰⠂
⢹⣿⠄⠃⠀⠀⠁⠈⠀⠀⠀⠀⠀⠂⡨⠀⠀⠀⠀⡊⠀⠈⢀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⡀⠁⠈⠠
⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠠⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠠⠀⠀⠘⠄⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡤⡻⡷⡕
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣵⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣭⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⣀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢠⣀⣤⣯⣔⠂⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠡⠀⠀⠀⠀⠀⠀⡈⣩⣶⣾⣶⡃
⠀⠀⠀⠀⢐⣿⣿⣿⣿⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⡀⠀⠀⠀⠀⢠⢆⢻⣿⣿⣿⢁⡀
⠀⠀⠀⠀⢈⡹⣿⣿⡿⢁⠐⠀⠀⠀⠀⠀⠀⣤⢰⣿⣿⣿⠗⠗⠀⠀⠀⠀⠀⠈⢀⣽⣿⠟⡈⠁
⠀⠀⠀⠀⠀⠀⣨⠁⢀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡶⠉⠝⠣⡦⠄⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡛⠗⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡌⠝⠮⣍⢀⢸⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠉⠠⠈⠂⡄⢸⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⣀
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣔⠀⠤⠤⠤⢄⣾⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠆
⠈⠢⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣇⣼⢸⡎⡇⢻⢸⡄⢀⣇⡰⢁⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠈⢧⠈⣷⣼⢻⡿⣼⡈⡇⣼⢹⣧⡞⡇⡎⢰⠃⢀⡜⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⣧⡿⢿⢇⣷⣿⣧⣱⡏⣏⣼⡀⡇⣷⠇⢀⠎⠀⡰⠂⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣼⣇⣽⢿⣀⣻⢁⡟⠋⠹⣷⢻⣤⢋⡠⠊⠀⣀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠈⠙⣿⡃⠈⠻⣿⡟⠀⠀⢀⣿⠞⠉⢁⡠⠔⠉⠀⠀⠀⠀⠀⠀⢿⣯⣩⢿⡿⠋
⠀⠀⠀⠀⢠⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠛⡇⠀⠀⣿⡇⠀⢠⡪⠋⠁⠻⢗⠐⠒⠒⠂⠉⠀⠀⠀⠀⢨⡶⠴⠂
⠀⠀⠀⣰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠻⡀⠀⢸⣷⢰⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⢠⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣾⣿⣿⡄⠀⠀⠀⣇⠀⢰⣺⠁⠀⠀⠀⠀⢀⣔⠟⣿⡿⠗⣦⡀⠀⠀⠀⢠⢰⠀⡆⡆⣠⡀
⠀⠀⡸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣻⣿⣟⣏⣧⠀⠀⠀⠉⠂⠈⢹⠀⠀⠀⠀⠀⣾⡷⣤⣿⣗⡎⣷⡷⠀⠠⡈⢆⢆⣆⣷⣿⠜⡠⢂
⠀⠀⠙⣿⠒⡿⣽⣿⣻⣿⢿⣿⣿⣿⡿⣽⣽⣿⣿⣿⣿⣿⣿⣻⠀⠀⠀⠀⠀⢄⠰⠀⠀⠀⠀⠀⢸⣯⣼⣻⢻⣳⣬⠇⠀⠐⠨⢱⣾⣾⣿⣷⡾⠢⠐
⠀⠀⠈⡉⡎⠣⡙⠫⡾⣄⣾⡾⣟⣻⣿⡿⣿⣟⣿⣾⣿⢹⣯⡏⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⡀⠠⠝⠛⠛⠛⠛⠉⠁⠀⠀⠀⠈⠹⠛⢿⡟⠻⠗⡀⠉
⠀⠀⠀⠁⡽⢾⡓⠽⢡⣎⡾⡗⡯⢻⣟⢯⣾⢽⣻⡿⣯⣟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠲⣟⠣⣿⠿⡕⠺⡥⡜⢩⡎⢴⠘⣛⣽⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠽⠁⣷⠒⣾⡛⣳⣉⠿⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡋⣘⡀⡦⡀⠀⠀⠀⠀⣀⡴⢖⠾⢒⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⣼⣳⣾⣳⣾⣽⣟⣷⡀⠀⠀⢰⣿⣿⣆⣧⣥⣸⠄⠀⠀⠀⠀⣴⣞⣿⠿⢉⠟⢢⠄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣽⣦⣦⠃⠀⠀⣸⣿⣯⣷⣢⡠⣎⠁⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠈⣿⠿⣿⣿⣿⡿⠟⠀⠀⠀⠻⣿⣿⣿⣿⣧⡷⣴⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠁⠀⠀⠀⠀⣠⠞⠁⠀⣬⣉⡅⠀⠀⠀⠀⠀⣠⠞⠿⠿⣿⣿⣿⡽⠏⠀⠀⢀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⡠⠃⣀⣫⣱⠉⠉⠀⣹⡀⠀⢀⠔⠁⠀⠀⠀⠀⠁⠀⠀⠀⢀⣾⣿⣷⡙⠇⣙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣐⠁⢀⣿⣿⣟⣻⣆⣶⣦⡇⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣮⣷⣮⣅⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⢀⠋⠀⠀⢿⣿⣿⣿⣯⣟⠟⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⣿⣿⣿⣿⣿⣟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠆⠀⠀⠀⠀⠉⠉⠙⡉⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠈⠁⠈⠙⠻⠿⠿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⡀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⡄⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⡀⠀⠀⠀⠀⠀⠆⢸⡄⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠁⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⢀⠀⠀⠀⠀⠀⢰⠀⠰⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢀⠂⠀⠀⡌⠀⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠉⠀⡀⠀⠀⠗⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⢸⢀⠀⠀⠀⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⢸⠁⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠘⠀⠀⠀⠂⢠⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡀⠀⠄⡜⠀⡎⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣸⢁⡎⢰⠃⢰⣡⠋⢀⠇
⠀⠀⠀⣀⣀⣀⣴⣤⣀⠂⡤⣉⡸⣷⠶⠋⢸⣁⢺⡃⣊⣃⣂⠆⠒⠱⠒⠢⠤
⠀⠀⠀⠀⠉⠁⠀⠙⠋⠉⠁⠉⠁⠀⠀⠈⠁⠐⠂⠁⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣬⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⣶
⠀⠀⠀⠀⢾⡷⠐⠟⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠡⠁
⠀⠀⠀⠈⠁⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣿⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⢒⣒⠂⠀⠀⢀⡤⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠈⡁⠀⠀⠐⠀⠀⠙⡁⠁⠀⠈⠃⠤⠤⠀⠒⠃⠐⠀⠉⠑⠠⢄⡀
⠀⠀⠀⠀⢀⠔⠁⡠⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡝⠢
⠀⠀⠀⢠⠂⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⢀⠴⠊⠒⡀⠀⠀⠀⠀⠰⡁⠐⠀⠁
⠀⠀⠀⡎⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⡕⠀⡆⡀
⠀⠀⠀⠱⡀⢣⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣀⠠⢀⠀⠀⠀⠙⠲⠐⠊⠀⠀⠀⠀⢠⠎⠀⠠⢠⠁
⠀⠀⠀⠀⠙⡦⠑⡄⠀⠀⠀⠀⢰⠀⠀⠀⡌⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣶⡥⠂
⠀⠀⠀⠀⢸⠁⠀⠇⠀⢠⢄⣀⠴⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⠒⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⡅
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⢄⠀⠀⠀⠈⡀⢧
⠀⠀⠀⠀⢸⢆⣬⠄⠐⠁⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢣⢀⠇
⠀⠀⠀⠀⢸⢀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⢆⠀⡠⠒⠉⠀⠀⠀⠀⡠⠚⠇⠈
⠀⠀⠀⠀⡌⠨⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰
⠀⠀⠀⢠⠁⠃⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠃⠀⡀
⠀⠀⠀⡌⢰⠀⠀⠀⠀⠀⢀⠎⠀⢀⠤⠤⡄⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢎⣀⡠⠔⠢⡀⢀⠀⠁
⠀⠀⢰⠁⡆⠀⠀⠀⠀⠀⡎⢀⠔⠁⠀⠀⠈⠒⠊⠢⠀⠀⠀⠈⢣⠀⠀⠀⢀⠴⢦⠀⢀⠔⠀⠈⡀
⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⡠⠊⠀⠀⠀⢀⣀⡠⠀⣎⠀⠀⠀⡇
⠀⠀⡇⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠎⠀⠀⡔⠉⠂⠀⠀⢀⣀⡈⠉⢦⣀⡥⡆
⠀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡐⢡⠊⡌⠢⢀⠌⡠⠁
⠀⠀⠹⡀⠘⠦⣀⣀⣀⣀⣀⣀⠤⠄⠀⣘⡆⠨⠬⠥⠤⠿⠭⠭⠭⠉⢁⣐⠒⠷⣬⣭⡤⠞⠥⠒⠁
⠀⠀⠀⠉⠒⠀⠤⠄⠀⠀⠀⠒⠈⠁

⠀⠀⠀⠀⢀⣠⡤⠤⠶⠞⠛⠩⠭⠉⢙⢒⡒⠒⠒⠤⠤⠤⠤⠴⠦⢤⣀⡤⢤⣀⣀⣀
⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⠀⠒⠂⠀⠀⠠⠤⠀⣀⡀⠀⠈⠉⠉⠒⠦⠤⣄⣀
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⣀⠀⠉⠳⣄
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈⢱
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸
⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡞
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠔⣁⣠⡼⠃
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⣼⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡟
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣀⣀⠀⠀⢀⠀⠀⠀⡇⠀⢠⡇
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠾⠁⠀⡣⠀⠟⠀⠀⢰⠀⠀⢸⠁
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⢀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⡸⠀⠀⣼
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⢸⡂⠀⠀⡠⠁⠀⠀⢀⠇⠀⢀⡇
⠀⠀⠀⠀⠉⠓⠶⠤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⡸⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠶⠶⠤⠤⣤⣄⣀⣀⣀⡀⠀⠀⣀⣰⠷⠚⠉

⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠾⠿⠛⠉⣉⠛⢓⠒⠂⠤⠴⠦⢤⡤⠤⣤⣤⠤⢤⣄⣀⣀⣀⣀⣀⣀
⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣄⣀⡀⠠⠤⠄⠀⣀⣀⣀⠀⠀⠉⠉⠉⠉⠓⠒⠲⠤⣄⣀
⠀⡰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠂⠀⠀⠀⠀⣀⠀⠀⠉⠛⠲⣄
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠈⠳⢤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠙⣦
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣠⡟
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠉⣴⠖⠚⠋
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⢠⠃⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡏
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣾
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠖⠒⢄⠀⢠⣦⠀⠀⠀⣌⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠈⠃⠀⠀⣸⠀⠈⠁⠀⠀⠀⡇⠀⠀⢸⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢲⠀⢰⡀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⢸⠁⠀⠀⣿
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⢟⠀⠀⠀⢀⠎⠀⠀⠀⠀⡞⠀⠀⢠⡇
⠀⠀⠀⠀⠈⠛⠶⢦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠁⠀⠀⠀⠀⢠⠃⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠳⠶⠶⢤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣛⣛⣛⣶⣶⣶⣶⣶⣶⣖⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢯⡭⠟⠀⢀⣶⣛⣳⡄⠉⠉⠙⠓⠶⢤⣀⠀⣀⡤⠶⠚⠋⠉⠉⠀⣀⣀⣀⣀⠀⠈⠉⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣀⡤⠶⠚⣋⠉⠉⠉⠉⠉⠉⠛⠲⢦⣤⣀⣸⠋⠉⠁⢀⣀⣤⠶⠖⠛⠉⢉⣉⣉⣉⣉⣛⣳⣶⣤⣀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣰⢋⡴⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠈⠙⠲⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠖⠲⠖⠖⠛⣦⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣯⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣈⣉⣉⣉⣉⣉⣉⣉⣿⠀⢀⡇⠋⢷
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠙⠳⣄⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⢤⣤⠞⠁⠀⣼⠃⠴⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⠀⠘⡇⠛⠀⠀⠀⢠⠴⠶⠟⢦⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠶⠶⢾⠁⠀⡴⠞⢁⡴⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠻⠀⠀⠀⠘⡦⠀⢠⡟⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣋⣉⣻⠀⠀⡇⣙⢊⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠙⠶⠛⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣽⠶⠾⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢾⠀⠀⡷⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢷⡀⠀⢀⣠⢦⡀⠀⠀⠀⠀⣠⠞⠁⠀⢸⠀⠀⡇⣿⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠙⠓⠲⠒⠛⠁⠀⠀⠀⢸⠀⠀⡇⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠓⠚⢦⠀⢠⡶⣦⠀⠀⠀⢸⢦⡀⡇⠿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⠾⠁⠈⠛⠋⠀⠀⠀⢸⣠⠟⡇⠴⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡇⣤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣰⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⠙⣀⣁⣈⣳⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⠀⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠈⠹⡏⠃⠙⡀⠛⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⡇⣿⢸⠇
⠀⠀⠀⠀⠀⠀⣀⠀⢠⡤⠴⠄⠀⠀⠀⠀⠀⠙⠋⠛⠲⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⡤⢤⡀⠀⠀⠀⠀⠀⢽⠾⣿⣝⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⢤⠶⠂⠀⠀⢹⣤⡴⣷⠀⠀⠀⠀⠀⠘⢧⣅⢙⢷⡴⠄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠞⠀⢸⠀⢸⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⣷⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⢸⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣻⡆⠀⠀⢀⣶⠛⣦⠀⠀⠀⠀⢾⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⢡⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠃⠀⠀⠈⠿⣿⡽⠃⠀⠀⠀⠀⠻⣬⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣥⣤⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠘⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⡶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⣠⣀⠀⣀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠈⠉⠉⡏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡴⠖⠲⢤⠶⡿⢦⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣙⣛⣛⣁⠀⠀⠀⠀⠀⠀⢀⣾⡷⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⢀⣾⣿⠁⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⢳⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠉⠁⢸⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⣿⣿⣿⣿⣿⣿⣿⡇⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣣⣴⣾⠿⢛⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣾⣿⡟⣡⣾⣿⡿⠟⢛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⡿⣴⣿⡟⣡⣾⣿⣿⠿⠿⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠷⠿⠿⣿⣶⣦⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⢟⢻⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣏⡽⣿⣷⣶⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣒⣒⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⠸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⣯⡗⢾⠀⠹⣿⣽⣿⣿⢛⣿⣛⡟⣯⢞⡷
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⣛⣿⣷⠿⠛⠉⠀⢸⡆⠀⠈⠻⠿⠟⡟⠿⢿⠟⠋⣸⠇
⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⠻⡆⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠅⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡆⠀⠀⠀⣸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣷⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢿⠀⠀⢀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣯⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠖⢫⣿⣿⣿⢿⣿⢿⣿⣿⡿⣿⣼⣾⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⢿⣽⣿⣿⣿⣿⣾⣿⡿⣿⠉⠹⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣛⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣯⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⠁⠉⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣠⣶⣤⣀⠀⢸⣿⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣿⣿⣿⣿⣿⣶⣾⣿⣿⠒⠒⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⢟⣿⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⣭⣭⣭⣭⡽⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠛⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠴⠤⠴⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡅⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣀⣀⣀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⠐⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣠⣾⣿⣿⡿⣿⣿⣿⣦⣌⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣡⣴⣿⣿⣿⢿⣿⣿⣷⣤⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣴⣿⣿⣿⣿⣍⡁⢉⣽⣿⡿⣿⣷⡌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢡⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⣿⣏⠃⠸⣿⣿⡇⣿⣿⣿⠆⢋⣽⣿⡄⢻⡄⠀⠀⠀⠀⠀⠀⢀⡟⢠⣿⣯⡙⠻⣿⠟⠛⠛⠻⢿⡿⢛⣹⣿⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣧⣶⣶⣬⡙⠃⠛⣉⣴⣶⣦⣽⣿⣷⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣷⠃⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡛⠻⠟⢋⣡⡄⣤⣍⠛⠿⠟⣻⣿⡿⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⡿⣆⠀⠀⠀⠀⢀⡿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠹⣿⣏⡄⢸⣿⣿⡇⣿⣿⣿⡆⣬⣽⣿⠃⣾⠁⠀⠀⠀⠀⠀⠀⠈⣷⠘⣿⣯⣥⣶⣿⣶⣤⢤⣴⣿⣷⣦⣹⣿⠏⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠙⢿⣿⣿⣿⣋⡄⣈⣹⣿⣿⣿⡿⢁⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡙⠻⠿⣿⣷⣿⡿⠿⠛⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣈⠛⠿⢿⣿⣾⡿⠿⠟⣋⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠦⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡩⠷⠒⠒⢼⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠱⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⠳⣄⠀⠀⡎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢧
⠀⠀⠀⠀⠈⠓⠬⣓⠢⢄⡈⢣⡈⠓⢶⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀
⠀⠀⠀⢀⡤⠤⠤⠀⣉⣦⠈⠉⠁⠀⠀⠙⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠌⡃
⠀⠀⠀⠀⢀⣉⣉⠉⠠⠦⠀⠀⠀⠀⠀⠀⠘⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⡟⠛⠛⠶⡒⢤⡀
⠀⠀⠀⠀⠙⢒⣒⣐⡒⠂⡄⡴⠒⠒⠒⠒⠒⠛⠒⣲⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⣧⣠⣤⣴⠡⠾⠇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⠛⠛⠛⠛⠉⠉⣩⣿⡇⣯⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠷⣿⡟⢏⠉⣲⠛⠦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠈⣿⠁⠀⡰⠋⠲⢄⡀⠀⠀⠈⣧⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠞⠀⠀⠀⠀⢘⣦⠞⠉⠀⠀⠀⠀⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡜⡇⠀⢀⡤⠤⠄⠀⠤⠤⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⡄⢰⡇⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⡖⠒⠤⠤⠄⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠎⠀⢠⠘⠇⡇⠀⢻⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠀⢸⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣸⠁⠀⠀⠀⢸⠀⡄⡄⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇⡇⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡆⠀⠀⠀⠀⡌⠀⢰⢁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡁⠀⠀⠀⢠⠃⠀⢸⢸⠀⠀⢣⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢇⣀⣀⡠⠋⠀⠀⢸⠘⡀⠀⠀⠙⠒⠒⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⢫⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢇⣀⠀⠀⠀⠀⠀⠀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣛⣭⣝⡳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣫⣵⡾⣛⣭⣝⡻⢶⣭⡛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⣭⣶⢟⣫⣵⣾⣿⣿⣿⣿⣶⣭⣛⠷⣮⣙⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢤⠾⠋⢡⣾⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⢻⣿⡆⠙⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⢛⣛⡻⢮⣙⣘⣷⣝⢿⣿⣿⣿⣿⡿⢟⣛⣵⢞⣛⣛⠷⣄⣀⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸⣿⣿⣷⣭⢉⡿⢋⣼⣿⣿⣿⣿⣷⣭⣭⣶⣿⣿⣿⡇⠈⠉⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣼⣿⣮⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣶⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠾⠟⠉⠭⠉⢛⡒⠒⠒⠲⠶⠤⠤⠤⢤⠤⠤⢤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠀⠤⠤⠤⠤⠤⠄⣀⣀⡀⡀⠈⠉⠉⠉⠒⠲⠤⢄⣀⡀
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠐⠒⠠⠄⢀⡀⠈⠉⠓⢦
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠈⠛⢦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸
⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢠⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠚⣡⠤⠴⠋
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢸⠇
⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⡏⠀⠀⣾
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡏
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠃
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣼
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⠀⢀⣄⠀⠀⠀⡇⠀⠀⡏
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠛⠁⠀⢈⠆⠘⠋⠀⠀⢸⠁⠀⢰⡇
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠀⢤⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⡸⠀⠀⢸⠃
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠙⡇⠀⠀⢀⠜⠀⠀⠀⢀⠇⠀⠀⡾
⠀⠀⠀⠀⠈⠓⠶⠤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠁⠀⠀⠀⠀⡼⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠲⠶⠦⢤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⣥⠖⠋⠁

⠀⠀⠀⢀⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⡀
⠀⠀⠀⣿⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣿
⠀⠀⠀⣿⠀⠈⠋⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠙⠁⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⢀⣤⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣠⡀⠀⣿
⠀⠀⠀⣿⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢀⣿
⠀⠀⠀⠈⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠖⠚⠛⠉⠉⠉⠀⠀⠄⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠚⠋⠉⠀⠀⠀⠀⢀⣄⠀⠃⠀⠤⠀⢀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠐⠂⠀⣠⡀⠀⠀⡈⠋⠀⠿⣟⠀⠀⠹⠇⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣧⡀⠀⠘⠃⠀⠀⠀⠙⠃⠀⠀⠛⣤⠀⠀⣬⠄⠠⢦⡀⢲⡆⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠉⢹⣿⡀⠀⠀⠀⠀⠘⣷⠀⡀⠀⠀⠈⠀⠘⢿⣷⡒⠈⠃⠀⠀⠘⠗⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠸⡏⣿⡄⠀⠀⠸⣶⡉⠀⠀⢀⠀⠺⡆⣀⠐⢉⣀⠂⠀⠀⢿⡶⠏⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠻⢇⡀⠹⣆⠀⠀⠨⣷⡄⠀⠀⢱⣶⠁⠈⠒⢷⣆⠀⠀⠀⠀⠁⢠⡄⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡝⠀⠀⠘⠀⠀⠘⣷⡄⠀⠀⠀⢱⣤⠀⠈⠀⠀⠀⠀⠁⢴⣷⠀⠀⠀⠈⠁⠀⠀⠃⠸⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⠛⠀⣀⠀⠀⢾⡇⣄⠘⠏⠀⢹⣧⡄⠀⢀⠛⠀⢀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⢤⠀⠸⡆⠀⠀⠀⠀⠈⢷⣄⠀⠠⠄⢠⢷⠀⠚⠻⠏⠀⠀⠀⠉⡿⢁⣤⡈⠀⠰⣌⡴⠞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⠁⠀⠀⠱⣄⠀⠀⢻⠀⠀⢻⣆⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⣀⣀⠜⠛⠉⠉⠠⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣂⠀⠀⠀⠈⠀⠀⠘⠛⢷⣄⠁⠀⣀⣻⣽⡷⠒⠉⠻⠁⠀⣠⣄⣀⣀⡤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢿⣯⣀⣤⢀⠀⠙⠀⢿⣷⠋⠉⠀⠀⠀⢠⡀⣀⡤⠞⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣶⣶⣼⣏⢀⣠⠶⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣲⢶⣤⡀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⢤⣀⡀⠠⢿⣷⣄⢺⣿⡅⣋⣮⠻⣩⣷⡄⠀⠀⠀⠀⠐⣤⠡⡀⠢⡀⠀⠀⠀⠀⠆⢀⠈⠳⡀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣵⣿⣿⣿⡆⠘⠛⡿⠟⠋⠨⣿⣿⣿⣿⣿⠏⠀⠀⠱⣀⢀⠀⢃⢸⡄⠀⡀⠀⠀⠀⠠⣎⡠⢀⢿⠁⠀⠉⠀⢰
⠀⠀⠀⠀⠀⠰⣿⣽⣿⣿⠃⠀⠀⡇⠀⠀⠀⠀⠀⡏⠉⠁⠀⠀⠈⠀⠀⡘⢳⣼⣿⣷⣏⠀⠱⠀⠀⠔⢿⠁⠆⡎⢠⠄⠀⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢙⡆⠀⠐⢇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡀⢼⡄⠄⢀⣿⠷⠹⠀⠀⡀⢤⣳⣼⣀⣷⠷⠤⠀⠂⠀⢈⠆
⠀⠀⠀⠀⠀⠀⡤⠀⠀⠈⠁⠀⠀⠿⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⡁⠈⠉⠀⢨⡿⠃⠀⠀⢸⠁⠀⠛⢹⢡⢿⣿⠖⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⢠⣀⣶⡿⠃⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢠⠄⣀⢀⣤⡁⢀⣴⣣⢞⡾⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡨⢿⢿⣿⣻⡿⡼⣟⣷⣾⣷⣟⣣⣜⡹
⠀⠀⠀⠀⠀⢱⣿⠺⠿⣿⣿⣿⠟⠻⠙⠏⠿⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣠⣶⣄⠈⠺⠋⠀⠀⠀⡀⠀⡀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠙⠛⠁⠱⡄⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⣤⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢱⡄⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⠀⠀⡜⠀⠀⠉⠚⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣈⢧⡤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠎⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣴⣀⣽⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠃⠀⠀⠀⠀⠀⠀⢀⣶⣦⡙⠛⠉⠀⠀⢸⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣇⠀⠰⣿⡆⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⢀⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣽⠁⠀⠀⢩⠁⠀⠀⠀⠉⠃⠀⠀⣛⡟⠀⠈⢿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠂⠀⠀⠘⢿⠆⠀⠀⠀⢈⣛⠀⠀⠀⣋⠀⠀⠀⠀⢂⠀⠀⣴⠎⠀⠀⢀⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⠀⠀⠀⠀⣾⡦⠀⠀⠀⢸⡿⠀⠀⠐⡄⠀⠀⠀⢰⠜⠀⢠⣤⠀⠀⢠⢾⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⣌⠀⠀⠀⠀⠨⡇⠀⠀⠀⢸⡇⠀⠀⢀⠧⠀⣀⠀⣾⠃⠀⡸⠋⠀⣠⠟⠊⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠸⡄⠄⢰⡿⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴⡀⠀⡰⠋⠀⠀⠈⠁⡯⠤⠾⢡⠀⠠⠉⠀⢸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⡄⣼⣧⠀⢀⢀⡀⠠⠷⠐⠊⢧⡀⠀⡼⠃⠒⠒⠒⠋⠀⠳⠠⡤⠄⢌⡦⣠⢆⠐⠂⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⡁⠀⠀⠀⠀⢴⡶⢤⡀⠀⣠⠾⡃⠀⠉⠀⠤⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠀⠤⠤⠶⠟⠛⠛⠓⠤⡀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠓⠆⠀⠀⠈⠁⠈⠀⠘⠁⠀⠀⠠⢤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠂⠀⠠⠄⠀⠀⠀⠓⠀⣄⣀⠐⠶⠒⠖⠀⢀⣄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠙⠋⠀⠀⠈⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡖⠁⠀⢠⢤⠀⠀⠀⠀⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢚⠀⠀⡜⢀⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣭⠃⠀⠙⠋⢁⠀⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠸⣦⣀⡼⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡞⠀⠀⠀⡀⠀⡉⠑⠒⡄⢀⠀⠀⠀⡰⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢒⠏⠻⠤⠕⠀⢾⣿⣿⣿⢖⠝⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣴⣶⣤⣴⣿⣿⣿⡷⡍⣴⡁⠀⠀⠀⣀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣯⢟⠞⠱⡄⠈⠓⣖⠉⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡻⢿⣿⣿⣿⡿⡣⢯⠶⠚⠙⠦⣤⡼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣹⠊⠀⢀⢞⠞⡔⠛⠿⣬⡉⠁⠀⡶⠒⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠰⣇⢀⡰⡻⠃⠀⢸⠀⢀⣨⣿⡆⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⡤⠒⠉⠀⠀⠀⠀⠙⢦⠌⡐⣟⠀⠀⢠⠃⠀⢻⣿⡿⠙⠋
⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡸⣢⠾⠁⠀⡰⠁⠠⣾⣿⡟
⠀⠀⠀⠀⢠⡏⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠜⠁⠀⢀⠜⠀⠀⣤⣾⣿
⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡡⠋⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠀⠀⢀⠤⠒⠤⣾⠝⠀⠀⠀⠐⠛⠛⢻⠉⠙⣯⠉⠙⣇
⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⢀⠔⠡⢄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⡀⠀⡇⢇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠄⠀⠑⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢣⢀⡇⠘⡄⢸
⠀⠀⠀⠀⠀⠠⠴⠚⣉⠄⢀⠅⠀⡠⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠁⠀⢣⢸
⠀⠀⠀⠀⠀⠀⠠⠭⠒⠭⡀⢄⣪⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠈⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠐⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢀⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢀⣴⡶⡶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢹⡄
⠀⠀⠀⢀⣴⢿⣼⠁⡄⢼⢉⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢸⡇
⠀⠀⢀⡟⢗⠸⣿⠀⢻⣾⢸⡇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠁
⠀⢀⡾⠀⢸⠀⢻⠀⠀⣿⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡟
⠀⢸⠇⢠⠌⡇⠈⣷⠀⢿⣸⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇
⠀⣿⠀⠘⡆⢣⠀⢸⠀⢸⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⡇
⢀⡇⠀⠀⢳⣼⠀⢸⡀⢸⣿⠁⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠶⠶⣾⡇
⢸⡇⠀⡄⢸⡏⠀⠘⠃⣾⢫⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢸⡇
⠀⣧⠀⡇⣼⠁⢀⠀⣰⠇⡿⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢸⡇
⠀⠹⣦⡇⡇⢠⠏⢀⡟⢸⠇⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⠇
⠀⠀⠙⢿⣇⡀⢠⣾⣀⣼⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠂⠀⠀⠠⢸⡇
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⢠⡟
⠀⠀⢀⣠⡶⠾⠿⢧⣀⠉⢷⡀⠈⠙⢦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⠁⠀⠀⠀⠀⢀⡾⠃
⠀⣰⡿⠟⠳⠶⠶⠶⠍⠻⠄⢀⣀⣠⠞⠁⠈⠉⠙⠛⠛⠿⠛⠓⠚⠛⠛⢁⡉⠀⠀⠀⠀⠀⠀⣠⡞⠁
⢸⣯⣤⠤⠶⠶⢦⣄⠀⢠⡾⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠓⠀⠈⠁⠀⠀⠀⠀⣀⡴⠋
⣿⡋⠀⠀⠀⠀⢀⣭⣴⣏⡀⠀⡀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⣿⣇⡄⠀⢀⣠⠞⣻⠏⣩⠁⢀⡿⢶⣤⣀⣀⣀⣀⣀⣤⣀⣀⣀⣠⣄⣀⣤⡴⠟⠋⠁
⠸⣿⡀⢀⡿⠁⣰⠇⢰⡟⠀⣾⠀⠚⣽⠟⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠙⣷⣞⠀⢠⡟⠀⢸⡇⠀⢁⣤⣶⣯
⠀⠀⠈⠙⠷⠾⣷⣤⣼⡷⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣴⢋⠁⡇⢏⢢⠸⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣸⡟⢸⡇⣥⢸⢸⠀⡇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡇
⠀⠀⠀⠀⠀⡿⠁⢸⠇⡿⣨⠸⠀⠁⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢧
⠀⠀⠀⠀⢸⣧⡄⠘⠃⢹⢸⢸⠀⢰⢸⣽⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⡄
⠀⠀⠀⠀⢸⢿⡇⠀⠁⡾⠀⢸⡀⡌⢸⢸⠀⠀⣴⣿⡍⡁⠏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⡇
⠀⠀⠀⠀⢸⣾⡆⠀⠆⢣⣆⠈⠇⢛⡎⡜⢀⡞⢡⢹⠂⢹⡀⡃⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠻⣇⠀⠀⠎⢸⠠⢀⣼⡿⠁⣸⠁⡸⠀⢧⠘⣧⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⣠⣶⠿⠋⠀⠀⡏⠀⡇⡆⢸⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⢸⡇⢸⢀⡿⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠈⠉⣟
⠀⠀⠀⠀⣀⣠⣴⢤⣤⣄⡀⠀⠀⠀⠀⠀⡇⢸⡼⢀⡀⡸⢹⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⣤⡾⡏⢁⡇⡜⠀⠘⣹⢦⡀⠀⠀⠀⠹⣾⡇⠚⣠⠃⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡟
⢀⡞⣰⠇⣷⢘⡼⣡⣃⠘⣡⠆⢳⡀⠀⠀⠀⠈⠛⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠇⠀⠀⢸⠃
⣾⠀⠥⢾⢟⣥⣜⠉⡠⢎⣁⡀⠈⡇⠀⠀⠀⠀⢀⣠⡴⠶⣖⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⢀⡟
⣿⠀⣠⡗⢡⢾⠛⡆⠁⠱⣄⠀⢠⠇⠀⠀⣠⡾⠋⢭⣙⠲⠌⠂⠀⡽⠓⠒⠶⠶⣶⠤⠤⠤⠆⠚⠉⠀⠀⠀⢠⠞
⢻⣼⠟⢰⣣⢸⡀⢇⢄⢣⠈⢁⡞⠀⠀⢰⡇⠀⠀⠀⠄⠀⠴⠋⠉⠀⠀⢀⡀⠀⠀⠄⠀⠂⠈⠁⠀⠀⢀⡴⠋
⠈⠻⣄⡏⡇⠞⢡⠸⠸⢈⣠⠎⠀⠀⠀⢸⡀⠀⠀⠀⢶⡿⠒⢠⣄⡀⠀⠈⠁⢀⠀⠀⢀⡀⢀⣠⡤⠖⠉
⠀⠀⠉⠻⠷⠦⢾⡤⠖⠋⠀⠀⠀⠀⠀⠘⣇⠀⡞⢀⡎⢰⠁⡞⠘⣿⠟⠉⠛⠙⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡘⠀⢛⣀⡵⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⢲⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡒⠒⠒⠒⠒⠒⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⢹⠀⢠⠤⢼⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠈⠉⠉⠉⠉⠉⠉⠉⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⢸⣴⣶⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠐⠀⢸⠘⢟⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣴⠏⠀⢀⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴⣿⣷⡀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣾⡿⠁⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠾⣿⣿⣿⡿⠦⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡿⠁⠀⠀⠀⢸⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⢘⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣿⣿⡟⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⣀⣀⣀⣀⣀⣸⠀⠘⠒⢺⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁⢀⣀⣼⣿⣿⣃⣀⣀⡇⢸
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⠇⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠐⢶⣿⣿⣿⣷⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠹⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⡤⠤⢤⣶⣶⣶⣶⡶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⡼⠓⣛⣿⣯⡙⡍⣉⠉⡍⢍⠙⣓⠺⣝⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⣸⠿⢭⣤⣈⣏⣀⣉⣔⣲⣌⣭⡾⠤⢗⣫⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣯⣍⣙⡒⠒⠲⠶⠶⠶⠶⠖⠒⠒⣋⣉⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠏⣇⠙⢮⡉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⡟⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞⠀⢻⠀⠀⠉⠲⣄⣀⠀⠀⠀⠀⠀⢀⣠⠼⠗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢾⡁⠀⠘⡇⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠺⠿⡶⠚⠁
⠀⠀⠀⠀⠀⠀⡄⠀⢀⣤⣤⣀⠀⢠⣞⣉⣙⠺⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣀⠀⢙⡶⣫⠶⠦⣌⣳⢚⣹⢹⡟⣿⣷⠦⣤⠼⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠂⠀⠀⣯⣍⡭⠞⡥⠖⠶⢤⡼⢋⡡⢫⣷⣿⣅⡠⢬⡛⣩⠴⢭⣳⡀⠀⠀⠀⠀⣀⡼⢣⡄⢀
⠀⠀⠀⠀⠙⢦⡤⠞⣱⠚⣹⡏⣰⢋⡶⢡⠋⡏⣽⡖⠲⠭⠗⣲⡶⠏⢝⢟⢛⡛⣫⠝⡴⡢⠢⠀⠑⠂
⠀⠀⠀⠀⠀⢀⣝⡻⢥⣞⡴⢧⣹⣼⣿⣜⣦⡳⢝⠻⣏⠙⣻⡥⣀⠀⠀⠐⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠐⠂⠁⠀⠒⠆⠉⠙⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠴⢶⣦⡤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠴⠶⠶⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⣠⠴⠛⠉⠳⣖⣃⣠⣤⣯⣭⣽⣄⣀⠀⠉⠙⠳⣦⣄⡤⠶⠛⠉⠁⢀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠈⠙⠳⣄⡀
⠀⠀⠀⣴⠃⠀⣠⢖⡋⠩⠄⠀⠀⠀⠀⠀⠈⠉⢙⡷⠦⣴⣇⠀⢀⣠⡤⠖⠛⠉⠁⠀⠰⠶⠶⠶⠶⠬⠿⠷⢦⣄⠈⢻⣦⡀
⠀⠀⢰⡇⠀⣼⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⡄⢨⣭⣭⣭⣭⣭⣭⣽⡆⠀⣿⣿⣆
⠀⠀⠸⡇⠀⢻⣈⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⢶⡾⠃⠀⡟⢀⢹⡀
⠀⠀⠀⠻⣄⡀⠈⠳⡌⣧⠀⠀⠀⠀⢀⢀⣄⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢉⣉⣉⡿⠋⠀⣀⡼⠁⡘⢸⡇
⠀⠀⠀⠀⠈⢹⠀⠀⡇⣤⠀⠀⠀⢾⡉⠉⢈⡇⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠬⠭⠭⡇⠀⣾⢡⡠⠞⢁⡞
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠈⠀⠀⠀⠸⣇⣠⠟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣇⣀⣿⢰⠀⡞⠉
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡏⠁⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⢦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡼⠉⠉⡇⠀⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠒⠋⠙⠶⢤⣤⡤⠴⠋⠀⠀⠀⡇⠀⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠤⣄⠀⢀⣤⡀⠀⠀⠀⣧⡀⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⢀⡼⠀⠙⠶⠃⠀⠀⠀⡇⣽⣿⢠⡀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⣿⢠⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡷⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⡟⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⣏⣉⣉⣉⣉⣉⣀⡈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⣿⢸⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠃⢀⡇⠀⣿⢸⢀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡿⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⠡⣌⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞

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

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣒⣒⣒⣒⣒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⢠⡖⢒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⢶⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⢸⣧⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣼⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⢸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡏⡥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠤⢬⢹⡇⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣧⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣼⡇⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡇⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡏⡭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⢹⡇⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠃⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠤⠤⠭⠭⠭⠭⠭⠭⠭⠭⠤⠇

⠀⠀⠀⣠⠞⣉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⡙⢦⡀
⠀⠀⠀⣿⢸⠁⢹⠀⡇⡒⠒⠀⠀⠀⠀⠀⢸⠁⡏⠈⡇⢹⠠⠤⠄⠀⠀⠀⠀⠀⡇⢸⠁⣿⢸⡇⡒⠒
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠂⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⢨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣄⣿⢸⡇⠂
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠥⠤⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⢐⣀⡀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢨⡇⠥⠤
⠀⠀⠀⡟⢸⠀⢸⠀⡇⡁⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⠐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢰⡇⡁
⠀⠀⠀⣷⢸⠀⢸⠀⡇⣂⣀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⠨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢰⡇⣂⣀
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠆⠀⠀⠀⠀⠀⠀⢸⠀⣷⣾⡇⢸⢈⠉⠁⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠆
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠅⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢨⡇⠅
⠀⠀⠀⣿⢸⠀⢸⠀⡇⡋⠉⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠰⠒⠂⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⡋⠉
⠀⠀⠀⣿⢸⠀⢸⠀⡇⡂⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⡂
⠀⠀⠀⣿⢸⣷⣿⠀⡇⠍⠉⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢘⠒⠂⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠍⠉
⠀⠀⠀⣿⢸⣿⣿⠀⡇⠅⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠅
⠀⠀⠀⡏⢸⣿⣿⠀⡇⠒⠒⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠨⠤⠄⠀⠀⠀⠀⠀⡇⢸⣿⣿⠐⡇⠒⠒
⠀⠀⡴⢃⣼⣿⣿⣄⠳⡄⠀⠀⠀⠀⠀⢠⠞⣠⣿⣿⣧⡘⢦⠀⠀⠀⠀⠀⠀⡴⢃⣼⣿⣿⣄⠳⡄
⠀⢸⡇⣾⣿⣿⣿⣿⡆⢻⠀⠀⠀⠀⠀⡟⢰⣿⣿⣿⣿⣷⢘⡇⠀⠀⠀⠀⢸⠃⣿⣿⣿⣿⣿⡆⣿
⠀⠈⢧⠙⢿⣿⣿⠟⢡⠟⠀⠀⠀⠀⠀⠹⡌⠻⣿⣿⡿⠋⡼⠁⠀⠀⠀⠀⠈⢧⠙⢿⣿⣿⠟⢡⠏
⠀⠀⠀⠙⠶⠤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⡃
⠀⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⢸⣿
⠀⢀⢀⣐⣐⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠇⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠂⠀⠁⡄
⠰⠀⠔⠀⠸⡇⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⢨⣇⠀⠀⠀⢀⡢⠃
⠀⠈⠒⠄⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠈⢤⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠂⠀⠂⠀⠀⠀⠀⢰⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢁⠒⣄⠏⠀⡌⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⠆⠐⢲⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⡈⠙⠀⠶⢁⠀⠀⠁⢱⠄⠀⠀⠀⠦⠤⠄⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⣁⣀⣀⠀⠀⠀⠀⠀⠠⠀⠄⣨⠐⠡⠎⠀⠀⠀⠂⠀⣄⢀⡀⠀⠀⠉⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠉⠛⠋⠓⠻⠤⡤⠤⠴⠺⢴⣞⠉⠉⠑⠒⢁⠀⠐⠈⠉⠦⣀⠀⠀⠠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠲⢉⠁⠀⠀⠈⠀⠀⠐⣢⠁⠀⠀⠀⢆⠏⠉⠙⠀⠀⠀⠈⠐⠒⠐⠒
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⠋⠀⢁⠖⠒⠓⠲⠂⠀⠀⢀⠀⠀⣀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠉⠃⠀⡀⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠔⠀⡠⠄⠒⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⣀⠀⠀⠀⠀⠀⠤⠀⠀⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⣸⣟⠋⢛⣄⣿⣿⣶⣶⣄⡀⣠⣿⣿⣦⡤⢦⢤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⣿⣶⠆⣿⣿⣿⠿⠫⣿⣿⣭⣿⣿⠅⣿⣿⣟⣿⣄⣀⣿⢾⣿⣿⣿⣧⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠘⠁⠙⣿⣟⣻⠀⣿⡿⢺⣯⣿⡿⢬⠙⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣃⢈⣴⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠃⠙⠂⠀⢴⣯⡁⣿⣿⠀⣿⡯⢸⣻⣿⠩⣾⢰⣿⣿⠀⣿⣿⣿⡿⣿⣿⢿⣿⣰⣾⣿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⣰⠃⠻⡛⠀⠀⢀⣃⢛⣾⣧⣟⣵⠀⣾⡇⢸⡊⣍⠘⣫⣬⣭⣽⠀⣽⣯⣷⣾⣿⣷⠸⣿⣿⣿⣿⣟⡛⣛⡿⡀
⠀⠀⠀⢀⡔⠉⠛⢲⣿⡶⣦⣿⣿⣸⠃⣿⣓⣿⡅⢿⠇⢸⣗⡌⠏⣻⣿⣿⣿⠀⣽⣿⣿⣿⣿⣯⠸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⣾⣴⢶⣶⣿⣿⠫⢸⣿⢿⡭⡀⣿⣾⣿⣀⣸⠀⢨⣽⢃⣯⣿⢿⣿⣿⠘⣿⣿⣿⣿⣟⣿⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢾⣷⣭⣾⣿⣿⣯⣶⣹⣿⣏⣤⣶⢬⣭⣿⠉⣓⣢⣬⣭⣤⣍⣭⡁⣾⠿⢠⣿⣿⢵⡞⣿⡧⠼⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠘⣿⡏⢶⡟⣿⣿⢿⣿⣿⢿⣻⡿⣿⣟⣁⣾⡿⣿⡿⣛⣹⡿⣿⣃⣟⢴⣬⡤⢱⣿⣿⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⢿⠿⠿⠛⠿⣟⣿⣾⢻⣶⣿⣯⣻⢿⣉⣱⣚⣋⣵⣧⣛⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⠿⣿⣟⢯⠌⣑⠞⠒⣩⣠⣼⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⡿⢛⣽⡾⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⣆⣶⣼⣶⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣻⢻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⢿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⣿⠿⠿⠿⠟⠛⠉⠁⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣋⡉⠉⠉⠉⣉⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣯⡟⢻⣶⠏⠀⠻⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠁⠀⣀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠖⢻⣧⠘⢦⠀⠀⢀⣴⣿⠙⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⣿⠐⢧⡙⣦⣤⣤⡴⠟⣹⠏⠀⣷⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⠀⣿⠀⠀⣧⣘⣿⠤⣠⡾⠋⢠⡴⠿⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⣼⠧⠾⠦⡄⠈⠉⢻⡷⠯⠶⠾⠟⠀⠀⢀⣤⠄⠀⠀⢹⡇
⠀⠀⠀⠀⢀⡾⠋⠀⢠⡞⠁⠀⠐⢚⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠘⣧
⠀⠀⠀⢠⡟⠀⢠⣴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣿⡤⢶⡟⠁⠀⠀⣠⣴⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⢀⡿⠁⠀⢸⣷⣀⣴⣿⣿⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠆
⠀⠘⡇⠀⠀⠸⣿⡿⢹⠏⢹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢣⡀⠀⠀⣀⠔⠁⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠸⣿⠀⢿⣄⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡟⠛⠂⣠⣤⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⠾⠗⠚⠉⠁⠀⠈⣿⡄⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⠀⠀⠀⠀⢀⣤⣴⣿⡇⣀⣤⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣀⣴⠟⠉⠉⠛⠿⠿⠿⠟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣴⣿⠟⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠉⣿⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⠀⠀⣼⠃⠹⣇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⠀⢹⡆⢰⡏⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⠀⢸⣇⡿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⠸⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣾⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣯⠞⣹⣿⡇⠀⠀⠀⠀⠀⠀⡴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢰⡏⣰⣿⣿⣷⣶⣶⣶⣶⣶⡞⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠇⠀⠀⠀⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⡿⠛⢛⣿⣆⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⡶⠟⢻⣿⣿⣿⣟⣛⣛⠛⠛⣿⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠇⠀⢺⣿⣿⡀⠀⠀⠀⠀⠀⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠶⣶⣾⣿⣿⡷⣄⠀⠀⠀⠀⢳⣶⣶⣶⣤⣤⣤⣤⣤⣉⣉⣉⣿⣧⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣟⣮⠳⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⣿⡇⢹⡟⣿⢹⣿⣿⠸⣧⢹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣼⣿⡇⢸⡇⣿⡘⣿⣿⡇⣿⣾⣿⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇⢸⣧⢸⡇⠿⢹⡇⣿⢻⣿⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⡿⢿⣿
⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠇⢸⡿⢸⣿⢰⠺⡇⣿⢸⡏⠀⢹⣿⣿⡟⠋⠉⠙⠛⠛⠛⠛⠿⣿⣿⣆⠾⠛⠁⠀⠈⠙
⠻⠿⠿⠿⠿⠿⠿⠛⠻⠿⢿⡟⣠⣼⠇⣸⣿⣾⡇⣇⣻⡘⣧⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⣥⣿⣾⡿⠿⠿⢿⣿⣿⣷⣿⡧
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⣴⣿⣿⣿⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣾⣿
⣿⣿⡇⣀⠁⠀⠀⠀⠀⠀⠀⢀⣸⢿⣾⣿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢹⣿⣿⣘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿
⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠙⠟⣛⡉⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣤⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡻
⢁⡄⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡟⢿⣿⣷⠀⣤⠀⠘⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠆⠀⠂⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣬⡄
⠀⡀⠀⠀⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢰⡆⠈⠙⠃⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⡄
⠄⢀⣄⣠⠍⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢤⣾⣿⣿⡗⠄⠀⠀⠀⠀⠀⠀⠀⡘⢿⢿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⡀
⣿⣿⣿⣿⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⢉⣽⠿⢿⣮⣁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠛⢡⣾
⣿⣿⡿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡼⣿⣿⠿⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠉⠛⡁⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠻⠗⠘⢿
⡉⠩⠚⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠻⠂⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡄⠈⡅⠀⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⢀⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣉⡻⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⣶⣶⣦⢉⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢁⣴⣶⣬⡋
⠀⠀⠀⠀⠀⠡⡄⢿⣿⣷⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣸⣿⣿⣿⣿⣿⡆⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣿⣿⣿⣿⣷⠛
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⠇⠒⠀⠀⠀⠀⠀⠀⠠⠰⠾⠹⢿⣿⣿⡿⣿⠇⠖⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠬⠻⠟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠙⣻⠟⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠠⠆⠰⠀⠀⠀⠀⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⢠⣴⣿⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠺⢶⡆
⠀⠘⢻⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠶⢀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣢⣼⣧⡄
⠀⠀⠆⠽⠙⠿⢿⣿⣧⡠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢶⠆⣠⣤⣥⣴⣷⣾⡄⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣽⣿⣿⣿⠃⠁
⠀⠀⠀⠀⠀⠐⠛⡈⣹⠿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⣿⣿⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠙⡿⠿⠿⢫⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠛⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⡅
⠆⠀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⣃⡀⠶⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⢐⣂⠐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇

⠀⠀⠀⠀⢀⣠⣤⡴⠶⣶⠟⠛⠛⣻⣿⣟⡛⠳⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⢤⣤⣀
⠀⠀⣠⠞⠋⠁⠈⠛⣛⣁⣠⣤⣼⣧⣤⣬⣷⣀⣀⠀⠀⠈⠉⠛⣶⣤⣤⠴⠞⠋⠉⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠈⠙⠳⣦⡀
⢀⡞⠁⠀⢀⡴⠞⠛⢉⣭⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣷⠶⣤⣰⡏⠀⠀⠀⢀⣠⡴⠞⠛⠉⠉⠀⠀⣤⣤⣤⠤⡤⢬⣭⣽⣷⣶⣄⠀⠈⠻⣦⡀
⣼⠃⠀⣰⠏⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠛⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠛⣷⡀⠀⢿⣿⣦⡀
⣿⠀⠀⣿⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⣘⣛⣛⣛⣛⣛⣛⣛⣛⣻⡇⠀⢸⡏⠏⣷
⢿⡀⠀⠹⢦⣄⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣉⣉⣉⣉⣉⡽⠟⠀⠀⣼⠀⣠⢸
⠈⠳⣄⡀⠀⠈⢷⡈⣷⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠈⢛⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⣉⣉⣉⣽⠋⠀⠀⣠⡾⠃⢀⠉⢸
⠀⠀⠈⢹⡇⠀⢸⡇⣤⠀⠀⠀⠀⣾⠋⠙⠁⢘⡇⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⣭⣭⣭⣿⠀⠀⣿⢩⡀⡴⠋⢀⡾
⠀⠀⠀⢸⡇⠀⢸⠇⠛⠀⠀⠀⠀⢸⡇⠀⢀⡟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣿⠀⠀⣿⢠⡍⣠⠶⠋
⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⠶⠚⣿⢸⡇⣿
⠀⠀⠀⢸⡇⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣿⠀⠀⣿⢸⡇⣿
⠀⠀⠀⢸⡇⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⣦⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣿⠀⠀⣧⢸⡇⣿
⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠞⠋⠙⢦⣤⣀⣀⣠⡴⠞⠁⠀⠀⠀⣿⠀⠀⣿⢸⡇⣿
⠀⠀⠀⢸⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢸⡇⣿
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠉⠉⢹⡆⠀⣾⣛⣷⠀⠀⠀⠀⣿⢻⡄⣿⠘⠃⣿
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⣠⠾⠃⠀⠈⠉⠁⠀⠀⠀⠀⣿⣰⠇⣿⠰⠆⣿
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⢠⠀⣿
⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣷⢸⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣧⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⠀⡇⢸⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣇⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⠀⠀⡇⢸⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣿⠀⠀⡇⢸⠀⣿
⠀⠀⠀⢸⡇⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⡇⢸⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣏⣉⣉⣉⣉⣉⣉⣀⣀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⡇⢸⠀⣿
⠀⠀⠀⢸⡇⠀⢸⣯⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡤⠤⡤⠼⠃⠀⣿⠀⠀⡇⢸⠀⣿
⠀⠀⠀⢸⡇⠀⠘⠷⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⠴⠏⠀⠀⡇⢸⠀⣿
⠀⠀⠀⠘⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠸⠀⣿
⠀⠀⠀⠀⠈⠻⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⣩⣍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅⠸⠦⠀⣿
⣀⣀⣀⣀⣀⣀⣈⣙⣳⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣴⣞⣃⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣯⠈⠉⠉⠙⠓⠒⠶⠦⠤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣍⣉⣉⠛⠒⠲⠶⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠷⠈⠙⠒⠦⢤⣄⡀⠀⠀⠀⣿⢸⠀⠀⠉⠉⠉⠛⠒⠒⠶⠤⢤⣬⣉⣉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⠲⠦⠤⠤⠤⠄⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠀⣿
⠀⠀⣀⡤⠖⠚⠛⠛⠒⠦⣄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸⠁⣿
⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⢀⡴⢺⠀⣿
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⢸⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠴⢤⠘⠟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⢸⠀⣿
⢳⡀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠳⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣀⡴⠋⠀⣠⠞⠉⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠳⢄⡀⠀⠀⠀⠀⠀⢀⡠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠞⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⢉⡗⠒⠒⠚⠋⠹⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⣠⠎⠀⠀⢠⠤⢿⡼⠯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⢰⠃⠀⠀⠀⠀⡀⠀⠀⠈⢳⡀⠀⡴⠃⢠⢇⡤⠞⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⡞⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠙⡼⠑⢃⡏⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢰⠃⠀⢸⠁⣀⡤⠖⠛⣿⢻⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠆⡾⠤⠆⡿⠈⠙⠓⠒⠲⠿⠤⢤⣌⣉⣉⡙⠓⠒⠲⠦⠤⢤⣄⣀⣸⠀⣿
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠶⠤⠤⣤⣉⣀⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠷⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⣆
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇

⠀⠀⠀⠀⢀⣠⣴⣤⣤⣄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣠⢴⡶⠶⣤⢄
⠀⠀⠀⢺⣿⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⣼⢻⠘⣿⢃⠠⠊⠷⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠐⡇⣆⡀⠈⠁⢤⠁⢺⠇
⠀⠀⠀⠀⠈⠻⠟⠛⠛⠻⠋⠀⠀⠀⠀⢛⣡⠀⢀⠀⢀⠂⣀⠀⠀⠀⠀⠀⠠⣤⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠳⠒⠀⠂⣼⠖⠉⠀⠀⠀⠀⢀⠄⠀⠀⠈⣗
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢱⡆⠀⡦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠇⠰⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⠲⠰⠞⠀⠀⠀⡇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠴⠆⢤⡀⠀⢶⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢰⢈⢶⡅⠀⠀⠀⢀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⢘⡃⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠓⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣄⠀⠀⠠⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⡄⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈⠃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⢠⡈⠢⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⠶⠀⠇⠀⠁⠂⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠃⠀⠀⡇⠀⠀⠛⠀⠀⠀⣠⠶⠆
⠀⠀⠀⠰⠃⠐⠉⠦⠀⠀⠀⠀⠀⠑⠄⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠆⠀⠀⠀⠁⠔
⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⡡⠀⠀⠀⠀⠜⠀⠀⠀⠀⣢⠀⢀
⠀⠀⠀⠀⠠⠀⠀⢁⣤⠁⠈⠀⣀⡀⢀⠛⠀⠀⠀⠀⠐⠄⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡘⠀⣀⠀⠀⢠⡤⠀⠣⡀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⠈⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢫⣩⡿⠟⠿⠿⠿⠟⠻⠿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⡿⠿⠿⠿⠿⢿⣿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠛⠉⠉⢀⣀⣀⣀⣀⣤⣤⣀⣀⣀⣀⡀⠉⠉⠛⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠁⣀⣠⣴⣶⣾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⣷⣶⣦⣀⣀⠈⠙⠻⣿⣦⣄
⠀⠀⠀⣠⣾⡿⠋⠀⣠⣴⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣦⣄⠀⠙⢿⣷⣄
⠀⢀⣾⡿⠋⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠻⣿⣦⡀⠙⢿⣷⡀
⢠⣿⡟⠀⣠⣿⠟⠁⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⠿⠿⠀⠀⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⢻⣿
⣼⡿⠀⢰⣿⠏⠀⢠⣤⠀⠀⠈⠉⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⢠⣤⠀⠀⠀⠀⠹⣿⡆⠀⢿
⣿⣧⣀⣾⣿⠀⠀⠈⠋⠀⠀⠀⠀⢀⣿⡇⠀⠙⠋⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⣤⣄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣿⣷⣀⣼
⠘⢿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢛⣿⡿
⠀⠈⢿⣷⠀⠀⠸⠿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⠿⠿⣿⣶⣤⡀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣾⡿⠁
⠀⠀⢸⣿⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⢀⣼⣿⠛⠁⣀⣠⣄⣀⠈⠻⣿⣦⠀⠈⢿⣷⣄⡀⠀⠀⢀⣤⣿⠟⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠁⠀⢾⡿⠛⠛⢿⣷⡀⠘⣿⣇⠀⠀⠙⠻⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣄⠀⠀⢰⣿⠆⠀⢠⣤⡄⠀⠈⠛⠃⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⡀⠀⠀⠀⠀⠀⣀⣽⣿⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢾⣶⠀⠀⠀⣤⣶⡿⠟⠛⠿⢿⣿⣦⡀⠀⠛⠛⠁⢀⣤⡀⠀⠀⣤⡄⠀⠀⠀⠀⢠⣤⡀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣀⠀⠀⠈⠻⢿⣶⣶⣶⣾⡿⠛⠁⠀⠀⠛⠃⠀⠀⠀⠀⠈⠛⠁⠀⣿⡇
⠀⠀⢸⣿⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⣶⣶⠀⠀⠀⠀⣿⡇
⠀⠀⣾⣿⠀⠀⠀⠀⠀⣰⣾⠿⣷⣆⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⢠⣶⠀⢀⣾⡿⠛⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⡟⠀⠀⠀⠀⠀⢿⣷⣤⣿⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⢿⣧⣤⣴⣿⠇⠀⠀⠀⢰⣶⡆⠀⢻⣿
⠀⠀⣿⡇⠀⢠⣤⡄⠀⠀⠙⠛⠋⢀⡀⠀⠀⢰⣿⠇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠈⠉⠁⠀⠀⠀⠀⠀⠿⠟⠀⠀⢿⣿⡀⠀⠀⠀⢠⣿⡇⠀⠀⣶⡆⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣦⣤⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⡆⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠉⠉⠉⠀⣀⡀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⢰⣿⡇
⠀⠈⣿⣧⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣼⣿
⠀⠀⠘⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠃
⣶⣶⡶⠶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⠀⠀⠀⠀⡇⢰⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠱⠗⡠⠊
⠀⠀⠀⠀⠀⠀⣰⠳⢦⣨⠇⠀⠀⠀⢳⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢀⡏⠀⠀⢧⣀⠀⠀⠀⠘⡆⠀⠀⣇⢀⡞⠉⠉⠙⠓⠒⠶⢦⣀⡴⠊⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠉⢦⠀⠀⢹⡀⠀⢹⡈⢳⠀⣠⢤⣤⣄⣀⡸⠊⠀⣀⡴⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⡀⠈⡆⠀⣀⠷⠚⠺⣇⡏⢠⡟⠾⣄⡴⠋⢀⣴⡚⢧⡀
⠀⠀⠀⠀⠀⢸⡀⠀⡎⠀⠀⠈⢂⠹⡋⠁⢀⠴⠒⠋⠀⠸⣧⡾⢋⣀⣜⡧⠤⠽⠔⠛⠲⡄
⠀⠀⠀⠀⠀⠀⢧⠀⢣⠀⠀⠀⠈⡄⢳⠀⡎⡤⠀⠦⠤⣀⠈⢱⠉⠀⢀⣀⣀⣀⣀⣀⠀⢻⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⡆⠈⢆⠀⠀⠀⡀⠀⢿⢰⠁⠀⠀⠀⢀⠀⢸⠀⡎⠁⠀⠀⠀⠀⠈⡆⢸⠋⠁⠀⠈⠙⠲⣄
⠀⠀⠀⠀⢀⡔⠒⠻⡀⠈⠳⠤⠴⠃⠀⠈⠇⠀⠀⠀⠀⢸⠀⡽⠀⢰⠀⠀⠀⠀⠀⠀⢣⠈⡆⠀⢀⣄⠀⢠⠘⡄
⠀⠀⠀⠀⠀⢳⡄⠸⠥⣀⣀⣀⣀⣀⣀⠀⠈⣆⠀⠀⠀⡇⢰⢦⡆⢸⠀⠀⠀⠀⠀⠀⠘⡄⣧⠜⡅⠈⡇⢸⠀⢹⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠠⠤⢀⣀⣀⣀⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⠔⠁⠀⡆⠀⢧
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⢢⠀⠀⣇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡟⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠳⡤⠇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡀
⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢠⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣬⠐⠀⠀⠀⠠⠤⠂⠀⠀⠀⢠⠖⡆⢹⡟
⠀⠀⠀⠀⢀⠀⠁⡘⠈⠀⠂⣀⠐⠁⠀⠀⠀⠀⠣⠋⠀⠀⠉⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠙⣾⣷⣀⡀⠈
⠀⠈⠀⠖⠂⠙⠒⠓⣷⣶⡏⠁⠀⠀⠀⠀⠶⠂⠀⡀⠀⠀⣠⡅⠀⠀⠀⠂⠀⢀⣀⠄⠀⠀⢁⣾⡏⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢘⣿⢿⠂⢸⣾⢋⣄⠹⠇
⠀⠀⣤⠀⠁⠀⠀⠚⣿⣿⠳⡾⠃⠀⠀⠀⡀⠀⠸⣦⡗⠀⠀⢀⣀⣠⣤⣖⣿⡽⣼⣆⣣⣀⡾⣛⠀⠀⠀⠀⣤⠀⠀⠳⠀⠀⠀⠸⣿⡔⠀⠈⠻⣿⣁⡄
⠀⠀⢀⠀⠘⠀⠀⢰⡟⣃⠀⠂⠰⠆⠀⢰⣿⠆⠀⡖⣌⢟⡯⢋⣉⣹⣾⣿⣿⢿⣿⣾⣿⣿⣷⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡤⠈⠠⢄⠀⠀⠉⠉⠁
⠀⠀⠀⡀⠀⢀⡤⠚⢉⠝⠀⢀⣴⣄⠀⠀⠉⣠⣼⡶⣿⣆⢻⣾⣿⢿⣿⣿⣿⣽⣿⣿⡞⡅⢿⡭⠀⠀⠀⢠⣿⡦⠀⠀⠀⠀⠠⡀⠀⠀⠀⠃⠀⠀⠀⠴⣢
⠀⠀⠡⡿⡧⠉⠀⠀⠊⠀⠠⠼⠿⠏⠑⠀⠀⢃⢸⠑⠈⠝⣦⣭⣿⣾⣿⣿⣿⡿⣿⣿⣷⣷⣿⣿⡄⠀⠀⠀⠘⠀⠀⠀⢀⡀⠐⣁⡴⠀⣜⠀⠀⠀⠀⠙⡇
⠀⠀⠀⣴⣱⣦⣤⠄⠤⠀⢀⣀⠀⠀⠀⠠⠤⠉⠳⠤⠄⢿⣟⡻⣿⣷⣿⣿⣿⡀⣨⣿⣿⣿⣿⡿⠷⠄⠀⠀⢀⣴⣤⠀⠀⢹⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠠⠈⠁⠁⠀⠀⠀⠀⠚⠹⠀⠀⠀⠀⠠⠚⠓⠓⣿⣿⣿⣽⡟⠨⢾⣽⣻⣿⣿⣿⣿⣿⡟⢃⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢷⣶⣤⠔⣑⣖⠲⠀⠀⠿⡅
⠀⠀⠀⠀⠈⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⢲⣄⣠⣾⣟⣳⣿⣿⣿⠇⠛⣿⣥⠀⢸⠿⡿⣷⣉⣞⣤⠂⠀⠊⠀⠀⠀⠀⣀⠀⢀⡀⠉⠁⠀⠀⠀⠄⢀⠀⠄⣤
⠀⠀⢀⣿⣿⣷⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⣽⣹⡟⢻⣏⠀⠸⣾⠀⣽⣥⣿⣿⡇⢹⣟⣻⠙⠏⠑⢤⣀⣠⡶⠀⠀⢈⣰⣿⢷⡄⡇⠀⢢⢠⠆⠀⢐⢰⠈
⠀⣰⣿⣿⣿⣿⣿⡆⠀⠠⡾⠛⠈⡆⠀⠀⠀⠀⠩⢛⣿⣾⠿⢠⣠⣟⣡⣵⣶⣿⣿⣷⣾⣿⡟⠀⠀⠀⠀⠉⠹⠇⠀⠀⢰⣻⣿⣾⡃⠇⠀⠘⠘⠅⢠⢸
⠀⠈⠛⠛⡟⠛⠋⠁⠀⢹⢷⡔⠒⠍⠀⠀⠀⠀⠀⠀⢻⣲⣾⣿⡿⣿⣯⣼⢾⣿⣿⣿⡿⡟⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⢨⡿⣬⣟⣁⠀⠀⢀⠠
⠀⠀⠀⢀⣅⣀⠀⠀⠀⡿⢬⣄⠀⠁⠀⠀⠈⠀⠀⠀⠀⢸⡿⢿⣿⠻⢡⠃⡾⣿⣿⣿⡾⠓⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⡇⠐⠀⠠⠀⠂
⠀⢀⠞⣙⡁⠉⠁⡀⠈⠁⠨⡷⢣⠀⠀⣴⣆⠀⠀⠀⠀⠀⣳⣘⣿⣤⣿⡾⠠⣟⡇⣿⣀⢹⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⠃⠠⣤⡄⠠⠀⢠⠄
⠀⢸⣷⡶⢮⡀⡈⠃⠀⠀⠀⠁⠀⠀⠀⠩⠉⠀⠀⠀⠀⠠⠿⠙⠻⠏⠙⠳⡦⢿⣿⣟⠛⠉⠉⠉⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠙⠓⠚⠛⠃⠀⠀⠀⠀⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⣀⠀⠘⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣆⣀⡀
⠀⠀⠀⢴⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠟⠋⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣏⣩⣽⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣷⣴⣶⣦⣄⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣮⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⢀⣤⣤⣶⣄⠙⠿⣿⣿⡿⠿⣁⣤⣦⣤⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡀⠀⢀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠉⠿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⠀⣸⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠘⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠛⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣶⠷⣿⡏⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣴⣶⣶⠿⠿⠿⠋⠀⠀⣿⡆⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣠⣴⣦⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⣀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠀⢀⣿⠇⢸⡟⢻⠀⠀⠀⣉⣴⣿⣿⡿⠀⠀⠀⠀⠀⠁⠀⠀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠻⡟⠀⠘⣡⡶⠶⠾⠟⠛⢿⣿⣿⣄⣀⠀⠀⢀⣀⣠⡄⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢇⡀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⡿⠃⠀⢀⣾⣿⡿⠉⢢
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣨⣿⡿⠁⠀⢈⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⢹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⢹⣟
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⣿⠃⠀⡀⣾⣧
⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣿⠀⠀⢳⢘⣿⡀
⠀⠀⠀⠿⣿⡟⠁⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⠏⠀⠀⠀⠻⠿⡇
⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⡶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠾⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣿⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠠⠤⠤⠴⠶⠞⠛⠉⠁⠀⠀⣿⠀⢀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠤⠴⠖⠛⠋⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡗⠒⠀⠀⠀⠀⢰⡋⠉⡆⠀⣠⠞⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠚⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡏⠉⠉⠉
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⣇⣀⣀⣀⣀
⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡧⠤⠤⠤
⠀⠃⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢸⠀⠀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⣴⣶⣿⣷⣶⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⣿⡟⠉⢻⣿⣿⣿⣷⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⣿⣿⣶⣿⣿⣿⣿⣿⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⢿⣿⣿⣿⣿⣿⣿⠟⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⠛⠿⠟⠛⠁⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣤⡤⠔⠚⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⣿⣽⣉⢵⣿⡇⠴⢪⡉⣿⣗⣤⠀⠀⠀⠀⠂⠤⣤⠂⡀⠠⡀⠀⠀⠀⠀⠘⡆⠀⠐⢦
⠀⠀⠀⠀⠀⠀⣖⣮⣷⣿⣧⡈⠹⣿⣯⣴⠿⠃⣾⣯⣼⣶⣿⣿⡇⠀⠀⢄⠀⠀⠛⢆⠙⣄⡙⠀⠀⠀⠀⠀⢃⡘⢀⠀⣷⡞⠉⠳⠂⠀⡄
⠀⠀⠀⠀⠀⠀⣽⣿⢿⣿⣿⠇⠀⠀⣛⠀⠀⠀⠙⠛⢻⠿⠟⠋⠀⢀⡄⠚⠓⠺⣄⠘⣦⣿⡤⠒⠂⡄⠀⠀⣨⣷⠎⣰⢹⠃⠀⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠻⣿⠀⠀⢀⡟⠀⠀⠀⠀⠀⣨⠆⠀⠀⠀⠈⠀⠀⢸⡐⠈⡛⣿⣿⠻⡄⠀⠆⠀⠀⡇⣿⠀⣅⣸⠔⠋⢀⠀⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡅⠀⠀⣱⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⢀⠔⠺⠧⠹⠑⣼⣿⠀⠀⠀⢠⠚⠲⣷⣿⠦⣿⣍⣋⣁⣈⣀⠀⠚
⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠿⠃⠀⠀⠀⡇⠀⠀⠁⢹⢰⣏⣿⠃⠀⠀⠈⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⣀⢀⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠸⣬⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣠⣤⣄⣠⣶⡀⠠⣼⣷⣛⣿⣿⡆⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠴⢼⡿⣿⣟⣿⣯⡮⢫⣾⣾⣛⣿⣩⣮⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣷⣿⠿⣿⣿⣿⣿⠿⠻⠟⠿⠻⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡁⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣠⣄⠀⠻⣶⠟⠀⠀⠀⡇⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠘⠽⠿⠓⣄⠀⠀⠀⠀⠀⡇⠀⠑⡾⠃⠀⠀⣠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠆⠀⠀⢰⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⣰⠁⠀⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠑⢀⠇⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢏⣽⠇⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠾⡆⢀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠻⠍⠡⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣶⡖⠀⠀⢰⣿⣷⣾⡗⠀⠀⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⡟⠛⠉⠀⠀⠐⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠚⠻⠂⠀⠀⢸⣿⣿⡇⠀⠀⣀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠗⠀⠀⢰⡶⠀⠀⠀⢴⣿⣿⠀⠀⣿⣿⠇⠀⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⢸⠀⡐⠀⠀⠀⢺⣷⠀⠀⠀⠀⠘⠿⠀⠀⠀⠸⠇⠀⠀⠀⠐⢀⠀⠀⣠⣏⠁⠀⠀⢞⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢠⠀⢸⠀⠃⠀⠀⠀⢈⣷⡀⠀⠀⠀⢸⣿⠆⠀⠀⡎⠇⠀⠀⠀⢇⡹⠀⠀⣉⠀⠀⠀⣠⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⢸⢸⠀⠀⠀⠀⠈⢛⠃⠀⠀⠀⢸⣿⠀⠀⠀⢸⠀⠀⠀⠀⣾⠀⠀⢸⠟⠀⠀⣸⣯⠏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⡆⠈⣸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠃⠀⢀⣴⠍⠁⠐⠦⠀⡟⢀⣠⠎⠀⠀⢼⠃⠀⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠈⣧⢡⢀⣿⠀⠀⠀⠀⠀⢰⣇⢀⣠⡴⠛⠒⢤⡾⣁⣀⣀⣠⡤⢠⣏⠉⠉⠘⢢⡄⠀⠀⠘⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⣿⠀⠐⠂⠒⠂⠉⠉⠀⠀⠐⢦⡀⠾⠁⠈⠀⠀⠀⠀⠀⠁⠛⠊⠉⢱⣿⠛⠷⢆⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⢀⣠⣤⣀⠀⠈⠳⣄⣀⠀⠀⠀⠐⢿⣛⣳⠀⢀⡾⠋⢃⠀⠀⠈⠐⠃⠰
⠀⠀⠀⠀⠀⢀⠤⠤⠄⠐⠂⠀⠒⠒⠛⠋⠁⠉⠉⠓⠦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⠀⠀⠀⠈⠁⠀⠘⠀⠀⠀⠐⠶⠒⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠰⠀⠀⠀⠀⠈⠁⠤⠤⠤⠐⠛⠛⠛⠂⠀⣤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⣿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡋⠋⠛⠛⠛⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣿⣿⣿⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⣿⣿⣿⡿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⡉
⠀⠀⠀⠀⠀⢠⣾⣷⣆⠸⡆⠀⡿⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⣿⣿⡏⠀⣷⠀⡇⣀⣤⣤⣶⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣶⣧⡘⣧⡀
⠀⠀⠀⠀⠀⠸⣻⣿⡟⢀⡇⠀⣿⢻⣿⣟⣻⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡶
⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠀⠀⣿⣸⣗⣷⣹⣽⣿⣿⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡘⣿⡄⠀⠀⠀⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠙⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡘⣷⠀⠀⠀⠀⠀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡘⡇⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠓⠒⠶⠶⠶⢤⣤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠤⠤⠶⠶⠶⠟⠛⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⣾⠂⠀⠈⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⡏⠀⠀⠀⢃⠀⠀⠁
⠀⠀⢀⣀⣤⣤⣤⣤⣴⣾⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠸⠟⠿⠟⣿⣿⠿⠿⠿⠟⠻⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠈⠛⠿⢿⠿⠿⣿⠟⠛⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠠⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢁⠀⠛⠋⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⡀⠀⠀⠀⠀⣿⣟⠀⠐
⠀⠀⠀⠀⠠⣤⣤⣼⣶⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡄⠀⠀⠀⠀⠀⣒⣶⠶⠶⢶⣖⠀⢠⠀⠀⢸⡏⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠴⠶⠾⣿⣿⣿⣿⣶⣤⣤⣶⣴⣧⣤⠾⠯⠍⠀⠀⠀⠀⠀⠀⠀⡪⠭⠭⠭⠭⠵⠀⠈⢧⡀⠀⠻⠶⠂⠀⠀⠹⣿⣄⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣽⣯⡭⠭⠳⠶⠞⠓⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⡤⠤⠄⠀⢤⡀⣄⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⢹⣿⡂⠀⠉
⠀⠀⠀⠀⠀⠨⢋⢻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠰⡿⣿⣿⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⣰⣻⠏⣧⣄⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠲⢂⡀
⠀⠀⠀⠀⠀⠀⠳⠏⣸⣿⣇⣔⡈⣒⠘⠃⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⢴⠋⣨⠻⡍⠧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣢⣽⠟
⠀⠀⠀⠀⠀⠀⠀⣼⠰⠂⠁⠷⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⣏⣈⠉⣀
⠀⠀⠀⠀⠀⠀⢀⠽⣗⣊⣨⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣿⣿⣟⠀⠀⢀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠂⠀⠀⢀⣁⣸⡠⢘⡀⠀⡀⠀⣤⠀⠀⠈⠀⠀⠹⣿⣻⣦⠶⠖⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢾⢃⢮⣅⣫⣅⡏⢀⠀⠉⠃⡀⠀⠀⠀⠀⢙⡿⣿⣿⣤⣶⣤⣤⣤⣤⣤⣆⣒⢒⣯⣿⣾⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣡⠀⡸⢨⣿⣈⡸⢏⡷⣿⣏⡄⢀⢹⣥⣴⡄⢶⡿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠳⢾⣭⣸⣶⣋⡜⢠⣷⣈⣅⣿⣿⣿⣷⣯⢺⣷⣩⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠾⠟⣻⠛⡟⠻⣻⢼⣟⡷⢏⣿⣯⣿⡿⡿⠾⠿⠿⠛⠋⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠠⠀⠀⠈⠘⠓⠸⠈⠀⠊⠘⠀⠆




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐

⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⣤⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢴⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠉⠉⠉⠉⠙⡇⡇⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⠀⢸⡇
⠀⢀⣠⣤⣤⣛⣀⢀⣀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⡇
⠀⠀⣀⣀⣀⡶⠉⠉⠀⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⢀⡤⡇⡇⠀⠀⠀⠀⣿⠛⢻⡟⢻⡇⠀⣠⡄⠀⠀⢀⡀⠀⠀⠀⡟⠛⡟⠛⢻⡇
⠀⠈⠁⠀⠉⠭⠙⠋⠁⠀⠀⠀⠀⢸⡇⡇⠀⢀⡴⢋⡤⡇⡇⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⠘⣿⣿⠂⠀⠀⠹⣦⠀⠀⡇⠀⡇⠀⢸⡇
⠀⠐⠉⠉⠉⣛⠒⠒⠂⠀⠀⠀⠀⢸⡇⣇⡴⠋⡴⠋⠀⡇⡇⠀⠀⠀⠀⣿⣀⣸⣧⠼⠧⠤⠸⣿⡠⠤⠤⠤⢹⣷⠠⠧⢤⣧⣀⣸⡇
⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣼⣧⣭⣭⣭⣭⣭⣭⣥⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠘⢿⠟

⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⠆⠈⢉⣭⣍⡛⢦⡀
⠀⠰⢿⡷⠲⠶⠶⠶⠶⠶⢤⣄⠀⠀⠻⠶⠶⠶⠶⠶⠃⣿⠀⠀⠀⠀⠀⠀⠚⠛⠒⠂⢠⡏⣈⠀⠀⢿⣿⣿⡿⠇⢻⡄⠚⠛⠒⠒
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠴⠶⠤⠄⣼⠸⣿⣿⣶⣿⣿⣠⣀⠀⠈⡇⠴⠶⠤⠤
⠀⠀⠀⠀⢸⢸⡟⢻⡇⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠛⢠⣀⠀⠀⠀⠀⣠⣤⣀⣀⠸⣆⠙⠿⠟⠀⣿⣿⣿⢠⣰⠇⣠⣤⣀⣀
⠀⠀⠀⠀⢸⢸⡇⢸⡇⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠻⠟⠃⠀⠀⠀⠀⠀⠈⠀⠀⠙⢦⣄⠀⠀⠻⠟⠁⠿⠿⠂⠀⠀⠈⠁
⠀⠀⠀⠀⢸⣘⣛⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⢿⣯⢿⣿⡿⣿⣏⣿⣷⢿⣿⢹⣿⣽⣿⣿⣿⣹⡿⣽⡿⣫⣾⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣟⠿⣿⣿⣷⣝⢿⣿⡻⣿⣿⣿⣞⣿⣷⢿⣿⠿⣿⣸⣿⣉⣯⣬⣭⣽⣷⣯⣼⣟⡵⣟⣫⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣯⣛⣿⣯⣽⣯⣭⣿⣾⣷⣾⣶⢶⡿⡿⢿⡿⠿⢿⡇⢸⠟⠃⢸⠟⣿⢻⡾⠿⠿⣟⣛⣫⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣿⣿⣟⣻⣿⠻⣏⠸⠟⣿⠋⢙⣿⣿⡇⢠⠀⡆⢸⠁⠏⣸⠁⣏⠘⠃⣻⣧⠈⡾⠿⠿⠿⢿⣿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⣿⣻⡷⠰⡏⢰⣀⣧⣀⣼⣿⣿⣷⣾⣾⣷⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⡹⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⠿⣛⣙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡿⣿⣿⠿⣿⠿⢿⣿⠻⢿⣿⣷⣮⣭⣛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣫⣽⣾⣿⣿⣻⣮⣿⡍⣩⠈⠏⢀⣼⠁⠆⢸⠃⣦⡟⢠⠀⡞⠁⡤⡷⠀⣻⣟⢿⣷⣮⣝⠿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣫⣷⣿⠿⣫⣼⠃⢡⣾⣦⣤⣾⣴⣶⣾⣶⣶⣷⣾⣶⠿⠶⠞⡛⣚⣋⢿⣷⣍⡻⣿⣿⣦⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣛⣵⣿⢟⣬⣛⣛⢛⣛⣛⡋⣩⣉⣭⣥⣶⡄⣶⣦⢾⣷⡽⣿⣝⣿⣧⡻⣿⣿⣮⣙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣫⣾⣿⢻⣿⣼⣿⣿⣇⣿⡏⣿⡏⣿⣿⣻⣿⡞⣿⣷⣹⣿⣞⢿⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⠷⠟⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⡿⣫⡷⣿⠛⣿⣿⣿⡿⣿⣿⣶⣿⣿⣿⣫⣴⢶⣿⣿⣿⡀⠌⢷⡼⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⢼⣷⡿⢻⣤⣻⣷⣅⡘⢸⠁⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠸⢿⣷⣾⣵⠿⣴⡥⣿⣏⣰⡘⢿⠟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠗⣿⣿⣿⣷⠶⢯⣷⢾⣿⣽⣿⣶⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣷⣭⣿⣟⣛⣛⣿⣽⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣷⣉⡟⠛⣭⣽⢿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡖⣈⡭⢶⣾⣿⣿⣛⣛⢿⣿⣿⣿⣿⣿⣿⢿⡎⢍⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⠩⠥⠄⠜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣷⣥⣤⡤⣭⣝⣷⣿⣌⣯⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⠂⠀⢀⣆⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣮⣬⣷⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢳⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣾⣷⡶⣿⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣧⠤⠤⠤⠦⠀⠀⠀⠀⠄⠠⠶⠶⠶⠶⣾⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠧⠤⠤⠤⠤⠠⠀⠀⠀⠤⠀⠀⠀⠀⠀⠙⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠋⢀⣙⡹⣿⣿⣿⣿⣿⣿⣿⣿⣔⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢽⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⣬⠏⢫⣿⣿⣿⣿⡿⣛⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣀⣾⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣇⣳⣻⣿⣿⣻⣿⣿⣿⡿⠿⠛⡛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣯⣽⠟⢋⣯⣿⣿⣧⣿⣿⣾⣷⣭⣭⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣨⣾⣿⣿⣯⣷⣾⣿⣿⣿⣿⣼⣿⣿⣿⣧⣫⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣯⣿⣿⣿⣿⣿⣿⣿⡟⠃⢹⠘⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡾⣯⡿⣯⣿⢿⢿⣿⣿⣿⣿⣇⣦⡞⠗⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠛⡛⠿⣻⡿⠋⣽⡏⣿⡎⠉⠑⠛⢳⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣧⢹⣿⡟⠀⣰⣿⡇⣿⣿⣦⣄⠀⢰⣿⣿⡿⣹⣿⣿⣿⣿⣿⣷⠐⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡟⢠⣿⡀⢀⣿⣿⣟⣘⣿⣿⣿⣷⣿⣿⣿⣶⣿⣿⣷⣷⣿⣿⣿⡧⢻⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣧⢻⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣾⢿⣿⣿⢁⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠇⣿⣷⣾⣿⣟⣿⣿⣿⣿⣿⣿⣿⣁⣿⢓⣿⣽⣿⣿⣯⣿⡏⠸⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣽⢾⣟⣹⣿⣿⣾⣿⣷⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⢻⡇⣾⡟⣻⣟⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠈⣿⡟⢩⣿⣿⣿⡇⣸⡿⢛⣿⣿⣿⠵⢻⡟⣷⢀⣿⣿⣿⣿⣿⢻⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣧⣿⣧⣾⣿⣿⣬⣿⠟⢧⣽⣿⣿⣿⣷⣾⠇⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣦⣯⣭⣬⣾⣿⣯⣤⣴⣾⣿⣿⣿⣿⣿⣵⣶⣿⣿⣿⣿⣭⣬⣩⣿⣿⣿⣿

⡏⠛⠛⠉⠉⠉⠉⠉⢉⡉⢉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⡀⣀⠀⠀⠀⠀⢰⡆⢰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄
⡇⠈⠒⠃⠚⠀⠈⠉⠉⢸⡄⢸⠀⠉⠉⠁⠀⠐⠂⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠁⠀⠀⠉⠒⢦⡀
⡇⠐⠂⠃⠜⠀⠁⠈⠉⢰⠆⢸⠀⠉⠉⠀⠀⠐⠀⠟⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡐⣁⡀⣠⢤⠤⡄⡄⣤⢠⣄⢙⣆
⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡔⠋⠈⠉⠁⠀⢀⣀⣀⣀⡀⣀⣈⣀⡉⣣
⡇⠘⠓⠃⠛⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⠘⠃⠛⠀⡇⠀⠀⠀⠀⠀⠀⣥⠐⠐⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⡇⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⢠⣌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐
⡇⠀⠈⠿⠃⠀⠀⠐⠀⢨⠀⢸⠀⠐⠒⠀⠀⠻⠌⠂⠀⡇⠀⠀⠀⠀⠀⡀⢰⠀⠈⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⠀⠳⡄⠀⢀⣀⠠⠤⠐⠀⠀⠀⠀⢀
⡇⠀⠼⠣⠇⠀⠒⠒⠒⢈⠀⢸⠀⠒⠒⠂⠀⠿⠧⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠈⠓⢤⣀⣀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄⣤
⡇⠀⠀⠛⠃⠀⠒⠒⠒⠀⠀⢸⠀⠒⠒⠂⠀⠙⠛⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⣀⣤⣲⣄⣭⡟⠁⠉⠻⠀⠀⠁⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠄⢸⢀⡤⢄⠀⡇⠀⠀⢸
⡇⠀⢴⠴⠇⠀⠒⠒⠒⢈⠀⢸⠐⠒⠒⠂⠀⣤⠼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠌⠳⣄⡼⠉⠐⢦⠇⡇⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡘⠀⠘⡇⠀⣄⠀⠀⢣⠀⣀⣴⣷⡄⠂
⡇⠀⠸⠿⠇⠀⠒⠒⠒⢀⠀⢸⠐⠒⠒⠂⠀⠷⠽⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠀⠈⠻⡀⠏⠀⠀⠈⠻⣿⣿⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢈⠀⢱⣎⠀⠰⠀⠀⠈
⡇⠀⢤⣴⠇⠀⠒⠒⠒⠀⣶⣾⠐⠒⠒⠂⠠⣤⡄⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⠀⠀⢻⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⠀⠀⠀⠁
⡇⠀⢰⢴⠆⠀⠒⠒⠒⠀⣿⣿⠀⠒⠒⠂⠀⡤⠤⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃
⡇⠀⣰⠄⠆⠀⠀⠀⠒⠀⣿⣿⠀⠂⠐⠀⢀⡆⡖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠰⢄⠆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠠⠀⡶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⢀⠆⠀⠤⠤⠤⠀⣿⣿⠀⠤⠤⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢸⡅⠐
⡇⠀⠀⢴⠆⠀⠤⠤⠤⠀⣿⣿⠀⠤⠤⠄⠀⠀⣤⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⢀⣨⠟⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⡇⠀⠀⠀⠀⠀⠀⣤⣟⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⠿⠭⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡄⠴
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣭⣥⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣾⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣟⢿⣿⣿⣿⡟⢋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡟⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣏⣭⣿⣿⣿⠀⠉⠻⠿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣾⡟⠃⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣏⠘⠛⠛⠻⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣁⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠺⠛⣿⡿⡿⠋⠉⠙⠻⣿⣿⡿⠟⠃⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⢀⠀⠀⠁⠀⡃⢀⡀⠀⣴⡟⢉⣠⣴⣾⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣶⣤⣈⠙⠿⣿⣯⡁⠠⣤⡾⠃⠀⠀⠀⠀⠀⠈⠿⢀⣈⣴⣿⡿⠛⠉⠀⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡉⠙⠻⣿⣷⣤⡈⠻⣿⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡏⠀⠺⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠷⠦⡌⠙⢿⣿⣆⣹⠇⣀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣿⣿⣿⣿⣍⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣦⡤⠀⣾⣿⣿⣿⠀⣹⣶⠾⠃⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠁⣦⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⣿⣿⢠⣿⡏⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⠀⣼⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣄⣉⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⢀⣶⣶⣷⣦⠊⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⢠⣿⠛⣿⠟⢿⡆⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠻⡶⢿⢶⠞⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⢲⠂⣶⣶⣦⠐⣖⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⡖⣤⠀⡄⢱⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠉⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠛⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠓⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣛⣛⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣸⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠉⠀⠀⠀⣀⣤⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⣟⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣯⣽⣶⣦⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣐⠾⣭⡋
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⢻⠋⠙⢷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⡟⠉⠉⠉⠉⠉⠙⢻⡟⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⡇⠀⠈⢷⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣸⠇⠀⠀⠈⣧⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠰⣇⢴⣯⡻⣦⢠⣴⡿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⣾⣿⢿⢿⣿⣝⢷⣅⠀⠀⠀⢉⣿⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣈⣇⣀⡀⣠⣤⣤⣤⣽⣀⡀⣤⣭⣤⣤⣤⢀⣸⣃⣀⣀⣀⣘⣛⣻⣞⣙⣿⣿⣓⣙⣳⣀⣀⣸⣟
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠛⣻⣻⣻⣟⣛⣛⣽⣿⣭⣽⣿⣜⢃⣿⣿⣭⣿⣿⣙⣟⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⣛⣛⣛⣛⣛⣛⣻⡛⠋
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣿⢻⡿⢿⡟⣿⢹⡟⢻⡿⣿⠿⣿⣽⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⡟⢸⡇⢸⡇⡙⣾⡇⣾⡇⣿⡆⣿⣼⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⣵⢹⣇⢸⣇⣇⣯⠀⣿⢧⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣜⣻⡿⠇⣿⣝⣧⣛⣽⡹⢿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣠⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⣠⣄⣈⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⣿⣿⣋⡉⠋⠉⠀⠀⠀⠀⠀⢀⣴⣦⣀⣀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠈⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣭⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠉⠀⠈⠻⣿⣿⣿⣿⡟⠁⠀⠀⠀⢠⣤⣤⣤⠙⢿⣿⣿⡿⢟⣡⣤⣤⣀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠛⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⡀⠀⢠⣾⣿⣿⣿⣿⠀⠀⠀⠈⠉⠹⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⢀⣸⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠿⠃⢸⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠉⠀⠈⠙⠋⠙⠛⠒⠓⠒⠘⠃⡂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠄⡌⠀⠀⢀⡄⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣐⢻⣅⡀
⠀⠀⠀⠀⠀⠀⠀⢃⣑⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠰⡿⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠻⡏⢡⣀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣄⠀⠀⠤⠬
⠀⠀⠀⠀⠀⠀⠀⣏⡾⠠⣾⣿⢁⠀⠐⡈⠠⠤⠀⠀⡒⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠕⠿⣶⠙⠂⢐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠐⠠⡀⡀⠶⢢⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡃⠀⠡⠰⡀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⢾⡏⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠈⠀⣀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⣾⣤⣀⡀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣯⠬⠤⣤⠀⠀⠂⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢺⣯⣬⡍⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠐⠀⠀⠀⠄⣀⣂⣄⡄⢁⡐⠀⡄⠀⣠⡄⠀⠀⠀⠠⠀⠐⣿⣛⣥⣄⡲⠖⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢼⠃⠼⣉⡰⣫⠁⡊⡁⠈⠈⠣⠀⠄⠀⠀⡐⠀⠀⠛⢿⣿⣶⣀⢴⣂⣠⣠⣠⣤⣄⡤⠤⠶⢞⣭⣥⣼⢿⣯⠑⣆⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠈⢸⡔⣲⡋⠋⠿⡋⣧⢶⡑⢄⠂⠘⢳⢂⢐⡀⠂⡄⡻⢼⣽⣟⣫⣽⢯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢱⣍⣨⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⢦⠩⣟⣱⠆⢐⢃⣏⠝⢚⢐⣶⣮⢷⢏⣆⢄⢿⡘⠦⣙⣫⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡠⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠜⠙⢚⠯⣥⢚⣧⣼⢿⣦⡞⣾⣿⣟⡟⣷⡿⣮⣿⣿⣟⡿⣿⣿⣿⣿⣽⣟⣫⣿⠿⠿⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠉⡉⠀⠏⠀⠤⠘⠨⠟⡏⡇⣏⠡⡛⢻⠞⠿⠝⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠈⠀⠸⠀⠠⠃⠠⠁⠀⠀⠁⠀⠀⠀⠂

⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣴⣶⣤⣶⣶⣤⣤⣶⣾⠿⣿⡿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣈⠄⠸⣿⣿⢻⣷
⠀⣠⣴⡎⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠁⣶⣆⠘⠛⣫⣿
⣶⣶⡿⠂⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣿⣾⣷⣿⣧⣿⣿⣧
⠻⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣩⣿⣮⣤⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⣥⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉⢙⣺⣿⣶⣶⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣟⣿⣿⣿⣿⣿⣿⣧⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣽⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⣽⡿⡇⠹⣿⡿⣿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⣿⣿⡿⠿⣿⡟⠿⠀⠀⠸⠿⢿⣧⣸⠝⢻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⡿⠟⢻⣿⣿⣶⡆⣦⡀⣈⢀⣰⣾⠶⢶⡶⠀⢀⣽⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡘⣿⣄⠉⣹⣻⠍⣠⣿⣏⠀⠀⢛⡉⠈⢰⣦⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⡿⣿⣿⣿⣷⣾⢼⢿⢏⣿⡇
⠀⠀⠀⠀⠀⠀⡇⣤⣿⣆⠙⠻⢿⠿⠛⢉⣀⢀⣐⣃⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠻⣧⣤⣽⣿⠿⢿⣿⣎⢦⣴⡇
⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣾⣇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣖⣦⣽⠿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⣿⣦⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⠛⠋⠁⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣭⣫⣍⣩⣏⣫⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⢀⡈⠁⠐⠃⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠈⠉⠰⠖⠀⢈⣁⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡉⠰⡦⠀⠛⠀⠒⠀⠒⠀⣉⡈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠁⣤⣄⠀⣠⣶⣿⣷⣦⡀⠙⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡁⠀⣿⣿⣿⣿⣿⣧⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢈⠃⢈⣠⣍⡀⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣿⣿⣿⣷⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠤⠀⠀⣀⠀⠈⠁
⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⡀
⠀⠀⠀⠀⣰⠃⠀⠠⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⣏⠀⠘⠿⠃⠀⠀⠀⠀⢸⣿⡇⠀⠙⠃
⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡿⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠈⠁⢸⣿⣇
⠀⣀⣀⣺⣟⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣛⣓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣃⣀⣀⣀⣀⣐⣿⣛⣀⣀⣀⣀⣀⣀⣀⣘⣛⣛⣀⣀⣀⣀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠿⢛⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⠛⠛⠿⠿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡉⣶⣿⣛⣉⣉⣉⣉⣠⠤⠤⠤⠤⠤⠬⠭⢭⣉⣉⣙⣛⣳⣶⠌⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣝⠻⠿⢶⣿⣭⣽⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣭⣭⣽⣶⠿⢛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣷⣶⣦⣬⣭⣉⣉⣙⣛⣛⣛⣛⣛⣛⣛⣛⣉⣉⣭⣥⣶⣶⢿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠳⣄⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣤⣀⣀⣀⣠⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⢀⡤⣴⢲⣤⣤⣴⣶⡶⠿⠿⠿⠛⠛⠛⠛⠿⣶⣤⠀⠀⠀⠀⢀⣀⣠⢴⣿⠟⠛⠳⠶⢤⣄
⠀⠀⣀⣻⣶⣀⣿⡛⠯⣄⣀⣀⣤⠤⠤⠖⢒⣒⣚⣩⣽⣿⣋⣻⣭⣭⣭⣶⡾⠿⠋⠀⠀⢠⣤⣤⣌⡷⠀⢀⡀⣀⣀⡀
⠀⠀⠧⠜⣠⡼⣿⡿⣷⣦⣧⣴⣶⣶⣿⣿⣿⣿⢻⣿⠿⠏⠉⠉⠉⠉⠁⠀⣀⣤⡴⠶⠶⠞⠛⠛⠋⢡⡞⠋⡽⣤⠄⣹
⠀⠀⠀⣾⠋⠀⠿⢿⣿⣿⣿⣿⡿⠿⠛⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣀⣀⣠⣤⠶⠶⢶⣝⠋⠓⠒⠒⠋
⠀⠀⠀⠙⠷⢤⣤⣤⣍⠉⠉⠉⠀⠀⢀⣠⣤⡴⠦⣤⣄⣀⣶⣖⣚⣛⣷⣶⣫⡿⠸⢯⡍⡁⢀⡀⠀⢸⣇⣀
⠀⠀⢠⠤⠖⠒⡦⠤⡝⠷⣿⣟⣿⡴⠛⣥⠶⠶⢦⣤⣄⠉⠉⠉⠉⢠⡽⠿⣧⡀⠀⠀⠻⣿⣯⣥⣤⣀⣬⡍⢻⣆
⠀⠀⠈⠓⠒⠶⠤⠼⠗⠀⠀⠈⢠⣤⣤⣝⠛⠻⣷⣀⣈⣛⣛⣶⠄⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠦⣄⠀⠀⢀⣀⡀⠀⠀⠀⣴⠿⠷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⣿⢑⠂⣿⠀⣰⠏⣉⢻⡆⠀⠀⢷⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⡟⢉⠉⡇⠉⠻⡟⠁⠀⠻⣄⢈⣼⠇⠀⠀⠀⣿⠁
⠀⠀⠀⣠⣤⣄⠀⠻⣾⡤⠃⠀⢸⠇⣠⣶⣤⡀⣿⠀⠀⠀⠀⢈⡇
⠀⠀⢼⡇⠃⠸⡆⢀⡏⠀⠀⠀⢸⢸⣿⠀⢀⡇⢸⠀⢀⣠⢤⣈⣷
⠀⠀⠘⢿⡦⠛⠀⢸⡇⠀⣠⡶⢻⣤⣹⡷⠞⠁⢾⠀⣾⠀⣧⣹⣧⠀⠀⠀⣠⡾⠗⣲
⠀⠀⠀⢸⠇⠀⠀⢸⡇⢠⠏⠀⣾⠀⠿⣦⠀⠀⣸⠀⠙⠿⣶⠛⢹⠀⠀⠀⢿⡄⢀⣽⠃
⠀⠀⠀⣿⠂⠀⠀⠸⣧⠏⠀⠀⡃⠀⡇⠸⣆⠀⢿⠀⠀⠀⡿⠀⢸⡄⠀⠀⠈⠙⡟
⠀⠀⠀⣿⠀⠀⢀⠾⡇⠀⣠⣤⠇⠀⡇⠀⢻⡄⣿⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢠⠞⠀⣧⡞⠁⢸⠷⣶⡃⠀⠀⢻⡟⠀⠀⢀⡇⠀⢸⡇⠀⠀⠀⢸⠇
⠀⠀⠀⣿⢠⡟⠀⣰⡞⠀⠀⢸⠀⣹⡇⠀⠀⢠⣇⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣼⠆
⠀⠀⠀⢸⣿⠁⣼⠛⡇⠀⠀⢸⠀⣹⠿⡀⠀⣾⢻⡆⠀⢸⠃⠀⢸⠇⠀⠀⠀⣿
⠀⠀⠀⢸⣿⡴⠇⠀⣿⠀⠀⢸⡇⢸⠀⣷⢀⡟⠀⢷⡀⣾⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣏⣷⠀⠀⣿⠀⠀⠸⡇⢸⡇⠈⣿⠁⠀⣼⠋⠉⠳⠀⣾⠀⠀⠀⢀⣿
⠀⠀⠀⠈⢉⠙⡆⠀⣿⠀⠀⠀⣇⠸⡅⠀⡾⢆⠀⢇⣀⢀⡽⠀⣿⠀⠀⠀⢠⡇
⠀⠀⠀⢀⣿⣰⡇⠀⢹⡄⠀⠀⢹⠀⣦⢀⡇⠈⠣⡀⢹⠃⠀⠀⣿⠀⠀⠀⣸⠃
⠀⠀⠀⢸⣣⣏⠀⠀⢸⡇⠀⠀⢸⠀⢿⣘⠁⠀⠀⠀⣆⠀⠀⠀⡟⠀⠀⠀⣿
⠀⠀⠀⠘⡏⢸⡀⠀⠀⠄⠀⠀⢸⠀⠘⣿⠀⠀⠀⠀⣟⣲⡐⠚⣧⠀⠀⢀⡏
⠀⠀⠀⢠⠃⢘⡇⠀⠀⠇⠀⠀⢸⡀⠀⡇⠀⠀⠀⠐⠸⣇⠂⠀⣹⠂⠀⢸⠇
⠀⠀⠀⢸⠀⠈⡷⡄⠀⡟⠀⠀⢸⡇⢸⡇⠀⠀⠀⢀⡄⠈⠁⣤⠉⠀⢀⡟
⠀⠀⠀⢸⡆⢸⡇⠹⣆⡛⠀⠀⠀⡇⢸⡇⠀⠀⠀⢸⡅⠀⠀⡟⠀⠀⢸⠃
⠀⠀⠀⠈⣧⢸⠇⠀⠙⣷⠀⠀⢠⡄⢸⣧⠀⠀⠀⠸⡇⠀⢠⡇⠀⠀⡿
⠀⠀⠀⠀⠙⣿⠀⠀⠀⠻⣧⠀⢸⡇⢸⣿⠀⠀⠀⣦⠀⠀⣸⠁⠀⢸⡇
⠀⠀⠀⠀⢠⣿⡆⠀⢀⡃⠸⣆⢸⠇⢀⣿⠀⠀⠀⣿⠀⢀⡿⠀⠀⣸⠃
⠀⠀⠀⠀⢸⠁⢿⠀⢸⡇⠀⠻⣾⠀⢸⣿⠀⠀⠀⢿⠀⣼⠇⠀⢀⡿
⠀⠀⠀⠀⢸⡆⢻⠀⢸⡇⠀⢠⣿⢠⡆⢹⠆⠀⢸⡄⢀⡿⠀⠀⣼⠃
⠀⠀⠀⠀⢘⡇⣼⠀⢸⡇⠀⢸⣙⡷⣀⣼⡆⠀⣼⠁⢸⠁⠀⣸⠃
⠀⠀⠀⡀⠀⡗⢏⠀⡸⣇⣤⣉⣏⣀⠘⠹⣧⢤⣻⣠⣈⠛⠛⠋⣷⣄⡀⡀
⠀⡴⣦⣄⡵⢊⠙⣩⣧⡙⢷⢈⠃⠀⣠⠊⣿⣦⣾⡿⡿⣦⢀⠀⠀⢸⣿⣖⠒⠂
⠀⠀⢈⠛⠧⣀⡀⣿⡀⠙⢧⡀⢉⠬⠇⢸⣡⠏⣽⡀⠘⢳⣦⣀⡠⢃⠙⣝⠳⠤⠤⣴⣃⣀⡀
⣈⡿⠃⠘⣤⢁⣸⠌⢿⡀⠈⠃⠙⢀⠰⠋⠀⣴⢇⠀⢰⠎⠸⢧⣀⠎⢢⡐⢄⠀⠀⠈⠈⠛⠢⣄⣠
⠀⢁⢠⡔⡻⢒⠟⢀⠀⣙⠢⠈⢳⠞⡧⢬⣤⠻⠀⡗⠂⢰⢇⠐⠙⡧⣈⠀⠘⠣⣀⠀⠀⠘⠃⠈⠑⠖⠒⠻⢤
⠀⠁⠲⡠⣧⠈⡛⠆⡰⢹⠃⠀⠠⡎⠠⡞⢡⡀⣴⠃⠔⠋⠘⠦⣤⢇⠸⠗⠠⢠⢉⡍⠙⠓⠶⣄⠀⢀⣁
⠀⢤⣶⢇⢾⡀⢱⣄⠘⢡⡗⠀⢀⠰⡾⠋⣦⠁⠈⣀⡤⠲⡏⠒⠋⢈⢀⡷⡀⠀⢀⠉⢦⠀⠀⠈⠃⠈⠁
⠀⣀⢾⠀⡛⠀⠈⠻⡄⢣⠇⠀⠈⣸⡄⠢⠙⡆⠀⢱⠀⠸⡧⡀⠀⠈⠛⠁⣼⠢⠛⠀⠤⣀
⣴⠇⣦⠸⡳⢦⡀⠀⠀⠸⠐⣀⠞⢨⠀⠇⠘⠀⠛⢸⠀⡰⢃⠋⠀⠀⡤⠚⡧⣀⠀⣿⠀⠉⢒⠲⠄⢤
⡷⢍⠉⡀⠁⠘⠂⠀⠀⠀⠀⠀⠀⠛⠄⠀⠀⣷⠀⠉⠁⡵⠈⠀⠀⠀⣠⢼⠧⢀⠚⠀⢆⠂⣼⢰⢤⣤⠴⠒
⢡⡾⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠳⠂⠀⠀⠀⢠⣌⠿⢺⠀⠀⠈⠀⠰⠸⠂⠔⠚⢷⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⣾⠟⢹⣿⠙⣿⣦⡀⠀⠀⠀⣀⣀⠀⠀⠀⣠⣾⠟⢹⣿⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠿⣷⡄⠘⠛⠁⠀⢸⣿⠀⠀⠙⠃⢀⣴⡿⠛⠛⢿⣶⡘⠛⠁⠀⢸⣿⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢹⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢠⠞⠓⣦⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⣾⣷⣆⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⢀⣿⡇⢸⠀⠀⣿⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⣿⣿⣿⢸⣿⡀
⠀⠀⠀⠀⣴⡿⠋⠁⢸⠀⠀⣿⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣿⣿⣿⣿⠀⠙⢿⣦
⠀⠀⠀⣸⡿⠁⠀⣠⠞⠀⠀⠛⢦⠀⠀⢻⣷⠀⠀⠀⠀⣼⡟⠀⠀⣠⣿⣿⣿⣿⣦⠀⠀⢻⣧
⠀⠀⠀⣿⡇⠀⢰⡇⠀⠀⠀⠀⠘⡇⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⢹⣷⠀⠀⢳⣄⠀⠀⢀⡼⠃⠀⣸⡿⠀⠀⠀⠀⢻⣧⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⣼⡿
⠀⠀⠀⠀⠻⣷⣄⠀⠈⠉⠋⠉⠀⢀⣴⡿⠁⠀⠀⠀⠀⠈⢻⣧⣀⠀⠉⠙⠋⠉⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣴⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣦⣴⣶⡾⠟⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣾⣿⣿⣿⣿⠿⢷⣀⠀⠀⠀⠀⠀⣠⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠃⠛⠟⠙⠋⢱⣿⣿⡽⣷⠀⠀⠀⢰⠃⢠⠏⠀⡔⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠄⢀⠄⠀⣀⡈⢿⣷⣿⣀⠀⠀⡎⢀⠃⠀⡘⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠚⠛⠶⠀⠐⠛⠉⠁⢸⡿⢩⠁⡇⠀⠁⢈⡀⠴⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡺⠀⠇⠀⠘⠃⠀⠀⠀⣀⡼⠃⠀⢀⣿⠥⠃⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣴⠆⠐⠶⠀⣘⠷⠆⢀⡏⠀⠀⠀⢸⠛⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠁⠒⠒⠢⡥⠀⠀⢀⠎⣀⡀⠀⠀⢈⠏⣽⣀⡴⠏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⣄⣀⣀⣤⠔⢹⣏⠉⠉⠓⢦⠎⡼⣡⠏⠀⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠒⢾⣿⣧⡀⢿⡗⣾⢻⣀⡆⠀⡜⣸⠋⠁⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠠⣀⢻⣏⣿⣼⣷⠏⣸⠿⢦⡔⢰⣃⣀⣀⣠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠈⢸⣿⣾⣿⢟⢤⡿⠀⠰⢡⠏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀⠀⠀⠀⠘⣷⣿⣿⣴⠟⠁⢠⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡻⣿⣧⡀⠀⠀⠀⠘⢿⡋⠀⡠⢤⢆⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣤⡀⠀⢀⡎⠉⠋⠀⠈⠛⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠶⣼⣷⣆⠀⠀⠀⠀⣾⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠉⠉⡓⣶⠶⠞⠁⢀⠞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣸⣶⠇⠀⠀⠀⠋⠀⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠊⠉⠉⠻⣄⠀⠀⠀⠀⠀⠀⣠⠆⠈⠉⠉⣧
⠀⠀⠀⠀⠀⢠⠖⠒⢦⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣠⣿⣻⠇⠂⠀⠀⠀⠀⠇⠀⠀⠀⢀⠀⣱
⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠘⣿⣿⣿⣿⣿⣿⡏⠹⣿⡟⠀⠀⠀⠉⠉⠉⢙⣿⠸⠀⠀⠀⠀⠀⠘⣁⠀⢀⠜⠶⠃
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣹⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠀⠀⠀⣀⠤⠂⠈⠉⠉⣠
⠀⠀⠀⠀⠀⠈⢦⣞⡀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢿⠀⡆⠀⡄⠀⠀⢸⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠸⡆⢧⠀⠐⣄⠀⠘⣎⢿⡦⣤⣤⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣶⡾⠟⢿⠿⠋⠁⠀⠀⠙⠦⠳⠦⠈⠦⠀⠈⠶⠝⠒⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⠌⢵⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣶⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⢶⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠳⢄⢀⣈⡉⠀⠀⠁⠀⢀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣼⠀⠀⠀⠀⠈⠉⠀⠙⠒⠉⠒⢤⠇⠀⠀⠀⠙⠖⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡶⠧⠄⠀⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠠⠽⠤⡄
⠀⠀⠀⠀⠀⠀⠘⢶⠤⠤⠄⠀⠀⠀⠀⠀⠧⣄⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡰⠈⠉⠑⢆⠀⠀⠀⠘⡄⠀⢰⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠣⣈⢀⡰⠈⢂⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⠹⡄⠀⠣⡀⠀⣇⡀⢸⠘⠋⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⢄⠀⠈⢦⡀⠸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠑⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠈⠓⠦⢤⣀⣀⣀⡠⠤⠞⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⣀⣀⣀⡠⠴⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉

⠀⠀⣠⣶⣶⣶⣶⣦⣄⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣠⠴⣶⠶⢶⣤⢄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡾⢿⠸⢻⡷⣆⠤⠚⣷
⠀⣿⣿⣷⣬⣻⣿⣿⣿⣿⡿⠀⠀⠀⢸⣏⡈⢁⡘⠓⢁⡰⠀⣶⣷
⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢰⣶⠓⠈⠀⠀⠈⢉⡄⠫⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⢋⡀⠀⢀⠏⠀⠀⠀⠀⠀⠈⢦⡴⢀⡀⠀⠀⣤⠀⣴⠆⠀⠀⠀⠀⠀⠀⠻⠛⣛⢶⣄
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠁⠈⢲⡷⠚⠁⠀⠀⠀⠀⠀⢀⠆⠀⠀⠉⢀⣿⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⡾⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⡀⠀⢀⠴⠶⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⡍⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠃
⠀⠀⠀⠀⠀⠀⡃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠿⣤⡰⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡆⠋⢼⠀⡿⠀⣠⢦⡄⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠸⡆⠀⠆⢀⡇⠈⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠋⠀⠃⣴⠀⡗⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡤⠶⠂⢤⠀⠀⢳⡄⢀⡼⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠌⠀⠀⠀⠀⠀⠀⢀⡌⣱⡴⠆⠀⠀⠀⠀⣈⡄⡄⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⠏⠀⠀⠀⠀⠀⠀⠀⠈⣦⠁⡙⢀⠐⠀⠀⡎⠁⢀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠤⠀⠀⠀⢇⡀⠉⠀⣐⠂⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠜⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡠⠈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢾⠀⠐⠀⢀⠎⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡆⠀⢠⠎⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠁⠀⠆⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⣆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣀⣀⠀⠀⠈⢠⠇
⠀⠀⠀⠀⠀⠠⠊⠁⠀⠁⠠⡎⠀⠀⣄⠘⠢⠤⠄⠒⠒⠈⠉⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠐⣤⡤⠃⢸⠂⠸⠃⠀⠰⡆⠒⠈⢉⠁
⠀⠀⠀⠘⣁⠀⣀⠆⠰⠋⠀⢀⠘⠂⠘⠠⠦⠀⢀⠀⠀⢀⣸⣀
⠀⣠⠆⣀⢤⡀⠀⠀⠀⠀⠀⠂⠣⡀⠀⠀⠐⠀⠀⠀⠐⠋⠁⠁
⠀⠉⠀⠉⠀⠙⠀⠀⢁⠐⣆⠀⠀⠉⠆⢀⠄⡄⠀⠀⠀⠠⠄⠂
⠀⠀⠀⠀⠄⠀⠶⠀⠀⢠⠘⠷⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠸⡀
⠀⠀⢀⠉⠙⠁⠀⢠⠀⠈⣠⠀⠀⠀⠘⢀⠕⠀⡁⠀⠈⠐⢢⠀⠀⡀
⠀⠀⣈⠀⠈⠁⢀⠈⠁⠜⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠈⢠⡍⠄⠀⠀⠓
⠀⠀⠀⠀⠀⠐⠛⠋⠀⠀⠀⠺⡆⠀⠘⠀⠀⠀⠀⠀⡀⠃⠠⣞⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣴⠀⠰⡀⠀⠴⠁⠀⠀⠟⠛⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠰⠚⠛⠁⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧

⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠅
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠄⠤⠀⠀⠠⠀⠠⠤⠠⠀⠤⠤⠤⠤⠄⠤⢠⡤⠐⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠉⠀⠀⠀⠈⠑⢄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⢠⡰⢴⡄⡄⢺⠆⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⢸⠀⠀⠀⠀⠀⠀⠀⠈⢇⢠⢘⡌⠀⠀⡍⠑⢸⠀⠀⠀⢂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠨⢸⠀⠀⠀⠀⠔⠀⠀⠀⠀⠢⡄⠁⠀⠀⠁⡦⠃⠀⠀⠀⠀⠣⡀⠀⢐⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⢸⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⢨⠀⢈⡀⣦⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠄⠳⠐⠂⠀⠀⠀⠀⠀⡀⠀⠄⠸⠀⠀⡇⠠⠀⠀⡀⠀⠀⠀⡆⠀⠈⡇⠡⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⠀⠁⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠎⠉⢢⡀⡇⢀⠞⠀⠀⣠⠀⠀⠀⠠⠀⠀⠀⠀⣆⠀⠀⠱⣄⠀⢀⡴⠉⢱⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⢸⢆⠀⠙⠶⠁⠀⢀⠔⢩⠀⠀⠂⠀⠀⠂⠀⠀⡏⠳⣄⠀⠈⠦⠊⠀⢠⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⢸⠈⢢⡀⠀⢀⠔⠁⠀⢸⠀⠀⠀⠃⠀⠀⠀⠀⡇⠀⠈⠳⡄⠀⢀⡴⠁⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠈⠘⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⠉⢀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⡀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⡼⠄⠤⠠⠤⠤⠤⠤⠤⠵⠤⠤⠤⠤⠤⠤⠤⠤⠾⠤⠤⠤⠤⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢸⣀⡠⠄⠂⠂⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠂⠤⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈