Soup Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⣀⣴⠾⠛⣉⣥⣤⣴⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠀⠀⢀⣠⠾⢋⣥⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠿⠛⠛⠛⠛⠛⢷⡶⠶⠛⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⠳⠶⠶⠶⠶⠖⢚⣡⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠉⠛⠳⠶⠶⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠋
⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⡆
⠀⠀⠸⣧⠒⠂⠒⠒⠐⠒⠐⠒⠂⠒⠒⠐⠒⠀⠒⠂⠒⠂⠐⠒⠂⠲⠖⠐⠶⠂⠶⠰⠆⣼⠃
⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣶⠶⠶⠶⠶⠶⠿⠿⠟⠿⠛⠛⠛⠛⠛⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⣀⣀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠻⣦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠾⢡⢀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡾⠋⠠⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢉⠙⣦⠀⠀⠄⠀⢀⣾⠋⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠛⠀⢸⡇⠀⠀⢠⡟⠁⠀⠀⠀⠐⢯⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⠀⢠⡟⠀⠀⠀⠊⣀⡴⠞⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠤⠢⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠊⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣿⠀⡀⣼⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⠽⣇⠸⢿⠀⢿⡇⠞⠒⠒⠒⠢⠤⠤⣤⣒⠤⠊⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡤⠖⠚⠉⢁⠀⠀⠘⢦⠻⢇⢻⡔⢀⡀⠒⠀⠠⠄⠀⠀⠁⣠⠞⠻⢷⡦⡀
⠀⠀⠀⠀⠀⠀⢀⠊⠞⠁⢀⠄⠀⠨⠀⠀⠀⠀⢂⡙⢦⣿⣗⠀⡀⠀⠈⠂⠀⠀⢀⡼⠋⣀⠠⠀⠉⢪⢢
⠀⠀⠀⠀⠀⠀⠈⢆⠢⢀⠀⠀⠈⠀⠠⠀⠀⠐⠀⠁⠄⠛⠿⠷⠀⠒⠀⠠⢁⠐⠸⠃⠀⠀⠀⠀⠂⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠈⣟⠮⣵⣀⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠂⡀⠈⣀⡯⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠈⠉⠛⠒⠾⠥⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠤⠗⠒⠊⠉⠀⢰⠇
⠀⠀⠀⠀⠀⠀⣠⠶⠖⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠿⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠱⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢡⠞⠁⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⣘⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⡾⠃⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢍⣉⠛⠓⠶⠤⠤⠤⣤⣤⣤⣤⣤⠤⠴⠶⠶⠟⣛⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠉⠁

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

⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡀
⠀⠀⠀⢀⡏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⡆
⠀⠀⠀⠈⢧⣄⣠⠆⠀⢸⠇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣳⣄⣠⣤⣄⣀⣀⣀⡀⣠⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣧⣤⣶⡾⠿⠟⠛⠛⠛⠒⠛⠛⢻⡋⠉⠙⠛⠛⠛⡿⠿⢷⣶⣦⣤⣀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⢦
⠀⠀⠀⠀⣠⢴⣫⠝⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠛⢿⣶⣄⣠⠶⠋⢁⣀⣤⣶⡿
⠀⢀⡴⡿⠟⠉⠀⠀⠈⠣⣀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣾⣤⣤⣀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⢠⣾⠿⠛⠉⠁
⢠⣯⠎⠀⠀⠀⠀⢀⣠⠴⠟⠻⠯⠉⠀⣤⡴⠀⢈⡁⠀⠀⣀⠀⠀⠀⠈⠙⣿⠒⠦⣤⣀⠀⣰⠋⠀⣰⣿⣇
⣾⣿⠀⠀⢀⡤⠞⠉⠀⠀⠐⠲⠆⣠⠤⢤⡄⠀⠫⠭⠇⠘⠾⠃⡴⡆⠀⠐⠯⠗⠀⢠⡼⠛⠁⠀⣰⣻⣿⣿
⣿⡹⡄⢠⠞⢡⢺⡆⠀⠀⠀⠀⢀⢬⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠖⣻⢆⠀⠸⡆⠀⢀⣴⣿⢫⢏⣿
⢻⡻⣿⣯⣀⠈⢉⢤⠸⣹⡆⠀⠈⢦⡗⠀⠀⠀⣴⣶⠆⠀⣴⣻⠇⠀⠈⠉⢁⣀⠀⢀⣽⣶⠟⢛⡷⣫⠟⡿
⠈⣧⣄⠙⠻⢿⣶⣯⣤⣀⣀⣀⡀⠀⠐⠏⡷⠀⠀⠀⠀⠀⠀⠀⠠⣒⣶⡆⠓⠚⠀⣈⣑⣿⣾⠿⠋⠁⣸⠇
⠀⠸⣞⡳⣦⣄⡀⠉⠙⠛⠛⠻⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣒⣛⣺⣿⡭⠽⠟⠛⠉⠁⢀⣤⢾⡏
⠀⠀⠹⣤⠇⣠⣿⢷⠲⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠚⣽⣩⡟
⠀⠀⠀⠹⣗⠿⣯⡾⢡⡞⡭⢹⡆⣩⢿⡻⣟⠛⢒⡶⢶⡖⠒⢲⡶⢶⠒⢛⡿⠿⡍⢡⣿⡏⣷⣹⠞
⠀⠀⠀⠀⠘⠧⣄⠉⠙⠓⠿⠯⣤⣇⣻⣥⠟⢰⡏⣞⣃⡿⠀⡯⣞⣸⠇⠸⡼⣷⣃⣸⢦⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⢷⠦⣄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠚⠛⠛⠛⠋⠉⠉⣉⡥⠖⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠉⠛⠓⠶⠦⠤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠴⠖⠚⠋⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡴⢊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⠂⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣀⡁⠔⠈⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣿⡿⡆⠀⡿⡇⢸⠶⢶⣶⣶⣤⡤⠃⢀⡔⠉
⠀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠋⢉⡉⠀⠀⠤⠀⠀⠀⠀⡇⡐⡲⢃⠆⠀⠀⠠⢠⠜⠁⡰⠛⠻⢷⣶⣄
⠀⠀⠀⢰⣿⡟⠉⠠⠐⠈⠀⠀⠐⠀⣠⣥⡀⢋⠀⠞⠨⠾⢁⠁⠒⠚⠀⠀⠁⠀⠌⠀⠀⠠⠀⣘⣿⡇
⠀⠀⠀⠘⢿⣿⣦⣦⣀⣘⣛⡁⠈⠤⠄⠐⢀⠲⠖⠈⠠⠤⠀⠀⠛⠡⠀⢶⣂⣀⣠⣥⣴⣾⠿⠟⡟
⠀⠀⠀⠀⠈⢦⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⡜
⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⣠⠖⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⢄⡀
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠱⡄
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⣿⡿⢻⠿⠻⠿⡿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠉⠓⢦
⠀⠀⣀⠴⠖⠒⠂⠄⡀⡖⠛⠉⠑⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠒⠭⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢠⠤⡴⠖⠒⠖⠛⠛⠛⠛⠓⠒⠒⠒⠤⠤⣠⡔⠉⠀⠀⠀⠀⢀⣀⠤⠒⠋⠉⠁
⠀⠀⠘⠑⠂⠠⠀⠀⢀⣀⣀⡀⠀⣾⡶⠔⠂⠈⠉⠉⠛⠛⢲⣤⣤⣄⣀⣀⡀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠈⣧⠤⠄⠀⠀⠂⠈⠀⠀⠀⡉⠉⠀⠉⠙⠻⠶⢦⣌⠑⣄
⠀⠰⢤⣤⣤⣶⣤⣤⠴⠿⠛⠒⠛⠛⠛⠃⠁⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡆
⠀⠀⢠⠸⣿⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠣⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠸⡦⡙⠷⢤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢄⢀⠀⠀⠀⠉⠑⢒⡠⠔⣫⢾⠃
⠀⠀⠀⢧⠙⠢⢄⣈⠙⠛⠲⠶⠤⠤⢤⡀⣀⣀⣈⣀⣀⣀⣀⣀⣤⠤⠴⠖⠒⣋⣡⠴⠚⠁⡼
⠀⠀⠀⠘⣆⡀⠀⠀⠉⠁⠐⠂⠀⠠⠄⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⠒⠋⠉⠉⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⢀⡀⠤⠀⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠠⢈⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⠠⠖⢋⣁⣀⣤⣤⠶⠖⠀⠀⠀⠀⠐⠂⠠⠴⣶⣦⣤⣤⣤⣤⣀⡀⠈⠑⠢⢈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢄⣪⣵⣶⣿⣿⡿⠿⠉⢁⣠⠴⠶⠖⠒⠀⠰⠦⢤⡀⠀⠝⢬⡿⠿⣿⡿⠿⠿⠿⣶⣦⣅⣠⡙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣰⣿⣿⣿⣿⡿⠻⠉⠀⢐⡊⠉⠍⠈⠈⠭⠁⢒⣀⠤⠀⢀⠀⠀⠀⠃⠂⠁⠶⠒⠒⠒⠲⢿⣿⣿⣿⣆⠙⣆
⠀⠀⠀⠀⠀⢀⣀⣼⠁⣼⣿⠿⠏⠁⠫⠥⠈⠉⠁⠤⠤⠤⠄⠀⠀⠠⠤⠀⣀⠀⠘⡏⠉⠉⠉⠐⠢⡀⣠⣤⠀⠀⠀⢄⠙⢬⡛⣛⣷⠘⣆
⠀⠀⣠⠖⠋⠉⠁⡿⠀⣿⠃⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⢨⡋⠀⠀⠀⠀⢀⣠⣴⡈⠁⢀⠚⠂⠀⠀⠉⠉⣩⣿⡆⣿⡀
⠀⢸⠃⠀⠀⠀⣴⢿⠀⢷⣀⣀⣀⣠⣤⠀⠀⠉⠁⣠⣤⡍⠀⣤⠄⠀⠀⠀⠠⠄⠀⠀⠧⠴⠚⠁⡀⠊⠙⡀⠻⠿⠃⠀⠀⠀⠀⣾⣷⡇⢸⢧
⠀⠈⢧⠀⠀⣼⠋⠈⣧⡘⣿⣯⣿⣯⣍⣀⠀⠀⠘⠛⠡⣬⣤⠰⣿⠏⣠⠀⠀⠰⠛⠀⢀⡀⠀⠀⢱⠀⠠⠴⠀⢀⣄⡠⠔⢣⣾⢿⡿⢠⣾⡀⠱⣄
⠀⠀⠀⢳⡼⠃⠀⠀⠈⢷⣜⠻⣿⣿⣿⣿⣷⣤⠀⠀⠀⠙⡋⠀⠀⢾⣿⣿⣿⡷⠀⠐⠿⠓⠀⠀⠈⠀⠀⠀⠀⠛⣉⣠⣶⣛⡷⠋⣠⡿⢿⠍⣤⠘⣆
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⠷⣌⠙⠿⣮⣍⣙⠉⠀⢈⣿⣿⣿⣦⠀⠙⠿⠿⠇⠀⠀⠀⠀⠐⠒⠐⠀⠀⣀⣤⣾⣿⡿⠟⢉⣤⠞⠁⠀⠀⣸⠈⠳⢿⣀
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡈⠙⠛⠲⠿⢭⣽⣿⣿⣴⣦⣤⣀⣤⣤⣶⣾⣿⣷⣶⣾⣿⠿⠶⠚⠉⣁⣤⠾⠋⠁⠀⠀⢀⡴⠋⠀⠀⠀⠉⢷⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⢤⣄⣀⣀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣠⣤⡶⠞⠛⠉⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⣻⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠒⠒⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜⣹⠃
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠶⠖⠒⠶⠶⠶⠖⠒⢛⣡⠾⠋
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣤⠶⠶⠻⣿⣿⣿⠋
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠐⡫⠀⡀⠀⠄⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠒⣄⠀⠈⢂⠀⠀⠀⠊⠈⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⢰⠁⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡠⠀⠀⠀⠀⠀⠁⠀⠀⢀⠄⢀⣀⡿⠋⠀⠀⢀⠂⠀⠀⠀⠀⠀⡀⠂⠁⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡄⠀⢀⠄⠁⢠⠜⠁⠀⠀⠠⠀⠁⠀⠀⠀⠀⠠⠀⠀⠈⢀⠔⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠁⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠸⣬⠀⢨⣴⠁⡰⠁⠀⠀⠀⠀⠀⢀⠀⠀⡠⠈⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣠⠥⠔⢺⡀⢛⡆⢘⡟⠘⠉⠉⠉⠉⠉⠓⠒⢶⠼⠊⢀⣴⡉
⠀⠀⠀⠀⠀⠀⠠⣂⠥⠒⠋⠁⡀⠀⠠⠀⠻⣜⠿⣞⣷⠋⣠⠈⠉⡁⠒⠒⠀⠀⢀⡔⠉⠑⠫⣒⢄
⠀⠀⠀⠀⠀⢊⡞⠁⢀⠄⠂⠁⠂⠀⠀⠀⠘⣈⠻⣾⣿⡄⠠⠀⠀⠀⢈⠉⡄⠀⠔⠁⠍⠀⢄⠈⠣⠑⡀
⠀⠀⠀⠀⠈⢆⠢⣀⠁⠀⠀⠀⠠⠄⠀⡀⢂⠈⠀⠘⠿⠿⠦⠐⠂⠠⠌⡤⠀⠶⠀⠀⠀⠄⠀⠆⡠⡰⠃
⠀⠀⠀⠀⠀⠘⣷⡮⣕⡠⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠠⠀⠈⠂⠀⠈⣁⠭⣾⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣶⣯⣝⣳⠶⢬⢥⣄⣦⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣤⡤⢤⠤⣶⣚⣯⡵⢾⠏
⠀⠀⠀⠀⠀⠀⠠⠘⢿⠆⣉⠛⠛⠛⢶⠶⠾⢾⣥⣭⣬⣽⣼⣵⣾⣷⡾⠶⠿⠻⡟⡛⠉⣇⣸⠟⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠈⠻⣭⡀⠥⢀⡀⡃⠀⠀⠀⢀⡇⠠⣢⣳⠅⠶⠦⣘⠶⠸⣧⣂⣀⡴⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠁⠢⣄⡀⠀⠈⠻⣟⠁⠙⣦⣼⣀⡠⠞⠉⠈⢹⣏⠀⠀⠀⢺⢱⣦⣈⡽⠋⠀⠀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠂⠈⠷⢾⣱⠠⣂⣐⣈⠈⢀⣌⣿⣀⠁⣀⣘⣮⡾⠉⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠉⠀⢉⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠚⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡖⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠴⠶⠶⠶⠦⠤⣤⣤⣤⣖⣶⠟⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⣿⣭⣴⣶⡾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⢀⣴⣿⣿⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢾⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠶⠆⠀⢠⣤⠀⠀⠚⠰⣠⠏⠈⠙⢷⣄⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⠋⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠘⣇⡆
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡙⠃⠀⠀⢀⠀⠀⠾⠃⠀⡀⠀⠗⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠰⡍⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠿⣮⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣺⣿⣿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣝⡢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⣪⣵⣿⣿⣿⣿⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣯⣥⣐⣒⡀⠠⠄⠀⢀⣀⠀⠠⠤⠤⠤⣖⣚⣋⣭⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠇⠁⠉⠙⠻⡿⣿⣿⡿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣇⠁⠀⠀⠀⠀⠀⠁⠐⠁⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣿⣻⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠿⠭⣯⣽⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣴⣾⣇⢁⣴⣄⣿⣿⣿⣽⣶⣶⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⢰⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠴⢡⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢫⢃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡵⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤⠤⠀⠂⠈⠛⠛⠉⠉⠉⠉⠉⠹⠃⡜⠓⠛⠛⠈⠉⠉⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⢠⣞⠑⠈⠀⠀⠀⠀⠀⣀⡀⠤⠄⠀⠀⠀⠠⠁⡸⠠⠄⠤⠤⠤⠄⠀⠀⠀⠀⠀⢸⣡
⠀⠀⠀⠀⠀⠻⣻⣀⢀⡠⠒⠂⠀⠤⠀⠠⠤⠀⠀⠒⠂⠈⠈⢁⣀⠀⠒⠂⠀⠐⠒⠂⠙⠶⠒⠉⡇
⠀⠀⠀⠀⠀⠀⢩⠉⠛⠋⠀⠀⠀⠤⠤⠠⠄⠀⠀⠀⠀⠀⠠⠌⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠋⠉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⣀⠀⠀⣀⡀⠀⠀⣀⣀⡀⢠⣶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠛⠓⠒⠒⠒⠒⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠘⠷⣦⡀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⣿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⠀⠉⠀⠂⣿⠃⠒⠛⠀⠀⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢛⣀⠀⢀⠒⢂⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⢀⠐⠀⠄⠠⠦⢤⡄⠀⠀⠐⠀⠀⠀⠀⠀⢀⡦⠀⠴⠦⠀⠠⠟⠀⠢⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁⠀⠀⠀⠁⠈⢁⡀⠀⠐⠛⣁⡀⠀⠀⠈⠙⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⠀⠉⠁⠀⠀⠀⠀⠀⠠⠸⠆⠀⠨⠀⠀⠊⠀⠀⠀⠀⡀⠐⠊⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠒⠒⠂⠀⠤⠤⠤⠤⠄⠒⠒⠒⠈⠀⠈⠁⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⢢⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⠀⢀⣀⡼⠃⠀⠀⠀⠀⠀⢸⣀⡀⢘⡆
⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⡏⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⠀⠀⠀⢀⡤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣶⠶⠿⠟⠛⠛⠛⠛⣿⠛⠛⠛⠛⢿⠿⣶⣦⣤⣀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡄
⠀⠀⠀⠀⠀⣠⣴⡿⠟⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢾⠀⠀⠀⠉⠙⠻⣶⣤⠴⠋⣁⣤⣶⡿⠃
⠀⠀⠀⣠⣾⠛⠁⠀⠀⠈⢳⣄⡤⠤⠴⠶⠒⢶⠞⠓⠒⠲⠶⠤⢼⣄⣀⡀⠀⠀⠀⡼⠁⣠⡾⠛⠉
⠀⠀⢰⣿⠁⠀⠀⡠⠖⠋⠉⣤⡍⢀⣈⡛⠁⢴⢢⡀⣶⡆⢀⡀⠀⣽⡎⠉⢓⡦⠞⠁⣰⣿⣿
⠀⠀⢸⣿⡀⣠⠊⠴⡄⠀⠀⠀⢀⣙⠐⠋⠀⠀⠈⠀⠀⠀⠛⠁⡠⢶⡄⠀⢯⠀⠀⣰⡿⣻⣽
⠀⠀⠸⡟⢿⣧⣀⠉⢥⠪⡷⠀⠈⢬⠇⡀⠀⣴⡶⠀⢠⣾⠆⠀⠉⢉⣀⠀⣨⢷⡟⣻⣾⠟⣹
⠀⠀⠀⢷⣤⡉⠛⠿⣷⣦⣤⣤⣄⣀⣘⣝⣀⣀⠀⠀⠀⠀⠐⣽⣶⣙⣋⣤⣬⣽⠿⠛⠁⣰⠇
⠀⠀⠀⠈⢿⢻⠓⣦⣄⣀⡈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢀⣠⣴⢯⡏
⠀⠀⠀⠀⠈⢷⢼⣿⡹⢠⢟⡻⡗⢲⣶⣶⠤⠤⣤⣤⠤⢤⣤⡤⠤⣴⣶⡒⣻⣟⣦⢯⠟
⠀⠀⠀⠀⠀⠈⠳⣌⠙⠚⠾⠾⣅⣗⣯⡼⢀⡏⣗⣸⠂⡮⣖⡽⠀⣷⣦⣃⣿⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡦⣤⣀⡀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⣉⡥⠔⢊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡈⠉⠛⠓⠒⠒⠶⠶⠶⠖⠒⠒⠋⠉⠁⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠶⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠶⠁⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⡐⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠔⠀⠀⠐⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⢆⠀⡆⠀⠠⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⠄⠀⠈⡄⢃⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣀⣧⣨⡄⣤⡄⢀⣄⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⢔⠒⠺⢋⠉⢃⠈⠃⡈⠀⠐⠰⢸⡏⠀⠻⠀⡀⠑⠀⠀⠓⠚⠻⣍⡙⢟⠭⣖⣲⡦⢤⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⢀⣤⣖⠒⠉⠅⠀⠀⠉⠀⠀⠀⠠⠀⢀⠀⠠⢀⡥⢀⠀⡞⡀⠄⡀⠀⡀⠡⠈⠒⢄⡠⢄⢀⠉⠓⢦⣀⣁⣿⢿⠧⣝⡷⣦
⠀⠀⠀⣴⠏⠁⠒⠂⠀⠀⠐⢤⡠⢠⡂⠄⠈⠒⠀⠛⠒⠓⠀⢀⡝⠉⠉⠀⢀⠀⠈⢩⡉⠉⠀⠁⠐⡛⠚⠗⣹⣾⣟⣳⣄⠰⢬⣃⡾
⠀⠀⠀⠙⣟⣢⣤⣤⣽⠔⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠰⠂⠀⠀⠀⠆⠀⡀⠀⠐⠀⠀⢀⣴⡾⠛⠄⠀⣈⣻⡷⣾⠋
⠀⠀⠀⠀⠈⢿⣛⣛⠛⠛⠶⠶⣤⣤⣤⣀⣀⢀⣚⠀⡀⠐⠀⠂⠀⠀⣀⣈⣁⣀⣀⣀⣀⣀⢠⡤⠾⠿⠿⠶⠚⠛⠋⠉⠁⡶⡏
⠀⠀⠀⠀⠀⢸⠃⠊⠁⠒⠒⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣇
⠀⠀⠀⠀⠀⢸⠁⣰⢶⠶⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡤⢢⠀⠀⠘⢢⠔⢡⣏
⠀⠀⠀⠀⠀⢸⢶⠈⡷⠊⢀⡀⠐⢄⣠⠜⠀⠀⠀⠰⡐⠖⡐⠀⠀⠀⣄⠶⣰⠀⠀⠀⠦⣉⠜⠀⣀⡉⠒⠁⠀⠞⣱⡄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠲⠯⢌⠆⠀⠀⠀⢀⢞⡱⡄⠈⠋⠀⢠⠖⢄⠈⠛⠁⢀⢔⡒⡄⠀⠀⠠⡱⠖⠂⠀⠀⠉⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠘⢦⣠⣤⣤⡄⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠑⠊⠈⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣭⣀⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠭⣭⣛⣓⣒⡦⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⠶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⠒⠒⠒⠒⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢠⡀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⡀⠀⠀⢹⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠨⠃⠀⠀⡘⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠊⠐⡑⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠐⠠⠀⠀⠀⠀⢠⠊⠈⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠤⡄⠈⡄⠀⡐⡑⠀⠀⡠⢈⠊⡁⡈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠄⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠻⢰⠀⡆⢐⠄⠀⢀⠊⠀⠘⣸⢧⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡐⠁⡠⠂⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠒⢺⠀⠟⡟⠀⠠⠋⠉⠉⣉⡽⠋⠀⠀⠘⠲⠲⠤⠤⢄⣀⠀⠀⠀⡀⠊⠀⠐⠒⠈⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢠⠁⠀⢠⠀⠉⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢉⠳⠂⠀⠀⠀⢀⠐⠊⠉
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⠾⡀⡆⣇⠐⠂⢲⠃⠀⠚⠉⠀⢀⡁⠀⠒⠀⠠⢀⡀⠐⠁⠀⠀⠀⠠⢐⠵⡁
⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠀⠀⡠⢤⡀⠀⢷⡁⠸⣆⠀⣟⠐⠁⠈⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⢀⢀⡊⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠐⡁⠀⠀⠈⣁⠀⠀⠀⠀⠀⠡⠤⡙⠀⠙⠀⠋⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠓⢀⠠⡲⠃⠁⠀⠑⣄⣠⢿
⠀⠀⠀⠀⠀⠀⢯⠛⠶⣧⣀⠀⠀⠁⣀⠀⠀⠀⠁⠈⠁⠤⠐⠖⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⣀⣤⠶⠛⠁⡇
⠀⠀⠀⠀⠀⠀⠘⡆⢀⡴⠟⠛⡿⠶⣦⣤⣤⣄⣀⣈⣂⣀⠀⠀⠀⠐⠘⠂⠀⠀⠀⠀⠁⢀⣓⣀⣀⣀⣤⣤⠴⠶⡛⢟⣉⣀⠀⠘⡺
⠀⠀⠀⠀⠀⠀⠀⠘⣾⠂⠄⠀⡔⢠⢀⡄⠁⠈⡍⢩⠏⡭⢿⠿⡛⠛⡻⡛⣟⠛⠛⣛⠛⠛⠻⡝⢉⡩⠋⡄⣀⠴⣠⠊⠈⡉⢳⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠛⠚⡃⠀⣀⠀⢧⢠⠖⠇⣺⠈⣤⠄⢰⣠⡆⠘⡀⣑⣤⢀⡡⣄⣈⢧⣄⣐⡃⣴⣡⡔⠹⣔⣀⣨⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡡⠋⢀⡈⠙⡿⢯⡂⡬⠘⢶⡭⢐⣉⣆⡑⣾⠅⣆⣰⡾⢀⢀⠈⣺⠀⢃⣰⡟⠋⠉⠑⠬⡅⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠁⠸⢰⠘⣿⠗⡊⠱⣴⡋⣐⠀⠘⡞⢻⡿⡑⢄⡀⣀⣰⠧⢻⣾⠿⡄⡠⠂⠑⠀⣹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡂⢏⢉⣩⣾⣆⡚⣴⣿⣇⡡⣄⡴⠁⢾⣿⠈⣨⠁⢘⠀⠁⢐⣏⠼⠒⡻⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⢨⠻⠺⢷⣌⣵⣠⡎⢫⢁⣾⣏⡰⠧⣺⣞⡴⠟⠛⡒⠈⢹⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢎⡘⠢⠸⠙⡋⢻⣷⠙⣿⡾⡟⢫⣀⣴⡿⠙⢲⢄⢀⡦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢦⣦⣭⣾⣿⠉⢹⡇⢡⣾⠟⢛⣓⣤⡽⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣉⣙⠛⠛⠛⠛⠛⠛⠛⣉⠉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⡿⣿⢟⠿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣛⣿⣛⣛⣟⣛⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠰⠖⠆⠰⠀⠠⠤⢀⣀⠀⠀⣠⡄⢤⠄⢤⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣋⣀⣤⣤⣤⣤⣧⣤⣤⣠⣀⣀⣀⠀⠉⠑⠓⠿⡞⢿⡛⣷⠺⣶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⡦⣤⣀⠀⣝⣯⣙⣏⣙⣟⢻⡛⣄
⠠⡤⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣟⣿⣿⣿⢏⠿⠂⠙⣷⣿⣿⣿⣭⣬⣽⣏⣹⣉⣆
⢂⠣⠀⠀⠀⠈⠙⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣴⣶⣮⣶⣴⢟⠟⠋⢻⣿⣴⣤⠼⣦⣽⣌⣦
⠈⢆⠠⠉⠌⠁⠀⠈⠿⣆⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣯⡷⠣⢊⣤⣾⣿⣿⣿⣷⣾⣦⠬⣤⣵⡀
⠀⠈⢢⠐⠀⠆⠀⠤⠠⣼⣷⣤⡀⠀⠀⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⢁⣴⣷⠿⠂⠀⡋⠀⠀⠻⣿⣿⡿⣿⠿⣦⠤⢦⡼⣄
⠀⠀⠀⠈⠉⠓⠻⣿⡛⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣯⣄⣀⣀⣀⣹⣆⠀⣚⣛⣭⡼⠿⡦⠼⢦⠤⢦⡬⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⡤⠀⠀⠀⠀⠀⠈⢉⣠⣶⣶⠾⠿⠛⠛⠛⠉⠭⢚⡩⠔⠋⠀⠀⢡⠿⣖⣻⣗⢻⣖⠚⢶⠖⢶⡖⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⣾⣻⣿⠿⠿⠉⠀⠀⠀⠠⠤⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢁⣾⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠖⠒⠒⢀⣀⣐⡒⠒⠒⠦⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⣴⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣩⣥⣶⣶⣞⡃⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⢻⣿⣓⡶⢤⣀⠀⠀⠀⠀⠀⣠⡶⣺⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⣵⣾⣿⣿⡿⠿⠟⠓⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⡷⣮⣝⢦⣀⣠⣬⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⣾⣿⡿⠛⠉⣁⡀⠀⣠⡄⠀⠔⠒⠓⠶⢴⢠⣶⡆⠐⠂⠒⠒⠂⠀⠀⠈⠙⠻⣿⣷⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣸⢹⣿⠟⠁⠀⠀⠘⠛⢡⡖⠁⣠⣄⠀⢀⣠⣴⣶⣶⠤⡄⠊⠀⠀⠠⣒⠍⠉⡐⢂⣬⠾⠛⡉⣼
⠀⠀⠀⠀⠀⠀⠀⢿⣖⠍⠀⠀⣀⠁⡀⠀⠘⠇⠘⢩⡤⠑⢦⣽⡟⠙⣾⣄⣉⣱⡆⠀⠈⠿⣦⡄⣉⠿⠖⠀⢘⡵⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠑⢤⠀⠈⠀⠈⠂⠀⢤⡤⠟⠁⠤⢄⠉⠀⠀⣀⣝⡄⠘⢛⣀⡀⠰⠁⠀⠀⢀⣠⣴⡿⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⣀⣠⣴⣶⣿⡿⣟⢂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠶⠶⠶⠖⠒⠚⠿⣿⣿⣿⣿⣿⣿⣿⡋⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣿⣿⣽⣿⣿⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠟⣁⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣛⣿⠟⠉⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⢰⠠⢄⣀⣀⡀⠀⠀⠀⠀⣈⣛⣉⣭⣿⡟⣩⠀⢀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⡋⠉⠉⠉⠉⠙⠻⣿⣿⠿⣟⣧⣾⣿⣿⣿⡿⠂⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⣶⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⣾⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⢔⣩⣾⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣵⣾⣿⣿⡿⣟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣷⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠭⠭⠽⠯⠿⠿⠿⠿⠛⠛⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠤⠤⠶⠒⠒⠲⠶⠦⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⣀⠤⠒⠒⠊⢉⠉⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⡀⠀⣀⠀⣀⣀⠀⢀⠈⠑⡄
⠀⠀⠀⠀⠇⢀⠐⠠⠮⠤⠂⠉⢀⣐⡈⠀⠀⠄⠈⠐⠐⠉⠀⠀⢀⠀⠀⠄⠚⠉⠀⠔⡃⠀⠀⣀⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠉⢄⠀⠀⠀⠘⡀⠀⠐⡨⠗⠊⠁⠀⠈⠉⠢⡀
⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠠⠒⠄⠠⠀⠀⠀⡆
⠀⢰⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠀⠀⠀⠀⢰⡠⠂⠀⠀⠀⡰⠃
⠀⢘⠀⢴⣆⠀⠃⢀⠐⠁⠊⢀⠀⢀⠈⢂⠂⢢⠂⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⢸⠀⠀⠁⠀⡄⠀⠀⡸⠀⠈⠀⠘⠀⠀⠀⠘⠀⠀⣁⠀⠀⢯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠁
⠀⠀⢧⠀⠀⠀⠐⠀⠈⠀⠑⠀⠂⠀⠀⠀⠄⠀⣀⡀⠀⠀⠀⠀⡀⢀⠀⠈⠀⠀⠀⠀⣠⣀⣠⠇⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢠⠞⠁⠀⠠⢈⠇
⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⢀⡠⠊⠀⢀⡠⠔⢊⡥⠒⠢⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡠⠔⠈⣁⠴⠊⠁
⠀⠀⠀⠀⠀⢠⡤⣄⠀⠉⠒⠢⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⣤⠴⠚⢁⡠⠔⣋⠀⠐⠫⠗
⠀⠀⠀⠀⠀⠀⠉⠁⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⡀⠄⠊⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠒⠉

⠀⠀⣀⣤⣀
⠀⠀⢿⠀⠈⠳⢄
⠀⠀⠈⠳⣄⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠑⢄⠀⠀⠀⠀⣠⠶⣤⣤⣴⡇⠄⣟⣉⡿⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠳⣼⡉⣳⡷⠤⡇⢂⣀⣿⡟⢠⠀⣱⠖⡚⠻⢟⠳⡤⠶⠦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠞⡩⠽⠲⢄⠈⠻⣁⠽⢦⡈⢣⡀⡨⠛⠉⢻⠟⡉⠈⡱⠞⠁⢣⡀⣀⣸⠢⡈⠉⢦
⠀⠀⠀⠀⠀⠰⡇⠀⠧⣠⣊⠉⠻⢾⢁⠤⡴⠋⠉⠙⣇⣀⡤⠷⠠⡴⠚⡓⢌⠀⡰⠋⢀⣈⠦⠃⣠⠟
⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠈⠉⠉⠛⠒⠒⠳⠤⠤⠤⠼⠧⠤⠶⠤⠞⠒⠒⠛⠚⠋⠉⠉⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢶⣒⠋⠭⠒⢂⣈⡀⢤⠀⣀⣀⣀⡤⣤⢤⠄⣀⡀⠉⠉⠛⠒⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⢞⣫⡴⠞⠋⠁⣀⠐⣨⡤⠆⣹⢭⣋⡿⢶⣻⣵⣶⣦⣏⠉⠛⠏⡏⠛⡟⡖⡶⣄⠉⢷⣦⣀
⠀⠀⠀⠀⠀⣠⣟⠵⠛⠛⣇⣤⡦⢚⣕⡺⢏⣖⡺⠗⠻⢞⣫⠯⣗⣻⡔⢮⡿⣦⣄⣠⣄⣀⣠⣆⣩⠞⣻⡓⣌⢿⡳⡄
⠀⠀⠀⠀⢠⣿⡏⢀⡰⢭⡿⣤⢙⣆⣉⣿⠻⣿⡿⡿⣷⣾⣻⠽⣟⡼⠿⢏⣸⠏⠈⣿⣿⣿⣶⣓⣻⣍⡽⢣⢼⠘⢉⣿
⠀⠀⠀⠀⢸⡿⣷⢌⡻⣯⣽⡯⣟⠛⢷⡾⣷⠿⣲⡏⡼⢿⣭⡿⢭⣿⣳⣯⣤⠞⠋⠒⣿⡖⣫⠿⠭⣯⣵⣾⣗⡷⣺⣿
⠀⠀⠀⠀⠀⣿⠘⠻⢯⣗⡺⠭⠟⡕⣶⣶⠒⣫⣝⣺⣳⣿⣿⣿⣷⣯⣽⣟⣯⣀⠶⣎⡹⣗⣓⣶⢿⣝⣯⣽⣿⡟⢱⡇
⠀⠀⠀⠀⠀⢸⡆⠀⡀⠀⡏⢛⡲⠤⣝⣚⣣⣟⣨⣟⣿⣙⢍⡁⠻⡩⢟⣿⣞⣯⣵⣿⣬⣷⣾⣿⡿⠷⡟⠋⣧⠇⣾⠁
⠀⠀⠀⠀⠀⠀⢷⠀⢇⠀⠁⠈⠭⣛⣒⠢⠬⠉⠉⠙⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⢋⠩⠭⠁⠀⠀⠈⠁⢸⠋⣸⠏
⠀⠀⠀⠀⠀⠀⠘⣧⠘⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠃⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⢚⣩⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠦⠤⣤⣤⣤⣤⡤⠤⠴⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⡀⢣⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡈⢿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡠⠂⢠⢊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠘⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠠⣀⠎⣈⠄⠂⢈⠜⡁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠘⡀⠀⣀⠀⢀⡠⢒⡁⠀⢀⣀⣀⣈⢉⠒⠤⡀⠀⡉⣸⡁⠈⡻⠀⢛⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣱⡰⣷⢹⡿⠥⡿⠋⠀⠡⠐⠀⠀⠀⣀⠨⢍⡙⠐⢌⠲⡦⠃⠀⠘⡤⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣵⢿⢿⡌⢻⣼⣷⣿⣿⣵⣞⣄⣠⣀⣄⣀⣘⣉⣀⣨⣄⣀⣐⣄⣙⠀⢀⡔⣡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠤⠤⠤⠤⠤⠄⠀⠠⠠⠤⠤⠤⠤⠬⢽
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⠠⠤⠤⣀⡀⠀⠀⡤⠀⠀⠀⠀⠀⠐⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠠⣒⠀⣀⢀⡀⠀⠀⠀⡔⠈⠀⣀⣀⣀⡀⠈⠢⠀⠀⢀⣀⣀⠀⠀⡠⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡀⠤⠤⠌⠂⠀⡌⠀⢠⠞⣡⠔⠢⡈⢣⠀⡆⠀⠑⠀⠠⠄⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠂⠀⠀⣂⠄⠀⡇⠀⢀⠀⣇⠸⠤⢃⠜⢠⠃⠀⠸⢉⡁⠈⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠘⠃⢠⠄⠀⠘⠄⠘⢄⡀⠉⢋⣁⠔⠁⠀⠠⡂⠀⠁⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠉⠐⠂⠀⠈⠒⠤⡌⠉⠁⠀⠀⠀⠐⠃⠉⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⡛⠀⠚⠂⠀⠺⠂⠐⠗⠐⠏⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠁

⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⣶⣶⣶⣶⠖⠀⠲⣶⠀⠀⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⠀⠀⢠⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⡿⠟⠃⠀⠘⠿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⣿⡇⠀⡄⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣡⣶⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⠀⠀⠀⠀⠀⠀⢛⣥⣶⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠛⣩⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⢀⣀⣀⣀⣀⣀⣀⡈⠻⢿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⣿⣿⡏⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠁
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⡿⠛⢉⣤⣾⣧⠀⠀⣿⣿⠇⠀⣤⣾⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠁⠀⣴⣿⣿⣿⣿⠀⠀⣿⣿⠀⢠⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠸⣿⣿⣿⣿⣿⣿⣧⡀⢀⣴⣶⣿⣿⣿⣿⣿⡏⠀⠀⢻⣿⡄⣼⣿⣿⡇⢠⡀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠁⠀⠀⠈⠛⠁⠙⠛⠛⠃⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠛⢛⣿⣿⣿⣿⣿⡇
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⡇
⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠚⠻⣷
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠰⠆⠒⠐⠂⠂⠀⢹⠋⠹⠉⠀⠉⠐⠰⠷⣒⠖⢠⣠⣤⣤⡔⠋⠁⠀⠀⠀⠘⠁
⠀⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⢀⡀⢤⢀⠀⢠⡿⠄⠀⠀⠄⠠⠅⠄⢢⣤⡙⣲⣴⡟⢑⠿⢳
⠀⠀⠻⣦⣄⣀⡤⠚⠉⠈⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠄⠀⣠⡶⠏⢁⣻⣦⡿⠋
⠀⠀⠀⠙⡯⢍⠉⡒⠲⠶⠤⠤⣤⣤⣄⣀⣀⡠⠤⡤⠤⠤⠤⠄⠔⠛⠛⠒⠉⠉⠁⠺⡅
⠀⠀⠀⠀⠆⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⡠⢪⡇
⠀⠀⠀⠀⣶⠹⠊⡠⡀⠢⠜⠀⣀⠠⣐⠄⠀⠀⢔⡢⠀⡀⠂⠅⠀⠄⡐⠁⠐⠴⠀⠘
⠀⠀⠀⠀⠱⣀⣈⣉⠀⠀⠀⠐⠒⠃⠀⠀⠬⠥⠀⠀⠰⠓⠂⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠨⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣦⡤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⠒⠈⠉⠉⠉⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠊⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠠⠐⣀⣀⣠⡤⠄⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⣀⠀⠀⠌⠑⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⢋⣀⣤⣶⣾⣿⠿⠋⣁⡤⠖⣒⠒⠀⠲⠤⠤⠙⠩⠿⣿⠿⠿⠿⢶⣦⣄⣈⠢⡀
⠀⠀⠀⠀⠀⢀⡞⣠⣿⡿⢿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠤⠄⠀⠁⠀⠈⠉⠉⠉⠻⡿⣿⣧⡙⡄
⠀⢀⡠⠔⠒⢺⢠⡿⠉⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⣠⡘⠃⠠⠄⠀⠀⠒⢺⣧⢹⡀
⠀⡞⠀⠀⢠⣿⡘⣧⣀⣀⣀⡀⠀⠉⢀⣤⠍⠠⠄⡀⠀⠀⠄⠀⠸⠀⠒⢁⠐⠁⠀⠿⠂⠀⠀⠀⣼⣿⢸⣧⡀
⠀⠘⢄⢠⠟⠀⢳⡘⣿⣿⣷⣤⣀⠀⠉⠁⢶⡆⠛⢣⣶⣀⣘⠉⢠⣄⠀⠀⠂⠐⠂⢠⡤⠔⣩⣾⡿⢃⣾⡄⠻⡄
⠀⠀⢸⠏⠀⠀⠀⠙⢮⡙⢿⣟⡻⠓⠀⠀⣴⣶⣄⠙⢿⣿⡏⠀⠈⠁⠀⠀⠀⠀⢀⣠⣶⣾⠷⢋⡴⠋⠉⢡⠳⣼⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠲⢬⣙⠛⠲⠾⢿⣿⣿⣤⣄⣀⣠⣤⣴⣶⣶⣦⣶⠧⠿⠛⣉⡤⠖⠋⠀⠀⣠⠏⠀⠀⠙⢦⡀
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⢤⣤⣀⣈⣉⣉⣉⣉⣁⣀⣤⡤⠶⠚⠋⠁⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⢈⣷
⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣤⣤⣤⣀⣀⠤⠔⣱⠃
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢀⣠⠤⠶⣶⣶⡖⠉
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⣤⣤⣀⣤⣤⣤⡤⠤⠤⠖⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⢷⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠟⠃⢀⡀⠤⠀⠒⠂⠂⠐⠀⠂⠠⠄⣠⣉⠃⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⠔⠊⠁⠀⠀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⡔⠁⢀⠄⠀⢠⠀⠀⢠⢔⠁⠈⢤⣁⢠⣪⢄⡀⠀⠈⢂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡜⠀⠀⡡⣮⢹⡧⠃⠗⠀⠀⠀⣀⠠⠁⠈⠃⠋⠁⠀⠀⠈⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⡇⠀⠀⠈⡡⠀⠀⠀⢤⠄⠰⠀⠀⠙⠆⠀⠁⠀⣀⠀⠀⢀⠖⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡀⠈⠀⠂⠀⠐⠦⣀⡠⠄⠠⢄⣀⡀⠀⠀⢀⠀⠐⡁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⣬⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠩⠤⠒⠂⠩⠔⠒⠉⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⢦⡋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠞⢁⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠈⢙⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⡔⠁⠈⡩⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠳⠲⠶⠖⠒⠛⠛⠛⠁⠀⣠⠎⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠒⠒⠊⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡄⠀⠄⡄⠀⠀⠀⣠⢞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠈⠓⠒⠒⣊⡤⠞⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⣠⢔⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⠠⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠄⠀⠀⡞⢁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠠⠚⢁⡠⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠂⠀⢇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠴⠔⠂⠁⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠁⠘⣆⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢀⡠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⡿⡇⠀⡿⢿⠀⡿⠿⢷⣶⣶⣶⣦⠞⠁⢀⠔⠁
⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠛⠛⠉⢉⣁⣀⠀⠠⠄⠀⠀⠀⡏⢀⢧⡞⡠⠧⠤⠤⠀⢀⡠⠋⠀⡴⠛⠿⣷⣶⣤⡀
⠀⠀⠀⢀⣴⣿⡿⠋⠁⡠⠔⠂⠉⠁⡄⠀⢀⣒⡀⠐⠂⢀⡼⢐⣵⠫⠊⠠⠶⠶⠀⠀⠚⠀⢀⠊⠉⠐⢢⠀⠙⣿⣿
⠀⠀⠀⠈⢿⣿⣧⣀⣊⠀⠀⠦⡶⠀⠴⢀⡈⠉⠉⢀⣅⣌⠐⢈⣀⠀⠀⠤⠴⠀⠀⣀⡀⠀⠀⠀⢂⣀⣤⣶⣾⣿⡟
⠀⠀⠀⠀⠈⢿⠛⠿⢿⣿⣿⣶⣶⣶⣤⣭⣥⣤⣤⣤⣬⣥⣤⣬⣭⣤⣥⣤⣥⣬⣤⣿⣷⣾⣿⡿⠿⠟⠛⠉⢁⡞
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⢀⡴⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠑⠦⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣶⣶⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠖⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠠⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠀⣀⡀⠰⠤⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣒⣭⣴⣶⣾⠁⠀⢈⣉⣁⠀⠀⠀⠀⠀⠙⣿⣷⡲⢤⣀⠀⠀⠀⣠⣖⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣿⡿⠟⠛⠋⠉⠀⣀⣀⣀⣀⢀⡀⠀⡀⣀⣀⠙⠻⠿⣿⣮⣕⣤⣶⠞⠁
⠀⠀⠀⠀⠀⠀⣜⣿⡿⠋⠁⠠⠶⢀⠔⠋⠀⠀⠀⣨⣵⣝⣃⠀⠀⠀⡀⡔⠀⠄⢀⣫⠿⢛⡇
⠀⠀⠀⠀⠀⠸⣯⠛⠀⠀⡐⠀⠀⠻⠐⢛⡭⠢⣹⡟⠻⣷⢙⢢⡄⠐⠺⣧⡜⠻⠿⠃⢘⣾⡇
⠀⠀⠀⠀⠀⠀⢿⠙⠦⠀⠁⠈⠀⠠⠶⠟⠁⡤⠌⠀⢀⣬⡅⠈⢭⣤⠰⠁⠀⢀⣠⣶⠏⡼
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⠯⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣩⣿⣿⣿⣿⣿⠗⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠒⡦⢤⡤⠤⣤⣤⣬⣴⣾⢟⣽⣦⣤⣄⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⣤⣤⣤⣬⣿⣿⣷⣿⣿⣿⠟⠁⠀⠀⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠋⠉⠀⠀⠀⢀⣠⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢶⠶⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⡦⠐⢄⣀⠀⣀⠀⠀⢀⣠⡤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠁⠀⠀⠀⠀⠸⣷⣺⡄⠈⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠈⠀⢶⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡓⠾⢦⠄⠀⢀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠇⠀⣁⠈⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣲⣤⠀⠀⠛⠿⠆⣀⠀⠲⠆⣀⠀⠀⠀⠀⢀⠀⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⢴⡶⠶⣒⡲⠸⠟⠁⠈⠀⠀⠀⢠⠿⣷⣄⡀⠻⠛⠚⠓⠂⠂⢶⣿⣿⣿⠍⢁⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠸⡀⡘⣶⣿⣶⡤⠀⠀⠀⠀⠀⠙⢭⣉⠐⠁⠀⠉⠀⠊⢡⠨⡁⢀⠙⢿⠟⠀⢠⠖⠂⠀⠀⠀⠙⢋⡡⢶⣌⣉⠂
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠑⢈⠷⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⣀⣀⣠⡷⠈⠀⡀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢀⡿⢢⡍⠷⠊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠧⣘⣆⠀⠀⠀⠀⠀⠀⡀⠈⠉⠒⠐⠒⠒⠾⣧⠶⠆⢠⣤⠹⣆⠀⠀⠀⠀⠀⠀⠀⣠⡎⣀⠟⢃⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⢦⣤⣉⠓⠤⢔⣁⣀⣠⣴⠁⠀⠀⠀⠀⠀⠀⣿⢲⣿⠌⢫⣄⠻⣷⣦⣀⣀⣤⠴⠛⠋⣀⡤⣾⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠒⠦⠤⠉⠉⠙⠑⠀⠀⠀⠀⠠⠠⠿⠮⠵⠿⠟⠉⠓⠀⠈⢀⣡⣤⡶⢒⠽⠃⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠂⠐⠂⠒⠒⠒⠟⠛⠛⠀⠀⠄⢠⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⢀⣴⡞⠉⠹⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⢑⠙⠒⢤⡀⠀⢰⡏⡞⠀⠀⠀⢀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠃⠀⠀⠘⠀⠀⠈⢳⡆⣿⣼⠇⠀⠀⠀⠀⠂⠀⠱⡄⠤⣤⡤⠤⠄⣀⣀⡀
⠀⠀⢀⣀⠤⠐⠒⠂⠉⠁⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠘⣾⡿⣼⠀⠀⠀⠇⠀⢈⠀⠀⢣⠴⠃⠙⣖⢄⠀⠀⠈⠉⠐⠒⠢⠤⣀⡀
⠔⠊⠁⠀⠀⢀⣀⣀⠀⣤⣤⢼⡀⣤⠮⡤⠤⠤⠤⠄⠤⢼⣳⣿⠀⠀⠊⠀⠀⠘⠀⠐⢹⠀⢀⡠⠜⠋⠉⠉⠈⠉⠉⠒⢄⡀⠀⠈⠙⠢⢄
⠴⠖⠒⠈⠉⣀⠠⠤⣴⠁⠈⢉⣿⠃⠀⠈⢦⠀⠀⠈⡆⠀⠙⣿⠀⠀⠀⢀⠀⡐⠀⢀⣸⢎⣁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢹⠉⠒⠒⠤⠄
⢰⡤⣀⡔⠉⠀⠀⠀⠈⢻⣦⣼⣿⠀⠀⠀⠀⢣⡀⣤⠃⠀⢀⢏⠀⠀⠀⠀⡐⠁⣴⠟⠁⠀⠀⠀⠀⠀⠀⢂⠀⠀⢠⠀⡠⠞⢠⣤⣀⣀
⠈⠈⢱⠁⠀⠀⠀⠀⠀⠀⠱⡽⣷⠀⠄⠂⢁⡠⠟⠈⠢⢤⣪⠤⠥⠤⣤⣊⠀⣼⣏⠀⠀⠀⠀⠀⠀⡀⠀⠈⠂⡠⠚⠙⣶⣪⣍⡥⢶⠏⠉
⠀⠀⣶⠄⡀⠀⠀⠀⠀⠀⠀⠹⣼⡄⢀⡔⠁⠀⠠⡀⠠⠀⠈⠢⡀⠀⠐⠉⢉⡿⠁⠁⠀⠀⣀⣤⣾⠶⠶⢶⣶⣥⡔⠚⠉⠁⠀⠀⣾⡄
⠀⠀⢸⡆⠈⠀⠀⠁⠀⠀⠀⠀⠹⣇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⠀⠸⡇⠀⠀⢠⢞⡕⠋⢀⠀⠀⠀⠀⠉⠳⠤⣤⠘⠀⢀⣿⢸
⢷⡀⠘⡇⠀⠀⠀⠈⠀⠀⠀⠀⢀⢿⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢿⠒⠒⢦⡀⢀⣴⣯⡏⠀⠀⠀⠑⠀⠀⠀⠀⠀⢠⠇⠀⠀⡼⣸⣼⠤
⠒⠬⣁⡛⠠⠄⣀⡀⠀⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⢰⠀⠀⠀⠘⢾⣱⠇⠈⠀⠀⠀⠀⠀⠀⠀⠄⣀⣎⣀⠤⠜⣛⠥⠔⠊
⠀⠀⠀⠈⠉⠐⠂⠬⠍⣁⣒⡒⠤⠤⠇⣄⣀⣒⠀⠀⠘⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡃⡀⢀⣀⡈⣀⠠⠤⣄⣒⡀⠭⠕⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠂⠀⠸⠏⠉⠉⠥⠠⠌⠁⠉⠉⠿⠛⠁⠙⠒⠒⠂⠈⠉
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠉⠑⠒⠀⠤⠤⠄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠔⠂⠉⢠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠒⠒⠒⠒⠂⠒⠲⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠒⠒⠒⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⢀⠎
⠙⣤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠊
⠀⠈⠢⡀⠉⠁⠒⠂⠄⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⠐⠂⢉⠜⠁
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠂⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠂⠈⠉⠉⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠤⣤⠤⠄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣠⡤⠄⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⢤⣤⢤⣤⣤⣤⣤⣠⢤⡤⠶⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⡇⠁⣴⢍⣠⠀⢓⣡⢶⣈⠁⢲⣼⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠡⠛⠓⢧⠐⠦⣈⡡⠀⡉⠡⠌⠍⠁⣀⢻⡖⠀⠂⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠲⣄⡀⣐⡀⠀⠀⢘⡇⠀⠀⠀⢀⠀⠀⠚⠋⠁⢉⡛⢄⡲⠝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⣀⣹⡦⠬⠫⠀⣤⣘⡀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢀⣀⣤⠷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢩⠞⠉⠁⠀⡰⠂⠈⠛⠉⠉⠤⡀⡆⠀⠀⡤⠀⠈⠀⣀⣀⣀⣀⣀⣀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⠀⠀⣀⠟⠀⠀⠀⠀⠐⠀⠀⠈⠑⠀⡘⠀⢀⡤⠋⠉⠀⠠⠀⠠⠭⣣⠄⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠈⣇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡇⢀⡎⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡦⠤⣤⡏⠀⠀⢀⣤⡤⢤⡀⢠⠀⠀⠐⠃⠀⠀⠁⠘⣆⠠⠀⠀⣀⡀⠀⠁⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⠄⢈⢹⡀⠀⠞⠁⠀⠀⠙⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠈⡸⠇⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⡄⠀⠀⠸⢷⣄⠢⣀⢆⡠⢃⣴⢻⡾⠋⠙⢛⠋⠀⠀⠀⠀⠀⠀⠀⠘⢻⠁⢄⡡⠰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠩⢬⠑⣀⠒⠒⠭⢁⣜⠃⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡾⢀⠐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠀⠋⠈⠁⠉⡐⠢⠤⣞⢋⣠⡆⠀⡄⠀⠀⠀⠀⠀⢀⠆⠀⡾⠛⠘⡇⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡄⠈⠁⠀⢌⣭⣍⡀⠰⠟⢀⠀⠀⠐⠒⠋⢀⡺⠂⡶⠁⠻⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⠈⣥⠀⢰⠀⠻⠧⠛⠀⠀⠠⠃⠀⠀⢂⠰⠃⣠⣤⣼⠃⢠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠐⠣⠀⠈⡛⠢⠀⠀⠀⠁⠀⠀⠴⢁⠿⣧⡀⢁⠀⠿⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀⢀⣀⣀⣠⠠⡤⠺⡇⠐⠀⡄⠈⠉⠙⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⠿⠷⠶⣮⢻⣄⠀⠠⠟⠈⠓⠉




⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⣠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣡⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣄⣀⡀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⣋⣉⣡⡤⠤⠶⠖⠒⠒⠒⠛⠛⠛⠒⠒⠒⠶⠦⢤⣈⡉⠛⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⣉⡤⠶⠚⠋⢉⣀⣠⡤⠤⠶⠶⠖⠒⠒⠒⠚⠛⠛⠛⠉⠉⠙⠛⠛⠿⣷⡄⠀⠹⡄
⠀⠀⠀⠀⣠⠞⠉⣠⠶⠋⠁⢀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⡿
⠀⠀⠀⡼⠁⢀⡞⠁⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⢀⡼⠃
⠀⠀⢸⡇⠀⠘⣇⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠋⠀⣀⡴⠋
⠀⠀⠀⠳⣄⠀⠈⠛⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⠒⠋⠉⠀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⠀⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠁⢀⣀⣀⣤⠤⠶⠚⠋⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠉⠉⠉⠀⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣄⣀⣀⣀⣤⣤⠤⠶⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡄⡚⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⢠⡶⢟⠛⠿⠖⠩⣶⣤⡂⠘⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠉⠲⠾⠽⣽⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠐⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠒⠒⠒⠒⠒⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠋⣁⣠⣤⡶⠶⠶⠶⠒⠲⠶⠶⠶⢤⣄⣈⠙⠲⢤⣀⠀⠀⢀⠀⠀⠀⣀⣴⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡈⡻⠆⠁⢀⣠⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⡾⠛⠁⠀⠀⢀⡴⠒⡛⢛⡭⠟⠽⠿⠉⣻⣶⣤⣤⣀⠀⢀⠀⠉⢀⣠⡾⢿⡏⠁
⠀⠀⠀⠀⠀⢀⡴⠃⣠⡾⠋⠀⢀⣠⣶⣿⡿⡘⢱⣧⣍⡓⣉⣠⡯⠙⣟⣁⠠⡟⠷⠄⢀⣠⠶⠋⠁⠑⠄⠙⢦⡀
⠀⠀⠀⠀⢠⡞⠀⣴⠋⠀⢀⣴⡿⢫⠍⡽⡋⠀⣠⠀⠟⢳⠀⣩⠗⣚⠈⢩⣅⠉⣠⠴⠛⠋⢀⡀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢠⡟⢀⡾⠁⠀⢠⡾⢩⣤⡔⠛⢳⢿⣿⡟⢀⠰⡶⢊⢡⡎⠀⢘⡈⠩⠇⣿⣴⣄⣸⡞⡛⢆⡀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⡾⠀⣼⠁⠀⢠⡟⡤⣷⣶⡀⣸⠟⣁⣠⠏⣸⡔⠡⢤⡞⠁⢠⣤⣽⠟⣰⡧⠞⠉⣿⣿⡁⠀⣷⠀⠀⠀⠀⠈⣷⡀
⠀⠀⣸⠃⢰⠇⠀⠀⣾⠠⢁⠬⠁⣏⡉⠉⠉⠁⠀⢼⡁⠲⣶⠾⠿⠛⣋⣤⢞⡅⢀⡶⢌⠋⣈⢧⣶⣿⡀⠀⠀⠀⠀⢸⣧
⠀⠀⣿⠀⡾⠀⠀⢰⠓⢰⡞⢳⡄⠙⠃⣄⣀⣈⠛⡞⠳⠦⠬⠴⣶⡿⠋⢴⣿⣀⡘⠓⠀⠛⢯⢸⡏⢙⠌⠀⠀⠀⢠⠀⣿⡀
⠀⠀⡇⠀⡇⠀⠀⠀⠀⠘⣿⣷⣹⣷⣆⠈⠛⠻⣧⡈⢲⠟⠉⣉⣩⡇⡷⡘⢦⣠⣴⣂⡤⠂⣤⠷⢌⢻⣠⡄⠀⠀⠈⠀⣿⡇
⠀⠀⣇⠀⡇⠀⠀⣿⠴⣾⣿⣿⠁⠉⠀⢒⣀⡀⠈⣧⠌⢠⡊⢽⡟⢰⡧⡻⢈⠇⠠⣍⣤⠆⣤⣴⣯⡨⢾⡁⠀⠀⢠⠀⣿⡇
⠀⠀⣿⠀⠃⠀⠀⠈⠠⣿⠁⠙⡀⢲⠀⢀⡿⠿⣷⡇⣐⢚⣏⣆⠀⠀⡱⡡⠊⠁⢀⣶⡤⠞⢷⣇⣿⢳⣬⠁⠀⠀⢸⠀⣿⡇
⠀⠀⢻⡄⠀⠀⠀⠀⣄⣽⡷⢾⣤⣾⣿⣖⠃⣊⣛⣁⠁⡾⠁⢿⡀⢤⡜⢓⣶⣾⡿⠛⢛⣦⠀⡷⠃⣸⠏⠀⠀⠀⡆⢸⣿⠃
⠀⠀⠈⣷⠀⠀⠀⠀⠘⣷⠆⠈⣋⠉⢋⠀⠀⣼⠉⠻⣄⣀⣲⠛⠺⣦⠀⣾⠋⣩⢤⡈⠛⣯⠺⣦⣾⠅⠀⠀⠀⡰⢁⣿⡟
⠀⠀⠀⠸⣧⠀⠀⠀⠀⠈⢧⡄⢉⡀⠉⠐⢀⢿⣷⢾⡿⠋⣸⣿⣷⠘⣷⣄⠃⢃⣾⠏⠀⣢⠆⣿⠏⠀⠀⠀⣰⢃⣾⣿⠁
⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠳⣾⣛⣛⣿⡷⢧⢮⣤⣤⠾⠛⢠⡟⠐⡃⠿⡿⠻⡇⠀⣨⣥⠾⠁⠀⠀⠀⡴⢁⣾⣿⠃
⠀⠀⠀⠀⠀⠘⢷⣄⠀⢄⠀⠀⠈⠻⣴⣟⠛⠠⢴⣎⣠⣤⡬⠡⣘⣎⣉⡷⠔⢂⣰⣆⡵⠃⠀⠀⠀⣠⠞⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠑⢄⡀⠀⠀⠉⠉⠓⠾⢿⣯⣽⣷⣭⣉⣀⣍⠀⠀⠈⠊⠁⠀⠀⠀⡠⠞⣡⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡈⠒⢤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⡠⠔⣋⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣌⣉⠒⠢⠤⠤⢀⣀⣀⣀⣀⠠⠤⠔⢒⣋⣥⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣶⡖⢲⡶⢶⣶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⣄⣴⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠧⠤⠤⠤⢤⠤⠤⣄⣀⠀⢀⠔⠂⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡔⠁⠀⠀⠈⢣⠈⡏⡔⡽⠛⠢⣽⡄
⠀⠀⠀⠀⠀⢀⠔⠙⣿⣿⣿⣿⣿⣿⣿⠟⠛⠢⢄⡜⢀⠖⢉⠑⢦⠀⡎⡜⡜⡰⠒⢢⠈⣧
⠀⠀⠀⠀⠀⡎⠀⠀⣈⠿⠿⠿⠿⠟⡡⠒⣉⣉⠒⡇⠸⡀⠓⠀⡸⠀⡯⢊⠔⣁⠀⢠⠃⢿⢆
⠀⠀⠀⠀⠀⣇⠀⡜⢡⠔⠂⣐⡒⢊⡠⢋⡤⠤⠌⡙⢄⠈⣐⠊⢁⠼⡁⡜⢚⣥⣉⡉⣮⠎⢸
⠀⠀⠀⠀⠀⢫⠲⣀⠃⡔⢉⠤⠤⠤⢴⡳⢌⠒⠠⠜⠂⠡⢄⡽⠭⣢⣱⠯⣱⣝⣩⡿⢔⢁⡞
⠀⠀⠀⠀⠀⠸⡀⠈⠚⠴⣁⢔⣉⣉⠢⡘⠦⠉⠭⠭⣉⠑⢄⠑⣄⣿⡫⣦⠿⠹⣩⡷⠞⠁⠇
⠀⠀⠀⠀⠀⠀⢋⠢⢄⡀⠀⠉⠐⠂⠵⠌⢑⣒⣐⣒⣌⣳⣀⣣⣸⡼⠤⠿⠒⠈⠁⠀⡠⡞
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠑⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠉⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡤⠄⣀⣀⣀⠀⠀⣀⣀⣀⠠⢄⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠤⠤⠤⠤⠤⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣷⡄⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⣴⡾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢋⣴⣿⡁⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢀⣽⣷⣌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⢿⣮⡻⣦⡀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⢀⣴⡿⣫⣾⠟⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⡻⣦⡀⠀⣼⡇⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⡄⢀⣠⡿⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⡻⣶⣿⠁⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢹⣷⡾⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⡟⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠸⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⡆⠀⠀⠘⣿⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⠀⠀⠀⣿⡀⠀⠀⠀⠀⣼⡇⠀⠀⢸⡿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⡆⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⠀⠀⢹⡇⠀⠀⠀⠀⣿⠁⠀⢠⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⣸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣿⣿⡆⠀⣿⡄⠀⢸⣧⠀⠀⠀⢰⣿⠀⠀⣼⡇⠀⣿⣿⣝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣦⣄⣠⣾⢟⣽⠟⠁⢻⡇⠀⣿⡇⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⣿⠃⠙⢷⣝⢿⣦⣀⣴⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠹⣿⣷⠟⠁⠀⠀⢸⣧⠀⣿⡇⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⣿⡇⢰⣿⠀⠀⠀⠙⣷⣿⠿⠁⠀⠀⠙⢿⣦⡀
⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠘⣿⠀⣿⡇⠀⠸⣿⠀⠀⠀⢸⡇⠀⠀⣿⡇⢸⡟⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁⠀⠀⠀⣿⣇⣿⠁⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢹⣇⣿⠇⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀
⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠈⣿⠇⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠻⣷⣄⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠈⠻⣷⣄⣀⣀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣿⠀⠀⠀⢸⡇⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⣀⣀⣴⡿⠋
⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⣿⠀⠀⠀⢸⡇⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⡀⣀⣼⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡶⠿⠿⠟⠛⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠛⠻⠿⠿⢶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⡠⢖⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⡲⢄
⠀⠀⠀⠀⡎⡔⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡴⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠈⢢⢳
⠀⠀⠀⠀⢇⠘⠦⣀⣤⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⢦⣄⠴⠃⡸
⠀⠀⠀⠀⠈⠳⢄⣀⠉⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠒⠉⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠈⢏⠉⠓⠒⠤⠤⠤⣄⣉⣉⣉⠙⠓⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠚⢉⣉⣉⣉⣠⠤⠤⠤⠒⠚⠉⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠤⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣤⠤⠤⠤⠤⠒⠒⠋⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⣒⠀⠀⠠⠤⠉⠁⠒⠒⠢⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣪⣷⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣽⣲⢄⠀⣀⣺⠿⠋⠁
⠀⠀⠀⠀⠀⢀⣴⣷⣿⠟⠋⣁⠀⣠⠄⠐⠒⠲⢦⠠⡦⠀⠀⠒⠂⡀⠈⠙⠻⣷⣿⡟⠁
⠀⠀⠀⠀⠀⣼⢻⠏⠀⠀⠀⠁⣾⢠⠶⢄⠠⣶⠿⣿⠲⢅⡀⠀⣬⡀⠠⣴⡚⠉⢤⡇
⠀⠀⠀⠀⠀⢸⡃⢀⠀⠂⠀⠀⢀⣤⠾⠁⣈⠛⠀⢙⡃⠘⢇⡀⢨⠋⠀⠈⣠⣴⢻⠇
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠉⠀⠀⣈⣁⣠⣤⣶⣿⣿⢃⡟
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢉⣽⣿⣿⣿⡿⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⣻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠰⠶⠤⠤⠤⢴⣶⣶⣿⣫⣾⣤⣤⠄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠦⠤⠶⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠁⠀⠀⢀⣀⣴⣾⣿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⢤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠿⠿⠿⠿⠿⠛⠻⠛⠛⠉⠁


⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⢀⡾⠁⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣌⣿⠠⣼⠁⠴⠤⢤⡿⠇⢀⣀⣀⡀⠀⠀⠀⠀⠀⢰⡁⠀⣠⠋
⠀⠀⠀⠀⠀⢀⠠⠄⠒⠈⠉⠀⠀⠀⣯⠀⡇⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠐⠆⠀⠀⢀⡴⠊
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢇⠁⡼⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⡶⠻⡀
⠀⠀⠀⠘⣄⠀⠀⢀⠀⠐⡀⠀⠉⠁⠢⡻⠘⠂⠇⠀⠀⠀⠀⠀⠀⠀⠆⢀⡴⠞⠁⠀⣠⡇
⠀⠀⠀⠀⢯⠓⠦⣄⣀⡠⠀⠀⠀⡀⠀⠐⠈⢁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠍⣀⣠⠴⠊⣩⠁
⠀⠀⠀⠀⠈⢆⠀⠀⠈⠉⠙⠓⠒⠒⠶⠶⠤⠤⠤⠤⠤⠴⠶⠶⠒⠒⠋⠉⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠛⠛⠛⠛⠛⠫⠝⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡶⠶⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⠿⢛⣷⣀⠀⠀⢀⣀⣠⣤⣶⠶⠟⣛⣻⣭⣧⣤⡶⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡯⡆⠛⢀⡚⠩⡬⢻⣷⣿⣛⣭⣭⡶⠶⠿⠛⣛⣉⣩⣥⣤⣶⠶⠾⠟⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣩⡥⠛⠯⢀⡙⠣⠴⠀⣿⣟⣭⣥⡶⠶⠾⣟⣛⣻⣭⣽⣷⠷⠶⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⡚⡍⣩⡤⠷⣶⣞⣫⣭⣭⣷⠶⠾⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠿⠾⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣤⣤⡶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣠⣴⡿⠙⢛⡛⠉⠁⣀⠙⣛⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠟⠛⣋⡡⠆⡀⠉⠩⣤⡄⠙⠁⠉⣡⡝⠿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠶⠿⠉⣀⣤⠆⣉⠀⡈⠋⠀⠶⠶⠤⠋⢋⠀⠐⠷⣤⡌⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢋⣤⡿⠉⠁⠛⢃⣈⣈⠺⢿⡟⠋⠘⣉⠉⢧⣤⡃⠀⠙⣂⡀⠻⢷⣶⡆
⠀⠀⠀⠀⠀⠀⢰⣿⠛⠡⠟⠉⢀⣐⣛⣁⣀⣈⣉⣀⣐⣒⣤⣀⣛⣋⣀⣀⣈⣁⣀⣀⣉⣁⡀⠳⠈⣻⣆
⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣧⠉⠋⠀⠛⠃⠀⠚⠂⠒⠒⠀⠒⠒⠀⠲⠶⠀⠲⠖⠀⠶⠶⠀⠰⠶⠀⠐⠒⠂⠀⠛⠃⠘⢻⣿
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠿⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⡶⠶⠿⠛⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠷⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄⡒⠒⠈⠉⠉⠉⢻⠋⠛⠀⠋⠀⠒⠒⠢⠤⢄⣀⠀⢠⢶⡆⢳⡆⠒⣦⠠⢤⠤⢤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣢⣽⣾⣿⢋⣀⣠⣤⣤⣤⣤⣼⣦⣤⣠⣄⣤⣀⣀⣀⡀⠀⠀⠉⠙⠺⢿⡛⢻⡛⠛⣟⠛⢷⠶⢷⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠈⣻⢿⡋⠻⡟⠛⢷⠺⢷⠾⣆
⠀⠀⠀⠀⠀⠀⢀⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣟⣿⣿⣿⣿⣿⡿⣷⡬⠛⢷⣄⣸⣯⣿⣟⠻⡛⠛⢷⠲⢷⠾⢆
⠀⣯⠉⠐⠐⠲⠿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣵⣿⣿⠿⠛⠑⣋⡠⢀⠀⣿⣿⣿⢿⣿⣞⠿⠟⠛⢷⠶⢷⡾⢦
⢹⡈⢆⠀⣀⠀⡀⠀⠘⢟⡛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣷⣾⣯⡾⣛⠝⡍⠀⣿⣿⣴⣿⡶⠾⢷⠶⢷⣤⢷⡀
⠀⢱⡈⢆⠁⡀⠁⠀⠀⠈⠻⣷⡊⠹⢛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⢫⣵⡾⢃⠎⣀⣴⣿⣿⣿⣿⣿⣿⡾⣧⣤⣵⣄⣸⣄
⠀⠀⠱⡀⢦⠤⢤⡀⢀⣀⣀⣈⣿⣦⣀⠀⠀⠈⠉⠛⠋⠙⠿⠿⠿⠿⠿⠟⠙⠛⠛⠋⠉⣡⣫⡾⣋⠔⢁⠜⠉⠁⠀⠿⣿⣿⣿⣿⢿⣸⣧⣀⣉⣄⡈⣆
⠀⠀⠀⠙⠤⠤⢤⢤⣤⣤⣤⣬⣿⠟⠉⠓⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠉⠀⠀⠁⢸⣄⠀⠀⣠⣿⡿⠟⣣⣿⣏⠈⣫⡉⠉⣋⠉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠝⡻⣦⣀⠀⠀⠀⠀⠁⠐⠊⠭⠄⣀⣒⣒⡐⠒⠒⣂⣴⣶⣿⣿⣿⣿⣿⠔⣒⡶⠖⢖⡿⠂⠠⡤⢤⡖⢛⣯⣉⣻⡍⠉⢻⡉⠙⠻⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠩⠟⣢⠄⠀⠀⠀⠀⠀⠀⢀⣠⣬⣾⣿⠿⠿⠛⠛⠉⠉⠉⠠⠒⢋⣁⠤⠊⠁⠀⠀⠀⠰⡞⣿⣍⣽⣯⣉⣿⣍⣉⣻⡛⠛⢻⣟⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣶⡿⢿⣿⣿⣟⣛⣉⡀⢀⠀⣀⣀⣀⠤⠤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠐⠤⠤⠠⠀⠤⠤
⠀⠀⠀⡤⣦⢤⣀⡀⣀⡀⣀⣠⣤⢀⡀⡀
⠀⠀⠀⠐⠃⠏⠓⠛⠓⠛⠓⠓⠓⠛⠛⠚
⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠤⠤⡤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠠⠠⢀⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⢁⡄⣀⣠⣶⣤⣄⣀⣀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡭⠔⠒⠛⣿⠁⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣀⠀⠈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⠄⠀⠀⠀⠀⠸⠗⠊⠁⠸⡄⠀⠀⠀⠀⠀⠀⠉⠑⢢⡀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠚⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠈⠓⢦⣀⣀⣾⠑⢂⠉⠁⢃⣤⣴⣾⣋⡡⠁⠀⢀⠀⠀⠀⠀⠀⢘⠀⠀⢠⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡔⠛⠉⠉⠀⠉⠀⠀⢨⠀⠏⠉⠻⢧⠀⡀⠀⡖⠲⠄⠦⠻⠁⠁⠀⡨⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⢀⣴⡦⢼⣾⣾⣷⡿⣧⠐⠈⡄⠀⠀⠀⢀⣀⡀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣤⡔⣷⠀⠀⠀⠀⠀⠈⡾⠁⡀⠠⠈⠻⣿⣁⡸⠀⠀⠈⠢⢤⡴⠚⠛⠳⢶⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⠀⠸⣤⣤⡄⢰⣦⠘⢧⠀⠙⠂⠀⠀⠈⣮⣭⡄⠀⠀⣤⡆⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠈⡁⠀⠀⠀⠀⠀⠀⢳⡜⠀⣐⠀⢀⣿⣿⣷⢞⡟⢿⡧⠀⣴⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⢦⣄⣣⠀⠀⠀⠀⠀⢠⠀⠳⢀⠈⠀⢺⣿⣿⣟⣾⣷⡟⡀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠋⠁⠀⠀⠀⠀⠈⠀⢀⡾⠀⢀⣼⣏⡿⣻⡿⣚⣜⠄⠄⠀⠀⠐⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠀⠤⢄⣀⠠⠞⢤⣀⣉⣿⣽⢏⡻⡿⢿⣸⣼⢷⡶⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠛⢦⣀⠀⢀⠿⣿⢿⡿⠷⠻⠑⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⣸⣧⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣷⣶⣿⡿⠟⠛⠛⠙⢳⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠛⠛⠛⠻⠟⠉⠀⠀⠀⠀⠀⠀⠻⣦⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠗⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⡏⠁⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠉⠀⠀⢀⣠⡴⠶⠛⠋⠁⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠈⠉⠀⠀⢠⡀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣦⡄⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠿⠟⠋⠉⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⠟
⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠉
⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⣼⣿⡿⠃
⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⣿⡟
⠈⠛⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣠⣾⣿⣿⣷⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⣾⣿⡟⠉⠛⠿⠟⠋⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⣠⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁
⠀⡁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠙⠛⠳⢦⣤⣀
⠀⠀⠀⠀⣀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣐⡛⡩⢢⠍⡐⢪⣔⠒⠠⡀⠀⠀⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠡⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢀⠰⢚⠶⠆⠀⠉⡢⠊⠓⠝⠫⢀⣠⣐⠲⠄⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠄⠁⣈⡯⡄⣕⠊⠓⡶⠉⠁⠦⡁⠑⢦⡲⢭⣉⠄⠘⠟⠂⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠰⡀⠀⠀⠀⠀⣾⠁⠀⠀⢀⡾⢠⡻⠗⢮⡊⠙⠀⠠⠑⢬⣀⠈⠙⠄⢡⠋⠁⠹⠰⠧⠀⠀⡈⢑⢄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⠀⠀⠀⠀⡇⠀⠀⡔⠈⠡⣰⠃⢀⠀⠗⠁⠀⠀⠉⡘⠟⢂⡈⠀⠲⠂⠀⡋⠈⠀⡀⡠⢄⡄⠀⠓⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⢠⠀⠀⠈⡇⠀⠐⠁⢠⠖⢳⡀⠈⠀⠣⡀⢀⠃⣀⠄⠒⠰⡇⢄⣿⢃⠀⠸⢗⡨⠊⣅⠥⣴⣠⣦⠤⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢣⠀⠀⢿⡞⠔⣀⡆⠀⠀⡵⢦⣤⣐⣀⣀⡘⠁⡠⡄⢰⡇⠀⠈⠓⠲⢄⠎⠀⢀⡃⠈⠊⡧⠄⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠘⣷⢳⠿⢣⡀⠺⠁⠈⣔⠉⢁⠀⢣⣐⠁⠐⢀⡇⠀⠀⠓⠓⣦⣞⡖⡈⠕⠀⢐⢩⠀⠂⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠈⢷⣀⢂⡞⠣⠸⠃⠠⠀⠈⠁⠾⠻⢌⠨⡀⠀⣢⠦⠤⠤⢖⡖⠰⡕⡹⠀⢌⠈⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⡆⠀⠈⢻⢿⣅⢁⠃⢄⣁⡾⠓⠺⠀⠀⠀⡑⠒⠋⠁⠀⡁⢀⡀⠀⠀⡔⠀⠸⢂⡴⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠀⠀⠱⡝⠻⣦⣴⡁⢢⠖⠶⠃⠡⢂⢟⡙⠡⢗⡦⡉⢤⣉⡠⠈⣠⣤⠾⠋⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠘⢄⡀⠉⠛⠷⢦⣴⣭⣬⣴⣼⣁⣠⣭⣂⣥⣵⣶⠶⠟⡋⠁⠈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠐⡀⠀⠀⠀⠳⢤⠀⠐⠀⠀⠈⠉⠍⠉⠉⠉⠉⠉⠉⠁⠀⠀⠐⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣡⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢰⠀⠀⠀⠀⠀⠰⣦⣍⣓⠶⠦⣤⣤⣶⣦⡴⠶⠞⣋⣥⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⣀⠣⡀⡄⠀⠀⠀⠙⢿⡿⢿⣶⣶⣶⣶⣶⣶⣿⠿⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢞⡄⢫⠹⡄⠀⠀⠀⠀⠉⠓⠲⠶⠤⠤⠴⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡙⡄⢇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈
⣿⠿⠿⠿⣿⢻⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣻⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇
⠈⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡽⠉
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠤⡀⠀⡇⠀⠀⠀⢰⡟⠉⠈⠻⣦⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣀⣀⠀⠀⠸⠀⢣⠀⠀⠀⠈⠛⢲⠂⢀⡿⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⡏⢡⣽⠇⠀⢠⠀⠀⠑⠠⠀⠠⠐⢁⣤⡟⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠸⣧⠀⠑⠠⠐⠁⠀⠀⠀⠀⠀⢀⣴⠟⠁⠠⣊⠤⠠⢀⠀⠀⠀⠀⠀⡠⠊⠁⣠⡴⠒⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢑⣄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⡠⠊⠌⠀⣠⡀⠀⢰⠀⠀⠀⡔⠁⢀⡼⠁⢀⠔⠁⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢠⣤⣤⠩⠑⢄⠀⠙⠳⣦⡀⠀⠀⢀⣾⠋⡠⠊⠀⠀⢀⣬⡿⠏⢀⠎⠀⠀⢰⠀⢠⡏⣠⠖⢁⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠻⣦⣛⣈⡀⠀⠁⠢⣀⠈⢻⣄⠀⣼⠇⡐⠁⣠⡶⠛⠋⡁⠄⠂⠁⠀⠀⠀⡆⠀⠘⢊⣡⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣀⣉⣉⣛⠛⠶⣿⣭⡷⡄⠹⡯⣿⢰⢯⡿⠉⣠⢖⣋⣀⡀⠀⠀⣀⠀⡰⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢂⡡⠴⠚⠉⠉⠉⠁⣒⣭⡳⣌⣦⣿⡟⣤⡿⣠⣊⣀⡀⠀⠈⠉⠙⠒⠲⡼⠁⣴⣿⠁
⠀⠀⠀⠀⠀⠀⢀⠔⣈⠴⠊⠁⠀⣀⡠⠴⢒⠉⠉⠉⣀⠉⠻⣧⢻⣷⣿⡷⠁⠤⠉⠉⠉⠛⠒⠶⣤⠜⢀⣾⠟⠲⣅⢀
⠀⠀⠀⠀⠀⠀⠊⡼⠁⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢿⢿⣃⠀⠠⠀⠀⠀⠀⠂⡠⠂⢀⣾⠿⢦⡀⠈⢣⠱⡀
⠀⠀⠀⠀⠀⠀⣄⢣⣀⠞⠁⠀⠀⠉⠂⠀⠈⠂⡀⠀⠀⠀⠀⠈⠛⠂⢤⠃⠀⠀⠀⠀⠀⠀⠈⣁⠀⠸⠏⠠⣀⠙⡆⢸⢇⠇
⠀⠀⠀⠀⠀⠀⢸⡢⡙⢦⡐⠂⠀⡉⠒⠀⠀⠀⠤⠀⠀⢀⡀⠀⢀⠀⠀⠉⢀⠀⠀⠀⠂⠀⠀⠉⢄⠀⠀⠄⠀⣠⠟⡫⡎
⠀⠀⠀⠀⠀⠀⠈⢏⠪⠒⠍⡓⠦⣄⣀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣠⠤⢒⡩⠒⠁⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣌⡂⠄⡀⠉⠐⠠⠍⢑⡒⢢⠤⠤⠤⠤⣤⣤⣤⣤⣤⠤⠤⠤⠴⠒⠒⣊⠩⠄⠒⠈⠀⠀⠀⠠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡦⢁⠂⠤⢀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⠦⢤⣤⣀⣀⣐⣀⣀⣀⣀⣠⣤⠤⠴⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⢀⠉⣀⣈⠀⠤⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⠶⠿⠿⠟⠻⠛⠛⠛⠛⠛⠻⠿⠿⠿⠷⢶⣶⣶⣤⢤⣀⡀
⠀⠀⠀⠀⣠⠴⢚⡭⠖⠋⠉⢁⣀⣀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀⣀⠉⠉⠓⠮⣝⠲⢤⡀
⠀⠀⠀⢰⡇⠀⣏⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣈⠇⠀⣷
⠀⠀⠀⢸⡏⠲⢬⣓⡦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⣖⣩⠴⠊⣿
⠀⠀⠀⠈⡇⠀⠀⠀⠉⠙⠛⠛⠿⠿⠷⠶⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⢶⡶⠶⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⠒⠦⠤⣤⣀⣀⣀⣀⣀⣀⡤⠤⠴⠒⠋⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣴⣶⠋⠁⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠗⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠋⠀⡀⠲⡞⠄⠀⠀⠀⠀⠉⠛⠛⠋⠉⠹⣦⡀⠀⠀⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⠀⠀⠀⠀⢀⠀⠁⢀⡆⠲⠆⠠⠀⠀⠀⠄⠁⠈⠀⠀⡈⠛⠛⢳⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠆⣠⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠠⣄⢀⡠⠒⠙⠀⠈⠁⠀⠘⣧⠀⢁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⢿⣸⣡⣁⠀⣀⠐⠃⠰⠂⠀⠀⠀⣠⠎⠀⠍⠂⠀⠀⠑⠀⠄⠀⠀⠀⠈⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢮⠉⠉⠁⠹⢣⢇⡀⠀⡠⠁⠀⢠⡶⣓⢀⡆⠘⠇⠀⠀⠀⣀⠀⠀⠀⠐⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⣵⠒⠀⠀⠀⠀⡀⠋⠔⡀⠠⠒⠰⠛⠁⠈⠙⠂⠀⠀⠀⠀⠀⠜⠀⠀⠀⡇⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢃⢠⢩⡄⠀⠂⠀⠀⢉⣤⣈⣉⡖⠓⠂⠀⠀⠀⠀⢠⡱⣤⣄⡀⠜⠀⠀⠀⠀⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⢧⠣⡻⣄⠲⡐⢛⠿⠽⠏⡩⣟⡽⡃⠀⠀⠀⠀⢀⡹⣣⠊⣱⡄⠛⠄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⢳⡑⡌⢷⢸⠃⠀⠀⠈⠙⠂⠼⣱⡆⠀⠁⠀⠀⠀⠀⠛⣕⡤⢵⠈⢀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⣷⢹⡸⡼⡇⠀⠀⠀⠁⠀⡀⣜⡢⠚⠀⠠⠄⠀⠀⠀⢸⠅⡾⠀⣾⢰⠰⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢿⡎⠇⢧⠈⢶⣦⣄⡀⡀⢡⢸⣅⠂⠀⣠⡠⠤⠀⠐⠭⠞⢁⣤⢁⢹⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⠸⡌⢣⣀⠭⠭⢚⣡⢎⣚⡭⠵⣂⠀⣀⠀⡀⠤⠴⠒⠋⠑⣋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠦⠭⠛⠋⠁⠠⠌⢕⣚⣭⡤⢒⡡⣬⠭⢕⣒⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠷⣬⣭⣭⣭⣶⣾⣷⡤⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠿⣿⣛⢛⠻⡻⣛⢛⣛⡿⠿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⢟⣫⣭⣴⣼⣯⣩⣽⣜⠂⠓⠯⡜⡱⣋⢽⢒⡎⣭⣛⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⣫⠞⢋⢀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⣦⠙⢛⣉⣬⣥⣤⣤⣴⣮⡟⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠛⠅⣰⠋⢰⠋⢉⢀⠇⠀⠀⠐⠚⠀⢀⣤⡶⠟⣿⠉⡩⣤⣰⠶⢶⣶⢸⡇⣳⣝⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⡉⣼⠁⠀⠀⠀⠂⠋⡀⠐⠀⠀⢀⡴⠋⡕⢶⠀⣿⠀⡆⣝⠀⠂⣸⢻⢸⡇⣿⡶⣻⡽⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠐⠠⢀⡇⠀⠠⠀⠀⠐⠀⢼⣿⠄⣰⠟⡴⢖⡌⠨⡄⢸⡇⠣⠉⠠⠤⠁⠺⢸⡇⢿⣿⡽⢞⣎⢿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠈⠀⠄⢸⠀⠀⠀⢀⣀⣠⣤⣤⣄⣼⣏⡰⢜⣌⠖⠂⣃⣸⣷⡴⠶⠶⠞⠛⠛⠻⡟⠛⠷⣼⠋⠭⢆⢿⣿⡄
⠀⠀⠀⠀⠀⠀⣸⢻⡏⠐⠀⡀⠐⣘⣧⣤⠞⢉⡍⡤⢤⠀⣿⠉⡉⠛⢦⣩⡴⢻⢉⡄⢸⡆⠀⡚⢃⠋⢿⢸⡇⠀⢠⠀⠙⢦⡙⡜⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠁⣠⠔⠋⢉⠉⠻⡄⠋⢃⢓⢸⡆⢻⡄⠆⠀⠃⢻⡄⢠⢼⠠⠄⣷⠰⠦⣬⣉⡺⣸⠃⠀⠈⠀⠀⠀⠹⣦⢹⣿⡆
⠀⠀⠀⠀⠀⢸⣧⡟⣠⠈⢀⠐⠀⠀⠀⠀⠹⣆⠈⠀⢈⡴⠛⠻⣄⠑⢤⡌⡇⣀⣼⡦⠶⠿⠶⠶⢶⣍⣡⡏⠀⠐⠂⠀⣤⡀⣀⢸⡎⣿⠇
⠀⠀⠀⠀⠀⢸⣿⡇⠙⠄⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⠋⡀⢒⠠⠊⠀⢠⣼⣿⡉⠔⠀⠀⠀⠀⡰⢀⡼⣿⠰⣿⠖⠐⠂⣈⣀⡍⠀⡇⣿⡄
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠘⢦⡀⠀⠀⠀⠀⠀⣠⠞⠙⢦⡐⠃⠀⢀⡴⠟⡉⢹⡙⠲⠦⡤⠴⣶⠟⡋⢀⠈⢷⠀⠀⠂⠀⠈⠉⠀⡼⠃⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠐⢀⠑⢄⠀⢀⡤⠞⠁⡄⠲⣤⠙⠳⣤⠟⠀⠀⠀⢸⡇⡀⠤⢀⡀⣿⢸⡇⠤⣛⡈⣧⠀⢠⡄⠀⠀⠎⣕⢸⣿⠃
⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢀⠸⢂⡿⣏⡠⠚⢜⡦⡱⢈⡱⠇⡿⠀⠀⠀⠀⣼⡁⠀⠚⢠⢠⡟⠸⢧⣅⡃⣇⠸⡆⠀⠀⣠⠌⢄⡄⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠥⡸⡇⠀⠑⠦⣌⣙⠂⠃⠘⠃⢷⡀⠀⢀⡼⠁⣉⠛⠒⠒⣾⠗⠶⢤⣌⣉⡉⢀⣇⢀⡴⣡⠐⡾⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠠⠸⠆⠀⠤⡀⠉⠉⠉⠛⣛⡛⢷⡚⢋⡠⢖⡍⢻⠆⢀⡿⠀⠆⠤⣀⣉⠙⢻⡟⣩⠾⢏⣙⣽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡠⢠⡧⠀⠀⠁⠀⠀⠂⠘⠿⠃⠈⠳⣌⠁⣬⣂⣝⠃⣼⠃⡘⠄⠉⢑⣾⠀⣼⢡⡏⢡⢬⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣌⣠⡱⡄⠀⠀⠠⠄⣰⢀⠐⠂⠀⠈⠛⢦⣝⣋⣼⠇⢰⣜⠒⠤⢼⠇⣰⠇⣶⢘⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣘⢦⡀⠀⡼⡀⠠⠀⠀⠐⠒⠀⠀⠀⣹⢟⠳⢦⣬⣉⣒⠋⣰⡟⣸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣯⣖⡦⠤⠀⠤⠤⠤⠴⠒⢒⢚⠱⠜⢙⡓⠆⠭⣉⣭⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⠙⠿⣿⣿⣿⣶⣶⣤⣿⣴⣤⣮⣧⣬⣼⣶⣶⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⡀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢀⣀⠠⠖⠰⠖⠀⠐⠚⠛⠒⠦⣄⠀⠉⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡹⣏⠀⠀⠀⠀⠠⠀⠁⠀⠂⠀⠀⠀⠈⠈⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢣⣀
⠀⠀⠀⠀⠈⢦⡑⠀⡀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠙⢮⡂⠲⢦⣄⡀⠀⠀⠀⢀⣤⡉⠁⣒⡼⢖⣥⣀⣤⠖⠁⠀⠢⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠉⠋⢄⡀⠀⠀⠀⠀⠀⠀⠙⢤⣤⣤⡾⠉⠃⠘⠛⠙⠋⠋⠉⠉⠠⠽⡶⠛⢩⠋⣿⠃⠀⠀⠀⠀⢳⡐⢂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⠄⠚⢻⡟⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢠⠟⠒⠦⡄⠀⠰⠂⡷⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡎⠀⠀⠈⢳⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠈⠀⠀⠀⠘⡄⠀⠀⠰⣆⠀⠀⠀⠀⠐⠲⠶⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢄⠀⠰⡀⠀⠀⠀⠙⠳⣅⣀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠃⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢇⠰⠦⡀⠈⠢⢄⠀⠀⠀⠀⠈⠉⠖⠒⠒⠞⠛⠛⠉⠁⠀⠀⠀⣀⡤⠊
⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠁⠒⠦⠤⢼⣀⣀⣀⣀⣀⣀⡀⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠁⠀⠠⠤⠀⠀⠀⣤⠰⠇
⠀⠀⠀⣄⡀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠠⠵⠀⠀⠀⠘⢦⠈⣀⣀⣀⣄⣠⢰⠿⠃
⠀⠀⠀⠈⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⣠⡶⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⠋⠉⠉⣉⣉⣁⣀⣀⣀⣀⣀⣈⣉⡉⠉⠉⠛⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠋⢉⣀⣤⣤⠶⠶⠖⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠓⠶⢦⣄⡀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣀⣤⡴⠶⠛⠋⠉⠀⣀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠶⠒⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⠿⢿⣷⣄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⢀⣠⡶⠛⠉⠀⠀⣀⣤⠴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠛⠁⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⣼⠃
⠀⠀⠀⢀⡾⠁⠀⢠⡟⠁⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⣠⡾⠃
⠀⠀⠀⢸⡇⠀⠀⠘⣧⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁⠀⢀⣠⡾⠋
⠀⠀⠀⠈⢷⡀⠀⠀⠈⠻⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠛⠋⠁⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠈⠙⠛⠳⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠶⠖⠛⠛⠋⠉⠁⠀⠀⠀⣀⣠⡴⠶⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠶⠶⠛⠋⠉⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣛⠛⠛⠓⠒⠲⠶⠖⠒⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠋⠉⠙⠓⠲⣄
⠀⠀⠀⢀⣤⡶⠤⠤⣀⠀⢠⡴⠶⠒⠒⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⣠⡤⠚⠉⠀⠀⠀⠀⠀⠑⠮⠆⠀⠀⠀⢀⢈⣃⣀⣤⣤⡤⠤⠤⡤⣤⣤⣤⣄⣀⣀⡀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠚⠁
⠀⠉⠙⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢽⠋⠁⣀⢩⡤⠤⣤⣧⣤⡀⠀⠀⠀⠀⠉⠉⠙⠋⠄⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⡉⠁⠀⠂⠀⠠⠤⠤⠴⠤⡀⠸⣿⠛⠉⠀⠀⠀⠀⠀⣀⠈⠉⠿⢿⣷⡦⠤⣤⣤⣄⡀⠀⠀⢦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣸⣷⠶⠂⠀⠀⠈⠀⠀⠀⠈⠆⠠⠀⠀⠀⠀⠀⠛⠛⠻⠶⢬⣈⢢⡀
⠀⠀⠦⣤⣤⣴⣶⣥⣤⡤⠾⠿⠛⠚⠛⠛⠛⠛⠋⠉⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠈⠣⣷
⠀⠀⠀⡄⣿⣿⠻⠭⠁⠀⠀⠁⠈⠁⠉⠉⠁⠈⠁⠈⠉⠙⠻⠿⢍⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠒⠀⠀⢰⣿
⠀⠀⠀⣷⡈⠳⣦⡀⠀⠀⡀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠦⣤⠄⢀⣠⠖⣫⡟
⠀⠀⠀⢹⡹⣦⣈⠉⠛⠶⢤⣬⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠨⠂⠉⠀⢀⣀⣀⣤⠶⠚⢉⣤⠞⢩⡇
⠀⠀⠀⠈⣇⠀⠈⠙⠲⠤⢄⣀⡉⠉⠙⠋⠙⠛⠒⠒⠒⠒⠲⠶⠖⠒⠲⠖⠚⠛⢛⣉⣁⣤⡤⠶⠞⠉⠀⠀⣸
⠀⠀⠀⠀⠘⢦⠄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠙⠭⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡄⠄⠸⠦⠵⢦⣄⣀⣀
⠀⠀⢀⣴⡉⢉⣱⠶⠒⣉⡡⠤⢄⣲⡶⣂⣐⣒⡒⠒⠒⠢⠤⢍⣙⡒⠤⣄⡀
⠀⣰⠁⣠⠖⢉⣤⣲⣭⠴⠶⣟⠫⢱⠖⣿⠋⠉⢉⠿⣗⠒⠒⠲⠶⢭⣕⡢⣈⠢⢄
⢀⣿⠞⢁⣴⠟⣩⢴⡃⢀⣠⢼⡷⠊⠉⠉⠉⠲⣁⣠⠂⠀⣀⣠⣄⠈⢙⣿⣿⣷⡄⠱⡀
⣸⠃⢠⡟⠁⠀⠙⠚⣋⣉⡀⠀⠧⣀⠀⠀⠀⢠⡏⣲⠬⠑⢊⠥⠊⠙⣯⡇⠐⢿⣿⡆⠘⡄
⣿⠀⡎⠀⠀⢀⢠⠊⠁⠈⠑⡦⠁⣈⠑⠲⠢⠞⢱⣃⣠⠚⠁⡰⠃⠉⢹⣤⣤⣭⠟⣿⠀⢣
⢻⠀⢷⠾⣆⠀⢿⡀⣠⣴⡚⣧⡘⠭⠟⠀⠀⣀⣄⡉⠀⣴⠋⣀⡀⢀⡿⠀⠀⠀⠀⠸⠀⣾⡀
⠀⣄⠘⡄⠀⣳⣠⠽⣹⡗⠀⠀⠋⠳⡴⣄⠀⡃⠀⠉⢣⠈⠋⠋⠙⢉⣤⠀⠀⠀⡤⠃⣰⣯⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢒⠟⡇
⢰⠏⠢⡈⠲⣇⠈⠒⠉⢯⡀⠀⠀⠀⠑⢹⠂⠱⠤⠤⠎⠀⠀⢀⣠⣷⣴⢋⡠⠎⢀⣴⣏⠈⢻⣶⣑⢄⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⡇⠀⢇
⡜⠀⠀⠈⠒⢌⡒⠤⣀⠀⠉⠳⠦⡵⠷⡛⢀⣢⣀⣀⡈⢀⡖⠉⣀⡠⠜⢋⣠⡶⠋⣀⠈⠳⠀⡇⠉⢢⡓⣄⠀⠀⠀⡠⠉⠀⠀⢀⠀⡘⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠉⠑⠲⠭⣉⡒⠒⠣⠴⠾⠯⠤⠤⠭⠛⢛⢋⣉⠤⠒⠊⠉⠉⠣⡀⠜⢳⠀⠀⠇⠀⠀⠑⣌⠢⣀⡴⠀⠀⠀⣄⡠⠊
⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠃⢀⡜⠀⠀⠀⠀⠀⠳⣌⠙⠶⠶⠚⠉⠀⠀⠀⠀⠀⢠
⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⡀⠀⠀⣀⠴⠃
⠀⠀⠘⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⣿⡏⠳⡦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⠟⢩⡿⠀⠀⣀⣀⠤⠤⠴⣒⣒⣒⣒⣒⣒⠰⠦⢤⣀⡀
⠀⠀⠀⠸⣧⠀⠑⣀⠀⠉⠙⠒⠒⠦⠤⠤⠤⠤⠤⠖⠒⠚⠉⠉⠀⡠⠋⣨⣾⡿⠟⢋⡡⠴⠒⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⢤⣉⠓⠠⣀
⠀⠀⠀⠀⠈⠧⣄⠈⠑⠦⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠒⠉⣡⣴⡿⠋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠈⠑⠠⢄⣀⠀⠀⠉⠉⠉⠙⠉⠉⠉⣁⣀⡤⠶⠋⢹⡏⠀⠀⡐⠀⠀⠀⠀⢀⠤⠒⠉⠉⠉⠀⠉⠉⠒⢤⠀⠀⠀⠀⠱⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠻⠟⠛⠻⣏⠀⠀⠀⢸⠀⠀⠀⢁⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⢃⠀⠀⠸⡀⠀⠀⠈⢦⡀⠀⠀⠀⠉⠒⠂⠤⠤⠤⠤⠴⠒⠁⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡢⠤⠊⡄⠀⠀⠱⡀⠀⠀⠀⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠃⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠘⢆⠀⠀⠈⠳⢄⠀⠀⠀⠉⠛⠻⠶⠖⠰⡤⢖⣶⣶⠾⠟⠋⠁⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠢⡀⠀⠀⠀⠉⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠒⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠀⠄⠠⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡤⢖⡛⣷⣬⠋⠣⡿⢂⣨⣧⠹⠦⠤⣽⡖⠠⢄⠀⠈⢢⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⠀⣰⣫⣷⣤⡀⠙⠅⢈⣦⠄⣶⠳⢒⢁⡖⠄⠈⠁⠀⠠⠱⠄⠀⣇⠈⠑⢌⠢⡀
⠀⠀⠀⠀⠀⠐⠰⠉⠀⠉⢧⠻⢰⡀⠙⠃⠁⠛⠛⠠⠂⠀⠔⡀⠋⣀⠀⢠⠀⠈⠰⠆⠀⡰⠃⠐⡀⠀⡑⡀
⠀⠀⠀⠀⠀⢀⠐⣄⠀⠐⡀⠐⣌⡃⠼⢃⡀⠀⠛⠘⠀⠰⠈⢁⣀⣹⣀⠔⠀⠠⠄⡠⠔⠁⠀⠀⠈⠂⡰⠁
⠀⠀⠀⠀⠀⠀⠁⡤⡁⠂⠈⠓⠄⠉⠑⠒⠤⠤⣀⣀⠀⠚⠒⠒⢚⣛⠉⠡⠄⠒⠋⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⣿⡆⠸⣦⠀⠀⠀⢰⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣉⡀⠀⠀⠸⣿⡃⠀⣿⡆⠀⠀⢠⣿⡃⢉⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⠛⠛⣻⡆⠀⠀⠻⣧⣼⡿⠀⠀⠀⢸⡟⠛⠛⠋⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠘⠿⠿⠟⠁⣠⣀⣤⣤⣤⣤⣤⣤⣤⣈⡀⠀⠀⢀⣠⣴⠶⠿⠛⠛⠛⠙⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣀⣴⣶⡿⠟⣛⣋⣩⣭⣭⣭⣭⣍⣙⡛⢿⣿⠶⠛⢉⠀⠀⠀⠀⠀⣠⣾⡿⠋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣇⣀⣴⡾⠟⣩⣴⣞⣛⣋⢉⣓⡡⢀⣂⠬⢉⣹⠿⠋⣥⡤⠈⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣋⣴⠟⠳⢒⣐⢒⣒⣐⣒⡊⡑⣢⡼⠟⠁⠀⠀⠉⣿⣦⡀⠀⣠⣾⡿⠋⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢻⣿⣿⡭⣐⡀⢑⣒⡀⣒⣒⢐⣲⡾⠋⢀⠰⣿⣦⣄⡀⠸⣇⣱⣾⡿⠋⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣱⠟⠬⠭⠿⣶⡤⠔⡐⠤⢐⣰⡾⠋⡠⡂⢅⠈⠀⠙⣯⡉⣳⣾⣿⣋⡑⢦⡀⠀⠠⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⣸⡿⣰⠏⣭⠭⠭⠭⠝⢻⣷⣂⣡⠞⠃⠔⠁⡂⠐⠁⠀⠀⠀⣠⣿⣿⡋⢱⡈⠙⠛⢻⣶⠁⣰⣏⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢸⣿⢣⡿⠤⢀⠨⠭⠀⠭⠥⣭⡿⣧⣠⡴⠴⢦⣤⣀⠀⠀⣠⣾⡿⠋⠀⠙⢮⣷⠠⠂⠂⢁⣼⢻⣿⡄⠁⢉⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⠀⣻⡟⣼⡃⠍⢐⡚⠉⣐⣉⡾⢫⢔⣿⣿⣿⣿⣿⣷⣯⣷⢾⡿⠋⠀⠀⠀⠀⠀⠻⠟⠀⣠⢿⣹⡆⣿⡆⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⣿⡇⣿⢒⣀⡒⢒⣁⣶⠋⣰⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣄⠀⠀⠀⠀⠠⠀⣠⡾⠋⠐⢺⡇⣿⣧⢀⣉⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣿⡇⣿⣐⣒⡂⣰⠟⢁⣴⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣿⣿⣿⣿⡆⠄⢀⠔⢁⣴⠯⠁⠫⠭⢸⡇⣿⡧⢈⣉⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣁⠡⠀⢻⣷⢹⡆⢒⡾⠃⣰⣿⣿⣿⣿⣿⣿⣿⢿⣾⣻⣽⣿⣿⣿⣿⣻⡇⠠⢂⣴⠿⠥⠉⠤⠭⠉⣼⠃⣿⡇⢈⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣈⣁⠘⣿⡌⣷⠟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢿⢿⣀⣴⢟⣁⠭⡍⣀⣍⣭⢩⡟⣸⣿⠀⣉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢹⡿⠋⣰⢟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡵⣫⡼⣿⣿⠉⠙⠛⠋⠙⣿⣿⡿⣱⣿⢇⣘⡛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡴⣷⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⢿⢋⣨⡾⠫⣐⣐⣟⣛⠛⣫⣹⣋⣉⡝⣱⣿⠟⠈⠙⠻⠗⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⣿⡿⡟⣁⣴⠟⢓⡶⠄⣒⠢⢛⡛⠿⢿⣿⡶⢋⣼⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⠟⢋⣠⡾⠯⠍⠭⠍⠈⠭⢁⡨⢑⣒⣤⠾⣋⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠘⢯⢷⡿⣿⣿⣿⡿⣿⣯⠟⠏⣠⡶⣟⣣⣈⣩⣤⣉⣡⣤⣉⣥⠶⠖⣫⣵⣾⠟⠋⠀⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⣠⣾⡷⢿⣷⣾⣾⠿⢉⣨⡶⠿⠿⣷⣶⣤⣭⣭⣭⣭⣭⣭⣴⣶⢾⠿⠛⠋⠀⠀⠀⠀⠀⠀⠙⣦⣄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⠋⠀⠀⣉⣩⣴⠾⠛⠁⠀⠀⠀⠀⣰⠉⢩⡏⢩⡋⢩⡉⢈⠁⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠾⠇⢸⠇⠸⡇⠈⠆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⠚⠛⢳⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⢏⠀⠈⠖⠀⠀⠠⡭⠉⠈⠙⠻⡟⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠁⠀⠀⠔⠈⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡛⠉⢀⠀⠀⠁⡀⠀⠆⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠴⠀⠀⡤⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⢀⡄⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⢀⣸⡇⣠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⡀⣿⣄⡀
⠀⠀⠀⠀⢀⡴⠋⣼⠁⠠⡄⠀⠀⠓⠀⠀⠀⡀⠀⠀⠀⠈⠂⠀⠀⡄⠀⠀⠰⠀⠀⠀⠀⠀⠠⢠⣏⠈⣷⡄
⠀⠀⠀⠀⢸⣧⠀⢿⣝⠀⠀⠀⠘⠀⠀⠀⠘⠍⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢠⠟⡇
⠀⠀⠀⠀⢸⡈⠳⢦⣈⠓⢦⣄⡁⠀⠊⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠲⠀⠀⣀⣤⠞⢉⡴⠋⢰⡇
⠀⠀⠀⠀⠈⣇⠀⠀⠈⠓⠦⣌⣙⠶⠶⠶⢦⣤⣤⣌⣂⣀⣀⣀⣀⣀⣀⣀⡤⠴⠶⠞⣁⡤⠖⠋⠀⠀⡾
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠦⢤⣤⣄⣉⣉⣉⣉⣉⣡⡤⠤⠖⠚⠋⠁⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣈⠉⠓⠲⠶⠤⠤⣤⣤⣤⣄⣤⣤⣤⡤⠤⠖⠛⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⣤⣤⣤⣀⣤⣤⣤⣤⣤⠴⠶⠚⠋⠁

⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⣿⠋⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠻⣦⣀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠉⠻⣦⣤⣤⣴⣶⠶⠶⠶⠾⠿⠿⠿⠶⠶⠶⢶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣷⣤⡀⠈⠻⣧⣀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⢉⣻⣶⣄⠈⠙⠛⠻⣶⣤⣼⠏⠀⠀⠀⠀⢿⡄⠀⠀⠀⠘⣷⣄⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣴⠟⠉⠉⢻⣧⠀⠀⠀⠈⣹⣷⣤⣦⣤⣤⣤⡾⠃⠀⠀⠀⠀⠈⠙⠻⣶⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠉⠀⢀⣀⡀⠙⢷⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠿⢶⣶⡶⠟⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⢿⡟⢷⣦⣀⡀⠙⠛⠛⠂⠀⢹⡟⠀⠀⠀⠀⠰⠶⢶⡆⠀⠀⠙⠁⠀⠀⠀⠀⣀⣠⣶⠟⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠈⠙⠛⠷⢶⣦⣤⣤⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⠟⠛⠉⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠛⠻⠷⣶⣤⣀
⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡟⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⠃⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⡁⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠛⠉⠙⠛⠻⢿⣶⣄⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣁⠀⠀⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣤⣤⣤⠤⠤⠴⠶⠶⠶⠶⠆⠰⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⢤⣤⣤⣄⣀
⠀⠀⠀⠀⢿⠛⠲⠶⠶⠶⣤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⣤⠶⠶⠶⠚⢻⡇
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⢷⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣰⠇⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⣧⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣼⠁⠀⠀⠀⢠⡟⠀⠀⠀⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠘⣧⠀⠀⠀⠀⢷⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠀⠀⠀⠀⣰⠇⠀⠀⠀⢠⡟⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠈⢷⡀⠀⠀⠈⣧⠀⠀⠀⠀⣧⠀⠀⠀⠀⡿⠀⠀⠀⢠⡏⠀⠀⠀⢠⡟⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠻⣄⠀⠀⠘⣧⠀⠀⠀⢹⡄⠀⠀⢸⡇⠀⠀⢠⡟⠀⠀⠀⣰⠏⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠙⢦⡀⠀⠘⣧⠀⠀⠈⣧⠀⠀⣼⠀⠀⢠⡟⠀⠀⢀⡾⠁⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣦⣤⣿⣦⣤⣬⣷⣤⣤⣼⣦⣤⣿⣤⣴⣯⣤⣤⣴⣯⣤⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠤⠤⠤⠤⠤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡶⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⣀⣀⣀⣀⣀⠀⣀⣤⣤⡾⠁⠀⠀⠀⣠⡮
⠀⠀⠀⠀⠀⢀⣴⣾⡭⠂⢀⡀⣀⣤⣦⣤⠶⠶⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⠾⠟⠀⠀⠀⣠⣾⣯⡀
⠀⠀⢀⡔⣰⣿⣿⣷⣶⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⠾⣿⣿⠓⠀⠀⠀⠢
⠀⢀⢎⣾⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠠⣤⡤⠀⠀⠀⠀⠀⣀⠘⠿⠃⠀⠀⢤⠤⠄⠀⠈⠑⠂⡼⠋⠀⠀⠙⢷⣄⠀⠀⠀⢲
⠀⠀⣸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠁⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⡇⡆
⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠙⠻⠆⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣸⡇⣷
⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠃
⠀⠀⢀⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢷⠄⠀⠀⠖⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢟⡎
⠀⠀⠈⣧⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢫⣾⣿⣷⣶⣤⣀
⠀⠀⠀⠹⠱⣬⡳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢊⣵⣟⣻⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠻⣿⣦⣝⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣊⣥⣾⣿⣿⣿⡏⠀⠈⠉⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣬⣟⣓⠦⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⢒⣈⣥⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣗⡚⡟⠛⠛⠿⠿⣷⣶⣶⣤⣭⣭⣀⣀⣀⣒⣀⣐⣂⣉⣉⣭⣭⣴⣶⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣴⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⢁⠀⠀⠀⠀⠀⠙⠯⠛⠛⢛⠩⣿⢿⣿⢿⣿⣿⣿⣿⣿⡿⡿⠿⢽⢿⢟⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣽⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⣿⣿⡿⢿⠿⣷⡖⣛⣛⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠑⠚⠋⠉
⠀⠀⠀⠀⠀⠀⢴⣤⣽⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⣾⢻⡋⠛⠿⠁⡛⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠭⠭⠉⠙⠛⢛⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣬⣤⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⣄⡀⣄⢈⣩⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣾⣤⣦⣄⣠⣹⣣⣼⣷⣶⣶⣶⣭⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢠⠀⡄⡠⠰⠂⢠⠀⡂⢰⠂⡄⢠⡠⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠛⠀⠉⠃⠐⠃⠂⠘⠂⠀⠘⠃⠀⠃⠐⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠒⡆⠀⠁⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡽⠂⠀⠉⠀⠐⠉⠉⠀⠂⢸⡈⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢂⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠷⢾⣩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⡼⡷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣷⣖⣏⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡈⣮⣿⢵⣃⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢎⣩⠿⣿⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢛⠚⢶⣦⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠞⠀⢀⡏⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠰⠂⡀⢉⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⢐⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃⠀⣰⠦⡀⡘⢀⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠛⠛⢾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⡱⠦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡤⠔⠒⣡⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠛⠩⢧⣏⡀⢠⢿⣷⣶⣖⠟⠧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢀⣀⠀⠘⠲⢋⣽⣿⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⣴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣤⢀⣿⣿⢀⣀⠀⢀⣤⡘⣾⢧⠆⠀⠀⣠⣴⡟⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⡆⠁⣾⣿⣿⣿⣵⣴⣿⣿⣿⡁⠀⠀⣰⣦⣽⣿⠁⠂⠈⢱
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⠿⠿⠿⢿⣾⣿⣿⣿⣍⢣⡟⠉⠸⣟⣷⣤⠿⠀⠅⠾⠿⣧⣤⣶⡇⠶
⠀⠀⠀⢀⣦⢠⠂⠀⠀⠀⣠⡄⠿⢏⣉⢻⣿⣿⣿⣯⠐⢂⢀⣠⡶⠾⠀⠤⢀⣆⠠⣀⣽⠿⣧⣤⣷⡀
⠀⠀⠀⠈⠛⢛⣄⡭⣄⣀⠀⢀⣤⣾⣿⣿⣿⣿⡾⢃⣾⠛⠉⠀⠀⠀⠀⠀⠀⠈⠢⣸⠗⠈⣿⣃⡉⠁
⠀⠀⠀⠸⠿⣿⣷⣽⣿⣟⣟⡬⠿⠟⠛⣿⣿⣿⠀⣿⣽⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠘⣀⠊⠉⢻⣧
⠀⠀⠀⠀⠀⠻⢿⡿⠛⠻⠿⢬⡡⢄⣀⣼⣿⣿⣿⣿⣿⣄⠀⠂⠀⠀⢀⣀⢤⡔⠂⠔⣟⣄⣄⣺⠙
⠀⠀⠀⠀⢀⣤⣄⣀⠀⢸⣿⣷⣾⠶⠖⠛⠉⠉⠉⠉⠉⠛⠓⠦⢅⣤⣾⣟⣥⣶⣞⣳⣿⣏⣙⡃⠀⡀
⠀⠀⠀⠀⠈⠙⠿⠋⢁⣾⡿⠟⠃⠀⡴⠀⢠⠂⢾⡷⠀⠀⠀⠀⠀⠈⢙⡻⣿⣿⠿⢿⢿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⢀⣠⣦⡞⠋⠀⣠⣶⣿⡇⠀⡏⢽⣶⣶⣤⠀⠀⠀⠀⠀⠈⠉⠸⣿⡶⢀⣌⡿⠛⠛⠁⠪⠂
⠀⠀⠀⠀⠀⠀⠙⠛⢿⡤⢾⣿⣟⣿⣿⡀⢻⣽⣿⠿⣻⡻⢿⣿⣷⣶⣦⣴⠒⠉⠁⠈⠇⢂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠾⢿⣿⣿⣿⣿⣤⣉⣵⠿⠋⠉⠁⢿⣿⣿⣿⣿⠳⣇⠂⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠟⠋⠀⠀⠀⠀⠃⠀⠈⠈⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿
⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⠮⠭⠭⠭⠍⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠭⠭⠭⠭⢶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢀⡤⣖⡿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣷⡦⣄
⠀⠀⠀⠀⡴⢫⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣴⠦⠤⠤⠤⠤⠤⠤⠶⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢪⠳⡄
⠀⠀⠀⠀⡇⠸⣄⠀⢀⣠⣤⠶⠶⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠶⠶⣤⣄⡀⢀⡼⠀⡇
⠀⠀⠀⠀⠘⢦⡀⠙⠛⠥⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠜⠛⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠘⣗⠢⠤⢄⣀⡉⠉⠑⠒⠒⠢⠤⠤⠤⠤⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠤⠒⠒⠒⠉⢉⣁⣀⠤⠴⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠁

⠀⠀⠀⠀⠐⠛⡿⠄

⠀⠀⢀⠀⣙⢀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠘⠼⠷⠞⡟⠿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣆⢠⣀⡀
⣷⡦⠀⠀⠐⢲⣦⠤⣀⢠⣶⣀⣄⣀⡙⢀⠀⠀⣦⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⣴⣈⣠⢀⡀⠈⢹⠋⠉⠀⠀⠈⡋⡆
⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠈⠈⠉⠓⠚⠃⠉⠉⠉⠉⠉⠉⢈⠀⠀⠀⠉⠉⠉⠉⠉⠉⠑⠚⠛⠷⣦⡤⠄⠐⠾
⠋⣷⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠦⢤⣀⡀⠀⠀⠀⣀⢀
⣄⣀⠀⠀⠀⠶⡇⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠾⣿⣿⣷
⢷⡏⣀⣀⠀⠀⣳⡠⠷⢄⣆⣤⣴⣶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠉⠈⠁⢉⣩⣿⣷⣿⣶⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⢀⣤
⣷⣿⣿⣿⣿⣿⣿⣯⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⠋
⣿⣿⣿⣿⣛⣻⠷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣄⠀⠀⢀
⣿⣷⣽⡻⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠋⠉

⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣞⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⡞⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⡇⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣽⣿
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⡏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⣿⣿⣽⣿⣶⣿⣿⣽⣼⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢀⣿⣿⣼⣿⣿⣿⣿⣼⣧⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣿⡿⠛⠉⠉⠉⠙⢿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣿⠀⠀⠀⣶⠀⠀⠀⢸⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⣆⡀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⡿⢿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢹⣿⣿⡶⠶⠶⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⡇⢸⣿⣿⣿⣇⢀⣿⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⢹⣿⣯⠍⢽⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣿⣿⡷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣷⣶⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣿⣟⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⢆⡁⡶⠀⠀⠂⠀⢤⣀⡈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠠⠄⠀⠨⠄⢠⠤⠀⠀⠃⠀⠾⠁⡈⣃⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢂⠀⠄⠆⠀⠂⠈⡋⢂⠀⠀⠀⠀⠀⠀⠀⠢⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢳⠑⠆⠀⠀⠀⠀⠀⠐⠁⠀⠐⠃⠀⡃⠀⠀⠀⠀⢀⣀⠅⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠠⢉⡅⡜
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⡠⠔⠂⢉⡁
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢰⠙⠓⠤⣄⣀⣀⠀⠀⣀⣀⠤⠒⠟⢁⠀⠀⠀⠀⠀⢈⣄⢤
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠢⢄⠁⠛⠙⠀⠈⠁⠀⣠⡴⠒⠁⠀⠀⡠⠔⠊⠉⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠤⠴⠖⠀⠀⢂⣁⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠮⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡄⠤⢠⠀⠀⣤⢠⠄⡠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠂⠈⠠⠀⠀⢀⠔⠈⠀⠀⠀⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠢⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⢀⠀⠀⣀⡀⠉⠂⢄⠀⠱⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠐⠀⠀⡁⠀⠀⠀⠀⠀⠀⠌⠀⢀⡨⠤⠤⠤⣀⡈⠑⠢⣀⠱⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⡠⠖⠉⠁⠉⠉⠙⠢⢀⠀⠀⣀⢢⢀⡀⠠⠀⢀⠞⠁⠀⠀⠀⠀⠀⠉⠢⡀⠈⠡⣽⠀⠀⠀⠰
⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠑⠀⡀⠀⠀⢳⡀⠀⣀⡤⠠⣤⣄⠀⠀⠀⠈⠄⠀⠘⠁⠀⠀⡐
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡞⢃⡀⠄⠀⠀⠀⠈⠙⢦⠀⠀⢠⠀⠀⡇⠀⢠⠅
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠀⠄⡀⠀⠀⠀⢃⠀⠀⠀⣜⣠⠴⠃
⠀⠀⠀⠀⠻⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠆⠀⠀⠀⠀⠀⠢⡀⠆⠀⠀⠈⡂⠏⣁⠘⠃⠇
⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠄⠀⠀⠀⠈⣀⢉⡃⡉⠀⠀⢀⣿⣿⣿⣴⣷⣧⣾⠂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⢸⡥⠤⠀⠀⠀⠀⠠⠤⠤⠄⡤⢤⠄⠞⠀⠠⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠁⠂⠤⢘⠇⠀⢀⠀⢀⠴⠃⠠⢊⠋⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠒⠀⠒⠏⢈⣥⠅⢈⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠰⢣⠠⠃⢠⠖⠀⠀⠀⠀⠁⠀⠂⠉⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠄⠀⠀⠂⡳⡀⠀⠊⠀⠀⠠⠰⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢐⠆⠁⠠⢔⡂⣀⣀⣀⠒⠂⠠⠌⣁⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡡⢂⠬⣀⠄⠈⠂⠌⣐⣆⠈⡀⢋⣕⣃⢰⣥⡐⢌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡀⠩⠊⡔⣉⣀⣈⠀⠈⠐⠘⠀⠈⠙⠉⠀⠀⠀⠑⡝⢗⢕⢀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⠖⠂⠈⠁⠀⢀⠢⠐⠀⠀⠀⠀⠀⠢⠪⠀⠀⠀⠀⠰⡀⠁⢧⣃⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⡇⠀⠀⠀⡢⠀⠁⠀⠀⠀⠀⢀⠀⠀⣀⠀⠱⠀⠀⠀⠀⣛⠠⡍⢫⠀⢡
⠀⠀⠀⠀⠀⠀⠰⠃⠘⣧⠀⠁⠈⠢⡀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠈⠑⢅⠀⠀⡐⡃⠀⠈⢸⠁⡘⡄
⠀⠀⠀⠀⠀⠀⠀⢨⡀⢻⣗⠖⠂⢄⡠⠊⠉⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⣢⠐⠲⡄⠏⢠⡅⠁
⠀⠀⠀⠀⠀⠀⢀⠘⡘⢄⠙⢄⠠⠀⠈⠒⠬⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢑⡲⠋⡠⢡⠀⡆
⠀⠀⠀⠀⠀⠀⠈⡄⠐⡀⠐⢄⠙⠲⣤⣁⠀⠀⠀⠉⠁⠢⣄⣠⣴⣯⣾⣾⡃⠜⢋⠠⠊⢠⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠢⡀⠉⠒⠤⢈⡙⠓⠒⠤⠀⠼⠿⠿⠓⠚⢉⡁⠤⠒⠁⢀⠔⢡⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣈⠐⠤⣀⠀⠀⠉⠁⠀⠐⠂⠀⠀⠉⠉⠀⠀⣀⠤⠂⣁⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡉⠐⠐⠀⠀⠀⠀⠀⠀⠀⠂⠂⣁⣤⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⠤⢒⡤⠒⠀⠒⢴⣲⠤⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠠⡄⠀⠀⢠⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢀⡇⠀⠀⢀⡟⠀⠀⢀⡿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢿⡀⠀⠀⢻⡀⠀⠀⢸⡀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⣀⣀⣁⣠⣤⣤⣥⣤⣄⣀⣉⣀⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⣚⣋⣩⡥⠤⠤⠶⠶⠶⠶⠶⠶⠤⠤⢭⣍⣙⣓⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣋⡥⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢮⣙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡞⠁⠀⠀⢀⣀⣠⡤⠤⠶⠶⠖⢒⣶⣖⠒⠲⠶⠶⠤⠤⣄⣀⡀⠀⠀⠈⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡀⣠⠴⠚⣩⡥⠖⠒⠂⠐⠃⠰⠯⠶⠮⠷⠀⠉⣠⠶⣄⠀⠈⠉⠓⠦⣄⢀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢆⠻⣅⡀⠀⠀⠠⠀⢰⡏⢙⡆⠀⠀⠚⠀⠀⠒⠀⠙⠶⠚⢀⠀⠘⠀⢀⣨⠟⡰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠑⠦⣉⡓⠦⢤⣀⣀⡉⠉⠰⠆⠻⠯⠭⠿⠃⠰⠆⢀⣀⣈⡤⠴⢚⣉⠴⠊⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠉⠓⠲⠦⢭⣭⣉⣙⣛⣛⣛⣛⣛⣛⣋⣉⣭⡥⠴⠖⠚⠉⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠉⠙⠓⠲⠦⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⠴⠖⠚⠋⠉⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠲⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠖⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠚⠋⠉⠁