UFO Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡖⠛⠛⠛⠛⠲⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣛⣓⣟⣧⣤⣤⣤⢤⣤⣴⣟⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣯⠯⠤⠷⠴⠷⠶⠾⠶⠶⠶⠶⠷⢤⣯⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣭⣷⣤⣤⡤⣤⠤⢴⠶⠶⠤⠤⠤⠤⠤⣤⣿⣿⣿⣿⣭⣓⣦⣄⣀⡀
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡿⡶⣶⣶⣾⣿⡷⠻⣿⣾⣿⣶⠟⢶⣿⣿⣿⡟⠛⣿⣿⣿⣿⡛⣿⣿⣶⣯⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢯⣟⣿⣍⡉⠫⡉⠉⠉⡩⠉⡏⠉⠙⡯⢉⡏⠉⡹⠋⣉⣽⢋⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣟⡷⢬⣄⣀⣄⡀⢁⠀⢀⠀⢀⢀⣰⣁⣰⣏⣷⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠚⠀⠀⠙⠛⠿⣻⣿⠛⠛⠛⣻⠿⠛⠋⠀⠀⠚⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⣰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⣀⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⢉⣽⣿⡿⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢹⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠾⠁


⢀⣀⡀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⠉⠉⠙⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠲⠦⣄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⠀⠀⡇⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣙⣧⠀⠀⠀⣰⣿⣏⣠⣿⣿⠀⠀⣿⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⡿⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⠟⠁⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠙⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⣤⣤⠀⠀⠀⣀⡈⢧⠀⠀⠀⢤⣀⣠⠆⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢀⡬⠧⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⣿⣿⠃⠀⢾⣿⣿⠎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⡤⢞⠉⠱⡀⠈⠳⣄⡀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢴⠻⡗⠲⣄⠈⡽⠯⠶⠾⠷⠤⢤⣀⠀⠀⠀⠀⠀⠀⢿⣀⣠⠴⡎⠉⢆⠈⢣⠀⠘⣄⡴⢋⡽⠦⣄⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⡧⣄⣀⣈⠙⣷⠦⡈⠻⣷⡖⠶⠦⢤⣄⠀⠀⢙⣦⣠⡤⢴⠒⠛⡅⠑⡄⠘⡄⠀⢣⣀⠷⠚⣡⠔⠋⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⢀⣠⡏⠀⢀⠋⠉⡝⠉⢻⠛⠒⢻⠓⠒⡖⠛⢛⠉⠹⠉⠀⠸⠀⠈⡄⠀⠸⡀⢘⣄⠼⠖⢋⣡⠴⠋⠁⠀⡖⠓⡄⠀⠀⢀⡞
⠀⠀⠀⣠⠖⣉⣞⠀⠀⡜⠀⠀⡇⠀⢸⠀⠀⠘⠀⠀⡇⠀⢸⠀⠀⡇⠀⢀⣇⣀⡼⠴⠒⠋⢉⣠⠴⠚⠉⠳⣄⠀⠀⠀⠓⠚⠁⢀⡴⠋
⠀⡴⠋⠀⠀⠧⣌⠙⠒⠧⠤⣤⣅⣀⣸⣀⣀⣠⣀⣠⣧⠤⠼⠶⠖⠛⠋⢉⣉⣠⠤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡴⠋
⡾⠁⠀⠀⠀⣠⠎⠉⠓⠒⠲⠶⠤⠤⠤⠤⠤⣤⠤⠤⠶⠒⠒⠒⠛⠋⠉⠉⢧⠀⠀⠀⡴⠒⢲⠀⠀⠀⠀⠀⠀⠀⢱⣤⣾⣿⡇
⢣⡀⠀⢀⡜⠁⠀⠀⠀⡴⠒⢦⠀⠀⠀⠀⢰⠇⠀⢠⠴⠤⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠙⠒⠋⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡟
⠀⠉⠓⠾⣀⡀⠀⠀⠀⠳⠤⠞⠁⠀⠀⠀⡞⠀⠀⠈⠲⠖⠃⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠉⠓⣶⣤⣄⣀⣀⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣿⣶⣿⣿⠏⠀⠉⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠻⢿⣿⣿⣿⣿⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡿⣁⠀⠀⢠⡄⠀⠀⣈⢿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⠿⠯⠉⠀⠀⠈⠀⠀⠀⠉⠽⠿⠿⡟⣶
⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢻⢉⣁⣠⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣈⡉⡟⠒⠦⢤⣀
⠀⠀⠀⠀⡴⢋⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣌⡙⢦
⠀⠀⠀⠈⢷⣟⢿⣿⣿⣿⣿⣿⣿⣿⠉⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⠉⣿⣿⣿⣿⣿⣿⣿⡻⣻⡾⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠚⠮⣿⢿⣿⣿⣿⡛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⢛⣿⣿⣿⡿⣿⠵⠓⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣟⢿⣿⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⣿⡿⣻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠤⠤⠤⠤⠤⠤⠶⠚⠛⠋⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠛⠋⠉⠉⠙⠛⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⣀⣀⠤⠤⠒⠒⠒⠒⠛⠓⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣀⡀⠄⠒⠊⢉⣀⣠⠤⠤⠒⠒⠒⠉⠉⢉⣉⣉⣩⣭⣭⣭⣭⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠥⠒⠉⣀⡠⠔⠒⠋⠉⠀⣀⣠⡤⠴⠶⠚⡛⡏⠉⠉⢩⣇⣀⣀⣠⣇⣀⣠⣇⡀⣹⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⡤⠒⠉⢁⣀⡤⣤⠒⠊⠉⠁⢫⣇⣀⠤⠤⠚⠓⠋⠉⠉⠉⡹⠋⠀⣀⠤⡀⠈⢉⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⣁⡤⠶⡾⡉⠁⣀⣡⠧⠶⠚⠉⢫⠀⠀⠠⠔⢀⡭⠛⠀⡠⠊⠀⠀⠐⠐⠊⣠⠔⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⡠⠖⢛⣃⢀⣠⠼⠓⠋⡉⠤⠀⡄⠀⠀⢸⠀⠀⠀⣀⣀⣠⡤⠮⠤⢤⣀⠀⡀⡴⣍⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠚⠉⣢⣤⡔⠋⠁⠀⠀⠀⠈⠒⠂⠉⣀⡠⠤⠞⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⡿⢘⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⢶⡴⠚⠁⠀⠈⠉⠓⠒⠒⢀⣤⠴⠚⠉⠁⠀⠀⠀⢀⣀⡠⠤⠤⠦⢤⡄⠀⣠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠯⣶⠋⠀⠰⠥⠤⠛⠀⠀⢀⠔⠋⠀⠀⠀⢀⡠⠔⠒⢉⣀⠤⠔⣂⣤⠶⣛⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣖⡒⠒⠂⠒⢀⠒⡐⢫⡀⠀⠀⠀⢰⣯⣧⣭⣭⡼⢶⣓⣿⠽⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠲⠶⠶⠟⠛⠒⠲⠶⠶⠒⠚⠛⠛⠉⠁









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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⣳⣿⣇⣀⣀⣀⡀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣦⠝⢉⣀⣀⣀⣀⣀⣀⣀⡉⠻⣦⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣟⣻⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠹⣛⡻⠿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣺⣭⣾⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣶⡽⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⣿⣿⣿⣟⣿⣏⣽⣏⣉⣉⣉⣹⣿⣯⣿⣻⣿⣿⣿⣿⣷⣿⣿⣷
⠤⠤⠤⠤⠤⠤⠤⠤⠭⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠋⠉⠡⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢰⣿⣿⣿⣿⡟⣿⢻⣿⣿⣿⣿⡌⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭
⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⣸⢻⡿⡿⢸⠁⣿⠸⠛⡟⢟⠿⡷⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭
⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⢢⠉⡄⠁⠀⡈⠀⡟⠀⠀⠁⠘⠀⠋⠐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒
⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶
⣶⣌⠭⡭⠭⢭⠭⠭⠭⠭⠭⠭⢭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢭⠭⠭⠭⠭⢭⠭⠭⡭⢍⣴
⣿⣿⣷⡝⠿⢋⣾⣧⡙⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣡⣿⣎⢻⠟⣥⣾⣿
⣿⣿⣿⣟⣲⣿⣿⣿⣿⡆⠛⣻⡃⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⣾⣿⣿⣿⣷⣚⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⣍⠀⠀⣼⣿⣷⣄⣤⣤⣤⣤⣄⣀⣀⣠⡀⢠⣾⣿⣄⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡇⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡏⠉⢽⣿⣿⣿⣽⠿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣧⡤⠾⠿⠿⠿⢿⣤⣬⣧⣤⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⠿⣿⣟⣻⣿⣿⣯⣽⣿⣿⣿⣧⣾⣿⣿⣿⣷⣾⣿⣿⣿⣷⣼⣿⣿⣿⣯⣽⣿⣿⣟⣻⣿⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠈⠙⠻⠿⠇⣽⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠋⠉⠉⠉⢉⠉⠉⠉⣭⠍⠉⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣯⠹⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿⣟⣛⠛⠛⠛⣛⣛⣛⣿⣿⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣳⠺⠋⡿⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣻⣿⠏⠀⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣹⣡⣿⣿⣀⣀⣀⣀⣀⠀⡼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣐⣮⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⣷⣄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⠿⠿⠛⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠛⠿⢷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣭⣵⣶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣶⣭⣝⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣷⠳⠿⢿⣟⣛⣛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣽⣶⣦⣾⣟⣀⣀⣀⣀⣈⣿⣿⣶⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠂
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠭⠙⠛⠛⠿⠿⠿⠿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠙⠉⠁⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⡉⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣩⣭⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⣾⢿⣿⢿⢻⣿⡏⣿⢻⣿⡟⣿⡿⣿⡆⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢠⡿⡿⡮⡾⡏⡇⡇⣿⢸⢻⢻⢷⢳⢿⢳⠘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛
⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠘⠓⠉⠃⠃⡁⠇⡇⡇⠈⠸⠸⢨⠼⠈⡎⡆⢭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭
⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⠃⠘⠀⠸⠈⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠰⠠⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠉⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠘⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠊⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠉
⣾⣦⡰⠶⠶⠶⠶⠶⠒⠶⠶⠶⠶⠶⠶⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠲⢶⠶⠶⠶⡶⢆⣼⣷
⣿⣿⣷⣌⠛⣛⠟⣡⣿⣎⠻⣛⣛⣛⣛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⣻⠛⣴⣷⣌⢻⡻⢛⣵⣿⣿⣿
⣿⣿⣿⣿⠧⢤⣾⣿⣿⣿⣷⣌⠻⣿⣿⠅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠏⣡⣾⣿⣿⣿⣷⡤⢼⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣯⠀⠰⠿⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⠀⠀⣽⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⣰⣆⠀⠀⣾⣿⣿⣿⣦⣶⣿⣶⣶⣶⣶⣶⣶⣶⣾⣆⠠⣿⣿⣿⡷⢊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡾⣿⣿⣿⡍⠉⢹⣿⣿⣿⣿⣹⣷⠿⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣸⣿⣿⣿⣿⣿⣀⣀⣸⣇⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⢀⣀⠠⠤⠭⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠠⠀⠒⣈⡁⠤⠐⠒⠒⣈⣉⣉⣩⡭⠭⢭⡽⠶⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢈⠠⠐⠊⣁⣀⠤⠶⢖⠊⢉⣩⣇⠤⠴⠷⠶⠚⠓⠒⠷⢴⣋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⢈⡠⢴⠒⠉⣩⠤⠄⠒⡏⠉⢡⠀⡀⠆⢀⠔⠁⠠⠠⢋⠠⢪⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣩⠒⠹⣇⠤⠘⠋⡁⠀⡠⠀⠀⣃⢀⡀⠤⠤⠶⠥⠤⢄⢀⢴⣅⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢟⣁⡴⠒⠋⠤⠀⣀⠀⣀⡠⠔⠒⠉⠁⠀⠀⣀⣀⡀⠀⢀⣼⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠢⡜⠁⠰⠠⠖⠀⢀⡴⠊⠁⠀⣀⠄⠐⣂⠩⠀⣠⣴⣫⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠭⣀⣀⢠⣠⡙⣄⣀⣀⣘⣿⣼⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁






⣿⣿⠿⠿⢻⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢊⡩⠥⠬⠭⠭⢍⡱⠮⢤⣤⣤⣤⣬⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⣀⡤⠶⠿⠀⠀⠀⠀⠀⠀⠿⠷⢤⣌⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣾⣿⣿⡿⠷⠶⡾⠿⢿⠟⠉⠉⠻⡿⠿⠿⢶⠶⣿⣿⣿⣿⡶⡖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠛⠿⠿⢮⣦⣤⣶⣶⣖⡚⢀⡀⠀⢀⠛⢲⣶⣦⣤⣤⡤⠾⠿⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠀⠀⠙⠛⠿⣿⣛⣋⡉⢉⢉⣛⣿⡾⠻⠛⠀⠀⠀⠀⠐⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡿⠤⠤⠤⠤⠤⠆⠀⢀⠀⠈⠉⠻⠟⠻⠟⠉⠁⠀⠀⢀⡀⠠⠀⠄⢀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠐⢀⡀⠀⡀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣀⠀⠘⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣸⡇⠀⠀⣀⣤⣶⣿⣿⣫⣢⣀⣸⣄⣄⡷⠀⠔⠢⣀⢀⣿⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣚⣛⡻⣞⣚⣛⣿⣯⣛⣿⠥⠶⣬⣍⣛⣿⡿⣿⣿⣿⡏⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⠩⣯⠕⠒⡖⠖⠒⢲⣿⡭⠭⠤⠤⠄⢰⡧⢐⣺⣿⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⣿⣄⣸⣿⣄⣤⣚⣿⡇⢰⡄⣼⡄⢾⣿⣬⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡿⣁⠀⠀⢠⡄⠀⠀⣈⢿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⠿⠯⠉⠀⠀⠈⠀⠀⠀⠉⠽⠿⠿⡟⣶
⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢻⢉⣁⣠⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣈⡉⡟⠒⠦⢤⣀
⠀⠀⠀⠀⡴⢋⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣌⡙⢦
⠀⠀⠀⠈⢷⣟⢿⣿⣿⣿⣿⣿⣿⣿⠉⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⠉⣿⣿⣿⣿⣿⣿⣿⡻⣻⡾⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠚⠮⣿⢿⣿⣿⣿⡛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⢛⣿⣿⣿⡿⣿⠵⠓⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣟⢿⣿⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⣿⡿⣻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠤⠤⠤⠤⠤⠤⠶⠚⠛⠋⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠚⠋⠉⠉⠉⠉⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⠛⠛⠲⢄⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⢧⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢰⠿⣆⠀⠀⠀⢀⣼⡿⢻⣿⣷⠀⢸⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡆⠀⠀⣾⣿⣷⣿⣿⠇⠀⢸⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⣿⣿⣿⡿⠃⠀⠀⣸⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⠿⠃⠀⠀⠈⠛⠉⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⣀⠀⠀⠀⠀⢹⡀⠀⠀⣀⠀⣀⠄⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣸⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⣿⣿⠀⠀⣾⣿⣷⢳⡀⠀⠀⠉⠁⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⣠⢴⠋⢣⠈⠙⢍⠉⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠻⡗⠦⣄⢉⠿⠥⠤⠷⠤⠤⣀⡀⠀⠀⠀⠀⠸⣇⣀⡤⢞⠉⢣⠈⢆⠀⠱⣀⠔⣫⠦⣄⡀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⡼⠤⣀⣀⡙⣗⢦⡈⢻⣷⠒⠲⠦⢄⡀⢀⣙⣦⣤⠤⠖⠚⢫⠈⢆⠈⢆⠀⢣⣨⠖⢋⡥⠚⠁⠀⠀⠉⠳⢤
⠀⠀⠀⠀⢀⣠⡾⠁⠀⠎⠀⠉⠉⢹⠛⠚⡟⠛⢻⠋⢹⠉⠙⡁⠀⢰⠀⠸⡀⠀⢃⣈⡦⠴⠛⣉⡤⠞⠉⠀⢰⠋⢳⠀⠀⢀⠞
⠀⠀⣠⠖⢋⡴⣅⡀⡸⠀⠀⡇⠀⢸⠀⠀⡇⠀⢸⠀⠀⠄⠀⣇⣀⣨⡤⠤⠗⠚⢉⣁⠤⠖⠋⠉⠑⣄⠀⠀⠈⠙⠁⢀⡴⠋
⢠⠞⠁⠀⠈⢳⣤⣉⣙⠒⠲⠦⠤⠼⠤⠤⠧⠴⠞⠒⢒⣛⣉⣉⣡⣤⡤⠖⠒⠋⠉⡀⠀⠀⠀⠀⠀⠈⢳⡀⠀⣠⣴⠉
⡏⠀⠀⠀⣠⠋⠀⠀⠉⣉⠉⠙⠛⠛⠓⢲⠞⠛⠉⠉⠉⠉⠀⠀⠀⠀⢧⠀⠀⢰⣋⣩⠆⠀⠀⠀⠀⠀⣠⣽⣾⣿⡟
⠙⢦⣀⡼⠁⠀⠀⠀⢸⡉⢹⠄⠀⠀⠀⡞⠀⠀⣏⣉⡷⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⠁⠀⠀⣀⣤⣶⣿⣿⣿⣿⠟
⠀⠀⠈⠙⠲⠤⣄⣀⠀⠉⠁⠀⠀⠀⢰⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣇⣠⣤⣴⠖⠺⠿⢿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣶⣶⣦⣾⣤⣤⣤⠴⠶⠲⢶⣶⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⣒⡐⠒⢖⡠⣄⣃⣈⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣊⡠⠴⠿⠀⠀⠀⠀⠸⠷⠬⣑⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠿⠟⠛⠛⣛⣛⠉⠉⠙⣟⡛⠛⠛⠿⢿⣿⣶⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠛⠛⢿⣿⣷⠦⠦⠴⠶⢤⣿⢿⡟⠛⠉⠉⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡀⠠⠁⠠⠄⢀⡀⠙⠛⠿⠾⠞⠛⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⣠⣤⣾⣇⠀⢠⣀⣰⡄⡀⡀⠀⣸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣶⣚⣻⣋⣛⣿⣟⣿⡯⢬⣽⣲⣶⡾⣷⣿⣖⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣐⡟⠒⣖⠒⠒⣿⡟⠓⠒⡒⢸⡇⢼⡿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣿⣶⣶⣿⣯⣼⣼⣧⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣛⣉⣭⡉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⡶⠛⠉⠁⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢀⣴⠶⠖⣚⣛⠛⠓⣾⠏⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⣏⣉⣿⠀⠹⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣀⠈⠉⠁⡶⠶⣦⠙⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡙⠒⠋⠀⣴⠛⣶⠉⢙⣛⠶⠶⣤⣤⣤⣀⣀⣀⣀⣤⠟⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⡛⠶⣤⣌⡛⠋⠀⣿⣉⣷⠀⡶⠳⣦⠉⢩⣭⣭⡀⢴⡛⠳⣎⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⢻⣶⣤⣈⠉⠛⠳⠶⢭⣤⣀⣙⠛⠋⠀⠹⢦⡼⠃⠈⠛⠛⠋⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⣴⠏⠀⠀⠉⠛⣷⠶⣤⣤⣀⣉⣉⠙⠛⠛⢲⣶⣶⠶⠶⠶⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛⠁⠀⠀⠀⠀⠀⡿⠀⢰⡏⠉⠉⠉⠉⢿⡋⠙⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⠾⠃⠀⠀⠀⠀⠀⠀⢿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣺⣿⣿⣿⣿⣿⣿⣿⣿⣭⣵⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠄⠒⠒⠒⠒⠒⢿⣦⣴⠶⠶⠶⠾⠟⢻⠛⢻⡩⠭⠭⠭⡠⣀⣀⠀⡸⢮⠳⡀
⠀⠀⠀⡠⢔⢚⡺⣿⣿⣯⣥⣤⡤⠄⠀⠀⠀⠀⠀⠀⠉⠛⠓⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡉⠉⠚⠗⣒⣠⠤⣀⡀
⠀⠀⢸⢹⣿⣄⠙⠪⢍⣛⠿⣿⣧⣤⣤⣤⣄⡀⠀⠀⠀⠀⣄⣀⣀⣠⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⣿⣷⣮⠙⢲⣦⣤⣀⡀
⠀⠀⠈⠳⣝⢿⣦⣤⣀⠈⢉⠑⠫⢟⡛⢿⣟⡓⣤⣠⠬⠗⠛⠉⠛⠛⠛⠻⠷⠤⢦⠶⢢⣄⣀⠤⠶⣿⣿⣿⣶⣾⣿⣿⠟⡫⣁⣴⣿⣿⣿⣿⣿⣕⡦⣀
⠀⠀⠀⠀⠀⠙⠺⠿⣿⢿⣿⣿⣷⡶⢽⣾⣿⣿⣟⣉⣀⣠⣀⠀⠀⠀⠀⠀⢙⣿⣿⣿⡏⠀⠀⠻⢧⣤⣙⣛⠿⣿⡁⣴⣶⣾⣿⡿⢿⡟⠉⠉⠛⠻⠿⠿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⠿⣧⣱⡻⣿⣻⠏⣛⣛⡶⣳⣿⣿⣷⣶⣖⣲⣶⣦⡬⢥⣄⣚⣻⣿⣿⠟⠿⠁⠀⠈⠉⠚⢿⣿⣄⣀⣀⣀⣠⠤⠤⠖⣢⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⡇⠺⣷⣄⢤⣬⣉⣩⣿⣿⣟⢩⠿⠿⢿⣻⡷⢾⣯⣽⣻⠕⡒⢤⣀⠀⠀⡠⠊⠈⢉⣀⣀⣀⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠳⠮⣉⣻⣿⣬⢿⣇⠀⠀⠀⠀⢀⠺⠏⢙⣿⣷⡻⣿⡿⢻⣷⡱⠚⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡻⠥⠭⢥⣖⢒⣛⣛⣿⣯⡭⠿⠿⠑⠚⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠿⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠯⠤⠶⠶⠒⢒⣒⣛⣛⣛⣛⣛⣛⣛⣓⣒⣒⡲⠶⠦⠽⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠖⣛⣉⣭⠴⠶⠒⠚⠛⠛⠉⠉⠉⣩⠭⣍⠈⠉⠀⠀⠈⠉⠉⠉⠉⠉⢙⡛⠓⠒⠯⢭⣙⡓⠶⣤⡀
⠀⠀⠀⣠⠴⣫⠵⠚⠋⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣉⣀⣀⣀⣀⣀⣀⠀⠀⠀⠸⠯⠽⠃⠀⠀⠀⠀⠉⠓⠦⣝⠳⣄
⠀⠀⢰⣷⠟⠁⠀⠀⠀⠐⠯⠽⠗⠀⣀⣤⣶⣶⡿⠿⠿⠿⠟⠛⠛⠒⠒⠛⠻⠿⠿⠿⣿⣶⣶⣤⡀⠀⠀⠀⠀⢀⣀⠀⠈⠳⣼⡇
⠀⠀⠘⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⣾⡿⠋⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠛⢿⣦⠀⠀⠀⠻⠾⠃⠀⠀⣿⠇
⠀⠀⠀⠘⢦⡀⠀⠸⣭⠽⠃⠀⠀⢻⡀⠀⠀⣠⠶⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⡄⠈⠉⠛⠲⣄⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡐⠂⠀⣤⣽⠦⣤⡿⠦⡀⡆⠀⠀⡄⠀⢠⠀⠀⠀⢁⠀⠀⠀⠐⢻⠶⠋⠀⠿⢗⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠚⠷⠶⠶⡇⠀⠀⡇⠀⢹⡏⠁⢸⠀⠀⠉⠉⠀⣀⠀⠀⢸⡶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⡇⠀⠸⠀⠀⠀⠀⠀⣿⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠘⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⠀⠀⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⣸⠀⠀⠸⠀⠀⠘⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⠀⣧⠀⠀⠸⠀⠀⢸⠀⠀⠄⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠿⠿⠭⠭⠭⠭⠿⠷⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠥⠤⠶⠶⣲⣒⣶⣶⣶⣶⣶⣶⡶⠤⠬⢷⣄⣀⡀
⠀⠀⠀⠀⣀⡤⣖⣫⡭⠵⠒⠒⠛⠉⠉⠉⢩⣭⣭⠀⠀⠀⠀⠀⠉⠉⠉⣉⣙⠓⠛⠯⢭⣓⠦⣄
⠀⠀⣠⣺⠗⠋⠁⢀⡤⣤⠀⠀⠀⣀⣀⣤⣤⣭⣥⣶⣶⣶⣦⣤⣤⣤⣈⡛⠋⠀⠀⠀⠀⠈⠙⠢⣝⢦
⠀⠀⢿⠁⠀⠀⠀⠀⠉⠉⢠⣾⠟⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⢷⣦⡀⠀⠠⣖⡶⠀⠘⣿
⠀⠀⠘⢦⡀⠀⢾⡿⠆⠀⢸⡇⠀⢀⡤⠖⠚⠋⠉⠉⠉⠉⠉⡉⠉⠓⠶⣄⠀⢹⠇⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⢠⡽⠲⢼⠓⠄⡆⢀⣀⠀⢠⠀⢀⣀⠀⠀⠀⢻⠖⠁⢚⣟⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢻⠀⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⡆⠀⢸⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠿⠀⠈⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⡀⠀⠀⡄⠀⠃⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡄⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆⠀⠘⠀⠀⠃⠀⠂⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣀⣀⣠⣽⣿⣀⠀⠀⠀⠀⠀⢠⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡤⣀⣴⠿⣿⣳⣯⢞⡟⢛⠿⣦⣀⢤⣀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡞⣱⣾⣋⣉⣉⣶⣿⠿⢿⣿⣿⣟⣉⣙⣳⣮⢳⣦⣄⠀⠰⣤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠇⣾⣻⢉⣍⣩⣉⣹⡉⢈⣙⢍⣯⣉⣏⣩⡉⣟⡗⠸⣿⠀⠀⢘⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣬⣶⠾⢿⣛⠻⣟⠻⠏⠙⣋⣥⠺⠷⢮⢉⠋⠹⠟⣹⠟⢛⣻⠷⣶⣤⣌⣀
⡇⠀⠀⠀⠀⠀⣀⣤⣶⣿⡫⣿⣿⣭⣽⣯⣭⣯⣿⣷⣶⣿⣿⣿⣧⣾⣇⣿⢿⣷⣾⣶⣿⣭⣽⣭⣭⣿⣿⢛⣿⠲⢤⣀
⡇⠀⠀⣠⣴⣾⣫⣭⣾⣾⢴⡿⠷⣶⡶⠶⣶⠶⢶⠖⠚⠿⠛⠿⠖⠒⠿⠟⠻⠷⠒⠶⠶⢶⡶⠶⣶⣿⣿⣶⣿⣧⣤⣿⣷⣦⣄
⡇⠀⠀⠙⠛⠿⣿⣿⣚⡛⠛⠛⢿⣿⣁⣤⡤⠴⠦⠰⠃⠀⢻⠃⠈⠉⠀⠙⠁⢰⠆⣤⠖⢥⣤⣄⣸⠟⠛⠛⠛⣛⣿⣿⠿⠛⠋
⡇⠀⠀⠀⠀⠀⣠⣬⣭⣛⡳⠶⢤⣬⣍⣛⣃⠀⠃⠀⠆⠀⠈⠂⠀⠀⠀⠀⠄⠐⠀⢈⠄⢘⣋⣉⣥⡤⠴⢞⣛⠉⠁
⡇⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣷⣶⣦⣭⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣩⡍⣥⣶⣾⣿⣿⣇
⡇⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣴⡿⣿⡟⣿⠿⣿⣿⣿⣿⣶
⡇⠀⠀⠀⠀⠀⠀⠀⢻⢹⣿⣿⣿⣿⣿⣿⣿⡟⢸⠁⠐⣤⣈⣀⠀⠀⢀⣁⣤⣶⣿⡻⠐⠒⠀⠐⠒⠘⣻⣿⣿⡟
⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⢻⣿⣿⣿⢿⠛⠸⡇⠸⠀⠀⠻⠿⠏⡠⠀⡀⣼⣿⣿⣿⣷⡄⠀⠀⠀⢀⣴⣿⣿⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠰⣯⣄⠀⢀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠙⣿⣟⢠⣤⣤⢾⣿⣿⡿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣔⡱⣤⡀⠈⠛⢷⣤⣤⣤⣠⣤⠞⢫⠟⠁⠀⠀⠈⢿⣿⣿⣟⠠⠿⡟⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠈⢿⣦⣄⠀⠉⢹⠿⠋⢁⡴⠁⢀⣀⣀⡀⠀⠀⢻⣿⡏⠀⠀⢸⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠢⢀⠹⣿⣷⣤⡀⢃⣴⠋⠀⠀⣛⣹⡋⠀⠀⢀⡀⠙⠃⠲⠶⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠛⠻⠿⣿⣿⡟⠉⣿⣿⣿⣿⠿⠟⠛⠉⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠘⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢻⣿⣿⣿⣟⣻⣿⣿⣿⡟⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠻⣿⣿⣿⢻⣯⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣽⡿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠉⠘⠛⠻⠿⠿⠿⡟⠉⠁⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠋⠀⠀⠀⠀⢠⡈⠉⢻⡿⠿⠿⠟⠛⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⡶⠶⠶⠶⠶⠶⢦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠒⠒⠒⠚⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⢼⠏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⣀⣁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠋⠉⠉⠉⠉⠉⠙⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢤⣄⣀
⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠖⠚⠋⠉⠀⠀⠀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⢠⠞⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠳⡄
⠀⠀⠀⢸⡀⠀⡞⣉⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢬⣙⡆⢀⡇
⠀⠀⠀⠘⢿⡦⣍⡁⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡿⠃
⠀⠀⠀⠀⠀⠉⠓⠯⣽⣒⠦⠤⣄⣀⣀⠘⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⣀⣀⣤⠤⠴⣒⣯⠽⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠾⠭⠭⣿⣛⣒⣒⣒⣒⣶⣶⡦⠤⠤⠤⠭⠤⠤⢴⣶⣶⣒⣒⣒⣒⣛⣿⠭⠭⠷⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠿⠿⠿⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣋⣁⣀⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣈⣙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⣛⣛⣩⣭⣭⠤⠴⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠮⢭⣭⣛⣒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⣠⠶⣛⣩⠴⠖⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⣯⣭⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⡄⠀⠉⠙⠓⠶⣬⣙⠳⢦⡀
⠀⠀⠀⣴⣫⡵⠛⠉⠀⠀⢠⡤⢤⣄⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⣈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡙⢦
⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠈⠛⠛⠁⢀⣴⣾⡿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⣿⣶⣤⡀⠀⠀⠀⣠⠤⣤⠀⠀⠹⣾⠆
⠀⠀⠀⢻⡄⠀⠀⠀⢀⣤⣄⠀⠀⠀⣿⠟⠁⠀⠀⠀⢀⣀⣠⣤⣤⠤⠶⠶⠶⠶⠶⠶⠤⣤⣄⣀⠀⠀⡉⢻⣷⠀⠀⠀⠉⠛⠉⠀⠀⢀⡿
⠀⠀⠀⠀⠻⣦⣀⠀⠻⠶⠛⠁⠀⠀⠹⣆⠀⠀⢠⡞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠉⠳⡄⠀⣠⡟⢀⡀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⣈⠁⠀⣴⣮⠓⠦⣾⠓⠤⢸⠀⠀⢠⡄⠀⢰⠀⠀⢀⣀⡀⠀⠀⠀⠊⣿⠚⠉⠀⠻⣛⣡⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⢾⠀⠀⢸⡆⠀⢹⡏⠁⢸⠇⠀⠈⠁⠀⢀⡄⠀⠀⢻⠶⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⢸⡇⠀⠸⠀⠀⠀⠀⠀⢸⡇⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⠀⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⠀⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢸⡇⠀⠄⠀⠀⢰⠀⠀⢠⠄⠀⠀⠟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠘⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⡀⠀⠀⢸⠀⠀⢸⡇⠀⢀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠈⠀⠀⠈⠀⠀⠘⠀⠀⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠒⠒⠒⠲⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀
⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠁⠈⠙⠒⠢⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠒⠊⠁⠉⠉⠓⠲⠤⣀
⠀⠀⢀⡜⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠑⣄
⠀⠀⠸⣇⡀⢯⠴⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠙⠢⠽⣀⡿
⠀⠀⠀⠙⠿⣶⡤⣄⣀⠀⠀⠀⠠⣋⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠈⢣⣱⠀⠀⠀⢀⣀⡤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠚⠿⠽⢷⣖⣲⣦⣤⠤⠤⠤⣤⣄⣀⣀⣘⣊⣀⣀⣤⡤⠤⠤⢤⣤⣶⣒⣲⠿⠽⠟⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣀⠀⣠⣶⣾⣿⣿⡿⢶⣄⠀⣀⠀⠙⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⢪⣷⢛⣁⣺⣿⡿⢿⣷⣷⡛⠛⣾⡝⣦⣄⠀⠤⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢱⣻⣉⣍⣭⣩⣏⢨⡝⣽⣏⣭⣩⣉⣟⡊⠿⠀⠀⢸⠆
⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣛⣯⣙⣯⣙⣋⣩⣷⡟⢻⣧⣍⣉⢋⣽⣋⣭⣟⣷⣶⣤⣄⡀
⡇⠀⠀⠀⣠⡴⢾⣟⣇⣻⣟⣛⣛⣒⣒⣛⣛⣿⣿⣿⣟⣛⣿⣿⣿⣛⣛⣒⣒⣛⣓⣿⣟⣰⣏⠑⣦⣄
⡇⠀⠰⠿⣿⣿⡿⠿⠭⠿⣭⡟⢉⡙⢉⣍⢡⠌⢡⡌⠱⠄⠉⠄⠉⡄⣉⣉⣛⣉⠹⡿⠿⠿⠿⢿⣿⣿⠿⠆
⡇⠀⠀⠀⠀⢉⣙⣛⠶⢤⣌⣙⡛⠻⠀⠂⠠⠀⠀⠣⠀⠀⠀⠀⠀⠁⠘⠀⠛⢛⣋⣡⡤⢶⡚⠋⠉
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣭⣭⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣩⣩⣴⣶⣿⣿⡁
⡇⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⢧⡄⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⠿⠿⠻⠿⢿⣿⣿⣷⠆
⡇⠀⠀⠀⠀⠀⠀⢟⢻⣿⣿⣿⣿⡟⡿⢸⠀⠡⣶⣤⠄⠀⠀⣶⣿⣿⣏⠈⠁⠀⠉⢈⣿⣿⡿
⡇⠀⠀⠀⠀⠀⠀⠘⣎⠛⠙⠋⠁⠀⣹⠈⠀⠀⠉⠁⠊⠀⢰⣿⡿⠿⣿⡧⣀⣀⣠⣿⣿⣿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⢀⡀⢤⡀⠙⠻⣦⣀⣀⣀⣠⡴⢻⠟⠀⠀⠹⣷⣿⣿⡊⣿⡿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠹⣦⣄⠈⠙⡿⠟⢁⡔⠁⣀⣀⡀⠀⠘⢿⣿⠀⠀⣱
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠑⠀⢘⣿⣷⣦⣈⣴⣃⣀⣀⣭⣏⣀⣀⡀⠌⠋⠐⠚⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠙⠛⠻⠟⠁⣿⡿⠟⠛⠋⠉⠀⣠⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠈
⡇
⡇⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣟⡻⠉⠉⠉⠉⠉⠙⠻⡶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣛⣒⣟⣛⣄⣀⣀⣀⣀⣀⣠⣚⣛⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⣻⣉⣹⣧⣤⣽⣤⣤⣧⣤⣤⣥⣤⣬⣇⣉⣏⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡾⡿⠟⣟⡉⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠘⠉⣛⣿⣿⣿⣓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⢻⣛⣛⣛⣟⣻⣛⣫⣿⣫⣫⣏⣉⣝⣉⣉⣉⣙⣙⣙⣛⣻⣟⣿⣻⣟⣿⣶⣾⣿⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣻⣙⣿⣿⣿⣿⣿⣫⣙⣿⣿⣿⣿⣿⠩⣹⣿⣿⣿⣿⣩⣉⣿⣿⣿⣿⣿⣍⣹⣿⣿⣿⡧⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣭⣻⣦⣅⠀⠹⣤⠍⠉⠉⡭⠉⢩⠍⠉⠉⡭⢍⣽⠉⠍⡭⠉⠍⣡⡿⢓⣶⣿⣿⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡟⣳⣎⣢⡀⠑⡀⠀⠘⠀⠀⢰⠁⠀⠉⠀⡰⠁⣀⣟⢻⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣖⣿⣹⠃⠀⠙⢿⣿⣿⣿⣶⣷⣶⣾⣶⣾⣷⣿⣿⠿⠉⠉⠫⣽⢶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠓⠿⢦⣤⣤⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣰⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠶⠚⠛⣿⣿⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⣿⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⢠⡿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⡛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠈⠑⢄
⠀⠀⠀⣀⣠⣤⡤⠔⠒⠀⠉⠉⠉⠹⠶⣆⠀⠉⠋⠉⠙⠚⠛⠃⠀⠀⠤⠿⣣⣀
⠀⢠⣾⣇⠘⢟⣻⣿⣯⣀⣀⣀⡀⠀⠀⢀⣀⣀⣠⡆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⣿⣷⣶⠢⣤⣀⡀
⠀⠀⠙⠿⣷⣤⣄⣉⣙⣻⣟⣿⣷⡶⠒⠉⠉⠉⠉⠻⣶⣶⣲⠤⢔⡶⠿⣿⣷⣿⠟⣋⣥⣾⣿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠈⠙⠛⠿⠿⡏⣻⣿⣿⠶⣶⣶⣴⣤⣤⣤⣝⣛⣛⣀⡴⣿⣿⠷⠉⠓⠛⢿⣿⡋⠋⠀⣀⢈⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⣿⢤⣬⣭⢽⡿⠿⠿⠿⢿⣿⣿⡿⣷⣒⣤⡀⢀⡔⣉⣉⠩⠥⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⠿⣞⡷⣀⣄⠤⣬⣥⢾⣿⠩⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠿⠟⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠷⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣯⣥⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣬⣽⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⣛⣉⣭⡭⠴⠶⠶⠒⠒⠛⠛⣛⣋⡉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠒⠲⠶⠯⢭⣟⣓⠲⠦⣄⡀
⠀⠀⠀⠀⣠⠴⣛⡭⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣻⡶⠀⠀⠀⠉⠙⠳⢦⣍⠳⣤⡀
⠀⠀⢀⣾⡵⠋⠁⠀⠀⠀⣶⣒⣶⠄⠀⢀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⢻⡄
⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣶⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣦⡀⠀⠀⢰⣖⣳⠆⠀⠈⣿⠇
⠀⠀⠀⠹⣄⠀⠀⢀⡴⢶⣦⠀⠀⢸⡟⠀⠀⠀⢀⣠⡴⠶⠖⠛⠛⠛⠛⠛⠛⠋⡛⠛⠓⠶⣤⡀⠠⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠙⠳⢤⣀⡉⠉⠁⣄⠀⠀⠻⣤⡀⢰⣯⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣹⣆⣠⠞⢡⣤⡄⠀⢀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣜⣿⣃⣉⣻⡇⠉⢸⡇⠠⢴⡧⠄⢸⠀⠀⠤⠖⠂⠀⠀⠀⠀⣏⣀⣀⣬⠽⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡇⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⠃⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⠃⠀⠈⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⡀⠀⢠⡇⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠁⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡇⠀⠀⠘⠀⠀⠸⠀⠀⠘⠀⢸⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁

⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠿⠿⣷⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣷⣶⣤⣀⡀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣄⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⢿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠈⠙⠛⠦⣄⠀⠀⠀⠉⠂⠀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢷⣶⣤⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⡀⠀⠐⢶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠉⠛⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⣶⣴⠂⠀⠀⠒⡒⠒⠦⣄⡀⠀⠀⠀⠀⠀⢀⣸⣦⠆⠀⠒⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣸⣦⡆⡀⠚⠈⠀⠀⠀⠀⠠
⠀⠹⠉⠃⠀⢀⠠⣌⠙⠢⣄⠙⠳⠀⠀⠀⠀⠀⠹⠋⠃⠀⠓⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠘⠋⠁⠀⠺⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⢱⡀⠑⣄⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠡⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣠⠤⠤⢤⣄⣀
⠠⣿⣿⣡⠰⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣶⡂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢾⣿⡇⠀⠀⠹⣟⣒⠲⢦⣌⠁
⠀⠈⠱⠆⠀⡅⠈⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠁⠢⡑⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢳⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢄⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿
⠠⣿⣾⡳⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣋⡼
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠛⠛⠛⢿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠛⠁⠻⠦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠈⠓⠶⢦⣤⣄⣀⡀
⠀⣀⣴⣶⡛⠋⠉⠀⠀⠀⠀⠘⠓⠲⠶⠦⠤⠤⣬⣭⣭⣭⣭⣭⣭⣉⣉⣉⣭⣭⣭⣭⣭⡤⠤⠤⠴⠶⠒⠚⠁⠀⠀⠀⠈⠉⠛⣻⣶⣄
⣼⣇⡙⠛⠿⢿⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣿⠿⠟⠛⣁⣷
⣿⣿⣿⣿⣶⣶⣤⣤⣭⣉⣉⣉⣛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⣉⣉⣉⣉⣤⣤⣴⣶⣾⣿⣿⣿
⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠈⠛⢿⣿⣿⣶⣧⣽⣍⣿⣙⣟⢻⡟⢿⡁⠀⢹⣿⣿⡏⠀⠈⣿⣿⠁⠀⣹⣿⣿⡁⠀⢹⡟⣿⢛⣟⣻⣉⣯⣽⣷⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣟⣛⣉⣉⣉⣉⣉⣉⣉⣤⣤⣤⣤⣭⣭⣉⣉⣉⣉⣉⣉⣙⣛⣿⣷⣶⣦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢷⣴⡾⠛⣿⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠀⠀⠀⠀⢸⣇⠀⠈⠻⣦⡀⢀⣀⡀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣯⣅⣀⠀⠀⠀⠀⠀⠙⢳⣦⡀⠙⣷⡄⢻⡟⠓⠶⠞⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣴⠟⠛⠛⠋⠙⠁⠀⠈⢿⣄⣿⠄⠀⠀⠀⠈⠛⠳⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠋⢀⣀⠀⠀⢠⡶⠞⠋⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣦⣄⠀⣀⠀⠀⠀⠀⠀⠸⣿⡟⠛⠋⠉⠳⣄⠸⣧⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢷⣄⠀⠀⠀⠀⣈⣿⣦⡀⠀⠀⠘⢿⣿⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢷⠶⠞⠛⢩⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠃⠀⠀⠀⠀⠀⣠⡟⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣯⣉⡀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⡄⠀⠀⢀⣀⣀⡀⠈⠛⣷⡄⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣰⠿⠋⠉⠛⠛⠛⠛⠛⠂⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⡀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠈⢿⣆⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⢻⣆⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⢿⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢿⡆⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⢿⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⢀⡴⠆⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢰⣿⠁⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠸⣿⡀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠹⣿⣆⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠘⣿⡆⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣿⠃⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠼⠥⠤⠤⠤⠾⠧⠤⠤⠤⠴⠯⠤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢋⣩⣥⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⡶⢶⣤⣤⣭⣉⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⠾⠿⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠴⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡠⠤⣄⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡰⠃⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣀⣀⣀⡀
⠀⠀⠀⠀⡤⠖⠚⠉⢉⣥⣤⣶⣶⣾⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣶⣶⣤⣬⡉⠉⠓⠲⢤
⠀⠀⠀⠀⠑⠦⣄⣀⡘⠛⠿⠿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⠿⠿⠛⢁⣀⣠⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⢶⡤⠬⠭⠭⢭⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⡭⠭⠭⠥⢴⡶⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢒⠂⢄⠴⠂⠠⠈⠭⠀⠀⠐⡂⣠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⢗⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠝⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⣿⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠈⢒⠶⠀⠤⠤⠄⠀⠀⠤⠀⣀⠀⠤⠤⠤⣴⡤⠼⠛⠉⡠⢐⣒⣒⡲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠀⠈⠊⠈⠈⠀⠉⠁⠀⠀⠀⠀⢠⡶⠋⠀⠈⠹⡆⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠤⢠⠀⠀⢀⠉⡑⠂⠁⠀⠀⢀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡔⠁
⠀⠀⠀⠀⠀⢀⣀⣤⣴⡖⣶⣞⠭⠀⠈⠉⠐⠒⠨⠶⣶⣽⣤⣶⡿⢧⠀⢀⣶⣣⠟⠀⠀⠀⢀⣶⠤⠀⣀⠄
⠀⠀⠀⡠⢞⠟⢃⠔⠘⠉⠊⠁⠀⠀⠀⠀⠀⠀⣴⠄⣹⡿⣤⣽⣿⠿⠃⢿⠛⠀⠀⠀⠀⠀⠔⣠⠴⣾⡷⣿⣵⣦⣄
⠀⢠⡎⠔⠁⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⣹⣿⡿⣏⠙⢄⠰⣖⢀⣀⠀⠀⡴⠋⠐⡾⢿⠳⠈⠉⠛⡾⣿⢦
⠀⢸⡐⠀⠀⠀⠦⣼⡷⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠋⠀⠈⠀⠀⠀⠉⠈⠁⠀⣼⣿⣿⢿⣿⡆⢀⣤⣴⠀⠀⢩⡿⡇
⠀⠀⠑⢵⣄⠀⠀⠘⠛⢿⠻⠿⠂⠀⠀⠀⠀⠀⣀⣀⣀⣠⡠⠤⢤⡤⠠⠄⣀⡀⠠⠷⢻⠿⣿⠿⣱⠾⠛⠛⠁⠀⢠⣖⡼⠁
⠀⠀⠀⠀⠙⠓⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠻⠻⠿⠿⢿⣿⣿⣿⣿⣿⢃⡄⠀⠀⠀⠀⠈⠆⠀⠀⡀⡤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠤⠤⣤⣤⣌⣒⣒⣠⣤⣤⣤⣤⣭⣵⣶⣫⡤⠤⠤⠶⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢤⡀⠀⠄⠀⠀⣀⣀⣀⣀⣀⣄⠀⠀⣤⣠⣤⡤⠄⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠈⠀⠀⣤⡀⠀⠐⠂⠀⠀⣠⡄⠀⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡄⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢾⡶⠀⠀⣀⣴⣶⠿⠿⣶⣦⡀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄⠀⠀⠀⠀⢀⡘⢿⣿⣇⠀⠀⣸⣿⣿⣷⡀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⣀⣠⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠘⠋⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣯⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠉⠉⠈⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⡙⣿⣿⣿⣿⢿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢀⣤⣤⢀⣴⣤⢠⣤⡀⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢹⣼⣿⣿⣼⣿⣧⣿⡿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⢡⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⢈⣿⡟⢿⠏⣿⣿⣿⠃⣿⣿⢿⡿⣿⡿⣿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⠀⢹⡟⢸⡄⢻⡟⠈⠃⠹⠀⠈⠇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠒⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⢠⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣈⠓⠻⠷⢶⣦⣥⣀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣡⣰⣴⠶⠟⠒⠉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⣄⣀⣀⡉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠉⣀⣀⣀⣤⣴⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢋⣸⣿⡻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡏⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡴⢋⡠⠔⠊⠉⣿⣿⣷⣈⣩⣤⣶⣦⣝⣿⣿⣿⣿⡿⣩⣟⢿⣿⡿⢙⣛⡻⠿⠇⢿⣿⣿⣿⣿
⠀⠀⠀⠀⡸⢠⠋⠀⠀⠀⢀⣸⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⣼⣿⣿⣿⣷⣾⣿⣿⢿⡇
⠀⠀⠀⠀⡇⠆⣀⣤⣶⣿⣿⣿⣷⡀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁⠀⣼⣿⣷⣶⣤⡀
⠀⠀⠀⢀⣥⢸⣿⣿⣿⡿⠋⠁⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠙⠻⣿⣿⣿⣷⣤⡀
⢀⠀⣴⣿⣿⣆⢻⡿⠁⠀⢀⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣄⠀⠈⠻⣿⣿⣿⣿⣆
⡄⣼⣿⣿⣿⣿⣧⣑⠤⣀⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⢹⣿⣿⣿⣿⣇
⡇⣿⣿⣿⣿⣿⣿⠈⠑⠢⢭⣙⣛⢛⡿⠿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⡿
⠱⡹⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⠄⠊⠀⠈⠙⠛⠿⠿⠿⠋⠀⣠⣿⣿⣿⣿⣿⠃
⠀⠑⢝⢿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠑⠿⣻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠈⠁⠀⠀⠀⠈⠉⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⢱⡎⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⡀⠄⠒⠊⠉⠀⠀⠀⠒⠄⢏⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⣶⠀⢠⠊⠀⠄⠀⢠⠀⠀⢀⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢸⡤⠑⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠸⡄⠀⣀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣌⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠔⡛⠀⠀⢀
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠠⠺⡁⠈⠉⠉⠉⠁⠒⠒⠈⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠸⠃⠀⠀⢸
⠀⠀⠀⠀⢘⠓⢄⠀⠀⠀⠀⠀⠀⠈⡒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠀⣤⠤⠤⠔⣂⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠢⣀⠉⠂⠄⡀⠀⠀⠀⠈⠁⠂⠬⣁⠒⠤⠤⣆⣀⠄⠂⠀⠀⠀⡄⠀⠀⢠⠬⠕⠲⠤⣤⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠈⠁⠒⠠⠬⢁⡐⠂⠤⠤⢄⡐⣀⡪⠀⠀⠀⠀⠀⠉⠛⠶⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠍⡒⠤⢀⠀⠈⠉⠒⠀⠤⣀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠠⠬⠍⢙⣒⠒⠒⠒⠒⠒⠒⡻⠋⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡈⠁⠂⠠⢀⡀⠀⠀⠉⠐⠂⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⠀⠀⠉⠑⠒⠠⠤⢀⡀⠀⠀⠀⠉⠁⠒⠀⠀⠠⠤⠤⢀⣀⣀⠀⠀⠀⢀⣀⠠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢀⡀⠀⠀⠀⠀⠉⠀⠒⠀⠀⠠⠤⠀⢀⣀⣀⠀⠀⢀⣀⣀⡀⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡄⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⢀⣤⣶⠿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡿⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋
⠀⠀⠀⠀⠀⣄⣋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣁⠀⠀⠀⠀⠀⠀⠀⢀⣸⡋
⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠺⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣇⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣖⣭⠵⠶⠆⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⡋
⠀⠀⠀⠀⠀⠠⣿⣗⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠇
⠀⠀⠀⠀⠀⠠⣴⣦⠁⠀⠀⠀⠀⠀⠀⠀⣴⣷⠍⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠀⠀⠀⠀⠀⠀⠀⣤⣦⠍
⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠂⠀⠀⠀⠀⢀⣰⣋⡀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣶⣯⠀⠀⠀⠀⠀⠀⠀⢀⣠⣜⠁⠀⠀⠀⠀⠀⠀⢠⣤⡄⠈⠈⠀⠀⠀⠀⠀⣴⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠈⠟⠓⠀⠀⠀⠀⠀⠀⠀⠈⠟⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⣠⡿⠛⢿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⡏⠉⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢹⣶⣾⣿⣯⣿⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⣛⣛⣿⠿⡿⣶⣾⣭⣟⡿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⣍⣿⣷⣾⣿⣻⢿⣿⣶⣯⣻⣿⣿⣿⣿⣿⣽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠿⠷⠟⠿⠛⠉⠙⠛⠿⠿⠿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⢀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠀⠤⢂⣆⠃⠀⢰⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠀⠠⠀⠀⢈⣙⣋⡙⠂⠀⠁⠀⢀⠈⡘⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⡀⠀⠀⠀⢀⠀⠒⢀⠀⢀⡈⣰⠄⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⢀⡈⠐⠛⠀⠀⠀⠀⠙⠂⢀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⡈⠀⠀⠁⡐⠉⠀⠁⡀⠂⠉⠂
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠆⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡀⡑⡄
⠀⠀⠀⠀⠀⠀⠙⠒⠠⢀⠀⠀⠀⠀⠰⣄⣀⣀⣀⠀⢀⣀⣀⡼⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⡀⠐⠀⠠⠬⠥⣭⡭⠭⡥⠤⠴⠖⠶⠶⠶⠎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠂⠒⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣥⣤⣤⣤⣤⣬⣷⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⢀⣤⣶⢿⣻⣽⣾⣿⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣖⣺⣭⣭⣽⣿⣶⡶⢶⣶⣶⢲⣶⣶⠾⣿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⣰⣴⡾⣛⡽⢾⣿⣷⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠿⠿⠿⣟⣻⣿⣿⣿⣛⣿⠿⠿⠷⠾⠛⠛⠉⠁⠀⠀⠀⠀⢀⣠⠾⣫⣵⢿⣿⡿⣟⣯⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⣵⠿⣿⣿⡿⣻⣾⣿⣿⡿⠟⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣾⠿⠿⠛⠉⠁⠀⠉⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣧⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

















⠀⢸⣤⡀
⠀⣸⡿⢿⣶⣄
⠀⣾⣿⣯⣿⣿⣿⣶⣄⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣧⣿⣟⠳⢶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⡆⣠⠄⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⣀⠀⠀⠀⠀⠐⠒⢰⣦⣄⣾⡿⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢳⣤⣀⡀⢆⣀⣠⣦⣿⣿⠛⠂
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⠟
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣫⣤⣄⣠⣤⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣦⣤⣦⠀⠀⣀⣀⢀⡀⣬⡍⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻
⣤⣄⣤⣤⣬⣤⣬⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠁⠀⠀⠈⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⣀⡤⠊⠀⣸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢰⣄⣀⣤⣤⢴⣖⣋⣁⣤⡶⠚⠁⣠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⣶⣾⣿⣿⣿⣿⣛⣛⣉⣥⣴⣾⣿⣿⣵⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣟⢻⡿⣅⣀⡀⠠⡤⢤⠀⠀⣤⢤⠄⠀⣀⣨⠿⡟⣻⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢻⡟⠙⠛⢛⠻⣿⣷⡘⠷⣄⠀⠑⠀⡇⢸⠀⠚⠀⣠⠾⢋⡾⣿⢟⡛⠛⠛⢻⡟⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⢾⣿⣷⣬⣿⡛⣇⠛⠈⡆⣱⠈⢳⡙⣌⢎⠓⠶⠊⠀⠀⠑⠶⠚⡱⣡⢋⡞⠁⣎⢰⠁⠛⣸⢋⣿⣥⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣿⣿⠻⠿⠿⠶⣭⣭⣀⣝⣌⣓⣷⣄⣐⡂⢐⣂⣠⣾⣚⣡⣋⣀⣬⣭⠶⠿⠿⠟⣿⣿⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣶⣦⣴⣀⠄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣀⣤⣴⣶⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣙⣿⡿⣿⣷⢶⣶⣶⡶⠶⠶⢶⣶⣶⡶⣾⣿⢿⣿⣏⠍⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠻⣶⠮⠝⠿⣭⣭⠿⠫⠥⣶⠟⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠖⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀





⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⣀⣀⠠⠤⠔⠛⠲⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠄⠒⢈⡁⠤⠐⠒⠒⢈⣉⣩⠬⢤⠤⠼⢿⠛⢷⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡁⠄⠂⢉⣀⠤⠔⠒⡏⢉⣁⠼⠤⠤⠚⢒⠖⠛⠒⠚⠒⢾⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢊⡡⠴⣶⠉⣉⠷⠤⠒⠸⠉⠀⢴⡦⠴⠀⡠⠂⠀⠂⠔⡡⢒⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⣵⠊⢹⠦⠔⠋⠁⠐⡒⠖⠀⢀⣄⠤⠤⠆⠒⠚⠒⠶⣤⢖⣧⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢯⣠⠜⠋⠑⠂⠤⠀⢀⡠⠔⠒⠉⠀⠀⣀⣀⢀⣀⡀⠀⣠⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢺⣲⡎⠀⠂⠉⠃⠀⢠⠖⠁⠀⢀⣠⡐⡈⠡⢠⣢⣬⣾⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠬⣄⣨⣵⣙⣤⣄⣀⣙⣓⣶⠷⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⡏⢹⣉⡇⠸⣿⠇





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⣠⣶⣿⣿⢿⡿⠛⢿⡛⠙⣍⠉⢹⣅⣀⣹⣀⣀⣀⣸⣇⣀⣹⡋⠉⡹⠛⢛⠿⠻⣿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠙⠛⣭⣵⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣯⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⢟⣿⣿⣯⣽⣯⣿⣿⣿⣭⣿⣿⣿⠿⠿⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⠭⣥⣤⣉⣉⡉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠤⠮⠤⠤⠴⢯⣅⣤⣾⣿⣿⣿⣿⣿⣿⣶⣦⣽⣆
⠀⢀⣠⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣬⣉⣉⡙⠛⠻⠿⣿⣿⣿⣷⣦⡀
⠀⠈⠛⢧⣬⣷⣤⣷⣤⣬⣧⣀⣸⣆⡀⠙⡏⠉⠉⣿⠛⠛⠿⠿⣿⣷⣶⣦⣬⣉⠛⠻⢿⣦
⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⣤⣤⣠⣟⠀⠉⣹⠟⠻⠿⣷⣦⣬⣑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣉⣈⡹⠟⠷⣦⣍⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠿⠿⢿⣷⣶⣯⣭⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣍⣉⡿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣷⡯⠻⠿⠿⠿⢿⣿⣿⣿⣿⡿⠷⠖⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡿⠛⠉⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀
⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣼⣿⣿⣶
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⠛⠛⠋⠉⠉
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣾⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⣿⣿⣿⡿⠁⡰⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿
⠸⣿⣷⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⡰⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆⠀⠀⠀⠀⠀⣿⡇
⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⣹⣿⣿⣧⡀⠀⠀⢸⣿⠇
⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠛⠁⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⣼⣿⣿⢿⣿⣷⡄⠀⢸⣿
⠀⠀⠀⠘⢿⣿⣿⣦⣀⠀⢤⣀⡀⠐⠛⠛⠋⠉⣉⠠⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⣴⣿⣿⣿⠋⠀⠹⣿⣿⣆⣿⡏
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣤⣈⠑⠪⠍⢈⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠒⢉⣠⣶⣿⣿⣿⠟⠁⠀⠀⠀⠈⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣶⣦⣤⣄⣉⣀⡐⠒⠒⠒⢂⣈⣉⣠⣤⣶⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⣴⣾⣿⣿⡿⣿⣿⣻⣿⣿⣍⣿⣿⣿⣭⣿⣿⣿⣯⣽⣿⣿⣿⣽⣿⣿⣿⣹⣿⣿⡟⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠉⠛⠿⠯⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠉⠉⠉⢉⠉⠉⠉⠍⠉⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⠽⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠃⠀⠀⠀⠀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⡀⠀⠃⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠯⠭⠭⠭⠭⠽⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠖⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠚⠁⠉⠉⠉⠉⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠒⢄⣀⣀⣀⢀⠀⠀⠀⠀⠠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠀⠤⢤⠋⠉⢄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠠⠈⠀⠀⢨⠀⠀⠰⠆⠀⠀⠠⠲⠀⠀⠀⠀⠈⠳
⠀⠀⠀⢀⠤⠒⠈⠉⠁⠀⠠⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⢔⠰⠀⠀⠀⠀⢀
⠀⠀⣰⠁⠀⢀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⢠⠀⠀⠐⠍⠂⠀⠀⠀⠀⠀⠀⢄⡸
⠀⠀⢣⡀⠀⠀⠁⠀⠀⠀⠀⠐⠁⠁⠀⠀⠘⠀⠐⠀⠀⠀⠃⠀⠂⠀⠀⠀⠃⠀⠃⠀⠀⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠑⠊⠁
⠀⠀⠀⠉⠒⠦⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⣀⡀⠤⡄⢠⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇⡟⠉⠙⡆⡇⠀⠀⢃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⠉⠉⣿⢇⡧⣀⠀⣃⠡⠀⠀⠸⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣠⠖⠒⠒⠢⠤⣀⡀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⡇⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠘⢆⠀⠀⠣⠀⢀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠑⠦⡀⠀⠀⠑⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠈⠑⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠐⡏⠑⠀⠀⠀⠀⠀⢄⠀⠈⠁⠒⠤⠄⣀⠀⠀⠀⠀⠀⢀⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠉⠐⢂⣤⢤⣀⣀⣀⣀⣀⣀⡤⠞
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣶⣶⣶⣶⣿⣯⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⠙⠛⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⠴⠚⠋⠉⠉⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡴⠖⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⠟⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⠤⠤⠤⠤⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣴⡶⠶⣤⡀⠀⠀⠀⠉⠛⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠶⠞⠛⠉⠀⠀⠀⣿⠇⠀⠀⠒⠲⢶⣤⣀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⢹⣷⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢈⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡴⠋⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⢀⡴⠋⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣴⠞⠛⠛⠳⢶⣤⡴⠛⠁
⠀⣰⡟⠁⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠻⠤⠤⠠⠖⠛⢿⣷
⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⢤⣤⡤⠴⠶⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⡶⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠰⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠉⠛⠛⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣻⣇⣽⣵⣶⣶⣦⣤⣤⣤⣧⡴⠂⣠⣴⣶⠦⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠟⠉⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡠⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠷⢷⡶⠶⠾⢿⣶⣤⣀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⡿⠉⠠⠄⠀⠀⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⠬⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⢶⡶⠿⠷⠿⠶⠀⠀⠀⠀⠰⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢀⣀⠀⠀⢘⣷⣦⣴⣦⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣋⠤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠉⠀⠀⠀⠉⠉⠉⠛⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣀⣀⣀⠀⠀⠀⢀⣀⣀⣠⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠏⢹⡀
⠀⠀⠀⠀⠸⡟⠁⠓⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠁⠀⠈⠓⠶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠰⠗⠢⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠦⠎⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⠶⠚⠶⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡀⢠⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡇⠀⠀⠀⠀⠛⠉⠉⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠴⠛⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠶⠤⠤⠤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⣨⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠛⠳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠀⠀⢀⣈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⢀⣀⡤⠴⠒⠉⠉⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⢀⡠⠔⠚⠉⠀⠀⠀⣀⣀⡠⠤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⣠⠔⠊⠁⠀⢀⣠⠤⠖⠚⠉⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡠⠖⠉⠀⣀⠤⠖⠋⢁⣀⡠⠤⠖⠒⠛⠉⢹⠏⢀⣀⣠⡤⠶⠞⠛⠳⣄⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠔⠋⢀⡤⠖⠋⣀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡠⠞⢁⡤⠚⣁⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣼⡇
⠀⠀⠀⠀⠀⠀⢀⠜⡡⢊⡠⢚⡡⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣠⣫⢊⢔⡫⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⢣⡀
⠀⠀⠀⠀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠜⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣀⢠⠟⠁
⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⡬⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙
⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠲⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠿⠟⠛⠛⠻⠿⠿⠷⠶⠶⢶⣶⠶⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣶⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡄⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⣀⣤⣶⣿⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⣰⣿⣿⣿⣿⢿⣿⣷⣶⣄⡀
⠀⢀⣴⣿⣿⠟⠋⠀⠀⠙⢿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣴⣿⣿⣿⡿⠁⠀⠀⠉⠻⢿⣿⣷⣦⡀
⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣷⣦⣤⣤⣤⣤⣴⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿
⠀⠀⠉⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⡿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠐⠂⠀⠀⠤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⠠⠤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠈⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠵⠤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠠⢀⠈⢀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠂⠉⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⢀⠠⠂⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠌⠵⠄⠀⡤⢐⠠⣥⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠛⠉⢉⣘⣇⣼⡀⣇⡀⣸⣽⣷⣷⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣠⣤⡟⢢⣀⣰⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢻⣟⣻⡵⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣾⠃⠈⢷⠸⣿⣿⢝⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⢧⡀⠸⣆⠹⡆⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣍⡂⠹⣆⠙⢆⢀⡻⢿⣧⡀⠀⠀⠀⠀⠀⢀⣾⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⣷⣬⣉⣡⣼⣿⣷⣀⠀⠀⠀⣠⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣴⡶⠖⠚⣫⣾⣿⣿⣭⣝⣛⣛⣛⣺⣽⣏⣿⣐⣀⡀⠀⢀
⠀⠀⠀⢀⣀⠀⠀⢠⣾⣿⣿⣿⣏⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠾⠿⠛⢋⣹⣿⣿⠿⠃⣡
⠀⠀⠓⠾⠿⣿⣿⣶⡟⠿⠟⠿⠿⢷⣶⣬⣭⣉⣙⣛⡛⠛⠒⠒⠂⠀⠀⠀⣈⣉⣡⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢬⣁⡀⠀⡀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⠶⠾⠾⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⡟⣛⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠯⡤⠙⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡁⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡈⢿⣦⣀⣠⣤⣶⠾⠿⠿⠛⠿⠿⠷⣶⣦⣤⣀⠀⣠⣾⢛⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⡿⠛⠋⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠉⠛⠿⣿⣥⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⢠⣾⠟⠁⣤⢶⣄⠀⠀⠀⢰⡟⠉⠙⣧⠀⠀⠀⠀⣀⣀⠈⠙⢿⣆⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⣴⡿⠁⠀⠀⠻⠶⠋⠀⠀⠀⠈⠻⠶⠾⠋⠀⠀⠀⠸⣯⣽⠇⠀⠀⠹⣷⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⠿⠿⠛⠛⠛⣉⣉⣉⣉⣡⣤⣤⣤⣤⣤⣤⡴⠶⠶⣤⣤⣤⣤⣍⣉⣙⣛⠛⠻⠷⣿⣷⣄⡀
⠀⠀⠀⠀⠀⣀⣴⡾⠟⢛⣉⣤⡴⠶⠶⠛⠛⠛⠉⣭⢿⣿⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⢯⡉⠉⠛⠛⠳⠶⣬⣙⠻⢷⣦⡀
⠀⠀⠀⢀⣾⠟⣩⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⢟⣿⣯⣤⣤⣶⣶⣶⣶⣤⣤⣄⣻⠾⢿⣿⠾⠃⠀⠀⠀⠀⠀⠀⠉⠛⠶⣝⡻⣶⡄
⠀⠀⠀⣾⡏⣾⠃⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠿⠷⠶⠶⠶⠶⢯⣽⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⢿⣆
⠀⠀⠀⢻⣿⡇⠀⠀⠀⣼⣫⣝⠛⢻⠿⣦⠀⠀⣿⡇⣿⠁⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠻⣿⣿⣆⠀⠀⢠⡶⢚⣛⠟⡛⢶⡄⠀⠘⢿⣿⡆⢸⡆
⠀⠀⣦⠘⢿⣧⡀⠀⠀⠈⠛⠛⠿⠿⠚⠁⠀⠀⠘⢿⣮⣿⣿⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣿⣤⠾⣻⡿⠀⠀⠈⠳⣮⣷⣶⡯⠾⠃⠀⠀⣸⡿⠁⠈⠁
⠀⠀⠘⠷⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢹⣷⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣤⣤⣄⣀⣀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⣿⡿⠿⠿⠿⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡶⠶⠾⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢀⣀⣠⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣄⣁⡀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢦⣌⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⠀⠰⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠉⠉⠉⠑⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡖⠒⠢⡟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠊⠛⠛⠋⠉⠉⠁⠈⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⢠⢾⠉⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⠀⠀⠀⢀⣎⠙⠛⠺⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠃⠉⠀⠀⠀⠀⠀⠠⠈⠛⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⠄⠄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠉⠓⠒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠖⠢⠤⢤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣷⠶⠦⢤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠉⠛⣿⠛⠒⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇











⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡄
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠉⣠⣴⠞⠋⠁⠀⠀⢀⡤⠊⠀⠀⠄⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⡴⠞⠋⠉⣴⣷⡶⠟⠉⠀⢀⡠⠒⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠺⣿⣿⠋⠁⠀⠀⠀⠀⠘⠉⠃⣠⡴⠞⢉⡤⠞⣁⠄⠀⠀⠠⣷⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⣀⣤⣶⠟⡉⣠⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠾⣿⣿⡋⠀⠐⡿⠏⣠⠞⠁⠀⠀⠂
⠀⠀⠀⠀⠀⢲⣶⣶⣿⣧⣀⠀⠋⠀⠁⠀⢀⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⠀⢰⣄⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠘⠋⠉⠻⣿⠀⠀⠰⢿⣿⣿⡄⠀⠀⠀⢀⠠⠀⠀⠀⢾⢿⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⣶⠶⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⢀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠮⠼⠷⠖⠓⠒⠛⠓⠒⠚⠧⠶⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡾⡿⠷⠿⠷⠶⠒⢒⣒⣚⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⠶⠶⠦⠤⠤⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣶⡻⣟⡩⠯⠵⣶⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠸⢿⣉⣭⣶⣶⡿⢿⠻⠟⠛⠛⠛⠛⢛⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠉⢈⣤⠄⠐⠂⠈⠁⠁⣀⣀⡀⠁⠀⠀⠤⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠻⣶⣌⡒⢌⣂⣙⠛⠛⠢⡀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠶⣤⣌⣶⣄⣀⣀⣠⣥⣼⣿⣿⡿⠿⠿⠟⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠤⠀⠠⠄⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣤⣴⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡶⠆
⠀⠀⠀⢠⣆⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷⡶
⠀⠀⠀⣼⣿⣄⠀⣀⣿⣿⣿⣿⣿⣿⢿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣛⣁
⠀⠀⣘⣿⣿⣿⡤⢨⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⣻⣿⣿⣿⣯⣛⢋⡀
⠀⢀⣭⣿⣿⣿⡿⢂⣿⣿⣿⣿⣿⣿⣻⣿⠷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢸⣇⠀⢰⣿⡀⡀⠀⠿⣿⣿⣿⣿⣿⣟⣋⣁
⠀⠦⣽⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣟⠛⠃⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡇⢠⡆⢠⣿⣿⣤⠾⠿⢷⣿⠆⣴⣾⣿⣿⣿⣿⣟⣋⣁
⠀⠲⠿⠿⡿⠿⠿⢏⡡⠒⠎⠤⠉⢩⣙⡛⠳⢦⡄⠶⢿⣦⣦⢀⣀⣀⣄⣼⠷⠴⠶⠾⠛⠛⠛⠛⠉⠉⠉⠁⣰⣦⠀⣿⣿⣦⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠈⠛⠛⠶⢶⣦⣴⣾⣿⣦⣠⣶⡄⠈⣁⢀⣠⣿⡒⠠⠥⣬⠎⠁⠀⠀⠀⠰⠶⠿⠿⠶⢦⣄⣼⣧⣤⢼⠿⠿⠾⠟⠛⣛⡉⢉⣡⣤⡷⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⠿⠿⠛⠋⠹⠛⠻⢿⣿⣷⣦⠐⠚⠂⠀⠀⢀⡘⠛⣉⣭⣦⣶⣷⣾⣶⣧⣀⢀⣾⣿⣟⠫⠥⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠛⠛⠉⠉⠉⠉⠉⠉


⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣿⣿⠿⠻⠿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢡⡁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠁⠀⠤⠀⠂⠀⡀⠀⠀⠐⡄⠀⠀⠀⠀⠉⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⡀⠄⠂⠁⢀⠠⠀⠂⠉⠄⠀⠀⢠⠜⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⠄⠊⠀⡀⠔⠈⢀⠄⠀⢁⠠⠐⠁⠠⠆⠀⠂⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠔⠁⠀⡠⠊⠀⡠⠊⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⠁⠀⡠⠊⠀⡠⠊⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠊⠀⢀⠈⠀⢀⠄⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⡠⠁⠀⠠⠁⠀⠠⠂⠀⢀⠂⠀⢀⠊
⠀⠀⠀⠀⠀⠰⠁⠀⠠⠁⠀⠠⠁⠀⢀⠂⠀⢀⠂
⠀⠀⠀⠀⢀⠃⠀⢀⠃⠀⠀⠂⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⡀⠀⠀⡄
⠀⠀⠀⢀⠁⠀⠰⠀⠀⠀⠁⠀⠀⠆⠀⠀⡄
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠶⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣥⡄⠀⢰⡆⠀⢠⣬⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠿⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠽⣇⣀⡀
⠀⠀⠀⠀⠀⣠⠤⠒⢋⣉⣧⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣼⣉⡙⠒⠤⣄
⠀⠀⠀⢰⢋⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡙⡆
⠀⠀⠀⠈⠻⢽⣝⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⡶⠶⠶⠶⠶⢶⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣫⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣟⣿⣿⣿⣟⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣻⣿⣿⣿⣻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢽⣿⣿⣛⠉⠉⠉⠉⠉⠉⣛⣿⣿⡧⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡛⠋⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡡⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠨⠙⠛⠛⠒⠒⠒⠚⠛⠋⠉⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠻⠽⢀⡴⡄⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⣠⢄⠀⠯⠗⣠⢿⡇
⠀⠀⠀⠀⠀⡀⠀⠈⢷⡉⠳⠦⣤⣉⣁⠀⠯⠇⠀⠯⠗⠀⠻⠞⠀⣙⣋⡤⡶⠋⣡⡞⠁⢀⡀
⠀⠀⠀⠀⠊⠁⢠⠀⠀⠙⠲⣄⡀⠈⠉⡉⠛⠛⠛⠛⠛⠛⠛⠋⠩⠉⢀⣠⡵⠟⠉⠀⢄⠀⠙⠦
⠀⠀⠀⠀⠀⠴⠁⠀⢠⠆⠀⠀⠉⠙⠒⠶⠤⠤⣤⣤⣤⡤⠤⠴⠖⠚⠋⠁⠀⠰⡄⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡄⠀⢳⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⣸⠀⠀⠀⡇⠀⠀⠘⡇⠀⠀⢷⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠛⠛⠳⠿⠭⠭⠭⢽⣭⣤⡤⠤⠤⠤⠤⠤⠖⠚⢛⡆⢀⡠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⠖⠒⠒⢄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠷⠁⠀⠀⠀⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣳⣶⣿⠀⠀⠀⠀⠀⣸⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣽⢷⣿⡇⠀⠀⠀⠀⡴⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠂⡀⠈⠀⠈⠀⠀⠀⠀⠰⠀⢀⠻⢿⣛⣻⠁⠀⢀⡠⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⢟⣒⣭⠵⠞⠉
⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡾⠋⠀⠀⠀⠀⠀⠀⠀⣈⣴
⠀⠀⠈⠙⠒⠶⢶⣤⣤⣤⣤⣌⣧⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⣀⣼⣤⣤⣤⣤⣶⣶⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢿⣿⣿⣿⡯⢉⣉⣉⣉⣙⣛⣛⣿⣿⣭⣭⣽⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠐⠒⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⡓⠀⠀⠀⠀⠀⢠⣵⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣥⣿⡿⢛⣩⣭⣭⣭⣙⠻⣿⣧⣿⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣿⣿⠿⠧⠈⠙⠛⠛⠛⠉⠤⠿⢿⣿⣧⣬⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢪⠉⠁⣶⣾⣿⣅⣀⣀⣀⣀⡀⣀⣀⣹⣿⣿⡆⠉⢩⢡⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣤⡿⣿⣿⣯⡽⠿⢛⣛⣛⣻⣿⣿⠋⠉⠉⠉⠉⠉⢹⣿⣿⣛⣛⣛⠻⠿⣿⣿⣿⡿⣧⣄
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠞⠿⣟⣛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⢛⣛⡿⠟⠾⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠴⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⢀⣼⣿⡇⠄⠀⠀⠀⠂⠀⠀⢀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠸⠟⠻⢧⣀⣽⡦⠀⠀⠀⠀⠈⠀⢻
⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⡿⠋⠀⠀⠀⠀⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴
⣿⣿⣦⡀⠀⢀⣼⣦⡀⢠⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡤⡣⣀⠀⠀⣧⣠⣿⣄⠀⠀⣠⣾⣿
⣿⣿⣿⣟⣶⣿⣿⣿⣿⣾⡇⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠁⢀⣿⣿⣿⣿⣷⣞⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⠀⠁⣴⣿⣷⣀⣠⣤⣀⣀⣀⣠⣄⣠⡀⠀⡃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡆⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣷⣷⡄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⡏⠉⢽⣿⣿⣿⣻⢿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣬⣬⣧⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⡀⠀⣒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢺⣷⣦⣤⣄⡀⠀⠀⠀⠀⣀⣠⡤⡴⢿⠷⡿⣿⢾⣿⣷⣿⣷⣦⣤⣀
⠀⠀⠉⠛⠛⠭⢭⣿⣿⣿⣿⣿⣿⠿⠻⠛⠉⠀⠀⠀⠈⠉⠓⠒⠀⠀⠀⠀⠠⣴⣿⡏⠉⠉⠙⠻⠋⠉
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠉⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⢀⣤⣴⠶⣿⠍⣤⣠⣤⠰⣿⠿⢶⢾⢭⣷⢶⣦⡄⠀⠀⠀⠀⡀⣀⠀⠀⣀⠀⠈⣛⠛⣶⣤⣄
⠀⠉⠉⠛⠻⠿⡿⢿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⡖⢂⡉⠐⣲⣿⣶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣺⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠉⠉⠙⠋⠁⠀⣇⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣤⣴⣶⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡷⠂⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠉⠉⠠⣶⠀⠀⠈⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠘⡿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠂⠀⠉⠀⠀⠀⢠⣦⡄⠀⢲⣿⡖⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡀⠀⠀⠶⠀⠀⣀⣄⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⢠⡀⠀⢀⠀⠀⣻⣿⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⣛⡛⢻⡛⣻⠿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⢀⣠⣤⣴⣶⣶⢖⣀⣀⣀⣀⣀⠀⠒⠒⠢⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠒⠒⢤⡀⣯⣿⣿⠟⠶⠶⠚⠿⠿⠭⠍⠭⠭⠿⠷⠦⠭⠶⠾⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣧⣸⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡔⢄⠀⠘⠿⡼⣿⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⠀⠀⢣⢻⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⡚⠌⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⢻⣶⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠱⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⢀⣀⣙⣓⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⡦
⠀⠀⠀⡤⠦⠤⠤⠠⠀⠀⠀⠉⠙⠒⠦⢤⣹⡿⣿⣮⣿⣄⡀⠀⠀⠀⠀⠀⢀⣠⣕⣀⣀⣀⣠⣤⠤⠖⠚⠉
⠀⠀⠀⠉⠓⠲⠦⠤⠤⣀⣀⠀⠀⠀⢀⠀⠀⠀⠺⠿⠿⠛⠛⢟⣿⣄⣀⣨⣤⡴⠶⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⣂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣦⠉⠉⠉⠀⠀⢄⠀⠀⢣⣉⣲⡆⠑⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠙⠛⠛⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉













⡀⢀
⣿⣦⣶⡙⣒⣤
⣿⣿⣿⣿⣿⣟⡛⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⣿⣿⣿⣿⣿⣷⣿⣤⣦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢄⡀⠀⠀⠀⠀⠀⠰⢦⣾⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⣶⣶⣴⣶⣾⣿⣟⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠄⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣷⣤⣤⣀⣀⣀⣀⡠⣍
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠛⠛⠛⠋⠉⠉⠙⠛⠛⠛⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠛⠒⣿⠃⠰⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⡴⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠶⢤⣄⣀
⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣀
⠀⠀⣿⠙⠶⢚⡟⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⢴⣟⣁⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⢹⣆⡞⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠦⣄⣀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⣶⠀⠠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡄⣀⣻⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣿⣿⣯⣿⣿⣿⢿⣿⠋⠟⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣿⣾⣿⣿⣿⡿⠁⠈⢁⡃⠀⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣿⣿⣿⠛⠟⠁⠋⠀⠀⠈⠃⠀⠀⠀⢵⣾⡟⠳⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣯⣤⣤⣤⡄⠀⠀⠀⠀⠀⠙⠿⠟⠏⠀⠀⠀⠀⣀⠀⠀⣀⣀⣶⣶⣿⣿⣥⠀⠀⢻⣿⣿⣿
⣤⣤⣶⡆⢀⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣿⣿
⣿⣿⡿⣿⡿⠀⠀⠀⣀⣤⣦⣀⠀⠀⣀⣤⣀⣀⣀⠀⠀⠰⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⡗⢀⣠⣬⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡏⣠⣾⣷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢻⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣰⣦⡐⠉⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠐⠆⠈⠻⡟⠁⠀⣀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠁⠰⠆⠀⠐⠂⠀⠀⠀⠀⠉⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠀⢀⣠⡴⣶⡟⠛⠛⠛⠛⠛⠶⣦⣄⡉⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢨⣴⢛⣹⣇⣿⣀⣀⣤⣤⣀⣀⣀⣀⣉⣻⣦⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣥⣴⠶⠾⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣬⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣾⣛⣩⣭⣤⡶⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⠶⠶⢦⣭⣽⣛⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠛⠋⠉⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⠀⠀⠉⠙⠛⠶⢶⣤⣄⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⢠⡌⠉⠛⠳⠶⣤⣤⣀⣀⠀⢉⡽⠿⠛⣛⣫⣭⣭⣭⣭⣍⣛⠛⠳⢶⣄⠀⢀⣀⣤⣤⠶⠞⠛⠉⠁⣤⠀⠀⠻⣦⡀
⠀⠀⠀⣀⣴⠟⠁⠀⠀⣠⣾⣧⡀⠰⠆⠀⢠⠀⠉⠉⠛⠛⠷⠶⢾⣿⡇⣿⠀⠀⠀⢹⣿⣧⣴⠶⠿⠛⠋⠉⠁⠀⠀⠴⠄⠀⣠⣿⣄⠀⠃⠈⠻⣦⡄
⠀⠀⠰⣿⡁⠀⠲⠀⠀⢼⣿⣿⠆⠀⠀⠀⠀⣦⠀⠀⠰⠆⠀⣀⠀⣿⢸⡏⠀⠀⠀⠀⣿⠀⠀⢀⣦⠀⠈⠁⢰⡆⠀⠀⠀⠀⣿⣿⣯⠀⠀⠶⠀⢈⣿⠆
⠀⠀⠀⠈⠻⣦⡀⠐⠀⣺⣿⣿⣃⠀⠤⠠⣾⣿⠦⠀⠀⣠⠀⠀⢰⡟⣼⠃⠀⠀⠀⠀⢹⡇⠈⠻⡿⠛⠀⠰⣾⣷⡆⢀⡀⢀⣻⣿⣿⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣦⣬⣿⣿⣿⣿⡁⠀⠐⣿⣿⣗⠀⠾⣿⡷⠀⣼⢇⣿⠀⠀⠀⠀⠀⠈⣷⠘⠃⠀⠀⠀⢘⣿⣿⡇⠀⢀⣻⣿⣿⣿⣧⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣟⡻⠡⠄⣨⣿⣿⣯⡄⠀⠙⣠⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢹⡆⣄⠰⠂⢀⣬⣿⣿⣧⡀⡌⠿⣻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡙⠃⢠⣾⣿⣿⣿⣦⠀⠀⠀⢸⡇⣾⠁⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣤⣿⣿⣿⣿⣤⠐⢛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣛⣽⣿⣭⣃⠀⠛⠀⣾⢣⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠴⠀⢈⣫⣿⣯⣝⣂⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⣭⡀⠤⢀⡿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⠀⣨⡷⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡛⠶⣄⠀⢸⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢠⡴⢿⣿⠟⠁
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣾⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣴⠟⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡟⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠟⠁
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⢠⣼⠟⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁
⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣴⣶⣶⣶⣶⣶⣶⣶⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⢶⣶⢲⣿⣋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣙⣿⠖⣶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⣠⣖⠿⣿⣿⣿⣿⣎⢻⣧⣿⠟⣶⣤⣴⣆⣰⠶⠶⡶⣖⣴⡶⡶⠶⢆⣰⣦⣤⣶⠻⣟⣼⡟⣱⣿⣿⣿⣿⠟⣲⣄
⠀⠀⠀⠐⣿⢮⣝⡲⣭⣛⣻⣳⣮⣭⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣭⣵⣞⣿⣛⣭⢖⣫⡵⣿⠂
⠀⠀⠀⠀⠀⠙⠻⠽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣏⠉⠉⠉⠉⠉⠉⠉⠉⣹⣿⢛⣿⣿⢿⣿⣿⣻⣿⣿⣿⣯⠿⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠿⣿⣷⡘⣿⣷⡹⣷⡄⠀⠀⠀⠀⠀⠀⢠⣾⢏⣾⣿⢣⣞⣿⠿⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⡮⣻⣷⣾⠶⠶⠿⠿⠿⠿⠶⠶⣷⣿⣟⣵⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠆⠀⠀⢸⠀⠀⠀⡄⠘⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠁⠰⠀⢰⠀⢸⠀⠘⠀⢠⠀⢀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠉⠐⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢤⣾⠟⠛⠓⠒⠒⠒⠖⠒⠒⠒⠛⠛⢿⣷⡄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠂⠉⠉⠁⡠⡤⠀⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⠠⡤⣄⠉⠉⠁⠐⠒⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣤⣀⡀⠈⠀⠀⠠⠾⠋⠀⠀⠀⣹⠀⠀⠀⠙⠵⠀⠀⠀⠀⣀⣠⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣗⡒⠦⢤⡤⢤⣤⣤⣤⡤⣤⡤⠤⣤⠶⢒⣾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢦⣤⣤⣤⣤⣤⣤⣤⠴⢶⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡎⠀⠀⠀⠀⠚⢻⠃⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡜⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⢿⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡑⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠺⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠁⠉⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⣒⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠾⢁⡼⠃
⠀⠀⠀⠀⣀⠼⣿⣄⣀⠀⢀⣀⣠⣴⠾⣛⣥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⡴⠊⠰⡎⢹⣿⠿⠿⣛⡫⠥⠒⣉⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠿⠷⠤⢤⣀⡀
⠀⠀⠛⠦⠤⠄⠒⠂⠚⠻⠷⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣳⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢧⣀⣀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉⢀⣀⣬⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠉⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠶⠶⠤⠤⠶⠶⠖⠒⠛⠋⠉⠁⠀⠀⠀⠈⠉⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡻⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⣳⠄
⠀⠀⠀⠀⢠⣼⣷⡦⠀⠈⠙⠲⢭⣟⡶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⣞⣯⠵⠚⠁
⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠈⠉⠛⠶⣯⣭⣛⣓⡶⠶⠤⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⡤⠴⢶⣒⣛⣯⣭⡷⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⢭⣙⡛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⣋⣩⡭⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠲⠶⠶⠶⠶⠒⠚⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⢆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⢆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠈⢆⡀⣈⣷⠤⠶⠶⢶⡶⠶⢲⡶
⠀⣀⣠⣤⠤⠶⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠏⠀⢀⡾⠁
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢠⣞
⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢠⠏⠉⠳⣄⡀
⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⣏⠀⠀⠀⠈⠻⣦⡀⢀⡤⣶⣤⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⢧⡀⠀⠀⠀⠈⠙⢾⣧⠆⠀⠉⠁⣰⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠳⡄⠀⠀⠀⢠⡞⠁⢀⠀⠀⠀⢈⡻⣤⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⠞⠁⢉⡹⠟⠲⠦⣤⣀⣀⠙⣄⠀⠀⠙⣄⠀⠀⠈⠉⠳⡆⠷⢠⡿⠷⣥⠾⠃
⠀⠀⠀⠀⠀⣾⠀⢀⡴⠋⠀⢀⣴⠋⠀⠀⠀⠀⠘⢧⠉⠉⠛⠓⠚⠂⠙⢦⠀⠀⠀⠀⠻⠞⠛⠳⣄
⠀⠀⠀⠀⠀⣧⣴⣩⣤⡤⠴⠋⠙⢷⡀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠁⠄⠀⠹⣄⠀⠀⠀⣀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠻⣄⣾⠹⠉⠳⢦⣀⣠⠴⠊⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠈⢦⣴⢛⠋⠡⡀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⡆⠀⠀⠈⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⣦⡔⠻⠓⠲⠾⡧⠂⠀⠀⠱⠤⠠⠤⠼⢶⡄⠀⠀⠀⠀⠀⢸⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠶⠦⣤⣀⡤⠖⠛⡇⠀⠀⠀⠀⠈⢷⡀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠲⡀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠙
⠀⠀⠀⠀⠀⢿⡀⠆⠀⠉⠀⠀⠠⣇⡀⠀⠀⠀⠀⠈⢳⡀⠰⡀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠙⣄⠀⣾⣉⠀⠀⠀⠀⢆⠀⠀⢀⣿⠲⠶⠶⢤⡶
⠀⠀⠀⠀⠀⢘⡷⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⣷⠀⠅⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠈⢦⠀⠉⠉⠛⠳⣆⠈⢆⠀⣼⠇
⠀⠀⠀⠀⣴⡮⠤⠄⠠⠄⠀⢀⣶⣠⠤⠟⠀⠀⠀⠀⢠⡇⠘⠀⠀⠀⣠⡔⠒⠤⢀⠀⠈⢳⡀⠀⠀⠀⢣⡀⠀⠀⠀⠘⢧⡈⣵⡟
⠀⠀⠀⠀⠀⠀⠉⠻⣦⠘⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣇⣀⣴⠊⠁⠈⠙⠒⢤⣌⣁⡲⠷⠀⠀⠀⠀⠱⡀⠀⠀⠀⠈⠉⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠀⠹⣆⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⡞⡛⠑⠦⣄⡀⠀⣠⠔⢻⡀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢸⡇⠁⠀⠀⠀⠉⠛⠁⠀⡞⠱⡀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢘⡷⠊⠀⠀⠀⠀⠀⠀⠀⣇⡀⠱⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⡝⢷⣄⡠⢶⡶⠀⠀⠀⠀⠈⢧⢀⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠳⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡞⡋⠀⠀⠠⠀⣾⠁⠀⠀⠀⠀⠀⠘⣟⠲⢤⡉⠑⠀⠀⠀⢠⣖⣐⣒⣒⣰⡾⠃⠹⡄⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡎⠇⠀⣀⢀⡙⣦⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⡆⢷⠀⣰⠋⢩⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠶⠋⠛⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⢷⠀⠈⣷⡤⠟⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⣀⡷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣤⡴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣤⡶⠚⠛⠉⠉
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣡⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⠀⠀⠀⢈⣗⠁⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠺⡋⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⢿⣿⣿⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣳⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣦⣤⣀⣄⣀⣄⣠⣀⣄⣤⣴⣴⣾⣿⣿⣿⣿⣿⡟⠯⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣯⣿⠛⠟⠻⠟⠿⠻⠟⣿⣿⣿⣿⣿⣿⣿⣯⣟⠂
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠁⠀⠛⣻⣿⣿⣷⣶⣶⣶⣶⣾⣿⢿⠛⠉⠛⠫⢻⣿⣿⡆
⠀⠀⠀⠈⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠻⡿⠦⢄⣀⣀⣀⣀⣽⣿⡿⠃⠀⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠒⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠒⢀⣱⣿⣦⣤⣤⣤⣀⣀⣠⣤⣶⣦⣤⣤⣤⣶⣆⣠⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⣀⣀⣤⡾⠿⠛⠋⠉⢉⠹⣿⠛⠛⠋⠒⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣶⣶⣾⠟⠛⠋⠀⠀⠀⣀⠊⠡⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⣀⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢯⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠿⡗⣄⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠔⢠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠽⠋⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠕⢋⡤⢾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⡀⠀⢀⣤⠖⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡈⠟⢃⡡⢔⣫⣡⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⢿⠀⣀⠠⢀⣀⣥⡶⢊⡡⠜⠛⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣷⠀⠀⢠⣶⣶⠀⠀⠀⡠⠿⢒⣫⣥⣴⣾⣯⠥⢶⣋⣁⠤⠒⡋⠙⠋
⠀⠀⠀⠀⠀⠀⠐⢭⣿⣗⣀⣀⡛⣈⣉⣀⣡⣴⣶⣿⡿⠿⠿⠿⠛⠛⠛⠿⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡄
⢀⣀⣀⣤⣤⣤⣤⡤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⣀⣀⣀⡀
⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠓⡆
⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⣅⣀
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠈⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠲⣄⡀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢸⠃⠀⠀⢀⣠⠴⠚⠉⠀⠈⠉⠳⢦⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢸⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣰⠟⢦⡀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧⣀⡀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⣟⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⠖⠋⠹⣄⠀⠀⠀⣀⡼⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣴⣿⠿⠟⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⣀⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣠⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣽⣛⣛⣿⣿⣿⣿⣿⣿⣿⣛⣯⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣏⣡⣽⣿⣿⣿⣿⡉⢉⣿⣿⣿⣿⣯⣈⣹⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠁