Triomphe Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⢠⣆⣠⣄⡄⠀⣤⣤⡄⠀⣶⢠⡄⣤⣤⣤⣤⣤⡤⣤⣤⡄⢀⣤⣤⠀⢠⠀⣶⠀⡤⣤⣤⣤⡄⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠈⠁⠀⠉⠈⠉⠉⠉⠁⠉⠉⠁⠉⠉⠁⠀⠁⠉⠀⠈⠀⠉⠀⠁⠈⠁⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈

⠀⠀⠀⠀⠀⠀⡀⣀⣀⢀⢀⣀⢀⠀⠀⣀⡀⡀⣀⣀⣀⢀⢀⢀⣀⣀⡀⡀⠀⣀⣀⢀⢀⣀⣀⠀⠀⣀⡀⡀⡀⣀⢀⢀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠘⢻⣿⢻⣿⣟⣿⣟⣻⣿⣛⣿⣛⣿⣿⣻⣿⣻⣿⣟⣻⣟⣿⣿⣛⣿⣛⣿⣿⣻⣿⢻⣿⡟⣻⡟⣿⣿⣛⣿⡛⠁
⠀⠀⠀⠀⠀⠸⠿⠸⡤⠿⢿⡿⢿⣼⠟⢿⠿⠧⡼⠻⠿⠺⠤⠿⢿⠿⠿⡴⠟⠿⠿⢧⠼⠻⡿⠻⣤⠟⠿⠏⠿⣼⠿⠿⠇
⠀⠀⠀⠀⠸⣷⢿⣿⠾⣿⢷⢿⢿⢾⣿⡷⢿⣿⢷⡿⡿⡾⣿⡷⣿⣿⡾⣿⡷⣿⡿⣾⣿⡿⡿⡿⠷⡿⣿⡾⡿⠷⡿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⢠⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣯⣭⣽⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡄
⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡞⣷⣶⣶⣶⢖⣚⣲⡞⣓⡲⣶⣶⣶⣶⢳⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡏
⠀⠀⠀⠀⠀⠀⢸⣿⢩⣭⣭⣭⣭⣭⣭⡍⣿⡇⣿⡿⣫⡾⠟⠛⠉⠉⠛⠻⣷⣝⢻⣿⢸⣿⢩⣭⣭⣭⣭⣭⣭⡍⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⡇⣟⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣹⢺⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠇⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡧⠸⣿⣶⣶⣶⣶⣶⣶⣶⣶⢿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⢻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡝⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⠛⠛⢿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⠛⠛⠛⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢨⠵⡗⠚⢸⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⠗⠁⠚⠧⠄⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠠⠆⣾⣧⢶⡘⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⠠⣶⠴⣷⠰⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡰⠧⠚⢿⠘⢡⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⡀⠛⢸⢿⡻⢢⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠗⠒⠂⠒⠀⠻⢿⣿⠇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⣿⠟⠐⠐⠒⠒⠻⢿⣿⠇
⠀⠀⠀⠀⠀⠀⣽⣿⣧⢸⣿⣿⣿⣿⡇⣿⣿⡏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣯⢸⣿⣿⣿⣿⡇⣼⣿⣏
⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⡇⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⢸⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣧⣼⣿⣿⣿⣿⡆⣿⣿⣷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣿⣿⢰⣿⣿⣿⣿⡇⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠉⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠉⠉⠁⠈⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣭⣬⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣶⣷⣿⣿⣾⣶⣶⣿

⠀⠀⠀⠀⠀⢬⣇⣤⣬⣯⣤⣬⣯⣤⣬⣯⣤⣬⣭⣤⣬⣯⣤⣬⣯⣤⣬⣯⣤⣌⣫⣤⣬⣍⣤⣬⣯⣤⣬⣯⠄
⠀⠀⠀⠀⠀⢸⡅⠀⢈⡇⠀⢨⣿⠀⢈⣇⠀⢈⠍⠀⢀⣏⠀⢈⣇⠀⢀⢉⠀⢀⢍⠀⢸⣯⠀⠀⣯⠀⠀⣿
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣀⠀⠀⡀⠈⢈⠀⠀⣀⠀⠀⡀⠀⠀⡀⠀⢈⡈⠀⢈⠈⠀⡀⠀⢀⡀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠻⠟⠻⠻⠻⠻⠿⠟⠟⠟⠿⠿⠻⢻⠻⠻⠿⠟⠟⠟⠿⠟⠻⢳⠻⠛⠿⠟⠟⠟⠿⠿⠻⢻⠻⠛⠿⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⢤⣤⣤⣤⣤⡤⣤⡤⣤⣤⣤⣤⣤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⡇⢰⢲⠶⠲⠖⢲⠶⣶⡆⢸⡻⣣⠝⡉⠐⠂⠛⠃⠐⠈⠙⢥⡻⣻⠀⣶⠶⢶⠒⠶⠶⢶⢲⠀⡇
⠀⠀⠀⠀⠀⡇⢸⢉⣘⡐⠒⢋⢡⠈⡇⢸⡸⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⢂⠹⣸⠀⡏⣀⢉⠓⠒⣘⣈⢻⠀⡇
⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠉⠉⠉⠉⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡏⠉⠉⢉⣉⣉⣉⠉⠉⠉⢹⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠉⠉⠉⢉⣉⣉⡉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⡇⠀⠀⣸⣿⢝⣿⣗⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢐⣿⣟⢽⣿⡀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠐⡿⣿⢫⡻⣭⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡽⣫⢻⣿⡗⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠘⣿⣇⣾⢯⣹⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⣩⢿⡆⣿⡟⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⣵⣶⣭⣴⡿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣷⣤⣵⣶⡄⠀⠀⡇
⠀⠀⠀⠀⠀⣷⣶⣶⣿⣿⣿⣿⣿⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠛⠒⠛⠓⠛⠛⠛⠛⠓⠒⠛⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠓⠛⠛⠓⠒⠒⠒⠛⠒⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⢀⠀⣄⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠈⠃⠙⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣴⣶⣶⣴⣶⣴⣤⣴⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠈⠃⠈⠙⠈⠀⠑⠁⠐⠂⠁⠀⠂⠃⠙⠈⠋⠐⠝⠉⠈⠃⠑⠈⠈
⠀⠀⠀⠀⠀⠀⢀⠎⠀⢰⠉⢰⡞⢅⡶⠈⡶⠊⡷⠀⠱⠎⠈⠷⠀⢸⠇⠀⠶⠀⠲⠆⠰⠎⠃⠖⠋⣰⠂⢀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣿⣿⣿⢿⣿⣿⢿⣷⣾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠦⠧⠴⠠⠬⠤⠀⠄⣤⡤⣦⣠⠾⣦⣤⠸⠧⢠⣄⢦⣬⠥⠤⡄⠤⠦⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣠⣤⣀⡀⠈⣿⢏⣹⣿⣿⣿⣽⣿⣿⣿⣏⠛⡿⢀⡀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⠛⣿⢩⡉⢀⠀⣷⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⡿⠈⣌⠟⣙⡽⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠙⠀⢡⣿⣿⡿⠟⠛⠛⠛⠻⢿⣿⣿⣿⠀⠋⠉⠫⠛⠬⠴
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠝⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣃⡄⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⣄⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇⢴⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⡨⢋⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠃⠈⠓⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡅⠶⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣤⡤⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⣿⡰⡀⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡏⠀⠘⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠽⠟⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣠⣏⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣥⠶⣿⠉⣼⡏⠛⢦⣈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⡥⠖⠋⠙⡇⠀⢸⣤⠼⣿⢦⣄⠈⠓⢮⣙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣞⡭⠖⠋⠙⡇⠀⠀⣀⣷⠞⠛⢀⣸⠿⣆⡈⠙⢦⣀⠈⠓⢮⣓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠡⡄⠀⠀⢀⣷⠴⠚⠉⣃⡴⠚⢉⣳⣤⡈⠙⠲⣄⡈⠙⢦⣄⠈⠓⠮⣗⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⣷⠀⠀⠀⣧⡤⠞⠋⣃⡤⠖⣋⣥⠶⠚⠉⠁⠀⡉⠳⢤⡀⠙⠲⣤⡈⠙⠦⣄⣸⡟⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠋⠁⢀⡆⠀⠀⣿⡤⠖⠛⣃⣤⣶⣻⠥⠖⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠳⢤⡀⠉⠳⢤⣈⠙⠳⣤⣠⡄
⠀⠀⠀⠀⢠⣶⣿⠛⡏⠀⠀⡇⠀⣸⣧⠖⠛⢋⣠⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠸⡇⢀⣀⠀⠀⠀⠉⠳⢤⡀⠈⠛⠦⣤⣟⠃
⠀⠀⠀⠀⠈⠀⣿⢠⠇⣀⡴⠗⠋⣉⣥⣶⡿⠟⠋⠁⠀⠀⠀⠀⢠⡀⠀⣀⣤⠴⠖⠋⠉⠀⠀⡇⠀⠈⠓⢦⣄⠀⠀⠀⠉⠳⢤⣴⠋⠉⠁
⠀⠀⠀⢀⣀⣠⡷⠚⢋⣡⣤⣶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⣿⠃
⠀⠀⠀⠋⢻⣤⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢸
⠀⠀⠀⠀⠈⢹⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠋⠛⠲⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡄
⠀⠀⠀⠀⠀⢸⣧⣄⣠⡤⠶⠚⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢹⡄⠀⠃⠀⠀⠀⠀⢀⣠⡤⠶⠚⠹⡟⠓⠶⣤⣀⠀⠀⠀⠀⠀⢻⣸⡇
⠀⠀⠀⠀⠀⢸⡏⡏⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠘⠇⣀⣀⡤⠴⠒⠛⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠙⠲⠦⣄⣀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠶⣼⣅⣀⠀⠀⠀⠀⠀⠀⢻⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⣤⡀⠀⠉⠙⠓⣷⠦
⠀⠀⠀⠀⠀⢸⠃⠁⠀⣀⣠⡤⢶⡏⠀⠉⠙⠓⠶⢤⣄⣀⡈⡇⠀⠀⠰⢤⡤⠞⢳⣄⡀⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠻⡄⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠰⣼⠒⠛⠋⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⠉⠙⡟⠀⠀⠀⣸⠃⠀⠀⢈⡿⠋⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⢹⡀⠀⠀⢸
⠀⠀⠀⠀⠀⢹⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠞⠉⠉⠙⠲⣷⠀⠀⠀⣧⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⢰⡇⠀⠀⠀⠀⣧⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⠀⣼⠛⢲⡖⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣿⠀⠀⢠⠏⠀⠀⠀⣼⠁⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⢹⡀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠈⢳⡀⢻⡄⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⢹⡄⠀⠀⠀⢳⡄⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⣾⠀⣼⠃⢸⠁⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢀⠰⠇⠀⠀⢰⠏⣁⣠⡄⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣇⠀⠀⣷
⠀⠀⠀⠀⠀⣿⠴⠋⠀⣸⠆⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⣸⠓⠓⠛⠛⠋⣿⠉⠈⠀⠀⣿⠀⠀⠈⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠐⡿⠒⠒⡾⣿⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⠀⠀⣇⠀⠀⠀⠀⠀⢻⠀⠀⢸
⠀⠀⠀⠀⢰⡇⠀⠀⡇⠀⠀⠀⢻⠀⠀⠀⠸⡇⠀⠀⠀⠀⢰⡇⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣀⡀⢸⠃⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⢸⡆⠀⢘⣇⣀⣀⣀⡀
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠒⠿⠶⠶⠶⠤⠼⢧⣤⣄⣁⣀⡀⠀⠀⠛⠀⠀⠀⠀⢸⡀⠀⠀⠘⣃⣠⡤⠴⠖⠛⠛⠋⠉⠟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠶⠤⣤⣀⣸⡧⠖⠚⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣭⣽⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠶⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⠶⠿⠛⠛⠉⠉⠀⣀⣀⣤⡼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⠾⠿⠛⠋⠉⣁⢀⣄⢤⡤⣰⣂⣸⣥⣭⣷⣶⢾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣁⣀⣤⠤⣖⣖⣻⣯⣭⣶⣿⣿⣿⣟⣛⠿⠿⠗⠒⠚⠛⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢮⣽⣾⢶⣿⣟⣛⠿⠭⠿⢓⣚⣋⡭⣭⠀⣀⣀⣤⣠⣴⣶⠀⡌⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠁⡀⢰⣞⠿⣻⣿⣭⣝⡳⣻⠀⢸⣽⣿⡽⠿⠻⠀⣻⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣻⢻⡟⠁⢸⢡⣾⣿⣿⣿⣿⣿⣼⠀⠀⠀⢀⣀⣀⣀⣤⣌⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣀⣂⣿⣿⣿⣿⣿⣿⠋⢻⠟⠛⠛⠛⠋⠉⠉⠉⠙⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⢛⠋⠉⠉⠉⠙⢻⣿⣿⣿⡃⠀⠀⠀⠀⢠⣿⢘⣷⠀⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣹⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⢶⣈⠿⣽⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡂⣥⡀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢾⣿⡘⠀⠓⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠿⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠏⠉⣓⢘⡁⠀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⡁⢼⡇⠀⠀⠀⠀⠀⠀⠀⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠂⠸⠃⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⢸⢿⠄⢸⢿⠀⣾⢷⠀⡿⡧⠰⡿⡇⢸⢿⠆⢸⢿⠀⡾⣿⠀⡿⡇⠰⡿⡇⢸⢿
⠀⠀⠀⠀⠀⠀⢶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⡶
⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣬⣭⣭⣭⣭⣭⣭⣭⣭⣥⣭⣭⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠐⠗⠒⠒⠒⠒⠒⠒⠒⠒⢲⠒⠒⢒⣒⠒⠒⠒⠒⣲⡒⠒⠒⡖⠒⠒⠒⠒⠒⠒⠒⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠁⡅⢸⢀⡴⢊⠅⠐⠀⠀⠀⠨⠁⢢⠀⡇⢨⠈⠉⠉⠉⠉⠉⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⡇⢘⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠸
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣉⣈⣉⣉⣉⣉⣉⣁⣸⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣇⣈⣈⣉⣉⣉⣉⣉⣉⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠿⢿⢿⠿⠿⠿⠿⠽⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠏⠿⡿⠿⠿⠿⡿⠿⠟⣿⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⢀⣂⢀⣀⣠⣤⣤⣤⠦⠶⣿⣿⣷⣶⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠟⠻⠟⠛⠛⣻⣿⠙⡃⠀⠈⠀⠀⠀⢟⢻⡿⢿⢿⣯⣟⡂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢁⣀⣂⣀⡀⡤⠤⠤⠄⣿⣿⣳⠧⣤⣃⡘⠋⡗⡶⢤⣄⡢⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⢨⠉⠉⠐⡷⠀⡀⠀⠀⠀⠀⠀⠯⣤⣉⠙⠓⠲⠭⣵⣀⠄⠀⠀⠋⠙⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠉⠑⠒⠤⢀⣈⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⢄⣀⠀⢰⠦⢤⣬⣀⠀⠠⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠱⠾⣿⣷⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒⠒⠀⠀⠀⠀⣈⣁⣀⡀⣀⣀⣙⠒⠒⠀⠀⠀⠈⠁⠀⠀⠀⠍⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠛⣋⣉⣩⣭⣭⡭⠅⠀⠀⠀⠀⠛⠻⠿⠛⠻⣶⠖⠀⠀⠀⠀⢀⡀⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡿⠛⠉⠓⡆⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢱⠿⠟⠛⠉⠿⠿⠷⠶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣞⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢰⡀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠠⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠰⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠛⠒⠒⠂⠆⠀⠀⠀⠀⠀⠀⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠠⠤⢦⠠⠄⢠⠀⣤⣤⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠘⠁⠛⠛⠓⠐⠀⠀⠛⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇⠀⠆⡇⠀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⠁⠀⠳⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣄⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡭⠭⠹⠬⠭⠥⠤⠀⠀⠐⠒⠒⠒⠲⠄⠀⠀⠤⠀⠠⠤⠤⠤⠬⠅⠭⠭⠩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠂⠀⢐⠂⠀⢐⣀⣀⣒⣀⣀⣃⣀⣈⣀⣀⣁⣀⣈⣁⣀⣀⣁⠁⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⣿⣯⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣬⣥⣬⣭⣥⣤⣤⣤⣭⣬⣭⣬⣭⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⣶⡶⠶⣖⣦⣶⡲⣶⢶⡆⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⢇⣾⣿⣿⣿⣿⣿⣿⡇⠀⠈⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⢠⣿⠟⠋⠉⠉⠙⢿⣿⣇⠀⠘⠉⠉⠀⠀⠑⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠒⠒⠒⠈⠛⠛⠛⠻⢻⡟⠀⠀⠀⠀⠀⠀⢿⡟⠛⠛⠛⠛⠛⠛⠛⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢽⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣼⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⢸⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠩⠭⡥⠄⠤⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠤⠤⠤⠾⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣄⠀⠀⣄⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡇⡀⢀⠄⣤⠀⣠⣤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠈⠈⠉⠉


⠀⣁⣈

⠀⠀⠀⡀⠂⠀⠀⠐⠂⢀⠀⡂⢒⠀⠀⡀⠒⠂⠀⠀⠐⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠒⠀⢀⠀⢒⠀⠀⠀⠘⠀⠀⠀⠐⢂
⠀⠀⠘⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠴⠶⠶⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⠶⠂

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠐⠂⠛⠐⠒⠒⠛⠐⠒⠚⠓⠂⠒⠚⠃⠒⠐⠛⠒⠒⠀⠛⠐⠒⠚⠛⠐⠂⠚⠓⠒⠂⠛⠂⠒⠒⠛⠀⠐⠐⠛⠒⠂⠚⠓⠐⠒⠛⠓⠒⠂⠛⠠
⠀⠙⣿⣿⠿⡿⠿⡿⠿⢿⠿⠿⢿⠿⠿⢿⡿⠿⠿⠿⡿⡿⡿⠿⢿⣿⡿⣿⣿⡿⠿⢿⠿⢿⠿⠿⠿⢿⣿⠿⠿⣿⠿⢿⡿⠿⢿⠿⠿⠿⢿⣿⠋
⠀⠀⠈⣥⣄⣴⣌⣲⣠⣓⣆⣠⣤⣄⣠⣠⣀⣀⣼⣒⣂⣀⣠⣂⣀⣈⣃⣊⣀⣐⣰⣃⣀⣀⣴⣦⣀⣀⣀⣐⣤⣤⣀⣲⣊⣄⣀⣐⣄⣠⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⣀⣀⣀⣀⣀⣈⡁⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠇⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⢸⣏⢙⠉⣉⢉⡫⠭⠉⢻⣿⠏⠩⠽⡉⢉⡉⡋⣩⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⡇⣰⡒⠲⢂⡄⢂⡀⠂⠰⠀⠃⢸⠘⢠⡼⠛⣡⣴⣶⣾⣿⣿⣷⣶⣤⣈⠑⢥⠄⢭⠀⠀⠀⠂⠰⠀⡀⠠⡔⢒⠒⣶
⠀⠀⠀⠀⢀⡇⠐⢀⠀⢄⠀⠀⠀⠠⣰⠀⠀⣿⢠⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠳⡈⠀⠀⡆⣄⠤⠀⠐⠈⢠⠀⡌⠦
⠀⠀⠀⠀⠈⠧⠀⠠⠀⠀⠀⠀⠤⠤⠤⠤⠀⢿⡿⣸⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣷⠀⢡⠀⠀⠦⠤⠤⠤⠤⠤⠀⠠⠀⠠⠠
⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⡇⢿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠇⠨⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄
⠀⠀⠙⡟⠛⠛⠛⠛⠛⠛⠟⠛⠻⠻⠟⠿⠿⣿⠻⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⢛⡋⠿⠿⠿⠿⠟⠟⠛⠻⠛⠛⠛⠛⠛⢻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠆⢁⣼⣿⠀⠀⠀⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⣻⣷⡈⠰⣿⡟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢶⣯⠏⡁⢊⡭⠛⠀⠀⠀⣿⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠛⣭⡽⠂⠹⣽⣶
⠀⠀⠀⠀⠀⠀⢀⡹⣔⣤⣷⠈⢱⠂⠀⠀⠀⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢐⡎⠃⣶⡤⣦⠟⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣟⠀⠇⠰⢃⣀⡀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⣀⡀⠇⠘⠁⣻⣽⣧
⠀⠀⠀⠀⠀⠀⢸⠇⣟⣤⠻⣬⠋⠌⠁⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠩⠙⣥⡞⣠⣹⠹⡏
⠀⠀⠀⠀⠀⠀⢘⠇⠃⠿⠘⠛⢷⣆⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢐⠟⠃⠻⠍⠙⡇
⠀⠀⠆⠀⠒⠒⡟⠚⠛⠛⠓⠛⢺⣿⡶⠒⠒⣿⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠒⢿⣶⠒⠀⢠
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⢿⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣘⣛⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⢀⣋⣀⣈⣋⣀⣈⣋⣀⣈⣛⣀⣀⣛⣀⣀⣛⣀⣀⣛⣀⣀⣙⣀⣀⣛⣁⣀⣙⣁⣀⣙⣁⣀⣙⡀
⠀⠀⠀⠀⠀⠈⠅⠀⠀⡽⠀⠀⡯⠀⠀⡍⠀⠀⢭⠀⠀⣹⠀⠀⣭⠀⠀⣭⠀⠀⠉⠀⠀⢽⠀⠀⣯⠀⠀⠈⠁
⠀⠀⠀⠀⠀⡀⠈⢀⠈⠈⠀⠈⠈⡀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠁⠀⠀⠀⢀⠀⠁⡀⠁⠁⡀⠁⢀
⠀⠀⠀⠀⠻⠟⠿⠿⢻⠻⢻⠻⠟⡟⡟⠟⠿⠻⠻⢻⠻⠻⠿⠿⠟⠟⡟⠟⠟⠿⠻⢻⠻⠻⠿⡟⠟⠟⠟⠿⠻⠛
⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⣤⣤⢤⣤⣤⣤⣤⣤⣤⡤⣤⣤⡤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⡇⢰⢲⡶⡖⠖⢒⠶⠶⡆⢨⡹⢃⡽⠋⠅⠂⠙⠋⠐⠈⠙⢯⡘⢏⡅⢰⡖⠶⠒⠲⢲⢶⡖⡆⢸
⠀⠀⠀⠀⠀⡇⢸⣊⣘⡱⠖⢊⢠⡈⡇⠨⢰⢋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⡇⢸⢁⡌⡑⠲⢎⣃⣁⡇⢸
⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠉⠉⠉⠉⠁⢨⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠈⠉⠉⠉⠉⠉⠉⠉⠁⢸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡏⠉⠉⠉⣉⣉⣉⠉⠉⠉⢹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡏⠉⠉⠉⣉⣉⣉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⡇⠀⠀⣸⣯⢝⣿⣷⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣾⣿⡫⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⢫⡿⣯⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣭⢿⡝⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠐⣿⣇⣾⢿⣹⡄⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢠⣏⡿⣷⣼⣿⠂⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⣴⣶⣭⣤⣿⠂⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⣿⣄⣭⣶⣦⠀⠀⢸
⠀⠀⠀⠀⠀⢷⣶⣶⣿⣿⣿⣿⣿⣶⣶⣾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⣶⣶⣿⣿⣿⣿⣿⣶⣶⣾
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣏⣥⠛⣿⣝⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⡿⡿⠏⢁⣀⣤⣿⣿⣿⣯⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⡿⠟⠛⣃⣠⣴⣶⣿⣿⣭⣿⣶⣮⣿⣷⣿⣿⢿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⡿⠟⢋⣀⣤⣶⠿⠛⣫⣵⣾⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⠟⢛⣡⣴⣾⠿⢛⣩⣴⣾⣿⣿⣿⣿⣿⣛⣿⣯⡿⠿⣿⣿⣯⣿⣿⡻⢿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⢟⣋⣬⣶⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⣾⣿⢾⣿⣿⣿⣿⣿⣶⡍⠳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢟⣋⣽⣶⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣭⠿⠟⢛⣩⡤⠶⣒⡛⡇⣿⣿⣲⢝⣿⣿⡿⣿⣿⣷⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣿⣿⣿⣾⣿⣿⣿⣿⡿⢿⣛⣽⣿⠟⣿⣽⠇⣾⣯⣷⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⣮⣙⠻⢿⣿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣶⣍⠫⣿⡘⣿⣿⣿⣿⣿⡿⠿⠋⠃⣿⠻⣿⣿⣿⣿⣿⣶⣤⢢⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢛⠙⣿⢿⣿⣿⣿⣿⣿⣽⣿⣷⣼⡇⣛⣿⠋⠅⠀⠀⣀⣀⣤⣿⣤⣄⣈⠙⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣴⣶⣾⣿⠀⣿⣿⣿⣿⣿⣿⢟⣿⡿⠙⢿⣅⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣙⠓⠼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣀⣿⡿⣿⣿⣿⣏⣼⣿⡇⢶⣿⣿⣿⣿⣿⠿⡿⠛⠛⠉⠁⠀⣿⠀⠉⢛⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠟⠋⠉⢀⣸⣿⣥⣛⣻⢿⢿⣿⣿⡇⠀⣿⢫⠁⡀⢀⣶⣷⣦⣤⡀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣽⠻⠿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣴⣶⣶⣾⣿⣿⣿⡯⠉⢉⡿⠻⠿⠿⣧⣄⣿⢼⡤⠝⣿⣿⣿⣿⣿⡅⠀⠀⣿⠀⠀⢸⣿⣿⡻⣿⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⣋⣉⡉⠀⢸⣯⡇⠺⠂⣿⣿⣿⢶⣶⣍⣿⣿⠀⣴⣿⣿⣿⡏⣯⠀⠀⠀⣿⡀⠀⠸⣿⣿⣇⣹⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢲⣿⣿⠇⠀⢸⡏⡇⢀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘⣿⣿⣿⣿⣿⠀⠀⢰⣿⡇⠀⠀⡛⡟⡯⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡗⠀⢸⡟⡇⠻⠂⣟⢻⣿⣿⣿⣿⣿⡇⠐⠼⣿⣿⣿⡻⢿⡇⠀⢸⠸⣇⠀⠀⣿⣿⣇⠸⣿⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⡧⠀⢸⣷⡇⢀⠀⣿⣤⣽⣾⣿⡇⢹⡇⠀⠀⢿⣿⣿⣻⢦⠃⠀⢸⠀⣿⠀⠀⢹⠟⠻⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣧⡗⠀⢸⣧⡇⣾⠀⣿⣿⣿⣿⣿⡇⠀⣃⣠⣤⣿⣿⣿⣿⣿⣤⣤⣼⣤⣿⣤⠤⢰⠀⢀⣀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣧⠀⢠⣸⡇⠈⠀⢻⣹⢻⣿⣿⡇⢸⡿⠻⡏⠉⠀⠀⠀⢸⣿⡀⠀⠀⢹⡇⠀⣸⡆⣸⠏⠉⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⠀⢸⡷⠀⢸⣿⠇⠀⠀⢸⣯⣿⣯⣿⠀⢸⣷⣀⣿⣀⣀⣀⣀⣼⣯⣇⣀⣀⣨⣤⣀⣿⡏⢹⣀⣀⣀⣀⣈⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⢀⡀⢀⣾⣷⣶⣦⣶⣤⣤⣾⣿⣶⣶⣶⣀⣅⣉⣉⣉⣁⣿⣯⣭⣭⣭⣭⣍⣉⣩⣭⣿⣤⣦⣭⠠⢤⣥⣤⣼⠭⢭⣿⣯⠴⠾
⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⣿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠯⠯⠿⠿⠷⠾⠾⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠟⠻⠻⠿⠛⠛⠖⠚⠓⠀⠒⠶⠶⠟⠻⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣁⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⣡⠔⠊⠈⠑⢦⡉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢋⡤⠖⠋⠀⠀⠀⡀⠀⠀⠉⠲⢄⡙⢦⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠚⣉⠴⣾⡁⠀⠀⢀⠤⠒⠊⠀⠰⠄⢄⠀⠀⠁⠢⣌⠓⢄⡀
⠀⠀⠀⢀⠠⢋⡡⠔⠋⠁⠀⢷⡀⡔⠊⠁⠀⠀⠀⢀⠀⠀⠀⠐⠠⠄⠀⠈⠑⠦⡉⠢⣄
⠀⠀⠀⠀⠀⠉⠀⢀⣀⠤⠚⠁⠀⠀⠀⡠⠔⠓⡄⣦⣈⠒⠄⡀⠀⠀⠁⠰⡀⠀⠈⠓⢤⡑⠦⡀
⠀⠀⠀⠀⢀⠴⠒⠁⠀⠀⢀⠠⠔⠀⣁⠤⣚⣫⠷⠓⠿⠛⠦⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠒⢌⠓⢄⡀
⠀⠀⠀⠈⠀⠀⢀⠠⠀⠂⡈⣐⢲⠎⠐⠚⠉⠀⠀⠀⠀⠈⠐⠠⣉⡢⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠑⠢⣍⠲⣀
⠀⠀⠡⢀⠔⠀⣁⠐⡘⠁⠘⠋⠈⠀⠀⠀⢀⢠⠀⠀⢀⡀⠀⠀⠀⠈⠓⠠⣀⡄⢀⠀⠀⠀⠀⠐⠀⠀⠀⠑⢮⡑⠆⡀
⠀⠀⠀⠀⢨⠤⠀⠁⠀⠀⠀⢀⡀⠄⠂⠉⢀⡀⠀⠀⠀⠀⠁⠂⠀⡀⠀⠀⠀⠐⠢⢁⠄⡀⠀⠀⠀⠈⠀⠄⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠠⠶⡖⠛⠃⠀⠀⠀⢸⣟⡲⠤⡀⠀⠀⠀⠄⠀⠀⠉⠉⠢⢀⢀⠀⠀⡀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠠⢀⠀⡴⢒⡚⢻⣁⣠⡴⠶⣓⡢⠀⠀⠀⣙⠸⠧⣤⠆⠁⠀⠀⢆⡴⠖⠂⠀⠀⠀⠑⠪⠄⠀⡐
⠀⠀⠀⠀⠀⠀⠠⢇⠀⠴⡚⡋⠭⠐⠊⢉⣀⠤⠔⠂⠄⣀⠉⠒⠠⢌⠀⠀⠀⠏⠀⠀⠀⠉⠀⠀⠄⠀⠀⠉⠉⠦⢠⠐
⠀⠀⠀⠀⠀⠠⠬⠒⢊⣁⣤⡤⠶⠓⠊⠉⠀⣀⠀⣄⡀⠀⠉⠐⠢⢄⣀⡁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢡⡴⠶⠛⠋⠉⢀⡀⠤⠀⠂⠀⠈⠁⠀⠀⠀⠁⠀⠂⣄⠀⠀⠉⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡶⢠⡄
⠀⠀⠀⠀⡄⠠⠄⠀⠀⠁⠀⠀⠐⠢⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⣧⡄⠀⠀⠁⠀⠀⢀⠀⠀⢀⡐⠀⠀⠀⠬⣃
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢻⢾⠛⠐⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⡁⠘⠲⠤⣌⡀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⡃⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⣧⢈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠁⠘⡅⠀⠀⠀⠀⠀⠀⠐⡾⠁⢮⠇⠀⠀⠀⢀⣄⣀⢠⠀⠀⠀⠀⠀⠀⠀⠧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⣣⣴⠛⠀⣄⠀⠀⣼⡓⠻⣤⠀⠀⠀⠀⠀⠀⢰⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣟⠋⡅⠁⠉⠀⠀⡇⠈⠁⡄⠀⠀⠀⠀⠀⠀⠸⠀⡭⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⡛⠁⠐⠂⠀⠀⠀⡿⠀⠂⠀⠀⠀⠀⠀⢠⠀⢀⠾
⣁⡠⣀⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠶⠛⠀⠀⠀⣸⡀⠀⠀⣀⢀⣀⡤
⠟⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠈⠀⠀⠺⠀⠀⢠⠤⠄
⢸⣿⣿⣵⣀⠀⢠⣤⠀⠀⠀⠀⠀⣼⢀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀⢑⣠⡛⠓⠛⢣⠀⣘⠀⠛
⣯⠭⠽⠿⠽⠴⠦⠿⠿⠯⠿⠾⠧⠹⠛⠋⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣸⡟⢛⠻⠻⢿⡷⣶⢶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠧⠤⣤⣄⣀⣀⡀⠀⠉⠉⠙⡛⠻⠿⢷⠶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣦⣭⣭⣑⣛⣃⠐⠶⠶⣶⣤⣤⣄⣀⡈⠨⠙⠙⠛⠛⡟⠳⠶
⠀⠀⠀⠀⠀⣸⣿⣿⡛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣭⣭⣟⣛⣛⠿⠶⠶⣤⣤⣥⣁⡀
⠀⠀⠀⠀⢠⣿⣿⣿⠀⣤⣤⣅⣀⣀⡚⠉⢩⣽⣛⣛⣻⠿⠿⢿⣿⣿⣿⣷⣿⣶⣦⣴⣦⠽
⠀⠀⠀⠀⠀⢹⣿⡿⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⠿⣛⣉⣷⣙⠷⣶⣶⡖⢀⠉⠉⠍⠛⠻
⠀⠀⠀⠀⠀⢸⣿⡇⠘⠛⠛⠻⠿⠿⠿⠇⠘⢁⣾⣿⣿⣿⣿⣿⣮⠿⡇⢸⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⢸⣿⡿⢿⣷⣶⣶⣶⣶⣤⣤⣤⡼⠿⣿⣿⣿⣿⣿⣿⣇⣤⠘⠛⠿⠿⠿⠇
⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⢀⣤⣈⡉⠉⠉⠉⠁⠀⠈⢿⣿⣿⣿⣿⣿⢿⠶⢶⣶⣶⣶⣤⡤
⠀⠀⠀⠀⠀⢹⣿⡇⠀⢀⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⢠⠀⢠⣤⣦⣄
⠀⠀⠀⠀⠀⣿⣿⠇⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⠀⠈⣿⢿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢨⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣀⣀⣘⣷⣿⣻⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠰⠀⠀⣿⠿⣿⡇
⠀⠀⠀⠀⠐⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⡁⠀⣿⠀⠀⠈⠁
⠀⠀⠀⠀⢠⣿⡟⠀⠀⣸⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⠦⢇⣀⣾⢀⣐⣒⣆⡆
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉
⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣴⢦⣄⣀⠈⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⢾⡇⢸⣏⠉⡓⠶⢤⣀⠈⠉⠒⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⠹⠿⠷⣯⣁⡉⣿⡍⠙⠲⠦⣄⡀⠉⠛⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣼⣿⠗⠠⠄⢈⠉⠛⠻⢿⣍⣁⣻⢊⣉⢻⣶⣤⣀⠈⠙⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣴⠿⠋⣠⣤⠄⣀⠀⠈⠐⠢⠄⢀⡉⠛⠿⣤⣿⣇⠤⢌⣿⠒⢤⣄⡈⠑⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⣋⣿⠿⢁⣤⣾⣵⣟⣶⣧⣔⣁⡄⢀⠀⠀⠀⠐⠂⠀⠈⠛⠓⢾⣿⣖⣲⣿⣉⢛⣶⢤⣀⠈⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣻⡾⠛⣠⣴⣻⠿⣭⠟⣭⡍⢙⠛⠻⠿⣓⣠⠤⢆⡀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠛⠷⣼⣧⣆⣽⡟⠒⣦⣄⡉⠛⠂⠄⡀
⠀⠀⠀⠀⠀⠀⢠⣶⡿⢋⣤⣾⣿⣿⢟⣯⣶⡽⠟⠛⠂⠀⠄⡀⠀⠉⠘⠛⠒⠈⣖⣮⣤⠒⣠⠀⡀⠀⠁⠐⠀⠈⠙⠻⠿⣧⣿⣤⣹⡟⢲⣤⣈⠙⠂
⠀⠀⠀⠀⠀⠀⠻⣋⣴⣿⣿⣻⣯⠞⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠋⠻⣶⠇⣂⣔⡠⠀⠀⠀⠀⠈⠉⠙⠻⠿⣾⣿⣬⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣟⡋⠈⠁⠀⠀⢀⡁⠀⠀⠐⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡈⠭⢙⣛⠶⣦⣬⣤⣀⡠⢀⠀⠀⠰⠯⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⠀⠀⠀⣀⣨⡄⡇⠀⠀⡄⢠⣿⣖⣦⣴⣤⣠⡄⠀⠀⠀⣿⣹⣿⣦⣤⣤⣀⢀⠀⠀⠑⠪⡰⡎⠱⣳⣾⣰⣰⢆⡄⠉⡛⠿
⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠀⣄⣤⣿⣿⠿⣧⡇⢰⠀⠀⢬⣭⣛⣛⠻⠿⣾⣿⠁⠀⠀⣿⣻⡿⢋⣁⣤⠀⠘⠚⠙⠷⣤⣁⠀⠔⠄⣤⣙⡧⣾⣿⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⣿⡿⣿⡿⠗⠉⣡⠴⠿⠶⠶⣦⣤⣌⣉⡙⠛⠲⠶⠦⠀⠀⣿⠋⣠⣮⣄⡉⠛⠿⣦⣄⠀⠈⢻⣿⠇⠀⡀⠀⠉⠙⠓⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢆⣿⠟⠋⣁⡤⠖⠉⢀⢰⡆⠀⠀⠀⠀⠉⠉⠛⠛⠳⠶⢦⣤⣤⡏⣸⢷⣿⣿⣿⣷⣄⠘⢿⣦⡀⠀⢿⡆⠀⢙⣁⢒⣢⣤
⠀⠀⠀⠀⠀⠀⠀⠈⣀⡠⠖⢋⣡⣤⠄⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⠤⡬⡄⢹⣧⣿⣿⣮⣿⣿⡄⠘⣿⣿⡀⠸⡇⠀⣿⣿⣿⣿⣯⠰
⠀⠀⠀⠀⠀⠀⠀⠘⣡⣤⡶⢟⣫⣤⡄⠀⠀⢲⠀⠀⠀⠀⢠⣤⣀⣤⣴⡆⠀⡀⠀⠈⠀⣗⠹⣿⣿⢾⣿⣿⣇⣘⣩⣭⣧⣄⣧⣤⣄⣉⡉⠛⠛⠐
⠀⠀⠀⠀⠀⠀⠀⡿⠟⠉⣷⠟⢈⣾⡇⠀⠀⠉⠀⠀⠀⠀⢸⣿⣿⠿⠁⣾⣷⠆⠀⠀⠀⣿⣥⣿⣷⣿⣿⣿⣿⠟⢋⣉⣭⡼⣿⠢⠬⠭⣉⣛⡛⠷⡆
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠟⣰⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠞⠂⠇⣽⡃⠀⠀⠀⣿⣽⣯⣩⣥⢴⣶⣾⣿⡟⢛⣽⠀⡇⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣶⣿⣿⣿⡃⠀⠀⡀⠀⠀⠀⠀⢰⣸⠦⢁⣤⡤⠛⠃⠀⠀⠀⣏⢸⣿⡟⣵⣿⣿⣿⣍⠿⢲⡶⠀⡧⠀⠀⠀⢲⡿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⠂⠀⠀⡅⠀⠀⠀⠐⣿⢟⠀⣴⠞⢶⣄⡰⠀⠀⠀⣯⣸⣿⢡⣿⣿⣿⣿⣤⠞⠛⠛⠀⡇⠀⠀⠀⣾⣶⣀⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⣿⣿⣿⠀⠀⠀⡇⠀⠀⠀⣠⣿⣽⠟⢃⣿⣼⢛⡁⠀⠀⠀⣿⣼⣿⠛⣿⣿⣿⣿⠁⠀⣀⣦⠀⣇⠀⢠⣍⡙⢏⣧⠚⠓
⠀⠀⠀⠀⠀⠀⠘⠀⠀⣼⢽⣿⣿⣿⠀⠀⠀⠇⠀⠀⠀⢠⣿⣃⣠⠚⣿⣉⣦⠀⠀⠀⠀⣿⢻⣿⣠⣿⣿⣿⣿⣯⣠⡭⢦⠀⣟⠀⠀⡭⣥⣟⢛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⢀⣀⣠⣄⣀⣀⣀⢠⣿⣟⠛⠛⠛⠿⠿⠄⠀⠀⠀⣿⣾⡷⠀⣿⣿⣿⣿⠀⠷⠤⣋⠀⡟⠀⠠⣿⣿⢫⣾⣿⣄
⠀⠀⠀⠀⠀⠀⣴⣶⠶⡿⢸⣿⣿⣿⠛⣯⢽⡅⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠀⣿⣿⣿⣀⣿⣿⣿⣿⣷⡤⣀⠁⠀⣷⠤⠤⣿⡟⠻⠿⢿⣯
⠀⠀⠀⠀⠀⠀⣼⡇⠀⡇⣾⣿⣿⡏⠀⠈⣷⡏⠁⠀⠀⢀⣧⡇⠀⠀⠀⠀⠀⠐⠂⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣟⠀⠀⢾⡅⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⣿⣿⣮⣇⣿⣿⣿⣧⣤⣴⣿⣿⣶⣤⣤⣼⣧⣧⣤⣤⣤⣀⡄⣬⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠀⣿⡄⠀⠀⢀⣨
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠿⠛⠛⠟⠿⠿⠿⠛⠟⠿⠿⠿⠿⠟⠓⠒⠚⠲⠶⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠃⠀⠀⠀⠋⠁
⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠗⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠐⠊⠁⠀⣠⡈⢉⣲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠊⠁⣠⡤⠖⢋⣩⣥⡉⠲⢝⡠⡍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢙⣠⡴⠞⣋⣥⣴⡾⠟⠋⠽⠛⢷⣦⣉⠛⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣩⡤⢞⣫⣥⣶⢿⣛⠉⠀⠀⣀⣐⠀⢒⡦⣍⡛⠿⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣷⣿⣿⣿⣽⣺⣿⣿⢸⣾⣧⠷⠸⠀⠘⠛⠶⣽⣷⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣭⣴⢸⡞⠁⣠⠚⠛⠈⠉⠀⣀⣀⣤⣆⣀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠋⣸⠁⠰⠣⡄⡴⠒⠊⡩⡉⡀⠀⠀⠈⢟⡳⢶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠛⠀⢠⣴⢤⡄⠀⠛⢷⣠⠄⠀⠀⠀⣿⣿⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠀⢦⢀⠀⣿⣿⣇⠀⡸⢻⣿⣁⠀⠀⠀⢙⣉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢽⠀⡏⠀⢸⣯⣭⠉⠀⢱⡞⢧⠁⠀⠀⠀⠘⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠽⠿⠤⡇⠘⠈⠉⢛⠀⠒⠋⠉⠉⠉⠀⠉⠀⠀⢸⠀⡗⠒
⠀⠀⠀⠀⢀⢀⢀⠀⡀⡀⢀⡀⠀⠀⡀⡆⠈⢀⣀⢉⡠⢀⠀⠀⠠⠄⠀⣀⠀⡀⢀⡀⣠⠤⡄⢀⢀⡀⡀⣀⣀
⣤⣤⣤⣤⣬⣤⣬⣤⣥⣥⣬⣤⣭⣬⣥⣤⣥⣬⣤⣬⣤⣬⣤⣭⣤⣬⣤⣭⣤⣥⣤⣥⣬⣤⣥⣬⣤⣤⣥⣥⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣴⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⠙⢊⣠⠂⠀⡀⠀⠀⠀⠀⠄⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣤⡶⠉⠑⠀⠀⠀⠑⠂⢤⣄⠀⠀⣀⠀⠈⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⢠⡶⠆⠀⠀⡀⡀⢀⠀⣀⡉⠑⠦⢄⣀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠫⠆⠂⠀⠀⠀⠁⠑⠀⠀⠊⣩⣵⠲⠤⡈⠘⠒⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠐⠂⠠⠤⣀⣼⣿⣿⣷⡄⡇⠰⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣴⣾⡏⠀⠀⠀⠀⡤⠂⠀⠈⠿⠯⠷⣛⡃⢄⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⡏⠀⠀⠀⡘⠠⠐⠀⠀⠀⣄⣴⠀⠀⠀⠀⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⡄⠀⠀⠀⠄⠀⠐⠀⠀⠐⠟⠻⠀⠀⠀⠠⠉⡀
⠀⠀⠀⠀⠀⠀⠤⣤⡄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠐⠲⠂⠀⠀⠘⠱⠆⠀⠀⠀⠀⠀⠀⣸⣀⡀⠀⢀⠀⡀⠠⠀⣀
⢀⣲⡤⣄⠀⠀⠈⠘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣷⣩⠈⠉⠙⠒⠉⠑⢷⠀⠀⢂⡀⠀⠀⠈⠡
⠺⠿⠿⠿⠶⠞⠿⠻⠖⠷⠞⠷⠘⠀⠀⠀⠂⠘⠓⠘⠂⠃⠐⠒⠂⠂⠆⠶⠀⠀⠹⠻⠳⠟⠓⠟⠇⠩⢿⢶⡾⡤⠍⢹⡍⠦⡵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣪⠥⡯⣑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠊⢩⡠⠖⣗⠢⣑⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠅⣀⠼⢒⠩⣒⠹⠦⣑⠢⣑⠢⣙⠳⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⡋⢡⢀⡤⢐⣫⡴⠞⠛⠉⠀⠀⢰⠀⠑⠢⣁⠢⢝⠢⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣄⠤⢒⣩⠴⠚⠉⠁⠀⣀⡠⠤⠒⠈⢸⠀⠒⠄⡀⠑⠢⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡭⠶⠚⠉⠀⠀⢀⣀⡀⠀⢹⠀⠀⠀⠀⠨⡀⠀⠀⠈⠑⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠠⠄⠀⡔⠉⠀⠈⡆⠘⠀⠀⣀⠠⠜⡗⠤⣀⠀⠀⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠠⣼⡀⠀⠀⠀⡇⠒⠉⠁⠀⠀⠀⡇⠀⠠⣌⠑⠂⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠒⠊⠃⠀⠉⢐⡒⢦⠀⢐⠊⠑⡤⠀⡇⠀⡐⠀⢱⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡤⣄⢀⠀⠀⡐⠁⠈⢻⠀⡸⠀⢀⠇⠀⡇⠀⡇⠀⠀⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⡁⠘⠀⠀⡇⠀⠀⠘⠀⠡⠀⣨⢆⡀⠇⠀⡇⠀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⣁⡇⢠⠀⠀⡇⠀⠀⠀⢸⠃⠀⢹⠉⠀⠀⠀⠃⠀⠀⢸⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠈⠀⠀⠇⠀⠀⡇⢸⠀⠀⢸⠀⠀⢰⠀⢠⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠐⠚⠓⠒⠓⠂⠠⠧⠬⠤⢀⣘⡀⠀⢸⣀⡈⠤⠄⠚⠂⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⣀⠀⠀⣀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⢺⡧⡤⣽⡧⡦⣿⡥⡤⣿⡷⡤⣿⡥⡤⣿⡵⢬⣿⠴⢬⣿⢴⢬⣿⢴⢼⣯⢴⢬⣯⢴⢼⡗
⠀⠀⠀⠀⠀⠸⠀⠀⡰⡇⠀⡸⡂⠀⡸⡀⠀⡾⡂⠀⠾⠀⠀⠷⠀⢐⢷⠀⢀⢆⠀⢐⢇⠀⢸⢇⠀⠀⠇
⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣄⣀⣠⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣠⣀⣀⣄⣀⣠⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠙⣹⣽⣯⣯⣯⣯⣽⣽⣹⣭⣯⣯⣯⣿⣿⣽⣽⣯⣯⣿⣿⣽⣽⣭⣭⣯⣯⣯⣽⣹⣽⣽⣯⣏⠋
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⡇⣀⣀⣀⣀⣀⣀⣀⣀⠐⢖⣶⢲⣶⠶⠒⣿⣿⠒⠶⣶⡖⣲⡶⠂⣀⣀⣀⣀⣀⣀⣀⣀⢸
⠀⠀⠀⠀⠀⡇⣿⡝⡹⣉⣉⠭⠉⢿⠀⠚⣡⢋⠐⠈⠀⠀⠀⠀⠁⠂⡙⣌⠳⠀⡿⠉⠭⣍⣉⢏⢫⣿⢸
⠀⠀⠀⠀⠀⡇⠿⠶⠶⠭⠭⠶⠷⠿⠀⣰⢁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⣆⠀⠿⠾⠰⠬⠭⠾⠶⠿⢸
⠀⠀⠀⠀⢠⣧⣤⣤⣤⣤⣤⣤⣤⣤⣴⣇⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣸⣦⣤⣤⣤⣤⣤⣤⣤⣤⣼⡄
⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿
⠀⠀⠀⠀⠀⡇⠀⠀⣼⣿⢶⣶⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣶⡶⣿⣧⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⢰⣿⣯⢻⣿⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢘⣿⡟⣽⣿⡆⠀⢸
⠀⠀⠀⠀⠀⡇⠀⢰⣿⣧⣿⡾⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⢷⣿⣼⣿⡇⠀⢸
⠀⠀⠀⠀⠀⡇⠀⢘⣭⣝⣋⣹⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⣏⣙⣫⣭⡁⠀⢸
⠀⠀⠀⠀⠠⣧⣤⣼⣿⣿⣿⣿⣦⣤⣤⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⣤⣤⣴⣿⣿⣿⣿⣧⣤⣼⠄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣊⣟⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠍⠖⢹⣐⣧⡑⠬⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⣇⡠⠶⣊⢥⡛⢌⡑⢬⡙⠷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⢹⠀⣨⠤⢒⣡⣔⠯⠒⠊⢉⠒⢌⡒⠬⡑⠤⣝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⠉⡆⣸⠤⢚⣩⡴⠞⠋⠉⠀⠀⠀⣀⢸⠀⡀⠈⠒⢌⡑⠢⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠷⣒⣩⠴⠚⠋⠁⠀⠀⠠⡦⠒⠊⠉⠀⢸⠀⠈⠑⠤⣀⠈⢲⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡶⠛⠉⠀⠀⠀⢀⡤⢄⡀⠀⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠙⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠔⠂⠀⡔⠁⠀⠀⠙⠀⠃⠀⣀⡠⠔⠚⠛⠒⠤⣀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⣴⣀⠀⠀⠀⠰⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠤⡉⠑⠂⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠤⠒⠈⡇⠀⠈⠉⡒⠲⡆⠀⢳⠉⠑⡤⠀⠀⠀⢠⠃⠘⡄⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡤⣀⠀⠁⠀⢠⠋⠈⠙⡇⠀⡅⠀⢀⠆⠀⠀⠀⠈⠀⠀⢰⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡆⡧⠀⠀⠀⡄⠀⠀⠀⡇⠀⡇⠀⢸⠄⠀⠀⠀⠀⠀⠀⠘⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠃⡦⠀⠀⠀⡇⠀⠀⠀⠇⠶⠀⠂⢳⠛⠁⢸⠀⢸⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⡙⠁⠘⠄⠀⠃⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⢸⠀⠸⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠼⠤⠥⠤⠤⠤⠀⢸⣀⣀⣰⠀⠱⠀⠀⢸⠀⠀⢸⠀⠀⠆⢀⣀⡧⠤⠧⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⢤⢬⠖⠊⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠏⣤⠘⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢠⠀⣠⠞⠀⠀⠂⣰⣄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⢶⣿⡄⡀⠀⠀⢿⡀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⣮⠁⠀⠀⢀⡿⠀⠀⠀⢀⡈⠗⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡄⠀⣇⠷⣤⠀⠀⠨⠀⠀⠀⠀⠸⠁⣀⣁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣧⣾⣟⣏⢲⣿⠐⠄⠰⠀⠀⠀⠀⢠⠠⡤⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⡟⢟⠙⠛⣅⡿⠀⠀⠀⢦⠀⠀⣰⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢸⡇⠀⠀⠀⢨⡄⣤⡿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⢸⡇⠀⠀⠀⣈⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢁⣤⠬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⢰⡶⠈⡄⠪⠛⡃⣙⠀⢠⡀⠀⠀⠀⠄⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡠⡁⣶⠑⡏⠡⡘⢋⠿⢐⠙⠞⠳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠌⠀⢴⡮⠺⠇⠚⠁⠋⠠⠋⢁⡰⠦⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⡀⣴⠤⡄⠀⠁⣩⡿⠳⣦⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠲⠊⠀⠢⠀⠀⡀⠙⠛⠳⢶⡀⠀⠸⠟⣅⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠁⢸⡷⢴⡶⣤⠙⣶⡛⠀⢀⡄⠈⠃⠀⠀⢠⠄⣴⡆⠽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠆⠀⠰⢟⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⣿⣿⡀⠈⠀⢠⠀⠀⣰⢿⡜⢿⣀⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠀⠀⠀⡀⠈⠃⣀⠀⠀⠠⢠⠀⠀⠀⠀⡘⠁⠿⠛⠉⠉⠓⠴⣠⢞⣥⣾⣿⣾⣿⣷⠀⠀⠓
⠀⠀⠀⠀⢀⣔⣡⠄⠀⠀⠐⣄⠀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⠳⣷⢶⡾⡰⠆⠀⠀⠀⠀⠀⠀⣤⣛⡟⠀⠉⠈⠉⠉⠀⣤⠊
⠀⠀⠀⠀⠻⠿⠛⠀⠀⠰⣶⡌⠀⠰⠊⠀⠀⠼⠃⠐⠁⠀⣠⠋⠀⢀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⠀⢀⠸⠆
⠀⠀⠀⠀⣄⠀⠀⣀⢠⠶⠋⠀⠀⢴⠄⠀⠀⠀⠀⠀⢡⡶⠀⣠⠶⠉⠁⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⡆⠀⢀⡀⠀⠀⡿⠃
⠀⠀⠀⠀⢸⣉⣋⡹⠏⠀⠀⠀⠀⠘⢧⣀⣁⠂⠀⢠⡘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠀⠀⡀⢡⠟⠀⣀⡠⠟⠁
⠀⠀⠀⠀⠀⠙⠈⠋⠀⠀⠀⠀⠀⠀⠀⢨⠹⢳⣦⣨⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡤⠚⢩⣵⣦⢰
⠀⠀⠀⠰⠀⠚⠉⠉⠉⢠⣄⠀⠀⢀⠰⠏⢙⣁⡉⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠃⢀⡅⢨⣿⠼⢦⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⢤⣉⡷⠾⣯⣼⡷⣤⡜⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠶⠆⠛⣋⣙⠳⠻⠷⠄⠡⠂⠈⡙⠂
⠀⠀⠀⠀⠀⠀⢀⡖⠉⠁⠛⠲⠚⠋⡁⣐⠂⠨⠷⢴⡋⠙⠂⠙⠙⡶⠄⠀⠀⠀⠀⠀⠀⠀⣼⡆⠁⠀⠁⠉⠉⠀⠁⠀⠀⠀⠃⠀⠁⠘
⠀⠀⠀⣀⣤⠄⣎⡀⠉⢙⡂⡴⠚⣂⠐⠛⢲⠾⡀⠀⠀⢂⠤⣤⣐⣀⣼⠀⠀⠀⠀⠀⢀⡴⠂⠉⠠⠆⠀⠀⠠⠀⠠⠄⠀⣀⡀⢠⠁
⠀⠠⠂⠉⠀⡰⠀⣑⠟⢈⠁⠆⠘⠁⢶⡎⠀⠃⢁⡀⡤⠉⣧⢘⣀⡅⠸⠀⠀⠀⠀⢀⠢⠄⣸⠀⠠⠤⠤⠤⠤⠤⠤⠤
⠰⠀⡀⠀⣰⠃⣔⠉⠖⠉⠀⠀⠛⠂⠒⠠⠠⣆⠀⣄⠁⠀⢀⡜⢿⠌⢀⠦⠀⠀⠀⡜⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⢀⠀⡙⠲⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠷⠚⠂⢿⠤⠶⠌⠀⠈⠂⠰⠁
⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠡⠤⠤⠤⢀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣐⣒⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠻⢿⣿⣿⣿⣿⡿⠟⠛⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣶⣶⣶⣶⣄⠉⠛⠛⠉⣠⣶⣶⣶⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⣄⠀⣠⠄⠒⠂⠀⠀⠐⠒⠒⠈⠉⠀⠀⠀⢡⠀⢀⠖⢉⠤⠄⣉⡑⡄
⠀⠀⠀⠀⠀⢀⠤⠊⠀⢀⣤⠶⠒⠂⠘⡖⢂⠀⠀⠀⣀⣠⣀⠀⢀⡤⠶⠤⣄⠀⢸⠀⢸⠀⠇⠀⡠⠚⣉⢝⠠⢄
⠀⠀⡠⠔⠊⠁⠀⣀⠀⠸⠀⢀⣤⠤⠄⠘⡌⡄⠀⠚⠁⠀⠈⠁⠈⠀⣀⣀⡀⠀⢨⠔⠁⡸⠀⠀⢧⡞⣊⠥⠒⠉
⠀⢎⠀⠀⠀⣰⠟⠉⠁⠀⠀⠸⠁⠀⢀⡀⠘⣷⠀⢠⠶⠛⠓⠄⠀⠘⠉⠉⠉⠀⡖⠒⠊
⠀⠈⠢⡀⠀⠃⠀⣴⠞⠃⠀⠀⡀⣠⠎⢱⠀⢹⢃⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠈⡶⢄⠀⠁⠀⠀⣀⡭⠞⠁⠀⠸⠀⢸⠈⢆⠀⣠⢶⣒⠛⠿⠟⠻⠄⠸
⠀⠀⠀⢠⠀⠎⠑⠤⡀⠙⠤⣀⣀⣀⠔⠁⠀⡌⠀⠈⢦⡉⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠈⣼⡀⠀⠀⠈⠒⠤⠤⢀⣀⣀⡠⠔⠁⠀⠀⠀⠉⠢⣀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠈⠫⣁⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣈⠆
⠀⠀⠀⠀⢐⣉⡁⠬⠄⠒⠊
⠀⠀⠀⠀⠈

⠀⠀⠠⣄⣠⣟⠀⠀⠀⠀⠀⣀⣀⣺⣦⠀⠀⠀⠀⢀⣼⣃⠀⠀⠀⠀⢀⣟⣀⠀⠀⠀⠀⠘⢶⣷⠞
⠀⠀⠈⠉⡿⣍⠁⠀⠀⠈⢿⣿⣟⠁⠀⠀⠀⠠⣤⡿⠷⢤⡄⠀⠀⠀⢸⡯⡅⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠼⠑⠃⠀⠀⠀⠀⠚⠑⠋⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠏⠁⠀⠀⠀⠀⠀⠞⠘⠇
⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠻⣮⡉⣫⡾⠃⠀⠀⣠⣶⢿⣦⡄⠀⠀⢠⢴⣿⣿⡦⡄⠀⠀⠀⣀⣴⣀⡀
⠀⠀⠀⠙⣿⡛⠁⠀⠀⠀⣿⣿⣈⣿⡿⠀⠀⠈⠲⢿⡿⠗⠁⠀⠀⠀⢹⠿⣯⠁⠀⠀⢠⣤⣿⣽⣇⢀
⠀⠀⠀⠈⠳⠃⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠚⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠚
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣤
⠀⠀⢉⣶⡟⡛⢦⡀⠀⠀⢰⢿⡆⠀⣄⠀⠀⣤⠠⡆⢰⠄⣤⠀⠀⠀⣀⠚⣿⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠸⣇⠘⠃⣸⠃⠀⢠⡟⠛⣿⡈⠛⠁⠀⠈⢷⣿⣿⡾⠁⠀⠀⠘⠻⢀⣿⠀⠀⠀⠀⠈⢠⣿⡄
⠀⠀⠀⠈⠙⠋⠁⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⢀⣤⠶⣤⡀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣠⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢠⣄⣀⣀⡀
⠀⠀⢸⣿⣴⣿⣷⠀⠀⠀⣠⣾⣿⣦⡦⠀⠀⠀⣿⡀⣠⣿⡀⠀⠀⠰⠐⠖⡶⠟⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⢹⣷⣿⣿⠁⠀⠀⢸⣏⣛⣹⡇⠀⠀⠀⢈⡉⠉⠉⠉
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡶⠆⠀⠀⢠⡀⠠⣶⠄
⠀⠀⠀⠀⠘⣿⡿⠂⠀⢸⡇⠀⢀⡼⠻⠀⠀⢀⣠⡍⣤⡉⣷⠀⠀⠀⠐⣡⣰⣶⠀⠀⠀⠓⠦⠥⠤⡶
⠀⠀⠀⣀⣴⠟⠁⠀⠀⢸⣇⣞⣉⣀⣀⠀⠀⠋⠈⢦⣍⣤⠟⠀⠀⣺⣇⣿⣟⣿⠀⠀⠀⠀⠰⣷⠄

⠀⠀⠀⡀
⠀⠀⣄⢡⣠⣤⣩⣤⣤⣬⣥⣠⣄⣉⣠⣤⣬⣥⣤⣤⠥⣄⣀⠬⣤⣤⣬⣥⣤⣄⣉⣠⣄⣬⣄⣠⣤⣅⣤⣀⣌⣠
⠀⠀⠨⠉⠉⠉⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠐⣘⣃⣒⣚⣃⣀⣚⣓⣐⣚⣓⣐⣒⣛⣐⣂⣛⣒⣂⣛⣐⣒⣛⣒⣒⣛⣒⣒⣛⣒⣒⣚⣓⣒⣘⣃⣀⣘⣃⣂⣘⣃
⠀⠙⡿⠛⠛⠻⠛⠛⢟⠛⠛⠛⠛⠛⠛⡛⡛⠛⠟⠛⠿⡟⣿⠿⠛⡻⠛⠻⢛⠛⠛⠛⠛⠛⠛⠻⠛⠛⠟⠛⠛⢿⠋
⠀⠀⠐⠓⠛⠒⠛⠓⠓⠒⠒⠒⠒⠒⠚⠛⠛⠒⠚⠒⠚⠓⠒⠓⠓⠓⠚⠒⠓⠒⠒⠒⠒⠓⠒⠚⠚⠛⠒⠛⠒⠂
⠀⠀⠈⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣾⢳⠒⠒⠒⠒⠒⢲⣶⠒⠒⠒⠒⠲⣖⡂⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡇⣄⢒⠂⡄⢀⠀⠀⠀⠁⡏⠁⣤⢋⣡⣴⣶⣾⣿⣶⣦⣀⠈⢤⠘⠁⠀⡀⠂⠀⡀⢠⠰⡂⢢
⠀⠀⠀⠀⡇⠊⠄⠀⠄⠁⠀⠠⠀⠀⣇⡜⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢠⠀⠀⡀⠄⠂⠈⠠⠠⠠⠰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠈⠀⣿⢰⣿⣿⣿⠟⠛⠙⠋⠛⠻⢿⣿⣷⠀⠄⠀⠁⠉⠁
⠀⠐⠷⡶⢶⠶⢶⡶⢶⣶⣶⣶⣶⣶⣿⢮⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡥⠽⣿⣿⣿⣶⣶⣶⣾⡿⢿⠿⢿⡿⠂
⠀⠀⡏⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠉⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠉
⠀⠀⠁⠀⠀⡄⠀⣿⡋⢠⣾⡇⠀⠀⡇⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⢀⣷⡄⢸⣿⡂⢠
⠀⠀⠀⠀⠀⠸⡿⡛⣅⢱⣎⠁⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⢱⡞⣡⢛⢿⠇
⠀⠀⠀⠀⠀⢰⣽⠟⡟⢡⠑⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣌⢻⠻⣯⣦
⠀⠀⠀⠀⠀⢿⢻⣀⢦⣐⢞⠒⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠐⡓⣂⡴⣀⡟⣿
⠀⠀⠀⠀⠀⣸⠚⠿⠓⠛⣶⠀⠀⠀⡇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠟⠓⠿⠓⣇
⠀⠰⠀⠐⠚⠏⠉⠉⠉⠉⣿⡷⠛⠛⡇⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡷⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠧⠸⠿⢆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠠
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⠈⠁⠈⠈⠈⠉⠁⠈⠈⠉⠉

⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦
⢸⣿⡿⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠉⢣⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠙⢦⠀⠀⢸⣿
⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣀⣀⣞⢲⣶⣶⣶⣶⣶⣾⠟⠢⣀⠀⢸⣶⣶⣾⣿
⢸⣿⣿⡿⠋⠉⠙⢿⣿⠛⠛⠻⢿⣿⣿⠿⠛⠿⣿⣿⣿⠟⠉⠉⣹⣿⣿⠉⠉⠉⠉⣿⡟⠉⠀⠙⢻⣿⣷⣄⡈⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣧⠤⠤⠤⠼⡇⠀⠀⠀⢸⠿⠧⠤⠤⠤⠼⣿⣏⠀⠀⠘⡕⢨⡧⠤⠤⠤⠼⢿⡀⠀⠀⠀⢀⣿⣿⣛⡻⠿⠿⣿⣿⣿⣿
⣸⣿⣅⡠⠤⠤⠴⠂⠱⢤⢤⡤⠊⡔⠉⠉⠉⠉⣲⣨⣿⣿⣿⣛⣷⡊⡰⠒⠒⠒⢒⣤⣕⣀⣀⣀⣺⣩⠒⠒⠒⠒⠒⣆⣹⣿⣉
⡧⠤⣁⠉⡆⠀⢰⠊⣉⣉⣉⣙⣒⠒⠢⡀⠀⡎⢡⠤⠤⠤⠤⠤⠤⣍⠙⠆⠀⡎⠁⣀⡀⠤⠤⠤⢄⠈⢱⠀⠀⡰⠉⣠⠤⠤⡼
⡇⠀⢸⠀⡇⠀⡇⢰⠁⠀⠀⠀⠀⢹⠀⡇⠀⣇⢰⠀⠀⠀⠀⠀⠀⡜⢸⠀⠀⠇⢸⠁⠀⠀⠀⠀⠀⡇⡸⠀⠀⢇⢸⡁⠀⠀⢣
⠑⢄⣸⣀⣯⣷⣷⣈⡆⠀⠀⠀⠀⡎⢸⠀⠀⠸⡀⢇⠀⠀⠀⠀⢠⠃⣾⡆⠐⣾⣈⣆⡀⠀⠀⠀⡰⢡⠃⠀⠀⠘⡄⢣⠀⣀⠼
⣶⣶⡶⣷⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⠰⠀⠀⢈⠑⠀⠀⢀⠂⠀⠀⢐⠀⠀⡄⠀⠀⡩⠡⠀⢀⠤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠱⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⢀⣀⡀⠄⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⢀⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠤⠄⣭⣀⣈⢐⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠈⠙⠛⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⣰⠀⠀⣀⡀⠀⠀⠀⢀⣧⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠒⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡀⡀⠀⢀⢀⡀⣀⣵⣺⣣⢒⡒⡢⢀⣀⣯⣿⣭⣽⣽⣟⣫⣽⣿⣶⣶⣀⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡵⢚⣉⣦⣖⣍⣯⣸⣳⣦⣦⣶⣎⣴⣽⠉⢉⣯⡭⠀⠉⠀⠉⠉⠉⠁⢩⠀⢻⡿⡟⢁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣧⣿⣿⣿⣥⣷⣿⣿⣿⣿⣿⣿⣿⣿⢳⠇⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⣷⡂⡡⠄⢀⡀
⠀⠀⠀⠀⢀⣄⣶⣶⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠀⠀⢯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠹⣷⡃⠘⠊
⠀⠀⠀⠀⠰⣻⢿⣿⣿⣿⠁⠀⠀⠀⠛⠛⣿⣿⣿⣿⣿⣿⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⢲⣞⣥
⠀⠀⠀⠀⠈⣻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣸⡇⠀⢰⣾⣶⣶⣶⣶⣶⣶⣶⣿⣷⠀⣿⡆⠀⣺⣿⣏⣻⣒⢀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⠁⠀⢀⣤⡀⠀⠀⠀⢹⣿⣿⣿⣿⢹⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠁⠀⣻⣿⣿⣿⡇
⠀⠀⠀⠀⢐⣺⣿⣿⡇⠀⠀⠀⠘⠀⢀⠀⢰⣴⡅⠈⣿⣿⢸⣿⣟⡄⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⠀⢀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⡨⡿⣿⣿⠀⣤⠀⠀⠀⠀⠁⠀⠀⠀⣥⠀⢿⣿⣇⢻⣿⣷⡹⣻⣿⣿⣿⣿⣿⣿⣿⣟⣼⣏⡼⣰⣿⣿⣿⣿⣾⠂
⠀⠀⠀⢼⣭⣿⣿⡟⢀⠁⠀⠀⠀⠀⠀⠀⠠⠚⠯⣄⣨⣿⣿⣯⡿⣿⣿⣷⣾⣿⠛⢻⣿⣿⣿⣾⣿⣿⣽⣿⣿⣯⢛⢟⣴⠕
⠀⠀⠠⣴⣿⣿⣿⠁⠘⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢺⣿⣿⣿⣿⣿⣿⣟⠻⣷⣿⣓⣐⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⡩⠶⠄
⠀⠀⢴⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⡿⢿⣿⣿⣿⣭⣛⠻⣿⡿⠿⢛⣟⣽⣿⣿⣿⣿⣿⣿⣿⣻⢏⡑⠉
⠀⢠⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⡀⠐⠘⠁⠈⢻⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡎⠐
⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠠⠡⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⢯⣭
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠃⠀⠌⠀⠀⠀⠀⠀⠀⢿⣻⡉⠻⣿⣿⣿⣏⠀⠳⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣅
⠀⢼⣿⣿⣿⡇⠆⠒⡄⠀⠀⠀⠀⠀⠁⠸⡗⣠⠤⠀⡀⠀⠀⠀⠀⡘⣠⡾⢻⠉⣿⣇⣽⣶⣶⣿⠭⢉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⣾⣿⣿⣿⠏⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢋⡇⠀⠀⡁⠀⠀⠀⣞⠀⣻⣧⣈⣀⣿⣿⠘⠻⡇⠀⠀⠀⠡⣴⡿⢻⡏⠙⠻⣿⣿⣿⣿⣿⣧⡀
⠀⣿⣿⣿⡟⠀⠀⠜⠁⠀⠀⠀⠀⠀⠲⠀⠈⠀⠀⠠⢶⠀⠀⢸⣧⣿⣷⣿⠿⣧⣾⣿⣶⣶⣿⠦⠤⣤⣶⣾⡇⠈⠃⣀⡀⣿⣿⣿⣿⣿⣿⣷⣀
⠀⣿⣿⣿⠏⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠈⣿⠖⠉⡀⠈⠀⠀⠀⠙⣿⢿⣿⣧⣻⣿⣿⡇⣿⢁⣁⢀⡽⠟⢻⣹⣷⡜⠻⡇⠈⢻⠿⠉⠙⢿⣿⣿⣶⣆
⠀⣽⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⡟⠀⠀⠻⡆⠀⠀⠀⢀⣻⣿⣿⣿⣿⣟⣿⣜⣿⣶⣿⣿⠀⢡⣤⠛⠛⢇⠀⠐⢠⣿⠋⠉⠀⣼⠛⠛⠿⡿⣷
⠀⠹⠿⠿⠷⢦⢤⣤⣀⣀⡀⢀⠀⢀⣤⣽⡇⠀⠀⠀⠁⢀⣀⣀⣨⣿⣿⣿⡿⣿⠇⣿⡿⣿⣿⣿⣿⡄⡇⠘⠄⠀⢸⣖⠶⢶⠉⠇⠀⠀⣟⠒⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⡛⠟⠛⠋⠙⡛⠛⠉⣉⣁⣯⠞⠻⣾⣿⣍⣝⢯⣿⣿⣿⣿⣦⣤⣤⣾⣿⡿⡿⣷⢦⡴⢾⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣒⣶⣒⣒⣒⣒⡒⢰⣶⣿⣿⣍⡀⠓⢋⣶⣷⣐⠞⠳⠞⠛⣛⣿⠾⠁⠙⠛⠻⠒⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠈⠉⠉⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈

⣠⠤⣴⣶⣶⣶⣶⣶⢒⡤⢤⡀⠀⠀⠀⠀⢀⣠⣠⣤⣤⣤⣤⡤⠤⣄⣠⣀⠀⠀⠀⡀⣠⠴⣄⣀⣀⣀⣀⡀⠀⢀⣀⠦⢄
⢧⠀⠘⣿⣿⣿⣿⣿⠼⠂⢀⡇⠀⠀⠀⠀⢎⠀⠉⣿⣿⣿⣿⣿⡇⠍⠁⢨⠀⠀⠀⠉⠣⡀⢹⣿⣿⣿⣿⣿⣿⣷⠹⢀⠜
⠀⠑⠠⣽⣿⣿⣿⣿⡤⠖⠋⠀⠀⠀⠀⠀⠈⠢⣀⠘⣿⣿⣿⣿⡝⢀⠤⠃⠀⠀⠀⠀⠀⠠⡈⠛⠿⣿⣿⣿⠿⠗⢏⠅
⠀⠀⠓⠟⢿⣿⣿⡟⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠃
⠀⠀⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⡀
⠀⠀⠀⠀⠰⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣾⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⢀
⠀⠐⢶⣬⣉⡋⠠⠓⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢊⠨⣭⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡄⣤⣌
⠀⠀⠀⠹⢉⣾⠘⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⡗⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠴⠛⢻⡓⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣠⣶⣾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⣴⠿⠿⠿⠿⠿⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣠⠆
⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠛⠓⣦
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠒⠒⠛⠃
⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⠀⠀⢀⣀⣀⡀
⠀⠀⢠⣾⣽⡿⠿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⢀⣪⣿⣿⡿⠟
⠀⠀⣽⣿⣿⠆⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣶⡶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣲⣿⣿⣿⠏
⠀⠀⠈⠿⣿⣿⣿⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠉⣯⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠻⣵⣿⣿⡿⠏
⠀⠀⠀⠀⣿⣭⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣀⣿⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⠷⢁
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⣷⣔
⠀⠀⠀⣾⣿⡟⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⣿⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣿⡏
⣤⣤⣤⣯⣥⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠂⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⢠
⠀⠀⠀⣆⣀⠈⠀⠀⢈⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣧⣀⣀⣀⣛⣀⡼⡀
⠀⠀⠀⢿⣿⡟⠀⠘⣿⣿⠇⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⢿⣶⣶⣶⣶⣶⣶⠀⢠⣶⣶⣿⠂⠀⠸⡇⠀⠉⠉⠀⠀⠀⠇
⠀⠀⠀⠈⢿⣿⠄⢸⣿⠟⠀⠀⠀⢿⡞⠒⠲⠶⠶⠶⠶⠶⠖⠒⠒⠉⢸⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄⢰⣿⡿⠃⠀⠀⠀⠳⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠙⠣⠽⠁⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠃⠈⠉⠀⠀⠀⠀⠀⠀⠈⠐⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠃⠀⠦⣈⣀⣈⡠⠞⣁⠀⢸⡇⠀⠀⠀⠨⣅⣀⣀⡠⠽⢡⠀⠀⠀⣿⠀⠘⠀⠀⠀⡷⡍⣁⣀⣩⡵⠀⢹⠀⠸⠀⠀⠸⢯⣀⣀⣈⡭⣇
⠀⠀⠀⡆⡀⣾⠀⠰⠀⠀⠈⢀⢸⠇⠀⠠⣦⠀⠀⢸⡇⠀⠀⠈⠉⠀⠀⡇⠀⠘⠔⠀⠀⣇⠂⠀⠀⠀⠁⠀⢸⠀⠀⠠⣤⠀⠘⢸⠃⠀⠀⠉
⠤⠤⠀⠤⠧⡿⡤⠤⠼⠀⠤⠼⠼⣤⣤⣤⢤⣥⠤⡼⠧⠄⠠⠤⠴⠤⢠⡧⠄⠄⠠⠼⣶⠿⠤⠴⠧⠠⠤⠦⠼⣤⣤⡤⠼⢤⠤⡼⠤⠄⠤⠤⠧⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠉⠉⠙⠛⠻⠿⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠦⣤⣤⣸⣷⣤⣀⣀⣀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠛⢛⢿⠏⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠆⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠐⠀⠺⢽⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⣀⠀⠀⠀⠀⣄⡶⡀⢠⣀⠙⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠛⠋⢉⡙⠿⣿⢿⣿⣿⣿⡀⣤⠀⣰⣿⣿⣷⣿⣿⠿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢘⢿⠁⣤⠴⠒⢋⣉⣩⣝⡳⣝⣿⢿⢷⡜⠢⣿⣿⡷⢛⣁⣘⠢⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠉⠃⣨⣤⠶⣿⣿⣿⣿⣿⣿⣿⠿⢷⠀⠉⠈⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠀⡿⠿⣿⣿⠿⠟⡁⠒⠀⠀⠀⠀⠈⢻⣿⡿⠟⠛⣻⡏
⠀⠀⠀⠀⠀⣴⠟⠟⠓⠄⠙⡀⠀⠈⡆⠀⠀⡀⠄⠊⠀⣠⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡛⠿⣿⣧
⠀⠀⠀⠀⠀⢿⣇⠀⠀⢠⠀⠃⠀⢀⠧⠀⠀⠀⡀⠴⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡵⡤⡸⣿⡄
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⢀⣠⠔⢁⡠⠔⠂⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⢀⣀⣀⠀⠀⢸⣷⡸⣿⣿⡇
⠀⠀⠀⠀⣼⣉⠙⠧⡀⢸⡿⢁⡔⠉⡗⡀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⣿⣿⣦⣼⣿⣿⣻⣿
⠀⠀⠀⣼⡟⡟⠓⠄⠘⢌⡇⣼⣀⣠⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢿⣿⡟⣿⣿⡏
⠀⠀⠀⢰⣵⣇⠀⠀⠐⠪⡿⣛⣿⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣴⣋⣍⠳⡶⣻⣯⣠⡉⢷⣽⠟
⠀⠀⠀⠘⣿⣿⠀⠀⠀⢰⠳⣾⣷⣄⠈⢆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠿⠛⠛⠉⠛⢿⣾⠛⠿⣿⢟⡿⠁
⠀⠀⠀⠀⢹⣻⣦⠀⠀⢸⠀⠘⡿⣿⣷⠈⢢⡀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢤⣀⣀⣀⣘⣿⣤⣾⣿⣼⠃
⠀⠀⠀⠀⠘⣿⡽⣄⠀⡀⠃⢁⡇⢿⣧⢠⡀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠿⠿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢾⣷⣤⣿⡤⠞⠀⠘⣿⢦⣕⢤⡀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⢷⣿⠉⠁⠀⠀⠈⠛⠷⣦⣤⣄⣀⠀⠀⠀⠀⡀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾⣿⣶⣾⣷⣶⣾⣷⣶⣾⣷⣶⣿⣷⣶⣿⣶⣶⣿⣶⣶⣷⣶⣾⣷⣶⣾⣷⣶⣾⣷⣶⣾⣶⣶⣿⣶⣶⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢙⣿⡏⢹⣿⡋⣹⣿⡉⣹⣿⠉⣿⣟⠉⣿⣟⢉⣿⣏⢙⣿⡏⢹⣿⡋⢹⣿⡋⣻⣿⠉⣻⣟⠉⣿⣟⢙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣄⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠉⠙⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠋⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠠⡄⠀⠀⠀⠀⠀⢠⢻⣷⣾⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣶⣦⣄⠀⠀⠀⠀⠺⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠟⠻⠿⠛⠻⣿⣿⣿⣦⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣴⣶⣶⡄⣠⣶⣖⣾⣿⣿⣿⣷⠀⠈⠈⠀⢐⠄⠠⠤⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡍⠛⠁⠃⠉⠉⠉⢸⣿⣿⣿⡋⠀⠀⠀⠀⠈⠀⠇⠀⠀⠀⢰⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣒⣒⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣄⡈⠛⠛⢁⣰⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡿⡿⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢺⣿⣷⡀⠂⣿⣿⣷⣄⣠⣴⣿⠛⠛⠲⠤⣀⠀⠀⠀⢀⣾⡟⣸⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡅⠀⢩⣧⠑⢶⣿⣿⣯⣀⣀⣼⣿⣿⣿⡿⢻⣮⣤⣤⣄⣡⠟⠳⡶⠲⣾⣿⠀⡇⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⠃⢸⠁⢀⣾⣿⣿⣿⣿⣿⡿⣭⡤⣼⠿⢿⣯⣴⣿⣿⣿⣿⣿⣏⣀⣀⣠⣠⣽⣿⣷⡇⠀⣿⣿⡀
⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡡⠀⠀⣿⡤⣻⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠟⠋
⢱⠀⠀⠀⠀⠀⡀⢀⣾⣋⠁⠀⡄⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠉⢹
⠈⠀⠀⠀⠀⠀⢀⣾⠛⠈⠀⢰⣿⣿⣿⣿⣿⣿⣿⣤⣂⣠⣃⢸⡤⢯⣽⣿⣽⣼⣿⢻⣿⡿⠟⠻⠁⠀⠀⠀⠀⠀⠈
⠄⠀⢀⠀⠘⣠⣿⣇⠈⠀⢠⣿⡿⢿⠻⣿⣿⣿⣿⢿⣿⠿⠻⠿⠾⠿⣿⣿⣿⣿⣿⣿⣻⠆⠠⠀⠰
⠰⡆⠀⠚⡾⠋⠀⠁⢠⣿⣿⠟⠑⣾⠀⢹⣿⣿⣿⣆⢿⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠆
⡄⢰⢤⣞⠋⠁⣤⣤⣾⣿⣻⡀⢳⠋⠀⢸⣿⣿⣿⣿⣷⣿⣷⣦⣴⣿⣯⣿⣿⣿⣿⣿⠀⠀⠀⠀⠠
⠁⠸⣶⣶⢶⣤⠙⢤⣀⠈⠉⠛⠛⠶⢤⣄⣿⣿⣿⣿⠟⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡀⣤⡙⣿⣆⣿⣶⣄⡈⠙⠲⢤⣄⡀⠀⣅⣙⠀⠀⢃⡀⢀⣀⢢⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡜⠛⢀⣼⣿⣿⣏⠉⣛⠛⠒⠠⡤⡍⠙⠻⠛⠻⣿⣶⣿⡮⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⣧⡤⣿⣿⣟⢿⠏⠉⣠⣦⡀⠇⣶⠀⠀⠀⢠⣾⣿⣿⣿⡇⣹⣟⣛⣛⣛⣿⣿⣿⣧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⢿⢿
⣁⠀⢀⣡⡏⣿⡄⢤⡿⢹⣿⣿⣧⡆⠀⣠⣿⣿⣿⣿⣿⣧⣿⣧⣦⣶⣶⣿⣿⣿⣍⡇⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⣴⡏⢉⠀⣇⢠
⠙⠀⢻⣿⠀⠙⢿⣼⣑⣫⣼⣿⣿⡇⢰⣿⣿⡷⣾⣿⡷⢾⣿⣿⣿⣒⡺⢽⣯⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⢃⣡⣄
⣀⣄⣰⠛⠀⣀⡸⠿⢷⣶⣿⣿⣾⣿⣿⣿⣿⠀⣿⡿⡇⠈⠁⣿⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⣤⠐⢾⡄⠁⠀⢰⣿⣿⣿⣷⣿⣿
⣿⣿⠁⢀⡜⠛⠁⢠⣄⣻⣿⣹⣿⣿⣿⣿⣻⢿⣷⣥⣿⣿⢰⣿⣿⣿⣿⣿⡿⡏⠀⠀⠀⠀⠀⠀⣿⣧⡈⢷⡀⠀⣸⣿⣿⣷⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠠⠅⠀⣠⠄⡂⠀⠀⠀⠅⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡙⡃⠘⠀⠘⣛⣛⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣈⣉⣉⣉⣉⣉⡅⠠⠂⢸⣿⣿⠇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⠈⠁⢠⡆⣼⣿⣿⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠆⠨⣤⠄⣀⣴⣪⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣌⢛⠋⣡⠎⠙⡽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢴⠃⠀⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠀⠀⣀⣼⣿⣴⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⠋⢹⡏⠐⠛⠉⡿⠿⠧⣤⣄⣰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⣞⣹⡂⣀⢀⡀⠀⢸⡇⠀⣇⠀⢸⡇⢀⠀⠀⠀⠠⠀⠈⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⣠⣾⣿⢛⣿⡿⠿⣾⣿⣽⣦⣯⣼⣇⣿⣿⠀⣿⣿⡈⠀⠀⠀⠀⠀⠀⡀⠈⠹⣷⠜⠛⢦⣾⣄⡤⡄
⠀⠀⠀⢀⣶⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣏⣿⡿⡧⢱⣟⠇⠀⠀⢐⣦⡀⠀⣀⠀⢸⣿⡟⠲⣾⠁⢻⡌⢶⠚⠀⠀⡀
⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣶⡿⢻⣾⣿⠟⣿⣋⠏⠃⢆⠀⠀⠀⠟⠒⠶⠆⢀⣡⣂⣠⣤⡼⠛⢏⡉⠿⠉⣹⡟⢢⣼⡷⠒⢡⣾⡅
⠀⠈⠛⠻⣿⣿⣿⣿⣹⣿⣼⣿⣶⣿⣿⣿⡏⠋⣐⣶⣨⣤⡀⠈⠛⠛⣙⣿⣿⣿⡟⠁⠉⣿⣿⠈⠁⣄⣸⡉⣳⡋⢻⣷⢦⣾⡿⠁
⠀⠀⠀⠀⠛⢛⣿⣿⣿⣿⣿⣿⣦⣿⣬⣄⣁⣀⣀⡐⣀⣉⣉⣰⡟⠹⣿⣿⣿⣿⡷⠠⣾⣿⣿⣦⣤⣷⣾⣭⣿⣯⣽⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠚⠉⠛⠃⠉⠉⠉⢩⢀⠁⠉⠩⠤⠖⠌⠻⠄⠰⠶⠂⠲⠛⠛⠛⠏⠡⠤⢿⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠎⢀⡴⠀⣠⠏⡀⠀⢄⣀⠒⠦⠤⣄⣀⣀⣀⡤⠤⠀⠀⢀⣀⡠⠤⠀⠙⣆⠀⢢⡀⠑⢄⡀
⠀⠀⠀⠀⣀⠴⠋⣀⡴⠋⣠⡞⠁⠀⠉⠲⢤⣀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⢀⣠⠴⠊⠁⠈⠳⣄⠙⠦⣀⠙⠦⣄
⠀⠀⠀⠀⠀⠠⢚⣡⡴⢾⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣤⣤⣤⡤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠈⡗⢦⣌⣓⠆
⠀⠀⠖⠒⠚⠋⠉⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠙⠛⠒⠒
⠀⠀⠀⠀⠀⠀⢰⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆
⠀⠀⠀⠀⠀⠀⡟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢹⡀⠀⡆
⠀⠀⠀⢰⠀⡸⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢧⠀⡇
⠀⠀⠀⡘⡰⠵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢧⢡
⠀⢀⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶⡀
⢀⢛⠩⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⡍⡙
⢸⢸⠀⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⠀⡇
⢸⢸⠀⡇⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⠀⡇
⢸⠀⠀⢁⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠸⠀⠁
⠈⠀⠀⢸⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡆
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠳⢦⣄⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⣦⣤⣤⠾⠋⠀⠀⠙⠷⣤⣤⡴⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⣤⠶⠶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⣤⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠸⣧⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⡿⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣿
⢀⣾⠃⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠏⠀⠀⠈⠻⠶⠶⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠘⣷
⣾⠃⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠘
⢿⡀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢀
⠈⢷⡄⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⢠⡾
⠀⠈⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣠⡶⠋⠉⠻⢦⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⢠⡏⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣴⠞⠉
⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠈⠻⠶⢤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠶⠛⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠾⠛⠻⢦⣄⠀⠀⣠⡶⠟⠛⠶⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠷⣄⣀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣀⣤⡾⠃⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⣿⠀⠀⠀⠙⠷⣦⣤⣠⡴⠞⠋⠁⠘⣧⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠐⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠐⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠘⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⣀⣠⣴⠞⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠹⣦
⠀⠈⡤⠾⠿⠶⠶⠾⠛⠛⠛⠉⠁⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠈⠉⠉⠙⠛⠛⠻⠶⠶⠾⣷
⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣰⡏
⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢰⡟
⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁

⠀⠀⠀⠀⠀⢀⣤⣖⠋⠉⠉⠉⠉⠉⠉⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⣀⠀⣠⣴⡖⠂⠀⠀⠀⠀⠖⠀⠀⠀⢄⡀⢀⡀
⠀⠀⣤⣠⠤⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠤⡤⡄⠀⢻⠻⣤⣽⡿⣦⣤⣤⣤⣀⣠⣤⣤⣤⣶⣿⣥⠿⣻⠁
⠀⠀⢻⡇⠈⠙⠀⠈⠉⠃⠀⠀⠒⣤⣤⡤⣽⣾⣾⠉⠁⢸⠇⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠟⢻⠏⠀⠀⣠⠃
⠀⠀⠀⠳⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠇⢀⡠⠋⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⣴⣴⢆⣠⠔⠁
⠀⠀⠀⠀⠀⠉⢈⡦⡀⠀⠀⠀⠀⠀⣠⣮⣿⢿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢠⣧⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⢤⣤⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠁⠼⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠳⠅⣀⣀⡨⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣏⣀⠀⢀⣙⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠉⠀⠐⠂⠚⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⣉⣉⢉⣥⡀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⣂⣒⣒⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣹⠟⣛⣻⣻⣡⠿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠤⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⠄⠀⠈⢄⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠢⠀⠀⢂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠐⠄⠀⠀⠑⢄⠀⠁⢀⠀⠢⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⡌⠀⡀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠢⢄⠀⠑⢄⠀⠢⡀⠡⠀⢂⠀⠄⠀⠀⠀⠈⠠⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⠀⠀⠀⠑⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠢⠤⠤⠄⣾⣄⣤⣤⢤⠦⡤⠴⡼⠇⠀⢀⠀⠀⠀⠀⠀⢸⠀⠐⠖⠰⠤⠴⠤⠦⠂⠒⠒⠚⡅⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢣⠀⠀⠀⠀⢻⡳⠤⣤⣠⣄⣤⣀⣄⣤⣠⡤⣠⠤⡔⡆⠀⠈⠀⠐⠂⠀⠀⠘⠓⠆⢤⠀⠀⡘⠇⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⢦⡜⣧⠀⠀⠁⠈⡒⠉⠠⠀⠀⠐⢌⠲⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠰⠀⡄
⠀⢀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⠠⠤⣼⣧⡀⠈⠀⠁⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⢀⡇⠀⠀⠀⠀⠀⠀⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠂⠄⠀⣀⣀⣠⠬⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠋⡇⠀⠈⠐⠂
⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⢦⡇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣀⠀⠀⢀⡀⠀⣨⠤⠀⠈⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠁⠉⠉⡁⠀⠐⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠤⠀⠀⠀⠀⠀⠀⠠⠤⠠⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠶⠿⠛⠛⠛⠛⠛⠛⠿⠶⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣀⣠⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉
⣶⣶⣶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⢞⣱⠆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⡐⠠⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⡀
⠀⠀⠀⠀⠀⣀⣿⣿⣀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠉⣿⣿⡄⠀⠀⠀⠀⢀⡃⡐⠁⢠⡆⠀⢡⠐⡀⠀⠀⠀⠀⠀⣰⣿⣧⢸⣿⢹⡆
⠀⠀⠀⠀⠀⠙⠁⣿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⣿⡿⠁⠀⠀⠀⠀⠐⡎⢷⡀⠠⠇⠀⣸⢅⡖⠀⠀⠀⠀⠀⢿⣿⣿⡾⢋⡼⠃
⠀⠀⠀⠀⠀⢀⣰⣿⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⣿⣶⡀⠀⠀⠀⠀⠀⠹⣞⡅⠀⠀⠈⣃⠾⠁⠀⠀⠀⠀⠀⢀⣙⣲⣖⠋⡀
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⠶⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⢤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠀⠀⠀⠀⠰⠻⢶⡾⠛⠁⠀⠀⠀⢠⣤⣄⣴⣿⡿⠙⢦⣠⣤⠀⠀⠀⢸⡁⢹⣿⣿⣿⣿⠸⠁⣹⠀⠀⠀⠀⠀⠀⠀⣷⣿⠃
⠀⠀⠀⠀⠀⣠⣬⣧⣤⡀⠀⠀⠀⢀⣿⣿⡞⢋⣰⣿⣯⣿⣿⡀⠀⠀⠀⠑⠶⣿⣿⣿⣿⡧⠖⠁⠀⠀⠀⠀⠀⠀⢀⣸⣟⣀
⠀⠀⠀⠀⢸⣿⠛⡿⢻⣿⠀⠀⠀⠀⠀⠀⠈⠚⠙⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠻⢿⡇
⠀⠀⠀⠀⠸⠿⠾⠷⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠾⠇

⠀⠀⠀⠀⢀⢔⣮⣗⡢⡀⠀⠀⠀⠀⠀⠖⢸⣿⣿⣿⡆⠲⠀⠀⠀⠀⠀⢀⠠⠀⢀⡀⠀⠠⢀⠀⠀⠀⠀⠀⣄⣀⣀⣀⡀⣀⣀⣠⡀
⠀⠀⠀⠀⡦⣿⡛⢛⣿⢰⠀⠀⠀⠀⠀⠈⠊⢿⣿⡿⠔⠁⠀⠀⠀⠀⢐⠂⡆⠀⢸⡇⠀⠈⠄⠂⠀⠀⠀⠀⣿⣿⣿⢟⣵⣿⣿⣿⡇
⠀⠀⠀⠀⠰⣝⡿⢿⣫⠖⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠨⡄⢷⡀⠾⠷⠄⡼⠁⡍⠀⠀⠀⠀⣿⡿⣣⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢺⣿⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣻⣷⡆⠀⠀⠀⠀⠀⠀⠛⣦⡅⢀⣀⠠⣤⠞⠀⠀⠀⠀⠀⠉⠀⠁⠉⠻⣄⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⣿⠛⢛⣛⠇⠀⠀⠀⠀⠀⠀⣔⣉⣯⣉⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⠁⠀⠀⠀⠀⠀⠰⠒⠄⣶⣶⣶⣶⠠⠒⡆
⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⠀⠀⣆⢹⠿⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠿⢧⠀⠀⠀⠀⠀⠀⠀⢂⠀⣿⣿⣿⣿⠀⡰
⠀⠀⠀⠀⢀⣀⣼⣁⣀⡄⠀⠀⠀⠀⠀⠀⠈⢑⣶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠏⠔
⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠸⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⣿⡇⠀⠀⠄
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡴⠉⠀⠀⢿⡿⠀⠀⠙⢦⠀⠀⠀⠀⡄⠀⠀⠀⢀⣯⠀⠀⠀⠀⢀
⠀⢰⢰⠀⢀⢄⣀⣀⣀⡠⠀⠀⡖⡆⠀⠀⢸⠁⠀⠀⠀⠼⠇⠀⠀⠀⠘⡆⠀⠀⢸⡇⠀⠀⠀⠘⠛⠃⠀⠀⠀⣿⡇
⠀⠸⣔⠣⣀⠀⠻⣿⠟⠀⣀⡜⡰⠁⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠠⠀⢀⣿⠁
⠀⠀⠈⠳⠄⠀⣴⣿⣦⠀⠡⠞⠁⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣿⠇⠀⠀⡀
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⢠⢞⠏⠀⠀⣴⣷⣤⠀⠀⠻⡳⡀
⠀⠀⡇⡄⠀⠀⣶⣶⣶⠀⠀⢠⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠝⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠧
⠀⠀⠳⡈⠲⠀⠹⣿⠋⠠⠔⣁⠞⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠹⣌⠦⡄⠀⠀⠀⠀⠀⢠⠴⣱⠇
⠀⠀⠀⠙⠃⠀⣴⣿⣦⠀⠘⠋⠀⠀⠀⠀⠈⢧⡄⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠈⠳⠀⠂⠄⠄⠠⠐⠐⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠄⠰⠖⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⢀⡀⠸⣿⠇⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠈⣿⡿⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⣿⠃⢀
⠀⠀⠀⡐⠝⠀⠠⣷⠄⠀⠺⣂⠀⠀⠀⠀⢠⡞⠀⠀⠀⣼⡆⠀⠀⠀⢳⠀⠀⠀⠀⢀⢣⠂⠀⢠⣿⡄⠀⠐⡵⡀
⠀⠀⣸⣺⠀⠀⠀⠀⠀⠀⠀⣟⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⡂⠀⠀⠀⠀⠀⠀⠀⣀⢪
⠀⠀⢸⡘⡄⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢳⠴⡀⠀⠀⠀⠀⠈⢠⠆⡏
⠀⠀⠀⠻⣜⢢⠀⠀⠀⡖⣣⠟⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⣠⢤⣶⡟⠋⠀⠀⠀⠀⠈⠧⣕⡢⡄⠀⢠⢔⣫⠞
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠠⠷⠂⠄⠾⠂⠄⠾⠆⠀⠾⠗⡀⠰⠷⡀⠒⠷⠀⡀⠿⢆⡄⠾⢆⠄⡺⠷⠀⡰⠷⡠⠐⠷⠠⠄⠾⠤
⠀⠀⠀⠀⠀⠀⣯⠉⠉⣽⠍⠉⢹⡍⠉⢙⠩⠉⠉⣫⠉⠉⣿⠉⠉⣿⠉⠉⣝⠉⠉⠍⡋⠉⢩⡏⠉⠩⣯⠉⠉⣽
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠂⠀⠊⠁⠀⠈⠐⠀⠐⠙⠀⠀⠉⠀⠀⠉⠀⠀⠋⠃⠀⠂⠁⠀⠈⠑⠀⠘⠙⠀⠀⠉⠄
⠀⠀⠀⠀⠤⣷⣴⣤⣧⣴⣴⣧⣤⣼⣦⣤⢾⣦⣤⣾⣤⣤⣶⣤⣤⣶⣴⣤⣷⡴⣴⣧⣤⣴⣧⣤⣼⣦⣦⣼⣦⣦⣾⠦
⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⢐⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⡃
⠀⠀⠀⠀⠀⢸⢠⣤⣤⣤⣤⣤⣤⣤⣤⠀⠉⣭⠍⣻⠽⢛⠉⢿⡿⠉⡛⠫⣟⠩⣭⠉⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄⡇
⠀⠀⠀⠀⠀⢸⢸⡧⠇⡗⣈⣀⠒⠀⢺⠀⠘⣣⠞⡁⠊⠀⠀⠀⠀⠀⠀⠑⠈⠳⣜⠃⠀⡗⠀⠒⣄⣁⢺⠸⢼⡇⡇
⠀⠀⠀⠀⠀⢸⠸⠷⠶⠶⠤⠦⠆⠷⠸⠀⢀⠎⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠱⡀⠀⠧⠾⠴⠴⠤⠷⠶⠾⠇⡇
⠀⠀⠀⠀⠀⣸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣇
⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⡶⣴⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣴⣦⢶⣦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢠⡿⣟⢾⣾⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣷⡷⣻⢿⡄⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⣹⣿⠏⣾⣻⡧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢾⣟⣷⠹⣿⣏⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢹⠿⢸⡗⢦⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡼⢺⡇⠿⠏⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢰⣿⣷⣦⣾⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⣷⣴⣾⣿⡖⠀⠀⡇
⠀⠀⠀⠀⠀⢺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡗
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⢠⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶
⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠉⠀⠀⠀⠀⠀⠀⢠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⣀⣤⣴⣶⡀⠀⠀⣶⣆⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣰⣿⠟⢉⣉⣻⣷⣄⠀⠘⢿⣦⡀⠀⢻⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢰⣿⠁⢰⣿⠛⣿⡏⠻⢿⣶⣤⡿⠿⣷⣶⣽⣿⠛⣿⣦⣤⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣶⣾⣿⠀⢸⣿⣷⣿⣧⡀⠀⠀⠉⠁⠀⠀⢈⣽⣿⣾⣿⡏⠉⣿⣷⣦⣄⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠁⢸⣿⣠⣿⠏⠀⠀⠙⣿⣦⠀⠀⠀⠀⣴⣿⠏⠀⠀⠹⣿⣄⣿⡇⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⢠⣼⣿⣿⠃⠀⠀⠀⠀⠈⠻⣷⣶⣶⣾⠟⠁⠀⠀⠀⠀⠘⣿⣿⣧⡄⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⣿⣯⣨⣿⡿⠿⠿⠿⠿⠿⠿⣿⣏⣸⣿⠿⠿⠿⠿⠿⠿⢿⣿⣅⣹⣿⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⠛⢿⣶⣄⠀⠀⠀⠀⠈⣻⡟⠁⠀⠀⠀⠀⣠⣶⡿⠛⣿⣿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠉⠻⣷⣦⡀⠀⠀⢸⡏⠀⠀⢀⣴⣾⠟⠁⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢿⣶⣄⣼⣯⣠⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠉⣿⡟⢻⣿⠉⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿
⠘⠟⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣩⣿⡏⠉⠉⠉⠉⢹⣿⣍⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠿⠿⠿⠛⢻⣿⠻⣿⣦⡀⢀⣴⣿⠟⣿⡟⠛⠿⠿⠿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⢸⣿⠀⠈⠻⣿⣿⠟⠁⠀⣿⡇⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⣤⡄⠀⠀⠀⠀⢸⣿⠀⣠⣾⠟⠻⣷⣄⠀⣿⡇⠀⠀⠀⠀⢠⣤⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢀⣿⡃⠀⠀⠀⠀⢈⣿⣾⡿⠁⠀⠀⠈⢿⣷⣿⡃⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⣰⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠉⢿⣮⡛⠛⠻⣶⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⡄⠀⢶⠆⠀⣶⠄⢀⣄
⠀⢰⣿⣿⣿⡯⣳⣿⠃⠀⠀⠀⠀⠀⢸⡇⠉⣇⣷⠄⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢻⣄⣸⣧⣠⣿⣠⡾⠉
⠀⢸⣿⣿⣿⣯⣿⡏⠀⠀⠀⠀⠀⠀⠘⣷⣄⠈⠉⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠃
⠀⠈⠉⠛⠛⠓⠛⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠒⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠻⣷⣆⠀⠀⠀⠀⠀⠀⠀⢒⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄
⠀⠀⢀⣝⠉⣀⣀⠈⢻⣆⠀⠀⠀⠀⢸⣿⣄⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢹⣿⡄⠀⠀⣴
⣠⠾⠻⠟⠁⠿⠿⠃⢸⡿⠀⠀⠀⠀⠀⢻⣿⣀⣾⣿⣿⡗⠀⠀⠀⠀⠀⠀⠉⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣴⣿⣷⠀⠛⣿⠛
⠃⠀⠀⠺⣶⣤⣤⣴⠿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣸⣟⠀⠀⣸⣿⣇⡀⠉
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⣛⡴⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡄⠀⠘⢀⠀⠀⠀⠀⣀⣀⣚⣛⣛⣛⣂⣀⡀
⠀⣼⣹⣿⡅⢸⣿⣏⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠲⣶⣿⣿⡶⠂⠀⠄⠀⠀⠀⢻⡈⣯⣿⣿⣿⠏⣼⠃
⠀⢿⣚⣿⡇⢸⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠿⡇⠀⠠⠀⠀⠀⠀⠀⠙⠚⢿⣿⠟⠚⠁
⠀⠈⠻⣾⣿⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⢿⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈

⠀⠀⠀⠀⠀⣠⡿⡇⠀⠀⠀⠀⠀⠀⣤⡄⠀⣤⣿⣤⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣘⠿⠃
⠀⠀⠀⠠⠚⠁⢠⣤⣄⠀⠀⠀⠀⠀⠘⢷⣄⣀⣛⣀⣠⣄⠀⠀⠀⠀⠀⠀⠬⠿⣾⣿⢷⡴⠆⠀⠀⠀⠀⠀⠀⢰⡶⢶⣶
⠀⢀⣀⡀⣿⣷⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⠛⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⠀⠀⠀⠀⠀⣴⣤⣤⣾⣿⣼⣿⣀⣀⣀
⠀⠸⠿⠇⠿⠿⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢼⠃⠀⠀⠀⠀⠀⣿⡗⣼⣿⣿⣿⣿⣿⢂⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢰⠂⢺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠒⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢲⡆
⠀⠀⠀⠀⠀⢸⠁⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠘⡄
⠀⠀⠀⠀⠀⢰⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⣴⠃
⠀⠀⠀⠀⠀⢼⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿
⠀⠀⠀⢀⣠⣧⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣼⡅⢛⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⡆
⠀⠀⢀⣿⢱⣿⣿⢻⣿⣿⣿⡄⠀⠀⠀⣼⠃⣿⡇⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠟⢁⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠊⢹⡆⡀
⠀⠀⢸⡏⢸⣿⣿⣸⣿⣿⡟⡇⠀⠀⠀⣿⣄⣿⠇⢤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣾⣾⣦⣼⣷⣧
⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⠀⠀⠄⠀⠀⠀⠀⠛⣿⣿⡄⠀⠀⠀⠀⠀⣼⢳⣿⣷⢸⣿⣿⣇
⠀⠀⠸⠇⠘⠛⠛⠻⠟⠿⠷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣼⣸⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⠀⠀⣸⠃⢸⣿⣿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣤⣀⣿⣀⣤⣿⣿⣿⡤⠀⠀⠀⡿⠀⢸⣿⣿⢈⣹⣿⣿⣧
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣿⡟⢸⣿⣿⣿⣿
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣶⣶⣤⣤⣤⣤⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠛⠀⠛⠛⠃⠒⠛⠛⠛⠛
⠀⠀⢸⡇⢸⣿⣷⣾⣿⣿⣿⡇⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⢸⣿⣿⣾⣿⣿⣿⣿⠀⠀⢶⣤⣤⢀⣀⣀⣀⣀⣀⣤⣤⣵⠂
⠀⠀⢸⡇⣺⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣾⣿⣿⣠⣿⣿⣿⣿⠀⠀⢸⣿⡃⠀⢙⣟⡓⠀⢹⣿⣿⣿
⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⠀⣀⣿⣥⣀⣸⣿⣿⡇
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠹⣿⣿⣿⣿⣷⣿⣿⣿⡿⠁⠀⠀⠘⠷⣶⣤⣤⣤⣤⣤⣤⣶⠾⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠁⠀⠁⠐⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠂⠐⠈⠁



⠀⠀⡀
⠀⠀⠀⠨⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠡


⠀⠀⠈⠀⠀⠂⢀
⠀⠀⠀⠀⣀⣀⡸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⣀⣀⡀
⠀⠀⠘⠋⠃⠀⠀⠀⠀⠲⠠⠖⠐⠀⠄⠄⠒⠺⢛⣿⠟⠛⠛⢿⡛⠛⠒⠠⠄⠖⠒⠐⠂⠐⠀⠀⡀⠠⠀⠋⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⡄⡀⠈⠉⠀⠀⠀⣄⣾⠁⠀⠀⠀⠈⢱⣄⠆⠀⠀⠈⠉⠀⡀⡎⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⡆⣠⡍⠀⠈⠉⠀⠈⠉⠀⠘⣤⢀⡇⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠛⠒⠒⠶⠶⠦⠤⠴⠶⠶⠒⠓⠚⠒⠤
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⣴⣾⣿⣿⣶⡄⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣛⡻⣿⣦⣤⣾⡿⣱⡿⣶⡿⣿⣦⣤⣾⡿⣟⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⢀⣾⣟⣾⡟⠻⣷⣿⣿⣯⡾⠏⠀⠈⢻⣾⣿⣿⣵⡿⠛⢿⡞⣿⣇⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣛⣿⠿⣟⣼⡟⢀⣠⣤⣽⣿⣿⣷⣾⡷⢶⣶⣿⣿⣿⣤⣄⣀⠘⣿⣜⣿⣿⣿⣏⣉⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠋⠛⠛⢻⣿⣿⣿⠿⣛⣻⣽⣽⡿⠶⠾⠿⠿⠶⠾⣿⣯⣽⣛⡻⢿⣿⣿⣿⠛⠛⠋⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⢷⣿⢀⣤⣾⡿⣻⣽⡾⠟⠛⠉⣁⣤⣤⣤⣤⣶⣶⣤⣤⣤⣀⡉⠙⠛⠿⣶⣝⡿⣿⣦⣄⢸⣿⢿⣷⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣼⣿⣫⣭⣭⣭⣵⣿⣿⣿⢟⣵⡾⠛⢁⣠⣴⣾⠿⣟⣛⣯⣭⣯⣭⣭⣭⣭⣟⣛⡿⢿⣶⣦⣀⠉⠻⣶⣝⢿⣿⣿⣾⣭⣭⣭⣽⣻⣿⡀
⠀⠀⠀⠀⠀⢻⣿⢿⣏⠉⠈⣴⣿⢟⣵⡟⠋⣠⣶⡿⢟⣹⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠉⠉⠛⠻⠿⣮⣟⠿⣷⣦⡈⠙⢷⣝⢿⣷⡄⠉⢉⣿⣻⣿⠃
⠀⠀⠀⠀⠀⢈⣿⣿⣿⢀⣾⡿⣱⡿⠃⣠⣾⡿⣯⡾⠟⠉⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⢿⣶⠀⠀⠀⠙⢿⣮⡿⣿⣦⡀⠻⣷⡽⣿⣆⢸⡟⣿⣟
⠀⠀⣠⣶⣾⠿⣟⣽⣿⣿⢟⣿⠟⢀⣼⡿⣣⣾⠋⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠻⣮⡻⣿⣄⠘⢿⣜⣿⣿⣿⣝⡿⢿⣶⣤⡀
⠀⢼⣿⢻⣾⠿⠛⢁⣿⡟⣾⠏⢀⣾⡟⣽⡟⠁⠀⠀⠀⠀⢰⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⡽⣿⣆⠈⢿⣏⣿⣏⠙⠻⢿⣾⢻⣿
⠀⠘⣿⣏⢿⣦⠀⣾⣿⣼⡟⠀⣾⡟⣿⡟⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡼⣿⡆⠘⣿⡼⣿⡄⢠⣾⢏⣾⡟
⠀⠀⠘⣿⣷⣿⣶⣿⣧⣿⠀⣼⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⢿⣆⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⢹⣧⣿⣷⣾⢳⣿⡟
⠀⣠⣾⡿⣻⣿⢿⣿⣹⡏⠀⣿⡏⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣿⣇⠘⣿⢸⣿⢿⣯⡻⣿⣦⡀
⣾⣿⣻⡾⠛⠁⢼⣿⣿⡇⢰⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⠀⣿⣾⣿⡆⠙⠻⣾⡻⣿
⠻⣷⣘⠳⣦⡀⢸⣿⢿⡇⠸⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⠀⣿⢻⣿⠇⣠⡶⢋⣴⡿
⠀⠙⠻⣿⣾⣿⣾⣿⣾⣧⠀⣿⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⡇⢠⣿⣸⣿⣿⢿⣾⡿⠛⠁
⠀⠀⣠⣿⢟⣿⠟⣿⡟⣿⡀⢹⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣹⣿⠁⣼⡟⣿⡿⢿⣞⢿⣧⡀
⠀⢰⣿⢯⣾⠋⠀⢻⣿⣹⣷⠀⢿⣷⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣳⣿⠇⢰⣿⢹⣿⠃⠈⢻⣯⢻⣷
⠀⠹⣿⣼⣻⠷⣦⣄⣿⣧⢻⣧⠈⢿⣷⡹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣰⡿⣱⣿⠏⢠⣿⣧⣿⣏⣤⣶⢿⣻⣽⡿
⠀⠀⠉⠛⠿⣿⣿⣽⣿⣿⣯⡻⣧⡀⠻⣿⣟⢿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⣠⣾⣟⣽⡿⠃⣠⡿⣳⣿⣿⡟⣿⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⢠⣿⢿⣿⠀⢻⣷⣝⣿⣄⠘⢿⣷⣟⡿⣶⣄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣴⡾⣟⣵⣿⠟⢀⣼⣟⣵⣿⠋⢸⣿⢿⣷
⠀⠀⠀⠀⠀⣼⡟⣿⣏⣀⣀⣙⣿⣾⡻⣷⣄⡈⠻⣿⣾⣽⣻⠷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⡶⢿⣛⣽⣾⠿⠋⢀⣴⡿⣫⣾⣟⣁⣀⣀⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠹⣿⣯⣿⣿⣟⣛⢿⣿⢿⣧⣝⢿⣦⣄⠉⠛⠿⢿⣿⣿⣿⣿⣛⣛⣛⣻⣯⣭⣿⣿⠿⠟⠋⢁⣤⣾⢟⣵⣾⢿⣿⢟⣿⣻⣿⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠙⣿⡏⣿⡀⠙⠿⣿⣾⣝⣻⣶⣦⣤⣀⣉⠉⠙⠛⠛⠛⠛⠋⢉⣁⣠⣤⣶⡿⣟⣽⣾⡿⠟⠁⣸⡿⣿⡿⠙⢿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣤⣤⣶⠾⣿⣿⠿⣿⣾⣭⣽⣛⣛⣿⣿⠿⢿⣿⣟⣛⣛⣯⣽⣶⡿⣿⣿⡿⢶⣦⣤⣿⡇⣿⡇⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣿⣿⣿⣿⣷⣹⣷⠀⠉⢉⣻⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣋⠉⠀⢰⡿⣱⣿⣿⣿⣯⣯⣾⠿⠁⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠈⠉⠉⠁⠀⠈⢿⣷⠻⣧⣾⢟⣯⣿⣽⣿⣦⡀⣠⣾⣟⣽⣯⣟⡿⣦⣿⣻⣿⠇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣷⣽⣿⡿⠛⠉⢿⣷⣝⠿⢟⣵⣿⡏⠙⠻⣿⣿⣽⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⢻⣷⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠸⣿⣶⠶⠿⠿⠟⠛⠛⠋⠉⠉⠉⢿⣦⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⢠⣾⠋⠉⠉⠉⠛⠛⠛⠿⠿⠷⢶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⣼⡟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦
⠀⠀⠀⠀⠀⣰⣷⢸⡿⠋⠀⠀⠀⣠⣴⣶⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣦⣄⠀⠀⠀⠙⢿⡇⣾⣆
⠀⠀⠀⠀⠀⣿⡟⣠⣾⡗⠀⠀⢰⣿⡏⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠹⣿⡇⠀⠀⢺⣶⣄⢻⣿
⠀⠀⠀⠀⣾⡜⠳⢿⡋⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢰⣿⡇⠀⠀⠀⠙⠿⠞⢋⣾
⠀⠀⠀⠰⣿⡇⣼⣿⠇⠀⠀⠀⠀⠻⣿⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⣿⠟⠀⠀⠀⠀⠘⣿⣦⢸⣿⠇
⠀⠀⠀⢠⡙⠇⠿⣋⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣴⠿⠃⠀⠀⠀⠀⠀⠀⣙⠿⠼⢟⡄
⠀⠀⠀⢸⣿⡄⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢠⣿⡇
⠀⠀⠀⠈⠿⣇⣿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⣸⡿⠃
⠀⠀⠀⠀⣷⣦⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣨⣾
⠀⠀⠀⠀⠹⣿⣧⢿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡿⣰⣿⠏
⠀⠀⠀⠀⠀⢨⣝⡋⢱⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡎⢁⣫⡅
⠀⠀⠀⠀⠀⠀⢻⣿⣦⢻⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡿⣱⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣛⣓⠁⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠈⢚⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣝⠿⠸⣷⣤⡀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣠⣶⡇⡿⣫⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢭⣥⣤⣙⡻⣿⡀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⢀⣾⠿⣋⣤⣬⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⡀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⢀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣤⣭⣭⣤⣭⣭⣥⣬⣭⣥⣬⣭⣥⣬⣭⣥⣬⣭⣥⣬⣭⣭⣤⣭⣭⣤⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⣿⣇⣿⣿⣇⣿⣿⣇⣿⣿⣇⣻⣿⣿⣸⣿⣿⣸⣿⣿⣸⣿⣿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⣛⣛⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⣛⣛⣛⣛⣛⡛⣿⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⣿⢛⣛⣛⣛⣛⣛⡛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⡇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣶⣶⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢟⣛⡻⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⢟⣛⡻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣾⣿⣷⢿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡏⣾⣿⡇⢽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⢸⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡇⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢻⣿⡟⣼⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣇⢿⣿⡇⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣷⣾⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣷⣮⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⡍⣭⣭⣭⢭⣭⣭⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣭⣭⡍⣭⣭⡍⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢿⣿⡿⢾⣿⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⣿⡇⢿⣿⡇⢼⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢰⣲⢠⠖⠀⣶⢲⣴⣖⠀⢶⢲⣶⢤⣦⠖⣦⣶⣰⡆⣖⡦⣆⣴⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠘⠘⠈⠓⠀⠙⠊⠘⠓⠀⠘⠘⠋⠋⠉⠓⠃⠋⠃⠃⠃⠀⠃⠙⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣠⣾⠟⠀⠀⠀⢠⡄⠀⠀⣿⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠙⠿⠿⠟⠁⣠⣾⡟⠁⠀⠀⠀⠀⠈⢿⡄⠀⣿⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣶⣶⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⣿⡇⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣶⡀⣾⣦⠀⠀⢠⣶⢸⣿⡇⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣆⢻⣷⡜⢿⣧⠀⢸⣿⡟⣛⡻⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠉⣿⣿⠀⠀⠀⢀⡘⣿⣧⠹⣿⡌⢿⣷⡘⣿⠀⣿⡇⢹⣇⣿⣿⠀⠀⣰⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡆⠀⠀⠘⣷⡌⢿⣧⡹⣿⣎⢻⣷⡘⠆⣿⣿⢸⣿⣿⣿⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⢸⣿⡇⠀⠀⠀⠘⢿⣌⢿⣷⣘⣿⣾⣿⣿⣶⣿⣿⡈⣿⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⣿⡇⠀⠀⠀⠀⠈⢿⣆⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠀⠀⠀⠀⠹⠿⠶⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⡘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⡆⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⢿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⢢⠰⠀⡆⠦⠰⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡆⢰⢠⠆⡄⠴
⠀⠀⠈⠀⠀⠆⡆⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠨⠘⠀⠅⡇
⠀⠀⠠⠀⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⠁⠀⠃
⢀⣠⠂⠄⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⢄
⠀⣿⣿⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟⡻⣻
⠀⠘⡠⢂⢋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠐⡙⢾
⠀⡀⡀⠈⡌⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠃⢀⣸
⠀⠉⠘⡂⢄⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⢐⠆⠘
⣈⣉⠉⠦⠮⣀⡀⠲⠖⠩⡉⡱⠶⠀⠠⡀⡶⠊⠉⢉⣲⡆⠀⠀⢰⣖⠈⠉⠑⢦⣀⠀⠀⠶⢆⢉⠉⠲⠤⢄⢀⠱⠶⠨⣉
⠈⠁⢲⡔⠀⠉⠁⠀⢢⡄⠉⠁⠀⠐⣤⠉⠁⠀⠀⠲⡟⠀⠀⠀⠀⢻⡄⠀⠀⠈⠙⣤⠂⠀⠈⠉⠲⡔⠀⠈⠉⠀⢢⡄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠤⠤⠴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠙⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠆⠀⠒⠛⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⣶⣶⡟⠒⠀⠀⡀⠀⠀⠀⣴⢯⠘⡍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣿⠟⠣⠄⢂⠀⠄⠀⠀⠀⠾⣞⣾⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢈⣙⠧⠻⣀⠀⠀⣿⣷⡆⠀⠀⠀⠀⣌⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄⣸⡿⠿⢤⡄⣡⢐⣿⣿⣿⡀⠀⠀⠀⠀⠈⢈⣁⠁
⠀⠀⠀⠀⠀⠀⢀⣷⠾⠿⡁⠉⠀⢹⣶⠂⢹⣿⣽⣏⠀⠁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢤⣾⠻⠆⠀⢷⡎⠐⠛⠫⣴⣿⣟⠁⢭⣤⡃⠀⢸⠋⠀⠀⠀⠀⢀⣠⣶⡿⠈⢀
⠀⠰⠐⠀⣦⠛⠁⢹⣄⢸⣿⠧⠀⠠⣤⡿⡧⣭⡤⠓⢩⢧⠀⠀⠀⠀⠀⠀⠐⠛⠉⡅⠂⠀⠀⠀⠀⢲⠒⠀⠂
⠀⠀⠀⠀⡇⠁⠀⠉⠙⢻⠉⠓⢶⠈⢈⡏⠉⠉⠁⠀⢸⠀⠘⠖⠀⡄⠤⠀⠀⠢⢠⡄⣰⣷⣶⣤⣄⡘⠈⠁
⣀⣂⣒⣛⢓⣒⣒⣐⠒⠚⠒⣒⣒⣒⣚⣓⣀⣒⣒⡂⣖⣀⣀⡐⣚⣓⣒⣓⡐⠚⠚⠒⣓⣛⣛⣛⣓⣀⣒⣒⣂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⡄⠀⢻⣿⣿⠃⠙⠳⠶⠤⠤⠶⠛⠁⢹⣿⣿⠃⠀⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⢀⣄⠀⣠⣤⣄⡀⢠⣤⠀⢠⣤⠀⣤⣤⡄⢤⣤⣤⡄⣠⣤⣤⡀⣤⠀⢀⣤⣤⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢶⣏⢸⣿⢸⡏⠀⢸⡇⢸⣿⣇⡿⣿⠀⣿⠶⠆⠀⢸⡇⠀⣿⣤⣾⠃⣿⢠⣿⠀⠀⠀⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠴⠟⠘⠟⠘⠷⠦⠾⠃⠸⠃⠿⠃⠿⠂⠿⠶⠦⠀⠸⠇⠀⠿⠀⠻⠆⠿⠀⠻⠶⠴⠀⠶⠾⠟



⣿⠿⠿⠿⣿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻⣿⣛⣿⣿⣿⣛⣿

⠀⠀⠀⠀⣀⣀⣀⣀⠀⣴⠶⢦⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠛⠷⣤⡾⠛⠿⠶⠶⠶⢶
⣴⠶⠞⠛⠋⠉⠁⠉⠻⠃⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡾
⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡾⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣸⣀⣀⡀⣰⡉⠀⠉⡿⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⣿⠀⠸⡿⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣿⠀⢠⡟⠛⠛⠛⠛⠓⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠍⠀⡀⠿⠀⠀⠄⠀⠠⡄⠀⠀⠉⠙⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠃⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⡀⠀⠺⠂⠀⢀⡠⠄⠐⠀⠠⠤⠤⠤⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢠⡦⠈⠁⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣀⠀⠀⠀⠃⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡀⡀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢰⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠰⠂⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢡⠄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠔⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠰⠅⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠚⠁⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⠀⠀⠀⢀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡀⠀⢠⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠹⣄⠀⠀⠨⠄⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠁⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠗⢝⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⣄⠀⠀⠀⠳⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢸⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠈⢣⠀⠀⠀⠀⢠⡄⠀⠀⠀⠙⠳⠤⣀⣀⣀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⣠⠞⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢄⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⢦⡀⠀⠀⠀⠑⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⠀⠀⠀⠉⠀⠀⣀⡴⠞⠁⢀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠙⠳⢦⣀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀⠶⠛⠁⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠈⠉⠓⠶⠆⢠⣤⡴⠒⠦⣤⣤⣤⣤⠤⠶⠖⠊⠉⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣤⣀⣀⣀⣰⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠞⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣭⣷⣶⣶⣶⣾⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠄⡀⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠻⣄⠈⠁⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠿⠛⠂⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠸⢰
⠀⠀⠴⢲⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠤⠴⠶⠚⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⠤⠤⠄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⢀⠀⠐⢦⡀⣀⢸⠀⠀⠀⠀⠀⠃⣠⠖⠉⠀⣀⢀⣀⠀⠀⠢⡄⣠
⠀⢠⡇⠀⣀⠀⠤⠤⠄⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢰⠋⢧⡀⠀⠀⠀⠀⠀⠀⠉⣀⡼⢺⠀⠀⠀⠀⠀⠀⠳⠄⡀⠀⠀⠀⠀⠗⠠⢀⣷⡏
⠀⢸⣷⠋⡀⠛⠀⠀⢂⡴⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠛⠓⠒⠒⠒⠚⠋⠉⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠤⣐⣽⠟
⠀⠈⢿⡋⠃⠜⠀⠒⠒⠈⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠦⣌⣙⠒⠚⠛⣋⣡⢄⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢉⠷⠶⠲⠶⡛⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢭⣭⣭⣽⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⠐⠂⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⡸⡄
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠇
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⣀⡀⠀⠠⠚⠁⠁⠀⠮⣰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⠦
⠘⠁⣬⠁⠀⠀⠀⠀⠀⠀⠘⠮⣏⠄⠀⠀⠀⠀⠀⠀⠀⠉⠂⠚⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠔⡁⠈⠃
⠀⠀⠀⠒⠲⠶⠶⠶⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⠤⠴⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⠤⠤⠴⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣤⣤⣤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣿⣿⠿⢛⡩⠭⠅⠒⠂⠭⠭⣙⠻⢿⣿⣧⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢟⠵⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢝⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⡟⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡙⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣷⡙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣱⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣡⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⣿⣿⣷⣦⣭⣓⣒⣒⣒⣂⣩⣥⣶⣿⣿⡟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⣷⣦⣾⣬⢉⡉⣉⢩⣴⣦⣶⣦⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⣿⣿⣿⡟⢸⠇⢿⠘⣿⣿⣿⣿⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣿⣿⣿⣿⠃⣿⠀⠸⡇⢻⣿⣿⣿⣇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠿⠛⣿⡟⢸⡇⠀⠀⣿⠘⣿⡟⠻⢿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣴⡾⣷⣌⠁⣿⠀⠀⠀⢸⡇⢉⣴⡿⣶⣤⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠈⠻⣷⡇⠀⠀⠀⠀⣷⡿⠋⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

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

⣰⡶⠀⠲⣶⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⢸⢉⣉⢹⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠢⠀⠄
⢽⣇⠀⢀⣿⠅⠀⠀⠀⢈⣿⣻⡅⠀⠀⠀⠀⢸⠐⠒⠀⠀⠀⠀⠀⠀⣹⣿⣿⡏⠀⠀⠀⠀⠀⠄
⠀⠛⠓⠛⠃⠀⠀⠀⠀⠈⠁⠘⠓⠀⠀⠀⠀⠈⠉⠉⠛⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛

⠀⢶⣦⣶⠂⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠀⢠⡿⣥⠀⠀⠀⠀⠸⣧⣾⣷⣴⡇⠀⠀⠀⢲⡞⠰⠠⠤⠀⠀⠀⠀⠢⣿⣿⠕⠀⠀⠀⢰⣶⣾⣿⣷⣶
⠀⠻⣤⡾⠀⠀⠀⠀⠘⠁⠀⠀⠈⠑⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡜⢣⠀⠀⠀⠀⠀⠀⠈⠛⠁

⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⡇⢀⣤⣀⠖⠀⠀⠀⠀⠀⠀⢀⣤⣾⠀⠀⠀⠀⠐⠿⠈⠀⠀⠀⠀⠀⣤⣴⣧⣤⠀⠀⠀⠀⣼⣯⡊⠙⡆
⣇⢋⣀⣁⣀⠀⠀⠀⠸⠇⠿⠻⠟⠿⠀⠀⠀⠀⠐⣿⡖⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠙⠢⣝⠴⠃
⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⡖⠒⠒⣒⡒⡄⠀⠀⠀⣠⣴⣮⡭⣄⠀⠀⠀⠀⣙⡿⣏⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⢀⣼⣿⣿⣷⡶
⠧⠭⠤⠭⠤⠇⠀⠀⠘⢔⡻⠿⠃⠁⠀⠀⠀⢸⣿⠴⣻⡇⠀⠀⠀⠰⠮⠽⠯⠽⠆⠀⠀⠀⠻⢿⣿⡿⠃
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⣀⠔⠂⠉⠉⠉⠒⠦⡀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠜⠀⠀⡠⠁⠀⠀⠀⣀⠴⡒⣒⢐⡒⡢⢄
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠙⣿⣿⣿⡄⠌⠀⠀⡐⠁⠀⠀⢀⠜⣵⠝⠋⠀⠀⠈⠛⣴⠑⡄
⠀⢸⠀⢴⠀⠀⠀⠀⠀⠀⠀⡆⠈⡄⠀⠀⠀⠘⣿⣿⣿⡄⠀⡐⠀⠀⠀⠀⡘⣸⡃⠀⣠⣴⣧⣄⠀⠘⣧⢸
⠀⢸⠀⢿⡀⠀⠀⠀⠀⠀⢀⡯⢀⠃⠀⠀⠀⠀⠘⠿⠿⣿⡔⠀⠀⠀⠀⠀⢇⢽⡄⠀⢈⡿⢿⡇⠀⠀⣯⢸
⠀⠀⢣⡈⠳⢤⣀⠀⣀⡴⠟⢁⠎⠀⠀⠀⠀⠀⠘⣓⠄⢖⡋⠀⠀⠀⠀⠀⠘⣔⢿⣄⠀⠀⠀⠀⣠⡼⢂⠎
⠀⠀⠀⢙⢦⣄⣉⣀⣁⣠⡴⡁⠀⠀⠀⠀⠀⣴⡽⢓⠀⢐⠻⡷⡄⠀⠀⠀⠀⠈⠲⢍⣛⠃⠀⢛⣨⠔⠁
⠀⠀⠀⡜⣾⣿⣿⠛⣿⣿⣷⢣⠀⠀⠀⠀⢸⣽⠐⠀⠀⠀⠑⢸⣽⠀⠀⠀⠀⠀⢠⣿⣷⣾⢿⣶⣿⣧
⠀⠀⢰⠹⠛⣿⠏⠀⢻⣿⠛⠏⡆⠀⠀⠀⠸⣻⡔⢄⠀⢀⠔⣸⡻⠀⠀⠀⠀⢀⣿⠿⣿⠇⠈⣿⡿⢿⡆
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠑⠿⣲⣀⣲⡺⠟⠁⠀⠀⠀⠀⠈⠀⠀⠛⠀⠀⠙⠁

⠀⠀⠘⠁⣀⠀⠋⣀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⣀⠙⠀⣀⠈⠃
⠐⠶⠃⠲⠎⡱⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣤⠀⢠⡄⠀⣤⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢎⠱⠆⠀⠶
⡻⣧⣅⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠐⡘⢡⣎⡛⣦⡄⠉⠛⠛⠉⢠⣴⢛⣡⡬⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⣨⣼
⠁⠛⢋⠄⠉⠀⠀⠀⠀⠀⠘⣠⡆⠀⠐⠪⡭⠉⠁⠒⠂⢶⡶⠐⠒⠉⠉⢭⠕⠂⠀⢠⡌⠁⠀⠀⠀⠀⠀⠉⠠⡙⠛
⠇⠘⠃⠀⠀⠀⢠⡶⠀⠀⠰⠁⠛⠶⣄⣀⡀⠤⠴⠦⣄⣿⣿⣠⠴⠤⠤⢀⣀⣠⠶⠛⠈⢆⠀⠀⢶⡄⠀⠀⠀⠘⠃
⠀⠀⠀⠀⢀⣴⣿⠁⠀⢄⣳⡟⣻⠶⣤⣴⠆⠘⠛⣃⣀⠈⠁⣀⣘⠛⠓⠰⢦⣤⠾⣟⠻⣞⣡⠀⠈⣿⣦⡀
⠀⠀⠀⢠⣾⣿⠇⢀⠐⡼⠋⣭⣴⠆⠚⠙⠋⠃⠀⠉⠉⠉⠉⠉⠉⠀⠘⠙⠋⠓⠰⣦⣭⠙⢧⠂⠀⠸⣿⣷⡄
⠀⠀⠀⠘⣿⡟⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆⠂⠀⠀⢻⣿⠃
⠂⠀⢡⠀⣿⠃⠀⠐⡾⠁⠀⠀⠀⣀⣤⡶⢀⣾⠿⠿⠟⢠⡄⠻⠿⠿⢷⠀⢶⣤⣀⠀⠀⠀⠈⢷⠂⠀⠘⣿⠀⡈
⠀⠀⠀⠀⠛⠀⠀⠐⠀⠀⢠⣴⠏⢁⣤⠞⠁⠀⠀⠀⠀⠬⠥⠀⠀⠀⠀⠈⠳⣤⡈⠹⣦⡀⠀⠀⠃⠀⠀⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣯⣵⠶⢾⣯⠀⠀⣠⡀⢸⡟⠢⠔⢻⡇⢠⣄⠀⠀⣽⡷⠶⣬⣽⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⡄⠀⠀⠀⠈⠀⠘⠟⢋⡙⠻⠃⠈⠁⠀⠀⠀⢠⣤⣬⣽⠆
⠀⠀⠀⠀⠀⡃⠀⠀⢸⣇⡀⣼⠟⠁⣴⣦⣀⠀⠀⠀⠀⣩⣍⠀⠀⠀⠀⣀⣴⣦⠈⠻⣧⢀⣸⡗⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⣶⢨⣟⣻⠇⠀⠜⢫⡴⠶⠛⠉⠉⠉⠹⠏⠉⠉⠙⠛⠶⢦⡝⠣⠀⠸⣟⣿⡅⢶⡄
⠀⠀⠀⠀⣟⠀⠟⠁⠘⠿⠟⠀⠾⠊⢁⠠⣤⠐⠶⠞⠓⢢⡔⠚⠳⠶⠂⣤⠄⡈⠙⠷⠀⠻⠿⠃⠀⠙⠀⣲
⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⣠⢾⣤⣾⠁⠀⣀⠀⣤⠈⠑⣤⠀⣀⠀⠈⣷⣬⡷⣄⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⡞⠁⠀⠾⣿⢁⣤⢀⡀⠀⠠⠁⠋⢙⡋⠙⠀⠁⠀⢀⡀⣤⡉⠿⠷⠀⠈⢳⠀⠀⠀⢸
⠀⠀⠀⠀⢃⠀⠀⠀⠁⠀⠀⡴⢫⡆⠠⣾⠹⠒⠒⠂⠀⠨⠅⠀⠐⠒⠒⠯⣷⠌⢲⡝⣦⠀⠀⠈⠀⠀⠀⡘
⠀⠤⠾⣷⣿⠆⠀⠀⢠⠁⠀⣶⠏⢠⠖⠀⠀⣤⠀⠒⠓⣒⣒⠚⠒⠠⢤⠀⠀⠲⡄⠹⣦⠀⠈⡄⠀⠀⠰⣿⡾⠷
⠀⠴⠴⢾⡇⠀⠀⠀⠘⠀⠀⠋⠀⠀⣴⠷⠙⠁⠀⠋⠘⢻⡟⠃⠉⠀⠈⠋⠾⣦⠀⠀⠙⠀⠀⠃⠀⠀⠀⢸⡷⠦⠤
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⠃⡰⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⢆⠙⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗
⠀⠀⠀⣾⡇⠀⠀⠀⠰⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠄⠶⠄⠠⠶⠐⠀⠀⠀⠁⠘⠃⠀⠀⠀⠀⠧⠀⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠒⢺⡟⠀⠀⡀⠀⠀⣠⡞⠋⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠉⢳⣄⠀⠀⢠⠀⠀⢻⡗⠒⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⢀⡇⠀⡀⠋⠀⡴⣊⣡⣭⣉⣛⣛⣉⣭⣌⡑⢆⠀⠙⠀⠀⢸⣄⠀⣸⡇⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⢻⠁⣀⠃⢀⠚⣼⣿⡿⢿⢹⡿⢿⡏⡿⢿⣿⣧⠓⡀⠘⡀⠈⡟⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⢸⠁⣿⠀⣸⡇⠿⠟⢻⠁⠀⠁⠈⠀⠈⡟⠻⢿⢸⡇⠀⣿⠀⡇⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⢸⠀⣿⠀⢹⡇⠀⠀⣿⣛⣛⣋⣙⣛⣛⣿⠀⠀⢸⡇⠀⣿⠀⡇⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⢸⠀⣿⠀⢸⡇⠀⠀⣿⣿⣉⣉⣉⣉⣿⣿⠀⠀⢸⡇⠀⣿⠀⡇⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢸⣇⠀⢸⠀⣿⠀⢸⡇⠀⠀⣿⣿⡇⠠⠄⢸⣿⣿⠀⠀⢸⡇⠀⣿⠀⡇⠀⣸⡇⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⢸⠀⣿⠀⢸⡇⠀⠀⢹⣿⡅⢠⡆⢸⣿⣿⠀⠀⢸⡇⠀⣿⠀⡇⠀⣿⡇⠀⠀⢸⣿
⠀⢀⠀⣿⡇⠀⠀⠸⣿⠀⢸⠀⣿⠀⢸⡇⠀⠀⢻⣹⣇⠿⠇⢸⣿⣿⠀⠀⢸⡇⠀⣿⢀⡇⠀⣿⠇⠀⠀⢸⣿
⠀⢘⠀⣿⡇⠀⠀⠀⣿⠀⣿⡄⣿⡀⢸⡇⠀⠀⣾⣯⣶⣫⣼⣿⣿⣿⠀⠀⢸⡇⢀⣿⢈⣇⠀⣿⠀⠀⠀⢸⣿
⠀⣬⢀⣿⡇⠀⠀⢠⣿⠐⠿⣿⡛⠇⢸⣷⣶⣶⣿⣿⣿⣯⣿⣿⢿⣿⣶⣶⣾⡇⠘⢛⣿⠿⠂⣿⣄⠀⠀⢸⣿⡀⣀
⠀⣿⣿⣿⣧⠀⠖⠿⠿⣶⡇⠸⠷⠀⠀⡀⠀⢰⣿⡇⠀⣽⣿⠀⠸⣿⣆⠀⢀⠁⠀⠾⠇⢰⣶⠿⠿⠶⠀⣸⣿⣿⡿
⠀⢿⡿⢿⣿⣷⡦⠀⠀⣿⠓⠀⠀⠀⡌⠁⢀⣿⡟⠀⠀⣿⣿⠀⠀⣻⣿⡀⠈⢁⠀⠀⠀⠚⣿⠀⠀⢰⣾⣿⡿⠟⠐
⠁⢉⣭⡄⢨⣿⡇⠀⠀⠿⠀⠀⢀⡚⠁⢀⣻⣿⡋⠀⢘⣿⣿⡃⠀⢙⣿⣟⡀⠈⢓⡀⠀⠀⠿⠀⠀⢸⣿⡅⢡⣄
⣴⣿⣿⠀⠐⣿⠀⠀⠀⠀⠀⠠⠯⠁⠠⢼⣿⠯⠅⠀⠬⣿⣿⠥⠀⠨⠽⣿⡧⠄⠈⠽⠄⠀⠀⠀⠀⠀⣿⠂⠀⣿⣿
⠋⣿⣿⡇⠀⣿⠀⠀⠀⠀⡘⠛⠀⠀⠲⣿⡿⠓⠀⠀⠚⣿⣿⠓⠀⠀⠚⢿⣿⠖⠀⠀⠛⢇⠀⠀⠀⠀⣿⠀⢸⣿⣿
⠾⣻⣿⣿⠶⠋⠀⠀⢀⡚⠛⠉⠀⢙⣻⣿⠛⠋⠀⠘⠛⣿⣿⠛⠃⠀⠙⢛⣿⣟⠛⠀⠈⠛⢓⡀⠀⠀⠙⠶⣿⣿⣝
⠼⠛⠋⠀⠀⠀⠀⠀⠛⠛⠁⠀⠛⢻⣿⠟⠛⠋⠀⠘⠛⣿⡿⠛⠋⠀⠘⠛⠻⣿⡟⠛⠀⠈⠛⠛⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡽⠃⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠈⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⢻⡀⠀⡇⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢁⣀⠈⠳⡀⢣⠈⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠁⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠳⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⣿⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠛⠒⢦⡄⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⡙⣎⣧
⠀⠀⠀⠀⠀⠀⢠⢏⡄⣤⣀⡀⢸⣿⠉⠉⠁⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⣧⢹⣎⢧
⠀⠀⠀⠀⠀⠀⡾⡼⢸⣿⣿⡟⡼⢸⢠⡠⣶⣶⣦⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⣮⠳⣄
⠀⠀⠀⠀⠀⢸⢳⢃⣿⣿⣿⢱⡇⠸⡇⡇⢿⣿⣿⡜⣇⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⣿⢸⢸⣿⣿⡟⡾⠀⠀⢷⢹⢸⣿⣿⣧⢻⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⣿⣿⣿⣷⡌⢷⡀
⠀⠀⠀⠀⠀⠘⢮⣛⢟⣋⡼⠁⠀⠀⠸⡜⣆⢿⣿⡿⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣿⣿⣿⣿⣆⢻⡄
⠀⠀⠀⠀⠀⠀⢨⡿⠉⠁⠀⠀⠀⠀⠀⠙⠮⣽⣭⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⣿⣿⣿⣿⡿⠶⢿⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡤⠾⠛⠉⠁⠀⠀⠈⢷
⠀⠀⠀⠐⠶⠾⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠷⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣀⣤⣴⣾⣿⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠚⠉⢿⣿⣿⣿⣿⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⠾⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠀⠈⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠁⠀⠀⠀⠀⢈⣿⣦⡀
⠀⠀⢀⣴⠟⠁⠈⠻⣦⡀⠀⠀⠀⠀⠹⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠏⠀⠀⠀⠀⢀⣴⠟⠁⠈⠻⣦⡀
⠀⠀⠀⠻⣦⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠈⢻⣦⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⡟⠁⠀⠀⠀⢀⣴⡟⠁⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣹⣷⠶⠟⠛⠛⠉⠙⠛⠛⠻⠶⣾⣏⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠙⣷⡶⠟⠉⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠉⠻⢶⣾⠋⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⢠⣾⠋⠀⠀⣠⡶⠟⠋⠉⠁⠀⠀⠈⠉⠙⠻⢶⣄⠀⠀⠙⣷⡄⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡿⠁⠀⣠⡿⠋⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⢿⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⣼⠏⠹⣧⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢰⡿⠀⢀⣀⣤⣤⣤⣼⠏⠀⠀⠹⣧⣤⣤⣤⣀⡀⠀⢿⡆⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠈⠻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠁⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⣿⡀⠀⠀⠀⢸⡟⠀⠀⢀⡀⠀⠀⢻⡇⠀⠀⠀⢀⣿⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡄⠀⠘⣷⡄⠀⠀⣼⣧⣴⠾⠛⠛⠷⣦⣼⣧⠀⠀⢠⡾⠃⠀⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠈⢿⣄⠀⠈⠻⣦⣄⠙⠉⠀⠀⠀⠀⠀⠀⠉⠋⣠⣴⠟⠁⠀⣰⡿⠁⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣹⣷⣄⠀⠀⠙⠻⠶⣦⣤⣤⣤⣤⣴⠶⠟⠋⠀⠀⣠⣾⣏⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠶⠶⢶⣾⠋⠀⠙⠷⣦⣄⣀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣠⣴⠾⠋⠀⠙⣷⡶⠶⠶⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣠⡾⠟⠛⠛⠷⠶⠶⠶⠶⠾⠛⠛⠻⣷⣄⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣠⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⣀⠔⠂⠉⠉⠉⠒⠦⡀⠀⠀⠀⠹⣿⣿⣿⡄⠀⠀⠜⠀⠀⡠⠁⠀⠀⠀⣀⠴⡒⣒⢒⡒⡢⢄
⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠙⣿⣿⣿⡄⠌⠀⠀⡐⠁⠀⠀⢀⠞⣵⠝⠋⠀⠀⠈⠛⣴⠑⡄
⠀⢸⠀⣴⠀⠀⠀⠀⠀⠀⠀⡆⠈⡄⠀⠀⠀⠘⣿⣿⣿⡄⠀⡐⠀⠀⠀⠀⡘⣸⡃⠀⣀⣴⣧⣄⠀⠘⣧⢸
⠀⢸⠀⢿⡀⠀⠀⠀⠀⠀⢀⡯⢀⠃⠀⠀⠀⠀⠘⣿⣿⣿⡔⠀⠀⠀⠀⠀⢃⢽⡄⠀⢈⡿⢿⡇⠀⠀⣯⢸
⠀⠀⢣⠈⠳⣤⣀⠀⣀⡴⠞⢁⠞⠀⠀⠀⠀⠀⠘⣻⠄⢖⡋⠀⠀⠀⠀⠀⠘⣔⢿⣄⠀⠀⠀⠀⣠⡼⢂⠎
⠀⠀⠀⢑⢦⣄⣉⣀⣁⣠⡴⠁⠀⠀⠀⠀⠀⣴⡿⢓⠀⢐⠻⡷⡄⠀⠀⠀⠀⠈⠲⣍⣛⠃⠐⢛⣩⠔⠁
⠀⠀⠀⡜⣾⣿⣿⠛⣿⣿⣷⢣⠀⠀⠀⠀⢸⣝⠐⠀⠀⠀⠑⢸⣽⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⣿⣷
⠀⠀⢰⠽⠻⣿⠏⠀⢻⣿⠛⠏⡆⠀⠀⠀⠸⣻⡔⢀⠀⢀⠔⣸⡻⠀⠀⠀⠀⢠⣾⠿⣿⠏⠸⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠑⠿⣶⣀⣲⡾⠗⠁⠀⠀⠀⠀⠈⠀⠀⠛⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠠⡾⠋⠡⠁⢈⢹⠆⠀⠀⠀⠀⣿⣿⢻⠃⠀⠀⠀⠀⠀⠀⡇⢻⣿
⠀⠀⢰⡦⠤⠴⢶⡆⠀⠀⡘⠀⠈⠈⠉⠙⢇⠀⠀⠀⠀⠀⠀⣩⣗⡢⢀⡝⣈⠔⠲⣄⠀⠀⣿⣿⣨⠇⠀⠀⠀⠀⠀⠀⡇⢸⣿
⠀⠀⢸⡇⠂⠺⢻⡇⠀⠀⠁⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢠⠾⠀⣠⡶⢻⡃⠸⣿⠟⠺⠄⠀⡏⠛⠸⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿
⠀⠀⠘⣇⠀⠀⣸⠁⠀⠸⣄⡀⠀⠀⣀⣀⣸⠃⠀⠀⢠⡏⡀⠚⠿⠁⠘⣿⣶⢿⣁⠌⠀⠀⡄⢸⢸⠀⠀⠀⠀⠀⠀⣠⣧⣼⣿⣄
⠀⠀⠀⠙⢷⣾⠃⠀⠀⠀⠙⠷⠶⣶⣾⡿⠋⠀⠀⠀⠸⠏⠀⠀⠀⠀⠒⠈⢳⠴⠋⠀⠀⠀⡅⢸⠠⠀⠀⠀⠀⢀⡾⠋⢉⣿⣿⡟⣳⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠲⠂⠀⢀⠒⢋⣦⡀⠀⠀⠀⠀⣠⡴⢴⡦⢔⠀⠀⠀⠀⡇⣿⠘⡀⠀⠀⠀⢸⠁⠀⣼⣿⣿⡇⣹⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠢⢾⠉⢙⠇⠀⠀⢤⣐⢃⠎⢹⡖⠁⠀⠀⣸⠷⢿⠔⢧⠀⠀⠀⢸⠀⠀⠻⠿⣿⡗⠛⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢹⡆⢱⡞⢀⠀⠀⢰⣇⠀⢠⠀⠈⣆⠀⠀⢸⠀⠀⠀⠀⣾⠀⠀⣿
⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠰⢿⣭⣷⠂⠀⠀⠀⠀⠀⠠⣤⡄⠀⠁⠀⠈⠀⠀⠀⢠⣯⣿⡆⢸⡇⠀⢹⣆⠀⢸⠀⠀⠀⠀⣙⠀⠀⣿
⠀⠀⠀⠘⠯⠿⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣠⠀⣚⣫⡽⣧⡆⡀⠀⠀⠀⠀⣸⣿⣇⣙⣿⡏⠀⠈⣿⠀⢸⠀⠀⠀⠀⢩⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⠴⣚⡿⡬⢇⣰⡼⡵⢫⠇⠀⠀⠀⣿⡿⠉⠀⢸⡏⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡄⢐⡆⠀⠠⠼⠃⠀⠀⠀⠀⢰⣟⣺⣽⡫⢞⣸⢛⣿⣉⣻⢤⣟⢷⣀⠀⠀⠀⣿⣇⠀⠀⢸⡇⠀⠀⣿⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣩⠗⢣⣎⣸⣶⣾⣉⣻⡷⣦⣉⣗⣆⠮⣭⡶⠀⠀⣿⣿⠦⣤⣼⡇⠀⠀⣿⠀⢸⡇⠀⢺⣿⣿⡟⣿⢷
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣛⡶⠷⠳⢲⡳⢾⢭⠷⣿⣧⡶⠏⢿⣭⡮⢽⣒⡿⠿⠧⠀⣿⣿⣠⣴⣾⡷⣄⣰⣿⠀⢸⡇⠀⢸⣿⣿⣇⢿⣿
⠀⠀⠀⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠁⠀⠮⠗⠑⠚⠣⣇⣶⣷⠞⠛⠙⠻⠦⠀⠀⢿⣿⣿⣿⣼⡇⠈⢻⣿⠀⢸⣇⣀⣸⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠓⠒⠛⠉⠀⠀⠉⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠊⠥⣄⠀⠀⠀⢠⡄⠀⠀⢙⣀⢐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠢⠠⣤⠀⠀⠀⠀⠀⠰⠚⢽⠀⠀⠀⢀⠄⠐⡀⠀⠈⠀⠙⢀⡀⡠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡀⣐⢾⠁⢀⣀⠐⡀⠀⠀⣀⡠⡄⠈⠣⡀⠀⢐⡄⡠⠃⠀⠀⠈⠂⢀⢄⡢⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠞⠁⠉⠃⠈⡃⣔⠤⠓⠀⡕⢔⡕⠂⠀⡇⣠⠜⢅⡂⠀⠀⠀⠘⠊⠉⢪⠀⠀⠀⠐
⠀⠀⠀⠁⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠋⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢀⣤⣒⠲⢤⡀⠀⠐
⠀⠀⠐⠀⠀⡰⡩⠖⢮⠱⣀⡀⡒⢒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢀⠀⠀⠄⠈⠐⣀⣠⡗⠁⠀⠈⢂⢱⡀⠀⠂
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠓⠛⠛⢊⢈⠀⠀⠀⠒⠀⠐⠐⠒⢃⢐⢂⠂⠀⠀⢀⠒⠚⠉⠉⠁⠀⠀⠀⢸⢀⡇⠀⠒
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢊⠔⠁⡠⠪⠀⠀⠀⠀⠀⠀⠀⡌⣸⠁
⠀⠀⠀⠀⠀⠔⠇⠀⠀⠀⠀⠈⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢂⠕⡠⢊⠄⠔⣡⠀⠀⠀⠀⠀⡐⢠⠃
⠀⠀⠀⠀⠀⠐⡹⡀⠀⠀⠀⠀⠁⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⡡⢘⠅⡡⠊⠅⠀⠀⠀⢀⠜⡰⠁
⠀⠀⠀⠀⠀⠀⠘⡳⣄⠀⠀⠀⠀⡄⠀⡀⠀⠂⠀⠀⠀⠀⠀⠤⡊⢐⠅⡨⠂⠍⠀⠀⠀⡠⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠺⡧⣄⠀⠀⠘⢄⠀⢀⠀⠀⠀⠀⠀⠀⡨⢀⠅⡪⢂⠎⠀⢀⡠⣊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢕⡢⣄⡈⠣⡀⠀⠀⠁⠀⠀⠠⣢⡴⣛⣖⣡⣤⡶⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠻⠻⠷⢤⡀⠀⡠⢞⡩⠖⠟⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠀⡚⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⡚⡺⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠃⠓⠰⠆⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠈⠌⡈⡀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⣆⣀⣀⣀⣀⣐⣷⣶⣤⣶⣶⣶⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡴⠤⠦⠤⠤⠤⠤⠤⠼⠤⠬⠬⠥⣌⢊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣴⡷⢝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⡠⠀⣄⣀⣀⣈⡉⢀⣉⢉⢋⢉⢫⡋⡓⣪⢊⣵⣡⣇⡀⣀⠀⡀⢀⠀⡀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠈⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠁⠉⠀⠁⠊⠁⠁⠈⠉⠁⠉⠈⠈⠈⠁⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿
⣻⣿⣿⣿⣽⣟⠾⣭⣭⡿⢟⣿⣟⠶⣿⣽⡷⣿⣿⣿⣻⣿⣿⣿⣽⣟⠾⣭⣭⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠯⣭⣯⢿⣿⣟⢮⣭⣭⣿⢟⣿⣿⢿⣿⣿⣿
⢾⣿⣿⠿⡋⢙⠿⣳⣶⣻⢿⡋⣛⢟⣿⣾⡻⠟⠉⠻⢷⣿⣿⠿⠋⠙⠿⣳⣶⣺⠿⠋⠻⢟⣾⣿⡿⠟⠉⠻⢿⣿⣿⠿⡋⢙⡿⣶⣶⡾⢟⡉⣻⢟⣶⣾⣻
⣷⢰⣿⣿⡇⣾⣿⢰⣿⢰⣿⢠⡿⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣿⠘⣿⡐⣿⡐⣿⡇⢹⣿⣷⠀⣿
⡏⢸⣿⣿⢁⣿⡇⣾⡇⣾⠇⣼⢣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠸⣇⢻⣇⢻⣇⢻⣿⠸⣿⣿⡄⣿
⡇⣸⣿⡟⣸⣿⢰⡿⣰⡟⣸⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢻⡌⣿⡜⣿⡘⣿⡆⣿⣿⡇⢻
⠃⣿⣿⢁⣿⢇⣿⢣⡿⢡⡟⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢿⡜⣷⡹⣧⢹⣷⠸⣿⣧⢸
⢸⣿⡏⣼⠏⣾⢣⡿⣱⠟⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⢻⡜⣷⡹⣧⢻⣇⢻⣿⡘
⣾⡟⣼⢏⡾⣣⡟⣵⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡻⣌⢷⡽⣧⢻⣎⢿⣇
⣟⣼⣫⣞⡵⣫⣾⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡻⣾⢷⡻⣎⢿
⣾⣿⡿⣫⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⣷⢿⣾⣿
⣛⡹⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣝⣛
⣭⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣽
⣿⣏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿
⣹⣽⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣹⢿⢿
⣇⣧⣿⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⡟⡿⣾
⣿⢸⡸⡞⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣸⢱⢇⣿
⣿⡎⡇⣷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡟⣼⢸⡿
⢿⡇⣿⢹⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢹⡇⡏⣾⡇
⢸⣿⢸⢸⡟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⢸⡇⣿⡇
⢸⣿⢸⣇⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣸⢠⣿⢰
⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠐⠒⠐⠒⠤⢠⢀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣴⣿⡷⠾⢗⠒⣀⠠⠀⣀⠀⠈⠘⢷⣄
⠀⠀⠀⠀⠄⠂⢀⠀⠀⠀⠀⢰⣿⠟⠑⠈⡀⠀⠀⡀⠀⠀⠀⠀⠢⣄⠀⠙⢗
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣉⢉⣀⠀⠶⡾⠿⠿⣶⣥⡢⡀⠀⠀⠪⠑⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠈⠠⡠⠊⢁⠔⠳⢄⠈⠑⢄⠀⠁⠂⢍⠻⠦⠀⠀⠀⠀⢎⠀⠀⢆
⠀⠀⠀⠈⠀⠀⠰⡝⢀⠔⠃⠤⣶⣄⡑⠀⠀⠀⠀⠀⠀⠑⢄⠈⠀⠀⠀⠀⠢⠀⠈⠂⠀⠀⢀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣿⣷⣦⡙⣿⠀⠀⠠⠀⢀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢌⠻⠿⠟⣡⠞⠁⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠑⢤⠞⠁⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠞⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠦⠊⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣄⠅⠉⠀⠀⠀⠠⠙⡄⠀⠀⠖⠙⢴⠈⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠂⢠⣉⡭⠋⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠠⡄⠐⠂⠀⠘⠀⢠⡄⠀⠀⠐⢖⠒⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⡠⠼⣷⡀⢲⠄⢈⠠⣄⠀⠰⡆⠺⣄⠀⠈⢦⠀⡀⢾⢵⠉⠀⠀⠀⢀⣈⡠⡞⢶⠛⠁⠀⠀⡀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡥⠯⣒⠁⠀⠰⠕⢵⠋⠁⠀⢸⡨⠚⠑⢦⡇⠀⠀⠀⠈⠁⠀⠈⠾⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⣒⡒⢦⡀⠀⠈
⠀⠀⠀⠁⠀⠀⠞⡱⠒⠢⡙⣆⣀⠊⡉⣉⠉⠉⠉⠀⠉⠀⠀⠀⠀⡈⠈⠀⠁⠠⠠⢠⠈⠄⠑⣀⣠⠿⠉⠀⠀⠈⢆⢳⠀⠀⠉
⠀⠀⠄⠀⠀⢸⣸⠀⠀⠀⠒⠚⠛⠒⠐⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⡓⢒⠒⠂⠀⠂⠐⠐⠒⠚⠉⠉⠁⠀⠀⠀⠀⢻⢸⠁⠀⠒
⠀⠀⠄⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⡄⢈⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⢊⠔⡡⢀⠔⡡⣂⢸⠀⠀⠀⠀⠀⠀⠀⡌⢸
⠀⠀⠀⠀⠀⠘⢸⡀⠀⠀⠀⠀⠀⠃⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⢊⠔⡡⢊⠔⡁⠊⠀⠀⠀⠀⠀⠀⡘⢠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠡⢇⠀⠀⠀⠀⠀⠸⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢊⠴⢈⠔⡡⢊⠔⠃⠀⠀⠀⠀⢀⠜⣠⠏
⠀⠀⠀⠀⠀⠀⠐⢟⣆⠀⠀⠀⠀⠀⢆⠀⡊⠀⠢⠀⠀⠀⠀⠀⠀⠀⠐⡡⢚⡡⢊⠔⡡⠆⠀⠀⠀⠀⣠⢊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢮⡷⡀⠀⠀⠀⠈⡆⠀⠆⠀⠑⠀⠀⠀⠀⠀⠀⣫⢔⡡⢊⠄⡊⡐⠁⠀⠀⢀⠔⣠⠊⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡢⣄⠀⠀⠈⢀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠔⡡⢂⠕⣪⠊⠀⢀⡠⢐⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢝⣢⢤⣀⡑⣄⠀⠂⠀⠀⠀⠀⢐⣡⡺⣿⣭⡥⣴⣶⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠷⠦⣄⠀⢀⠬⢓⡥⠚⠛⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠕⢐⡺⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⡰⠖⣟⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠸⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠢⢒⠔⠉⡑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡁⠀⠀⠀⠀⠀⢀⠕⡈⠢⢀⢀⢈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠺⠿⠧⠄⠾⠋⠭⠤⠿⠿⢿⠿⠿⠿⠿⠿⠧⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠟⠛⠛⠒⠒⠒⠒⠒⠒⠚⠛⠚⠚⠛⢺⠌⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢌⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣿⣯⣺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⣀⠸⡀⠀⠈⠀⠀⠀⠰⠛⠛⠃⠚⢒⢂⠶⠖⡷⢋⡞⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠙⠉⠛⠐⠛⠓⠓⠓⠓⠛⠟⠺⠞⠟⠛⠛⠟⠩⠓⠩⠂⠩⠒⠝⠊⠔⠚⠊⠓⠉⠊⠂⠒⠑

⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠀⠠⠰⠀⠀⠀⠀⠖⠀⠀⠀⠰⠀⠀⠀⠀⠆⠀⠀⠀⠰⠀⠀⠀⠀⠆⠀⠀⠀⠒⠀⠀⠀⠀⠂⠀⠀⠀⠒
⠀⠀⠈⣤⣤⣤⣤⣤⡤⢭⣭⣭⣭⢭⣭⠭⢭⡭⢭⠥⠭⣤⡬⣭⠭⢭⣭⣭⣭⣭⣭⣬⡬⠭⠥⢬⣬⠭⠭⠭⢭⡭⠭⠭⠭⡽⠭⠯⠭⣤⡬⠥⠤⢤⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⣀⣲⣐⣒⣒⣶⣖⣒⣂⣲⣒⣒⣂⣲⣖⣒⣂⣲⣖⣒⣂⣰⣆⣒⣒⣲⣖⣒⣒⣲⣖⣐⣒⣲⣖⣐⣒⣲⣖⣒⣒⣲⣖⣐⣒⣲⣶⣐⣒⣒⣶⣂⣒⣂⣶⣀
⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠹⡟⠋⠉⡉⢙⠉⢫⢙⢉⠉⠉⠉⠉⠉⠋⠙⢉⣍⡙⠙⡙⣋⠉⠙⢻⡏⡹⠟⠛⠉⣉⢋⠉⢋⣩⣉⠫⠙⠉⠉⠉⠙⠉⠉⣛⠝⠉⡋⢉⠉⠉⢻⠏
⠀⠀⠀⠫⠱⠼⠧⠤⠷⠷⠬⠷⠾⠾⠷⠾⠾⠶⠶⠾⠿⠶⠶⠷⠾⠶⠷⠾⠴⠤⠶⠾⠶⠷⠶⠷⠶⠿⠷⠶⠶⠶⠶⠾⠿⠾⠾⠥⠾⠾⠤⠼⠥⠿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⢸⣷⡄⠆⣰⣤⠬⠒⠂⣁⣹⣿⣃⠀⠒⠥⣐⣆⠠⢀⡆⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢
⠀⠀⠀⠀⠀⡇⠨⢯⠉⠋⢵⠀⣄⠄⠃⠘⠀⡇⢸⡧⢀⣶⠛⣡⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⡈⠑⢖⠀⡆⠀⢸⠀⠃⠘⠀⢠⠀⢮⠙⡍⡭⠇
⠀⠀⠀⠀⠀⣧⠀⠃⠄⠠⠠⠀⢀⠀⠒⢤⠀⡇⢸⢠⡾⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⢢⡀⠀⢸⠀⣤⠐⠀⠈⠠⠤⠄⠰⠸⠇
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⢡⣿⣿⣿⣿⣿⡿⠿⠻⠟⠿⢿⣿⣿⣿⣿⣇⠈⡆⠀⠈⠂
⠀⠀⣀⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣸⣏⠸⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⠀⣩⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀
⠀⠀⠙⡟⠟⡛⣛⠛⡛⠛⢻⢻⢿⣻⣻⣟⣟⣟⣿⡟⢟⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠳⠛⣛⣿⣛⣻⣻⣿⠻⡟⡟⡟⡛⢛⠻⢻⠛⢻⠋
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠀⡀⠐⣶⡾⡔⢀⣴⡶⠀⠀⠀⢸⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠀⠀⠀⠀⠀⠀⢲⣦⣀⢡⣯⣶⠂⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⢢⣴⣻⡟⠠⢞⠛⠟⠀⠀⠀⢸⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠿⠛⣣⠄⠺⣟⣦⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣏⣩⢘⣄⠸⢯⠄⠀⠀⠀⢸⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⡹⠇⢠⡇⣌⣹⡃
⠀⠀⠀⠀⠀⠀⠀⣷⢽⡏⢹⠋⢠⠈⠂⠀⠀⠀⢸⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠉⣄⠙⡏⢹⡯⣾
⠀⠀⠀⠀⠀⠀⠸⡿⢻⡇⠠⡄⠩⡴⠶⠆⠀⠀⢸⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠶⢦⠌⢀⡅⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣯⣸⡻⣷⣝⣿⣀⡃⠀⠀⠀⣼⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢀⣾⣹⣸⡷⣇⣽
⠀⠀⠀⠀⠀⠀⣠⣏⣀⣉⣁⣀⣉⣻⣧⠀⣀⣀⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠨⠭⠀⠈⠉⠀⣹⣄
⠀⠀⠧⠀⠉⠉⢹⠀⠀⠁⠀⠉⠉⢹⣿⠛⠉⠉⣿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠙⣿⠿⠉⠁⠼
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢻⡄⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠆
⢀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣉⣙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⢿⣶⣝⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣫⣶⡿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣤⣴⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣦⣤⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠈⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠁⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⣀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣾⣿⣷⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣾⣿⣷⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠉⠹⣇⠀⠀⠀⠀⣸⠏⠉⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠹⣆⠀⠀⣰⠏⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠁⠈⠳⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⡤⠴⢤⡀⠀⣄⢦⠀⠀⠀⠀⠀⣠⠴⠦⣄⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⢸⢸⢸⠀⠀⠀⠀⠠⡇⠀⠀⠈⠕⡎⡇⠀⠀⠀⠀⠀⢾⣿⣿⣿⠷⢰⢸
⠀⠀⠀⠀⠀⢀⣈⡓⠦⠖⣋⣀⠈⠚⠀⠀⠀⠀⣀⣙⠦⠴⢚⣀⡑⠃⠀⠀⠀⠀⢀⣈⠻⠿⠟⣁⡈⠘
⠀⠀⠀⠀⢠⡏⠀⠉⠉⠉⠁⠈⣧⠀⠀⠀⠀⣼⠁⠉⠉⠉⠉⠀⢹⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠉⠓⠶⠤⠴⠖⠋⠁⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠛⠋


⣿⣿⠛⢿⣛⠿⣿⣻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⣴⠒⠒⠒⠒⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠉⠳⠋⠙⠓⢲⠆
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠋⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠶⢤⡄⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢀⠀⠓⢤⠄
⠀⠀⠀⠀⠀⠀⠁⡄⠐⠂⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠌⠀⠠⠤⠄⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢨⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠱⣄⣀⠀⠉⠊⠑⠒⢳⠀⣗⢒⠒⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠦⡀⢠⠀⠀⠘⠁⠉⠀⠀⠠⢊⡤⠖⠒⠊⠉⠻⣀⠇⠀⠀⠉⠉⠒⠦⢄⣀⠛⢄⠀⣀⠀⠀⠐⢀⠀⠀⠀⢙⠲
⠀⠀⠀⠒⠀⠀⠀⢅⠀⠀⠀⣠⠔⠊⡵⠛⠉⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠨⡁⠀⠀⠘⠁⠀⠀⢀⣨⠄⠒⠠⡄
⠀⠀⠀⠀⠰⠎⠀⠈⠀⠀⠀⠀⢀⠞⠀⠐⠀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠱⡚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⠈⢉⡏
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢒⠼⡆⢈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡆⠚⠀⠁⠀⠀⠀⢠⢤
⠀⠀⠀⠐⠀⡏⠀⠀⠀⠐⠓⣤⠘⠢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠀⢀⠀⠠⠂⠀⠓⡲
⠲⠦⠤⠀⠀⠥⠦⠀⠀⠀⠀⡏⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠠⠎⠙⡰⠂⠀⠀⠀⠀⡠⠄⣴
⠀⠀⠀⠰⠃⠀⠀⠀⠀⠲⢀⡇⡀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⢠⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠂⠈⠉⢉⠝⠀⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⣈⣧⠀⠁⢀⠀⠑⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡐⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢨⢦⡐⢄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡀⠀⠀⠀⢀⡠⢊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⠦⣑⠢⠄⡀⠀⠀⢀⠀⠀⠀⠀⣀⠠⠀⠀⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⣤⣉⣈⣤⡉⠉⠉⢁⣀⣠⠴⠚⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠁⠀⡠⠋⠀⠀⠋⠉⠁⠀⠀⠰⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠸⡄⠀⠀⠀⠀⠀⠐⠂⠁⠀⠄
⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣈⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⡶⠂⠀⠀⠀⠀⢠⡿⠟⠛⠛⠿⣿⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⠀⣾⣿⣿⣿⣿⣦⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠘⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⡿⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠈⠛⠿⠿⠋⠀⠀⠈⠛⠿⠿⠛⠁⠀⠈⠛⠿⠿⠛⠁⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣶⣶⣶⣶⣶⣶⣄⠐⣶⣶⣶⣶⣶⣆⠐⣶⣶⣶⣶⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠟⠛⠛⠛⠻⠿⢿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢐⣤⡶⠿⠛⠛⠛⠛⠛⠿⢶⣤⡂⢀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⡀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⠂⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⢈⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡣⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠈⠛⠷⣶⣤⣤⣤⣤⣤⣶⠾⠛⠑⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠙⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠙⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠈⠻⠃

⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠘⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣖⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢻⣵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠈⢻⣿⣭⣍⣻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢃⠀⠙⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣆⠀⠈⠻⣿⣿⣿⣿⣿⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡄⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡝⣦⡀⠀⠈⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡳⣄⠀⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣀⡈⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠚⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣿⠉⠉⢹⠉⠉⢹⡏⠉⢹⡏⠉⠉⡏⠉⠉⡏⠉⠉⣿⠉⠉⣿⠉⠉⢹⠉⠉⢹⡋⠉⢹⡏⠉⢹⡏⠉⠉⡏⠉⠉⣿
⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⡇⠀⠀⣿⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⠀⣿
⠀⠀⣿⠀⠀⢸⣀⣀⣸⡇⠀⢸⣇⣀⣀⡇⠀⠀⣇⣀⣀⣿⠀⠀⣿⣀⣀⣸⠀⠀⢸⣃⣀⣸⡇⠀⢸⣇⣀⣀⡇⠀⠀⣿
⠀⠀⢻⣄⣀⣼⠋⠉⠹⣇⣀⣸⠏⠉⠙⣧⣀⣠⡟⠉⠉⢿⣀⣀⡿⠉⠉⢻⣄⣀⣼⠋⠉⠹⣇⣀⣸⠏⠉⠙⣧⣀⣠⡟
⠀⠀⣾⠈⠉⣦⠀⠀⠀⠈⢉⡅⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢨⡉⠁⠀⠀⠀⣴⠉⠁⣷
⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⣿⠀⠀⡇⠀⡖⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢲⠀⢸⠀⠀⣿
⠀⠀⣿⠀⢸⡇⢠⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡄⢸⡇⠀⣿
⠀⠀⣿⠀⣼⠁⣼⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣧⠈⣧⠀⣿
⠀⠀⣿⢀⡟⢰⠏⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠹⡆⢻⡀⣿
⠀⠀⣿⣸⢣⣏⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣜⣇⣿
⠀⠀⣿⣉⣉⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣿
⠀⠀⣿⠀⣧⢿⡻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⡿⣼⠀⣿
⠀⠀⣿⠀⢹⡎⢷⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡾⢱⡏⠀⣿
⠀⠀⣿⠀⠈⡇⢸⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⡇⢸⠁⠀⣿
⠀⠀⣿⠀⠀⡇⠈⠁⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠁⢸⠀⠀⣿
⠀⠀⣿⣀⣀⣃⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣘⣀⣀⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣿⣷⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣵⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣛⠿⢿⣿⣿⣿⣯⣭⡵⠶⢛⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣤⣄⠀⢤⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣉⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠶⠒⠋⠐⠒⠂⣨⡿⠷⠖⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠶⠶⠶⠶⠶⠖⠛⠛⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⢦⣤⠞⠲⡴⠛⢲⡴⠛⢦⡤⢦⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡞⠉⠹⡁⠀⢹⠀⠀⠇⠀⢸⠀⠀⢸⠀⠀⡝⠉⠹⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠺⡁⠀⠘⡀⠀⢡⠀⠐⠂⠀⢀⠀⢸⠀⠀⡇⠀⠰⠀⠀⡜⠀⢀⡿⠒⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠻⣄⠀⠘⢄⠀⠹⡀⠀⠆⠀⢁⠀⣸⣀⣈⡀⣀⠁⢠⠃⠀⡘⠀⢀⠎⠀⢀⠜⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢴⠛⠛⢄⠀⠈⠢⡀⠈⢂⠀⣡⠤⠚⠂⠉⠁⠀⠀⠀⠀⠈⠉⠙⠒⠶⣀⡠⠂⠀⡰⠋⠀⣠⠞⠛⢲
⠀⠀⠀⠀⠀⢠⠖⠛⠦⡀⠀⠑⢦⡀⠘⣤⠔⠋⠀⠀⣀⠤⠔⠒⠒⠒⠒⠐⠒⠢⠤⣀⠀⠀⠉⠢⣎⠀⡠⠚⠁⢀⡠⠚⠓⣦
⠀⠀⠀⠀⣠⠼⠧⣄⠀⠈⠓⢄⡀⢁⠞⠁⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠈⠻⣄⡀⠔⠉⠀⣀⡤⠷⢤
⠀⠀⠀⢀⣹⣄⠀⠀⠉⠒⠄⡀⡸⠁⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⢢⡠⠔⠊⠀⠀⣀⣼⡀
⠀⠀⠀⢻⡀⠀⠉⠑⠂⠄⠀⡜⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢣⡠⠄⠒⠉⠀⠀⣸
⠀⠀⢰⠋⠉⠁⠒⠂⠤⠄⣸⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⠄⠒⠊⠉⠙⣦
⠀⠀⣠⠗⠲⠤⠄⠀⠤⢀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⡄⠄⠠⠤⠤⠲⢧
⠀⠀⠹⣄⣀⣀⣀⣀⣀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⡜⠁
⠀⠀⢾⡀⠀⠀⢀⣀⠀⠠⡆⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣀⠀⠀⠀⣹⠆
⠀⠀⢠⠟⠉⠁⠀⠀⠀⣀⣧⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠉⠙⣇
⠀⠀⠈⢳⠶⠒⠂⠉⠁⢀⠸⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠉⠐⠒⠲⢶⠋
⠀⠀⠀⠻⣄⡠⠤⠒⠉⠀⢀⠰⡄⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡎⠀⠉⠐⠢⢄⣀⠼⠂
⠀⠀⠀⠀⢼⡀⢀⡠⠔⠊⠁⢀⠜⢆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢇⠀⠉⠲⠤⣀⠀⣹
⠀⠀⠀⠀⠀⢸⡃⠀⢀⡤⠊⠁⢀⠔⠑⣄⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⡠⠞⢅⠀⠙⠢⣀⠀⠈⣏
⠀⠀⠀⠀⠀⠈⢙⡿⠁⠀⣠⠞⠁⢀⠔⠁⢙⠦⢄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠊⠑⣄⠀⠑⢄⡀⠈⢹⠻⡅
⠀⠀⠀⠀⠀⣠⠏⡸⠒⢾⠁⠀⡰⠊⠀⡠⠁⢀⡎⠈⠱⠒⠤⠤⣀⢀⣀⡀⢤⠤⢲⠊⠙⡄⠀⢢⡀⠈⠢⡀⠀⢹⠖⠛⢆⠙⣄
⠀⠀⠀⠀⡴⠁⡜⠁⠀⠘⠓⢺⠀⢀⡜⠁⠀⡔⠀⢠⠁⠀⠆⠀⢸⠀⢰⠀⠀⡄⠀⢆⠀⠸⡄⠀⠱⡄⠀⢹⠒⠛⠀⠀⠀⠣⡈⢣⡀
⠀⠀⢀⡞⢀⠌⠀⠀⠀⠀⠀⠈⠉⠹⣄⢀⡜⠀⠀⠎⠀⢠⠃⠀⡀⠀⢸⠀⠀⢁⠀⠘⡄⠀⠹⣄⢀⡟⠉⠉⠀⠀⠀⠀⠀⠀⠱⡀⠱⡀
⠀⢀⡎⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠴⠺⣄⣀⣼⠀⢀⣇⠀⢘⡀⠀⣸⣄⣀⠖⠢⠞⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢹⡄
⠀⡼⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠝⠋⣨⠖⠻⣌⠛⠣⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢣
⢠⠃⡘⠀⠀⠀⣀⣠⠤⠶⠒⠛⠛⡆⠀⠀⠀⠀⠀⠀⢀⠊⠀⡴⠁⠀⠀⠈⢧⡀⠑⡄⠀⠀⠀⠀⠀⠀⢠⡟⠛⠒⠶⠤⢄⣀⠀⠀⠀⢣⠈
⣘⣠⡤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⠃⢀⠞⠀⠀⠀⠀⠀⠀⠳⡄⠈⢆⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣄
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢠⠃⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢆⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⠇⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠘⡆⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⡎⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠏

⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢐⡀
⠀⠀⠀⠀⠀⢀⠀⡀⡇⣿⣾⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⣾⡇⡇⡀⢀
⠀⠀⠀⠀⠀⣸⢶⣳⣼⡛⠭⠽⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠺⠭⢝⣻⣴⣳⢺⡀
⠀⠀⠀⢀⠀⡘⢼⡿⠓⣯⠄⠀⠀⠀⠀⠐⢞⠋⠉⣀⣀⣀⣀⣀⡈⠉⠛⠖⠀⠀⠀⠀⠀⢬⡓⠻⣿⠜⡀⢀
⠀⠀⠀⢰⠺⢿⣿⣶⠎⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠉⠀⠈⠉⠙⠓⢦⣄⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⠿⢸
⠀⠀⡀⢠⡁⡿⠫⠝⠋⠀⠀⣠⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢴⡀⠀⠈⠛⠭⠻⡗⣡⠀⡀
⠀⠀⢹⠺⣾⣯⠿⡥⠀⠀⣰⡇⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⣷⡀⠀⠠⡽⢯⣿⡾⢺⠁
⠀⠀⠀⣕⣺⣇⣪⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢩⣂⣿⣂⡅
⠀⠈⢲⠻⣽⣷⢛⠏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢟⢳⣿⡽⢳⠋
⠀⠀⠀⢑⣘⣟⢵⣂⠄⠀⣀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡀⠀⢄⣲⢍⣟⣐⠁
⠀⠀⠐⣢⠳⢿⣝⡹⡁⠀⠸⣆⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⢀⡾⠀⠀⡹⣙⣽⠷⢫⠐
⠀⠀⠀⠀⠩⣬⢿⣞⡷⣇⠀⠙⠁⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠙⠁⢀⡗⣟⣾⢯⡬⠁
⠀⠀⠀⠀⠒⣞⡛⢻⣧⣱⣤⡗⠀⠀⠀⠀⠙⠷⢤⣄⣀⣀⣀⣠⣤⠴⠟⠁⠀⠀⠀⠐⣧⣴⣡⣿⠛⣛⠖⠂
⠀⠀⠀⠀⠀⠀⠉⠴⣞⢷⣟⡈⣄⡔⠀⠐⠶⣤⣀⠀⠈⠉⠉⠀⢀⣠⡴⠖⠀⠐⣄⡌⣈⣷⢟⡶⠌⠁
⠀⠀⠀⠀⠀⠀⠈⠈⠣⠤⣋⠷⣮⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣯⡶⢏⡣⠤⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢍⡉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣉⡯⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣤⣤⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣿⡿⠻⣿⣷⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣡⣬⣿⣿⣶⣶⣾⣿⣯⣤⣸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣀⣴⣿⣿⠿⢛⣋⣭⣭⣭⣭⣍⣛⠻⠿⣿⣷⣄⣸⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⢛⣿⣿⠟⣩⣴⣾⣿⠿⠟⠛⠛⠛⠿⠿⣿⣶⣬⡙⢿⣿⣿⠛⢻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣴⣿⡿⣡⣾⣿⠟⠉⠀⠀⢀⣴⠞⠛⢻⡆⠈⠙⠿⣿⣦⡙⣿⣷⣼⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣻⣿⡟⣰⣿⡟⠁⠀⠀⣶⠛⠋⠁⠀⠀⢸⡇⠀⠀⠀⠘⢿⣷⡌⢿⣿⡛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣿⡿⢰⣿⡟⠀⠀⠀⠀⠿⠤⣤⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⣷⠸⣿⣧⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢻⣿⡇⣼⣿⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⣿⡇⣿⣿⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣸⣿⡇⢿⣿⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣷⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⢡⣿⡿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣄⣻⣿⣆⢻⣿⣆⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⢀⣾⣿⢇⣾⣿⣃⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⠿⣿⣦⠹⣿⣷⣄⠀⠀⠀⠸⠦⠤⠤⠼⠇⠀⢀⣴⣿⡿⢃⣾⣿⢿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣙⣿⣷⣌⡛⢿⣿⣶⣤⣤⣀⣀⣠⣤⣴⣾⣿⠿⢋⣴⣿⣿⣁⣸⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡙⢿⣿⣶⣬⣙⡛⠻⠿⠿⠿⠛⣛⣩⣴⣾⣿⠟⢻⣿⡿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⡿⠿⢻⣿⣷⣶⣾⣿⠃⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠹⣿⣷⣾⣿⣿⣦⣠⣾⣿⣿⣶⣿⣿⠉⠉⠉⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠙⠛⠉⠀⠘⠿⢿⡿⣧⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠚⠛⠛⠙⢧⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⢠⡟⠙⠛⠓⠲⠶⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢻⡄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡤⠀⢤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠋⠉⠉⠉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠙⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⢤⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡄⠀⠀⣀⣈⣻⣋⣀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⠀⠘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣿⣃⡀⠀⣿⣿⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⣬⣽⣥⡄⣿⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣳⣶⣶⡰⠶⢶⠿⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⡇⣶⣶⡆⠸⠿⠇⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠻⠿⠿⠿⠳⠿⠃⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⠇⣿⣿⡇⣿⣿⡿⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠠⣿⣷⢀⣻⣟⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢙⣿⣿⡋⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣻⣋⡰⣾⣶⡎⣿⣿⠋⠀⣠⡀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡿⣷⠿⠿⠀⠈⣻⠁⠀⠀⠀⠀⠀⢠⣄⠀⣶⣶⡄⣶⣶⢠⣶⣶⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣧⣼⡇⣿⠉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠹⣦⣭⣯⣄⣭⣭⡄⣽⣭⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⠿⠿⠷⠟⠀⠼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡇⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠦⡴⠓⠶⠋⢲⠖⠳⡤⠶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠒⠺⡁⠈⡄⠀⢁⠀⠈⠀⢸⠀⠀⠁⠀⠎⠀⡱⠒⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠁⠙⢆⠀⠡⠀⠘⠀⠈⠀⢰⠀⠸⠀⠘⠀⠈⠀⠰⠁⢀⠎⠁⢹⣤⡀
⠀⠀⠀⠀⠀⠀⣠⠴⢇⠀⠑⢄⠀⢢⠀⢁⣀⠤⠄⠒⠘⠒⠚⠀⠓⠠⠅⣀⠁⠠⠂⢀⠔⠁⢀⠷⢤⡀
⠀⠀⠀⠀⢀⡤⠶⣄⠀⠑⢄⠀⠂⡠⠗⠈⠀⢀⡀⠤⠄⠀⠀⠠⠤⢀⡀⠀⠉⠣⢤⠂⢀⠔⠁⢀⡴⠧⣄
⠀⠀⠀⢠⡴⠧⢀⠀⠑⠠⡀⡱⠊⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠑⢅⠠⠐⠁⢀⠠⠷⣄
⠀⠀⢀⡸⠦⢀⠀⠉⠂⢀⡜⠁⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠣⠐⠊⠁⢀⠠⠮⣀
⠀⠀⢈⡦⣀⣀⠈⠀⠀⡔⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢱⠀⠈⢀⣀⡠⣏
⠀⠀⢫⣀⣀⡀⠀⠁⢰⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⣀⣀⣸⠃
⠀⠐⢇⠀⠀⠀⠈⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠀⢈⠗
⠀⠠⡏⠁⠀⢀⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀⠀⠈⢉⡦
⠀⠀⣜⠉⠁⠀⢀⠀⠸⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠉⢙⡄
⠀⠀⢈⠗⠈⠉⠀⡀⠄⢣⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡀⠈⠉⠐⣏
⠀⠀⠈⢱⠖⠉⠁⣀⠀⠊⢣⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⢄⡀⠉⠑⢖⠁
⠀⠀⠀⠈⠳⡖⠉⠀⡠⠐⠁⡱⢄⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⠑⠠⡀⠈⠐⡶⠋
⠀⠀⠀⠀⠈⢳⡶⠉⠀⡠⠊⠀⠔⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠀⠈⠢⡀⠈⠳⠶⡋
⠀⠀⠀⠀⢠⠎⡹⠲⡊⠀⡠⠂⢀⠌⠀⡘⠉⠒⠢⠤⢀⠀⣀⠀⡤⠔⡆⠙⡀⠐⠄⠈⠣⡀⠈⡲⠚⢣⠑⣄
⠀⠀⠀⡰⢁⠌⠀⠀⠉⠹⣀⣠⠊⠀⡐⠀⠠⠁⢀⠀⠰⠀⢰⠀⢰⠀⢠⠀⠰⡀⠈⢆⣀⡸⠉⠁⠀⠀⠁⡈⢦
⠀⠀⡜⢀⠂⠀⠀⠀⠀⠀⠀⠘⠤⠰⡁⢀⠃⠀⡈⠀⢀⠀⢸⠀⠀⠀⠀⢆⠀⡱⠠⠎⠀⠀⠀⠀⠀⠀⠀⠐⡀⢣
⠀⡘⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠒⠳⡤⠒⣄⣴⠢⡴⠓⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢣
⢠⠁⠈⠀⠀⠀⣀⣠⡤⠤⠴⡄⠀⠀⠀⠀⠀⢀⠊⢀⠜⠁⠀⠳⡀⠑⡀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⣀⠀⠀⠀⠡
⣘⣠⡤⠖⠚⠉⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠂⣠⠋⠀⠀⠀⠀⠘⣄⠘⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢠⠁⡰⠁⠀⠀⠀⠀⠀⠀⠈⢆⠈⠄⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⡄⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⡠⠤⠒⢶⠖⢳⡄⡠⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠑⡄⠀⠀⢨⠀⢎⠀⠀⢠⠏⢠⢊⠞⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠘⡄⠀⠘⡄⢸⠀⠀⠸⣀⡷⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣀⣠⣄⣀⠀⢹⠀⣰⠁⢸⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⠤⠒⠋⠉⠀⠀⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⠀⢟⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠉⠲⢄⡠⠴⢺⠁⠀⠀⠀⠀⣠⡶⠖⠒⠀⠈⠀⠀⣠⡶⠶⠖⠀⠘⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠀⠑⢤⡚⠣⡀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣤⠞⠋⠀⠀⠀⠙⢦⡑⣄⠀⠀⠀⢠⡾⠛⠙⠁⠀⠀⢀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢸⠃⠀⠀⣴⠞⠋⠀⠀⠙⢮⡦⡀⠀⠋⠀⠀⠀⠀⠀⠒⣋⣤⠔⣛⡻⠆⠀⡇
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠈⠀⠀⠐⡏⠀⠀⠀⣰⢄⠀⠑⣞⢦⡀⠀⠀⠀⢀⡴⢟⡭⠖⠈⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⣠⡤⠔⠒⠒⠦⣄
⠀⠀⠀⡴⠃⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⢀⢀⡏⠀⢣⠀⠘⡄⠙⢦⡀⠀⢿⠞⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡴⠻⡀⠹⡄⠀⠀⠀⠀⠑⡄
⠀⢀⠎⠀⠀⠀⠀⡟⠀⠀⣠⡶⠒⠀⠀⢀⢄⡾⠁⠀⠀⡆⠀⠰⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⡘⠀⠀⠹⡄⠙⣄⠀⠀⠀⠀⢸
⠀⠘⢄⠀⠀⠀⠀⠇⠀⠀⣿⠁⠀⠀⠀⣡⠞⠁⠀⠀⢠⠁⠀⠀⡇⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠁⠀⠀⠀⠀⣠⣖⠉⡆⠀⠀⠘⣄⠘⢆⠀⠀⠀⡸
⠀⠀⠀⠉⠒⠤⣄⣀⡀⠀⠁⠀⠀⢠⣞⣁⣀⣀⣀⠴⠃⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠈⢆⠘⢆⠀⠀⠈⢦⠈⠳⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠏⠉⣑⠂⠤⢄⣀⣀⠈⠀⠈⣀⣀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠈⢢⣈⣳⣀⣀⣀⡳⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡀⡿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⣀⠀⠀⠀⠀⠀⣀⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⣄⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡤⢚⣥⡮⡓⢄⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠤⠤⠤⠀⣈⣣⠀⠀⠀⠀⠀⢀⡠⢎⠀⡠⢒⢥⡺⣟⠥⣊⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢒⡩⠋⠀⠀⢀⡠⠖⠉⠀⠈⢦⠻⣅⠛⣊⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠊⠁⡠⠖⠉⠀⠀⣠⠴⠉⠀⠀⠀⢀⡠⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠋⠀⠀⣀⠴⠊⠙⢄⠱⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁


⣿⢿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠶⠶⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠶⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠋⣁⠠⠀⠒⠒⠒⠒⠠⠄⡉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣡⢀⣀⣀⡀⣀⣀⣀⣄⡉⠳⢤⡀
⠀⠀⠀⠀⢀⡞⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢦⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡄⢾⡷⠮⠉⠀⠀⠀⠈⠹⠾⠿⠠⣄⠙⣦
⠀⠀⠀⢠⠏⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢷⡀⠀⠀⠀⠀⣰⠏⣀⠻⠿⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠻⠿⢠⠈⢧
⠀⠀⠀⡟⠀⠆⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠱⠈⣇⠀⠀⠀⢠⡏⠀⣿⡾⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠻⣾⠇⠘⡇
⠀⠀⢸⡇⢸⢀⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠾⠀⡄⢻⠀⠀⠀⢸⠁⢸⣤⡆⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣦⣾⠀⣿
⠀⠀⠸⡇⢸⢈⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢛⠁⠃⣼⠀⠀⠀⢸⡀⢨⣋⡄⠀⠀⠀⠀⣹⣿⣿⣿⣿⠀⠀⠀⠀⠀⣌⣡⠀⣿
⠀⠀⠀⢧⠀⠆⢹⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⡏⡸⢠⡏⠀⠀⠀⠘⣇⠈⠿⢧⡀⠀⠀⠀⠟⠋⠁⠉⠛⠆⠀⠀⠀⣄⠿⠋⢠⡇
⠀⠀⠀⠘⢧⠈⢆⠳⢿⣦⢀⠀⠀⠀⠀⠀⢀⡀⡔⡷⠎⡰⢁⡞⠀⠀⠀⠀⠀⠹⣆⠙⢾⡧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠟⢀⡞
⠀⠀⠀⠀⠈⠳⣄⠑⢌⡉⠼⢇⣶⠀⠀⣖⡸⠵⢉⠡⠊⣠⠏⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠲⢿⠧⢶⣤⡀⠀⠀⣠⣴⠢⠿⠷⠂⣰⠏
⠀⠀⠀⠀⠀⠀⠈⢳⢤⣈⠐⠂⠡⠤⠤⠈⠐⢊⣡⡴⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡈⠛⠛⠉⠀⠈⠙⠛⢋⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡏⣼⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣧⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣭⣛⡒⠒⠶⠖⢒⣛⣩⣵⣶
⠀⠀⠀⠀⠀⠀⡾⢰⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⡆⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣸⢃⣿⣿⣿⣿⣿⡟⠀⠀⢿⣿⣿⣿⣿⣿⡘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠀⠸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⠇⡼⠟⠉⣿⣿⡿⠁⠀⠀⠘⣿⣿⡿⠉⠛⠧⠸⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⢿⣿⣿⠃⠀⠀⢻⣿⣿⠟⠻⢿⣷
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠘⣿⠇⠀⠀⠀⠈⣿⡏⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⢠⣿⠃⠈⢻⡄⠀⠀⠀⠀⣿⣿⡷⠀⠀⠀⠀⢸⠠⠄⠀⠇⠀⠀⠀⠀⣤⣴⣦⣤⠀⠀⠀⠀⠠⠁
⠀⠐⢿⣄⢀⣾⠇⠀⠀⠀⠀⢾⢹⣧⠀⠀⠀⠀⢸⣀⣀⣀⡀⠀⠀⠀⠀⠜⢿⠿⠇⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠘⢿⡾⠏⠀⠀⠀⠀⣀⠠⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣾⣷⣤⠀⠀⠀⠀⡀⣠⢤⡀⡀
⠀⠀⢠⡞⠳⡄⠀⠀⠀⠀⣿⠿⠿⠿⣟⠀⠀⠀⢠⣯⣘⣀⣲⠀⠀⠀⠀⠑⢿⡿⠋⠀⠀⠀⠸⠿⢿⣻⠿⠿⠃
⠀⠀⠈⠛⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑

⠀⢰⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⡤⣲⣤
⠀⢸⢀⠔⠦⠞⠁⠀⠀⠀⣀⣠⣠⡆⣿⠀⠀⠀⠀⠀⣩⠁⠀⠀⠀⠀⠀⣶⣾⣿⣷⠀⠀⠀⠀⣟⠛⢅⢈⡇
⠀⠸⠤⠤⠤⠤⠄⠀⠀⠈⠋⠙⠉⠋⠙⠀⠀⠀⠀⠈⠟⠏⠀⠀⠀⠀⠀⠛⠙⠛⠋⠀⠀⠀⠀⠈⠑⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠀⠀⠀⠀⢰⣶⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⢸⢀⢀⢠⡀⠉⠀⠀⠀⡄⢾⣿⣎⠱⠀⠀⠀⠀⣴⢟⣶⡄⠀⠀⠀⠀⢠⣾⣽⣄⠀⠀⠀⠠⣾⣿⣿⣿⡏
⠀⠈⠂⠋⠈⠁⠉⠀⠀⠀⠘⠤⠛⠁⠀⠀⠀⠀⢠⣿⣕⣼⣧⠀⠀⠀⠘⠛⠘⠃⠛⠃⠀⠀⠀⠈⠻⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡈⡀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠁⠉⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠈⠈⠈⠈⠋⠀⠉⠈





⠀⠀⠀⠀⠀⠀⠀⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠉⠘⠈⠁⠓⠈⠃⠉⠳⠈⠁⠱⡋⠙⠍⡉⠘⠉⡏⠙⠈⠏⠁⡟⠙⠉⠃⠑⠉⠉
⠀⠀⠀⠀⠀⠐⣀⡖⠀⣰⠂⢠⡖⢀⣶⢀⣶⠖⣶⠢⢱⠆⠀⡶⠀⢰⠖⠀⡶⠒⣶⠒⢰⠖⢒⡶⠒⣲⠖⢒
⠀⠀⠀⠀⠀⠀⠙⢶⣶⣿⣶⣾⣷⣾⣷⣾⣷⣷⣷⣶⣾⣶⣾⣷⣶⣿⣶⣾⣷⣶⣷⣶⣶⣶⣶⣶⣶⣷⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠩⠄⠠⠤⠉⢤⠈⡄⠠⠀⠈⢡⡤⠤⣠⣞⠦⡄⠬⠦⠠⠀⣦⣤⣥⣤⡱⡤⠮⠑⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⠀⠘⣻⢟⣻⣿⣿⣿⣲⣿⣿⣿⡟⠻⡷⢀⣀⣀⣀⣀⢀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⠿⠿⢛⠉⠀⠀⣿⢎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠿⠈⡉⠟⢛⡹⣿⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠒⠛⠉⠀⠒⠀⢫⣾⣿⣿⠿⠛⠛⠛⠿⣿⣿⣿⣷⠀⠛⠛⠛⠓⠉⠸
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣿⠏⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠛⠋⠉⠉⠛⠋⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠿⠻⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠰⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠙⠻⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠆⠈⠗⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡥⠦⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣠⣠⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣡⠀⡆⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠉⠸⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⡷⡧⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢹⠇⠀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣽⠤⠤⠴⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠋⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠛

⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣷⣷⣆⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⢰⣾⣷⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⢿⣿⣿⠛⢩⣿⡍⠛⣿⣿⣿⠀⠀⢸⣿⣿⡟⡉⠒⠝⢿⣿⣿⣿⠀⠀⢸⣿⣿⢰⣖⣒⣦⢹⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⡿
⠀⠀⣸⣿⡏⠠⣿⣿⣿⠄⢹⣿⣿⠀⠀⢸⣿⣿⣧⠳⠤⢊⣾⣿⣿⣿⠀⠀⢸⣿⣿⢸⣚⣚⢟⡸⣿⣿⣿⠀⠀⢸⣿⣿⣥⠀⠀⠀⣬⣽⣿⣿
⠀⠀⣿⣿⣧⠄⠙⠿⠋⠠⣼⣿⣿⠀⠀⢸⣿⣿⣏⣄⣧⠀⢻⣿⣿⣟⠀⠀⢸⣿⣿⣘⣒⣚⠊⠁⣿⣿⣿⠀⠀⢸⣿⣿⣽⣤⣀⣠⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣾⣾⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣾⣿⣿⡿⣿⣿⣿⢿⣿⣿⣷⠀⠀⢸⣿⣿⡟⠛⣿⠛⢻⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿
⠀⠀⣿⣿⣯⣻⣿⣿⣿⣿⣽⣿⣿⠀⠀⢸⣿⣿⣿⣄⠀⣠⣿⣿⣿⣧⠀⠀⢸⣿⡟⢿⣯⠁⣹⡿⢻⣿⣿⠀⠀⢸⣿⣿⢩⠁⠀⠈⣭⣿⣿⣿
⠀⠀⣿⣿⣷⢿⣿⣿⣿⡿⣿⣿⣿⠀⠀⢸⣿⣿⡿⢉⣤⡉⢿⣿⣿⣿⠀⠀⢸⣿⡇⠂⣀⣈⣀⡐⢸⣿⣿⠀⠀⢸⣿⣿⣮⣅⠀⢀⣵⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣷⣈⠛⣁⣾⣿⣿⣿⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⠀⠀⢸⣿⣿⣿⡏⠤⢙⣿⣿⣿⣿
⠀⠀⠹⢿⣿⣿⣿⡿⣿⢿⡿⡿⠏⠀⠀⠸⠿⣿⣿⣿⡿⢿⣿⣿⡿⠏⠀⠀⠸⠿⣿⠿⠿⠿⣿⣿⣿⠿⠏⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟

⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠀⠀⢸⣿⣿⠟⢋⡵⠉⡻⣿⣿⣿
⠀⠀⣿⣟⡍⢭⣿⡉⣿⢸⢿⢿⣿⠀⠀⢸⣿⢸⣿⠟⡻⠿⢣⣤⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠟⢻⠀⢸⣿⠀⠀⢸⣿⡇⣃⢋⢴⣿⣦⢸⣿⣿
⠀⠀⣿⣗⣃⣛⣛⣓⣛⣘⣘⣻⣿⠀⠀⢸⣿⢸⣧⣾⣿⣶⣿⣿⣿⣷⠀⠀⢸⣿⠉⣿⠀⢸⠀⢸⠀⢸⣿⠀⠀⢸⣿⣧⣙⢻⣆⡹⢟⣼⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣈⣉⣉⣉⣉⣉⣉⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣷⣮⣶⣿⣿⣿⣿
⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣄⠀⣬⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠟⡋⢷⣮⢿⣿⣿⠀⠀⢸⣿⡿⠉⠁⠐⠈⠙⢛⣿⣿
⠀⠀⣻⣿⣿⡏⠠⢟⡂⢹⣿⣿⣿⠀⠀⢸⣿⡿⠏⣤⠙⢤⠹⢿⣿⣿⠀⠀⢸⣿⣏⡿⡀⠱⠆⣻⣴⣿⣿⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⢽⣿⣿⣅⣈⣛⣁⣸⣿⣿⣿⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣮⣩⣶⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣦⣤⣤⣾⣿⣿⣧
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⠂⠀⠀⠐⠆⠈⠀⠀⢀⠀⠀⣀⢀⡀⠀⢀⠀⠀⠀⡀⢸⠇⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠁⠀⢰⠀⠘⠓⠀⠀⠏⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⠄⠌⠀⠀⢲⠴⢆⠋⠙⠀⠀⠀⠀⠀⡓⠀⠈⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠀⠀⠀⡄
⠀⠀⠀⠰⠒⠒⠃⠀⠀⠀⠰⣀⠀⠾⠀⠀⠀⠃⠀⠀⠀⠠⡤⠀⢉⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠆⠤⠀⠠⠆
⠀⠀⠀⠘⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢿⠀⠀⠀⠀⠃⢀⡸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⢀⡀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡄⠀⠒⠂⠀⠀⠋⠀⠀⠀⠀⢺⠀⠀⠺⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠇⠀⠀⠔⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⠀⠀⡇
⠀⠀⠀⠃⠀⠀⠀⠀⢠⠎⣉⠛⢦⠀⠀⠁⠄⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠄⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠛⢁⡼⠀⠀⠀⠀⠁⠀⠀⢉⢊⣿⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠂⠀⡀⠀⠀⠈⠚⠐⠒⠀⠂⠂⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠹⠉⣧⠀⠀⢲⠆⠰⡶⠀⠈⠁⠈⠀⠀⠐⠀⠀⠀⠤⠤⠴⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠳⣦
⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⢸⡇⠈⡇⠀⠶⠀⠀⠀⠀⠀⣀⠤⡀⠲⠀⠀⠂⠀⠐⠖⠓⠆⠀⠀⠀⠀⠶⠄⠁⢀⠀⠀⣀⠀⢛⠀⣀⣼⠃
⠀⠀⠉⠀⠀⡀⠀⠀⣿⣻⣿⠀⠀⠀⠀⡴⠟⠻⢤⡀⠀⡀⠉⠀⠀⠀⠈⠀⠁⠀⠠⠧⡀⠀⠀⢀⣼⠁⠀⢠⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⢀⠃⠲⠈⠄
⠀⠀⠀⠀⠐⠂⠁⠀⠈⠛⠀⠀⠀⠀⠰⣇⠀⠀⠀⡇⠀⠈⠰⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⢀⡀⠀⠀⠈⢈⡀⠁⠀⠈⠀⠃⠁
⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠈⣀⠀⠀⠀⠙⠓⠒⠊⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠐⠓⠄⠀⠀⠀⡎⠈⠌⠈⡀⠀⠀⠀⢀⣛⠆⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠞⠁⡤⠆⠅⠙⠈⡂⠀⠀⠀⢾⡉⠉⣿
⠀⠀⠀⠈⠀⠀⠀⠤⠀⠀⢠⠖⠺⠻⠀⠀⠀⠀⠐⣈⠀⠀⠄⠀⠈⠀⠐⠀⠀⠂⠀⠀⠀⠀⠄⠌⠀⠀⠣⣀⠈⢂⡀⠖⣠⠐⠁⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠐⠀⠀⠒⠒⠒⠀⠀⡀⠀⡄⠀⠄⠀⠯⠀⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠐⠀⣀⣀⣀⠈⠀⠀⠄⠈⠀⠀⠀⠀⢀⠀⠠⠤⠀⠀⠀⠐⠀⠀⠀⠀⠊⠀⠀⠀⠤⠀⠤⠀⠀⠀⠀⠙⠀⠀⠀⠀⠠⡀⠀⠀⢀⠄⠂
⠀⠀⠀⡀⡀⠀⠰⡉⠉⠁⠀⠀⠀⠀⠘⢇⢸⡏⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⣑⣒⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠁⢀⠀⠐⠀⢀
⠀⠀⠀⢈⡀⠁⠀⠘⠷⠂⠀⠶⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣼⡍⠈⣛⡦⠀⠀⣀⠀⠀⠊⠊⠘⠀⠀⠀⠀⠀⠼⠀⠀⠀⡀⢤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣛⣖⣃⠗⠀⠀⠁⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⢀⣄⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⣀⡀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠈⢿⣆⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣱⡿⠁⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⢻⣧⠹⣧⡀⠀⠀⠀⠀⠀⢀⣼⠟⣼⡟⠁⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠹⣧⡹⣷⡀⠀⠀⠀⢀⣾⢏⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠹⣷⡘⣿⡄⠀⢠⣾⢋⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡜⢿⣤⡿⢣⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣼⡿⣡⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⣥⣶⣷⣦⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠠⣀⠄⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣄
⠀⠀⠀⠀⠀⢢⡕⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⣥⢀⠀⠀⠀⠀⠀⢠⣿⠯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣦⡵⣿⣦
⠀⠀⠀⠀⠰⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡥⠀⠀⠀⠀⢀⣿⣓⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣑⣻⡇
⠀⠀⠀⠀⠬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠄⡄⠀⠀⢸⣟⣒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣾⣿
⠀⠀⠀⠀⠩⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠯⠁⠂⠀⠀⠸⣿⣒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣿⡏
⠀⠀⠀⠀⠈⠽⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⢍⠌⠀⠀⠀⠀⠻⣷⡪⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠳⢣⢤⠀⠀⠀⢠⢶⠻⠓⡠⠊⠀⠀⠀⠀⠀⠀⠙⢿⣷⣻⢻⢻⣿⣿⡿⢟⢏⣮⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣿⣿⠿⢛⣋⣭⣭⣭⣭⣭⣭⣙⡛⠿⣿⣿⣶⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⢇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣝⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣫⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣶⣤⣭⣭⣭⣭⣤⣶⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠛⠛⠋⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⣿⣻⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⣄⠀⢠⡄⠀⡄⢠⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⢸⣦⠀⠉⡇⠀⣿⠀⠀⠆⠀⡇⢸⠋⡍⢠⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡶⠀⢸⠞⢧⡀⠷⠤⠙⠒⠂⠀⠾⠃⠚⠀⠰⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠸⠀⠀⠁⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠉⣀⣀⢠⡞⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⠀⣀⠾⡡⠾⡵⠄⠀⢀⣤⣤⠀⣿⣀⣟⢈⣿⠛⣶
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⡄⣾⣡⣟⠁⣾⣏⣹⠟⠛⠛⠁
⠀⠀⠀⠀⠀⣰⣟⠀⠀⢀⣸⠇⠀⠀⠀⢀⡤⣰⠟⣿⠄⣿⠁⣩⡁⣿⣀⠿⠁⠈⢉⣀⣠⣤⣶⣶⣶⣿⠿⠂
⠀⠀⠀⠀⠀⠘⠿⣦⣤⣤⣤⡀⠀⣾⡇⢸⠀⣿⠀⣴⡄⠿⠤⠛⠁⣀⣠⣤⣶⣾⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣴⣶⠀⠀⠈⣿⠠⣿⣇⣿⡄⠛⠒⠋⢀⣠⣴⠶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⡾⠋⠀⠉⠉⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⣰⡟⠷⢸⣏⣬⣤
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⣤⠀⣠⡶⣶⣦⣿⣩⡤⠻⠶⢻⡧⣭⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⢉⣀⣀⡀⠀⠀⢀⡀⠀⣠⣶⠏⠈⣿⣾⡋⢸⣟⣩⡶⠻⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⣉⣽⡷⢶⣶⡟⢻⣿⡏⠘⠛⣿⣿⠈⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠛⣿⣥⣾⣏⢿⣤⡾⠋⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣿⠁⠀⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠤⠤⠀⠒⠒⠒⠀⠀⠀⠀⠐⠒⠒⠒⠂⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⢁⣀⣤⠤⢴⣖⣿⣿⣟⠛⠙⠛⠛⠛⠻⠿⣻⣷⣶⣶⣦⣤⣤⣀⣈⠁⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⢁⣤⣶⣿⠭⠥⠒⣋⣴⢡⣿⣿⣿⣟⡒⠒⠒⠒⠊⠉⠭⣿⣿⡙⠛⡇⠙⣿⡻⠿⠷⢦⣤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⣠⣾⣿⣿⣏⡙⠛⣛⡫⠝⡡⠻⣿⣯⡫⣈⠉⠑⠒⠂⠈⢑⡿⢟⣿⣷⣄⠙⠦⣈⠉⠙⠛⢋⣹⣿⡄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⣰⣿⢿⣷⣼⣯⣬⣭⣥⡔⠋⠀⠀⠙⢿⣷⠄⣈⣉⠉⠂⠈⡡⠴⣿⠟⠁⠀⠙⠦⢄⣉⠉⠉⢈⣁⣽⣿⣆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡯⣿⢺⠇⠀⣄⠐⢸⠟⠀⠀⠀⠀⠀⠀⠈⠒⠀⠤⠀⠀⠰⠒⠊⠀⠀⠀⠀⠀⠀⢸⣿⣟⢻⣯⣿⠛⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡟⢻⠏⣿⠀⠻⢻
⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⡗⢄⠀⢀⡿⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣿⡇⢸⠀⠈⡇⠀⡇⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠃⢸⠀⡿⠀⠂⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠇⣿⠀⢸⡇⢀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢠⠀⢠⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢸⢿⠀⣿⠀⣼⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⢸⢸
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⣿⢀⡿⠀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠐⠀⡄⠈⡀⢸⣸
⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⢠⡟⢸⡇⢰⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡇⢠⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⢸⠁⣼⢀⡏⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢱⠘⡇⣦⢻⡄
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣾⢀⡇⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⡀⣇⢹⣼⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣷⣾⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠈⠀⢺⣟⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣝⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢿⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢸⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣸⣷⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⡀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠋⠉⠁⠈⠁⠈⠉⠉⠋⠉⠉⠙⠉⠁⠉⠁⠉⠉⠉⠸⠀⠀⣸⣯
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⢠⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠤⠠⢦⠼⠿⠘⠛⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠙⢻⣤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠑⡀⢐⠀⠀⠀⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣶⣶⣤⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⣤⣬⡍⢩⣥⣤⡘⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⣡⣾⣿⣿⡇⢸⣿⣿⣷⣌⠛⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣶⣤⣭⣭⣭⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡇⢸⠁⣤⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣡⣶⣶⣦⣿⣿⡇⢸⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡟⢩⡉⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣷⣬⣤⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡉⢉⣉⡁⢸⠟⢙⠻⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣦⣍⡃⢸⣄⣋⣠⣿⣿⣿⣿⡿⠁
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡿⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⡇⢸⡿⠿⠿⠛⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣾⣶⣶⣶⣶⣶⣶⣷⡄⠀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⢉⣴⣿⣷⣄
⣟⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣠⣄⠻⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⠟⣠⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢀⣴⣿⣿⣿⣷⣌⠻⢿⣿⣿⣿⣷⡌⠻⣿⣿⠟⢁⣾⣿⣿⣿⡿⠟⣡⣾⣿⣿⣿⣦⡀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣦⡙⢋⣴⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⡟⢡⣾⣿⣿⡿⢋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣷⣦⠉⣰⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢠⡤⠄⢠⣤⠀⢠⡤⠤⢠⡤⠄⠀⣤⠤⣄⢤⡤⠄⣠⡤⠄⣠⠤⣄⠀⣠⠤⠄⢠⣄⢠⡄⣤⠤⣤⠤⣤⠀⣤⠤⣄⢠⣄⢠⡄
⠀⠀⠀⢸⡗⠂⡾⠼⡇⢿⣀⡀⢸⣛⡂⠀⣿⠺⡇⢸⣛⡂⢿⣀⡀⣯⣀⣼⠻⣇⣐⣦⢸⠙⣾⡇⣿⠀⣿⠀⣿⠸⣇⢀⡿⢸⡟⢿⡇
⣶⣶⣶⢾⣷⣶⣷⣶⣿⣶⣿⣿⣾⣿⣷⣶⣿⣶⣿⣾⣿⣷⣾⣿⣷⣾⣿⣷⣶⣾⣿⣷⣾⣶⣿⣷⣿⣶⣿⣶⣿⣶⣾⣿⣶⣾⣶⣾⣷⣶⣶
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⢤⣤⣤⣤⣤⢤⠄⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣀⣠⣠⣀
⠀⠀⠘⡆⣿⣿⣿⡏⡸⠀⠀⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⠀⠀⠀⠹⢿⣿⡿⠟⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠈⢻⣿⣿⠛⠁⠀⠀⠀⠟⠛⣾⢻⣿⣿⣿⡟⢿⡞⠻⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣹⣇⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢻⣿⡅
⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⠿⣿⣄⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⣠⣿⣆⡀⠀⠀⠀⠀⠀⠸⠻⡏⠈⡟⠻
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⣀⠉⣀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠉⠉⠉⣁⣀⡀⠀⠀⠀⡀⣀⣀⣀⣀⡀⡀
⠀⠀⠀⢀⣠⣾⣧⣄⠀⠀⠀⠀⠀⢰⡜⠋⠁⠀⠀⠙⠣⡦⠀⠀⠀⠀⢇⢸⣷⣶⣶⡇⣸⠁⠀⠀⠘⠀⣿⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⢘⡗⠀⠀⠙⣿⡇⠀⠀⢸⣇⠀⠀⠀⠈⠺⣿⣿⣿⡷⠁⠀⠀⠀⠀⠀⠻⣿⣿⡿
⠀⠀⠀⠀⠈⣸⡆⠁⠀⠀⠀⠀⢨⡅⠀⠀⢀⣿⡇⠀⠀⢸⡧⠀⠀⠀⠀⠀⠘⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠺⣠⠀⠉⠉⠉⠀⣠⠿⠀⠀⠀⠀⠀⢀⣶⣿⣷⡆⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦
⠀⠀⠀⠐⠚⠛⠛⠓⠀⠀⠀⠀⠀⠀⠉⠚⠓⠀⠘⠓⠉⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠃⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿
⠀⠀⠀⣆⣤⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⢶⣦⠀⣠⣶⠆⠀⠀⠀⠀⠀⢠⡼⠂⠀⠀⠷⡄⠀⠀⠀⠀⢰⠂⠀⠀⠀⠷
⠀⠀⠀⠈⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧⣿⠏⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠣⢄⣀⠴⠃
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣏⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣿⣷⠀⠀⠀⠀⢀⣠⣶⣷⣶⣿⣶⣶⣶⣄⡀⠀⠀⠀⢀⣸⣇⣀
⠀⠀⠀⠀⠸⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⠋⠀⠀⠀⠀⠈⠿⠟⠉⠀⠀⠀⠉⠹⠿⠁⠀⠀⠀⠸⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠠⠄⠤⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡜⠁⠉⠁⠀⠉⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢠⢤⠄⠂⠀⠀⠀⠀⠀⠂⠀⠀⠠⡀⢤
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⠀⠀⠀⠸⣶⣤⣤⣤⣆⣀⣀⣀⣐⣒⣒⣀⣀⣀⣀⣾⣼⣿
⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⢿⣿⣿⡏⠛⠋⠉⠉⠿⠩⠭⠉⠉⠈⣽⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠑⡣⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⠈⢿⣿⣷⣴⣇⠀⠀⠀⠁⠀⠀⠐⣰⡟⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣯⡛⠆⠀⠀⢀⠀⢶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣧⣵⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⡷
⠀⠀⠀⠀⠀⢀⡀⠂⢠⠊⠇⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⡄⢀⣡⡐⢀
⠀⠀⠀⠀⠘⠶⣽⣶⣊⣝⣶⡀⠈⣁⣒⣂⣭⡴⠂⠀⠀⠀⠀⠀⠐⠦⣀⢀⣉⠀⠀⠀⠀⢀⣀⣁⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠁