Amniote Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⣛⣩⡍⠉⣉⣉⣭⣭⣥⣤⣤⣼⣯⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣴⡾⠃⢀⣠⠔⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⠉⢀⣴⢋⣴⣶⣶⣿⡏⣿⡿⠟⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠃⢠⡾⣵⣿⣿⣿⡿⠿⢷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣽⠏⠀⣿⢰⣿⣿⡿⠋⠀⠀⠀⠿⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢳⣿⠀⠀⣿⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢿⣇⠀⠈⢿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡈⠻⣷⣄⡈⠛⠿⠿⢿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣈⠙⠛⠲⠦⣤⣤⣀⣈⣉⡉⣉⠉⠉⠹⠟⠉⠉⢭⣿⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣤⣤⣄⣀⡉⠉⠁⠀⠀⠀⣠⠒⠀⠸⣿⣶⣤⣀⠀⢈⣽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡷⠀⣀⣠⣦⣿⣯⣕⣲⣾⣯⣭⣭⣭⣭⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣯⣤⣾⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⠟⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣽⡷⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⢀⣀⠤⠤⠤⠤⠤⣀⡀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⡠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⢀⡤⠊⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠘⢆⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡴⠁⠀⠀⠀⠀⢀⡤⠔⠒⠂⢉⣉⣉⣁⠒⠒⠢⠤⣀⠀⠀⠀⠈⢆⠈⢦⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢠⠎⠀⠀⠀⢀⡤⢚⠏⢀⡴⠊⠉⠁⠀⠀⠀⠀⠉⠁⠒⠤⣍⡒⢤⡀⠈⢣⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢠⠋⠀⠀⠀⡠⠋⠀⡌⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣈⠶⡈⢣⡀⢣⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣰⠃⠀⠀⢀⡔⠁⠀⠀⡇⢠⠀⠀⠀⠀⢀⡤⠒⠓⠘⠓⠒⠢⠤⢀⣀⠄⠉⢢⡘⢆⢧⠀⢧⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠃⠀⠀⢀⠞⠀⠀⠀⠀⠁⢸⠀⠀⠀⢀⠎⠀⢀⣀⣀⡀⠀⠀⠀⠈⠉⠓⠦⡀⢱⡘⡆⢇⡈⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠏⠀⠀⢠⠋⠀⠀⠀⠀⠀⡄⠘⠀⠀⠀⢸⠀⠀⢎⣿⣿⡹⠀⠀⠀⠀⠀⠀⠀⠘⣆⢣⠸⡚⡅⠈⡄
⠀⠀⠀⠀⠀⠄⠀⡸⠀⡜⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⡆⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⡇⢃⠸⡀⠸
⠀⠀⠀⠀⠀⠀⠈⠇⢀⠃⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢁⠀⢡⠀⠀⠀⠰⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢷⢸⠄⢣⠀⢇
⠀⠀⠀⠀⠀⠀⢰⠀⠸⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠈⢆⠀⠀⠀⠑⠤⠴⠮⠐⠂⠉⠒⢦⠀⠀⠀⠀⠀⡗⢸⠀⡄⠈⡆⠘⣀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⡌⠀⡇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⡀⡇⢀⣵⠀⢋
⠀⠀⠀⠀⠀⠀⡇⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣦⢸⡧⡟⠀⠸⠀⢸
⠀⠀⠀⠀⠀⢠⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡈⠒⠢⠤⠤⠄⠰⡎⠀⠀⠀⠀⠀⢠⠆⢘⠀⠆⠀⠀⡆⠘
⠀⠈⠀⠀⠂⢼⠤⣼⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣉⠁⠐⠒⠂⡞⠀⠀⠀⠀⠀⢀⡞⠀⣸⢸⠂⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠉⠁⠀⠀⣀⣀⣀⣉⠉⠒⢤⠃⠀⠀⠀⠀⢀⠎⠀⢠⡇⡌⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⢀⡤⠔⠊⠉⣀⡀⠤⠤⠬⢍⡹⠀⠀⠀⠀⢠⠎⠀⠀⡜⣰⠃⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡔⠁⠀⢠⠊⢉⣀⡀⠀⠀⠀⢀⠇⠀⠀⠀⣠⠋⠀⢀⠞⢠⠇⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⡇⠐⠷⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠱⡀⠀⠘⣄⠀⠱⡄⠉⠑⠒⠉⠀⠀⠀⡴⠁⠀⡰⠋⡰⠁⠀⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠘⢄⢀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⢦⡀⠈⠢⣄⠀⠀⠀⣠⠞⠀⣠⠞⣠⠎⠀⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⢦⡀⠈⠑⠒⠊⢁⡠⠚⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠈⠓⠠⠤⠐⢋⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡌
⠀⠀⠀⠀⠀⡇⠀⡇⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠠⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠇
⠀⠀⠀⠀⠀⣠⠰⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢘⠁⠸
⠀⠀⠀⠠⠂⠹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡜⠢⣇
⠀⠀⠀⠀⠀⠀⠇⠀⠦⢀⣀⠀⣀⠀⠤⠄⠐⠀⠈⠁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠀⠀⠒⠀⠀⠐⠀⠀⠀⢰⠃⡘⠀⠑⠠
⠀⠀⠀⠀⠀⠀⠸⡀⢸⡀⠀⢀⡀⠤⠐⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠂⠤⣀⠀⠀⢀⠋⢠⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠱⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⠊⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡉⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡰⠶⢮⣡⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠭⢉⣐⣒⠒⠒⠀⠀⠤⠤⠤⠀⠐⢒⣒⣂⠉⠥⠄⠒⠈⠁⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣰⣶⣤⣀⣦⡴⣶⣶⣦⣄⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⢾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣯⢟⣙⠛⠛⠿⢿⣶⣷⣎⢩⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⣛⣥⣶⣶⣶⣶⣶⣮⣽⣾⡿⠶⠤⠀⡈⠙⠻⢾⡿⣷
⠀⠀⠀⠀⠀⢰⣦⠾⣿⣓⣽⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣟⠶⢤⡙⣟
⠀⠀⠀⠀⢀⣾⣫⣷⣿⣿⣿⢛⣷⣿⣿⣿⣿⣿⡿⠿⠟⣛⣛⣛⡛⠿⣿⣿⣿⣿⣿⣿⣶⣯⣌⢷⡀
⠀⠀⠀⠀⣦⡀⣿⣿⣿⡟⣰⣾⣿⣿⣿⡿⢛⣉⣤⣶⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣯⢻⣦⠴⢶⠦
⠀⠀⠀⢰⢋⡿⡿⠿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⠔⠭⠃
⠀⠀⠀⡝⣾⣗⠉⠛⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡆⡔⠃
⠀⠀⠀⠁⠿⠋⣲⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢥⠄
⠀⠀⠀⠲⠃⠀⣼⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰
⠀⠄⠀⠸⠏⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠿⠋⣤⣿⣿⣿⣿⣿⣿⠸⠲
⠀⠀⢀⡐⣀⠀⠀⢸⡽⣿⣿⣿⣿⡿⠟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣧⡙⠿⡏⠄⢁⣾⡭⠍⠉⠛⢿⣿⣿⠸⢶⣴
⠀⢀⣾⠟⠉⠱⡀⡌⣿⡟⣿⣿⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⡌⠀⠉⠁⠀⠀⠀⠐⣾⣿⡟⢀⣭⠍
⠀⢸⣟⠀⠀⠀⢹⡹⣿⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠏⠙⠉⠀⠀⠀⠀⠙⠛⠀⣸⣿⣿⠣⠆
⠀⠘⠃⠀⠀⠀⠀⠱⣵⣿⣏⣗⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠄
⠀⠀⠂⠀⠀⠀⠈⠀⠹⣮⠙⠘⣦⣄⢀⢀⡀⡀⠀⠀⠀⠠⠖⠋⠀⠀⠀⠀⠀⠔⠀⠀⣠⣿⣿⠏⢠
⠂⠀⠀⠀⠀⠈⠀⢸⡶⠈⠓⢘⢮⡻⣿⣿⣿⣷⣶⣾⣿⣷⠄⠀⠀⠀⠀⠰⠀⢀⠀⣰⣿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⣣⡟⠀⠀⠀⠁⢽⣮⡻⢿⣿⣿⣿⣿⣿⣵⡀⠀⣀⣠⣶⣾⣿⣿⠟⠃⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠻⣿⠡⠀⠀⠠⠀⢨⡛⠷⢯⣝⣛⣿⠿⠿⠿⠿⢿⡿⣟⣻⡭⠊⠁⠀⠄
⠀⠀⠀⠀⠀⠀⢸⠀⣿⡃⠀⠀⠀⠀⣼⣿⣤⠄⠀⠈⣉⡟⡟⠛⠛⠛⠩⠉⣀⣡⡐⠥⢻⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠯⠀⠀⠀⠡⠟⢛⠛⠂⠀⠊⡙⢂⠷⡠⠄⠑⠀⠚⢹⣿⣿⣽⡖
⠀⠀⠀⠀⠀⠀⠈⠔⠀⠈⠀⠀⠐⠀⠀⠄⢀⣀⠀⣀⠁⠈⠰⠛⠚⠁⠀⠀⢾⣿⣿⡻⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⠎⠉⠉⠚⠁⠂⠐⠀⠀⠀⠀⢉⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠾⠟⠁
⣶⣶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣯⣿⣿⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣟⡉⠁⠁⠉⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⣴⣾⣿⠟⠉⣿⣿⣿⣿⡿⢿⣧⣀⡤⢀⠴⠸⣿⣿⡆
⠀⠀⠀⠀⠀⢀⡀⡀⠂⢀⣀⠀⢺⣇⣾⣿⣿⣿⣿⠉⠛⢿⣾⣿⣿⣿⣯⣵⣾⣞⠬⣼⣡⣿⣿⠁⠀⠁⠀⡀⢀⣤⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠁⣀⣤⣿⣻⣷⢼⣼⣿⣿⣿⣿⣿⠀⠀⠈⢻⣿⡛⠻⠙⠋⠟⣿⣾⣿⣿⣿⣿⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢙⢢⡻⠝⠂⠉⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠙⠻⣀⣀⠀⠀⠀⠙⣹⣿⣿⣡⠈⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠉⢌⡑⠀⠀⢘⣿⣫⢈⠈⠀⢀⣼⣿⠇
⠀⠀⠀⠀⢀⠀⠃⠀⠐⡔⢯⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⠉⠁⠀⠀⠀⠀⠑⠄⠀⠘⡟⡇⠈⠀⠠⣿⣿⣿⡃
⠀⠀⠀⠀⠰⣜⡤⠀⢈⠽⣟⣿⣿⣿⣿⣷⣿⣿⣯⣿⣹⡆⠀⠀⠀⠀⠀⠀⠘⡀⠀⠙⠜⠀⠀⡄⢻⣿⠇
⠀⠀⠀⠀⠀⠻⣷⡑⢠⣄⣹⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠑⢾⣇⢸⣿
⠀⠀⠀⠀⠀⠀⠙⢷⣴⡦⢸⢿⣼⢿⣿⣿⣿⣿⡿⣿⣷⣤⣤⣀⡀⠀⣠⠞⠁⠀⢀⠀⠀⠀⠀⠈⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣼⠻⣿⣻⣿⣿⣿⡿⢿⣾⠷⠛⠋⢉⠀⢀⣀⠔⠀⠀⠀⠀⠀⡠⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⢿⣶⣾⣿⢿⣿⣿⣿⣿⠿⢷⣾⠒⠊⠉⠉⠀⠀⡀⠀⢀⣶⣾⠃⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⠙⠛⠁⠙⠛⠻⠇⢸⣏⠘⠟⠚⠚⣿⣮⣭⣷⣾⣿⣿⣿⣦⠀⠀⢸⣿⣷⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠀⠤⣥⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣸⣿⡿⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡿⢿⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⠟⠀⠈⢿⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠺⠶⢦⣤⣶⣾⣿⣿⠿⠟⠁⠀⠀⠀⠀⠙⠻⢦⣄⡀⠀⠀⠁⠂⠀⢩⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠤⠐⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣤⠴⠂
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁
⠀⠀⠀⠄⠀⠹⢻⠛⢷⣤⡀⠀⠠⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣤⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠴⠋⢁⣾⣿⠟⠀⠀⠀⠀⠀⠀⠁⠓⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣎⣿⣧⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣠⣤⣴⣾⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠿⢿⣿⣿⡿⠃⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⠻⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠙⠻⠿⠷⢮⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣷⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠟⠛⠛⠋⠉⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣈⡉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⠀⠀⠀⣀⣀⢠⣄⣀⠀⠀⠀⠀⠀⣬⡄⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⡰⠆⠛⠛⠛⢿⣿⣿⣿⣿⡿⠟⠉⠀⠹⠗⢀⣀⠀⢉⣁⣀⣀⣀⣀⣀⠀⠀⠀⠀⠑⡄
⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠀⠀⠉⠉⠉⠋⠳⢦⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⡀⠀⢠
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⣄⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣺⣆⠀⢀⡀⣮⡏
⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⠛⠛⠟⠧⡀⠸⣇⠘⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣇⠁⠠
⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⢀⠀⠀⠀⣠⣾⣷⠀⠀⠀⢠⣶⣇⣤⠆⠀⠀⠀⠐⠈⠁⠀⠻⡆⠀⠂
⠀⠀⠀⢀⠹⣦⡀⠀⠀⠀⠀⢫⠃⠀⣠⠞⠁⠀⠠⣤⣿⣿⣇⣠⣄⠀⣹⡏⠉⠁⡀⠀⠈⠀⠀⠀⠀⠀⠀⠿⡀⠈⡀
⠀⠀⠀⠀⢂⠈⠇⠀⠀⠀⠀⣿⠀⠀⣿⡀⠀⣀⣷⣿⣿⡟⠙⢉⣼⣤⣾⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇⠀⠁
⠀⠀⠀⠀⠀⡆⠐⢀⡀⠀⠀⢻⣆⠀⠈⠉⣋⣴⣿⣿⡿⠇⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸
⠀⠀⡖⣠⣤⠸⣚⡏⠁⠀⠀⠀⠈⠓⠶⣤⣽⣿⢿⣿⣿⠀⠀⢻⣷⡀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢿⡀
⠀⠈⠁⠈⠀⠀⠿⡟⢻⡄⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠃⠀⣾⠉⠁⠛⠃⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠀⠀⢷⣾⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⢻⣴⠀⠀⢠⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇
⠀⠀⠀⢠⡇⠀⠁⠘⢟⢧⣤⣄⠀⢀⠀⣀⣶⣿⣿⣿⣷⣦⣀⣀⣈⣹⣿⢞⣿⡯⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⢀⡀⢀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣻⠏⠁⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⢀⡇⠀⠂⢀⡀
⠀⣼⠁⠀⡤⣤⡀⠀⡀⠈⡦⢟⣻⣿⣿⣿⣿⣿⠿⣿⣟⠋⠀⣠⠴⣿⠀⠀⠀⣀⣤⣶⣾⡏⠋⠀⠀⠀⠈⠁⣸⠁⠘⠀⠊⠙
⠀⠈⠀⠀⠀⠀⠀⠀⢁⣴⣿⣿⣿⣿⣿⣿⣿⣶⣴⣚⣩⣤⣾⣧⣶⣿⣷⣶⣿⣿⣿⡿⠛⢀⠇⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⡿⠋⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠁⠀⢠⠂⠀⠀⠀⠀⠀⡹⠀⠠⠀⣰⠀⢀⣀
⠀⠀⢀⣠⣾⣿⡿⠻⢋⣼⣿⣇⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠟⠝⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⢀⠀⠈⠈⠁⠈⠉
⣠⣾⣿⡿⠟⠁⠀⠂⣼⣿⣿⣿⣧⣄⠀⠈⠀⢼⣿⣿⣿⣯⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⠂
⠿⠛⠉⠀⠀⠀⠘⠀⣟⣿⣹⣿⣿⣿⣿⣦⡀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣁⡿⠑⠻⠌⠋⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠴⠚⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢰⣶⢏⠁⠀⠀⠈⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠠⠀⠀⠀⠀⠀⢹⣿⣟⢣⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡄⠀⢀⡄⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠄⠧⠰⠄⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣄⣀⣠⣀⣴⣿⣿⣿⣿⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⢿⡟⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠅⢰⡖⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⡆⠓
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢠
⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡄⣄
⠀⠀⠀⠃⠃⠀⠀⠀⠁⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⢠⠇⠀⠀⠀⡄⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠂⠀⠛⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣿⠟⡻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⢿⣿⣻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣢⡙⢿⣷⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣶⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡩⣙⡫⠝⣛⢛⠻⢿⠿⣿⣿⣿⡿⠿⠇⠹⣿⣿⡀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠠⢀⡀⠀⡀⣀⠠⢄⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣒⣿⣿⣿⣿⣿⣿⣿⡶⠦⡀⠀⠀⠁⠃⠊⢡⣶⣿⡛⣭⣋⣉⣉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢚⣭⠿⠟⠛⠉⠙⠓⢒⠒⠓⠦⠵⠦⠖⠆⠰⣶⣦⣬⣥⣤⣤⣶⣶⣶⣶⣶⠤⡀
⠀⠀⠀⠀⠀⠀⠠⢎⡭⠶⠋⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠤⣒⣤⡶⠂⠉⠀⢈⡉⢻⣿⠿⠟⢛⣛⠻⡈⠁⠄
⠀⠀⠀⠀⠘⠁⢠⠞⠀⠀⠀⠀⠀⠀⠀⠐⣶⣦⡄⣠⠴⠊⠉⠀⢋⣠⡤⢞⣡⢚⣅⡴⠚⠋⠉⠉⠉⠋⣀
⠀⠀⠀⠀⠀⢠⡏⠐⠀⠀⠀⡄⠀⣠⡤⢾⣯⡵⠚⠀⠀⠀⠀⣴⠋⢀⣴⢟⣵⠖⠉⠁⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⣿⠀⠀⠐⡀⠀⠈⢐⡵⠚⠉⠀⠀⠀⠀⠀⣀⠀⢁⣴⡿⡡⠋⠁⠀⠀⠀⠀⣀⡀⢠⣶⠏
⠀⠀⠀⠀⠀⢻⠀⠀⠈⢀⡴⡵⠋⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⢟⢞⢞⠄⣂⣤⣶⠆⣽⠻⠋⢀⣼⠏
⠀⠀⠀⠸⡆⢨⢧⡀⠀⠉⢸⠃⠀⠀⠀⠀⠀⠤⠀⠾⢻⡿⡱⠳⢁⡕⠊⠁⠠⠋⠰⠋⠁⠀⠺⠋
⠀⠀⠀⠀⠙⢆⠀⠻⣄⢀⢸⠀⠦⠄⠀⠀⠴⢚⣚⣻⢟⠪⠊⢰⡿⠀⢄⠈⠁⡐⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠙⠨⢬⣁⡈⠓⡶⣶⠶⣛⡿⢿⡛⠁⠉⠀⠀⠀⢻⣄⣠⡄⠄⠀⢤⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠲⠌⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⢂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⢀⢠⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⠀⣉⣉⣉⣉⣉⣉⣉⣉⠀⠒⠒⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⣉⠤⠐⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠠⠤⣉⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⢀⠔⢈⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡁⠂⠄
⠀⠀⠀⠀⢀⠔⢁⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠤⠤⠤⠀⢀⡀⠀⠂⠠⠀⠀⠀⠀⠀⠈⠑⠠⡈⠐⢄
⠀⠀⠀⡰⢁⡴⠋⠀⠀⠀⠀⠀⠀⠠⠀⣁⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⢄⡁⠀⠀⠀⠀⠀⠈⠑⢤⡉⠢⡀
⠀⠀⡜⢠⠞⠀⠀⠀⠀⠀⠀⠠⠊⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡈⠀⢀⠀⠀⠀⠙⢦⠈⢄
⠀⡜⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠪⣵⣄⠑⠀⠀⠀⠀⠑⡄⠱⡀
⢰⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠈⢪⡣⡀⢀⠀⠀⠀⠘⡄⠱
⡄⢰⡇⠀⠀⠘⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠱⢡⠀⠀⠀⠀⠀⠸⡀
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⡇
⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡔⢁⣠⣶⠶⡾⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠠⠀⠀⠀⠀⡇
⢁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣠⠒⣲⣤⣭⣭⣥⡀⠠⢀⠀⡟⣰⡿⠟⣛⣛⡛⠻⣦⡄⠠⠀⠀⠀⠀⠀⠀⢸⠠⠁⡄⠀⠀⠀⡸
⠘⡄⢻⡀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡇⢸⠃⠀⠀⠀⠀⠉⠲⣤⡑⣷⣿⠀⣾⣍⣉⠙⢳⡌⢛⢆⠀⠀⠀⠀⠀⢠⡣⠃⡐⠀⠀⠀⡰⠁⠜
⠀⠱⡀⢳⡀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢇⢸⣧⡀⠀⠀⣶⣿⣿⣿⡧⠟⠋⢲⣀⣀⠼⠁⢨⡗⢸⠹⠀⠀⠀⢀⡔⠟⢁⠄⠀⠀⢀⠜⢁⠎
⠀⠀⠱⡀⠳⡄⠂⠀⠀⠀⠀⠡⠈⢂⠘⢄⠻⢿⣴⣹⣿⣿⠿⠿⠷⣷⣶⡿⠋⠁⠀⢀⡼⢀⠏⠀⠀⣀⢔⠕⠁⠀⠀⠀⢀⠔⢁⠔⠁
⠀⠀⠀⠈⢄⠙⢦⡀⠀⠀⠀⠀⠀⢄⠐⢄⠁⠠⠈⣙⠻⢦⣄⡀⠀⠀⢹⣿⣤⣤⡴⠟⡵⠋⠀⢈⡩⠓⢁⠄⠀⠀⡀⠐⢁⠔⠁
⠀⠀⠀⠀⠀⠑⢄⠙⠦⣀⠀⠀⠀⠀⠁⠄⠀⠢⢄⡈⠉⠓⠚⠽⠿⣿⣛⣋⡩⠤⠖⣉⡠⠔⠂⠁⠄⠂⣀⡤⠒⢁⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠢⢈⠙⠢⢄⡀⠀⠀⠀⠀⠠⢀⣉⠀⠒⠒⠀⠀⠒⠒⠂⠈⣁⠠⠄⣂⡨⠤⠒⢋⡡⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢈⡉⠒⠲⠤⠤⠤⣀⣀⣀⣀⣀⡀⠤⠬⠥⠔⠒⠊⣉⡡⠤⠐⠈⠁
⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠠⠤⠤⠤⠤⠤⠤⠄⠒⠒⠈⠉
⠀⣁⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠇⡼⠋⠈⠉⠻⣿⡌⠻⠈⢻⣿⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠸⠈⣿⡄⠐⠀⢻⡆⢹⣿⣿⣧⠀⠀⡆⣰⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠠⣤⣄⠀⢀⠠⠀⠹⢼⣷⠀⠰⣿⠇⢀⣿⣿⣿⣧⠀⣇⡏⢀⢀⣤⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠐⣿⡇⠀⡉⠀⠣⢯⡿⣿⣿⣿⣾⣿⣡⣬⣦⣷⡌⠀⠀⠀⡆⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⠠⠀⠀⠀⠘⠒⠒⡆⠀⠀⠀⠘⢦⣼⡇⡀⠘⠳⣄⠀⠀⠛⠛⠿⣿⢿⣿⣿⣿⣿⡇⠀⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡆⠀⠀⣀⡀⠀⠔⠀⠀⠀⠀⢢⠀⣿⠁⠷⣥⣤⠬⠍⡐⠀⠄⠠⢤⡀⠘⠛⠛⠻⢿⡄⠀⡾⠂⠀⠶⣤⡆
⠀⠀⠀⠀⡄⠀⠀⢸⣿⣿⣿⣿⣦⡼⠉⠙⠓⠂⠀⠀⠀⠀⠀⠀⡇⠀⡐⡑⢶⣄⠀⠈⢀⡀⡀⠀⠁⠀⠰⣦⡀⠨⣿⠀⣿⡀⠘⠘⠻⡇⠀⢠
⠀⠀⠀⠀⠇⠀⠀⠈⢿⣿⣿⣿⡧⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡀⠀⠈⠀⠉⠷⣄⡀⠙⣷⣀⣀⣛⣿⣿⣿⡃⠀⠀⢹⡟⠑⡒⢲⡆⢈⣸
⠀⠀⠀⠀⢸⡄⠀⠀⠆⢿⣿⣿⣷⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠈⠙⠻⠆⠀⠀⠀⠀⠱⠐⠉⠉⢿⢻⣿⣿⣿⣿⣿⣀⡄⣼⣇⡀⠃⠘⢇⠈⣿⡅
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⠅⠀⠰⠘⡄⠘⣿⣿⠿⠃⠉⣭⣾⡗⠌⠉⠿⣿⣖⠇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠈⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⣸⡀⠉⠁⣠⣴⣯⣟⠩⣮⣰⣆⢀⢤⣬⡀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠐⢀⠹⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢁⠳⠀⠰⣷⣬⣋⣝⣶⠟⣻⣹⣧⣀⣀⡀⠀⡈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠈⠢⠉⢿⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣢⡘⢦⣘⣿⣿⢯⡅⢘⡿⢿⣿⣿⣿⠳⠀⢧⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡄⠀⠢⣝⢿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠁⠁⠀⠀⠀⠀⡀⢠⣋⣩⠽⠶⢽⣷⣷⡀⠐⠃⢲⣿⠟⣯⣿⣇⣀⣿⣼⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠆⡀⠈⢿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⡤⠟⡋⠥⠒⠂⠐⠒⢿⡿⣦⠀⠀⠁⣿⣿⣌⣷⡄⣾⢰⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣵⣙⢦⠠⠛⢞⠿⣟⠻⣿⣿⣿⣿⣶⣤⠀⣀⡄⣫⠔⠋⠀⠀⠀⠀⠀⠀⠈⣿⣻⡀⠀⠀⢻⣿⡿⣻⣏⣿⠁⠋⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⡿⠛⢻⣿⣷⡄⠀⠈⠳⣻⣶⣄⠉⠻⣿⣿⣿⣿⢋⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡟⡃⠀⠀⢸⡽⣷⣿⣿⡿⡏⠣⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠙⠀⠉⠳⣄⡈⢻⣿⣧⣾⣿⠛⠂⠀⠀⢀⡀⢠⣤⣶⣤⢰⣷⠠⠄⠀⣼⣧⣧⣿⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠂⡀⠀⠀⠀⠀⠁⢀⣿⣿⣿⣿⢀⣠⣆⣀⣬⣤⣾⡿⡛⢩⣾⠃⠀⠀⢠⣧⣿⣾⣯⡏⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠙⠂⠀⠀⠀⠀⣡⣿⣿⣿⡿⡋⢹⣿⣿⡟⣙⣡⡞⣵⣿⠟⠀⠀⠀⡚⣸⠋⣿⠉⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣧⣰⣦⣀⡰⠶⣻⡟⣿⣿⠻⢿⣿⣿⣿⣿⣿⢯⣾⣿⢋⣄⣠⠞⣡⣿⣿⠟⠁⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣝⠛⠿⠾⠩⠔⠋⠀⠀⠀⠈⢻⣿⡿⢻⣿⣿⣿⣟⣩⣥⣴⣿⠟⠁⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣄⠀⠀⣀⣀⣀⣠⣤⡶⠟⣹⣶⣿⣿⡟⢟⣫⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣿⣯⣥⢈⣀⣀⣬⡿⠿⡿⠋⠁⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠋⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣦⣴⣶⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⢋⣁⣤⣴⣶⣭⣭⣉⡙⠛⠿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠸⠛⠛⣫⣿⣿⡿⢿⣿⣿⣷⣄⡈⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠀⠀⠀⠀⠀⠾⠋⠈⠁⠀⠈⠙⣿⣿⣿⣷⡞⠛⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿⣿⠇⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⠀⣾⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠃⠀⠀⠀⢀⣼⣿⣿⣿⠃⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⣸⣿⣿⣿⣘⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⣿⣿⣿⡀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣬⣿⠁⢨⣧⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣯⡄⠀⠀⠀⠀⢰⣤⣈⢿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⢻⣿⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣸⣿⡿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣿⣿⣿⡀
⠀⠀⠀⢀⣟⣩⣤⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⠘⣿⣿⡇
⠀⠀⠀⣾⣿⣿⡉⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣙⣋⣁⣀⠀⠻⠇⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣦⠉⠼⣿⡆⢻⣿⣷
⠀⠀⢰⣿⣿⣿⡇⠀⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⡉⢀⣏⡀⠀⢹⣿⣿⠀⠀⠀⠀⠀⢀⣦⠹⣿⠀⠻⡇⢸⣿⣿
⠀⠀⣾⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠈⠻⣿⣿⣿⣾⣿⣿⣿⠟⠻⠟⠀⠀⠀⠀⢸⣾⡗⠀⠀⠀⠀⡇⢸⣿⣿
⠀⠀⠙⠋⠛⣛⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⢢⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢈⣷⣶⣤⣀⡀⠀⠀⠀⠀⠈⠉⠛⠳⢄⣀⣥⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⣿⣿⣿⣿⡇⣤⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⣉⣉⣉⣉⠙⠓⢦⡄⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⢀⣸⣿⡟
⠀⠀⢸⣿⣿⡟⠇⢀⡘⠃⠈⠉⠛⠿⡿⠿⡿⠿⢿⠟⠛⠛⢻⣛⣿⡟⠓⣶⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⡀⠀⠀⠸⣿⣿⠃
⠀⠀⠀⠿⠿⠿⠆⠚⢡⣶⣶⣦⣦⣤⣤⣦⣤⣤⣴⣿⣶⡾⣿⡟⠉⠉⠛⠻⠷⣧⣜⣿⣿⣶⣄⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⠁⠀⠂⢠⣿⡿
⠀⠀⠀⣴⣶⣶⡇⠀⡘⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠻⠶⠦⠄⠠⠟⠀⠀⠾⠇⠀⠀⢀⣿⣿⠃
⠀⠀⠀⣿⣿⣿⡇⠀⠃⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠠⣄⠀⠀⠈⠀⠀⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠸⣿⣿⣷⠀⢻⣤⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠰⢩⣷⡀⠀⠀⠀⠀⠘⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⢻⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿
⠀⠀⠀⠀⠰⣶⣾⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢹⣿⣷⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢻⣿⣽⢿⣿⣦⣄⣀⣀⣤⣶⠟⢋⣤⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⡄⠀⠀⠘⣿⣿⣷⣴⣿⠛⠻⣟⣭⣷⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠘⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢹⣧⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⢻⣷⣤⠀⠈⠉⠉⠉⠉⠁⠀⠀⣸⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠘⠁⣿⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣾⣷⣴⣶⠀⠀⢠⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⠛⠿⢿⣿⣷⣶⣾⣿⣿⣿⣿⠿⠟⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠄⠉⠀⠀⠠⠤⠄⢀⣀⣀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠏⠀⠀⠀⠀⠀⠈⠁⠤⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠉⢀⣠⡀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠠⠀⠀⠤⠤⠀⣀⠀⠀⠀⠀⠶⣶⣤⡀⠈⠢⣀
⠀⠀⠀⠀⢀⠔⠁⢀⣴⣿⠿⠁⠀⢀⠠⠐⠈⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠈⠀⠀⠀⠀⠈⠻⣿⣦⡀⠈⠠⡀
⠀⠀⠀⡠⠋⠀⣰⣿⡿⠃⠀⠀⠄⠁⠀⢀⠤⠴⠚⠉⠉⠀⠈⠁⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠘⢄
⠀⠀⡰⠁⢀⣼⣿⡟⠁⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣈⣨⣤⣤⣤⣄⡀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠈⢯⣽⣧⠀⠈⢡
⠀⢠⠁⢀⣾⣿⡟⠀⠀⠂⠀⢀⣼⠉⠀⠀⠀⡐⢠⣾⣟⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⣿⣹⣧⠀⠈⢃
⢀⠇⠀⣼⣿⣿⠁⠀⢠⠀⠀⣸⠉⠀⠀⠀⠀⢇⠘⠛⠋⠹⣿⣿⠁⢹⣿⠟⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣭⣻⡄⠀⠀⡆
⢸⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⢹⠂⠀⠀⡰⢲⡀⠙⠛⠀⠀⢿⣿⠀⢽⡿⠀⣤⠠⣤⠞⠀⠀⠠⠀⢀⣿⣿⣿⣿⣿⠀⠀⢸
⡇⠀⢸⣿⣿⡇⠀⠀⢠⠀⠀⠈⢇⡠⠀⠈⠉⣠⠊⠀⠃⠀⣼⣿⠀⢸⣿⡄⠀⠠⠀⡀⠄⠈⠀⣠⣾⣿⣿⣿⣿⢿⡀⠀⠘
⡇⠀⣹⣿⣿⣇⠀⠀⠈⠂⠀⠀⠀⠙⠒⠒⠈⠁⠀⡐⠀⢠⣯⣿⡄⠈⢻⣿⣷⣤⣤⣤⣤⡴⣾⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠢⢀⡀⠀⠀⠀⠤⠊⠀⣠⣿⣿⣿⣿⣄⠀⠙⠻⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢱⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⣾⣿⣿⡿⡟⣿⣿⣷⡄⠀⠈⠙⠻⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸
⠘⡄⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢻⣿⣾⣯⣿⣿⣧⡀⠀⠀⠈⢿⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⡜
⠀⠑⠀⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⡟⢼⡛⢫⣝⣳⠿⣻⣿⣷⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⠃
⠀⠀⢢⠀⠈⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣿⢻⣿⡿⡴⣿⣶⣯⣷⡿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠃⠀⢀⠏
⠀⠀⠀⠡⡀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠻⣾⣿⣯⣝⣾⣿⣿⣿⣿⠿⠋⠁⠀⠀⣠⣿⣿⣿⣿⣿⡿⠃⠀⢠⠏
⠀⠀⠀⠀⠑⢄⠀⠘⢿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠋⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠙⠻⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠋⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠢⠤⢄⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠚⠉
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢢⠠⡄⠀⠀⠀⢾⣿⣶⣴⡃⢀⣎⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠤⢤⡬⣄⡹⠦⢀⡚⣿⣿⠟⠝⣲⣿⣝⢢⠀⠀⠀⠀⣠⠦⢀⡀⢴⣦⣹⣇⠼⠇
⠀⠀⠀⠀⠀⣰⡾⢉⣽⣿⣷⣶⡾⣽⣴⡶⣿⣿⣧⣿⠿⠋⠀⠀⠀⣿⣶⣾⣿⡀⡛⠛⠿⣀
⠀⠀⠀⠀⠇⠙⠃⡐⠻⣿⠛⢻⣿⣏⡉⣥⢁⡐⣴⣅⠸⣧⣤⡤⠀⠙⣻⠿⠿⠷⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⣿⣿⣷⣥⣿⣿⣿⣿⣿⣿⣿⣕⣌⠙⠓⡠⣤⡀⠀⢠⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢯⣭⣥⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⡿⣻⣿⣷⣦⢇⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⢠⡼⡯⣝⡿⠛⠉⣯⣿⣿⣿⣾⣽⡶⢳⣾⣿⢝⣿⣿⣿⣻⣻⠿⣷⡀⣀⣈⢠
⠀⠀⠀⠀⠀⠀⠀⡀⡤⢶⢠⣀⣤⣮⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⡂⣿⣿⣟⣟⣈⡁⠙⢿⣿⡽⠡⡀
⠀⠀⠀⠒⣫⡷⣾⢸⣋⣴⠾⢿⣿⠿⠛⢉⠛⣿⣿⣿⣿⣿⣿⡷⢦⣛⣯⠜⣿⠛⣽⣷⣀⣉⠁
⠀⠀⠀⠀⠴⠞⢳⠞⢛⡷⠗⠈⡟⠀⢁⣤⠉⠟⢿⣿⣿⣿⣟⠁⡈⠉⠀⠀⠋⠈⡿⠀⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡤⠀⠀⠀⠘⠀⠄⠄⠀⣿⣿⡿⠃⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡲⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡷⡜⢀⠀⠀⠶⠃⠁⢠⠀⠀⠻⠆⠀⢀⠘⡴⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⠆⢀⠛⠄⠷⣠⠐⠀⢸⢀⠐⢠⡰⠇⡜⢃⠀⠾⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠤⡷⠀⠛⣢⡘⡂⠀⣌⠃⠀⡀⡀⣰⢼⡌⠔⡀⠀⠀⢋⡄⠀⡚⣠⡚⠃⠐⣦⠄
⠀⠀⠀⠀⠀⠀⠈⠛⣠⠄⢀⠡⣀⠀⡀⢙⣆⠀⠘⡃⠁⣸⠃⡀⡛⠀⢀⣞⠁⡀⢀⡠⢁⠀⢤⡘⠋
⠀⠀⠀⠀⠀⠀⣠⡄⢠⡤⠘⠛⠉⢇⠈⢲⡁⠹⢶⡇⠀⣿⠀⢠⣷⠾⠁⣱⠊⢀⠏⠙⠛⠠⣤⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠅⠀⠠⠀⢶⡖⠾⢧⣘⠋⠛⠢⢽⣆⣿⣠⣾⠥⠚⠋⢛⡠⠿⠖⣶⠆⠠⠀⠀⠍
⠀⠀⠀⠀⢀⡆⠀⠚⠢⡠⢔⡒⠳⠦⣤⢬⣽⣿⣶⣶⣿⣿⣿⣷⣶⣾⣿⣭⢤⡤⠶⠓⣒⠤⠠⠚⠂⠀⣄
⠀⠀⠀⠀⠈⠱⠶⡄⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⡴⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠴⠶⠾⠿⠿⠿⠿⠿⠷⠶⠦⣤⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠈⣇⠀⠀⡅⠠⠄⠀⠆⠀⠀⠀⠀⢆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣆⡴⠃⠀⠀⠎⠀⠀⠀⠠⠀⠀⡏⠀⡠⢇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡸⠀⠁⠀⠀⠀⠈⠒⢇⠀⠀⠈⠢⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣈⡇⠀⠀⠀⠀⠀⠀⢄⢸⡇⢸⣰⠃⢃⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⡀⠀⠀⠀⢸⡇⠀⣼⠅⠀⠀⠀⢀⠀⠀⠀⢸⡇⣸⠏⣀⡆⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⢻⣧⠞⠁⠀⠀⠀⠀⠀⢣⡀⠀⢸⣧⠟⠉⠀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢾⣿⠀⠀⠈⠠⣀⠀⠀⣼⡇⠀⣼⡟⠛⠛⠫⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⢀⠀⠀⢸⣿⠀⠄⠀⠀⣸⠀⣠⣛⣇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠈⠙⠓⣶⠶⠶⣷⣄⣸⣿⠀⠈⢢⡴⠛⠛⠁⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠑⠦⣀⡨⠶⠋⠁⢀⠠⠈⠉⠻⣿⣆⢠⡿⠀⠀⠀⣠⣿⣿⣶⢶⣄⣀⠠⠄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠈⣷⣿⠏⠀⠀⠀⠉⠛⠛⢶⣒⣒⢛⠓⠒⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⣼⣿⠏⠀⠀⢠⠀⠤⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⢀⡴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡥⠖⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⢀⣀⣸⣄⠀⠀⠀⢀⣤⣤⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⢀⡄⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠
⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣶⣶⣷⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠈⠒⣦⣤⣤⣤⣀⣀⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠓⢄⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠉⠢⡈⠙⠳⠒⠚⠉⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢨⠀⠀⣰⠀⠀⡀⢀⠀⠰⡀⠈⣦⠆⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⠀⠀⢸⡇⠀⠇⠀⠰⡡⠊⣤⣄⢳⢀⡏⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⣠⣴⡟⠀⠤⢤⡀⠈⠀⠀⠀⠀⣸⢞⠇⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣧⣿⣿⠀⠐⠒⠦⠄⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢙⣟⡏⠀⠀⠒⠤⡀⠀⣠⠄⢠⣿⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣿⡟⠒⠤⠤⡀⢠⣾⣧⢄⡸⣿⡆⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢀⣀⣀⠤⠖⠋⢀⣠⠟⠋⢿⣐⠠⠀⠀⣾⡿⡇⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣼⡀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡇
⠀⠈⠓⠒⠒⠒⠊⠉⠀⠀⠀⣼⠮⢭⣥⣠⣿⡵⢃⣼⣿⣿⣿⣿⡛⢶⡦⢾⣤⣤⣤⣤⡜⠸⣿⠇⠀⠀⠀⠀⠀⠀⡏⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠷⠌⣿⡶⠿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣠⠈⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡄⠀⣼⠁⠀⣿⣿⣿⣿⣿⣿⠙⠿⠿⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢀⡟⠀⠀⢹⣿⣿⠿⠿⢿⡄⠀⠀⡇⠀⠀⠀⢀⣀⣼⣀⣀⣤⣴⠟⠛⠉⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠦⢀⡾⠁⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢰⠁⠀⠀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢰⢹⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠃⢈⣰⠟⠁⠀⠀⠀⠀⢸⣧⠀⠀⢠⠀⠀⠃⠀⠀⢸⣿⡿⠋⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠧⣤⣤⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⢠⠁⠀⠀⢀⣼⠋⠀⠀⠀⢋⠹⣿⡆⠀⠀⠀⡄⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⡀⠀⠀⣾⠃⠀⠀⠀⠀⠸⡄⢿⣧⠀⠀⠀⡇⠀⢰⡏⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠤⠀⠈⢷⠈⢀⠄⢻⣶⡄⠀⠀⠀⢀⠁⢸⡿⠀⢀⡀⢱⠀⠈⢇⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠀⠀⢨⡇⠀⠀⢸⣿⣿⠀⠀⠀⠈⠓⠚⡇⠀⠠⠄⠀⢧⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⠀⠀⣸⢿⣷⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⣧⣠⣤⣤⣤⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣤⡀⢀⣤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣀⠸⠿⠀⢿⣿⠺⢿⡶⢀⡄⣿⠇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣿⣾⢛⠹⠷⠞⠀⢰⣦⡀⣴⡇⢿⠇⠀⢸⠟⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠉⣤⡉⣿⠇⠀⣷⣦⢠⣝⠃⡙⠁⣠⣴⠆⣀⢼⡿⠇⢀⡄⢠⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣜⠃⣶⡠⢰⣬⣄⠈⣷⠄⡙⢀⡟⢣⣾⣶⠙⠋⣾⡿⢀⣴⣇⢻⠟⠛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⡉⠉⠀⠉⠁⠳⣍⠙⣇⣠⠊⣴⡄⠶⣶⢧⣸⣴⣦⠉⠀⠘⢿⡣⠔⠀⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣃⣙⣿⠗⢾⡛⠋⠻⢾⣿⡄⠛⠀⠀⣤⣤⣹⣇⣹⣴⠗⠫⣿⣶⠆⠀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⡜⠛⣋⡀⢈⣭⣝⠲⣾⣿⣋⢻⣇⠙⠛⠢⣉⣹⡿⢛⣭⣽⣓⠒⢰⡾⠀⣶⠶⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⢀⡘⢿⣆⡭⠴⣾⣿⣍⠽⠿⣿⣷⣤⣤⣾⣿⣷⣮⣭⡭⠁⠀⣶⡿⠃⠀⠘⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠏⠠⡶⠀⢀⡀⢺⣷⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠙⠍⠛⣻⣯⣅⠀⠘⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠈⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣌⠙⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢻⣿⢿⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠖⣛⠿⢛⢿⡥⡴⠋⣹⡈⢯⡛⢤⣘⠿⣛⡛⠛⠛⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠸⡀⠀⠃⠈⠂⠉⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⣠⠴⠶⠦⣤⡤⠖⠒⢄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠘⣁⡄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⡎⠀⠀⡀⠘⠢⣄⠀⠈⠣
⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡐⢄⡂⢄⢢⡘⣦
⠀⠀⠀⣿⣾⢏⡠⠀⣠⠔⠡⢾⣿⣡⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⢿⣖⠈⡂⣤⡘⢮⠃
⠀⠀⠀⠸⣯⠞⢠⣞⠁⢊⣴⣿⣿⣯⡻⠛⠀⠀⠀⠀⠀⠀⢹⡿⠿⣿⣿⣿⣷⡌⠳⣝⢦⡱⡄⠁
⠀⠀⠀⠀⠿⣶⡿⢑⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠹⣾⣿⣿⣿⣿⢝⢦⡙⣷⡝⡆
⠀⠀⠀⠀⠀⠙⠇⣼⠇⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡡⣵⡈⠿
⠀⠀⠀⠀⠀⠀⠀⠙⢲⣟⣿⣿⣿⣿⣿⠁⣤⠀⠀⣀⡂⠀⠀⡇⠈⣿⣿⡿⣿⣾⣿⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⣼⣿⡟⠐⣿⡀⠀⣒⡆⠀⠀⣤⠄⡘⢿⣿⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⣰⡇⠀⠯⠄⠀⢀⣿⣦⠀⠈⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣀⣤⠤⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢙⠘⠿⣯⠁⠀⠀⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈⡄⢠⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠘⢆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢣⠈⣷⠀⠀⠀⡀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⢸⣷⠀⠀⠀⣤⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣬⣿⣤⠉⣾⢰⣆⠻⣚⣛⠁⠀⢴⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⣀⣻⣿⡆⣶⠉⠓⣶⢣⡘⢿⢢⡭⢁⡀⣿⣌⢱⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡘⣿⡇⢉⠩⣤⣤⡈⠀⣧⢉⡋⢻⡞⣰⣾⡿⢀⢿⣿⢈⡁⣰⡄⣶⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⣬⡀⠟⣶⠀⢭⣬⡶⣤⡸⢳⡎⣷⢛⡋⠀⢸⣷⠀⢛⡁⣿⣇⡉⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⣦⡀⠠⣭⣤⡤⡉⣾⠀⢹⡇⠨⣤⣪⡁⣿⠛⢡⣦⠷⢈⣀⡍⣁⣼⣋⣛⠛⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡗⠙⢹⡇⡀⠉⠁⢘⢦⡼⢻⣰⣷⠨⣥⡙⣿⢠⡞⢿⣴⣣⠴⢺⡅⠈⠛⠿⠓⠀⢸⣿⣀⣴⠂
⠀⠀⠀⠀⠀⣀⠀⠀⠈⠛⠻⠃⠠⡼⠇⠸⠆⣿⡆⡙⢾⣜⠁⢶⣌⠁⣿⢈⣷⠿⣷⣤⡛⢂⠀⣜⢣⡄⢰⣦⡘⠟⣩⣵⠆
⠀⠀⠀⠀⠀⠸⣷⡄⣦⠀⣌⠛⠃⢹⡍⣛⣇⡪⣹⠇⣶⡙⣷⡜⠿⢶⣿⣿⠃⢠⡌⢉⠉⢸⣷⣉⠴⢛⢈⠿⡷⠘⣛⡋
⠀⠀⠀⠀⠀⠰⢾⡄⢉⠰⣿⡆⣷⣄⢷⡈⠉⢤⡬⠓⠾⣤⣘⣿⣆⣼⣿⡏⣀⠳⠕⣫⡄⣦⠙⡿⢚⣋⡘⢇⡜⢙⣋⠁
⠀⠀⠀⠀⠀⠰⠾⠿⠗⠑⢮⣄⣙⠿⠈⢳⣄⣀⣚⣛⡛⠀⠹⣿⣿⣿⣿⢣⣼⠷⡞⣩⡤⣉⡾⢳⡿⢟⣥⠟⣴⣿⢏⣾⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣃⣤⣭⡝⢛⣛⣛⢋⣹⠿⠿⠿⣶⣼⣿⣿⣿⠈⠿⣴⣧⣽⣿⣿⠷⠶⠶⠿⠧⠤⠽⢥⢠⣅⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠛⣠⣤⣉⠝⣭⠸⡆⠀⠀⠘⣿⣿⣿⣿⠀⣴⠟⠛⠉⢩⡍⣙⠲⣟⣧⡄⢾⣷⡌⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣶⡆⢿⡇⠁⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠛⢸⣿⢀⡓⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣾⠈⢂⠀⡀⣤⡵⠄⢄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠻⠿⡿⢷⣖⣋⠘⢓⡃⠀⠙⠓⢬⣋⠑⠁
⠀⠀⠀⠀⠀⢀⡄⢰⣦⠀⢢⡯⣻⢻⣭⣷⣄⠙⠷⣶⣤⣤⡀⠁⠀⢈⢄⡀⡂⠀⠀⡀
⠀⠀⢀⣀⡀⠂⣵⠛⠙⢷⡯⠀⣷⢈⠢⣿⣿⣧⣤⣠⣠⡋⡛⠻⠿⢿⠋⢰⠿⢿⣷⣶⣶⢷⣧⡀
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠸⠦⠝⢦⣁⠾⠻⠭⠟⡿⢿⡁⠞⡃⢼⠶⡶⡘⠓⡐⡞⢿⢋⢯⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠊⠙⠓⠓⠂⠴⠤⠥⠔⠁⠀⠀⠀⠈⠆⠏⠠⠐⠺



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀⠀⢀⠤⠔⠒⠚⠛⠃⠒⠢⠤⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠘⠿⢿⣿⡻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠦⢤⣀
⠀⠀⠀⠁⠈⡃⠀⠀⠀⠀⠀⠀⢙⣀⡀⢰⣄⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡂
⠀⠀⠀⠀⡀⠁⠀⢄⠀⠀⣰⣿⣸⠏⠁⣰⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⡀
⠀⠀⠀⠀⠆⠀⠀⢀⣳⣤⡍⠉⢠⢆⣴⣿⣿⣷⡄⢠⣾⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢷
⠀⠀⠀⠀⠀⠛⠻⠟⠟⠋⠀⠀⠉⠀⠀⠀⠀⠘⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠶⠾⠯⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠃⠀⠀⠀⠀⣰⣾⣄⣀⠀⠀⢀⣴⣶⣿⣧⠴⠀⠀⠀⠀⠀⢰⣿⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⣴⣿⣿⣾⠿⣾⣶⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⡀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠉⠉⠉⠛⠛⠋⠉⠉⠙⠻⣶⡀⣤⡀⢸⠅⢱⡀⠀⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡏⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢹⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠘⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠻⠀⣠⣶⡴⠃⡀⠀⣤⣦⣤⢤⡤⠀⠀⠁⠀⠀⠶⠶⢦⣉⣡⣠⠼⡿⢿⠿⠷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⢿⣿⣿⣿⡿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⠿⣿⡿⠛⠉⠁⠀⠀⣀⡄⢰⣶⣦⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⠀⢀⡀⠀⣀⣠⣠⣶⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿⣿⣶⣿⣿⣿⠿⣿⡿⠋⢉⣠⣼⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣆⣉⡉⢉⡁⠹⢏⣤⣶⠌⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠉⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡃⠙⠛⠃⠉⠉⠰⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠘⣿⡿⠿⢿⣿⣦⡄
⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⢀⣀⡤⢤⣄⣀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠟⠁⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣇⣴⣿⣶⣷⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⢿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣧⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣯⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⣍⣙⠻⠛⠛⣩⣼⡿⣱⣿⣿⣿
⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⡻⢷⣶⣿⡿⣋⣴⣿⣿⣿⠇
⣧⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣦⣨⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠀⠀⠀⠀⠈⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠽⠛⠻⠷⣶⣤⣄⣀⣈⣩⠍⠛⠛⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢀⣠⣿⡗⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠰⣦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣀⣀⣭⣅⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠑⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⣾⣿⡛⠻⣿⣿⣿⣶⣄⡀⠙⠲⠶⠶⠶⠖⠛⠩⠕⣒⣀⣀⠐⠲⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢻⣿⣿⣫⣼⣿⡿⠟⠛⠛⠛⠿⣶⣦⣤⣤⣤⣶⣾⣿⣿⣿⡿⢿⣯⡛⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⢀⣿⣟⣾⡟⣽⡟⣠⠾⠓⠀⣀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠁⠀⠀⠉⢿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠁⠻⠀⠸⡿⡿⣿⣱⣿⣗⢿⣼⣿⣿⣿⣿⣿⣿⣤⣧⣤⣤⣶⣶⣿⣿⣿⣾⣧⢸⣷⣅
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠳⠤⣿⣿⣿⣟⢸⣿⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣾⡿⡉⠙⠲⣤⡀
⠀⠀⠀⢀⣠⡴⠾⠿⠛⢻⣿⠗⠀⠙⠻⠆⠻⢿⣿⣿⡘⠛⠛⠙⠉⠛⠋⠹⠛⢻⣿⣿⠿⣿⡿⠛⠃⠘⠛⣿⡇⠙⠷⠂⢸⣿⣦⡀
⠀⢀⠴⠋⠁⠀⠀⠀⢠⣿⣿⣤⣄⡄⠀⢀⡄⠘⣿⣿⣷⣄⠀⠀⠒⠐⠒⣠⠢⣄⣥⠬⠓⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⢀⣼⡿⠉⠛⢦⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣟⡀⠀⠚⠁⣄⠙⢿⣿⣿⣷⡀⠀⢀⡞⠉⠳⠀⠀⠀⡒⠸⣆⢀⣀⣴⣿⢿⡇⠀⢠⡾⠋⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣷⠀⠀⠀⠘⢶⣌⠻⣿⣿⣿⡄⡏⢠⣶⣶⣦⡀⣠⣤⣤⣌⢿⣿⢻⠏⢸⡇⣰⠋⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⢿⣦⠈⣻⡟⠀⠳⣄⠉⠻⣿⠇⠿⠿⠋⢁⣾⠁⠘⠀⢸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠇⣴⣿⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠠⠟⠹⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣄⠀⠀⠀⠀⣿⡻⣦⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⢻⡧⠹⣿⣷⡶⠶⢶⠶⠶⠶⣶⣤⣴⡾⠁⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠙⠄⠈⠛⠿⣿⣶⣶⣶⣶⣶⣿⠟⠁⣠⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠿⣷⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣠⣾⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⡦⠀⠀⠀⠀⠀⠀⠀⡩⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⢄⡀⠀⠀⠀⢃⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠁⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡠⠠⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠀⠀⠀⠀⠢⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠁⠀⠀⠀⠀⠀⢀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠱⠀⠀⠀⠀⠀⠈⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠄⠠⠀⠀⠀⠌⠀⠀⠀⣤⠲⣦⠀⠀⠀⠀⠀⣠⡶⢢⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⡐⠀⠀⠀⠀⣿⣿⣿⠂⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⡐⠀⠀⠰⠀⢃⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠊⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠁⠀⠀⢰⣄⠀⠀⡀⠀⠈⢰⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⢀⠔⠃⠠⠄⢀⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⡀⠀⢀⠄⠈⠉⢀⠀⠀⠀⠂⠈⠐⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⠀⢔⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⠃⠉⠄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠋⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠑⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠁⠸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠢⠘⠀⠑⠀⠒⠀⠀⠀⠈⢀⠀⠀⠀⠇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢠⡇⠀⠂⠀⢀⠂⠒⠀⠚⠇⠈⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⢣⡀⠀⠀⠀⠀⣀⠀⠈⠆⠀⠀⠊⠀⠀⡄⠀⢀⠄⠀⠤⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣿⠿⠿⠿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣛⣿⣿⣿⣻⣿⣿⣟⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠎⣂⣠⣤⣦⣴⣭⣦⣤⣶⣄⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣷⣧⠼⠛⠛⢋⢉⣥⣍⣭⣩⣿⣽⡿⢿⣻⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠿⢋⣀⣤⣶⡀⢿⣅⣷⣯⣿⣴⠿⠷⠿⠶⠿⠗⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠟⣡⣤⢸⣿⣬⣿⠷⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢰⣿⣠⣬⢿⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠆⠘⣿⣭⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠒⠀⠙⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⠆⠀⠘⢿⣾⣦⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠰⠆⠀⠸⣿⣿⠳⣿⣦⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠂⠟⠀⠀⠙⣷⣎⠉⠳⢿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠀⠐⠃⠀⠈⠙⢷⡦⠚⠉⠳⢿⣿⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡆⠀⠘⠃⠀⠀⠾⢶⣤⡄⠶⠏⠢⢽⡟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡔⠄⠀⠠⠗⠀⠀⡙⣦⣤⣀⠰⣦⠛⢄⣿⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡶⠳⣤⠀⠦⠉⠙⠻⣄⣀⡀⠀⣤⡽⢟⣹⡿⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠛⣿⡇⠀⠈⢷⣆⢠⡄⠀⠿⣏⠁⠀⠆⠀⣈⠙⣷⣼⣻⣿⣿⠿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢰⡟⡇⠀⠀⠘⢿⣤⡀⢀⠀⠈⢷⡿⠆⠀⠉⠗⠙⢷⣿⢺⣿⣿⡿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢰⣘⣇⡏⠀⠀⠀⠈⣿⡄⠉⠀⠀⠛⢷⣘⣄⡑⠀⣠⡀⢻⣌⢻⣿⣷⡶⣎⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣦⠘⣧⠸⣧⡀⠀⠀⠀⢈⣷⠀⠰⠀⠀⣸⣿⠛⡏⠁⠙⠏⠉⢯⣿⣿⣿⣿⣜⣧⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⣆⣹⡄⢹⣆⠀⠀⠀⣀⣆⠀⠀⠀⠀⠈⢻⡐⢰⣄⠂⢠⡀⠘⣿⣄⣻⣿⣿⣀⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢷⣄⡙⠀⠀⠀⢺⣿⠀⠀⠋⠀⠀⢸⡟⠉⠋⢠⠉⠉⠈⣹⠟⣝⣿⢿⣿⣿⣿⣷⡀
⠀⢀⢤⡖⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣤⠿⣿⠀⠀⡀⠀⠀⢹⣷⣤⣦⠀⣰⡄⠀⢻⣶⡯⣿⣿⣿⢻⣼⣿⣟⠢⡀
⠀⢹⣶⢾⠁⠛⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⠀⠀⣸⡟⢩⠉⡄⢹⠉⠀⢿⣁⣅⣾⣿⣿⢿⡙⣧⠙⢷⣷
⠀⠠⠿⣾⣰⣦⡤⣀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢠⡴⠀⠀⠀⣿⣧⣾⡀⣡⡆⠀⢸⡞⢙⣧⣼⣿⣿⣽⣧⣽⢷⡆
⠀⠀⠘⢩⠻⢿⣷⣦⣿⣯⣟⠲⠶⣄⣂⡤⢀⣠⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠁⠀⣄⣴⠋⡠⣉⠀⡙⠁⢲⣿⠂⡿⣈⣿⣿⣿⠻⠈⠗⠀⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣱⣿⣙⣛⡷⣮⣭⣽⡿⢶⣿⣭⣽⡎⡩⣯⡳⣆⠀⢀⣠⡞⠀⠞⣠⣠⣼⢿⣧⡄⢀⣠⠎⠀⣠⡟⢁⣿⡟⢻⠿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠿⡹⢦⣿⡨⣿⣤⡈⢿⣴⣆⡀⠁⢹⣷⣦⣿⣿⣾⠻⣿⣍⠙⠳⠾⠿⢯⣤⣘⣉⡱⠟⢧⣤⠞⠃⠀⠊⠀⠀⠀⠘⠁⠋⠹⢙⡟⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⠉⠉⢦⡙⢯⡎⠹⢿⣧⣿⣼⡿⣿⡙⢉⣿⣿⣶⣿⣄⢾⣮⣲⣦⡀⠙⣿⣤⠶⠊⠃⠀⣤⡴⠂⢀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠘⣹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣿⣷⣤⡋⣛⡻⠷⡿⠷⣞⠛⢛⠷⣾⣽⣫⡛⠹⡟⣽⣆⠘⣷⢀⢀⠠⣼⢟⢀⡶⢈⡄⣸⠀⡖⢠⠆⠀⠀⠀⠀⠈⢛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠛⡿⠿⢿⣷⣶⣤⣴⣾⡿⢟⢻⣷⣾⣈⣙⠿⣬⢫⣿⡄⢛⣿⡞⠛⠃⢻⡌⣷⢹⡆⢻⠙⣿⢺⡿⠻⠀⠀⠀⠄⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⡀⣀⣤⠖⠁⠛⠉⠀⢸⣿⣁⣠⣜⣛⣻⣿⣿⣿⣷⣌⢻⣏⢘⡏⠁⢀⣰⣼⢷⣿⣿⣷⣿⣼⣿⣿⣷⠃⠀⠀⢀⠀⠰⢾
⠀⣀⢤⣱⣃⡀⣀⢠⣤⣴⣶⣶⡶⣾⣿⣿⣛⣁⠀⣁⠠⣄⡴⠟⢃⣉⣩⣭⣭⣭⣭⣍⣉⣍⣩⣾⡥⣿⡷⠞⠻⣙⣦⠛⣡⣿⣿⠃⠀⠀⢸⣟⠁⠀⢀⡈⣠⣛⣿
⣆⠛⣼⣯⠿⠿⢿⡿⣿⣯⣿⡞⢷⣛⣉⣙⣛⣛⣉⣙⣛⣯⣴⡞⠿⣋⡉⢉⡉⣻⣏⣽⠿⣭⣬⣉⣿⣿⣧⡼⠶⣋⣤⡟⣿⡿⠃⠀⠀⠀⢸⡇⠀⠐⠆⢡⣬⣿⡏
⠙⠷⠿⠾⠿⠿⣿⣴⣶⡶⠾⠿⠟⡛⣏⡛⡙⢿⠛⡟⢻⣟⠰⢛⣤⢉⣧⡸⠟⣫⢾⡧⣴⣦⣶⣿⣧⣴⢼⡶⣟⣯⣵⡿⠋⠀⠀⠀⠀⠀⡸⠃⢠⠋⣀⢶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠛⠛⠻⡿⣷⣿⣦⣿⣿⣨⣷⣺⣭⣶⣿⣷⢾⣷⣷⣾⣿⣶⣷⡾⠿⠿⠿⠚⠉⠀⠀⠀⢀⣠⣖⣿⠶⢋⣹⡿⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢄⣘⡟⠳⣿⣿⣵⣫⣿⣿⠟⠛⠿⠿⠿⠿⠿⠛⠋⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣀⡞⢋⣵⠉⢃⡾⢟⣡⣶⡟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢵⣾⣋⣥⣤⡞⢉⣌⣛⠿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢫⣴⣾⢁⣴⡟⢁⣤⡆⣿⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢉⡞⠛⣩⠟⠛⠁⡟⠋
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠓⠲⢤⡀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠒⠒⠲⠤⣄⡀⠀⠀⢀⣠⡤⠤⠶⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠒⠒⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⣠⡀⠀⠀⠀⠀⠙⠦⣄⠀⠀⢈⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠺⡁⠙⠦⣄⣀⠀⠀⠀⣀⣩⡗⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⢧⡀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠀⠳⣄
⠀⠙⢆⠀⠈⠉⠉⠙⠉⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⠤⢤⣀⣀⣀⣀⡤⠔⠋⠁⠀⠉⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠙⠲⠤⢤⣀⣀⣀⣀⡤⠤⠖⠚⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⡤⠄⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠙⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢻⣸⡇⠀⠀⠀⠀⠀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠀⠉⠉⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣔⣩⣁⠠⠅⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣭⣯⣤⣾⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡀⠀⣠⢜⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠧⣾⣥⣤⣼⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠠⠀⠄⣀⣁⠜⣁⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠧⣶⡏⣤⣤⡟⠛⠻⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡓⣿⣒⠭⢄⣊⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢑⣶⠧⢤⣄⡽⠛⠳⣾⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠮⣥⣽⡟⠓⢦⣴⠟⠛⢷⣦⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢟⣉⣭⡗⠚⠛⣤⣬⠿⠯⡉⠙⣶⡶⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⣤⡤⢖⡋⣁⣨⡿⢯⣁⡈⢙⣦⠼⠟⠉⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⡤⠀⠀⠀⠀⠀⠒⠊⢠⣤⡄⠈⠛⠛⢤⣄⣀⡼⠿⡍⠀⠁⠀⠀⠤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠉⠀⠀⠀⠞⠀⠠⣤⠔⠋⠁⠀⠀⣤⠄⠚⠙⠅⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠐⠁⠀⠠⣄⠈⠑⠀⡤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⢀⣴⠾⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢤⣀
⠀⠀⣞⠈⠁⠀⠠⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣔⠂⠤⠀⠀⠀⠋⠙⡆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣀⡀⣠⢄⡀⠀⢢⣿⣷⡖
⠀⠀⢫⠟⢶⣦⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠿⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠈⠛⡙⢿⡁⠀⠀⠉⠀⠀⠈⠑⠒⠚⠃
⠀⠀⠈⠀⠈⠾⠀⠿⠀⠈⠟⠉⠳⣶⣶⣶⣿⣶⣦⣤⣠⣤⣤⣴⣤⡀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡹⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠃⠀⢈⣿⣿⣿⣿⣿⣿⣷⠃⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢠⣤⠀⡰⠀⢠⠷⣴⣧⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡠⠘⣧⠄⠀⠽⠛⠉⠁⢀⣈⡹⢫⠇⠀⠀⠀⠀⠀⡻⠿⠷⠂⠚⠋⠉⢻⠁⠀⠀⢐⠃⠁⠀⠁
⠀⠀⠀⠀⠀⢀⠀⡟⣤⠟⣣⠀⢨⠔⠂⠀⠀⣶⠀⠰⣚⠯⠅⠀⠀⠀⠀⠀⢀⡸⠁⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠺⠏⠉⠁⠉⣀⠀⢐⡶⠂⠀⠒⠋⠀⠈⢀⣀⣀⣀⣤⣤⣠⡶⠟⠁⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⣄⠀⠲⠴⠴⠿⠂⢀⣤⣦⠀⣉⠥⠀⠚⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠘⠀⢰⡀⠀⣀⣤⡀⠀⣦⣙⡄
⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢣⡀⡸⠁⠙⠲⣼⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡟⢰⣶⣦⣀⠀⠀⠀⢠⣿⣿⣷⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⣿⣿⣿⡏⠀⠀⠀⠈⠻⡿⠿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⣠⠹⣿⣧⡀⠀⠀⠀⠀⠄⢤⠙⢿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡈⢻⣷⡀⠀⠀⠀⠄⠀⠀⠈⢻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢻⣿⡆⠀⠀⠀⠀⡂⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠻⣿⣆⠀⡒⠉⣀⡀⠀⠀⣰⠊⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣷⡀⠹⣷⣎⠀⠀⠚⠀⠀⠀⠣⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢹⣿⣿⣷⡀⢹⣿⣧⠀⠀⠀⠀⠀⣐⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⠆⢸⣿⣿⣿⣷⡀⠸⣿⣧⣘⠂⢊⠐⠟⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣿⠀⢸⣿⣿⣿⣿⣷⡀⠙⣿⣷⠄⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣿⠀⢸⣿⣿⣿⣿⣿⣷⡀⠘⢿⣧⣆⠀⣶⠒⠑⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠉⣛⡸⣁⣴⣶⠞⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣴⠟⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⡗⣿⠀⢸⣿⣿⣿⣿⢹⣿⣿⣿⢫⣿⣿⣧⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡟⣿⠀⢸⣿⣿⣿⣿⢸⢿⡟⢁⣾⣿⠃⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣗⣿⠄⢸⣿⣿⣿⣏⢸⠛⢁⣾⣿⠇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣶⣼⣿⣿⣿⣿⠈⠀⣽⣿⠏⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⣿⣿⣿⣿⡏⢀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡷⣿⣿⣿⣿⣿⡿⠀⣸⣯⡿⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣯⣿⣿⣿⠁⣼⣷⡿⠁⠀⠀⠀⢀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣿⣿⣿⣿⠁⣰⣷⣿⠇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣇⣿⣿⣿⠃⢰⣿⣿⢋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣹⣿⠏⣸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⡄
⠀⠀⠀⠀⠀⡀⢠⣼⣿⡿⠃⠈⢋⣰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢸⣅⠀⢀⠄⡀
⠀⠀⠀⠀⠙⢻⡟⣿⣿⣷⣖⣾⣧⡿⣻⣿⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⣧⣦⡴⡿⠏
⠀⠀⠀⠀⠀⠀⠠⠶⢾⣻⣿⢿⡏⢺⣿⣿⣿⣖⡲⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⢁⡤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣶⣶⣶⣖⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢤⣀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣻⣿⣆⣄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⠼⣿⣾⣿⣟⣿⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⢀⣠⣠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣶
⠀⠀⠀⢰⣾⠿⣿⢿⣿⣿⣞⣳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣛⠿⠛
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣽⣿⢯⣯⣯⣿⣿⣩⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⣙⣛⣿⣿⣿⣶⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣿⣯⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠟⠉
⠀⠀⠀⠀⢹⣾⣿⣿⢿⣿⣿⣿⡿⠿⣧⣿⡏⢻⣽⣿⣹⣿⡿⠋⢉⡽⣿⣟⣿⣿⣿⣾⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠻⣿⣾⣷⣿⣿⠝⠒⠚⠛⢷⣾⣻⡿⣹⣿⣽⠟⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠈⠏⠫⣷⣿⣿⡟⠀⠀⠀⠀⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠿⠿⠿⠧⠿⠿⠿⠿⠿⠶⠶⠦
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢀⡌⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢗⡖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⢤⠀⠹⣶⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⢸⣧⣁⠂⣀⣠⣴⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡶⠐⣦⠀⢷⣾⣿⣿⣿⢿⠟⢁⣴⣄⢹⣿⣿⣄⡙⢧⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣼⠃⠀⣸⣶⣾⣿⡟⢡⣤⣴⡛⠛⣽⠟⠋⢻⣿⣿⣿⣬⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⢀⣼⣿⠀⠀⣿⡑⣬⣿⣿⣾⠿⠟⠂⠀⠀⠲⠆⣼⡟⠋⠠⠰⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣠⣞⠁⠡⠢⢳⣿⣿⣦⡭⠀⠂⠉⠀⠀⠀⠴⢉⠀⠠⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣾⣿⣿⣿⡏⠀⠀⠈⣲⣍⣿⣿⣷⠀⠀⠀⠀⠀⢀⡇⢠⣶⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⣾⣿⣿⣿⣿⡅⠀⢀⣼⣿⣿⣿⣿⣿⣧⣀⣀⣤⡤⠞⢁⠾⠁⢀⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⢸⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣠⠈⠀⣴⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢀⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣟⣩⣤⣤⣄⣀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠉⠻⡿⠿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠛⠋⢉⣉⣅⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣆⠸⣤⡼⠟⠛⠿⠿⠻⠿⠛⣿⣿⣿⠿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠛⠛⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⣁⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠷⠓⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⠞⠛⠁⠀⠀⢀⠘⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣼⣿⣿⣿⣿⣿⡄⡀⠀⠀⠀⠄⠀⠿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤⣿⣿⣿⣿⣿⣿⣿⣷⢷⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠐⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠿⠿⠿⣿⣿⡿⠻⠟⣡⣤⣶⢶⣤⡀⠀⢀⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣠⣴⣿⣷⡄⠀⡁
⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⣴⡿⠋⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⡼⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠈⠀⠈⢠⠀⠀⠀⠀⠈⠀⠰⢖⡊⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠛⣀⣴⣶⣒⡂⠀⠀⠙⠋⠉⠉⠋⠁⡘⢀
⠀⠀⠀⠈⠐⠈⠀⢀⣠⣦⣤⣀⡀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢷⡼⣻⣿⡯⠿⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠿⣿⣶⣤⡀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⢀⠀⠚⠛⠉⠀⢱⡆
⠀⠀⠀⢀⢀⣾⣿⣿⣿⣿⣿⣷⣦⣼⣿⣿⣷⣄⡀⠀⠠⣿⣿⣿⣿⡿⠟⠀⢀⠀⡏⡜⠁⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣛⠠⢿⠉⢸⡿⠀⠀⠀⠀⠀⠀⠀⢴⣿⣧⠀⠀⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣽⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⠂⣉⣼⠇⠀⠀⠀⠀⠀⠀⣠⣼⢽⣿⣆⠀⠈⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⠛⠛⢻⣯⢦⣀⣰⣿⠀⠀⠀⠀⠀⠀⠐⢫⣍⣹⣿⣿⣷⡀⡄
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⢛⡀⠈⡻⠉⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⡿⣿⡟⢴⣿⣿⢰⡀⠸⣶⠏⠀⠀⠀⠀⠀⠀⣴⣤⢿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⡿⠟⣿⠛⣿⣿⡅⠀⠉⠁⠀⢡⣾⠋⠀⠀⠀⡀⠀⠄⢸⣿⣟⣻⣿⣿⣿⣿⣿⣋⣀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠃⢀⠁⢐⡏⣴⣿⣿⠗⠀⠀⣀⣤⡾⠋⠰⠦⠐⠊⠀⠀⠀⣨⠙⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⠀⠀⡠⠄⠒⠲⢤⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⢎⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠑⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠤⠶⠒⠛⠛⠛⠛⠛⠒⠲⠒⠚⠋⠁
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⡾⠒⠋⠉⠉⠉⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣸⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠳⢶⣤⣤⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣿⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣔⢜⣾⠏⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣶⣤⣴⣤⣤⣼⣶⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⢸⡀⠀⠂⠀⠠⠤⠤⠤⠤⠴⠷⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⢻⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢤⠛⠳⠴⠴⢦⣠⢖⡥⢽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⠚⠛⠲⡞⠋⠉⢋⠀⣐⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠸⡄⠰⣼⡇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠛⠋⢉⠇⠈⢿⡉⠙⠛⠒⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠃⠀⠀⢈⣈⣿⣿⠃⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⣩⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠶⠾⠾⠷⠷⠾⠿⠛⠋⠁


⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣔⠀⣿⣿⢈⣿⣿⠄⣤⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣮⣿⣎⣻⣟⣦⣿⣧⣿⠿⢦⣶⣿⣿⣿⣿⣽⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⡀⣀⣠⣤⡾⣿⣻⣜⡷⢻⣃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣤⣄⣴⢦⣤⣶⣶⣶⡶⢶⣿⣶⣿⣿⣟⣻⣽⣦⣿⢮⡻⢿⣋⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⢒⡿⢯⠽⢛⣒⣋⣉⣾⡉⠽⢷⠛⣙⠫⢿⣲⠟⠁⣟⡕⣷⣿⢻⣾⣷⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢼⣛⣹⣯⣭⠿⠦⠛⠋⣐⡉⣍⡿⢠⣥⠴⡶⣭⣿⣿⣿⣽⣿⣿⣿⣿⣾⣿⣾⣿⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⠷⣿⡉⢓⡶⣄⠿⢲⣟⣿⣉⣼⢩⡝⢷⣾⣶⣹⠿⣿⣿⣷⣾⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⡿⢿⡳⣷⣼⢉⣭⣿⣶⣿⣷⣻⣿⡟⣿⣿⣽⣿⣿⣟⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠤⠶⣟⢰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡫⣧⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⡆⠘⠛⠁⣀⣀⣠⣿⣴⣿⣿⣿⣿⢷⣯⣿⣩⣿⣿⡻⡿⠛⠛⠋⠉⠁⠀⠀⠈⠙⠛⠙⠋
⠀⠀⠀⠀⠀⠀⢠⣴⣿⠟⠁⢠⣥⡤⠊⣛⣽⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣃⣀⣘⣧⣤⣶⡿⠷⠿⠛⠛⠛⠛⠉⠉⠀⠀⠀⢀⣤⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡿⣿⡿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋






⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠹⡀
⠀⠀⠀⢾⠓⠦⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢷⠀⠀⡠⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⢆⠀⠈⠳⡀⠀⢀⡎⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡼⠀⡜⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢳⠀⠀⢀⡴⠒⠉⣹⠃
⠀⠀⠀⠀⠈⠳⣄⡀⢳⠀⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⢹⣸⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⠀⡰⠋⠀⢀⡴⠁
⠀⠀⢠⡤⠖⠒⠲⠭⢟⣦⣿⠀⣸⠁⡠⢦⠀⠀⠀⠀⠀⢇⠀⠑⢆⢸⡇⣠⠋⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢳⠀⢸⣰⠧⠴⠚⠁
⠀⠀⠀⠉⠓⠦⠤⠤⠤⠬⢿⡰⠃⡼⠁⢸⠀⠀⠀⠀⠀⠈⢦⡀⠈⢿⡛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⢱⡀⠘⣆⣾⠓⠚⠉⠉⠙⠒⣢⠄
⠀⠀⠀⠀⠀⢞⡉⠉⠙⠒⢬⣇⢸⠁⢀⡞⠀⠀⠀⠀⠀⠀⣀⡈⠓⠺⡇⣠⠖⢻⡆⠀⠀⠀⠀⢸⡀⠀⣇⢀⠞⣀⣉⣓⣒⡒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠶⠺⣟⡠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠙⢆⠀⡷⠁⠀⡼⠀⠀⠀⠀⠀⠀⢣⠀⢸⣾⣉⣀⣀⣀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⣦⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢱⡏⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢃⣤⡀
⠀⠀⠀⠀⠀⢸⢰⠞⠁⠁⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠈⣣⣼⡉⣠⠄⠄⠀⣀⠀⠠⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⢸⡤
⠀⠀⠀⠘⠦⠀⠿⠂⠘⠃⠀⠀⠀⢆⠰⠀⠀⢀⢰⡆⠀⠀⣩⣷⠀⠀⠀⠀⣤⡀⣄⢃⣨⡶⠗⠀⠉⠁⠀⢀⠈⠳⠄⢀⠀⣀⠀⠀⠠⢀⠀⠘⠟⠁
⠀⠀⠄⡀⠠⡿⠤⠤⠔⠂⠀⢶⠄⠈⢶⡀⠐⠁⡘⠀⠀⠀⠉⢹⠀⠀⠀⠀⠈⠛⣤⣏⡏⠒⣧⠔⠀⠀⠀⠀⠁⠤⢮⢠⣤⡄⠀⠈⠦⠐⢂⡴⠆
⣀⠀⠑⠼⣶⠆⠀⣀⡰⠀⠀⠀⠀⡀⠘⣧⡔⠀⠀⠀⠀⠀⠘⠝⡇⠴⠀⠀⠀⠀⠀⢸⣽⠻⠃⣤⠀⠀⠀⠙⢦⡀⠈⣥⡅⠀⠀⠐⠶⠂⠾⠆⠀⢀⡀
⠀⠀⢠⣤⠅⠀⢀⡴⠀⠀⠀⠀⠠⡈⣤⡼⠀⠐⠒⢰⠇⠀⢀⢠⣧⠤⡄⠀⠀⢰⡟⠂⢸⠀⣀⡰⠆⠀⠀⠀⠀⠐⠄⠘⡁⠀⠀⡀⢠⡀⢀⠀⠀⠘⠊
⠀⠀⢰⠉⢁⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⣸⣀⣁⠠⠂⠀⠀⠘⠈⡁⠀⠀⠀⠀⠀⠘⠟⣛⣯⡌⠁⠀⠀⡶⠀⠀⠀⠈⣿⠀⠀⠀⠙⠛⣁⢨⠦⡤⢀⠬
⠀⠀⠀⠀⠻⠀⣀⠀⠀⠀⠁⠀⢢⠀⠀⠉⠀⠀⠤⢺⠃⠀⠀⠻⠟⠀⠃⠀⢀⠀⠀⣀⠈⢃⠀⡠⢪⡞⠁⠀⠀⠀⠀⠘⠀⠀⠠⣠⠀⠻⠂⣤⠀⠾⠆
⠀⠀⠃⣀⠀⠀⠀⠑⢄⠀⠈⣦⠀⠁⢀⡄⣃⠠⢀⠂⠀⠀⠸⠽⣷⡰⠀⠀⠈⠙⠦⠬⠽⢶⣔⡤⠊⠀⠀⢠⠎⡆⠀⠰⠀⠀⢠⠼⠦⠀⠀⠉⣀⡀
⠀⠀⠀⡌⠀⠂⠀⠀⠀⢢⠀⠈⠳⣇⣸⠁⠉⢀⠁⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠁⢃⡅⣀⠀⠀⠀⠈⠚⠁⡤⣄⢠⠉⠋
⠀⠀⠀⠆⠀⠩⡀⠀⠀⠀⡆⠀⠀⠈⠏⠁⠀⠁⠀⠀⠀⠀⠑⠊⠋⠀⠀⠀⠀⡤⠄⠀⠀⠘⠀⠀⠈⢇⠀⡆⢸⠃⡋⢀⠀⠀⠀⠐⣀⣿⠁⠀⡤⢌⡆
⠀⠀⠀⠘⡀⠀⢧⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢐⠀⠀⠐⠀⡀⠀⠀⢰⠀⠀⠀⡀⠀⠲⣆⢠⠀⠉⠂⠀⠀⠀⠠⠈⠀⢹⠁⡤⡑
⠀⠀⠀⠀⠀⠦⠘⠦⠀⢀⠃⠀⠀⠀⡆⠀⠀⠀⢈⣋⠀⠀⠀⠀⠀⠈⠄⢁⢰⠀⠀⠀⠀⠈⠀⠀⠈⠙⠁⠀⡹⠊⢀⠀⠁⢠⠞⡇⠀⠀⠀⠌⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠁⠀⢀⠀⠀⠐⠀⢀⡀⠉⠉⠰⢄⠀⠰⡄⣄⠈⢠⡌⠀⢸⠇⠀⠠⠀⠀⠀⠑⢵⣀⢻⠀⢂⠀⠠⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢑⠀⠀⠀⠈⠳⠀⠀⠑⢬⡃⠀⠀⠫⡁⡀⢣⡀⠂⠃⠀⠀⠀⠀⠰⣄⠀⠙⢢⡦⠂⠀⡡⠀⢠⠊⠇
⠀⠀⠀⠀⠀⡀⠰⣆⠀⢀⠀⠀⠘⢆⣴⡄⠀⠳⠇⠀⠀⠀⠀⠀⢀⡀⠀⠳⣘⢁⣅⡀⠐⡦⠀⠀⠰⡈⢆⢂⠀⠡⡀⣸⠃⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠐⡀⢈⣆⠈⠀⠀⠰⠻⢀⡀⢠⠶⢀⠀⠀⠀⠀⡀⠀⠆⠐⠦⢜⡆⣑⣀⠄⠀⠀⠀⠀⠈⠒⠶⣄⠀⠡⠀⠀⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠋⠀⠈⢆⠱⣅⠀⠀⡸⠁⠀⠀⠄⠠⠘⡗⠁⠀⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠢⣇⠀⠋⢀⡶
⠀⠀⠀⠀⠀⠀⠃⠈⡒⢷⠀⠀⠾⠀⠀⠀⠦⠀⠑⢾⠇⠈⢀⡀⠀⠀⢆⢀⠁⠀⠰⣠⠁⠠⠀⠉⠀⠀⠈⠫⣆⡃⠀⠘⡀⢀⣨⠌
⠀⠀⠀⠀⠀⠀⣤⠀⡀⠹⡇⣀⠆⠀⠀⠀⠈⢌⠄⢸⡆⠀⠘⠁⠀⠀⢀⠑⣤⡔⡐⢡⡎⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠀⡟⠉⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡟⠡⢂⠏⠀⠀⠀⠀⠙⢾⡇⡴⠰⡄⠀⠇⠈⠃⢸⢪⡴⠊⠁⢀⡄⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⡧
⠀⠀⠀⠀⠀⠀⠀⣤⠈⡘⢿⠔⣃⠀⠀⠀⢂⠀⠀⢸⡟⢁⡠⠁⠀⠛⠷⣷⠀⡼⠀⠀⡄⠘⠀⠀⠀⠀⢊⡁⠀⠀⠀⠀⠀⠀⠐⠠⠀⢱⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⡇⠀⠁⠀⠀⠀⠈⠢⡀⢄⣧⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢰⠀⡠⠁⠀⠠⠀⡤⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠘⢢⡀
⠀⠀⠀⠀⠀⠈⠣⡀⠀⣀⡅⢐⠂⠀⠀⠀⠀⠄⢈⡀⡟⠠⠐⠹⠁⠀⠀⢰⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⢈⠁⠀⠀⠐⠃⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢈⠓⠻⠃⠍⣀⠀⠀⠀⠀⠀⠀⠈⢟⠐⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠃⠀⠀⠀⡤⡄⡠⠀⠨⠀⠀⠀⠤⠀⠀⠀⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢇⢰⣿⣠⠍⠀⠀⠠⢀⠤⠀⠀⡏⠀⠀⠀⠴⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠐⠼⠧⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⡠⢀⢍⠜⠀⠀⠀⠀⠀⠓⠀⠀⠀⡇⠀⠀⠀⠠⠀⠀⠀⠀⠀⢶⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠶⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣤⠀⢰⣦⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡙⠛⠃⠘⡿⠀⣾⣿⠃⠀⢀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠛⠁⠀⣶⡄⠈⢢⡉⠀⠀⠀⣟⡰⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⡀⠀⠸⡇⠀⠀⠀⠁⢢⠀⣧⡴⠚⠋⠀⠀⠰⣷⣞⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠐⢤⡃⠀⣴⡄⠀⠀⣷⠏⢠⡶⠀⠀⢀⣀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⢧⡀⢿⠇⠀⢸⡿⠊⠀⠀⢀⠰⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠗⠐⠊⣩⡍⠉⠛⣶⣤⡀⢸⡇⠀⢀⣠⠾⠒⢒⣀⠀⠴⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠂⣀⣉⣀⣀⠞⠁⠈⢻⣾⣿⣴⣯⠥⣤⠄⠈⠻⠷⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠁⠀⠀⢠⣄⠀⠀⢻⣿⣿⢀⣀⠀⠀⠀⣤⠲⠦⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠈⠀⠀⢸⣿⡇⠈⠉⣷⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷

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

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠠⠤⠀⠦⠤⠴⠔⠤⠀⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠄⢀⠀⠅⠠⠈⠀⠃⠘⠒⠐⠘⠛⠛
⠀⠀⠀⠀⠀⢠⠏⠸⢐⡴⡨⠂⠁
⠀⠀⠀⠀⠀⡾⠈⠂⡈⠊⠩⠐⠶⠒⡦⠐⢦⠤⠄⠠⢤⣀
⠀⠀⠀⠀⠀⣇⠀⡀⡐⠁⠈⠄⠁⢃⠐⠒⠀⠢⠠⠭⡀⢘⠙⢲⢤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠹⡗⠐⠤⠐⠐⠄⢀⠀⠀⠀⠘⠀⠓⠀⠀⠀⡁⠁⢐⣄⠀⠉⠀⠒⡀⠂⠐⠢⠀⠴⠊⠀⠈⠀⡂
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠠⠂⡀⠀⠀⣈⠄⠃⠀⠀⠢⠀⠀⠠⠀⠐⠀⠖⠀⠈⠋⠀⠀⣀⣤⣤⣤⡶⠻⠛⠙⠁
⠀⠀⠀⠀⡴⠒⠾⡄⠀⠀⠐⣻⡀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠀⠈⠀⠀⠀⠔⠋⢉⠛⠛⢿⣍⢌⠆
⠀⠀⠀⢠⠀⡜⢦⣀⣀⣀⡿⠃⠉⠓⠶⢤⣄⣀⣠⠐⠊⠳⣄⠈⠀⠁⠢⡀⠳⢄⠂⣈⠀⠛⠓⠽⢲⡦⣀⡀⠠
⠀⠀⠀⢀⡃⣠⡰⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⣤⠦⠗⠦⣀⣀⢘⣦⣄⠁⠺⠿⣖⡤⢄⣀⠈⡙⡿⣷⡄
⠀⠀⠐⠛⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠁⡩⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⠋⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠆⠀⠀⠀⠪⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⣉⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⡶⠶⠿⠿⠛⠛⠋⠛⠛⠀⠀⢤⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⢦⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢾⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠄⠀⢼⣿⡆⠀⠀⠀⠀⠀⣤⠔⠋⠁⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⠀⠀⠴⢠⣷⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢾⣯⢀⡄⠀⢀⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠿⣿⣿⣿⣿⣿⡿⣋⣼⠷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠛⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⣿⠛⠏⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣁⣾⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⡧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠉⠛⢀⠀⣹⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣦⢠⣄⣹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠈⠛⢷⣶⣶⣶⣶⣶⡶⠦⢴⣶⣶⣶⣶⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢶⡶⢶⠶⠔⠶⡄⢤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣛⠛⢁⣀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠉⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢲⡔⢂⠀⠸⠀⠀⠀⠀⠉⢠⠢⠀⠀⠀⠀⠠⠀⣝⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡏⢈⣽⣶⣶⡄⠘⠠⠃⣀⠀⠀⠀⢠⡀⠀⠀⠀⠂⠈⠀⠀⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣰⣿⣿⣿⡿⣗⠄⣀⣶⣿⣇⣼⣦⣦⣿⣿⣧⡀⠀⠀⡀⠡⠛⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠛⣋⠉⣙⡟⠿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡆⠐⣀⣿⢈⣿⠉⣿⣿⣷⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠛⠛⠯⠉⢋⡓⠀⠀⠀⠙⠛⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⢼⠋⡀⢛⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣷⣶⣆⠈⠀⠀⢀⣀⣴⣤⣵⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠳⠙⠿⠟⠛⣿⣿⠿⡿⠿⠛⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⠋⠉⠀⠀⠀⠞⠛⠁⠁⠀⠀⠐⠋⢈⣿⣿⣿⡯⠁⠀⠈⠃⠁⠀⠀⠂⡈⠹⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣿⠫⡄⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣿⣾⣿⣿⣿⣿⣶⡖⠦⠈⠀⠀⠀⠁⢰⣿⣿⠃⠁⠀⠀⠀⠀⠳⠀⠀⠀⠀⢰⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⠿⠿⣿⣿⡿⣿⣿⡶⢂⡀⠀⠀⢀⣤⣽⣿⣿⠃⠀⠀⠀⠀⢠⡸⣄⠀⠈⢷⠁⠈⣛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⢆⠠⠃⠋⡚⣳⣴⣿⣿⣿⣿⡦⠀⠻⢷⣿⣿⣿⣆⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠈⠀⠝⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡤⠐⠂⠈⠉⣉⣵⡾⢻⡿⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⢀⠀⠁⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠂⠀⠴⠛⡫⠟⠀⠈⠀⠀⠀⠀⠐⠃⠘⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⡄⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠀⠈⡠⠎⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠔⠈⠇⢰⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠔⠀⠀⠸⠀⠀⠀⠀⠚⠀⠷⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠗⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠⠄⠠⠀⠀⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠂⠀⠀⠁⠀⠂⠀⠠⡆⠀⠀⢀⠀⡠⠀⠀⢢⠀⠀⠀⠀⠀⠀⠄⠈⠀⡀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠰⠃⠀⢂⠀⠘⠒⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠗⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠆⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠆⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⣀⣀⣠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⠊⠀⠀⠀⠀⠀⠐⠟⠠⠌⣹⣻⣿⡯⡾⠻⠄⠉⠁⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣨⣖⣶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠁⠠⣾⣿⠃⢹⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⢨⡄⠀⡐⠢⠀⢨⠀⢛⣀⣠⣾⡟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠈⠁⣒⠒⢠⣆⠀⠀⣾⠋⣽⣿⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⡧⠁
⠀⠀⠀⠀⠀⢀⡡⠈⢉⢿⡽⠀⢀⣿⣶⣿⠈⠀⠀⠀⠀⠀⠀⠈⠂⢀⣠⡶⠊⠀⠰⣆
⠀⠀⠀⠀⠤⠥⠐⠲⢤⣷⡧⠀⣸⣿⣟⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣾⠋⠀⠀⠀⠀⢢⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠰⡒⣉⢐⠿⠟⠁⣠⠟⠉⠀⡇⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠁⠀⠀⠀⠀⢠⣾⠟⣆
⠙⠒⠀⠁⠈⠙⡓⢶⡶⠒⠈⠀⠀⠀⠀⠐⣶⣾⣾⣿⣿⣿⣿⡿⡌⠀⠀⠀⠀⠀⠀⠀⢸⡔⠀⠀⠀⠀⠀⠀⢀⠉⢃⠀⠀⠀⠀⠀⠀⣹⡏⠀⢠
⠀⠀⠀⠀⠀⠰⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⡉⠃⣻⣿⣿⣷⣿⠻⠿⡀⠀⠀⠀⠀⢠⠀⠈⡗⠄⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢁⣄⠋⣿⣿⣿⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⢹⣂⠀⠀⠀⠀⡄⠃⢇⠀⠀⠀⠀⢀⠔⡞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⣿⣹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠈⢻⣿⣿⡀⢄⠀⠀⠀⡼⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠈⠀⢢⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⣻⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢯⣷⠈⠀⠀⡼⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡏⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢠⠂⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢻⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⡠
⠀⠿⠦⡤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡀⠀⠀⠀⠐⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢧⠄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣷⠊⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⢧⠁⠀⢤⣱⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠉⠉⠉⠀⠀⠁⠀⠀⠀⢻⡀⠂⠀⠀⠉⠁⠠⠀⠀⠀⠀⣡⠣⠒⠀⠀⠀⠀⠀⠀⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠈⠉⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡿⠟⠋⠁⡀⠈⠛⠿⣿⠿⠛⠉⢀⠈⠙⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢀⣠⣴⣿⣿⣷⣤⣀⠀⣀⣤⣾⣿⣷⣦⣄⡀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠠⣀⠄⠐⠀⠈⠉⠐⠂⠤⡀
⠀⠠⠆⠀⠀⠀⠀⠀⠀⢄⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠂⠐⠐⠂⠠⠤
⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⡂⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢛⠋
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤
⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⠂⠔⠀⠀⠀⠀⠀⠀⢠⢤⡁⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⡀⠀⠑⠀⠀⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠃⠂⢀⡀⠀⠊⡁⠀⠀⠀⠀⢠⠋⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠀⠂⠀⠀⠀⢀⡏⠀⢠⢀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⡄⠀⠀⠀⠈⠐⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⣀⠄⠀⠂⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠄⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠠⠄⠰⡎⠀⠀⠀⠀⡘⠀⠀⡂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠃⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠀⢀⠀⠀⠘⡀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⢠⠀⡆⠀⣀⢀⡲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠘⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⡆⠀⡆⢦⠶⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁⠀⢠⡈⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠱⡀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈⠦⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠿⠿⠿⠿⠾⠾⠇⠒⠂⠒⠲⠂⠿⠟⠷⠿⠿⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣠⣄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡿⢡⣿⣿⣿⠃⠀⠀⠈⠁⠀⠀⢀⠄⠀⣀⣼⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣐⣶⡴⣾⣧⡄⢹⣿⡿⠀⠀⠀⠀⣤⣄⢀⣃⣴⣾⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⠟⢠⣏⠘⠏⠀⠈⠀⠀⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⢛⢳⣾⡿⠆⠀⠀⠀⠈⣷⣯⣯⠭⣿⣿⣿⣿⣶⣶⣿⣿⡿⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠿⠋⠀⠀⠀⠀⠀⣨⣷⣦⣬⣽⣿⣿⣿⣿⣿⣿⣿⡏⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⣠⣄⡀⠀⠀⠀⠿⠯⣭⣭⣿⣿⠃⠀⠛⠉⣻⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⣟⡶⠀⠀⠀⠀⢹⣿⣿⣁⠀⠀⠀⠀⠁⠉⢀⣐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀⠀⠈⠿⠃⠈⠀⠀⠀⠀⠀⠀⠿⠿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠴⠆
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠀⢀⣄⣻⣩⣫⣽⣧⣩⣘⡝⣤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠛⠓⠛⠛⠙⠒⠛⠃⠀⠀⠀⠀⠚⠒⠚⠀⠑⢄⠀⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠄⠀⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⠀⡠⠂⠉⠀⠀⠀⠋⠀⠈⠀⠀⠈
⠀⠀⠀⠈⡁⠀⠀⢀⡙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⣀⠀⠓⠊⠠⣄⠁⢀⡀
⠀⠀⠀⠀⠤⠀⠀⠈⠀⠀⢠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣧⠄⠀⠀⠰⣎⢱⠁⠘⡀⠄⠀⠀⠀⠀⠂⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⣠⠄⠀⠀⠀⠀⠀⠀⠈⠢⠀⡨⠋⠀⠄⠀⢸⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠄⡠⠒⡄⢀⠀⠔⠠⡀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⣠⡞⠀⠀⠀⠀⡖⠁⠀⠤⠀⢂⡈⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠄⡪⠙⠀⠉⡤⠤⠀⢀⡈⠢⡀⠀⠀⠈⠒⠐⠤⣤⣴⠟⠛⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡠⠀⠢
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡵⠀⠀⠠⠖⠀⠈⠀⠀⠀⠘⠃⠀⣷⠀⣀⣀⣀⣀⡼⠍⠀⠀⠀⠀⡀⠀⠀⠀⠸⠠⠀⠀⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⣠⠀⠆⢠⣀⠀⠀⢀⣀⣠⣀⡀⠀⢀⠇⠀⠬⡉⠉⠂⠀⠄⠈⠀⠔⠋⠀⠀⠀⠀⠀⠀⢀⣀
⠀⡀⠀⠀⠰⠀⠀⠂⠀⣤⠋⠀⠀⠀⠈⣿⠛⢿⡟⠙⠛⣝⡛⠛⠋⠀⠀⠀⠀⠉⠉⠉⠐⠖⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠑⢀⠀⠂
⠀⠀⠠⠈⠀⠀⠆⠀⠸⠁⠀⠀⠀⠀⠀⠈⢧⡀⠘⣦⡀⠀⠈⠲⡄⡀⠀⠀⡀⠀⠀⠆⢤⡀⠒⣄⠀⠃⠠⠠⠀⠀⠀⠀⠀⠀⠀⠳⠐
⠀⠈⠀⠀⠀⠂⣲⡸⠡⠀⠀⠀⠀⢄⣀⠀⠀⠙⠢⠌⡻⠷⣆⠀⠙⠸⣎⠉⠁⠀⠘⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⣄⠀⠀⠀⠀⡤⠓⠀⠉
⠀⠀⠀⠀⣰⠓⠁⠁⠃⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡄⢠⡔⠁⠀⠀⠀⠹⢄⠲⠀⠀⠀⠀⠀⠀⠂⠀⡄⠘⢄⠀⠀⠀⠑⠄⠀⠀⠀⠀⢀⡃
⠀⠀⠠⠋⠀⡠⠀⠐⠆⠀⠀⢀⠦⠀⣀⡆⠀⠀⠒⠀⠀⢁⡘⠉⠀⠑⢦⡀⠃⠀⠀⠒⠁⠘⠀⠲⠄⠂⣀⠜⠀⠀⠀⠀⠀⢳⠠⠄⠀⣀⠁
⠀⠀⢀⠀⡺⠀⠀⠀⠸⠄⠀⠠⢧⡄⠈⠀⠀⠀⠀⠀⠀⠘⠇⠠⠀⢰⣀⠑⠠⠠⠀⠠⠢⣀⢀⠂⣄⠀⠣⠀⠀⠀⠀⠀⢀⠓⠀⠀⠀⠉
⠀⠀⠀⠠⠁⠀⠀⢠⠶⠀⠀⠵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠤⡜⠂⠶⠃⠐⠻⠛⢻⢃⣒⠖⠁⠀⣸⠀⠀⠀⠀⠤⠁
⠀⠀⠀⢠⢀⠀⡀⠸⠀⢑⡤⠂⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠀⠀⠠⠺⠆⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣄⠀⡼⠁⠀⠀⣄⡠⢀⡀
⠀⠀⢠⠠⠏⠉⠁⠀⠀⠈⠀⠀⠀⢨⠑⢤⡀⠀⠀⠀⢀⠀⣀⠡⣲⠀⣧⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡼⠀⠀⡘⠉⠀⠈⠓⡄
⠀⠀⠁⠒⠀⠆⣀⠀⠀⠀⠀⠀⠔⠀⢀⠘⠑⢄⠀⠀⠀⠀⠆⠠⡉⠀⠈⠠⠃⡳⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⢀⡉⠉
⠀⠀⠖⡂⠀⠀⠄⡈⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⢑⠀⡀⠀⠆⠀⠙⣲⠀⢠⠀⠀⢀⡌⠙⢦⠤⠀⠀⢀⠦⠤⣄⠀⢠⠀⠀⠀⠀⠔⠒⠊⠁
⠀⠀⠀⠀⠠⠬⠀⠀⢠⡀⠠⠀⣄⡀⢀⠀⠀⠀⠈⠃⠑⡞⠀⠀⠈⠫⠀⡜⠀⢰⠊⠑⠞⠛⠐⠀⠀⡗⠀⠀⠈⠆⠐⡀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠘⠀⠀⠄⢀⠢⠄⠀⢢⡄⠁⠰⢂⠀⠀⠀⠀⠀⢸⠇⠀⡠⠴⠀⠐⠋⠙⠉⡀⠀⠀⠰⡧⣤⠤⠃⠀⠀⠀⢼⣀⣰⠆⠀⡇⠀⠠⠆⠀⠠
⠀⠀⠀⠂⠀⠀⠊⠀⠒⠐⠁⢈⢁⡤⢄⠀⠀⠄⠀⠘⢅⠀⢀⠇⡊⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⢀⡟⣃⠖⠋⠊⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠈⠀⠀⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠢⠀⠀⠀⢀⡀⠈
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠃⠘⠀⠀⠀⠈⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⢀⢐⣦
⠀⠀⠀⠀⠂⢕⠀⠁⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠑⠤⣂⣐⠀⡄⠘⠿⢶⣔⢂⢦⠲⠄⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⡀⣄⠤⠄⠄⠄⠤⠄⣄⣀⡀⡀
⠀⠀⠀⢀⠀⣀⢀⠀⠄⣀⢱⢌⡀⠐⠠⡨⣽⠌⠦⠀⢈⠈⡀⠀⠀⠀⠀⠒⠀⠀⠀⠠⠀⠀⠀⣤⠤⠀⠠⠀⠀⠁⡄⠀⠛⡈⠉⡙⠑⠈⠁⠂⢒⣤⡀
⠀⠀⠀⠁⠓⠲⠤⠭⠿⠯⡍⠿⠾⠷⢦⡈⠷⠀⠀⠀⠉⠰⢆⡀⠂⢀⡀⠀⠀⠀⠀⠀⠈⣃⡾⠁⠠⣤⣄⠰⠀⠰⠇⠀⣼⣇⠀⣃⠀⠠⠆⠀⣼⠇⡤⢁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⣀⣀⣀⣀⣴⠀⠀⠀⠀⢀⣀⣈⡁⢠⡀⠑⠂⡀⠀⠀⠲⢄⡝⣧⠠⣿⣿⡿⣇⡤⠿⠒⠉⠈⠁⠀⠉⠀⠈⠐⠚⠇⣠⣄⢘⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠧⢤⣤⣀⣋⣅⣼⡿⢿⡐⠶⠀⠉⣦⠈⠀⣤⠄⣼⣦⢛⣻⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠝⣹⣍⣤⣮⠏⠁⠐⠘⠐⠓⠒⠊⠻⠿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠠⣤⢀⡀⠉⠤⠈⡿⠴⢤⠀⠀⢄⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢐⣛⡣⠀⢋⠛⠈⣡⡄⡔⠒⠊⢠⣐⢎⡉⢁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣭⠀⣭⣉⢀⠣⠆⢀⣰⠀⠒⠁⠂⣦⡀⣚⣁⠜⠃⠾⠃⣦⣀
⠀⠀⠀⠀⠀⠀⢠⣤⣤⠆⡂⠩⢗⡉⠛⢠⡌⠻⢼⡩⠧⣽⠀⠘⠀⠿⠀⢤⡉⠈⡆⣙⠁⢷⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠹⢢⣅⢠⢨⠛⠖⡄⠰⠆⢺⠛⣶⣭⡓⣤⡃⣦⣻⠞⣁⠨⠶⢈⠷⠀⣄⣇⠤
⠀⠀⠀⠀⠀⠀⣤⡴⠘⢂⣑⠚⣄⠥⣤⢀⣤⢖⡻⣟⠰⡀⠀⣶⢶⡞⢱⣄⣿⣘⣦⣊⣒⠇⢠⡌⠔⠃
⠀⠀⠀⠀⠀⠠⠾⣤⠘⠿⢧⠇⢈⡱⢦⣌⡬⢻⡁⢻⣄⣴⠊⠷⣼⢠⡞⢭⢀⠖⣤⠐⢀⣵⡊⠁⡆⡒⠶
⠀⠀⠀⠀⠀⠀⢀⣀⠋⠴⠀⣂⡘⠁⣛⣉⢷⡖⠘⡙⣿⡑⠄⡽⣣⡟⡰⠦⣼⠳⠀⣶⢀⡘⣥⠰⠇
⠀⠀⠀⠀⠀⠀⠠⠝⢡⠤⣃⠤⣠⠴⠶⢤⣩⡟⢾⣅⢸⣧⣾⣵⠟⢸⣷⣎⡬⣕⠂⣽⣦⠉⡀⠠⠜⣤⡀
⠀⠀⠀⠀⠀⠀⠸⠇⣲⣀⠘⠂⣈⠋⣰⢿⣮⠳⣦⡙⣿⣿⣿⣷⡿⣿⡉⣱⡦⠝⢃⡄⢍⡚⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠨⠌⢁⠱⠀⣧⠡⠀⣀⠈⢻⣾⣿⣿⡏⠀⠈⠃⢀⡰⠒⠀⠠⡌⠑⢇⡐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠃⠀⣅⠷⠀⠰⡶⡉⠀⠀⢻⣿⣿⠀⠀⠀⠀⠨⠤⢀⣄⠃⠀⠠⠾⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠀⠀⠘⠁⠙⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⢟⣽⠿⣯⣭⡽⡿⠿⢿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⠶⣾⣯⠿⣟⣡⢄⣼⠛⠛⢿⣮⣙⣿⡛⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡯⠟⠻⠷⢄⢻⠖⠛⠍⠛⣻⡷⠾⣯⣤⡬⠉⣿⣗⣛⠒⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⢀⣾⣿⣿⣿⢿⣅⣺⣷⣮⡈⢠⣂⣶⣶⡀⢋⡠⡶⠿⣿⢧⡄⡀⠉⢩⠇⠀⢹⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⢀⣾⣿⣿⡿⣵⢿⡜⢉⡙⣿⠿⠚⠷⢾⣯⣟⣴⣆⠱⠤⠿⢿⣶⡛⠷⠈⠛⣧⡘⢲⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣿⣿⣇⣇⣽⣧⠙⢛⠁⢠⣶⣤⡈⣿⣿⣟⣿⣫⡀⣀⠀⠹⣯⣾⠆⠸⣿⣶⠈⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⣿⣿⣿⣿⣿⡇⡽⣿⣇⣻⠤⢻⢿⣿⠇⣴⡘⣟⡙⢳⡞⢋⠁⠈⢉⣿⣷⡀⠈⠻⣿⠛⣶⡝⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⠏⢩⡻⢽⡧⣿⡙⢿⣤⣾⠃⣦⣾⡏⣠⣿⠞⢺⣿⢿⣶⡶⢊⣉⢿⡇⡤⠰⣿⡄⢸⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣾⣆⠙⢲⣅⢨⣻⣆⣸⣽⣧⠒⣿⢆⣾⣭⣧⠀⣿⣿⣯⣇⣓⣺⣿⣩⠄⣸⣿⡝⢘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣹⣦⣄⠙⠷⣍⣥⣵⡟⠻⢻⡿⠀⡿⣦⡛⣏⡿⣿⣴⠦⢭⣾⣿⠑⣛⣽⢋⣶⡾⣼⠛⢿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣽⡿⣻⢦⡀⠙⢶⣍⠻⡿⠿⣿⢿⢿⡿⣧⣽⣿⣁⣽⣗⣋⣿⣯⣛⣭⣵⡾⣫⣿⣿⣶⣤⡈⠛⠻⠟⣿⣿⣿
⠈⣿⣿⣿⣿⣿⡝⣿⡷⣿⣶⣄⠉⠳⣔⡶⢤⣴⣿⣰⣆⣶⣾⣿⣿⣿⣫⡾⢷⣋⣼⢟⣾⣿⡿⠿⠛⠉⢉⣁⡀⢀⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣜⡦⢬⣽⣿⣷⣄⡈⠙⠺⠙⢛⣻⡤⠖⠒⠛⠓⠒⠒⠒⠲⠾⢥⣿⣿⠃⢠⣤⣶⣾⣿⣿⠃⣸⣿⣿⡏
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣶⡭⣉⣿⣿⣷⣦⣀⠀⠀⠀⢰⣶⣶⣶⣶⣿⣿⢻⡆⠈⠛⠋⠀⢺⣿⣿⣿⠿⠁⣠⣾⣿⡿⠁
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡻⢟⣭⣾⣷⣤⡆⠈⢿⣿⣿⣿⣿⣿⠟⠁⢠⣶⣴⣄⠀⢀⣂⣠⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⡿⠛⠋⠵⢞⣯⣶⣿⣿⣿⣿⣿⣷⡀⠈⠙⠛⢉⣉⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣯⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄⠀⠀⠀⠀⠀⣠⠊⠉⠒⠉⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡁⢘⣆⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⢀⣠⡤⣤⡀
⠀⠀⠀⢀⡴⠊⠉⢠⡰⢏⠃⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣘⣁⠙⣉⣏⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⣾⣿⢶⣤⠁
⠀⠀⠀⢺⢀⡷⠚⠃⠀⡜⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⣍⢤⣩⣀⣯⠀⠀⠀⠀⠀⠀⢸⠟⠈⢲⡷⠊⠀⠘⠃
⠀⠀⠀⢸⠻⢥⣤⠤⠊⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⢲⠓⠂⠚⠀⠀⠀⠀⠀⠀⠼⠥⣌⣡⣨⢽⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚

⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⠀⠀⡄⡀⠀⢰⣀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⢀⣠⡴
⠀⠀⢀⢤⠀⡊⠀⢈⡀⡏⣷⠀⠀⠀⠀⠀⠀⢠⡀⠀⡰⠟⢠⠖⠁⠀⠀⢈⠙⣶⠂⣿⡃⢀⣞⣥⠾⠤⠀⠀⠀⠀⢹⣧⠀⢀⢔⡿⠉
⠀⠀⢸⣸⣦⣰⠀⢸⣾⢟⠟⠀⠀⠀⠀⣀⣈⢳⣿⣠⢶⣟⣿⣀⠀⠀⠀⠋⠉⠙⠫⡹⡴⠟⠛⠻⢯⠀⠀⠀⠀⠀⠀⣯⣇⢎⡞
⠀⠀⠈⠳⠥⡟⠀⢰⠏⠁⠀⠀⠀⠀⠀⠈⢎⠳⡽⢡⢻⡟⣥⠴⠀⠀⠀⠀⠀⠀⠀⢧⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡏⣿⣼⡀
⠀⠀⠀⠀⠀⠱⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣇⣬⣬⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣇⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⣸⣿⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⠤⡄
⠀⠀⠠⠶⢤⡄⡾⢁⣤⡆⠠⠀⠀⠀⠀⠀⣴⣸⣿⣷⢟⣾⣶⡁⠀⠀⠀⠀⠀⢀⠮⠭⠥⠬⠭⢷⠀⠀⠀⢀⡴⠅⣤⢤⠒⠶⠻⣄⣄
⠀⠀⠀⠑⠂⠉⣧⠟⠭⠔⠀⠀⠀⠀⠀⠀⣿⣿⣾⡏⣼⣿⣹⣃⠀⠀⠀⠠⣦⠞⠀⠀⠀⢀⠀⢈⢣⣄⠀⢨⠧⡶⠿⢀⡀⢐⠶⠀⠘⢶⡄
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⡟⣾⡛⣿⢗⡿⡏⠁⠀⠀⠘⠓⠚⠙⢦⡜⠉⠑⠊⠀⠀⠀⠀⢱⠃⣠⠋⠉⠙⠋⠙⣦⠀⢣
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠟⠚⠃⠛⠂⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⡤⠢⡔⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢟⣻⠒⢤⠀⡴⢫⠝⠋⠉⠀⠀⠀⠀⠀⠀⣀⠤⢤⡀
⠀⠀⣀⣀⣃⢀⣀⡸⠊⠉⢿⠀⠀⠀⠀⠀⠠⡀⠀⡆⢀⡠⠔⠂⠀⠀⠀⠀⣀⠤⡤⡭⣿⣿⣧⡀⣲⢦⡀⠀⠀⠀⢀⠞⠁⠀⠀⠙⡄
⠀⠘⠢⡤⠈⣿⢉⡷⠠⢤⠚⠀⠀⠀⢀⣀⣀⡈⢦⣷⢞⣡⡤⠤⠀⠀⢠⣮⢴⠎⣰⠇⠈⠘⡆⢷⠈⠉⠁⠀⠀⠀⠸⠀⣄⠀⠀⢀⢱
⠀⠀⠘⠤⠚⢈⡉⡧⣸⠞⠀⠀⠀⠀⠉⠀⠀⠉⠻⣷⠋⠁⠀⠀⠀⠀⠈⠀⠈⡼⠀⠀⡇⡆⠙⣾⠀⠀⠀⠀⠀⠀⠘⠚⢻⠒⢲⠛⠋
⠀⠀⠀⠀⠑⠊⠳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⣀⡠⠄⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠤⢍⣩⠩⣽⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣣⡄⠀⠀⠀⠀⣾⣷⣾⣂⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠨⠿⠞⠛⣟⣻⣿⣿⣧⡤⣀⡀⠀⠀⠒⣻⣿⡿⠅⠀⠀⣰⣾⣿⣿⣿⣿⣶⣦⠀⠀⠼⣿⣿⣟⣻⡿⣿⣿⡉⠃
⠀⠀⠀⢀⣀⣤⣴⣿⣿⣟⢿⣿⣿⣟⡿⢷⣤⣀⠈⠇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⢸⡏⠁⠉
⠀⠀⠐⠋⠉⠉⡬⣭⣿⠿⣟⡯⢿⣿⣷⣾⠛⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⢃⣀⡀⣐⣦⠤
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⢠⡄⢀⣀⠀⠉⠛⠉⠉⢹⠏⠉⠙⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⣧⣿⣯⣵⡆⡻⢶⡀
⠀⠰⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⣠⣤⣶⣿⢐⣬⠶⢉⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⣀⣴⣶⣦⡄⠀⠀⠋⠛⠛⢿⡻⡿⠿⠿⠃
⠀⠘⡿⣽⣿⣿⡿⠄⠀⠀⠀⢠⣿⣾⣿⣿⣏⣿⣿⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⡟⠛⡃⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠘⠉⣉⣉⠁⠀⠀⠀⠀⠾⠘⠋⠿⢿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⣻⣿⣿⣭⣤⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠠⣽⠦⠀⠀⠀⣠⣤⣤⣤⣤⣠⣀⡀⠈⠉⠱⡿⠉⠀⠑⠀⠠⣤⣶⣦⣤
⠀⣠⣴⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣏⣻⣇⠤⠄⠀⠀⠀⠀⠀⣾⡿⢿⣿⣿⣿⣷⣦⡄
⠀⣿⣯⠽⠿⠿⠾⣿⣿⣿⣿⡟⠿⣿⣿⣿⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣄⣀⠀⣴⢶⡾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠈⠀⠀⠀⠀⠿⠿⠯⠟⣿⠧⢴⣿⣿⣿⣿⣦⠀⠀⠉⠈⠉⠻⣻⣧⡿⠋⠉⠈⠉⠉⠁⠀⠀⠀⠈⠁⠈⢻⡿⢩⠋⠻⠟⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⡄⠚⠀⠀⠀⠀⠀⠀⠰⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⢦⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣷⡄⠀⠀⠀⠀⠀⣠⣶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢨⣾⠈⣷⣤⠀⠀⠀⠀⢋⢸⣿⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢹⢧⡁⢸⣿⣷⣿⠏⠀⠀⠀⠘⢿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⣿⣿⣿⣿⣆⠀⠀⠀⢄⣻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⡷⢾⣿⣿⣿⠁⠀⠀⠀⠀⣸⡘⢿⡋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠄⣟⣰⣿⠒⠀⠀⠀⠀⢹⣷⣼⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠰⡧⢾⣷⡧⠀⠀⠀⢀⣼⡗⢰⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⠀⠀⢴⣾⣿⣗⡀⠀⣀⢠⣿⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⣮⡋⣿⣏⠶⠀⠈⢸⣿⣿⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢰⠟⡝⠛⣿⠀⠀⠀⣾⣽⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢷⣦⡧⠀⣿⠰⡟⠀⣿⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⡇⢸⣿⡇⠀⣿⠄⣦⢠⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⣇⠀⣿⠋⠀⢹⠀⣍⢸⡇⣺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣀⡇⠀⠺⠁⠀⠸⠀⠀⡀⠸⢦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⣰⡇⣤⣴⣂⣀⡄⢠⢸⣷⣦⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠁⠀⢨⠅⠀⠀⠈⠶⠿⠯⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣆⣠⣤⣄⣀⣀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⡤⠖⠚⠉⠉⠀⠀⠀⠈⢳⠀⠀⠀⠀⠉⠙⠓⠦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠇⢀⣠⠤⠴⠖⠒⠒⠛⠛⠛⠛⠛⠛⠒⠲⠦⣤⣀⡀⢻⠲⣄
⠀⠀⠀⠀⠀⣠⠞⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣉⣡⣤⣤⣀⠀⠀⢠⣈⡉⠻⢤⣈⠳⣄
⠀⠀⠀⠀⢰⢃⡴⠋⠀⢀⡴⢋⣠⠤⠶⠚⠉⠉⠛⠋⠁⠀⠀⠀⠈⠳⠦⠤⣤⡉⠂⠀⠈⢳⡹⡄
⠀⠀⠀⠀⣿⡞⠀⠀⠐⢁⡴⠋⠀⠀⠀⠀⢀⠴⠊⠉⠉⠉⠲⡄⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢹⣇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢯⣀⡀⠀⠀⣀⣠⠇⠀⠀⠀⠀⣀⡼⢁⡇⠀⠀⢰⡿
⠀⠀⠀⠀⢹⠙⢦⡀⠀⠘⢤⣖⣂⡀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠟⠩⠖⠉⠀⢀⡴⢫⡇
⠀⠀⠀⠀⠈⣧⠀⠙⠳⠦⣤⣀⣂⠙⠶⠤⠴⠿⠿⠧⢤⣤⣤⡤⠴⠚⠁⠀⢀⣠⠴⠚⠉⢀⡞
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠦⠤⠤⠤⠤⠤⠤⠶⠖⠚⠋⠉⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⢿⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⡀⠈⣿⠀⠀⢸⡟⠀⠀⣀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣀⣄⣰⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠍⠒⠒⠨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠒⢲⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠛⠚⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠋⠉⠉⠓⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⡤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠆⠡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠁⠀⠀⢀⡈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠄⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠌⠀⠀⡠⠤⠤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠈⠆⡃⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠐⠄⠠⠂⠀⠀⠀⠀⠀⡠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡠⠂⠀⡠⠀⢄⠀⠀⠀⠴⠀⠀⢈⠀⠀⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠉⠁⠀⠀⠀⢄⢀⡸⠀⠀⠀⡀⠀⠀⠀⠁⠐⠈⡀
⠀⠀⠀⠀⠈⠫⡐⣠⠄⣀⠀⢀⡀⢰⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠈⠠⠐⡠⡀⠀⠆⢀⠆⠀⠀⠑⠂⠀⠐⠄⣀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠴⡉⢠⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠴⠶⠟⠠⠶⠾⠂⠂⠊⠳⠠⠶⠾


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⣠⡀⠉⠓⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠤⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡠⠒⡄⠀⠀⡴⠁⠀⠀⣠⠔⠠⠚⠁⠈⠓⠤⣀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⣰⠊⠉⢈⡽⠅⠴⠃⡠⠊⠀⠶⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠈⠑⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⡏⠀⠀⠳⣄⣀⡠⠊⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⢷⠀⠂⠀⠀⠀⠐⠀⠀⠀⢀⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠜⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠈⠢⣄⣀⣀⣀⣀⣀⣀⠔⢁⣴⢁⡤⠤⢼⠀⠀⠀⠀⣠⠞⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⣀⣠⣔⡩⠖⠉⠈⠁⠀⠀⠀⠣⠤⠒⠯⢤⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⣀⣥⠀⠀⠀⢰⠁⠈⠉⠒⠒⠤⠤⠤⢤⣤⣤⠤⠤⢤⠤⠤⠤⡄⠀⠀⠀⠀⡎⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠣⡀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⢱⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⢤⣀⠤⠜⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡎⠸⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⠁⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⢀⠤⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⢀⣀⡀⡜⠀⠀⡧⢄⣠⣄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠈⠉⠁⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠂⠚⠉⠘⠢⠔⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠊⠀⠀⠀⠀⠀⠀⣀⣴⣄⣀⡀⠀⣀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⡀⠀⠀⠀⠀⢀⣸⣯⢉⡙⠁⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂
⠀⠀⠀⢀⡀⠤⠀⠒⠂⠚⠁⠹⠛⠆⢴⣏⣲⣿⣿⣿⣯⣧⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣲⣊⣄⢰⣿⡛⠁⠈⠉⠙⠐⢣⣤⡀⣄⢚⡃⠀⢠⠌⡧⠴⠛⠀⠀⠀⣤⠀⠀⡄
⠀⢀⣀⠀⣠⣀⣠⡀⢀⠀⠲⣼⣿⣿⣿⣿⡿⢆⣙⢶⣤⣆⡸⣷⣿⣯⣿⣦⠿⠷⣦⡄⣶⣬⣥⣀⣀⣾⣷⣊⡀⠂
⠀⠀⠁⠚⠒⠛⠿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡶⠿⠿⣿⣿⣿⡿⢻⣿⣽⣥⣔⣒⡋⠁⠐⠋⠉⠋⠁⢛⣿⣿⠛⠁
⠀⠀⠀⠄⠠⠀⠂⠀⠒⠝⠛⢀⠛⠀⢜⢻⣅⢿⣯⣼⣿⣻⣷⣸⣿⢿⣷⣶⣷⣶⡆⢠⣄⣤⣄⣼⠛⠉⠁⠀⢀⠉⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢓⣦⣀⡚⠓⠙⠛⢛⣿⡿⢷⣇⡿⠋⠈⢿⡿⣿⠿⣿⡷⡿⠀⠉⠀⠀⠀⠀⣀⣈⡀⠀⠀⢀
⠀⢐⣂⡄⠀⢀⣀⡀⠀⠠⠉⢩⡭⢉⢽⡶⢦⣼⣿⡟⢁⡿⢋⡀⣸⣠⡏⢀⣤⣬⣟⣂⣄⠂⠀⠀⢠⣬⠣⠘⣫⠄⣤⣇⣩⡁
⠀⠈⢻⣶⢆⣍⠿⠖⣛⡉⣀⣮⡟⢥⣿⣧⣦⣿⣿⣧⣼⡧⣿⠛⢻⣿⢻⡏⠀⠸⠆⢴⠀⠀⠴⢡⣶⣷⠮⣿⣿⣾⣿⡶⣸⠿
⠀⠀⠀⠀⠉⠐⡿⣾⣿⣷⣶⣿⣿⣷⣾⣿⣿⣎⣿⣿⣿⣇⣿⣴⣿⣧⣾⣯⣤⣾⣧⣶⡭⠾⢗⣋⣿⣿⣿⣿⡿⣛⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠿⠛⣫⠿⣿⣿⣿⣿⠟⢿⣿⣿⣾⣿⠻⣿⢟⣿⣿⣿⣿⣷⣎⢻⡁⢠⡾⠉⠘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⡿⠙⡃⠐⠀⠉⠛⠻⠟⠛⠛⠛⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⢀⣀⣤⣤⣴⠶⠿⠿⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢠⢖⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢿⠸⣠⠇⠀⠀⠸⣆⠈⣩⡆⠀⠀⠀⠀⢀⠄⠀⠀⠀⣿⠀⢛⡛⠁⠀⠀⠀⠀⠀⠀⢀⡤⠘⢁⡀
⠀⠀⢈⠐⢋⡶⠀⠀⠀⠈⢈⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡈⠚⡏⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⠴⠛⠁
⠀⠀⠸⣦⡈⢠⢴⡆⠛⣩⠃⣞⣴⠃⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠇⠀⠀⠀⠀⠁⠀⠀⠸⣷⡄⣰⡦
⠀⠀⠰⣄⠙⣠⠶⠀⢤⣘⠧⢁⣴⠇⠀⠀⠐⠀⠀⠀⠀⣄⡻⠄⣀⡶⠀⠀⠁⠀⠀⠀⠀⣤⣽⣡⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠚⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡶⢀⣤⣄⠀⠀⠀⠀⠀⠀⡀⠀⡀⢠⠆⠀⠀⠀⡈⠗⠾⠇
⠀⠀⠰⡆⢚⡋⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⠰⠈⠛⠋⠀⠀⠀⠀⠀⠊⠸⣀⠁⢋⣠⠄⠀⠀⣻⢇⢴⣦
⠀⠀⢀⠐⢋⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⣠⡄⠀⠀⠀⠀⠀⢀⠀⠈⠁⠋⢡⠆⠀⢀⡹⠀⢰⠇
⠀⠀⠘⠇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⡾⠍⡉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣗⡏⢠⡶⠘⣛⠁⢉⡙⠁
⠀⠀⠰⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠋⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⣈⣹⣏⣤⠀⠙⠳⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣶⣾⣷⣿⣿⠶⠁⡺⣿⣿⢿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣹⣿⠿⢉⣉⣿⣿⣤⣬⢻⣿⡭⢄⣻⣿⣿⣿⣁⣤⡖⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣥⣼⣿⣯⣷⣦⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠠
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣭⣿⣯⣛⣛⣩⣴⣤⣀⡀
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠈⣹⠟⠚⠛⣿⣟⢻⣲⣷⣿⣿⣿⣷⣿⣿⣿⣿⠆⣙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣦
⠀⠀⠀⠀⠀⡀⣾⣿⣿⡟⠛⢉⠺⢛⡛⢻⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⢷⣼⡿
⠀⠀⠀⠀⠀⠚⠫⠋⣀⣠⣄⠴⣾⣿⣿⢀⡘⢯⣻⡟⢿⣿⣿⣿⣿⡿⢿⣿⡟⡫⣿⣿⡽⣿⣯⡳⢿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠁⢠⣽⣿⣿⣤⠀⠀⠙⢻⣦⣿⣿⡿⠉⠃⠀⠛⠀⠀⡓⠻⡆⠈⣿⠁⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠈⢻⠇⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣴⣀⣀⣠⣠⣤⣤⣴⣶⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣤⡶
⠀⠀⠀⠀⠂⡞⠿⡿⢿⠟⠛⢿⣿⡿⡿⡿⠿⠋⠉⠁⠠⠬⠭⠿⠿⡿⠿⠿⢿⡿⠻⠟⠻⠻⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠴⠤⠤⠤⠤⠄⠤⠈

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣾⣏⣀⡀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⡿⠀⠀⠀⠀⢻⡄⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⢻⣏⣹⠃⠀⠀⠀⢀⡾⠁⠀⠀⠀⣷⠀⠀⠀⣀⣺⡄⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢀⡼⠛⠛⠛⠉⠛⠀⠀⠀⠀⠸⣧⡀⠀⠀⢀⠘⠷⠶⠟⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢠⡟⣀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⢾⡄⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣠⠟⢉⡽⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡠⠀⠀⠷⣤⡄⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣾⣧⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⣠⣷⡀⢀⣀⡘⣷⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠙⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣤⣤⣤⠴⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⢤⣤⣄⡀⠀⠀⠈⠻⠶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢴⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠘⠞⢁⣤⡾⠃⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢀⡠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠦⣤⣤⣤⣤⣤⣤⠶⠞⠛⠁⠀⠀⠀⠀⠀⢀⣀⣭⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣷⣄⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠾⢛⣛⣿⣿⡶⠶⠶⠶⢶⣝⡲⠶⣤⣂⣤⣤⣤⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⡼⠇⠀⣠⢴⡆⠉⠙⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠂⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⠀⠀⠀⣿⣿⠋⣿⣆⡹⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⠀⢸⡄⠀⠀⠟⠁⠀⣽⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢸⣿⣿⣿⡆⢸⡇⢀⡆⠀⣠⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⠀⠀⢻⣿⣿⣟⢀⣬⣱⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⣀⡀⠈⣿⣿⡿⠸⠏⠁⣤⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣿⣇⡀⢿⣿⣿⣄⣠⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣾⣿⣿⣹⡟⣋⣠⣶⡿⠻⠋⣹⣿⣿⠋⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣋⡻⠿⣹⡧⠀⠶⠖⠉⠀⢀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣇⠙⣿⣷⣬⣇⠀⡀⢢⡀⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠈⣿⠟⠉⠉⠙⣄⠰⢶⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⠿⠶⢷⣆⣨⣧⣼⣿⣿⡿⠿⠿⠛⠛
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡀⠀⣸⠇⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠘⠋⠉⣤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⠀⢸⡅⢰⣿⠄
⠀⠓⣤⡄⣿⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠸⣿⡟⠁⣀⡴⠀⣠⡞
⠀⠀⣿⠁⠙⣯⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣴⣾⣶⠚⠉⠠⢞⠛
⠀⠀⠀⠰⠃⠀⠀⠀⢀⡀⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡌⠻⣿⣿⣿⣁⠤⣠⣾⡵⠊
⠀⠀⠀⠀⠀⠀⢤⣴⣀⡷⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠙⣿⡄⠘⣿⣿⣿⡿⠟⠋⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⡟⠁⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣧⡸⣇⠀⣾⠉⠁⠀⡀⢀⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⣠⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣶⠏⣠⡄⠟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠀⠛⠀⠀⠀⢀⡀⠀⠘⣿⣿⡟⣿⣿⣟⠰⣿⣀⣠⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣴⡎⡹⣶⣄⢿⡟⠀⠈⢻⠋⢁⣴⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣷⣠⡆⠉⠘⣧⠘⣿⣿⠁⢠⠀⠀⣠⣟⣱⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣣⣄⠀⢠⣄⠸⣿⡄⠸⡆⠀⣹⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢿⣏⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⡋⠉⠙⠁⠈⢿⣧⢿⣷⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⢰⣦⣴⣄⠀⠉⠠⣿⣧⣿⠇⠀⠀⢀⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢀⣾⣀⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⡌⢻⣿⣿⣷⣤⣀⠙⣿⠇⢀⡄⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣼⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡃⣼⠏⠻⣿⣿⠱⠀⢾⣠⡎⢡⡾⠃⠀⠀⠀⠀⠀⢦⣄⠀⢀⡁⠀⠙⡏⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠀⠀⠀⠙⠟⠀⠗⠀⢿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄⠀⠀⢶⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣦⡀⠀⣾⠀⠀⢴⣧⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⣄⣸⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣴⡟⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⠿⣿⣿⡅⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠙⠟⠀⠀⠀⡀⢻⡷⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡟⢻⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠐⡇⠀⣾⡇⢸⣿⡀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠘⣟⣧⡴⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣄⡀⠀⠀⠉⠻⢼⡆⠻⡇⠈⣿⠁⠀⠀⠀⣴⠀⠀⠈⠁⠀⢠⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣈⠳⣎⣁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠄⠃⢰⣧⠀⠆⣠⣴⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣮⣿⣦⡄⠀⣦⠀⠀⠀⢀⣴⣷⡆⠘⣿⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠙⢿⣿⣿⡇⠀⡷⢀⡄⢠⡾⠋⢹⣿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠙⢿⣧⣤⣥⢀⣴⠟⢀⣶⣾⣿⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀⠀⠀⠉⢿⡇⠈⢡⡄⢸⣿⠀⣈⢠⣶⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠈⡇⢀⣾⡇⢸⣿⣦⣈⡵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡀⠀⠀⣿⠈⢿⡇⠀⡻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣷⣄⠀⣿⡆⠘⠃⠀⢃⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⢿⣿⣿⠿⢇⣿⠀⢀⠀⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢿⣿⡆⠈⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠛⣿⡄⢰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈
⢸⣆⣀⠀⠀⠀⠀⠀⠀⣀⣠⡖⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣻⣭⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠃⠀⠀⠀⠀⠂
⣼⠅⠀⠀⠀⠰⠒⠂⠀⠙⠁⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⠡
⣤⣀⣀⣀⣀⣀⣀⣀⣐⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰⣔⠖⠚⠒⠀⠀⠀⠀⠀⠀⠤⠤⣀⡴⠀⠀⠀⠀⠀⠦
⢉⡉⠙⠉⣙⡛⠛⢋⡉⠀⠀⠶⠆⠀⠈⠀⠀⣿⡀⠀⠀⠀⠀⢸⢿⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠂⠢⢴⣰⡄⢀⠐
⠀⠰⠀⠀⠻⠁⠀⠀⠁⠀⠈⠁⠲⠀⠀⠀⡐⣿⣟⠀⠀⠀⠀⢸⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠀⢸⠿⢁
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠲⣇⣸⣇⣠⠐⣲⣥⡬⢀⠆⡠⡀⣀⠀⠀⠀⢀⠀⠀⠀⢀⡀⢀⣐⣀⣈⠁⠀⠐⠒
⢀⣀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠤⠤⠓⣶⡖⢾⣿⣯⣿⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣦⣜⣄⣀⣀⣾⣿⠀⢁⣠⠀⢀⣀⠀⠀⣀
⠹⣿⠀⢸⢺⣁⠀⢨⣷⡄⠀⠀⠆⠈⣿⢇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠀⠈⢻⠀⢸⢟⠀⠀⡄
⠀⠉⠀⠀⠈⣣⣥⢼⣿⠆⠙⠳⢖⠠⣉⣉⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⣀⠘⠃⢲⣦⣕⠀⢀⣤⣤⠐⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠩⡀⣬⣻⣒⣊⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠁⠀⠘⠋
⠴⢡⡤⣤⡀⢨⠉⠴⠗⣷⠿⠷⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣦⣴⣦⣤⣤⣐⡄⣈⣈⡛⢉⣛
⡀⠒⠀⠐⡂⠒⠒⠒⠙⠛⣻⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⣀⡠⣒⣖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠽⠊⠉⠉⠡⠈⠁⠉⢢⡬⠛⢷⣦⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠧⠀⠀⠀⠀⠀⠻⡷⠖⠈⠐⢦⡙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣖⣤⣀⣀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣧⣿⣿⡟⣿⣿⣿⡿⢶⠽⠖⠉⠁⠀⠀⡈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣿⣿⠿⣷⠀⡀⣀⡴⠀⢰⠀⢹⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠋⣀⠿⠊⠉⠉⡀⢀⠟⠀⠜⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⣄⣶⠟⠁⠀⢀⡤⠤⠤⠾⣚⡵⠆⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡛⠋⠁⠀⠠⠔⢿⣶⣶⣶⡿⠏⠀⠀⠀⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⢸⢹⠛⠛⢃⠔⠀⢰⡧⢨⡁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⢤⡤⠤⣤⡠⠴⠆⠉⠉⠛⠈⣟⠁⠀⠀⠀⠀⠀⣪⡇⠀⠠⣯⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⢀⢀⡀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠀⠀⡁⢀⡏⠇⢠⠄⣿⠏⢡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣥⣼⣶⣿⠾⠿⠛⠛⡟⠃⡀⠀⠀⢠⠄⠀⠈⠹⠶⠤⢴⠶⢶⠼⠥⢚⢷⡤⠘⢀⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣿⠋⣩⣀⣠⣤⣴⣾⣿⡇⡘⠀⠀⢸⠉⢀⡄⠚⠀⠀⠀⠸⣷⡞⠀⠀⣼⣿⣧⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡟⢠⣶⠿⣟⡯⠿⠿⢿⣿⠀⡇⡇⢠⣼⠄⠘⠷⢆⠀⠀⠀⠰⠋⠀⠀⣰⢟⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⡟⡡⠊⠀⠀⠀⠀⠀⢿⠀⢧⢻⡼⠏⠟⠛⠘⠿⠉⠀⠉⠙⠒⢶⣌⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣟⣸⠁⠀⠀⠀⠀⠀⠀⣿⡄⠘⣾⣡⠆⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢿⡆⠀⠀⠀⠀⠀⠀⡟⢃⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⡄⠀⠀⠀⠀⠀⢹⠈⠀⠀⢹⣷⣦⡀⢀⣀⣀⣤⣶⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡙⣧⡂⠀⠀⠀⢠⡟⠀⣀⣠⣾⠙⣿⣿⣦⣤⡶⢻⣟⣉⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡳⣕⢄⠀⠀⠆⡄⠈⢰⡿⠀⠀⠉⠉⠁⠀⠀⣿⠟⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣵⣤⣀⣳⠀⣾⠁⠀⠀⠀⠀⠀⣴⣿⣷⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠠⣟⣤⣀⠀⠀⠀⠀⠙⣿⣇⣾⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⡲⡟⠉⠉⠙⠂⠀⠀⠀⠺⡃⣾⠇⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣄⠐⣾⠀⠀⠀⠀⠀⠀⠀⠈⣻⠙⠷⡼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣟⣩⣻⣷⡦⣄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠜⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠛⣉⠄⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠙⢀⡴⠋
⠀⠀⠀⠀⠀⢠⣾⣿⢟⡁⠀⡰⠋
⠀⠀⠀⠀⣰⠿⣿⠟⠘⠁⢰⠁
⠀⠀⠀⠀⠏⢾⣿⣏⡀⠀⢢
⠀⠀⠀⠀⡄⣙⢿⣿⣑⣄⡐⢦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⣿⠃⠐⢾⣿⣿⣕⣳⢦⣽⡀⠗⢲⡆⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡀⢁⡾⠁
⠀⠀⠀⠀⠸⣦⣀⣀⠠⠙⣿⣿⣻⣶⣭⣟⠳⠶⢤⣜⣋⠷⢤⡄⠀⠀⠀⠀⠀⠀⣀⠰⠾⠟⠛⣡⣴⣶⠟⠁⠈
⠀⠀⠀⠀⠀⠈⠉⠀⠄⣦⡾⠙⠉⠛⢯⢯⣾⠿⣶⣨⣽⣛⡺⢧⣛⡶⢂⣠⣠⣼⣁⠎⠽⣦⠟⢊⠑⠂⠈
⠀⠀⠀⠀⠀⢀⣮⠷⠚⠿⣄⡠⡀⠀⠀⠀⠋⠘⠛⠱⠽⠿⠟⢓⠋⠙⠋⠛⠉⣠⡶⢶⣦⣤⣦⣵⡶⢶⣖⡲⠶⢟⡶
⠀⠀⠀⠀⠻⠛⢂⣀⡀⠀⠈⠻⢧⣄⡰⢀⠀⠀⠀⣀⠄⠉⠀⠀⠀⠀⢀⣆⠈⠀⠀⠀⢀⣉⠩⠤⠲⠖⠶⠒⠋
⠀⠀⠀⠀⠀⠐⡶⠿⠁⠀⠀⠀⠀⠈⠙⠺⠶⡶⠋⠁⠀⠀⢀⣀⠀⠀⠀⠘⠷⣶⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠱⡤⠴⠚⠛⠓⠂⠴⠶⠶⣾⡇⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠇⡶⢽⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠒⠦⠤⠤⢶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⠚⠉⠲⡼⠀⠀⠀⣀⠀⣀⣀⡀⡀⢠⡄⡀⠀⠀⠀⠳⢋⡁⠉⢦
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⡏⠀⡼⡗⡄⠀⠀⠊⠁⠀⡀⠈⠉⠀⢀⠀⠀⡘⠀⠀⢰⠙⠛⠀⢸
⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠸⡀⠀⡅⢉⠀⣀⠏⢠⣶⣮⢱⠄⡒⣥⣴⡄⠘⢀⠀⡏⠀⠀⢠⠏
⠀⠀⠀⡰⠃⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⡴⢋⠀⠈⠉⠡⠠⠀⠀⠉⠉⠁⠀⣈⠓⠁⠀⢀⠏
⠀⠀⣰⠁⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠽⡈⠳⡄⡀⠀⢸⠄⠀⠀⠠⠀⡼⠃⢴⠦⠖⠋
⠀⢠⠇⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠁⠀⢣⠀⠹⠁⠀⠀⠄⠔⠀⠀⢸⠀⣠⡟
⠀⡜⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠈⢲⢴⣄⠠⠤⠴⠤⠤⢀⣾⡴⠋⠑⣦⡀
⢀⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡄⠀⠀⠀⠀⠀⠀⡷⠈⠳⠦⠤⠤⠶⠋⠀⠄⠀⠀⠐⢏⡀
⣘⠃⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢘⠃⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠯⣄⡀
⢹⠃⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠘⢷⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⠀⠀⠀⠀⠀⠉⠓⣄
⠘⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣧⠬⠳⣤⠄⠀⠀⠀⠀⠈
⠀⢧⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣠⠼⠶⠶⠖⣒⣢⣽⣿⣏⡀⠒⠄⠏⠀⠀⠀⠀⢀⡴
⠀⠘⡄⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠆⠘⢧⡀⠈⠑⢶⡏
⠀⠀⢳⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢧⡀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠳⡄⠀⠀⢣⡀
⠀⠀⠰⣧⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠳⡄⠀⠀⢣
⠀⠀⠀⠱⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⡀⣇⢀⠼⢸
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠙⠲⠤⠤⢬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢠⡿⠇⢠⢂⠔⣁⣧⡼⠒⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⡴⡟⠤⠤⠧⡞⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡴⠟⢻⠁⠀⠀⡞⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠶⠶⠟⠒⠦⢤⣀⣀⣼⠃⠀⠀⠀⣰⠋⠀⣀⣞⡀⠀⠈⠛⠉⠉⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠴⠳⠖⠚⠁⠀⠈⠓⠤⡤⠶⡄⠀⢢⠈⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣤⠀⠀⠀⠀⠀⠀⠀⠑⢦⠈⢲⢸⡄⡜⠀⣸⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢤⣤⠤⠒⠉⠉⠉⠱⡄⠈⠇⠀⣧⠇⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣠⠦⠜⠁

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣄⣀
⢀⣾⣷⣀⢠⣶⠖⡀⠀⠀⠀⠀⠀⣬⣣⣒⣻⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⢀⠰⠞⠁⢀⠀⠀⠀⠀⠀⠀⣿⣭⣧⠼⠿⢟⠃
⢹⢿⡿⣿⣵⣥⡩⠒⠀⠀⠀⠀⢓⣑⣢⣶⠗⠆⠀⠀⠀⠀⠶⠀⣀⠈⡲⢤⡆⠒⡀⠀⡤⠆⠀⠀⢠⣶⣿⡿⣿⣶⣷⣿⣿⣄
⠠⠉⢲⣿⣿⢬⡟⠛⠃⠀⠀⠀⣷⣾⣶⣶⣾⠆⠀⠀⠀⠀⠀⠀⠖⣠⣄⣁⣴⠖⠒⢤⣀⠀⠀⠀⠸⠟⢠⣾⣿⣿⡿⠉⠛⠁
⠀⢂⡾⠜⣿⣷⡽⠲⠢⠀⠀⠀⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿
⠀⠸⠁⢈⡿⠹⡗⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟
⠀⠀⢀⣶⠅⢺⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⣀⣤⣴⣴⣦⣤
⠀⠀⠚⣯⠀⣿⠇⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⣾⠟⠘⠁⠀⠀⠀⠀⠀⠀⠀⣦⡀⣦⢀⣴⠟⠀⠀⠀⠀⠀⣠⢿⣿⣿⣷⣽⣖
⠀⠀⠀⢀⣸⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣇⣿⣿⣿⣥⣾⠃⠀⠀⠀⠀⠞⠋⣿⢻⡉⠙
⠀⠀⠀⠀⠱⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣼⣿⣿⡧⢾⣒⠀⠀⠀⠀⠀⠀⠀⣿⡄⠁
⠀⠀⠀⠀⠀⠃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⣽⣿⣿⣱⣆⠈⠁⠀⠀⠀⠀⠠⠶⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠈⠈

⠈⠙⢿⣿⣧⡻⣿⠋⠃⢨⠇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠠⠀⠈⠙⢿⡕⢿⣤⢱⣯⣰⡆⠀⢂⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⣰
⠀⢤⠤⢄⡀⠙⡜⣿⡿⣄⡿⡟⢢⠀⡇⠀⠀⠤⡆⠀⢶⣄⠃⠰⡃
⠀⠀⠆⠀⠑⠀⠷⢻⣧⣾⠀⣿⣎⣠⠇⠀⠀⣃⠀⠀⠀⣽⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⡽⠋⠁⠀⠐⢶⣿⣁⠋⣟⢺⡀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⢀⠀⠈⠀⠈⠓⠛⠉
⠐⠀⠀⠀⠺⡏⢡⣿⣿⡇⠀⢤⠀⠀⠀⡟⢻⡸⣶⠆⢀⢸⡇⣤⠄⠀⠀⠀⠀⠉⠉⠙⠉⠉⠀⠈
⣀⠄⠀⠠⠤⠀⠸⣿⣛⣯⡦⢼⠁⠀⣄⣷⡴⣿⠟⠐⠻⢸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⠀⢸⡷⠦⢙⠈⣧⠧⣔⢦⠀⢸⡂⢀⣀⠀⢀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⡷⠀⠸⣆⠂⠈⠁⠀⠀⠫⡁⠀⠈⢢⡀⠈⣤⣿⠿⠷⠚⠉⠃⠺⣿⡷⠀⣆⢀⣴⠀⠈⢲⡾⠃
⠀⠀⠀⠀⠀⢠⠟⠀⣴⣿⡆⢆⣀⠀⠀⠠⠄⠀⠛⢟⠀⢀⣧⢿⣶⣦⠀⠀⠀⠀⢹⣟⣿⣿⣿⣽⠀⠀⣈⣋⣽
⠀⠀⠀⠀⠀⡆⠀⠀⠘⠻⣿⣎⢯⡁⠀⠒⠈⢠⣦⣜⣿⠋⠀⢀⣿⡿⠦⢄⣀⣴⣿⣿⣻⣍⡀⢽⢇⣡⣾⣋⣩
⣀⡀⠀⣀⡠⠁⠀⠀⠀⠀⠈⠙⢦⡹⣃⣆⠠⣦⡀⠻⠷⣀⣀⡴⠃⠘⠻⣿⡝⠋⠉⠀⠠⢠⠋⠁⢈⣿⣿⠿⢧
⢨⡟⠻⣍⡀⡀⠀⠐⠄⡀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠙⠿⡄⠀⠀⠀⡿⡄⠀⠀⠀⣼⠿⣂⣀⣼⣿⣧⣄
⠈⠓⠤⡽⠛⣇⣀⡔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢶⣿⣷⡽⣶⣶⣿⣿⢿⡛⠙⢷⣍⠙⣿
⠀⠀⠐⠫⣀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⣲⣾⣿⣿⢿⡅⠙⢷⡀⠹⠆
⠁⠀⢀⠀⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⠿⢧⡀⠙⣦⠀⠛⠀⠀⠀⣠
⠤⠀⠒⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢦⡈⢳⡀⠙⣆⠈⠁⠀⠀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⡀⠓⠀⠻⡄⠙⠆⠀⠠⠶⣶⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⡀⢀⠈⡀⠰⡄⠹⠄⠓⡄⠉⠀⢀⣴⣿⣿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠿⢿⣿⣿⣷⣦⡐⠌⠲⠄⢠⡈⢦⠈⠀⠀⠀⠐⢿⣿⣿⢏⠞⠘⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⡌⢀⠰⡄⠑⠀⠀⠀⠀⠀⠀⠈⣿⡁⠀⣄⣤⡄⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠳⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⢹⡆⠀⠀⢹⡇⠀⠀⠱⣄⡄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠉⢣⡀⠀⢸⡎⠀⠀⠀⠹⡀⠀⠰⣀⡄⣱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡇⠀⠀⢳⣗⣾⡧⠀⠀⠀⣀⢿⣠⣤⣋⣀⣼⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆⠀⠀⠃⠀⠀⠸⣿⣿⣇⢠⣤⣾⣯⣤⣿⠃⢹⠀⡞⠁
⠀⠀⠀⠀⠀⠀⣠⠀⢀⣀⠸⢤⡀⠀⠀⠀⠀⢹⠛⣿⠋⠁⠀⢻⣿⡿⠀⠈⢿⠇⠀⠀⡰⣂⡀
⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⢳⡀⠀⠀⢀⡀⠀⣿⡇⠀⠀⠸⢹⡗⠀⠀⡛⠀⠄⠀⠉⢿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢻⣤⠄⣻⠃⠀⢻⣄⡤⠀⠀⢸⡇⠀⠉⡏⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⣠⣄⣀⣠⡧⢼⣷⠞⠀⢸⣯⠀⠀⠀⣸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠙⢻⣿⣿⣯⠀⠀⢸⣧⠀⠀⢀⡟⠀⢀⣼⠦
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⢀⠀⠀⠀⡈⢻⣿⡆⠀⢼⡇⠀⠀⣾⡇⢀⣾⠇⠀⠀⠀⠀⠀⠠⢴⠠⠄
⠀⠀⠀⠀⢸⠀⠙⠀⠁⠀⠙⠃⠀⠈⠓⢸⣿⡇⠀⠈⠇⠀⢸⣿⣿⣿⠃⠀⠀⠾⠀⠠⠀⠀⢾⡄⠄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⣀⣿⣿⣿⠃⠀⣠⣴⢶⠿⠏⠀⣤⠀⠀⢰⣇⡀⠀⠀⢨⡅
⠀⠀⠀⠀⠀⢠⣹⣯⣩⣹⣿⣗⣱⣟⣿⣿⣿⣟⢢⣴⣿⠟⠀⠀⣴⣒⣷⢆⣰⣿⣿⣧⣀⢀⣨⣷⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⢻⣿⣿⣿⣿⠟⠏⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠿⠿⢿⢿⣿⣼⣧⣴⠖⠊⠐⠶⠶⠖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣼⠿⠿⣷⣶⣶⣾⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣍⣍⣈⣠⣀⣀⣀⣰⡦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠻⠿⠟⠋⠋⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡖⢢⠤⢀⡠⣴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢡⡿⣇⠛⣛⣿⢿⣿⣿⠏⠻⢻⢷⣤⣀⣵⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣬⡫⠩⠝⠂⠍⣄⠘⣛⣦⣶⢶⣾⣿⣿⢻⠿⠿⢿⣾⣷⣄⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠐⠘⠀⠒⣶⠤⠤⣹⣾⣷⢾⡉⣾⡵⠻⢿⣤⢠⡄⢉⣽⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⣾⣿⠩⠉⢀⠀⠁⠐⠂⠳⡄⠠⠙⡃⠹⣿⡴⣦⡟⠻⡆⢳⣌⢽⣿⣿⣿⠥⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢴⣼⣤⠄⢿⣇⣆⢠⢈⠀⠃⠀⡀⣀⠀⡀⠤⠈⣧⠀⠀⢁⢰⣆⠞⣼⠼⣽⣿⣿⣷⣿⣗⡃
⠀⠀⠀⠀⠀⠀⠀⠰⢇⣻⣶⠾⠛⠻⡶⠂⠀⠤⠀⣠⣙⡷⢂⠀⡠⢄⡐⣄⣸⣛⡿⡻⠁⠐⠢⢻⣿⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⡂⠀⠀⣀⡆⠄⣒⣠⣾⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣟⠳⢈⣷⣾⣿⡛⠿⣟⣻⣿⡇⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣾⣯⣠⣾⣷⣿⣯⢿⠿⠿⣿⣿⣿⣿⣿⣿⢿⣿⢻⣿⢿⢿⣿⣿⡿⠿⠷⣾⣽⣿⣿⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⣿⣿⣿⣿⣛⠛⠻⣿⣄⢨⣿⣿⣥⣴⠟⣻⣿⣾⣿⣿⣿⣦⣧⣼⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⢰⠾⡟⠛⠿⠟⠛⠛⠛⠻⣿⣾⠋⣿⡏⢁⣴⣿⠟⢩⣽⣿⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢿⠟⣹⡿⠋⠀⢾⢿⣿⣿⣿⡿⣯⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢀⠀⢰⣿⠁⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⢀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡸⢺⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣷⡜⠿⢿⢷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠁⡤⢭⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠰⡀⣤⣣⠒⠈⠁⠈⠉⠑⠲⣄
⠀⠀⠀⠀⢀⣀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⢀⡈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢀⠔⠁⠀⢀⣀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠠⠂⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⡇
⠀⠀⡇⣀⠔⠊⠀⠀⠁⠢⡀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠒⠒⣒⠒⠒⠤⢀⡔⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆⠀⠀⠀⠀⠘⠋⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠢⢄⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⡀⠀⡐⠀⠀⠀⠀⠀⠀⠎⠀⠠⢀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠁⠀⠈⠃⠉⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠀⠀⠀⠀⠀⢰⠀⠀⢀⠘⠄⠀⣿⠁⠈⠹⠖⢦⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠄⠀⠤⠆⠈⢂⠈⠁⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠚⠀⠉⡁⡀⠀⠀⠁⠂⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡠⠴⠊⠉⠑⠰⠆⠀⠈⠢⠁⠁
⠀⠀⠀⠀⠀⠀⠤⠅⠀⢁⠇⡀⠒⠀⠀⠀⠀⠘⠢⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡼⠀⠐⠈⠁⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⢐⠄⠀⡰⠈⠎⢀⠈⠑⢎⠉⠉⠁⠐⠛⠶⠾⣄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠑⢄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠙⠀⠀⠘⠀⠀⠀⠁⢸⠙⠲⢄⡀⠀⠈⠀⠀⣠⣠⠊⠀⠀⠀⣀⣀⠤⠶⠞⠤⣀⠀⠀⠖⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠊⠀⠀⠸⠀⠀⠀⠉⠒⠤⠤⠐⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣠⠔⢁⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢁⣀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂⡀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠞⠁⠀⢘⣉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠐⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡁⠐⠒⠀⠀⠀⠀⡆⣈⠥⠒⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠊⠀⠀⠀⠀⠀⠀⠀⠉





⣀⡀⣀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠒⠶⠀⠒⠉⠉⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠰⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠊⢹⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡀
⠀⠀⠀⢀⠀⢻⡀⠜⢀⣴⠇⣠⣾⣧⣀⣼⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠓⠤⣤⡤⠦⠶⠞⠛⠛⠉⢛⣿⡀⠀⠀⠀⠀⠠⣄⣤⣤⣄⣠⡞⠁⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣄⠀⠀⠀⠀⣿⢿⢿⣿⣯⠀⠀⠀⠀⠀⢀⣾⡿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⢼⠁⠀⠈⠉⠛⠳⠀⠀⢀⣤⣿⣿⣧⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠉⠝⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠄⠘⠇⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠀⠀⠉⠚⢆⠇⠀⠀⠀⣴⠋⠀⠀⠀⣀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠛⠛⠙⠛⠓⠈⠁⠙⠙⠛⠛⠀⠘⠘⠛⠛⠛⠀⠐⠛⠛⠙⠓⠛⠛⠛⠙⠛⠒⠂⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠄⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠐⠂⣀⣈⣀⣢⣀⡀⠀⠔⠒⠂⠀⠀⠚⠀⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⠤⡖⠢⠤⠒⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠐⠦⠤⠤⠴⠖⠛⡷⡈⠉⠉⠉⣀⠤⠂⠀⠂⠤⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠈⠢⢄⡀⠀⢘⣪⡦⠔⢊⠤⠀⠈⠒⠤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠆⠀⠀⠀⠠⠾⡀⠀⠀⠀⠀⠋⠉⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⠀⠀⠀⢳⢤⡀⡠⠔⢄⣀⡜
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢉⠐⠠⡀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠆⠈⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠰⠕⠊⠉⠀⠀⠀⠈⢢⠀⠀⠀⠀⢸⠀⠀⢀⡤⠚⠚⠀⠀⠀⠀⠀⢸⠠⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⡆⠀⠀⢠⠀⠣⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢸⠀⠀⢠⠁⠀⠀⢈⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⡇⠀⠀⣹⠀⠀⠀⠳⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢽⠀⠀⠀⠀⠐⠉⠉⠉⠉⠑⡄⠀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠣⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡸⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⡠⠀⠀⠀⠸⢀⠔⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣔⣊⡚⠒⡲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⠤⠤⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⢠⠛⠪⣶⠥⠤⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠚⠂⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠈

⠀⡴⢚⣉⠉⠲⣄
⣼⢰⠋⠉⠣⠀⠈⠳⣤⣀⠀⢀⣠⣤⠶⠶⠶⠲⠶⠶⠶⣤⣄⣀
⡇⡎⠀⠀⠀⠳⢄⣀⣨⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⢀⣠⡤⠖⠚⠛⠉⠉⠙⠛⠶⣤⡀
⣇⠇⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⠟⠒⢋⣩⡤⠴⠶⠒⠒⠛⠓⠲⣤⡀⠈⢿⡄
⢻⢠⠀⠀⢠⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣄⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⣿
⢸⣼⡆⠀⡟⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣸⣀⡀
⠀⡟⣷⣀⡇⡜⢉⡇⠀⠀⢀⣶⠀⠀⠀⠀⢀⠖⠢⢄⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠏⠉⠛⢷⣄
⠀⡇⢃⡹⠀⣇⢿⡇⠀⠀⢸⢻⠀⠀⠀⠀⡌⢠⣘⣾⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠙⣷⣀
⠀⠹⡌⠣⣀⢈⣟⣀⡈⠁⢸⢸⡄⠀⠀⠀⠙⠬⠿⠛⠓⠒⠀⠀⠀⢀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠈⢿⢷⡄
⠀⠀⢻⢦⣌⡟⠁⡀⠉⠀⠘⡄⠳⣄⠀⠀⠀⠀⠀⠐⣲⣤⠀⠀⠀⠈⢷⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⡄
⠀⠀⠀⠑⣼⠉⠈⠉⠃⠀⠀⠙⣆⠈⣳⣤⣠⣤⠶⠋⠁⠘⠀⠀⢀⣴⠛⣧⠀⠈⠳⠶⠒⢉⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⠹⡄
⠀⠀⠀⠀⣿⠒⠒⠲⠀⠀⠀⡴⠋⠉⠉⠉⠀⠀⣀⣤⣦⡤⠴⠚⠉⠀⠀⠙⠷⣦⣤⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠳⢬⣶⡤⢤⢤
⠀⠀⠀⠀⣿⠤⠤⢄⠀⠀⣼⠧⠤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡈
⠀⠀⠀⠀⢿⣀⣀⣀⠀⢰⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠸⡇⣀⡈⠀⢸⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⣿⠉⠉⠁⢸⠀⠘⣧⡀⠀⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢸⡉⠉⠀⢸⡀⠀⠈⣷⣤⡈⠳⣄⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠈⡏⠉⠂⠘⡇⠀⠀⢹⡄⠉⠓⠛⠓⠶⠤⡤⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠶⠒⣿⠉⢧⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣏⠈⠁⠀⡇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⢠⣏⣉⡁⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡰⠓⠂⠄⠀⡼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⡗⠀⠀⠀⠀⠀⠀⠀⣟⣁⡐⢂⣀⡼⠃⡾⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠻⠖⠻⠦⠼⠃⠀⠀⠀⠀⡿⠄⡠⡀⣀⠀⠀⢻⠀⡟⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠉⠀⠀⣧⠛⢂⠄⠄⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠦⢴⣤⠵⠖⠋⢠⣇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣃⠴⡄⠀⠀⠀⣀⡼⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣦⡘⢃⣤⠄⠀⢀⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣶⣄⠿⢃⡀⠀⠈⠛⠘⠉⢠⡄⠸⢟⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⣙⣃⠀⡈⠉⠐⠛⠋⠀⠀⢠⠀⠀⣿⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠁⠀⠀⠙⠲⡀⠀⠀⢀⣿⠖⠉⠀⠀⠀⠀⠀⢠⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡻⠂⠀⠀⠀⢠⡼⣦⠀⣼⠁⠀⣿⣆⣴⠀⠀⠀⠈⠋⠒⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠸⠃⠀⠀⠸⠛⠀⠘⣷⡿⠘⠿⠮⠙⠋⠀⠀⠀⠀⠀⠀⢠⣶⠟
⠀⠀⠀⠀⠀⢀⣤⣀⠿⡆⠀⢸⡆⠀⠀⠀⢰⣿⢀⣀⠀⠀⣿⡇⠀⠀⢀⣀⠤⠒⠂⠈⠉⠁⠀⢴⡜⠿⠆
⠀⠀⠀⠀⡀⠈⠉⣽⠶⠂⠀⠀⠁⢀⡀⠻⢷⠅⠿⠃⠀⠀⣿⣷⠀⣠⠞⠁⠀⠀⣠⡄⠀⠀⠀⠈⢁⠀⣠⡄
⠀⢀⣤⣀⠿⡆⠀⣀⠀⠐⠈⠉⠉⠉⠛⠓⠶⢦⣤⡀⠀⠀⢻⣿⣼⠃⠀⠀⠀⢷⡝⡷⠆⠀⠀⠀⠻⢎⣯⡀⠀⠰⠿⣡⣤⡀
⠀⠀⠉⣩⡶⠀⣼⠟⠇⠀⠀⣀⣀⠀⠀⡀⠤⠶⠾⢿⣷⣄⠸⣿⡿⠀⠀⠀⠀⣠⣤⣤⣤⣤⡤⠤⠤⠬⠅⠀⠀⠀⢶⡌⠉⠁
⠀⠀⠀⠉⠀⠀⠈⠀⠀⠺⢶⡌⠉⠃⣼⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⠀⣀⢾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠀⠀⠁⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠂⠐⠈⠀⠀⠀⢀⠜⢂⠀⠀⠀⠑⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁⠒⠁⠀⠀⠀⠀⠀⢀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠢⠀⠁⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠐⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠠⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠔⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠦⡐⠄⠀⠄⠀⠀⠀⠈⠂⠀⠠⠂⠐⠀⠀⠐⠁⠀⠀⠀⠢⠀⠠⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⡀⠂⠠⠀⡐⠀⠀⠁⠀⠀⢀⡀⠀⠀⠈⠀⠀⠂⡀⠀⠐⢈⠠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⡀⠠⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠀⡀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠈⠶⠀⠀⠀⠀⠀⠀⠑⣀⠀⠐⠂⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠢⡀⠀⡌⠀⠀⠀⠐⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⡀⠀⠀⠀⠀⢈⠤⠇⠀⠀⠀⡠⠆⠌⠀⠀⠀⠀⢀⠄⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡞⠉⠐⢀⡀⠀⠀⠈⢀⠈⠀⠀⢀⡄⠂⠉⠉⢃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠄⠂⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡟⠛⠛⠻⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣛⠻⣦⣈⠛⠶⣤⣤⣄⣨⣽⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢋⡾⢻⡄⠀⠉⠛⠷⠶⣤⣬⣥⣤⡴⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡛⢷⣌⠙⠛⠃⠀⠀⣠⡶⣆⠀⠀⠀⢠⣄⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉⠻⣦⡈⠻⣦⣀⠀⠐⢿⣤⡿⠀⠀⣰⡏⣿⡀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢤⣄⠀⠈⠙⠷⣤⣉⠛⠶⣤⣤⣀⣀⣀⣈⣹⣿⠶⠞⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛⢶⡙⣧⡀⠀⣀⣈⡙⠻⠶⣤⣤⣉⣉⣉⣉⣥⣤⠶⠛⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢈⣿⣌⡛⢛⣉⣉⡛⢦⣀⠀⢀⣩⣭⣍⡉⠀⠀⣠⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣴⠏⣿⠈⠉⠉⠉⠉⠻⣆⠉⠛⢋⣥⡴⣬⡻⠶⠞⣋⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡝⠛⠛⠀⠀⢠⡞⢷⡀⠈⠛⠛⠋⠁⠀⠀⠙⣳⠛⠋⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⡀⠀⠀⠻⣦⡾⠃⠀⠀⣰⠿⣆⠀⠀⣾⠻⣇⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠷⣦⣄⣀⡀⠀⠀⠀⠻⠶⠟⠀⠀⠙⣛⣫⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣌⡉⠙⠛⠛⠲⠶⠶⠶⠞⣛⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣶⣶⣶⣶⣾⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⢀⠀⠀⠀⢲⣤⡄⠀⠀⢀⠀⠀⡸⣆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⣿⣶⠘⣧⣿⡟⠀⠀⠈⠿⣿⣄⢐⣾⡷⢳⣽⡍⠀⠀⠀⠀⠀⢻⣖⠀⠀⣤⣶⠧⠀⠀⠀⢀⠀⢀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⡤⣤⣠⡿⣌⢡⡄⢶⣷⣤⢨⣉⡱⣎⣠⡔⠊⠀⠀⠀⠀⠻⣤⣀⠘⣿⡦⠴⠛⣧⣴⠀⠀⠙⣾⣇⠈⣯⣵⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣴⡓⢿⡀⠘⡯⣞⠛⢿⠃⠠⣍⣡⣔⠛⣵⠋⠸⣿⡍⠀⠀⣰⡦⡿⠶⣏⢠⡟⢀⠀⠀⠸⠻⠄⢠⣄⡄⢉⣌⣰⡞⠟⠃⢰⣠⣆⡤
⠀⠀⠀⠀⠀⠀⡼⠟⠋⠀⢉⣭⡿⡿⢟⢧⣈⣯⣟⣯⠃⠀⠙⢻⡄⢀⠙⠠⣄⣤⠀⣀⣼⣄⡎⣻⠹⡿⠶⠂⠀⢀⡰⠀⠽⠻⡟⢣⠶⣀⢀⣤⣊⠃⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢠⣄⣰⠀⠀⠀⠀⠀⡿⡿⠙⣷⢼⡟⠀⠀⠀⠀⢻⣜⣿⣤⡽⢿⠆⡁⢀⠈⠁⢿⡘⣶⡂⣀⡴⣮⡛⢃⣀⣺⣡⡴⠞⣉⠁⠀⠨⠗⠢⣴⣧⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣽⠿⡃⢦⣾⣧⠄⠀⢀⠙⣲⣆⢺⣴⣧⢀⡄⠹⣿⢿⣟⢪⡴⣷⠿⣄⠀⡀⢹⡝⠁⣼⣟⣋⡉⢚⡿⠋⠀⢙⡘⠿⢾⡀⠀⢸⣷⠃⠉⠟⠷
⠀⠀⠀⠀⠀⠀⡼⠟⠋⢉⣿⢶⣿⡁⠀⠒⠻⠟⡿⠛⢿⡁⠀⢠⣿⣶⣿⠉⣽⢦⡶⣤⣤⠀⠀⠽⣾⢀⣔⠉⣀⠿⢳⢎⣴⠃⠀⢿⡿⠁⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠻⠃⠈⠙⣦⡷⣏⠀⢠⠄⠈⣷⠀⠀⠀⠀⠹⣦⡆⠈⠞⢳⢾⡇⠀⢀⡟⠋⠈⠽⣋⣠⢸⠋⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣁⣾⢷⣄⣀⣬⣷⡀⣹⣿⣶⠀⢿⡄⠀⠀⡽⡏⢻⡇⢀⠴⢶⡍⢉⡝⢺⡇⠀⢠⣴⣍⣇⢸⠀⠀⠀⠀⢀⢠⣤⠄
⠀⣀⣶⣼⣦⢤⣤⣄⡤⢿⣿⡿⠟⠋⠀⠈⠻⢶⡶⠿⠶⢾⣷⣦⣤⣅⣄⣼⡇⣸⠻⠎⢀⣻⡛⠀⣷⢢⣴⠚⠟⠙⣿⠀⠀⣳⢆⢸⠟⠁⢀⣦⡞⠀⢠⡀⣴⣀⡀
⠘⠉⠋⣸⠗⠋⢀⣹⠳⠈⠸⠆⠀⠀⠀⠀⢻⡻⠀⢀⢀⣤⣤⡋⠉⠉⠉⠛⢿⣷⡄⢤⡬⠉⣥⣠⣟⠁⠉⣉⣢⠀⡏⠰⣾⣈⣋⡇⠀⣀⣼⢷⠶⢖⣉⡛⠉⠈⠁
⢀⣀⡞⣋⣓⠀⠈⠏⠃⠰⣆⣀⡀⢲⡤⣶⣫⡝⢳⣿⠀⡴⣿⠏⠀⠀⣄⣲⠺⣉⣿⣎⣹⠏⣙⠈⣧⢼⡔⣊⢉⡾⢡⣸⡏⣙⣫⣷⡋⠀⠛⠻⠦⠈⢿⡦⠄
⣘⣽⠻⠁⣯⠀⠀⠀⢀⣏⣹⣻⣥⣿⠾⠧⣿⣛⣹⣏⣀⡀⠘⠂⠀⠀⠚⣫⡾⢹⠇⢻⣇⠰⣿⢷⡇⣰⣽⡛⢻⣇⣹⡿⠛⣉⣡⣸⣷⢀⣠⣦⣶⡀⠀⠛
⠉⡎⠃⠀⠀⠀⠀⠀⢶⣾⣣⡝⠁⣞⡆⠀⠀⣉⢹⡭⠛⠛⠻⠶⢦⣴⣿⣤⠈⣏⣻⣾⢻⣧⢀⣾⣥⡿⢟⡏⢹⡟⠻⢞⣰⡜⢻⣽⠖⢺⣿⢩⢧⣇⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠘⠁⠾⡿⠀⠙⠇⠀⠈⡽⠟⠃⠀⠀⢠⣠⡄⠈⢛⣿⣶⠋⠓⠀⠀⢻⣿⠋⣷⡤⠘⣿⣿⠘⢶⣸⣻⣶⠻⡄⢐⣤⠖⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣝⢫⣽⣍⠛⣦⡀⠀⠀⢰⣿⣆⠘⠿⠀⢠⡿⠤⠚⣹⡦⠈⢿⡉⠉⢩⢾⠃⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣯⠐⢻⡏⠀⠈⠻⣦⣀⡿⠀⣿⣄⣀⣀⣿⣁⣰⡧⠄⢹⢢⣼⣧⣗⢀⣼⣤⠤⠴⣚⣟⠻⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⣿⣉⣿⣿⣽⣫⡴⠶⠂⡉⢸⣯⢀⡟⠀⠀⡀⠀⠋⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡷⣰⠄⠻⣷⣄⣼⣿⣿⡇⢡⡈⢯⣥⡄⠙⠛⠛⢻⣾⠖⣰⡮⠷⠍⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣼⣿⣱⣦⣿⣿⣿⣿⣿⡇⠺⣟⡄⡿⠀⠲⣿⠋⣸⣿⣉⠀⣻⣆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣤⣸⣿⡎⢸⣿⣿⣿⡇⠈⠉⢙⠇⠀⢸⣠⡾⣅⠀⠞⣗⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⣹⣾⣿⣿⣿⣿⣶⣶⣾⠶⣤⣾⠻⣟⠟⠃⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡝⠀⢹⣿⣿⣿⣿⡏⢠⢷⡂⠸⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⣿⣿⣿⡇⠀⢫⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢄⣰⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣤⣾⣿⡿⡧⣀⢙⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⢙⣿⣿⡸⠅⡹⣾⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⢠⡄⢉⡿⣶⡁⢸⣟⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠁⠀⠀⠀⢀⣿⢻⣿⡟⠛⠷⠬⠿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣵⣿⣿⠿⠟⠛⠋⢁⣀⣠⣾⡟⠀⠀⠀⢀⣤⡾⢏⣼⣽⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣛⠿⢷⣶⡲⠿⠿⠿⠛⠛⠛⣧⣤⣶⣶⣿⠍⣠⣿⡿⠃⢀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⣿⣿⣷⣤⣽⣷⣦⣄⠀⠀⠀⠈⠛⢿⣿⣿⣷⣽⠊⠀⢀⠄⠄
⠀⠀⠀⠀⣰⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⣀⠀⠀⡐
⠀⠀⠀⢰⣿⣿⡟⠃⠈⢳⣌⠘⠲⠺⣍⠙⠛⣿⡟⣿⠀⠀⠀⠀⠀⠀⠈⢧⢸⡏⠉⠁⠉⠁
⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠈⢻⡄⠀⢸⠀⠀⣿⣧⢿⡆⠀⠀⠀⠀⠀⠀⠈⠆⣿⠀⢀⣀⣄⢀
⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⢀⣾⣷⣾⡟⠀⠀⠘⣿⢺⡇⠀⠀⠀⠀⠀⠀⠀⣀⡬⠛⠋⢩⠀⠉⠀⠀⠠
⠀⠀⢸⣿⣟⠀⠀⠀⠀⢀⣼⣿⣿⠟⠀⠀⠀⠀⢹⢼⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⠒
⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣽⠿⠉⠀⠀⠀⠀⠀⠈⠙⣳⣶⠷⠶⠖⢀⡐⠊⠁⠀⠀⠠⣀⠀⠀⠄
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠙⢿⣇⣀⣀⣀⠀⢀⡀⠐⠉⠡⠀⡠⠶⠒⢀⠁⠀⠀⠀⠀⠀⠀⠉⠀⢰
⠀⠀⢘⣿⣯⠀⠀⠒⠶⠶⣾⡭⠋⠛⠚⠓⠀⡀⠀⠐⠚⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⣆
⠀⠀⠀⢿⣿⣇⣄⣤⣠⡀⠀⠀⢀⠄⠚⠋⠉
⠀⠐⠂⠸⣿⣷⣍⠁⠀⠈⢦⡰⢎⣠⣀
⠀⠀⠀⠀⠙⣿⣿⣷⣤⣄⣀⣹⣮⣩⣙⠂
⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠛⠋⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣴⢲⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⣬⣦⣤⢤⣍⣻⣿⣿⡶⣿⠾⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⠰⠦⣿⣉⠀⠐⣶⣶⣿⠳⠞⣿⣿⠿⠋⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⣻⣿⣧⡳⣾⣿⡞⢲⠛⡻⣿⣤⣶⡾⣤⣤⠖⠺⣿⣯⣣⣤⢠⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⣿⣿⣿⣿⣿⣿⣶⣬⠵⢿⣿⣧⣠⣃⣽⣶⠀⠀⢘⠛⠛⢲⠿⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢿⠻⠟⠁⡟⠻⣿⣿⣽⣷⣿⣿⣿⣿⣿⣷⣴⣾⣦⡄⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠠⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⡀⣤⣤⣥⣥⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣷⣿⣿⠛⣿⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⣤⢀⣀⣤⡟⢿⡿⠷⡌⢤⡛⠻⢿⢿⣿⣿⣿⣿⣻⣿⣿⡿⢝⠻⠿⢣⣖⢾⢾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠛⣻⣿⣟⣿⣿⣷⣦⣾⣿⣶⣦⣼⣿⣿⡿⣷⣿⣿⠐⣀⣉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠇⠈⢋⣻⣿⠇⢿⡻⣿⣿⣿⢫⣾⣿⣿⣿⡟⠻⠉⠀⢴⣴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣬⣿⣿⡿⣧⢹⣿⣿⣿⣿⣵⡆⢀⣷⣼⣿⣶⣤⡐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣄⡀⢀⢀⡛⣿⣿⣿⠹⣿⢠⣿⢿⣿⣿⣿⣇⠀⢿⣯⡍⠍⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⢿⣿⣿⣮⣭⣯⣥⡉⠀⣿⣧⣶⣄⢋⣭⣿⣍⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠹⠷⠇⢡⣿⠉⠹⡟⠀⣿⣷⠛⠷⠻⣼⣿⣻⣿⣃⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⡉⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠒⠢⠀⡈⠙⠟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣭⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠺⠛⠃⠃
⣿⣿⠿⠿⡿⢻⠿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠟⠛⠛⠛⠛⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠛⠛⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠁⠀⠀⢠⡀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⣠⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣀⢠⡿⠀⠀⠀⠀⠀⠀⠸⣿⣤⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠛⠳⠶⠶⠶⠞⠛⠛⣿⡇⠀⠀⢀⣤⢤⡀⠀⠀⣀⡤⣤⡀⠀⣠⢀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣴⡏⣀⡴⠃⢀⣾⢋⣀⡼⠃⢰⡟⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢸⣿⠉⠁⢀⣠⢾⡏⠉⠁⢀⣴⡟⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⢉⣿⠿⣤⣀⣠⡶⠶⢶⣄⣀⣴⠟⠁⠀⠀⠀⠙⠓⠚⠋⠁⠈⠛⠒⠚⠉⣾⠁⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠛⠁⠀⠈⣿⠛⠳⠶⠞⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠠⠾⠁⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣖⡀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⡟⠉⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠹⣧⠀⠀⠀⢀⣴⠞⠉⣶⡀⠀⢀⡶⠀⠀⢠⡶⠀⠀⢠⡟⠀⠀⢠⣿⡶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢹⡆⠀⢠⡿⠁⠀⠀⢸⡟⠋⣿⠁⠀⢀⡾⠀⠀⢀⡿⠁⠀⢠⡿⠋⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇⠀⠀⣠⠟⠀⢸⡇⠀⣠⢾⡃⠀⣠⢿⡇⠀⣠⣾⠁⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠁⠈⠛⠛⠁⠈⠛⠛⠙⠃⠀⢀⣀⣿⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣤⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠋⠉⠹⣤⣀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠞⠀⠸⠖⡇⠀⢾⣨⠇⠀⢸⠀⠀⢇⡸⠂⠸⣇⡄⠀⢸⢾⠀⠀⠾⡆⠀⢸⠞⠀⠸⠇⠇⠀⠈⡟

⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂⠀⣀⣀⣤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⡆
⠀⠀⠀⠀⠸⠿⣿⣿⣶⣶⠄⠀⠀⠀⢴⣞⣷⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⣣⠸⢠⡇
⠀⠀⠀⠀⠀⠀⢨⢿⣿⢯⣿⣶⣄⠀⠀⠉⠉⠀⢋⡴⢁⠀⠀⠀⠀⢀⡄⡇⢘⢨⠷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⡙⢲⣦⢀⡀⠀⢲⠏⡀⠀⠀⠀⣸⡇⡿⠈⢿⢀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣖⣂⠁⢘⣧⡤⠀⠀⠀⠹⣂⠀⠀⢠⡇⠁⡀⡆⠈⢸⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠗⠁
⠀⠀⠀⠀⠀⣀⣠⣬⠟⠉⣩⣭⠝⠋⠏⢀⠁⠀⠀⠻⠈⡇⠀⠀⣿⢠⡏⠇⡇⡇⡄⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⡥⠋
⠀⠀⠀⠀⢸⠛⠻⠞⠛⠉⠡⡤⣤⣤⡄⢋⠀⠀⠈⠀⠀⠁⠀⣰⡇⠈⠀⠀⠁⡇⠛⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠊⣵⠎
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠓⢷⣺⣂⢂⠀⠃⠀⠲⣿⡗⡄⡄⣆⡄⣧⡄⣛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠵⢋⡄⠀⠀⠀⠙⠋⠓⠃⠇⠃⠻⢻⢿⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡇⠀⠀⠀⠀⠀⣀⡤⣮⠀⢂⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢒⡉⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠙⢆⣀⡠⠔⠞⠙⠧⢘⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢈⠿⠦⣬⣃⡀⠀⡆⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⠋⠀⠀⠀⢹⣿⣦⣿⣀⠀⡸⠀⢀⣾⣏⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡰⣦⡞⠀⠀⠀⠀⠸⡏⣻⠟⠁⣸⣇⣨⡾⢻⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡛⣧⣿⣷⠀⠀⠀⠀⠀⣑⣡⠀⢀⡿⠈⠉⠀⣾⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣄⠙⠄⠀⢠⣲⠽⢛⢯⣶⣾⡇⠀⠀⠀⣿⠀⠠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣿⠀⠙⠉⠁⣡⣶⣷⡿⠏⠛⢧⠀⠀⠀⢁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠛⠩⠽⠛⠀⡼⠁⣿⡆⠀⠀⢸⠧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⠀⠀⢰⣯⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣦⠀⡼⠒⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣷⠄⠀⢃⣅⡾⢡⠆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⢞⠿⡇⣸⡇⢨⡟⠀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠑⠊⡞⠡⠛⣱⡟⠀⠸⢐⣀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⠛⣱⡟⢯⠃⣾⡆⢩⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠛⢴⣾⠣⠉⠙⠀⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⢀⠜⣽⠋⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠜⠁⠠⠒⠉⠉⠁⠒⠤⠀⢀⣀⣤⡶⠏⠀⠀⠀⠒⡲⠖⠐⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⢀⢀⡴⠅⠀⠀⠠⢤⣤⣶⣶⢿⢿⠭⠭⠓⠒⣚⠡⠄⠊⠁⠀⠀⠁⠀⠄⠀⠀⠀⢠⣤⣦⡾
⠀⠀⠀⢀⣠⣴⡏⠀⠀⠀⣀⣤⠶⠚⠉⠀⠀⠀⠀⢀⠔⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢝⣿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⡴⠒⢒⣲⡖⠉⢉⣈⠉⠉⠋⠀⠀⡄⠐⠀⠆⠀⠀⡍
⠀⠀⠀⠀⠀⡸⣋⠄⠀⠀⠁⠀⠀⠀⠀⠀⠁⢐⠀⠀⠠⢀⡂⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠄⠀⠀⠀⠘⠀⠀⢀⠄⡉⢆⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠁⠔⠁⠠⠁⠀⢀⡀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢾⣦⡀⣀⡜⠊⠅⢰⠂⠰⠀⠀⠀⠀⠚⠋⠁⠀⡀⠠⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣦⣿⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⢀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠐⢂⡀⢬⣿⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡞⠀⠀⢠⢃⠄⣀
⠀⠀⠀⠀⠀⠀⠀⡁⢠⠊⠁⠀⠀⢈⠡⠴⢒⠂⠒⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⡾⠉⡿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⢛⣶⣧⡀⠂⢀⠀⢀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠓⣄⢸⣶⢸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢉⡿⠿⡇⠄⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣆⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠞⠀⠐⢠⠄⠀⠀⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠤⠤⠄⣀⠀⠀⢀⡀⠀⢀⡀⠈⠢⣀⣀⣀⣀
⠀⠀⠀⠀⢠⠊⠤⠄⠀⣾⠀⠀⠌⠁⠀⠀⠀⠀⠀⠱⠊⠁⠀⠀⠀⠈⢢⠀⠘⠆⠀⢢⠘⡆
⠀⠀⠀⠀⢁⠘⢄⡀⣠⠇⠀⠰⠀⠀⢠⢤⡄⠀⠀⠀⠀⠀⡠⣤⠀⠀⡆⠀⠀⢰⡤⠜⡠⠁⠀⠀⠀⠀⠀⠀⢀⠄⣒⡒⢄
⠀⠀⠀⠀⠀⠑⠂⠠⠿⠀⢀⠔⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠛⠋⠀⠀⠉⠄⠀⢸⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢺⠸⡁⠙⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠻⠿⠟⠃⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⢃⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⢈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡀⠀⠀⠀⠀⠑⠂⠐⠁⠀⠀⠀⠀⣠⠊⣰⠟⠁⢠⠐⠉⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠮⣐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠊⡀⠜⠙⢤⣠⠁⠠⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡏⢹⠐⠂⠀⠭⠭⡤⠀⠲⣖⠉⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⠈⡄⠀⠀⠀⢰⠁⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢡⠀⠰⡀⠀⡀⢸⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⠦⡀⠈⠢⣁⣸⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠔⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⢄⠈⣳⡦⢄⣹⠀⠀⠰⠤⠀⠂⠁⠀⢀⡠⠊⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢆⢀⡴⡁⢀⡠⠋⠀⠐⠀⠈⣄⢀⠶⡒⠀⠀⠐⠂⡝⠣⠀⠤⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠘⠘⠴⢀⡀⢠⠀⠘⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠣⠒⠔⠒⠁⠈⠉⠑⠁⠀⠓⠢⠰⠤⠖⠂⠀⠀⠀⠱⡀⠀⠀⢢⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⣤⣶⣶⣿⣶⣾⣷⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣟⣦
⠀⠀⠀⢸⣿⡄⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠸⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡄
⢀⣀⢀⣀⠉⠻⠿⠿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⢿⣿⣿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠿⢿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣤⣤⣦⣤⣠⣤⣤⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠋⠙⠃
⠀⠀⠈⠘⠻⢿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣻⣿⣿⢿⣿⣟⣿⣿⣽⣿⣿⣯⣽⡷⣦⡀
⠀⠀⠀⠀⠀⠁⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣯⣿⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡟⣙⣫⣥⣿⢿⣿⣿⣿⣿⡿⠃
⠀⢰⣀⣀⣦⣬⣦⣼⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡋⠿⢻⣤⣀⣿⣿⠟⠛⠁⣰⣿⣿⣿⡿⠁
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⠿⠿⠿⠿⠟⠛⠛⠛⠿⢿⣛⣟⣙⣿⣷⣴⣾⡿⠾⠛⠉⠀⠀⠀⢠⣾⣿⣿⣿⣿⢷⡄
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠉⠉⠉⠙⠋⠙⠋⠙⠚⠛⠚⠛⠛⠛⠛⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠟⠀⠛⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠐⠂⠂⠚⠋⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⣀⠀⠄⠠⠤⠦⠄⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡆⡀⠀⠂
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⡤⠜⠁⠀⠀⠀⠈⢁⣀⣤⡤⡤⠤⣠⣠⡄⠘⠉⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡒
⠀⠀⠀⢀⠈⠀⠀⠀⢀⡀⠄⠐⠀⠁⠂⣁⠤⠒⢊⣛⡩⠤⠤⠴⠶⠶⠴⡎⠀⠀⠀⢈⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠈
⠀⠀⠀⠐⣄⡀⠈⠀⠀⠀⠐⡀⠔⣚⠭⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⡄
⡀⢀⡀⢀⡈⠐⠲⠒⠶⠒⠑⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⠀⠀⢠
⢣⣤⣤⡀⠈⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⣉⠑⢦⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢻⡄⣧⡀⢦
⠈⠟⣻⢿⣦⣄⡐⠈⠉⠐⠒⠊⠸⠖⣀⢷⡶⠄⠰⠢⠄⠤⠖⠾⠟⠙⠋⢨⠁⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣀⠤⠄⠤⠤⣠⣿⠀⠁⠈⠁
⠀⠀⠀⠈⠹⡿⣿⣦⣀⠁⠂⢤⣀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣄⡈⠑⠂⠒⠀⠄⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠤⡀⠈⣁⠀⠀⠐⠂⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⠻⠿⣿⣷⣷⣷⣶⣶⣿⣿⣆⡀⠀⠸⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠤⠘⠡⢀⡰⢒⢴⠇⢀⠀⠨⡐⣸⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣠⢞⠋⣉⣽⣿⡿⠿⠟⠟⠋⣀⠐⢀⠀⠀⡀⢀⡀⠤⠀⠐⠂⠀⣠⠐⠂⢅⡀⣁⣦⡾⠿⡃⠀⠀⢀⡜⠃
⠀⢰⣤⣄⣶⣶⣶⣾⣿⡿⠟⠁⠊⢛⢉⡅⠠⠄⠔⠒⠚⢂⣭⠀⢉⠀⡀⣤⡀⢄⡀⠀⠲⠀⠒⠩⣔⣠⠭⠶⠟⠉⠀⠰⢧⠀⠀⠞
⠀⠢⠉⠉⢉⣩⣩⣄⣀⣐⣄⣠⣄⣈⣏⡔⢒⣚⠓⠚⠓⣋⡁⣀⡁⠂⣙⣓⣄⣀⡴⠷⠶⠿⠗⠚⠁⠀⠀⠀⠀⠠⣶⢢⣄⠤⠐⠢⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠍⠈⠀⠀⠀⠀⠀⠀⠀⠛⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢀⠀⠀⠂⠐⡠⢖⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡠⢃⣝⣦⣔⣢⣴⣷⣸⣆⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠈⠴⠎⡀⠀⠀⠀⠀⠠⢠⢋⣊⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠾⠾⠶⣾⣶⠎⢠⠀⠀⣶⠠⢎⣿⣼⣿⣿⡿⠿⠛⠛⠻⣿⡿⠋⠀⠀⠀⢀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠐⢿⠃⠀⢀⠀⣡⣰⣸⣿⣿⣿⠏⠀⠀⣀⣠⡀⠸⡇⠀⣠⣶⣿⣿⣧⠸⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠰⣛⣿⠿⣟⣿⣿⣿⣿⠁⠀⡀⠀⢻⢟⣿⣿⡿⢁⠀⢩⣽⢾⠿⣿⣦⣷⣾⢻⠋⡝⢿⣿⡇⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣧⣼⣿⣿⡟⣿⠂⠀⠀⢰⣷⣿⣿⣿⣶⣾⣤⣾⠋⡉⢳⡜⣿⣿⡿⢿⡶⢶⣛⠟⢧⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠴⡿⣿⣿⣿⡿⡟⢺⣿⡿⣮⣥⣿⣧⡼⣿⣿⢒⣟⡛⣓⡁⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢛⠙⣿⣉⠻⣿⡄⠀⠀⠴⣿⣿⣿⣷⡘⢸⡪⡙⠆⣥⣿⡿⠉⠀⠀⠀⠀⠙⢿⣳⣿⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢁⡊⠙⠚⢻⡇⠀⡐⠟⣿⣿⣿⣿⣿⣈⠄⠀⠩⢻⣿⠄⠀⢠⣄⡀⢰⡇⠀⠙⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣦⡄⣸⠿⠀⠀⠘⠿⠇⠘⠁⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠺⢿⣿⣿⣿⣿⣿⣯⠿⠌⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⣿⣿⣿⣿⣿⣿⣿⣧⠀⠹⢠⣌⣦⠀⠀⠀⠀⠀⠀⡹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⢩⣾⣿⣿⣿⣿⣿⣶⣶⣬⠉⠃⠀⠀⠀⣠⣶⣿⣟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⡿⣿⣿⣿⣿⣿⣿⠅⢔⠦⢤⣴⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⡿⢿⣿⣿⡇⠈⠉⢲⣤⣍⡙⠋⠀⠈⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⣿⣿⣧⠀⠀⠀⠹⣿⣿⡷⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠿⢷⣦⣤⡼⠝⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠰⣧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠁⣠⢎⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠓⠊⠛⠩⠏⠉⠚⠓⢓⡒⣒⣒⡢⠤⠤⠤⠴⠚⠉⠁⣠⣤⡶⠟⠷⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠙⠈⠡⠀⠈⠀⣀⣸⣦⣶⢶⣿⣻⣿⠿⠛⣻⣿⣿⣶⠶⣾⡿⠛⢙⡅⠆⢊⣴⠏
⠀⠀⠀⡐⠂⠉⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠋⡠⠀⠠⢀⣤⣼⣟⣿⣿⠿⣿⣿⣿⣿⡃⡬⢀⣙⣿⣯⣉⣀⢭⣵⢤⠽⠗⢸⡷⠃
⠀⠀⠀⠹⠿⣷⡄⣀⠈⠒⠂⠐⠒⠢⠤⢤⣤⣀⡀⣠⡴⠋⠀⠂⠀⣠⣶⣿⣿⣻⣿⣽⡚⣏⡹⣿⣇⣻⡿⠣⠀⢲⡿⣿⡶⢻⡔⣿⠸⣜⡼⠚⠁
⠀⠀⠀⠀⠀⡛⠛⢿⣿⣦⢤⣤⣄⠀⠈⠙⠚⠚⠉⠁⠉⠳⣦⣨⣞⣿⣁⢌⢿⣇⠄⣿⣟⢚⠅⡸⣿⢽⣿⣎⠒⡌⠥⢻⣟⣼⡧⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⣿⣿⣷⣦⣄⢀⣠⣌⡀⠀⠐⠙⢿⣿⡗⣍⣞⢋⢻⣿⡸⢿⣽⠪⢻⡿⣹⠿⠿⠗⢾⣷⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⣛⣿⣿⣿⣟⡩⠍⠻⢦⡀⢀⣈⡔⣜⠜⣶⢿⣿⣿⠧⢬⣿⣯⣽⣷⠶⠶⠒⣶⠾⠿⠿⠃
⠀⠀⣤⣦⣤⣴⣷⣿⣿⣿⣿⡻⠿⠿⠿⠟⠉⠀⠀⠀⠀⠈⢹⡚⡗⣐⣷⢛⣻⣳⣿⣦⣦⣭⣿⣏⠁⠛⠋⠟⠛⠟⠁
⠀⠰⣆⣉⣉⠛⠛⠛⠉⠁⠁⠀⠀⠀⠀⣀⣀⣀⣀⣤⣠⢔⣣⠵⣿⣿⣻⣿⠿⣿⢯⣅⢵⡁⣈⣿
⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠯⠭⢭⠭⠶⠿⠓⠚⠋⠀⠀⠀⢠⡇⡓⠻⣾⣩⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣺⣅⣷⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂