Consonant Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢰⢦⠀⠀⠀⠀⠈⢦⣸⠁⠀⠀⠸⡇⣀⣸⡇⠀⢀⠀⠀⣿⢠⠞⠁⠀⢸⠋⢳⡄⠀⢠⠴⠆⠀⠀⠈⢹⠟⠃
⠀⠀⠀⢸⡤⠷⡄⢠⠀⠀⢠⠟⢦⠀⠀⠀⣏⠉⠈⣇⠀⠘⡇⠀⣿⢻⡄⠀⠀⢸⠒⢾⡀⠀⠘⠦⣄⠀⠀⠀⢸
⠀⠀⠀⠸⠀⠀⠀⢸⡆⠀⠘⠀⢀⠤⣄⠀⠛⢀⣀⠉⠀⠀⣧⠀⠏⠀⠹⠄⠀⠾⠤⠼⠃⠀⢀⣀⣸⠃⢠⠀⠉
⠀⠀⠀⠀⢀⠔⠀⠀⠓⠂⡀⠀⡇⠀⠘⡆⠀⢸⠤⠀⠀⠶⢋⠤⣄⠀⢦⠀⡄⢀⠄⢰⠉⢱⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡀⢰⠄⠀⡄⢱⠀⠙⠤⠞⠁⠀⠐⠒⠀⣀⠀⣏⠀⢀⠀⢸⡜⢱⡜⠀⠘⢤⡜⠀⢱⡰⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠉⠁⡀⠀⢈⠁⡠⠁⢐⡠⠠⠁⠈⡀⠀⠀⠐⠈⠁⣐⠃⠈⡁⠘⠁⠊⠂⢀⢀⠀⠘⠁⡀⠀⠓
⠀⠀⠀⠀⠀⠈⠢⠁⠀⠩⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠁⠀⠀⠐⠀⠈⠉⠂⠀⠐⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢂⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀
⠀⠀⢀⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡴⠋⠀⠄⠀⢀⢀⢀
⠀⠀⠈⡘⢇⡘⡀⠀⠀⠀⠀⠀⠀⠀⢨⡸⠀⠀⠀⠀⠀⠀⢉⠆⠀⠀⠄⢀⡀⡆⠀⠀⠀⠀⠀⠀⢀⡠⠔⠤⠎
⠀⠀⠀⠀⠂⡑⠠⠀⠀⠀⠀⠀⠀⢀⠍⠂⣀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠄⠀⠀⣰⠀⠀⠀⣄⠊⢁⡠⠔⠂
⠀⠀⠀⠀⠊⡝⢇⢡⠀⠀⠀⠀⢠⠃⠀⡜⠁⠀⠀⠀⠀⠀⢄⢃⠀⠀⠀⢁⡠⠚⠁⠀⠰⠜⠓⠃⠉
⠀⠀⠀⠀⠂⠰⠈⠉⠀⠀⠀⠀⠇⢀⠞⠀⠀⠀⠀⠀⠀⠀⠈⢦⡍⠞⢆⡸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡀⠤⠀⠀⣠
⠀⡄⠀⠈⢠⠀⢰⣽⠀⠀⠀⠀⣰⣙⡠⡶⡦⢄⣀⣀⡀⠀⠀⠘⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢰⣡⠠⠤⠈⢦
⠀⠈⢉⡄⠈⡀⠀⢏⢦⡀⠀⠰⢃⣭⠫⡊⢌⣳⡜⡹⠁⠀⢐⡆⠐⢀⠟⣭⢂⣄⠀⠀⠀⠀⠸⡀⠀⢀⢀⠌
⠀⠀⠨⣄⢧⡁⠀⠀⢱⡇⠀⠣⠦⠞⠖⢵⡋⠴⢡⠃⠀⠀⠀⠉⢂⡞⢛⢹⠈⠈⠀⠀⠀⠀⠀⡨⠑⠂⠹
⠀⠀⢠⢾⣼⡀⢠⣞⡵⠃⠀⠀⠀⠀⠀⠀⠉⠑⠃⠀⠀⠀⠀⠃⠺⠋⠔⠉⠘⠋⠔⠀⠀⠀⠘⠒⣂⡀⡜⠃
⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠁


⠀⡈⠀⠡⡆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⡜⠀⠴⠀⠀⠀⢀⠀⠀⠀⠀⠀⣧⠄⠸⠀⢀⡀⡀
⠀⠠⠧⠈⠛⠟⠉⢩⠀⠀⠀⢠⠁⠀⠤⠀⠠⠄⠀⢃⠀⠀⠀⢀⠀⠀⡇⢠⠂⠀⠅⠀⠀⠀⠀⢣⣥⡠⠠⠄
⠀⠀⢠⠇⡠⠚⠀⠆⠀⠀⠀⠘⡄⠀⠄⠀⠀⡠⠀⡈⠀⢀⡔⠁⢀⢘⣱⠂⠀⠘⠀⠀⠀⣤⡶⠮⢤⣤⣢⢴⠄
⠀⠐⠛⠛⠠⢾⠧⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⢀⠔⠁⠀⠀⠂⠒⠛⣁⠀⢠⠢⠆⠀⠀⠀⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⢀⠀⠦⣠⢂⡀⡄⠀⠀⠀⠀⢀⠤⣀⠀⢄⡠⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⢀⣑⠂⠒⣀⣣⠂⠀⠀⠀⠈⠒⠉⠿⢫⠑⠂⠀⠀⠀⠀⠀⠀⠠⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⣢⡀
⠀⠖⠺⣿⠟⠻⠿⠗⣯⠀⠀⡔⠂⡤⠼⠹⠛⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠤⠉⠀⠀⢛⠂⠩⡇
⠀⠄⡐⠄⠠⠔⠰⠣⠁⠀⠀⠣⠤⠤⠦⠀⠤⢄⠀⢃⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⢈⡴⣢⡀⣀⣠⠠⣔⠁
⠀⠐⠛⣣⠆⡐⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠜⠀⠀⠀⠀⠀⢰⠀⠀⢀⠔⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠐⢀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯
⡀⠓⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⢧⡀⠈⠳⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡶⠃
⠈⢳⡄⠀⠈⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢹⣆⠀⠀⠈⠢⣀⣴⠀⠀⠀⠀⠀⢀⠀⠙⣆⠀⢀
⠀⢠⡀⠘⢧⡀⡀⠀⠙⢶⣀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⠀⠀⠀⠀⠁
⠀⠈⣧⡀⠈⢻⡅⠀⠀⠀⠙⢶⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠁⠀⠀⠀⢀
⠀⠀⠈⢧⡔⠀⠹⣦⡀⠀⠀⠀⠉⠚⠀⠀⠀⠀⠀⠐⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀
⠀⠀⠀⠈⢳⡄⠀⢈⠻⣷⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⢶⣤⣤⣤⣤⣤⣾⠟⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠙⣶⡁⠀⠈⠻⡏⠂⠀⠀⠀⠀⠀⠀⢀⠧⣾⡿⢂⡦⣿⣿⣟⣿⢯⡧⡄⠒⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠸⠇⠀⠀⠀⠀⠀⠀⣴⠏⣠⠏⠀⠈⣸⣿⡟⠀⠿⠄⠀⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣴⠀⠀⠀⠀⠀⢠⠞⠁⠀⠃⠀⠀⠀⣻⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⣎⣡⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡛⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣠⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣶⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠒⠊⠁⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡉⠙⠛⢁⡁⠉⠠⢍⡉⠉⠀⠀⣀⣀⣤⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⠀⠊⢠⠂⠐⣄⠹⡆⣸⣿⣿⣟⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠻⡄⣠⠟⢠⣷⣿⣿⡿⣿⢛⣿⡿⡿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⡂⠀⢀⠤⣿⣿⣿⣿⡝⠼⡿⣿⣟⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏⣳⣄⠀⣰⣿⣿⣿⡯⠃⠀⠠⣿⡏⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢷⣷⡌⣶⣿⣿⠿⡝⠁⠀⠀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈⠹⣗⢾⣻⡯⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠑⠙⠀⠃⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠒⠊⠉⠉⢁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⢀⠀⠀⢀⡼⠁⠀⠀⠀⠀⡀⠀⠀⠁⠠⣀
⠀⠀⠀⠀⠀⠀⣀⠄⠚⠁⠀⠀⠀⣰⠀⢀⣠⣾⣤⣴⠟⠁⡀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⢀⣠⠴⠛⠋⠉⠁⠀⠠⣞⣰⠟⠀⠀⣀⣴⣿⣷⣀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣀⠄⠒⠁⠈⠛⠛⠛⠉⠀⠀⠀⣀⢀⣠⡴⠀⠈⣻⡶⠚⣛⢫⣿⣿⣿⠯⠤⠤⠤⠀⠀⠀⢸
⢀⢰⡃⠀⠀⠀⠀⢀⣠⣶⠤⠶⠒⠉⠉⠉⠉⠁⠀⢈⣿⣿⢢⣿⣼⡟⠟⠉⠂⠀⠀⠀⠀⠀⠀⠘⡀
⣦⠀⠙⠒⠖⠒⠚⡉⠀⠀⣀⣤⣀⣀⣤⠖⠀⢀⡠⡎⣹⠃⢎⣿⣿⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠘⡷⠶⠶⠤⠐⠚⠛⠋⠉⠉⠉⢉⠉⠁⠀⣠⠚⠃⠀⠀⠀⠈⢻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠙⠲⠴⡤⣤⣤⠴⠶⠶⠾⢶⣿⣤⣤⠆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⡀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢤⡐⡀⠀⠀⠀⠀⠀⠀⠘⢮⠉⠓⠆⠀⠀⠀⠀⠈⠁⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⢀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⢤⢀⢄⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠜⠋⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⣿⡉⠀⠀⠂⠀⢀⣀⢠⡄⠖⠒⠉⠀⠀⣀⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⠛⠉⠛⡡⠖⠉⠒⢤⡀⠑⢀⣤⣶⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⠀⡏⢰⠛⠲⠛⢧⢹⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣇⠈⣳⠀⢰⣃⢸⣿⣿⣿⣿⣿⣽⣿⣿⡽⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⠄⣀⣀⡴⣿⣿⣿⢿⢿⡛⠽⢻⣿⡗⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⢦⡀⠀⣰⣿⣿⣟⣿⠛⢀⠀⢚⡿⠥⣈⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡙⣦⣿⣿⣿⣿⠅⢀⠀⠀⣻⠋⠐⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡐⢿⣿⣸⡿⣿⣟⣾⠂⠀⠤⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢨⡻⡇⢿⠞⢻⠅⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⠁⠀⠙⡆⠸⠀⠂⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠾⠀⠀⠀⠁

⠀⠀⠀⡶⠒⠲⣦⠀⠀⠀⣠⠖⠲⠦⠀⠀⢰⠖⠲⢆⡀⠀⠀⢴⠖⠒⠒⠀⠀⠀⣴⠖⠶⠆⠀⠀⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⣶
⠀⠀⠀⡷⠚⠚⢾⠀⠀⠐⣇⠀⠀⠀⠀⠀⢸⠇⠀⢀⡇⠀⠀⣸⠓⠒⠂⠀⠀⠸⣇⠀⠒⣶⠀⠀⢸⠗⠒⠚⡇⠀⠀⣤⠀⠀⣿
⠀⠀⠀⠛⠓⠒⠋⠀⠀⠀⠈⠛⠚⠉⠀⠀⠘⠒⠒⠋⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠙⠓⠚⠉⠀⠀⠘⠀⠀⠀⠃⠀⠀⠈⠓⠚⠁

⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⣄⠀⠀⠀⠀⠀⣄⠀⠀⠀⣠⠀⠀⣄⠀⠀⠀⡄⠀⢀⣤⣤⣤⡀⠀⠀⠀⣀⣤⣀⠀⠀⠀⣤⣤⣤⣀
⠀⠀⢸⡇⢀⣾⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⣟⠳⣤⡞⢹⠆⠀⡟⠳⣄⠀⡇⠀⢸⣇⣀⣈⡷⠀⠀⣼⠁⠀⠈⣷⠀⠀⣿⣤⣤⡽
⠀⠀⠸⡷⠋⠙⢷⡄⠀⠀⣧⣤⣤⡄⠀⠀⠏⠀⠀⠀⢸⠆⠀⡇⠀⠈⠻⡇⠀⠸⠏⠉⠉⠀⠀⠀⠙⢦⣤⡿⣇⠀⠀⡇⠀⠙⢦


⠀⠀⠀⣴⠒⠒⠒⠀⠀⠐⠒⢲⡖⠒⠀⠀⠰⡆⠀⢠⡆⠀⢰⡆⠀⠀⠀⣶⠀⠐⣦⠀⣀⠶⠀⠀⠲⣄⠀⣠⠖⠀⠐⠒⠒⢲⡶⠂
⠀⠀⠀⠙⠒⠒⢦⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢹⣄⡞⠀⠀⢸⣇⡔⠳⣄⣯⠀⠀⣠⠿⣯⡀⠀⠀⠀⠈⣿⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠒⠒⠚⠋⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠘⠋⠀⠀⠈⠋⠀⠘⠉⠀⠈⠙⠀⠀⠀⠀⠛⠀⠀⠀⠐⠛⠓⠒⠒⠂

⠀⡀⢀⡀⠀⠠⠈⠀⠀⠀⠄
⡆⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⠀⠠⢄
⠻⣦⡀⠀⠀⡀⠘⢿⣄⣀⡐⠀⠀⠀⠁⠢⢀
⠀⠈⠻⣶⣏⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠙⠷⣄⣤⠊⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠹⢷⣾⡞⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠳⣦⣀⣠⡠⠀⠀⠀⠀⠙⠢⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⢿⣤⠀⠀⢀⡀⠈⠛⠋⠁⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠢⡝⢿⣶⣿⠟⠀⠀⠀⠀⣠⣴⠶⠋⠁⠀⠀⠀⢀⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣶⡟⠉⠂⢙⠆⣹⠟⠀⠀⠀⠀⢾⣿⡄⠀⠀⠀⠀⠀⣠⣾⣿⠁⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢠⠦⠙⠻⣿⣆⣴⠁⣸⣿⠀⠀⠀⠀⠀⣸⣿⣿⣷⣶⣶⣶⣿⣿⢿⣧⣤⠔⠃⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠸⣧⣤⣤⣿⣹⠟⢿⡟⠉⠀⠀⠀⠀⢀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣦⢉⠉⠉⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡟⢄⣾⠁⠀⠀⠀⠀⠀⣼⠏⠴⠘⢿⣿⣿⣿⡟⠛⠻⠃⠉⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣏⠀⠀⠀⠀⢀⡼⠃⠀⠀⠠⢾⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⡀⠀⠀⠌⠀⠀⠠⠀⠀⣜⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⡀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠈⠉⢢⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠠⠶⠤⣤⣛⠀⠹⡀⢀⣀⣤⣴⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠻⠂⢠⠟⣁⡤⠠⣌⠳⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣿⠀⢻⡄⠀⠈⠀⣿⣾⣿⣿⣿⡿⣿⣿⣿⣻⣿⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣄⡙⠦⠰⠚⣰⣿⣿⣿⣿⣿⡿⣿⣿⣿⣉⣵⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣏⠁⢰⡟⣹⣿⣿⣿⣿⡏⠀⠁⢿⣿⠉⠿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⣷⣄⢡⣿⣿⣿⣿⠏⠀⠀⠠⡼⠧⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣾⣿⣿⣿⣿⠇⠀⠀⢀⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⠀⠿⣿⠻⠟⣿⣷⠋⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠀⠀⠍⠁⠞⠀⡹⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡄⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡀⠀⠀⢠⣤⢠⠄⡀⢀⡀⢀⡄⡀⠀⣠⠤⣤⡀⢀⡀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢋⡟⠀⠘⠛⠛⠛⠛⠛⠙⠛⠛⠛⠀⠙⠒⠋⠙⠛⠛⠃⠙
⠀⠀⠀⠀⢰⣄⣰⡆⠀⠀⡁⠀⠀⠀⣀⣰⡆⠀⠀⠀⠀⠀⠀⣼⢸⡆⠀⠀⠀⠀⠀⠀⣀⣀⣾⠀⠀⠀⠀⠀⠀⣀⣀⣰⠀⠀⠈
⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠛⠚⠃⠀⠀⠀⠀⠀⠀⠙⠘⠃⠀⠀⠀⠀⠀⠀⢿⠿⠛⠀⠀⠀⠀⠀⠀⠿⠛⠙
⠀⠀⠀⠠⠠⠀⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠤⠤⠄⠤⠠⣬⠠⠤⠤⡥⠠⠠⣤⠤⠤⠤⠤⠄⠤⠤⠤⠄⠤⠤⠤⠤⠄⠤⠠⠤⠧⠠⡄⠤⠈⠤⠤⠤⠤⠀⢄⡤⠤⠍⠤⢠
⠀⠀⠀⠀⢰⣆⣿⠀⠀⠀⡀⠀⠀⣿⣶⣴⠆⠀⠀⠀⠀⠀⠠⣖⣲⠂⠀⠀⠀⠀⠀⢠⣶⣷⡖⠀⠀⠀⠀⠀⠀⢺⢧⡖
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡂⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⣤⠀⠐⠃⠀⠰
⠀⠀⠀⠀⣿⣷⠾⠀⠀⠀⡀⠀⠀⣿⡷⠾⠁⠀⠀⠀⠀⠀⠸⡇⠾⠀⠀⠀⠀⠀⠀⢷⢶⠆⡟⠀⠀⠀⠀⠀⡟⠿⠻⠾⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠨⠐⠀⠠⠐⠂⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠆⠀⠀⠂
⠀⠁⠀⠀⢀⡀⣤⠀⠀⠀⡃⠀⠀⢀⡀⣀⡀⠀⠀⠉⠀⠀⢀⡀⣀⣀⠀⠀⠉⠀⠀⠀⣀⢀⡀⠀⠀⠈⠀⠀⠀⡀⢰⣀⠀⠀⠈
⠀⠀⠀⠀⠘⠓⠛⠀⠀⠀⡆⠀⠀⠘⠛⡿⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠛⠛⠇⠀⠀⠀⠀⠀⠀⠛⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⡅⠀⠁⠀⠀⠈⠈⠁⠀⣀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠀⠀⠈⠀⠈⠈⠈⠀⠀⠀⠉⠀⠀⠀⠈⠀⠠
⠀⠀⠀⢠⣄⣤⣤⢤⠀⠀⠀⠀⠀⣤⢠⣤⣤⠀⠀⠀⠀⠀⡶⢤⣤⣠
⠀⠀⠀⠈⠉⠉⠉⠈⠀⠀⠀⠀⠀⠉⠀⠁⠉⠀⠀⠀⠀⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡀⠠⠀⠀⢠⠀⠀⠃⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠄

⠀⠀⠀⠔⠐⠢⡀
⠀⠀⠸⣿⠂⠀⠈⠦
⠀⠀⠀⠸⣷⡇⣠⠀⠑⠄
⠀⠀⠀⠀⠘⢿⡁⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠈⢿⣦⠄⠀⠀⠀⢄⠀⢀⡴⠂⣶
⠀⠀⠀⠀⠀⠀⠀⠻⣶⡷⠀⠀⠀⠀⣸⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣯⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡀⠀⢀⡿⠀⠀⠀⠀⡀
⠀⠀⡠⠂⠉⠙⠐⢤⣀⠀⠈⠻⠞⠃⢸⡆⠀⠀⠈⠉
⠀⣸⠀⠀⠀⠈⠃⠀⠈⠳⠆⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠐
⠀⢹⣄⠀⢠⡃⠀⠀⢹⠇⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢢⡀
⢀⠊⠉⠛⠻⣿⣿⣷⡟⠀⠀⠀⠀⠰⣄⠀⣾⣿⣶⡀
⢰⡄⠀⣀⠄⠠⠙⢯⡀⠀⠀⠀⠀⠀⠛⣶⣿⣿⣿⣿⣒⠀⠠⠀⠀⠀⠀⠄
⢀⠻⢶⣿⣤⣀⢀⣼⠧⡀⠀⠀⠀⠀⢨⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠄
⢸⣶⣠⠈⠙⠛⢿⠛⠀⠀⠀⠀⠀⠀⣸⠇⠚⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠙⠻⢶⣶⣶⣾⠐⠀⠀⠀⠀⠀⠀⠏⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣗⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣆⣀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⣠⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣾⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣤⠀⠀⠀⠀⣀⣀⣀⡤⠤⠶⠚⠋⠁⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⢛⣛⢛⡭⠛⠓⠢⣍⠈⠁⢀⣀⣠⣤⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣬⡄⡏⠀⡰⠢⡀⠈⣧⢠⣿⣿⢿⡽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⢿⢈⠇⠀⣼⣾⣿⣿⣻⢚⠽⣯⢿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡦⠄⣁⡠⢼⣿⣿⣿⣿⡵⢿⢿⣿⡋⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠝⣆⠀⢠⣾⣿⣿⣿⠞⠀⠀⢼⣿⡀⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡞⣯⣌⣱⣾⣿⣿⡿⡋⠀⠀⠀⡿⠃⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⢹⣿⡸⣟⢿⣯⣿⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⣽⡇⠳⢘⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠹⠇⠀⠀⠁⠈

⠀⠀⠀⠀⡀⠂⠠⠀⠢⡀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡠⠤⡄⠀⢄⠤⢄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠇⠀⡀⠀⠈⢳⡄⠀⠀⠃⡀⡤⡀⠀⠀⠀⠀⠐
⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠇⢀⠐⠁⠀⠘⠑⡐⠭⠕⠀⠠⡄⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⠀⢌
⠀⠸⠄⠠⡀⢀⢠⠀⠀⠐⢀⡲⠌⠀⠈⠂⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠁⠀⢠⡰⠀⡪⢄⣡⠅⠰⡲⠀⠐
⠀⠀⠀⢈⠈⡁⠆⡈⠩⠀⡉⠀⠀⠔⠀⠄⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠂⠋⠁⠀⢀⡀⢐⢲⠈⡥⡄⠰⡖⠀⠆⠊
⠀⠀⠀⠈⠀⠉⠀⠙⠀⠀⠀⠐⢂⠄⠣⠀⠄⣀⠀⠀⠀⠀⠀⠈⡀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠊⠀⣈⢀⠀⠀⠈⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠠⢠⠀⠀⠀⠀⡀⢐⠀⠊⠉⠀⠀⠀⠈⡔⡄⠀⠀⠀⡀⠀⢠⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠇⠠⠌⣠⠀⠀⠀⢐⠈⡄⠀⠀⠀⠀⠀⠀⢁⣸⠀⠀⠀⠈⢳⣜⡃
⠀⠠⠠⠀⠐⠂⠁⢀⠔⠀⠐⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠁⠻⠆⠬⢢⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠒⠠⢅⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠈⣲⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢋⠩⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⢁⠴⠂⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠖⠺⣷⣁
⠀⠀⠀⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠈⡂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠡⢛⣮⢤⠴
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡅⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠉⣠⠃
⠀⠀⠀⣀⡰⡀⠀⠀⠀⠀⢑⠀⠀⡀⠀⠀⠈⠐⠀⠀⠀⠁⠀⠈⠚⠀⠀⠀⠜⠂⠰⠒⠪⡘⠂⡀⠀⠀⠀⠀⠀⠈⠓⠈⠂
⠀⠀⠀⠈⠀⠀⠁⠂⢔⠆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⡀⠀⢀⠄⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠊⠀⠄⠚⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠠⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⢰⡭⠀⠀⢀⠘⠌⠀⠀⢀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⠀⠆⠀⣈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠊⠀⠂⢀⢠⠀⠀⠀⠀⣁⡘⠀⠀⠀⠀⠀⠠⡤⠤⠀⣤⠄⠀⠀⡀⠂⠂⡀⠐⠀⢬⠀⠃⠀⠞⠀⠂⠓⠘
⠀⠀⠀⠐⠁⠈⠀⠀⠠⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠑⠒⠲⠒⠒⠆⠁⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠂⠄⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡒⢄⠀⠀⠀⢠⢈⢙⣿⣷⣦⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋
⠀⢠⠄⡀⠀⠃⢠⠂⠀⠖⠀⠁⠐⠊⡤⠓⠦⠑⠄⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠈⠙⠤⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠉⠚⠁
⠀⠀⠀⠀⠐⠂

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠙⠀⠀⠀⠘⠂⠀⠀⠀⠀⠉⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠠⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠤⠐⠄⠀⠀⠀⠀⠀⠀⠁⠂⠀⣤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠔⠀⣀⠐⠀⠀⠄⡈⠠⠕
⠀⠀⠀⣄⠁⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⢋⢴⣌⠐⠁⠀⠀⡄⠇⠀⠍⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠉⠀⠀⠀⠀⠀⠂⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⠀⢠⢠⠈⠁⠀⠀⠀⠰⠠⡀⠀⠀⠀⠀⠀⠀⡀⠐⠐⠀⠀⠑⠀⢰⣶⠀⠀⠀⠀⠠⠀⠠⠀⢠⢀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠠⠀⠄⠁⠄⠀⠀⠀⠨⣽⠀⠀⠀⠀⠀⠀⠐⡑⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⡆⡧⠸⠸⣏⡆⠀⠀⠀⠴⠀⠆⡈
⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠩⠆⠀⠀⠀⠀⠀⠌⠁⠀⠀⠊⠁⠀⠰⠀⠀⠈⠀⠀⠀⠀⠻⠿⠿⠆⠀⠀⠀⠘⠤⠑⠂⠈⠁⠀⠀⠈⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⡠⠠⠤⡀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⢀⡠⣄⡀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⢀⢤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⠠⠁⠀⠸⠀⠀⠀⠐⢂⠄⢄⠀⠀⠀⠀⠀⠸⡹⢬⡚⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⡈⡖⠀⡴⡜
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠾⡶⠀⠅⠀⠀⠀⠀⠼⠀⠀⠂⠀⠀⠀⠀⢇⣦⢁⡴⠀⠀⠶⠀⠀⠰⢈⠀⠀⠀⠡⣷⣄⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡀⠘⢕⠂⠠⠀⡀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌
⠀⠀⠀⠆⡀⠠⡡⠃⠀⠀⢠⢂⠰⠊⣀⣏⡅⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠐⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠰⠐⠠⠁⠀⠀⠀⠀⢢⠱⢚⢫⠃⡅⠀⠀⠀⠠⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠜⠀⠀⠀⠀⠀⠒⠀⠠⣄⡄⠀⠀⠀⢲
⠀⠀⢶⠀⠈⠁⠀⠠⠀⠀⠀⠀⠠⠒⠀⠉⠁⠀⠴⠂⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠂⠀⠀⠁⣦⠀⠀⠀⠀⠒⡡⡟⠁⠈⠓⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣀⡀⠀⠀⣀⣀⢀⣄⠀⠀⡀⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⣠⡤⢀⣄⠀⠀⣀⣤⣤⡀⠀⢀⣀⣀⣄
⠀⠀⠀⠀⣇⠀⠀⠀⣿⠀⣿⠀⠀⣇⣀⣀⡀⠀⠀⣿⣀⣀⣀⠀⠀⣿⣀⢸⣀⠀⢠⣖⣶⣞⡃⠀⢸⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁
⠀⢠⠀⠀⢠⠀⠀⡄⢠⡄⣄⠀⠀⠀⣠⡀⠀⠀⠀⢠⡄⢠⡄⠀⠀⢀⣤⣤⡀⠀⠀⠤⣤⠤⠀⠀⠤⡤⠠⡤
⠀⢸⣟⣋⣿⠀⠀⣿⣿⢿⡿⠀⠀⡴⠋⠳⠄⠀⠀⡼⠧⠼⢧⠀⠀⠸⢤⣤⠟⠀⠀⡴⠉⠳⠄⠀⣰⠷⢠⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠐⢲⣶⠶⠀⠀⢐⣒⣲⣶⠀⠀⢰⣖⣒⣶⡀⠀⠀⢲⠒⡖⠂⠀⢰⡾⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠙⠶⠀⠀⠀⠀⠀⠛⠀⠀⠘⠓⠒⠒⠀⠀⠀⠾⠶⠓⠂⠀⠘⠶⠶⠃

⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⠒⠀⠀⠀⢸⢾⠀⠀⠀⢸⠿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠺⡇⠀⠀⠒⡇⣿⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁

⠀⠀⠀⠀⣠⡆⡆⠀⠀⠀⡀⠀⠀⠀⢀⠀⣆⡀⠀⠀⡀⢰⣰⠄⠀⠀⡀⢠⡄⠀⢀⣠⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠙⡇⡇⠀⠀⠙⠋⠁⠀⠈⠛⠛⠏⠀⠀⠚⠛⠻⠹⠆⠀⠉⠛⠻⠁⠀⠚⠛⠃⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠹⠩⢼⠀⠙⠏⢴⡇⡇⠀⠛⠟⢻⠀⠀⠀⠻⠷⠂⠀⠀⠴⠶⠶⠀⠀⠀⣿⠀⠀⠀⠴⠶⢾⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⢀⡀⠀⠀⠀⢀⣤⣤⣄⠀⠀⣤⣤⣤⣤⠀⠀⣤⣤⣤⡄⠀⢀⡀⠀⢀⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠶⠶⠿⠀⢠⣿⠀⢸⡇⠀⢸⡟⠛⢻⡇⠀⢀⣼⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘⠓⠒⠒⠂⠀⠙⠛⠛⠛⠀⠀⠛⠛⠛⠓⠀⠀⠛⠛⠛⠃⠀⠘⠛⠛⠛⠃⠀⠘⠁⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣤⠀⠠⠶⣶⡶⠆⠀⠰⢶⠴⢶⠆⠀⣶⠆⡶⠂⠀⢰⣄⣤⣄⡄⠀⠀⣰⠀⣰⠆⠀⠶⡶⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡿⠀⠀⣰⡿⢥⡀⠀⠀⡾⠀⡟⠀⠀⣿⠀⣿⠀⠀⢸⣏⣟⣏⡇⠀⢠⢏⢠⡯⠀⠀⣸⠇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠈⠉⠀⠀⠉⠁⠉⠁⠀⠈⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣭⣽⡇⠀⢺⣉⣉⡉⠀⠀⠈⣿⢻⠋⠀⠀⠶⢿⣶⠆⠀⠐⠖⣿⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⣍⣉⣉⠀⠀⣤⣧⣼⡄⠀⠀⣴⠟⠷⡄⠀⠀⠸⣭⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣀⠀⠀⠀⢸⣄⣿⠀⠀⠀⣿⣶⠄⠀⠀⠀⣷⣾⠀⠀⠀⠠⢤⣿⠀⠀⠀⢠⣿⡇⠀⠀⠀⠶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠉⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⡟⢹⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⢸⣇⠀⠀⠀⢠⢸⣧⡇⠀⠀⢀⡆⣿⠀⠀⠀⢤⢰⠀⠀⠀⠐⠶⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠚⠚⠛⠒⠀⠀⠘⠛⢻⡏⠀⠀⠈⠉⢹⡟⡇⠀⠀⠉⠉⣿⠀⠀⠒⠛⠚⠓⠂⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⡏⠀⠠⣤⠀⣿⡇⠀⠠⣤⢼⡇⠀⢤⡤⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⡇⠀⠀⠃⠈⣿⡇⠀⠀⠘⢸⡇⠀⠀⠃⠛⠀⠀⠈⠉⠉⠉⠁⠉⠉⢹⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣦⢀⣦⠀⣦⣠⢠⠖⠆⣤⢠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠚⠙⠚⠋⠛⠘⠶⠋⠙⠞⠛⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⠀⠛⢦⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⡃⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⢹⣿⣿⡿⠀⣿⠀⠀⢸⣿⣿⠖⡀⠀⠀⠀⠀⠛⣄
⠀⠀⠀⠀⢀⡀⠀⣼⣧⠂⠀⠀⢈⣿⣿⣿⣿⡿⣿⣶⠀⣿⣿⣿⣤⡖⠀⠒⠄⠐⠋⡀
⠀⠀⠀⣰⣿⣧⣀⣿⣿⣦⣴⣤⣌⣹⣿⣿⣿⣧⡄⠈⠢⠘⢻⣿⣿⣧⣶⣤⡀⢶⡀⠃⠀⠀⠀⣀⣤⣶⡿⠿⠿⣷⡄
⠀⠀⠀⣿⣿⡏⠙⢿⣿⣿⡿⠛⢿⣿⣿⣻⣿⣿⣿⠶⠐⣶⣾⣿⣿⣿⣿⣯⣤⠞⠁⢸⣀⣴⣾⣿⣟⠵⠂⠒⡲⢤⣿
⠀⠀⠀⢹⣿⣿⡇⠘⣿⣿⣿⣦⠀⠙⠻⣿⣿⣿⣿⡾⡟⠉⠀⠙⠿⠯⠉⠉⠙⠂⠀⣼⣿⣿⣿⠛⠁⠀⢀⣮⡵⠊⠁
⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣿⡶⠈⣳⠟⠋⠉⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠁⠀⠀⢀⡜⠁
⠀⠀⠀⠀⢸⣿⣿⣳⣿⣿⡿⢿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣴⣷⡶⢀⡀⢀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠙⠛⠁⠀⠀⢀⣶⣏⣾⣷⣿⣿⣿⣿⣿⣴⣾⣿⡿⠁⠀⠀⠀⣶⣾
⠀⠀⠀⠀⣼⣿⡷⠟⣁⡘⠀⠀⠀⠀⠀⣠⣶⣿⡾⢿⠟⣛⣯⣴⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⣿⣿⣿⠟⠃⠛⠃⢀⣀⣤⡾⠟⠋⠁⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⢸⣿⣿⣯⣤⡶⠾⠛⠛⠁⠁⠀⠀⠀⠀⣠⠾⣛⣽⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⢸⣿⡿⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢡⣴⣿⣿⣿⣿⣿⡿⢿⣧⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⢸⣿⣧⡿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⡿⠃⠈⢻⡀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⢸⣿⣿⠃⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠙⡄⠀⠀⠀⢀⡀⣠
⠀⠀⠀⠈⣿⣿⣿⡬⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠘⢀⡿
⠀⠀⠀⠀⣿⣿⣿⣤⡖⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧⡜⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⢸⣿⣿⣃⡻⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢻⣄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⢸⣿⣿⡀⣠⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣟⡇⣀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⡀⢀⠀⠀⠀⠀⠀⠈⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣁⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⡿⣯⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠊⠁⠀⠀⢀⣶⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⠤⣤⣴⣀⠐⠒⠂⠻⡁⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠛⣿⣋⣉⣩⣶⠖⠂⢉⣉⠙⠷⣄⠀⠙⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣙⣉⠉⡿⠁⢰⡖⠉⠉⢳⣆⠘⣧⢠⣿⣿⣿⣿⣿⣸⣿⣿⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⣧⠀⢸⡃⠐⠖⢚⣿⠀⣿⣿⣿⣿⣿⣿⡟⢋⣿⣿⣿⣭⣤⣟⣿⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⣿⣦⠀⠻⠶⠶⠞⠁⣰⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡅⣬⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠲⠦⢀⣴⠾⣻⣿⣿⣿⣿⣿⣿⠋⢈⡋⢘⣿⣿⡴⣿⠶⢁⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⠸⣯⢰⣿⣿⣿⣿⣿⣿⡿⠂⡁⠄⢙⣹⡟⠶⣁⠶⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡷⡿⢷⣀⣿⣿⣿⣿⣿⣿⠎⠀⠀⠁⠀⢻⡟⠋⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢟⣿⣿⡖⢻⣿⣿⣿⣿⡟⣡⠴⠖⠒⢀⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠹⣿⣿⣿⣿⠿⣿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠐⢾⣿⣧⠘⣤⠛⠻⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⢿⣀⠀⠁⠀⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢳⠄⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠏⢀⡀⢹⡿⣯⠙⠿⢻⡟⠛⣟⠙⠿⣏⢉⣿⠟⠙⠻⠛⣿⠋⠛⠛⠛⠛⠛⣿⣟⠛⠛⠛⠛⢛⣻⣿⡟⠛⠛⠛⠛⠻⣿
⣿⣷⣤⣴⣾⣿⣾⣷⣿⣶⣷⣾⣿⣷⣿⣿⣷⣾⣿⣿⣷⣶⣿⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⡄⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠂⠀⠀⠀⠀⡅⠀⢢⠀⠀⠸⠀⠀⡀⠀⠀⡇⠀⢸⠀⠀⢸⠁⠐⠂⠀⠀⢈⠀⠒⠀⠀⠸⡁⠀⢠
⠀⠈⠀⠀⠃⠀⠀⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠘⠉⠙⠃⠀⠀⠘⠀⠀⠀⠀⠀⠈⠉
⠀⢰⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡄⠀⠔⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⢠
⠀⢨⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠈⡇⠀⠈⡆⠀⠈
⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠉⠉⠃⠀⠀⠁⠀⠀⠁⠀⠈⠁
⠀⣠⠀⠀⡀⠀⠀⡆⠀⢠⠀⠀⢠⠄⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠂⠀⠀⠀⠀⠐⡆⠀⢰
⠀⠫⣀⠀⠃⠀⠀⡅⠀⠀⠀⠀⠘⠀⠀⡃⠀⠀⡇⠀⢀⠀⠀⠠⢄⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠂⠀⠀⠀⠠⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠀⠀⠐⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡟⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠛⠛⠛⢻⣿⣿⣴⣿⡗⣤⣤⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⢨⠅⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠘⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣶⣶⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⢹⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡔⣶⣿⣷⡆⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣛⡁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠛⠀⠉⠉⠉⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠇



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠐⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠘⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠀⡇⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⢰⣤⡄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⠤⠤⠀⠀⡃⠀⠀⠀⢀⡠⢴⣿⣶⣶⣿⣿⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⡠⠄⠒⡦⠤⠤⠴⠶⠷⠶⠶⠶⠆⠀⢀⠇⠀⠀⠀⣇⠠⠄⠼⡞⠉⠉⠙⢻⣽⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⢻⠋⢉⣉⣉⣙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⣩⣷⡶⠶⠿⣯⡉⠑⡆⠀⣿⣿⣤
⠀⠀⠀⠀⠀⢰⣶⣴⣶⡿⣧⣼⣀⣈⣉⣉⣉⣉⣑⣢⣤⣀⣀⣠⣤⣄⡀⠀⢣⣀⣀⣛⣹⠒⠀⠉⢙⣁⣀⣧⣼⣿⠏⠉
⠀⠀⠀⢀⣀⣾⣷⠟⠉⠉⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠇⠀⠀⠈⡀⠀⠀⡌⡀⠀⡿⠻⣿
⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡇⠀⢃⠃⠀⡇
⠀⠀⠀⠀⢈⣿⣻⣄⣀⣀⣠⣾⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣼⠋⠀⠈⢳⡄⠘⢸⠀⢸⢸⠀⡇
⠀⠀⠀⠀⠙⠋⠛⢿⣯⡿⠿⢿⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠙⣦⡈⡇⠈⡎⡄⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢸⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢈⣧⢧⠀⢳⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣠⠟⠘⡸⠀⠸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠅⠆⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⠉⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢢⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻


⣏⠉⣹⣛⣛⣛⣛⣛⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠛⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣼⠏⠀⠙⠳⢶⣤⣤⣀⣀⣀⣤⣴⠶⣆⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⡏⢠⡶⠶⠶⣦⡀⠀⢩⡽⠷⠶⢦⡀⢹⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⣿⠀⠀⠀⢸⡟⠛⢻⡇⠀⠀⢸⡟⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠘⠳⠶⠲⠛⠁⠀⠈⠳⠶⠶⠟⠁⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⣤⡀⠀⠀⠀⣀⣤⠶⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠙⠛⠛⠛⠉⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠟⠙⢷⡄⠀⠀⠀⠀⠀⢠⡾⠋⠻⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠛⠛⠋⠉⠁⠹⣆⠀⠀⠙⢦⡀⠀⢀⡴⠋⠀⠀⣰⠏⠈⠉⠉⠛⠛⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⣶⡟⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠳⣄
⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣴⣿⣭⣿⣦⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆
⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠛⠛⠷⢦⡀
⠙⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠟⠁⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠹
⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣦⣦⣤⣴⣶⣶⣶
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠘⠀⠁⠀⠁⠀⠀⠀⠀⢠⡞⣟⢷⡀⠀⠀⠀⠀⣠⠴⣶⡶⢤⡀⠀⠀⠀⢠⣶⣾⠒⢲⠀⠀⠀⠀⢀⣀⣀⠴⠦⡀
⠀⠀⠰⠴⠶⠶⠆⠀⠀⠀⠀⠈⢷⣶⣿⡇⠀⠀⠀⠀⠀⠫⠿⠯⠃⠀⠀⠀⠀⠘⡏⠙⣶⡾⠀⠀⠀⠀⠘⠛⠈⠢⢴⡃
⠀⠀⠛⠛⠀⠚⠛⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⣸⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⢰⡶⠶⣶⣶⡆
⠀⠀⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⢸⣿⣽⠀⠀⠀⠀⠀⣽⣟⣿⣿⣿⡅⠀⠀⠀⠀⠸⠀⠝⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣾⠛⠻⣷
⠀⠀⠁⢰⠚⡆⠀⠀⠀⠀⠀⣰⠞⢾⣻⣢⠀⠀⠀⠀⢾⣿⣿⠆⡠⠁⠀⠀⠀⠀⢸⣸⣿⡇⠀⠀⠀⠀⠀⣷⣈⣤⣿
⠀⠀⠂⠀⠛⠐⠒⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠿⠶⠾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣀⢀⡀⡀⣀⢀⠀⠀⠠⠀⣀⢄⡀⠀⢀⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇⠀⠀⠀⠀⢠⣤⡤⠞⠛⠃
⠀⠀⠁⠉⠉⠉⠈⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣷⡤⣤⡇⠀⠀⠀⠀⠈⠋⠓⢦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⠉⠉⠀⠀⠀⠀⠉⠉⠛⠛⠶⣤⣀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠛⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣰⣿⣄⣀⣀
⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠈⠉⠉⠛⠷⢦⣄⡀
⠀⠀⠀⠀⢀⡾⠁⠀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠾⠛⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢀⣿⣥⡶⠛⠉⠀⠀⠈⠉⠉⣿⠛⠉⠉⠉⠉⠀⠘⠛⢛⡟⠛⠛⠛⢛⣿⠛⠛⠂⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣄⡀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠛⢶⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⠛⠓
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡦
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠰⠟
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣴⣾⣶⣄
⠀⠀⠀⠀⠀⠀⢸⣿⢻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⠈⠙⣿⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣸⣿⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⡟⠁⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣟⣛⣿⣛⣛⣿⣭⢯⣽⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢿⠿⠛⠁⠠⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⣀⣀⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⠿⢿⣿⣿⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠋⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⠇⠀⠀⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⣿⣤⣄⠀⢿⡿⠀⠀⠀⢸⡿



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⡿⣿⢸⡇⠇⡟⣽⠋⠿⠛⣳⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠁⠀⠀⠀⣀⣥⣴⣾⣿⣿⣯⣽⣶⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠛⠉⠁⠀⠀⠀⠲⢶⣶⣶⣶⣭⡙⢿⣿⣿⣿⣿⣭⡶⢾⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡆⢿⣿⣯⣵⠶⣿⣛⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢃⣾⣿⣿⠿⠟⠛⣉⣥⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣀⣀⠀⠀⠀⢠⣶⣦⣤⣀⠀⠀⠀⢰⣿⣃⠚⣩⣥⣶⣶⠏⣈⡁⡴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⡾⠋⠉⢛⠀⠀⡀⠀⠀⠈⠉⠉⠻⡄⠀⣿⣿⡿⠼⠛⠋⣉⠀⣀⠤⣛⣠⣟
⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢷⡀⣀⡉⠙⠆⠀⠀⣀⣀⣀⠀⠀⠣⠀⠙⣿⣇⠀⠾⠿⡛⠀⣰⠏⣩⣭⡙⡆
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠀⠙⠈⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⢀⣿⣿⠀⠢⠭⢆⢰⠇⠀⠙⢻⠃⠇
⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣇⠐⠉⢁⣚⠀⠠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⠇⠀⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣦⡰⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢀⣤⢠⣄⡀⠰⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡻⣠
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⢤⡤⢠⣤⢤⣀⡀⢻⢀⣤⡄⢠⣤⣄⣀⣀⠳⢤⡀⠀⠀⢤⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⠁
⠀⠀⠀⠈⠛⠛⠛⠛⠓⠀⠀⠀⢰⠙⠃⠃⠘⠃⠘⠛⠀⠀⠉⠙⠿⡬⠀⠀⠲⠿⠭⢭⣁⡀⠉⠧⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠾⠿⠀⠿⠿⠿⠶⣶⣦⣄⡀⠀⠘⣿⣾⣿⣿⣿⣿⠟⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⡳⠶⠆⠀⠿⠿⠶⣶⣷⣤⣌⡉⠐⠶⡞⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠟⡇
⠀⠀⠀⢀⣤⣴⠞⠋⠀⠀⠀⢈⠶⠶⠆⠰⠶⠶⢶⣶⣤⣌⣉⠃⠰⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠛⠁⠀⠑⢦⡀
⠀⠀⠸⡿⠋⠁⠀⠀⠀⠀⠀⣸⠶⠶⠀⠰⠶⠶⣶⣶⣤⣉⡙⠃⠲⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠶⠆⠀⢶⣶⣦⣤⣤⣉⡉⠛⠀⠶⣤⠷⢶⡄⣀⣀⣠⣤⣶⣾⡿⠋⠀⣿⠿⠛⠉⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠆⠀⣶⣤⣤⣬⣍⣉⠛⠓⠰⠦⣼⡀⠀⣇⢿⣿⡿⠿⠛⣁⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣷⢰⡆⣴⢀⣄⠉⠉⠙⠛⠂⠰⢶⡾⠁⠀⠙⢿⢶⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣇⢿⡌⣿⡀⠀⠀⠀⠀⠀⣰⡇⠀⣀⡴⣉⢆⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣷⣼⣷⣬⣴⢤⣴⣾⣿⠶⠋⠁⡧⢗⠛⡳⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡵⢂⢤⡜⢻⣿⣿⠀⠀⠀⢳⣤⡆⠀⠀⠈⠳⣄⠀⠀⠀⣠⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡁⣀⠸⣇⠈⢠⠀⠈⣿⣿⠀⠀⢠⣾⢿⣇⠀⠀⠀⠀⠈⠓⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⠘⡷⢿⠋⠁⠀⠀⠸⣿⠀⣀⣶⣿⣝⣿⠀⠀⠀⢀⣤⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⢸⠀⠀⠀⠀⠀⢻⢀⣿⣿⣿⣿⠿⠀⣠⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣤⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠋⠙⠇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡯⡄⢠⠀⠙⢿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⡿⠷⠆⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⣟⣛⣛⣛⣛⡃⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⢂⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠾⠷⠆⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⡅
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⠀⠀⠀⠀⠠⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠴⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣿⣿⡇⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣯⣭⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠻⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣇⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣷⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠁⠛⠀⠄⠙⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠣⡀⠀⢻⡀⢠⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⣀⣀⣀⡀⠀⡄⠀⡄⠀⠀⠘⠂⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢱⣦⡀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⣾⣽⣿⣿⣿⡎⠀⢐⡃⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠘⠳⣦⠀⠀⠀⠀⡜
⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⢸⡇⠀⠀⣀⡆⠀⠀⠀⠀⠀⢀⠀⢠⡆⠂⠀⠀⠓⢀⡈⠂⠀⠦⠀⢰⠃
⠀⠀⠀⠈⢿⣿⣿⣿⠄⠀⠀⠧⡞⠉⠉⠁⠀⠀⠀⠀⠀⠘⠊⠛⣷⣆⠈⠂⠀⠀⠉⠠⠄⠀⢠⠇
⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⣀⡀⠠⣤⠄⠀⠟⢻⣾⠃⠀⢠⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢹⣿⡿⢀⣀⡤⠄⣐⣓⣤⣦⣭⣭⣥⣴⠄⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⣠⠿⣛⣩⠶⠚⠋⠉⠤⠀⣸⣿⡉⠁⣀⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠓⣿⠉⣡⣶⣿⣿⣿⣿⢗⣿⣿⣿⣿⣷⣶⣧⣄⠀⠀⠻
⠀⠀⠀⠀⠀⢀⣿⠀⣹⡿⠿⣛⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣿⣿⣷⡶⣾⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⢿⣿⡟⠋⠉⠉⣱⡋⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⡴⠋⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢀⣶⠾⢶⣦⠀⢀⣴⠾⠷⣦⡀⠀⣶⡶⠶⣶⣄⠀⠀⣶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⣿⠀⢸⡏⠀⠀⢹⡇⠀⣿⡇⠀⠈⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⣸⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢽⡇⠀⣿⡇⠀⠀⣿⡇⠀⣿⣀⣀⡀⠀⢦⣄⡀
⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⢀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣸⡇⠀⣿⡇⠀⠀⣿⡇⠀⣿⠉⠉⠉⠀⠀⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣤⡀⢸⡇⠀⠀⣼⡇⠀⣿⡇⠀⠀⣿⠇⠀⣿⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠆⠀⣿⠀⠀⠀⢸⣧⡀⣀⣿⠀⢸⣧⡀⢀⣾⠇⠀⣿⣇⢀⣰⡿⠀⠀⣿⣀⣀⣀⠀⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠉⠛⠛⠉⠀⠀⠙⠛⠛⠉⠀⠀⠀⠛⠛⠛⠋








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⠶⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⡇⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣾⡧⠀⠀⠀⢸⣿⣿⣉⣉⣉⣀⣉⡉⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⣾⡎⢿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠿⠿⠿⠿⣿⣿⡿⢃⠀⠀⠀⢸⣿⣿⣿⡙⣿⣿⣿⢃⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠘⠛⠛⠘⣿⣿⣿⡇⠀⠀⠀⡟⣴⣿⣿⣿⣷⣶⣶⣶⡜⠃⠀⠀⢸⣿⣿⣿⣷⡙⣿⢃⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣰⣿⣿⣿⣷⠹⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡿⢋⣤⡛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⣀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣴⣿⣿⣿⣤⣾⣿⣿⣿⣷⣬⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣦⣝⣛⣡⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠿⠿⠿

⡏⠉⠉⠉⠉⢉⠭⠭⣉⠭⠉⡭⡍⡩⠭⢍⢩⠭⠭⣉⡩⠉⠩⢍⠉⡭⠍⢉⠍⣉⠭⠭⣍⡩⢍⠉⠉⢩⢭⠭⠭⢭⡍⡍⠉⣉⠭⠭⡉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⢇⠪⢵⢺⠀⠤⠇⡏⠐⠉⠂⣷⢰⣒⠎⡇⠃⠈⠁⠀⢱⠰⡌⢰⡇⡜⠉⡎⢷⠘⡤⢤⡆⡸⠀⣛⣻⡇⡇⠀⢇⠪⢵⠚
⡇⠀⠀⠀⠀⢄⠯⠦⣹⠀⠒⠄⡧⡰⠤⢃⢿⢠⣄⠮⡀⠀⠀⠀⠀⠀⡆⢁⠇⢣⠱⠤⢃⠃⠇⠋⡌⢰⠁⠀⠅⢠⡇⠧⠠⣄⠯⠆⡁
⡇⣤⣤⣤⣤⣬⣥⣬⣭⣭⣤⣭⣥⣭⣬⣥⣬⣬⣤⣭⣤⣭⣭⣤⣤⣤⣬⣭⣤⣤⣭⣭⣥⣤⣬⣭⣭⣥⣤⣥⣤⣬⣥⣬⣭⣭⣥⣬⣥⣤⣤⣤⣄
⡇⡇⠀⠀⢂⠂⠀⠀⠠⠄⠀⣿⠀⠀⡦⢶⠀⠀⡠⠀⠀⢸⠀⠀⠀⠰⡔⠀⠶⠀⠀⠀⡇⠀⠐⡀⠠⠀⠠⠤⠀⠀⣿⠀⠐⠀⠐⠂⠠⠀⠄⠀⢸
⡇⡇⠀⢀⠨⢀⠀⠌⢠⡀⠀⣿⠀⠀⠏⠀⠄⠠⡛⠇⠀⢸⠀⠀⠀⠐⠀⠀⢠⠀⠀⠀⡇⠀⠠⠁⠐⠀⠣⠀⠀⠀⣿⠀⠀⠄⠠⠀⠠⠒⡀⠀⢸
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⡠⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡐⠢⣵⣧⡎⠤⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡤⠤⠄⠤⠀⣀⠀⢸
⡇⡇⠀⠄⠀⠀⠀⠀⠀⠄⠀⣿⠀⢡⠀⠻⠛⢏⠐⠒⠀⢸⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠟⠀⠀⠃⠚⠟⠀⢸
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠃⠰⠀⠓⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠡⠡⠰⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⣀⠠⡄⢤⠄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸
⡇⡇⠀⢀⠀⠀⠀⠀⢀⠀⠀⣿⠀⢴⣀⠀⢀⡀⠘⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠀⠄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⡇⠀⠘⠁⠀⠀⠀⠛⠀⠀⣿⠀⠀⠉⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⡇⠀⠀⠀⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸
⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠆⠀⠠⠀⠀⠀⢸
⡇⡇⠀⠀⠀⠒⠒⠀⢄⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣟⠓⠒⠑⠖⠛⠂⠀⡇⠀⠐⠓⠊⠃⠀⠀⠀⠀⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸
⡇⡇⠀⠀⠀⠀⠀⠀⢹⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠇⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠀⠀⠀⢀⠀⢸
⡇⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⠀⠁⠀⠀⠈⠀⢸
⡇⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠥⠤⠤⠤⠬⠥⠤⠤⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼

⠀⠀⠀⠀⠀⢀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⢶⣄
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣰⠏⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣤⣸⡇
⠹⣴⠞⠛⢉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠈⢻⡇
⠀⠉⠛⠛⠉⠙⣦⠀⠀⠀⠀⢀⠀⣀⣤⣴⣶⠖⠚⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⣤⣀⣀⠀⠀⠀⠈⢷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡏⠀⢠⡇⠀⣾⠛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⣧⡀⠀⠀⢸⣄⣠⣾⡇
⠀⠀⠀⠀⠀⢾⣀⣠⡼⢧⠞⠁⠀⣿⣿⣿⣿⣶⣶⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠈⠁⢸⡇
⠀⠀⠀⠀⠀⠈⣉⣁⣀⢠⢤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⡀⠘⠇⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠰⣏⠁⠉⣿⡘⣇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⠏⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⢀⡼⠃⢀⣀⡀⠀⢀⡀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⡉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸⠁⣴⠟⢩⡇⢀⣾⡇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⣄⠀⠀⠀⠀⠈⠛⠁⠀⣼⠁⢘⡟⠃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣸⠃⠀⣼⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⡟⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣧⠀⠀⠀⢀⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢹⡄⠀⠀⢸⡿⡆⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣤⣷⣤⣤⣼⢧⠿⣤⣤⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣕⣺⣏⣂⣿⣑⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⢟⣿⠿⠿⠿⠿⠿⢿⡿⠿⣿⣟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣭⣭⣭⡎⣿⣷⣜⣿⣛⣛⣛⣛⣻⣟⣛⣿⣻⣟⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠿⠿⠿⢃⢹⣿⠟⣿⣿⣿⢻⣿⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⣿⣿⣿⣿⣿⡆⣸⣷⣜⣿⣿⣱⣿⣿⣣⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⢣⣿⣿⡿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣙⡍⣶⣿⣿⣯⡘⣿⡿⣣⣿⣿⢃⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣛⠿⣣⣦⣼⣍⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣚⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⡟⡛⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⢸⡆⠀⠀⠀⣿⣿⣿⡇⢤
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡘⣿⠀⠀⠀⣿⣿⣿⡇⠼
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣤⣾⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⣄⡀
⠀⠀⣰⠟⠁⠀⠀⠈⠉⠙⠻⠿⣿⣷⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠛⠉⠉⠈⠉⠙⠳⣄
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣾⣿⠋⠉⠉⡉⠛⠾⣓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣆⣠⡤⠶⣶⣷⣦⣀⡀
⠀⠀⠀⠀⠿⡃⠀⣾⣿⣿⣷⡄⢈⣄⠑⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⢢⡿⠃⢠⣴⣶⣤⡀⠙⡙⣿⣄
⠀⠀⠀⠀⠸⢗⠢⡟⠟⠿⠿⠴⡾⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣶⡿⢇⡀⢻⣿⣿⣿⠇⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠉⠉⠛⠚⢿⠻⠺⠚⠁⠨⠂
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⣡⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠉⠉⠛⠻⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠿⠉⠉⢀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣙⠿⠆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢈⣉⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⣰⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⢹⡟
⣶⣷⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⣿⣿⡇⢠⣶⣄⢰⣿⣷⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⢠⣿⡌⢿⣿⣧
⣿⣿⡇⢸⣿⣿⢸⣿⠋⣤⣭⣭⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⢸⣿⣿⠀⣾⡏⣡⣬⣭⣭⣤⣄⡀
⣿⣿⡇⢸⣿⣿⢸⣿⣆⠙⠿⠿⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⠀⣿⣷⡘⠿⠿⠿⢿⣿⣿
⣿⣿⣷⣈⠛⣃⣄⡛⢋⣰⣶⣶⠆⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⣃⣌⡛⣋⣰⣿⣿⣿⣷⣶⡶⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣥⣴⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣴⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢁⣾⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣁⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⠏
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⠻⠿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⠿⠟⠋




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⢰⣾⣶⠀⣠⡶⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣉⣹⣿⡞⠛⠛⣾⣋⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢸⣿⣿⠙⠿⣭⣁⣀
⠀⠀⠀⢀⣀⡀⢀⣴⣦⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⣿⠀⠀⠈⠙⠋
⣠⣤⡀⢸⣿⣿⢸⣿⡿⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣴⣶⡄⣾⣿⡇⢸⣿⣿⠀⣠⣶⡆
⣿⣿⡇⢸⣿⣿⢸⣿⡀⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⣿⣿⡇⢸⣿⣿⠀⣿⣿⡇
⢿⣿⠇⢸⣿⡿⠸⣿⡷⠈⣙⣛⡛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⣀⠻⠟⢃⠙⠿⠃⣼⣿⣿⠀⣿⣿⡇
⣦⣤⣶⣦⣤⣴⣷⣤⣴⡾⠟⢛⣃⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣥⣴⣶⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⣴⣿⣿⣿⣿⣿⣿⡇
⢻⣿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣟⢸⣿⣿⣿⣿⣿⣿⣿⠁
⠈⢿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⢲⡄
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⣰⠒⠒⠲⠶⠶⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡀
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠠⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠑⠺⡆
⠀⠀⠀⠀⠀⡞⡠⠀⠀⠀⠀⠇⠀⠲⠔⠀⠀⡌⠀⠢⡀⠀⠀⠀⢸⠦⡀⠘⠻⠃⣴⣶⡇
⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢴⣿⠄⠈⠆⠀⠀⠈⠉⠀⠀⣤⣄
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢠⠃⠀⠀⢀⣴⣾⣿⣿⠀⡞⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⢨⠛⠋⢰⣿⡆
⠀⠀⠀⠀⠀⠳⡴⠀⠀⢠⣿⡀⠀⠀⢈⣉⣩⠝⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣲⢼⠉
⠀⠀⠀⠀⠀⠀⠘⣧⡔⠁⠀⠈⠒⠦⠤⠥⠒⢋⣤⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢰⢛⡀⠠⣻⣳
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡳⣬⣻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠯⣁⣠⠇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⡀⠀⠀⠀⣀⣀⣀⡤⠤⢶⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠑⠛

⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡄⢠⡍⡄⠀⠀⠄⠀⢀⠀⠀⠀⠀⡀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡄⢀⠀⠀⠀⠣⡔⠈⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠈⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⣤⣁⣀⠔⠄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣄⡄⠀⣠⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⠸⠙⠞⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠟⠙⠻⠸⡊⠠⡀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠜⣔⠀⡀⠀⣰⣥⣠⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠈⠀⠂⠀⠀⠀⠡⠐⠒⢫⡆⠀⠀⠀⠀⠀⠀⠀⠉⠱⠶⠚⢉⡉⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠠⠀⢴⠘⢇⠀⠀⠀⠤⠀⠴⠾⠷⠄⠀⠠⠾⠶⠱⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠠⢀⠣⣀⠘⠀⠠⠀⠄⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⢉⣠⡆⠀⠀⠀⠀⠀⠁⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣃⣈⣀⣀⣀⣓⣤⡀⠇⠀⠀⠀⠀⠀⠈⠡⠤⠤⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠄⠀⠀⠃⠠⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠆⠀⠤⠸⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢃⠀⠤⠀⠀⠀⠘⠀⣀⠐⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠊⡀⠠⡀⠀⡠⢘⠇⠀⠄⠂⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣢⡀⠈⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⢠⠀⠀⠶⠥⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣀⠁⠂⠍⢈⡉⡐⢄⠀⠁
⠀⠀⠀⠀⠀⢀⠐⢎⣪⢗⡋⢉⡱⠂⢄⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠉⠉⠈⠉⠈⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠉⠀⠁

⠀⠀⢠⣤⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣴⣶⣶⣶⣶⣶⣶⣴⣦⣶⣶⣶⣦⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣶⣶⣶⣦
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣤⠶⠆⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⣶⠀⢀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠠⣿
⠀⠀⢾⡇⠀⠀⠀⠀⠀⢿⣤⡀⡇⣄⣀⢀⣤⣤⣄⡲⢦⠦⢸⠀⠀⠀⠀⢻⢇⡘⡏⠀⣠⣤⣄⣠⣄⡠⣄⡴⡄⡤⣶⣶⣤⣿⠠⣾⣿⣋⠀⠀⢰⡟
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠳⠾⠿⠷⠋⢯⠿⣧⣼⡾⢿⡀⠰⢎⡳⠆⠀⠀⠈⡞⢳⠀⢨⣯⣀⣷⡿⡏⣳⢎⣹⠈⢶⣯⣭⡍⣿⠰⢶⠿⠿⠃⠀⢸⣇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠈⠁⠀⠘⠁⠀⠀⠀⠀⠘⠃⠀⠀⠉⠉⠉⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠛⠁⠀⠋⠈⠛⠁⠘⠃⠉⠉⠀⠉⠀⠃⠀⠀⠓⠋⠀⠐⠉⠙⠒⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠄⠤⠠⠄⠠⠤⠤⠀⠀⠰⠐⡶⠆⠀⠰⠀⠀⠀⠠⠄⠤⠄⠠⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠋⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣒⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⡤⠽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣔⠂⠀⡒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⠀⣉⢁⡴⡤⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠠⠄⠀⠰⠬⠰⠹⣱⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠒⠸⠏⠋⠀⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠒⠢⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⠀⠀⠀⠒⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣺⣻⠇⠂⠀⢶⣶⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣾⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣘⣀⣎⣀⣘⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢿⣷⣶⣶⣶⣦⣶⣶⣦⣶⡶⢶⡶⣶⣶⣶⣤⣴⣤⣤⣤⡶⣦⣤⣴⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⠀⠀⠸⣿⣿⠀⠀⢰⣿⡆⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⣿⣿⣦⡀⢸⢋⣥⣤⣬⣭⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⡇⢸⣆⠛⠿⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⡿⠷⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⠿⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣼⠇


⣏⣈⣯⣭⣟⣯⣿⣹⣽⣻⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣶⣶⣶⣶⠂⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣛⣀⣿⠿⠿⠛⠁⠲⠿⠿⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠛⠋⠉⠀⠠⠔⠀⠀⠀⠀⠀⠂⠈⠛⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⠤⠄⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⢄⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠚⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢰⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⣿⠀⠀⠀⠸⠟⠋⢡⠀⠀⠀⠰⣶⡄
⠀⠀⠀⠀⠀⢀⠄⠀⠿⠀⠀⠈⡟⠋⠙⢿⣿⠁⠈⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠻⠀⢄⠀⠀⠀⠀⣠⣦⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⡅⠊⠉⠀⠀⠀⠀⠱⣦⣤⣾⣿⢆⣀⢀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⢀⠀⠰⣿⠂⠈⠁⠰⠿⠃
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠱⡄⠀⠀⠹⣿⣿⡇⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢉⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠙⠧⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣴⣿⠏⢡⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠤⢀⡀⠀⠀⠀⠀⠀⢀⣀⡄⠊⠀⠀⠀⣿⡇⢠⣿⣿⠏⠈⠦⢤⠀⠀⠀⠀⠀⢰⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡀⠀⠀⠈⢉⠉⢉⠙⠛⠛⠛⠄⠀⠀⠀⠙⠲⠾⢿⠋⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠙⠢⣀⠀⡘⠀⠘⠀⠀⠀⠀⡀⢂⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⣠⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢹⠓⠤⢠⣇⠀⠀⠀⢃⡀⣦⡀⠤⠄⠒⠁⠈⠐⠒⠒⠈⠙⠢⠜⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠸⠉⠉⠉⠉⡁⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⡆⠀⠀⠀⢡⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠃⢀⡦⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠈⠁⠒⠂⠤⡀⠀⠀⠀⠐⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⡜⠁⠀⠀⡀⣀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⢀⡬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠀⠀⠛⠘⠃⠀⠙⠢⠊⠀⠀⠀⠍⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢤⠤⠄⠒⠊⠀⠀⠈⠓⠒⢠⡤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⡿⠄⠀⠀⠀⠀⠘⠿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇⠀⠀⠿⠿⠀⢀⣀⣸⣿⣀⣀⠀⢀⣀⣿⣿⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⣾⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⡇⠀⠀⣠⣄⠀⠘⠛⢿⣿⠟⠛⠀⠘⠟⣿⣿⠟⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣴⣿⡟⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⣿⣷⠀⢸⣿⡇⠀⠀⣿⣿⠀⢸⣿⡟⠛⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⣴⣿⡟⠀⢸⣿⣧⣤⣤⣿⣿⠀⢸⣿⣧⣤⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠁⠀⠀⢸⣿⡏⠉⠉⣿⣿⠀⢸⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠘⠿⢿⣿⠿⠟⠀⢸⣿⣷⣄⣠⣾⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡟⢿⡿⠛⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⡇⠀⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠙⠻⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⣿⣿⠟⠟⠛⢿⣿⡟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⡆
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⡆⢴⡄⠤⡤⠦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⢶⡶⠰⠒⠂⠀⠀⠀⢰⠄⠀⠋⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠑⠂
⠀⠀⠀⠀⠀⠀⢸⡄⣾⠟⣳⠀⠀⠀⢸⡄⢮⠟⢈⡆⠀⠀⠀⢈⠀⠀⠍⡆⠀⠀⠀⢰⠀⠪⠏⢱
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠚⠁⠀⠀⠀⠈⠁⠀⠐⠉⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠐⠁
⠀⠀⠀⠀⠂⠀⠀⠀⣄⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣘⡏⣀⠀⠀⠀⠀⠀⠐⣺⢉⣀⠀⠀⠀⠀⠀⠂⣟⣁⠀⠀⠀⠀⠀⡀⢘⠋
⠀⠀⠀⠀⠀⠀⠘⠗⡏⠁⠀⠀⠀⠀⠐⠃⣏⠁⠀⠀⠀⠀⠀⠑⠀⡅⠀⠀⠀⠀⠀⠀⠧⢸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠁⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣈⢳⡀⠀⠀⠀⠀⣀⣀⣙⡆⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠺⢭⣭⠙⠃⠀⠀⠀⠈⠢⠤⣤⠀⠀⠀⠀⠀⠀⠢⠤⠄⠀⠀⠀⠀⠀⠰⢄⡙
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠒⢳⠀⠀⠀⠀⠀⠀⠘⢻⠂⠀⠀⠀⠀⠀⠀⠤⢰⠀⠀⠀⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠠⣦⣆⣿⠈⢷⠀⠀⠀⢀⣀⢸⠀⢢⡀⠀⠀⠀⢀⡀⠴⠠⣄⠀⠀⠀⢀⠀⢸⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠉⠀⠛⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠺⢽⣄⠀⠀⠀⠀⠀⠨⢙⣤⡄⠀⠀⠀⠀⠨⠃⡇⠠⠀⠀⠀⠀⠀⠠⠍⡇⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠟⡁⠀⠀⠀⠀⠀⣋⠸⢀⡀⠀⠀⠀⠀⢀⠀⠃⢀⡀⠀⠀⠀⠀⡀⠸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⣴⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⣿⠀⡧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⡄⠈⠳⡄
⠀⠀⢠⡴⣄⣿⠀⠇⠸⠉⡟⢦⠀⠀⠀⠀⠀⠀⡇⠀⠙⣆⠀⠀⠀⠀⠀⠀⢸⡇⠐⠲⡗⠒⠀⡿
⠀⠀⠀⠳⣌⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠳⣄⣀⢁⣀⣼⣇
⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡇⢀⡴⠶⠶⠾⠗⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛



⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣿⢰⡆⣤⢀⡀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠰⠿⣿⡄
⠀⠀⠀⠘⢶⣀⣿⣾⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠉⣿⠟⢻⣿⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠂⠄⠀⠈⠁⠀⣿⠟⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠹⠟⢹⡿⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡱⠁⠀⠆⠀⣴⠀⠀⠘⠀⢹⣿⣿⣿⠇⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠠⠀⠈⣿⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠰⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⣿⢟⡉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠈⠀⠰⠀⠀⠈⠀⣽⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⢀⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠄⠀⠃⠀⠀⠀⠀⠀⠀⡐⠐
⠀⠀⠀⠀⠀⠀⡄⠄⡄⠀⢀⡀⠀⠀⠀⠀⠈⠒⠲⠒⠊⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠙⠚⡀⠀⠀⠀⠈⠀⢄⠀⠀⣀⠠⠴⠊
⠀⠀⠀⠀⠀⠀⠑⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠔⠈⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠁⠄⠰⠀⠄⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢁⡀⠀⠈⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢠⡔⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⢀⡀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣌⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢣⣭⡸⣿⣶⣿⣣⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣭⣭⣿⢟⣫⣽⣶⣤⣄⡀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⢳⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣩⣿⣽⣿⣿⣿⠀⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⡹⣿⣿⡿⣿⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⡇⣿⢸⣿⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⣿⣾⣿⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠼⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⢀⠄⠈⠉⠐⠤⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠠⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠤⣀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠳⡀⠀⠀⢠⠞⠀⡘⠉⠉⠀⣾
⠠⠒⠊⠉⠉⠓⠢⣄⠀⠀⠰⡄⠀⢀⡤⠤⠤⠶⣌⡓⠲⠄⡀⠀⠀⠀⠀⠀⡏⠉⠉⠁⠒⠦⡀⠀⠁⠀⢀⠏⠀⠀⠁⠀⠀⢀⠇
⣧⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠸⡜⠁⠀⠀⠀⠀⠀⠙⢆⠀⠀⠑⠄⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠅⡠⠊⠀⠀⠀⠀⠀⠀⠃
⠈⠢⡈⠀⠀⠀⠀⠀⠀⠀⠘⠒⣼⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠢⠆⠀⠀⠀⠀⠈⠀⠀⠜⡜⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠘⠧⠄⡀⠀⠀⠀⠀⠀⠀⠹⣗⠀⠀⠀⢀⠈⠀⢀⡿⡀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠈⠀⢹⣦⣤⡎⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⢣⡀⠀⠁⠀⠀⠀⠀⠀⠈⠈⡟⠀⠈⠀⠀⢀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⡁⠀⡀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠒⠒⠀⠀⣰⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠴⠣⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠐⣀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢘⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠤⠀⡀⠄⠀⣀⡀⠠⠤⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠄⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠉⠙⢦⡀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣠⣤⠖⣿⢠⣀⡄⢻⠀⠀⠀⠀⠀⢸⠀⢿⢰⣷⡗⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠧⠾⣄⣿⠰⠛⠆⢸⠀⣠⡤⣄⡀⢸⠀⠒⠘⠙⠁⠐⠀⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠈⠉⠀⠀⣠⠏⠀⡷⠴⠾⣿⠈⠛⠛⠛⠛⠛⠛⠛⢻⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠒⠋⠁⠀⠀⢳⣀⣀⡏⢀⣀⣀⡀⠀⠀⠀⡃⠸⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⡟⠳⠞⢹⣯⣤⢦⡍⣧⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡟⠀⠀⢈⣧⣀⣼⡃⣿⠀⠀⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠯⠿⠛⢲⡾⠉⠉⠛⠋⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣷⣤⡀⠀⠀⣀⣤⣾⣿⣶⣦⣄⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣈⣽⣿⣿⣿⡟⢛⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⣿⡟⠛⠉⠉⢛⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠘⣿⣷⡀⢀⣴⣿⡿⠟⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⠟⠋⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣧⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⠀⠹⣿⣧⣀⣤⣶⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠋⣀⣠⣤⣴⣶⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣷⠶⠾⢷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠹⣿⣧⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠹⣿⣷⡀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⠀⠹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠙⠻⢶⣦⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢀⣴⣿⣿⣿⣿⣿⠟⠁⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠙⢿⣿⣿⣿⣿⣷⣄
⣾⣿⣿⣿⣿⠟⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣿⣿⣿
⢿⣿⣿⣿⣿⣧⡀⠀⠙⢿⣿⣿⣿⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠻⢿⣿⣿⣿⠟⠁⠀⣠⣿⣿⣿⣿⣿
⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠹⠟⠁⠀⣠⣤⣄⡀⠈⢻⡿⠋⠉⠀⠀⠉⠛⢿⣿⣤⣼⣿⠟⠋⠁⠀⠈⠉⠻⣿⠋⠀⣀⣤⣤⡀⠀⠙⠿⠁⠀⣠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠈⠻⣿⣿⣿⣿⣿⡀⠀⠀⣠⣾⣿⣿⣿⣷⠀⠀⠀⢀⣴⣿⣷⣦⠀⠈⢿⣿⣿⠏⠀⢠⣶⣿⣶⣄⠀⠀⠀⢰⣿⣿⣿⣿⣦⡀⠀⠀⣸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠻⣿⣿⣿⠃⠀⣼⣿⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠃⠀⣸⣿⣿⡀⠀⢻⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⡄⠀⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⣠⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⠰⢿⣿⣿⠷⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣧⡀⠈⠻⠋
⠀⠀⠀⠀⠀⠀⢠⣀⣠⣴⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣀⣀⣀⠀⠉⠁⢀⣀⣀⡀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⣿⡿⠃⠀⠘⠿⣿⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣷⣄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⣿⡿⠋⠀⢀⣠⣄⡀⠀⠹⣿⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠁⠀⢠⣾⣿⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⡄⠀⠈⠻⣿⣿⣿⣿⣿⣦⠀⠀⠙⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢻⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⣿⣿⣿⠟⠁⠀⣄⠀⠈⠻⣿⣿⣿⣿⠃⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠸⣿⣿⣿⣿⠟⠁⠀⣠⣾⡿⠀⠀⠀⠈⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢀⢀⠀⢀⡄⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⢀⢀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠀⠈⠉⠀⠉⠈⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠈⠈
⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠅
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠌⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠈⠀⠈⠈⠋⠀⠘⠛⠈⠀⠐⠁⠉⠀⠁⠛⠁⠈⠛⠘⠉
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣑⣑⣈⣊⣂⣑⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢁
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐

⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠈⠓⠒⠈⠂⠘⠃⠉⠛⠀⠋⠐⠛⠛⠓⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠘⣀⣋⠘⠓⠀⠘⠓⠃⠀⠘⠐⠃⠈⠀⠃⠛⣑⡀⠈⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠓⣄⣠⠤⠖⠒⠚⠛⠒⠲⢤⣀⡴⠋⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠔⢻⠀⢀⣠⣄⡀⠀⠀⠀⠀⣀⣄⡀⠀⢸⠓⢄⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡏⠀⡏⢠⣤⢱⠀⠀⠀⢜⣤⣄⢘⡄⠈⣇⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠑⠮⠽⠋⠀⠀⠀⠈⠻⠥⠚⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠒⠒⠊⠁⠀⡠⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠢⡀⠉⠑⠒⠢⠤⠤⢄⣀⡀
⠀⠀⠀⢀⡴⠋⣁⣀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⣤⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣤⣄⠀⢹⠀⠀⠀⠀⠀⢀⣀⡀⠉⠳⣄
⠀⠀⢠⠋⣠⠊⠁⠀⢙⡦⣀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠛⠋⠀⠀⣀⣠⡔⠲⡏⠁⠀⠈⠙⠢⡌⢣
⠀⢠⠃⢰⠁⠀⠀⠀⠮⣀⡼⠋⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠚⠉⠀⠈⠷⠚⠁⠀⠀⠀⠀⠀⠘⡄⢣
⠀⡎⠀⡇⠀⠀⡠⣄⠀⠈⠀⠀⠀⢀⠀⡀⠉⠉⠒⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⠶⠒⠊⠋⠁⠀⠀⠀⠀⣠⣄⠀⠀⠀⠐⡀⠀⠀⣆⡇⠀⢱⠈
⢠⠁⠀⡇⠀⠀⠓⠒⠀⠀⠀⠀⠀⠘⢤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠛⠐⠀⠀⠀⠦⠃⠀⠀⠀⠀⠀⢸
⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠒⠀⠀⠀⠀⢨⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⢇⠀⠀⠀⠀⠀⠰⣊⡗⠀⠀⠀⠀⢠⠒⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⠀⠀⠀⠀⢰⢀⠀⠀⠀⡰⠁
⢸⠀⠀⠘⡄⠀⠀⠀⠀⠀⠬⠃⠀⠀⠀⠀⠈⠚⠃⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⣄⠀⠀⠀⠀⢰⠖⠁⠀⠀⠀⠀⠸⠫⠀⠀⣰⠃
⠈⡆⠀⠀⠹⣄⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠇⠀⠀⠀⠳⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠹⡀⠀⠀⠘⢄⡠⠟⠀⠀⠀⠀⣎⡱⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠹⣄⡠⠋⠀⠀⠀⠀⡰
⠀⠀⠙⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⣠⢤⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠘⣆⡴⠋⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠉⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠒⠉⠀⠀⠀⠀⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⠖⠤⢤⣀⣀⠀⠈⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉⠀⠀⠀⠀⢀⣀⡠⠤⠔⠲⣍
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠠⠄⠰⠤⠄⠴⠄⠢⠄⠤⠤⠀⠤⠤⠤⠄⠠⠤⠤⠤
⠀⠀⠀⠀⢀⠀⣀⠀⡀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠠⠀⠃⠀⠀⠀⠘⠀⠀⠀⠠⠂⠃⠀⠀⠀⠈⠀⢄⠀⠀⠀⠋⠀⠈⠀⠘⠀⠄⠀⠀⠀⠋⠀⠀⠀⠀⠃⠠⠀⠀⠄⠚⠀⠀⠀⠀⠃⠠
⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⡀⡘⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠲⠀⠀⠀⠀⠀⠀⠠⠆⠀⠑⠀⠀⠀⠀⠀⠀⠀⠖⠀⢑⠀⠀⠀⠀⠀⠀⠀⠖⠀⢐⠂⠀⠀⠀⠀⠀⠀⠴⠀⠐⠆⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠈⠀⠆⠀⣀⠀⡠⠐⠀⠀⠀⠀⢂⠀⣀⠀⠰⠀⠀⠀⠀⠀⢆⠀⣀⠀⠠⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠄⠀⠀⠀⠁⠰⠀⢀⡀⠀⠆⠈
⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⢀⣀⠈⠀⠈⠁
⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣁⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣉⣉⣉⣉⣉⣉⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣴⡄⣠⣤⣤⣌⣭⣍⣩⡁⠀⠀⠀⠀⢸⣿⢣⣿⡿⣱⣿⡜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣰⠖⠒⠒⠒⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣷⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣠⣴⣶⣶⣶⣶⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣻⣟⣛⣛⡋⠄⠀⠀⠀⠀⠀⠀⠀⢠⣴⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⣴⣿⢻⣽⣿⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⢰⡟⢛⣦⠻⠓⠈⠁⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⣷⣿⢀⣀⡀⠀⠀⠘⠻⢿⣋⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠾⠶⠿⠿⠿⠆⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⣿⡇⢤
⠀⠀⠀⠀⠀⠀⠠⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⠀⠀⢸⣿⣿⡇⠼
⠀⠀⠀⠀⠀⠀⠠⠭⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠥⠤⠤⠬⠭⠭⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢿⣿⣝⣂⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣭⣛⣻⡟⠋⠀⠀⢀⠑⢄⠀⠀⠀⠀⠀⢀⡀⢠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣶⣷⢮⣭⢭⣍⠁⠀⠀⠀⠈⠁⠘⡆⠀⠀⠀⠀⢯⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢸⣿⣿⣿⣿⡌⣿⠀⠀⠀⠀⢠⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⡻⢿⣿⠆⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠋⠀⠀⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠻⡿⣀⣀⣀⡀⠀⠀⠀⢿⣿⡿⢡⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡱⣶⡝⣚⢫⣵⠆⣿⣷⣄⠀⠀⠀⠀⠀⠐⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣱⣛⣡⣭⣬⣛⣇⣿⣿⣿⣆⠀⠀⠀⠀⢰⣿⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣠⣿⣿⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⢹⣿⣿⣿⠀⣰⣿⣟⠟⠀⢀⡤⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⣿⡏⠀⢻⣿⣿⣷⣿⣿⣿⠁⠀⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣭⣝⣛⡿⠿⢿⠿⢃⠀⠀⠻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠂⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣀⠀⣀⢀⡀⡀⠀⢀⢀⡀⠀⡀⣀⢀⢀⡀⡀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠹⠼⠻⠄⠼⠳⠿⠄⠸⠾⠗⠧⠏⠵⠾⠻⠄⠧⠟⠦⠟⠇

⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⡄⢀⣀⠀⠀⠀⠀⠀⡠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣤⣤⣤⣀
⠘⣷⠀⣸⡏⣴⠟⠙⢷⡄⣿⢀⣿⠀⣿⡀⢰⡟⠛⣧⠀⢸⣷⣟⡙⠗⠀⠀⠀⠘⠠⠀⠀⠄⠱⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠛⠛⠀⠀⠀⣿
⠀⠹⣧⣿⠀⣿⠀⠀⢸⡷⣿⢸⣿⠀⣸⡇⣾⣧⣤⣿⠀⢸⣷⡍⠛⣷⠀⠀⠀⠠⡀⠫⠝⠀⡘⠀⠀⠀⣼⠃⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⡇⠀⠹⣧⣤⡿⠁⢿⡿⢿⣦⡿⠀⠸⢷⣤⡶⠀⢿⠿⢷⣴⠟⠀⠀⠀⠀⠈⠃⠐⡏⠀⠀⠀⢸⡇⠀⠀⠀⣶⠋⠉⠛⣷⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡇⠀⠀⠀⠘⣧⠀⠀⠀⠻⢦⣤⣤⡟⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢠⠋⠠⠶⠀⢱⠀⠀⠀⠀⠀⢘⡧⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢶⣒⡶⢎⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⣤⣤⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢄⣁⡤⠊⢀⡀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡈⠈⠀⠀⠀⠀⠒⠒⢲⠀⠀⠀⠀⠉⠉⠉⠙⠁
⠀⠀⢀⣀⣀⣤⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠢⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠉⠁⠁⡰⠂⡠⠁
⠀⡔⠉⣀⡁⠀⡇⠀⠀⠂⠀⡀⠀⠀⠀⣀⠀⢀⠀⠀⠀⡠⠤⠤⡀⠀⠀⠀⠀⢡⠀⠂⠤⠤⠤⠄⠒⢁⡠⠊⠀⠀⠀⠀⠀⠈
⠀⡇⠐⢥⡽⠀⡇⠀⠐⠀⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⡞⢀⠤⡄⠘⡄⠀⠀⠀⠀⠁⠢⡄⠤⢤⠤⠒⠁
⠀⠘⠦⠤⣄⣀⡇⠀⠀⠡⡀⠀⠡⠠⠀⠀⢀⠌⠀⠀⡇⠸⠤⠆⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠢⠀⠀⠠⠂⠁⠀⠀⠀⠑⠤⢤⠤⠞⠀⠀⠀⠀⠀⠀⠰⠆⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡁
⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⡖⢲⠀⡖⠒⡆
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠀⠐⠀⡇⠈⠛⠀⠀⡇⠀⠀⠀⠀⠀⡼⢰⠒⣆⠏⠀⠀⢰⢡⠖⡆⡞⠀⠀⢠⠃⡖⣶⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⠀⠀⠈⠒⠒⠦⠴⠇⠀⠀⠀⠀⠀⠃⠈⠋⠙⠀⠒⠀⠁⠈⠛⠉⠁⠘⠀⠈⠀⠙⠋⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣦
⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠠⣶⠶⣶⠟⠃⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⢠⣤⣤⣿⣤⣤⡀⣿⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⣀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠟⣿⠙⠿⠀⣿⠀⠀⢸⣯⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⡆⠀⠀⠙⠋⠀⣿⠀⠀⠐⣿⣤⣤⣼⣯⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠛⠛⠛⠛⠛⣿⡄
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠿⠀⠀⠈⠛⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠙⢿⣦⣤⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢹⣿⡆⠀⠀⠀⠀⣠⡴⠒⠲⣶⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢠⣿⢃⣠⣾⠟⠉⠉⠉⠉⠉⠉⠉⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢻⣿⡀⠀⠀⠀⠻⠋⠀⠀⣸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠰⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⢿⣷⠀⠀⠀⣠⡶⠚⠉⢹⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠘⣿⣧⡀⢸⣿⡇⠀⢀⣼⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠀⠙⠛⠛⠉⠈⠛⠛⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠠⣤⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠄





⠀⡉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣄⡀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠙⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⠀⠀⠘⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣤⣄⠀⠀⠀⠈⠿⠿
⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⣶⣾⣿⡿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠘⠷⣦⣤⣀⣀⠀⠀⠀⣀⣀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣾⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⠉⠉⠉⠙⢻⣿⣦⡀⠀⢀⣀⣤⣴⣾⣿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣷⣾⠿⠛⠋⠉⠀⠀⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠉⠻⢷⣤⣀⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⢉⣿⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢀⣠⣾⣿⡿⠋⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠙⢿⣿⣦⣄⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡅⢠⡄⢤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣩⣤⣥⣬⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⢤⣤⣤⢤⣤⡤⢤⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠄⠤⣤⡄⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⢀⠔⠀⡖⠲⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣾⠀⠀⠀⡖⣿⣦⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⣿⠀⠀⠀⠙⢦⣸⠁⡴⠋⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⡇⠀⠀⠀⠀⠀⠀⠙⠓⠋⠀⣀⡀⠀⠣⣀⠼⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠼⠀⠀⠀⠀⠀⠀⢠⢖⣒⢢⡀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⠠⠤⡆⣦⢴⠠⠰⢤⠀⠀⢀⣄⣴⠶⣶⣠⣄⢻⡘⠿⣡⠇⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡉⢻⣤⣟⢋⣹⣇⠈⠉⠁⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⡿⣿⢤⣽⣿⣿⣿⣧⣤⣤⣤⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠒⠛⠓⠀⠀⠀⠐⠚⠚⠚⠛⠓⠛⠛⠛⠓⠛⠛⠛⠛⠚⠛⠛⠓⠂⠀⠀⠀⠚⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⡿⣻⢿⢿⣿⣻⣷⡗⢿⠛⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠛⠛⠛⠋⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⢸⠚⠛⣲⢷⡛⠚⠛⣒⢒⠓⠓⠛⠚⠒⠓⡞⠓⠓⢻⢚⠖⠓⠚⠒⠛⠓⠛⠓⠚⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢺
⢸⠀⠐⣅⣐⡦⣾⡷⣟⢹⢸⠋⣶⣊⣳⠈⡷⣋⡆⢸⢧⡀⡏⣶⣊⡱⢦⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠈⠩⠀⠈⠁⠀⠀⠈⠀⠀⠮⠛⡇⠀⣩⣤⣠⣤⡅⣠⣤⡌⠁⢿⢀⡖⣠⣄⢠⢀⡄⣠⣠⣤⣰⣆⣤⡄⠀⢸
⢸⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⡀⢧⡤⠻⠼⠻⠇⠿⣳⣾⡇⠀⠈⠿⠀⠧⠾⠈⠿⠻⠇⠻⡽⠹⠇⢽⠆⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠐⡀⠀⡀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⢀⠀⠀⠀⠀⠀⡇⠐⠐⣁⠆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠣⠀⠀⠰⠀⠣⠀⠐⠂⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠤⠔⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⢠⣴⣦⡄⢠⣶⣦⡀⢀⢸
⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠘⠿⠿⠃⠙⠯⠿⠁⠀⢸
⢸⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠶⠠⠶⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡈⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠄⠨⠀⠀⠀⠀⠀⠒⠪⠇⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠠⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢤⡤⠀⣴⡴⠀⠀⠀⠀⢸
⢸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡊⠠⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠈⠁⠀⠀⠀⠈⢁⠂⢄⠀⠀⠀⢸
⢸⠀⠀⠀⠰⢀⣀⠀⠀⠐⠀⠀⠀⠀⠲⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⡴⢁⠔⠁⠀⠂⣀⣁⡒⠀⢄⡁⠀⠑⡄⠀⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠂⠜⡤⠊⡁⠤⠠⠬⣉⠢⡙⠄⢡⠘⡀⢸
⢸⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡈⠘⢠⢊⠔⣈⢉⡐⢌⠂⠑⡀⠀⡆⠀⢸
⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠌⠀⠀⠀⠈⠆⠄⠡⠄⠇⠡⠀⢸
⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣐⡦⠾⠖⣷⠈⠉⠉⠃⡾⠻⡿⣻⠿⠡⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣵⣤⣀⡉⠛⠶⢿⠸⠠⡀⢠⣿⢽⠯⣒⠕⠋⣠⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⡉⠉⠡⠤⠤⠴⠈⠀⠰⠄⠭⠈⠤⠤⠬⠁⠐⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠡⠐⠂⣀⢊⣈⣀⠀⠀⠄⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣨⡤⣥⡄⠀⠀⡀⣁⠙⠀⠀⠈⢡⣀⠐⢖⣢⣶⣷⣾⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋⠁⡠⣒⣥⣶⡛⡇⠚⡁⠘⠸⡏⠑⢤⡈⠑⠟⠫⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡭⠶⠿⠿⠿⢶⣶⣢⣕⠿⠍⠿⠳⡿⠶⠇⢦⡥⠤⠧⠷⠷⠿⠛⠂⠁
⠀⠀⠀⠀⠀⠀⢶⠋⡔⠁⠀⢀⡤⣠⣀⠀⠈⠙⢝⢽⢦⠀⣄⠀⠰⠎⢀⡀
⠀⠀⠀⠀⠀⣼⠏⠘⢀⡼⢟⡩⠚⠁⠙⢷⣦⣇⠈⠆⢲⡇⠀⠈⡩⢂⠤⠊
⠀⠀⠀⠀⢠⣿⠠⠧⠔⠚⠁⠀⠀⠀⠀⠀⠉⠛⢤⢼⣀⡿⠀⠘⢢⠌⣒⠄
⠀⠀⠀⠀⣸⠃⠀⣆⠀⠠⣀⡀⠀⠀⢀⡀⣀⠀⢸⡄⠀⢱⡄⠖⠓⠊⠁
⠀⠀⠀⠀⠙⢔⠰⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡇⠔⡜⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠄⡀⠠⠔⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⢲⢄⣀⣀⣀⡴⡿⢦⢄⡀
⠀⠀⠀⠀⠀⢀⠔⠡⠤⠀⢸⣄⣀⠑⠁⣰⡏⠐⠁⡈⠲⢄
⠀⠀⠀⠀⡰⠁⡀⢁⠀⠐⢽⡝⠂⠀⠐⢚⡯⢐⠠⡂⠄⠀⠣⡀
⠀⠀⠀⠐⠀⠄⡠⢀⠀⡀⣸⡇⠀⠀⠀⢰⡪⢐⠨⢂⠅⡂⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠐⠈⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣄⣀⣀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣛⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⡷⠾⠿⠛⠛⠛⠛⠛⠛⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⣀⣀⣠⣤⣶⣾⠿⠟⠋⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⣿⣿⠀⠛⢶⣤⣀⣀⡀⠀⠀⢀⣠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⣀⡈⠙⠛⠿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠈⠉⠙⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣤⣴⣶⡶⠶⣿⣿⡿⠶⢶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⣿⡏⠁⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⣿⡇⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⣿⣷⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⡄⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⡀⠀⠀⣀⣤⣶⠿⠋⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⣤⣶⣾⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣤⡀
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠉⠉⠛⠛⢿⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠾⠿⠛⠉⠁⠈⠉⠓
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡠⢄
⠀⠀⠀⠀⠀⢠⠿⠉⠀⣀⣄⣉⠓⢬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠔⠒⠛⠛⠻⠦⣤⡀
⠀⠀⠀⠀⠀⠈⠁⠀⢾⣿⣿⣿⠀⣐⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣃⣿⠁⢰⣿⣿⣷⡀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠍⠉⠉⠠⢬⣛⣿⡟⠀⠠⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⢰⡟⠀⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⢠⣶⡟⠋⠀⠀⠀⢠⡿⠀⠀⠀⠉⠛⣶⣤⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡀⠀⣾⠃⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⣿⣇⣀⢀⣉⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣶⠶⠶⠾⠷⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⠀⠀⠉⢻⣿⣿⣇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣷⣀⡀⠀⣀⣼⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⣿⣿⠏⠁⠈⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⣄⡀⣀⢠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣤⡆⡄⣦⣤⢠⣴⡆⠀⠀⠀⠈⠈⠈⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢠⠰⢂⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠸⠦⠰⠰⠦⠇⠆⠖
⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⣤⢀⢀⣀⣄⡀⣀⣠⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⡀
⠀⠀⠈⠉⠙⠋⠉⠁⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠐⢠⡄
⠀⠀⠀⠀⠰⢴⢴⢶⣶⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠆⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣶⠀⣤⣄⡤⡄⣤⣤⣤⢀⡄⣆⡄⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠
⠀⠀⠀⠀⠘⠸⠐⢲⡿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⠿⠿⠿⠿⠿⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣄
⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀
⠀⠀⠀⢠⣾⠟⠁⠀⠀⢀⣤⣴⣶⠿⠟⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠿⢷⣶⣤⣄⠀⠀⠀⠙⢿⣦
⠀⢀⣴⡿⠃⠀⣠⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣤⡀⠀⠻⣷⣄
⢠⣾⠏⠀⣠⣾⠟⠛⠿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠟⠛⢿⣦⡀⠈⢿⣦
⢿⣿⠿⠿⢿⣧⣄⠀⠀⠀⠉⠙⠛⠻⠿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⡿⠿⠛⠛⠉⠁⠀⠀⢀⣤⣿⠿⠿⢿⣿
⠀⢻⣧⡀⠀⠈⠛⢿⣶⣤⣤⣴⣶⠿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢷⣶⣤⣤⣴⣾⠟⠋⠀⠀⣰⣿⠃
⠀⠀⠙⣿⣄⠀⠀⠀⠈⠙⠻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠿⠛⠉⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠷⣶⣶⣶⣶⣶⣶⣶⡶⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣶⣤⣤⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣠⣤⣴⣶⠿⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠉⠉⠉⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣆⣨⣿⣿⣿⣿⣿⣿⠟⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣠⣾⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠋⠛⠛⠿⠿⠿⣿⣿⣿⠿⠿⠿⠛⠛⠉⣀⣤⣶⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⣶⣾⣉⣉⣉⠉⣉⣁⣀⣉⣉⠉⢉⣉⣉⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⡿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠟⠋⢁⣠⣶⣿⣶⣄⡈⠙⠻⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡄



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⣠⠶⠶⣄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠶⣤⣨⡿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣤⣤⣤⠶⠋⠀⠀⠀⠀⣀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠲⢦⣤⣀⣀⣀⣀⣀⣸⡆⠀⠀⠀⠀⠀⠙⢷⡀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢳⣴⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢠⠀⠀⠉⡍⢩⠉⣿⠉⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠞⠁
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠀⢸⡆⠀⠀⠀⠀⠀⠀⠾⠋⠀⣠⠟
⠀⠀⠀⠀⠀⠈⠳⣤⣀⣸⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⣄⠀⠀⠀⠀⠀⠈⠛⠓⠀⠀⣰⡇⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⣴⠟⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠻⣆⡀⣀⣼⠛⠛⠓⠶⠦⣤⣀⠀⠀⠀⠀⢧⡀⠀⠀⠀⣠⡟⠉⠛⣶⠖⠓⢶⣄
⠀⠀⣠⠞⠋⠁⠀⠀⣰⠟⠓⠶⣌⡉⣉⡁⣠⡴⠞⠛⣦⠀⠉⠛⢦⡀⠀⠀⠙⠓⠒⠛⠋⠙⠛⢾⠃⠀⠀⠀⣹⣄
⢀⡾⠁⠀⠀⠀⠀⠀⠘⣷⡄⠀⢨⡟⠉⢹⡇⠀⠀⣴⠋⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠦⠤⣴⠿⢮⠗
⣼⠁⠀⠀⠀⠀⠀⠀⢼⡃⠀⣀⡴⠟⠶⠞⢧⣄⡀⠈⣷⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣾
⡿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠀⡀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⡿⠀⠀⠀⠀⢰⣤⡀
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣼⠃⠀⠀⠀⢠⡅⠀⠀⠀⠀⢸⠈⠁
⣿⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣼⠃⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠸⣇⡼⠃⠀⠀⠀⠀⢰⡏⠀⠀⢀⡴⠶⣾
⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⢻⡄⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⡾⠀⠀⠀⠸⣆⣀⡾
⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠉
⣿⠀⠀⠀⠀⠀⠛⠓⠶⢤⣤⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⠦⣤⡴⠶⠛⠛⠲⣦⡀⠙⢦⣄⠀⠀⠀⢀⣤⠞⠁
⠈⠷⣤⣀⣀⣀⠀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣈⣷⠀⠀⠈⠙⠛⠛⠋⠁
⣶⣶⡾⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢛⡁⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⡛⠀⠃⢸⡇⢸⣿⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣴⡆⣾⣿⡇⣿⣦⠙⢡⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⡃⣈⢁⣹⣧⣼⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣭⣥⡙⢿⡇⣿⣥⣾⣦⣽⡿⠀⠀⠀⣀⣀⡀⠀⠀⠀⢸⣿⣧⣾⣷⣿⣾⣿⣷⣾⣿⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣤⣿⣿⣿⣿⡿⠃⠀⠀⠘⠿⠿⠟⢷⠀⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠰⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⢶⣄⡒⢒⣰⡆⣰⣾⣿⠿⣷⣦⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡌⣿⣿⣿⣿⣧⢠⣤⣤⣶⠆⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢡⣿⣿⣿⣿⣿⠌⠻⠿⠋⢼⣿⡇⠀⠀⠀⠸⠇⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⢨⣭⠄⣉⣙⠻⠟⣡⣶⣷⣌⣉⣴⣶⣮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣾⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣟⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛







⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⠊⠙⠳⣄
⠀⠀⠀⣷⡶⠆⠀⠀⠳⡄
⠀⠀⠀⠘⣿⣄⢠⡀⠀⠈⢦⡀
⠀⠀⠀⠀⠘⢿⣷⣸⡶⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠈⢻⣴⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⡆⠀⠀⠀⠘⢀⠀⠀⠀⣠⡶⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣌⣤⣌⠂⠀⠀⠠⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠘⣿⠋⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢆
⠀⠀⠀⡠⠐⠂⠲⡖⢀⡀⠀⠀⠀⠹⣷⣠⡶⠋⢻⡄⠀⠀⠀⠀⠛⠲⡀
⠀⢀⠞⠁⠀⠐⠀⣴⠀⠙⠳⣦⣄⠀⠌⠉⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠈⢀
⠀⣸⠂⠀⠀⠀⠀⠀⠁⠐⢀⠀⡙⠃⠀⠀⠀⠀⠀⠸⣿⡃⠂⠀⠀⠀⠀⠀⠣⡀
⠀⠻⣧⡀⠀⣠⡇⠀⠀⠀⠘⡿⠁⠀⠀⠀⠀⠀⠀⡀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠘⢦⡀
⢀⡔⠉⠻⠿⣿⣿⣷⣾⣷⣾⠇⠀⠀⠀⠀⠀⢴⡀⠀⣈⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠁
⣆⠀⠀⠀⠀⠀⠉⠻⣿⣿⠁⠀⠀⠀⠀⠀⠀⢠⣿⣀⣝⣿⣿⣿⣷⣦⣀⠀⡀⠀⠀⠀⠀⠈⢀
⠹⣇⠀⠠⡴⠋⠁⠒⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣶⣮⣍⠉⠀⠀⠀⠀⠀⠈⢡
⠀⠙⠿⣶⣿⣦⣤⡀⣴⣶⠛⠧⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⣿⣏⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⣿⣶⣀⣉⠋⠙⡿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠐⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠈⠻⣿⣇⣀⣀⣤⢠⣯⡀⠀⠀⠀⠀⠀⠀⠀⠰⡮⠀⠀⠰⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠉⠙⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠈⠀⠀⣿⢿⣿⣿⣇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣴⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣦⣀⡀⠀⠀⠀⠀⢀⣴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣴⣠⣠⡴⠉⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⠀⠀⣀⠀⢀⣀⡀⣀⣠⣤⠔⠒⠒⠉⠁⠀⢀⡀⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠛⣻⡛⠛⣛⡤⠍⠀⢭⣉⠉⠉⡅⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⡛⠁⡾⠋⠀⢀⣄⠀⠈⢳⡄⠀⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣴⡇⠀⢰⣧⢨⣦⠀⠀⢿⣰⣿⣿⣿⣿⢳⡿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⠀⠈⠇⣠⠋⠀⢠⣿⣿⣿⣿⣿⡾⣉⠾⣛⣷⣛⣽⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠠⠄⣀⣠⣴⣿⣿⣿⣿⣿⣿⡷⣿⣾⣿⣿⣏⣚⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠻⣦⠀⢘⠋⣰⣿⣿⣿⣿⡞⣿⠁⠈⣁⣿⣿⡏⢙⡯⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣟⣛⠷⣀⣰⣿⣿⣿⣿⣿⠿⠀⠀⠂⢉⣼⠟⠑⠪⢐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⢹⣿⣶⢹⣿⣿⣿⣿⢻⡟⣀⡀⡀⠠⡸⠉⠀⠐⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠈⣿⣯⡴⣿⢻⣿⣿⢿⠆⠀⢠⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⢳⣿⠃⢻⡄⠹⣿⣴⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡀⠀⠘⣿⡗⠀⠀⠸⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣽⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣄⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⢤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣀⣀⣠⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡆⠐⠀⠘⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⢿⡟⠡⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⣴⣂⠀⠀⠀⠀⢿⣿⣇⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⣄⢸⡇⢀⡀⠀⠀⠁⢀⢽⠇⠀⠀⠀⢘⣿⣦⣄⠀⢀⣠⣀⠀⠀⠥⠀⣴⣿⣿⠟⠁⣀⠀⠀⣿⣿⡇⠀⠀⢠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡇⠀⣀⣴⠀⠀⢠⣀⡆⠀⠀⠀⢰⣮⣟⡯⡱⠟⠉⣰⡀⠀⠀⢰⣿⣿⠃⢠⣶⠤⠀⠀⢿⣿⡇⠀⠀⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡷⣿⡇⣿⡿⣿⠀⠀⢸⣿⡇⠀⠀⠀⣼⡇⠨⣿⢠⠀⢐⣿⣿⡇⠀⠈⠻⢿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠻⢿⡇⠉⠀⣿⠀⠀⢸⣿⡇⠀⠀⠀⡝⠋⠐⢿⠈⡀⢐⣿⣿⡇⠀⠀⠀⠀⢤⣾⣷⣤⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⣿⣸⣷⣾⡇⣾⣿⣿⠀⠀⠀⠉⡀⠀⠀⠀⣿⣷⠀⠀⠀⡂⠠⣿⣿⡇⠀⠸⢦⡄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠈⠠⠃⣿⢙⠻⢿⡇⠁⠀⣿⠀⣠⠰⠃⠑⠀⠀⠒⣿⣿⠀⠀⠀⣤⡾⣿⣿⡇⠀⠀⠀⡀⠔⠚⡃⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣷⣾⡇⣶⣿⣿⣾⣿⡷⠉⠀⠀⠀⠈⣿⣿⠀⠀⠊⠀⠀⠙⡿⡇⠀⠀⠀⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⠙⢻⣋⠉⠉⢤⣿⣿⡟⠀⣄⡀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢹⣿⣿⠀⢸⣿⣿⠀⠘⠻⡇⠠⣦⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡅⠿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣦⡤⠈⠀⢸⣿⣿⠀⢸⣿⡇⠀⠰⣿⣶⠖⠀⠀⠀⠈⠁⠀⠀⠀⢰⠈⣿⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⡇⠀⠀⠀⠙⠿⠀⠈⠙⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣤⡞
⠀⢠⣄⠀⠀⠀⠀⢀⣀⢰⣿⡇⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⡇⠀⢴⣾⡿⠛⢹⣿⡇⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⣿⡇⠄⠂
⠀⢸⣿⡇⠀⠉⠁⣀⣤⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠐⠋⠁
⠀⢸⣿⡇⠀⣴⣿⠟⠋⠉⠉⠓
⠀⢸⣿⡇⠀⠈
⠀⠈⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⠀⠀⠀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠿⠛⠋⠉⣹⣿⣿⣿⡿⠋
⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇
⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆
⠀⣿⣿⣿⣿⠃⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀
⠀⣿⣿⣿⣿⡀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡄
⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠶⠾⠿⠛⠛⠛⠛⠿⠷⠶⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠶⠿⠟⠛⠛⠻⠿⠶⠶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣴⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⢠⣾⠃⠀⠀⠀⠀⢠⣾⠋⠻⣷⡄⠀⠀⠀⠀⠈⠻⣶⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠘⣷⡀
⠀⠀⣾⠃⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠘⣷
⠀⣼⡏⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣼⡿⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢹⣇
⢀⣿⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢀⣾⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿
⢸⡏⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸
⣾⡇⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣬⠿⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⡴⠦⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠸⣇⠀⢈⡿⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢿⡀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿
⣿⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠻⣦⡀⠙⠛⠉⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⣿
⢿⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⠿⣦⡀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸
⢸⣇⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣸
⠈⣿⡀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠙⢿⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢀⣿
⠀⢹⣧⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⣼⡏
⠀⠀⢻⡄⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠂⠀⠀⠀⠀⠀⣠⣴⠶⠾⠟⠛⠿⣦⡀⠀⠀⠀⠀⠙⠻⣆⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⢰⡟
⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠈⢿⣦⣠⡿⠁⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠰⣿⣿⡶⠶⠾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠋⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⣶⣶⣶⣶⠶⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠛⠛⠉⠁
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⡄⠸⠀⠀⢰⠙⣾⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⢀⠰⡉⢻⡄⢻⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢴⡏⠱⡅⢸⣇⠸⣇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣷⡈⢻⣄⠹⡄⣿⠀⣿⠀⢧⢀⡔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣧⠈⢷⡄⠻⡆⠘⡏⠀⡼⢢⠸⠞⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣷⡈⢿⡄⠻⡆⠘⣆⡇⢸⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣷⠈⠋⠀⠀⠀⠈⠀⢸⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠒⠉




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

⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⡿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠾⠿⢷⠷⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⢿⢾⡾⣶⣶⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢨⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠾⠶⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠾⠿⣿⠾⡿⣿⠿⠟⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⡦⢀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢷⣾⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣡⣤⡤⠀⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⡄⡀⢀⡀⠀⠀⠀⠂⠀⢰⣿⣿⡿⠟⠁⠸⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⠄⢀⠠⠔⠀⠀⠸⠿⠿⠇⠀⠀⢀⡀⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢇⡳
⣠⣤⡬
⣿⣿⠇
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⣠⠆⠀⢀⣀⣀
⠛⠋⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠻⠿⠟⠋⠀⠀⢸⣿⡿⣿⣿
⠀⠀⠸⣿⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶⠶⠒
⠒⠂⠀⠈⠉⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠠⢤⣤⣤⡤⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠉⠁
⣤⣤⣤⣦⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣤⠤⠶⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⠷⠄
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⡇⠀⠀⠀⠀⢀⣤⣴⠟⠘⢠⠁⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠋⠀⣸
⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⣾⣿⡇⢠⠀⢸⠀⠀⠊⠉⢆⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠁
⠀⠹⣿⣷⣦⡀⠀⠀⠐⢶⣄⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⢻⣇⣾⣄⠺⣼⣄⠀⠳⠀⡜⠈⠋⢀⣇⣀⣀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣶
⠀⠀⢻⠉⠻⣿⡆⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠻⢿⣿⣿⠟⠻⠗⠒⠚⠁⠀⢴⠟⠁⠀⠉⠹⡆⠀⠀⡾⠁⠀⠀⠀⡿
⠀⠀⠈⢧⠀⢹⠇⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⢴⣿⡆⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣄⠀⢸⡇⠀⠀⠀⢠⠇⠀⠀⣀
⢀⣀⣀⠘⣆⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠙⢶⣄⠉⠀⠀⠀⢺⠿⠿⠃⠀⠀⠀⠰⠾⠿⠶⠀⠀⠀⠀⢀⡀⢉⣠⢶⣀⡿⠀⠀⠀⣠⠎
⠸⣿⣿⣷⣿⡆⠀⠀⠀⠈⠛⠷⣤⡀⠀⠀⠀⠈⠓⠄⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠈⣁⣼⠟⠀⠀⠀⢠⡏
⠀⠙⠿⢿⡿⠇⠀⠀⠀⠀⠀⠀⠈⠙⠷⢤⡀⠀⠀⠀⢠⡏⠀⠀⣠⡤⠖⠒⠒⠦⣄⠀⠀⠀⠀⠀⠨⠿⠋⣀⡉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⢸⡇⠀⣼⣯⣶⣿⣿⣿⣿⣾⣷⡄⠀⠀⠀⠀⠀⠖⠂⣿⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠈⠛⠒⠒⠶⠶⠤⠤⢤⣀⣀⣀⡀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠉⡠⠃⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⣧⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⡠⠛⠋⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠙⢿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣼⠁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣴⡟⡡⠊⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠿⠤⣀⠈⠉⠉⠉⠁⢀⡠⠖⠉⡇⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⢻⡞⠁⠀⠀⢻⡄⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⢠⠋⠉⠀⠀⠉⠉⢁⣀⡀⢸⣧⣀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢷⡀⠀⣾⣿⣷⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣀⠀⣾⠀⠠⢤⡀⠻⠀⣭⡀⠀⣸⣿⡈⠁⠀⢠⡿
⠀⠀⠀⠀⠀⣀⣈⣧⠀⠈⠛⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠚⢁⣰⣿⣀⡠⠊⢰⣁⠔⣌⣵⣾⣿⣿⣿⣶⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠻⢿⡿⠇⠀⠀⠀⠀⠀⠠⣧⠀⠀⠀⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⣀⣀⣀⠠⠤⠒⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⣿⣿⣿⣷⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⢿⡿⠿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠀⠀⠀⠀⠈⠻⡄⠀⠈⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠛⠛⠅⠻⠈⠘⠩⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠿⠽⠞⠛⠛⠷⠧⠋⠻⠽⠻⢆⠠⠀⠈⠋⠒⠉⠒⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠘⠘⠳⠛⠟⠎⠳⠿⠌⠳⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠸⠀⠀⠀⠀⠀⡶⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⢀⣾⣷⡄⠀⠀⣴⣦⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣄⣰⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⣿⣿⣿⠏⢹⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠹⢿⡿⠃⠀⠈⠿⠋⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠇⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡎⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢸⠆⠀⠀⠗⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⡇⠃⣴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣴⡶⠶⣄⠀⣀⡎⢠⡄⣾⣄⣤⡼⠀⣥⣽⡄⣧⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠃⠀⠀⠀⠙⠉⠀⠀⣇⠁⣦⠀⠀⠀⣂⡀⣷⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢻⡇⡿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠾⠿⠀⠀⠐⠖⠶⠆⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⡏⢉⣉⣉⠉⣿⣿⡿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠟⢁⣄⠉⣁⣼⣿⣿⣆⣈⢁⣤⡈⠻⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣅⠰⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠂⣼⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠺⠟⠀⣾⣿⡿⠋⣠⣶⣦⣄⠙⢿⣿⣷⡀⠛⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⣿⣿⣿⠁⣼⣿⣿⣿⣿⣧⠈⣿⣿⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣟⢙⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠇⡆⡔⣦⡔⡆⡖⡰⡄⡖⡖⡆⡖⡖⡆⣤⠖⣤⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠙⠃⠁⠉⠁⠁⠁⠁⠁⠁⠁⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⣀⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠛⠛⠛⠛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⠷⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠟⠉⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⠏⢀⣤⣾⠿⠋⠉⠉⠛⠛⠻⠿⢿⣿⡶⠶⠿⠿⠿⠛⠛⠋⢩⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⢾⣿⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠉⠀⠀⠙⠻⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⡶⠿⠟⠛⠿⣶⣤⡀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿