Matchbox Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⡤⠤⠖⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠔⠒⠚⠉⠉⠁⠀⠻⡉⠀⠀⠀⢠⣴⣶⡇⠱⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠐⠒⠊⠉⠉⢀⣀⣀⠠⠄⠀⠀⢀⡄⠀⠀⠀⠹⣖⣚⣉⣁⢻⣿⣷⣆⠘⠄
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⣿⣷⣶⣾⣣⣴⡶⠃⠀⠀⠀⠘⣦⢒⠂⠍⢹⣿⣿⣦⠈⢆
⠀⠀⠀⠀⠀⠀⡇⠐⡀⠀⠀⠀⠀⣯⣴⣤⡈⣉⣽⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠈⢷⣒⣭⡥⢽⣿⣿⡀⠈⣧
⠀⠀⠀⠀⠀⠀⢻⠀⠠⠀⠀⠀⠀⢿⡿⠋⣴⣿⣿⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠤⠖⠚⠉⠉⠉⠀⡜
⠀⠀⠀⠀⠀⠀⠈⢇⠀⢡⠀⠀⠀⠀⠙⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢃⠀⠀⠀⡀⠀⠀⣀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⠤⠤⠔⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⣧⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣛⣛⡭⠭⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⣿⣿⣿⣿⠿⠿⢟⣛⣫⠭⠵⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡭⠿⠗⠒⠋⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢊⡥⠄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢉⠤⠊⠁⠀⠀⠀⠈⠀⠒⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡁⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡀⠀⠀⠀⠉⠑⠒⠢⠤⣤⠀⠀⠀⣀⠤⢒⣨⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⠃⠀⠀⠀⠀⠀⣠⠔⠊⢀⠠⠂⢁⠤⠂⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠀⡡⠔⠊⠁⠀⢀⠤⠒⠊⠛⠂⠐⠒⠉⢁⢠⠄⠀⢀⠌⠀⠀⠀⡠⠒⠉⡀⠔⠊⡡⠔⠊⢡⠖⠂⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢁⡠⠐⠉⠀⠀⠀⢀⠞⢁⡤⠔⠒⠲⠄⠤⠄⠂⡡⠂⠀⡠⠊⢀⡠⠐⢁⡠⠔⢉⠠⠒⠉⠀⠀⣤⠁⢀⡔⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⣤⠊⠁⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⢊⠤⠒⢁⠤⠊⣁⠔⠊⠁⢀⣠⠁⠀⡐⠀⠀⠈⣁⠔⣊⠼
⠀⠀⠀⠀⠀⡀⠄⠀⠁⠀⠀⠀⠀⠀⠈⠉⠒⠢⠄⣀⠀⠀⠈⠙⠲⠦⠄⠀⠀⠀⠀⠂⡠⠔⠊⡡⠐⠈⡠⠐⠈⠀⡠⠀⠀⠰⠁⠈⠙⢁⡠⢔⡩⠒⠉
⠀⠀⠀⡴⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⣀⠀⢀⠠⠒⢉⠠⠒⠁⠠⠐⠀⠀⠀⠀⡜⠉⠁⠞⠂⢀⠤⢒⠡⠂⠁
⠀⠀⠀⠀⠚⠡⢉⡐⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠔⠈⡀⠄⠈⠀⠀⢀⠞⠒⠠⠼⠀⠀⡀⠔⢊⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⢀⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢉⠀⠂⠀⠀⡴⠒⠀⣤⢆⣀⠀⢀⡠⠒⣉⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠊⠑⠂⢀⢤⡤⠒⡁⠄⠈⠀⠀⠀⢤⠮⠀⡀⠐⠁⡀⠤⠊⡁⠐⠈
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡇⠈⠀⠠⣴⣁⠀⡠⠃⠀⢀⠠⠒⢁⠠⠐⠁
⠀⠀⢠⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⡰⠁⠀⠉⡁⠔⢊⡡⠔⠊
⠀⠀⠀⠑⠂⠨⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠐⠈⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⢐⡂⠤⠀⣀⠀⠀⢸⡄⠧⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠐⠂⠀⠀⠀⠀⠉⠀⠂⠂⠀⠤⠀⢀⣠⣴⡆
⠀⠀⠀⠀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣏⣉⣋⠉⠂⠐⠀⠤⠀⢀⣀
⠀⠀⠀⠀⢇⠈⠉⠐⠲⠠⠀⣀⠀⠀⠉⠀⠈⠁⠈⠁⠀⠀⠐⠂⠀⠀⣀⣤⣾⣿⣿⣟⡿⢯⣬⣉⣓⡒⠂⠠⠤⡀⣀⠉⠉⠒⢲⣤
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠄⣀⡀⠠⢀⡀⣀⣴⣾⣿⣿⣿⣿⠿⠿⣦⣦⣤⣄⣉⡉⠓⢒⡂⠰⡤⠠⣼⠟⠃⢘
⠀⠀⠀⠀⠈⠭⠝⡲⠤⣤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠙⡕⠶⠎⡿⢿⠿⢿⣿⣿⣷⣲⠤⢤⣀⣈⢉⡽⠻⠟⣓⢷⡖⠺⠛⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠥⣒⠂⠤⣈⠀⠀⠀⠀⠠⠈⠒⡗⠀⠀⠀⠀⠈⠉⠉⠛⠷⠞⣻⣥⡀⢈⣶⡾⠉⠕⠀⠀⠀⠂⣈⡾
⠀⠀⠀⠀⢠⣔⡂⠀⠀⠀⠀⠀⠀⠀⠁⠒⠩⢅⣒⠲⢤⣀⡁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡋⠀⠀⠂⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠾⠯⣭⣭⣁⢀⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠭⢽⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⡔⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠿⣾⣶⣶⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠓⠲⠤⢄⣀⠀⠀⠀⠂⣀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢯⣍⠲⠌⠉⠛⠑⠦⠅⣐⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣄⡐⠀⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠆⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠟⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⢋⡡⠔⠀⣀⠔⣊⣽⢟⡷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠶⣿⣿⣿⣿⣯⣶⣑⡭⠴⠚⣉⡠⢔⡪⢗⣫⠿⢛⣶⣦⣄⡀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣟⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣴⣫⠕⣊⠥⠚⠉⢀⡠⢴⠮⣚⠝⠋⡗⢸⣿⣿⣿⣿⣿⣿⢟⡿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣾⣷⣷⣺⣟⠍⡪⠅⠈⠀⠀⢸⡇⢀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢸⡧⢸⡿⡿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠷⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⣀⠀⣀⣀⣠⠤⠴⠒⠚⢹⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠒⢒⣋⣉⡭⠥⠖⠒⣄⠈⢯⡁⠀⠀⠀⠀⠀⠀⢸⠀⠈⠣⡀
⠀⠀⢀⣀⡤⠤⠖⠒⢋⣉⣩⠤⠔⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⢣⡀⠙⣄⠀⠀⠀⣀⣠⡼⢢⡀⠀⠙⢦⡀
⣾⠉⠉⡠⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⢳⣽⣿⣿⣾⣻⣭⣼⠄⠀⠀⠑⢄
⡏⠳⡄⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠙⢿⡻⣿⣷⣭⡮⠭⢳⡄⠀⠀⠱⣄
⣿⠀⠙⣆⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠻⣾⠿⣿⡷⣖⠋⣙⣆⣀⣀⡬⢧
⣿⠀⠀⠘⢦⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠤⠶⠒⢒⣉⣠⡤⠼⣿⡿⠿⠟⠛⠋⠉⠁⠀⠀⢸
⢻⠀⠀⠀⠀⠳⡄⠈⢦⠀⠀⢀⣀⣠⠤⠴⠖⠒⠚⢉⣉⣁⣤⠤⠶⢒⣚⣩⣭⠥⠶⠒⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⢀⣀⡠⠤⠔⠛⣄⠀⠙⣉⣉⣤⠤⠴⠖⣒⣛⣉⣭⠥⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⢧⠀⠀⠀⠀⠀⠈⡏⡩⠥⠴⠒⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼
⠀⠈⢳⡀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢻⢸⣀⡤⠤⠖⠒⠛⠉⠉⡠⠤⠒
⠀⠀⠀⠱⡄⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢴⣒⣯⠥⠞⠀⠀⠀⠀⢀⡠⠔⣺⣀⣀⡤
⠀⠀⠀⠀⠙⢆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢖⣒⣫⠭⠽⠒⠚⠉⠉⠀⠀⠀⠀⠀⣠⠴⣺⣽⡾⠟⢉⣀⣀⡀
⠀⠀⠀⠀⠀⠈⢧⠀⡟⡇⢀⣀⡤⠤⣖⣚⡯⠽⠒⠚⠉⠉⠀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠶⠿⠖⠛⠛⠛⠒⣚⠁⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠳⣧⠭⠿⠒⠚⠉⠉⠀⠀⠀⢾⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⣴⣶⣶⠶⠾⠿⠛⠛⠛⠛⠉⠑⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢋⣩⢝⣫⣽⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⡾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡶⣭⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⣾⠎⠻⠗⢲⣦⣍⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠰⠿⠉⢧⠀⠀⠘⡇⠋⢹⣲⣤⣭⣒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠄⡾⠳⠀⢄⡈⢆⠈⢂⠱⢐⡶⣀⠬⠤⠨⣟⠒⠬⣕⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡱⠋⠋⠒⠤⣀⠈⢪⡴⠗⠊⠘⠋⠀⠀⠀⠀⢹⠀⠙⠿⢅⣀⠨⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡨⠞⠛⠒⠤⣀⠒⢠⢿⠟⠀⢀⡠⡄⠄⠐⠘⠃⠀⣼⡶⠔⠒⠀⠚⢋⣧⡑⠢⣑⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠄⣽⠿⠁⠀⠀⠀⠀⢙⡎⠚⡠⠴⡾⠆⡀⡥⠂⠀⣀⠈⢡⡗⠤⠄⠀⠀⠀⣀⠈⠗⣾⠑⡿⣷⠄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⣵⣤⢃⠀⠀⠀⠀⠀⠈⠀⢹⡆⢰⣫⣖⣼⢋⠁⠀⡠⠁⠀⢑⠟⠂⠀⠀⠀⠀⠀⢚⡗⡡⢊⣾⣿⣷⣦⣄
⠀⠀⠀⠀⠀⣴⣤⡾⠊⠅⡻⢟⢧⣂⣄⠂⠈⠡⠐⠉⢙⣦⣜⡩⡛⢁⡠⠊⠀⣀⡔⠉⠒⢤⡀⠀⢀⡀⢋⢞⠊⣠⣾⣿⣿⣫⢋⡅⡉⠲⢤⡀
⠀⠀⠀⠀⠀⡃⡬⢙⡲⣄⡀⠀⠩⢑⠩⢿⠴⢀⠀⣠⡟⠉⣹⡿⠭⠥⠤⡶⠋⠳⡘⠀⠀⠀⠈⡀⣎⣽⠓⢁⡴⣿⡻⠿⢿⢃⣼⠃⡔⠀⠀⠉⠳⢶⡤⣄⡀
⠀⠀⠀⠀⢠⡇⣷⡀⡭⠲⠨⣒⢤⣀⠈⠂⠝⠛⢶⣿⢄⣴⠋⠀⠀⢸⠀⠉⠀⠀⠱⡄⠀⠀⠰⢹⡮⠊⣠⠮⣉⡒⠪⢭⣓⠮⣅⡘⠁⡄⠀⠀⠀⢺⢭⣲⡿
⠀⠀⠀⠀⢸⡿⣿⡝⣦⠒⠂⡀⣈⡝⢩⣒⠤⡈⠐⠉⢛⠡⢍⣢⡤⠀⣇⠀⠀⠀⠀⠘⠂⠐⣾⠟⢁⣼⣍⡑⠢⢭⣓⠦⣄⡉⠉⠛⠷⣦⣄⡀⠀⠀⡜⣽⡇
⠀⠀⠀⠀⠀⣛⣿⣧⡜⢖⠉⠁⡘⡀⢤⠲⠓⠈⢕⡒⠤⣁⠐⠠⣉⠺⢏⣠⠆⡀⠀⣀⣙⡿⠋⣠⡛⠢⢍⣛⠲⢤⣈⡑⠲⢬⣑⠲⢤⣀⠉⢹⢶⣮⠊⢸⡇
⠀⠀⠀⠀⠀⠉⠙⠻⢯⣷⠯⣙⢻⡀⡄⣇⠚⡄⢃⠈⣑⡦⣍⠒⠤⣈⡂⠍⡚⠣⢾⣢⡟⢡⡞⠻⠯⣷⣶⣤⣉⠒⠬⣝⡻⢶⣦⣭⣒⢤⡽⣦⡟⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣑⡲⠼⣦⢻⡈⣇⢉⣢⢰⡌⠱⠢⣭⣑⠒⠥⣄⣼⣰⡯⣍⡑⠲⠤⣉⠙⠺⢭⣒⠤⣌⡑⠪⢍⣛⣏⣵⠋⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣬⣃⠲⣝⡁⠈⣇⢀⡈⠤⢰⠐⠋⢑⡦⣿⢛⠶⣤⣈⠑⠒⠤⣉⡓⠢⢄⣉⠒⠭⣓⡢⣾⡿⠁⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢬⣚⠩⣅⡀⣰⡤⠗⢫⣙⡇⣿⣼⡄⠀⠉⠙⠲⠤⣀⡈⠑⠒⠤⣍⣑⣶⡿⠃⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠭⣶⣷⣚⣩⡇⠇⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣼⣿⠋⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣂⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡿⠐⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⣃⡇⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣤⣀⡀⢳⡖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡄⠀⠀⢀⣾⣿⣦⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠸⣿⣿⠏⠀⢠⣿⣿⡿⢠⣶⣶⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡹⠀⠀⠀⢯⠃⢀⠔⣩⠟⠉⠀⣾⣿⣿⢇⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⢇⠀⡘⣸⡔⣡⠞⠁⠀⠀⡰⡻⠋⢁⢾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⢹⠈⣿⣷⡷⢋⡜⠁⠀⠀⣠⢎⡔⢁⡴⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠁⠀⠀⢸⢀⡸⢋⠴⠁⠉⠓⢦⠜⣡⠊⡰⢫⠞⠁
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣽⣾⣿⣶⣤⣀⡼⢋⢴⠃⢷⠀⠀⡰⢋⡔⢩⢟⡴⣅⡀
⢼⣿⣿⣿⡖⠒⠒⠶⠶⠶⠖⢒⣂⣀⣠⠶⣫⣶⣿⣿⣿⣿⣿⣿⣾⣿⣷⢾⣜⡇⢣⠞⣡⢋⠔⡱⠋⠀⠀⠉⠓⠦⣄⡀
⠀⠉⠉⣴⣶⣶⣤⣀⣀⣀⣀⣠⡴⣋⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣟⠹⠿⠿⠿⣿⣷⢾⣵⣫⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠙⠛⠉⠁⠀⢀⣠⢞⣭⣾⣿⣿⣿⣿⣧⡙⢿⣿⠿⣿⠡⣍⣶⣶⡶⠆⢸⣿⣿⣿⣭⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⣠⠴⣫⣶⣿⣿⣿⣷⡝⢿⠿⢿⠟⣦⡀⠲⣤⡁⠀⠙⠃⣹⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⢀⣠⠞⠉
⠀⠀⠀⢀⡴⢚⣵⣾⠟⠛⠿⣿⣿⣿⠟⠤⣄⠛⢦⡙⠻⢶⡀⠙⢦⣬⣿⣿⣿⣮⣿⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠶⣤⡴⠋
⢀⣠⠞⣩⣶⣿⡟⠛⠃⠻⣶⣄⡙⠳⠌⣳⠆⢑⢀⡉⠂⢀⣤⣾⣿⣿⣿⣿⣿⣿⢿⢟⣼⣟⣻⢿⣿⣿⣿⡿⢟⣥⣾⣿⡇
⡿⢤⣝⠻⢿⣿⣿⣦⡙⢦⣄⠙⠻⢦⣄⠙⠦⣈⢃⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⣿⣿⣿⣿⠟⣫⣶⣿⣿⣿⣿⡇
⡇⠀⠈⠙⢲⣬⣙⠻⢿⣶⣝⡻⣦⣄⡙⠳⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⡿⣋⣵⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⢸⡇⠈⠙⠲⢬⣙⠻⢷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠙⠲⢬⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣿⣿⣿⠟⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣠
⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣙⠻⢿⣿⣿⣏⣰⣿⡿⢋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⠴⠋
⡇⠀⠀⣀⡼⠷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣙⠻⢟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⣇⡤⠞⠁⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣤⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⢤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣸⣿⠿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣷⣴⣤⣷⣴⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣀⡤⠤⢄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⠿⡿⣧⣘⡋⠍⢠⣤⣄⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⢯⣄⣐⣀⠠⠤⠖⠚⢉⣉⣠⣤⣴⣶⡚⠛⢿⣿⣶⣶⣭⣝⣂⠀⠱⣄
⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⢺⡍⠁⢭⣽⣿⣶⣶⣬⣭⣍⣛⡛⠫⠠⢦⡸⣿⣿⣿⣿⣿⣿⣿⣷⡲⠮⣤⣂⡀⠄⡀
⠀⠀⡶⣞⠉⠉⠀⠀⠀⠀⠀⣿⣦⡀⠙⢿⣿⣿⣿⣿⣧⣀⠹⠿⣿⣶⣶⣶⣶⣶⢾⠛⣿⢛⣿⣿⣦⡈⠳⣍⠙⠃
⠀⣸⠁⠈⠳⣄⠀⠀⡀⠀⠄⣸⣿⣿⣦⡀⠉⠻⣿⣿⣿⢿⡿⢿⡛⢍⣉⢧⠸⣮⡓⢑⢯⡑⣦⣭⠻⣿⣦⡈⠳⣄
⠀⡏⠀⠀⠀⠈⠻⣶⣖⣛⣭⣥⠶⠿⢿⣿⣦⡀⠈⠻⣿⣷⣍⠂⠙⣦⡉⣳⢝⢮⣛⣛⣽⣮⡦⠴⠖⠛⠋⡉⠀⠌⡇
⠀⠙⢦⡀⠀⠀⠀⠈⠻⣿⠴⠖⠚⠋⡉⠙⢿⣿⣦⡀⠈⠻⣿⣷⣵⣽⠶⠾⠛⠛⠋⢉⠁⢀⡀⣂⣤⢥⣶⣶⣿⢧⠃
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠲⣦⠥⠶⠒⠋⡉⠉⠿⣿⣦⡀⠈⣉⠀⠤⡀⣐⣠⣬⣴⣶⣿⣿⣻⡽⣞⣯⣿⣿⠟⡞
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠲⣦⡬⠴⢖⣚⣫⣭⣴⡞⣡⢶⣻⡽⣟⣿⣿⣿⣿⣿⡿⠯⠷⣛⡉⠬⠕⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⠣⣞⣯⢷⡻⠽⢚⣋⡯⠭⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣾⣿⣿⣿⣿⣿⣿⣟⣈⠭⠴⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⠟⠛⠋⠉⣷⣶⣶⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⠉⠓⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠔⢡⠔⠒⠢⠤⢀⡉⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠠⡢⠚⡇⠀⢀⣀⠀⠀⠈⢉⡶⠀⠀⢉⠛⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡼⢁⠔⠐⢥⣶⣄⠀⠡⢤⣤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠂⠤⢉⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⣠⠏⠠⠂⠀⠀⠈⠙⠛⢦⣔⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⣰⢿⢤⣀⡀
⠀⠀⠀⠀⢼⣥⣀⡑⠀⠄⡀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣼⣛⡒⣬⢿⣽⣟⣦
⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣤⣁⡐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⣾⣑⠢⣽⢯⣿⣫⠏⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣁⠐⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⢠⣞⡓⠦⡽⣿⡿⣳⠋⠀⡿
⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠁⠈⡀⢀⠊⣰⣟⡒⢬⡿⣽⡿⡽⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣁⣴⡛⠢⢭⣿⣿⣟⡞⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣷⣿⡏⠙⠛⠿⣿⣯⠟⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣇⠀⠀⠀⠀⢹⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠙⠲⠦⣄⣸⡾⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⣻⡤⠂⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢞⢛⡑⠒⣄⡐⠈⠐⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢑⠀⠀⡜⠁⣿⠒⠢⣌⡐⠂⢄⡔⠁
⠀⠀⠀⠀⠀⠀⡠⠃⠀⣜⣀⠎⣠⣖⠙⠳⣦⣀⡉⠣⠆
⠀⠀⠀⠀⢀⠜⠀⡠⠔⠸⣡⠚⢦⣈⠷⠶⣦⣈⡙⠁⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠰⢁⠀⠀⢱⢥⠶⢮⣙⠺⢭⣙⠲⢦⠈
⠀⢠⣀⢠⠀⣤⠜⣡⣎⡉⠳⢬⣙⠲⢬⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘
⠀⠀⠈⡈⠳⠅⢸⣤⣌⡉⠿⢶⣬⣉⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢠⠁⠀⠀⠀⠈⠉⠙⠳⡦⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢄⠒⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡘⠀⠀⠀⠀⠀⡂⣀⡤⠒⠘⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣼⠇
⠀⠀⠀⠀⠈⠐⠠⢀⠀⠐⠀⢘⠀⠠⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠁⠈⠈⡤⠆⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠞
⠀⠀⠀⠀⠈⠀⠀⠀⠈⠐⠶⣼⡀⢸⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠠⠄⡀⠀⠀⠀⠀⠔⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢈⠑⠢⠄⣀⠀⠀⠀⠀⢑⠢⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠀⠀⡁⠂⠤⣀⠁⠀⠀⠀⠁⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠤⢉⡒⠤⢄⠀⠀⠀⠀⠈⢀⠠⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠐⠠⢈⠀⠂⢄⡀⠀⠀⠀⠠⠓⠀⢀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠐⠂⠈⡁⠒⢤⣀⠀⠀⠁⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠇⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠂⠄⡠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣘⠋⠀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⠉⠉⠉⠉⠉⠉⠉⡉⠩⢡⣞⠉⣉⠉⢉⣉⡉⠉⣩⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⣿⣿⣷⣾⣿⣷⣼⣿⡟⢡⣿⣿⣿⣿⣷⣿⣿⣿⡾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠷⠶⠾⠿⠿⠿⠽⠿⠿⠯⠿⠴⠯⠿⠷⠿⠿⠷⠾⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡾⣻⣇⣿⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣰⠟⣼⣿⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠄⠀⠂⠀⠀⠀⠀⢀⠈⢀⣾⢋⣼⣿⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠁⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣡⣿⣿⣿⣯⡞⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣴⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣫⣾⣿⣿⣿⡵⠃
⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⣿⣿⣿⢿⠟⠁
⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣾⣿⢿⡻⣳⠋
⠀⠀⡟⠒⠚⠛⠛⣻⢛⢛⣛⡛⣛⣛⣛⣛⢛⢛⡛⡛⣛⡛⡟⠛⢛⣏⣾⣿⣻⡟⡵⠁
⠀⠀⡏⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢏⡾⠁
⠀⠀⡇⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣣⠏
⠀⠀⣇⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡼⠁
⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣁⡀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣡⣴⣾⣿⣛⣿⣿⡿⢿⣦⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣡⣴⣾⣿⣿⣿⠿⠛⠛⠿⠿⠿⣷⣝⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣡⣴⣾⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⣷⣄⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⣁⣴⣾⢿⣻⣿⡿⠟⠋⠁⢰⡶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣿⣷⡌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠁⠀⠀⠀⣿⡏⢰⣿⠋⠁⠀⠀⠀⠀⠈⡇⠀⢹⣦⠶⢦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣟⢿⣦⡈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠙⣦⠀⠀⠀⣰⣿⠇⣼⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⠃⠀⠙⣆⠀⠀⠀⠀⠀⢀⣾⡟⢀⣿⡟⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⢀⣠⠶⣛⣷⡄⠀⢿⣯⡸⣿⡅⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⠀⠀⠀⠀⢻
⠀⠀⣀⡴⠞⠋⢁⠀⢀⣤⣞⣫⠴⠚⠉⣀⡽⣦⡀⠻⣿⣜⢿⣦⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣠⣴⡿⣇⣼⣿⠇⣀⣤⠞⠋
⣴⣿⠉⠀⣴⣿⣿⣿⡿⠛⠉⠀⣠⡴⣞⣭⠞⠋⣳⣄⠈⢿⣷⡹⣿⣄⠀⠀⠀⠀⠀⠙⠓⠶⠖⠛⠁⠀⢀⣤⣶⡿⣻⣽⡾⠟⣩⡴⠞⢉⣠
⣿⠘⢷⡘⣿⣿⣿⣿⣧⣤⣴⣟⡵⠞⠉⣠⡴⣻⡵⠛⣦⡀⠙⣿⣮⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⢿⣫⣷⠿⢛⣡⠶⠛⢁⣤⣾⣿⣿
⣿⠀⠀⠹⣎⠉⣽⣿⣿⣿⡿⢁⣠⡶⣫⡵⠛⣡⡴⠞⣩⣷⣄⠈⢻⣷⣝⠿⠿⠿⣷⣦⣠⣴⡿⣟⣭⣾⠟⢋⣥⠖⠋⣁⣴⣾⣿⡟⣻⣿⣿⠀⢸
⣿⠀⠀⠀⠈⢷⡙⠛⢻⣿⣿⣿⡷⠛⣁⡴⠟⣩⡴⠛⣡⡴⢛⣦⡀⠙⢿⣶⣶⣶⣤⣙⣫⣵⡾⠟⣉⡴⠞⠋⣀⣴⣾⣿⡿⢻⣿⣷⣿⡿⠛⠀⣸
⣿⠀⠀⠀⠀⠀⠹⣦⠻⣿⣿⣿⣷⣾⣯⡴⠛⢁⣴⢞⣩⠶⣫⠿⣳⣄⠀⠀⠀⠈⠙⠿⠛⣡⡴⠛⠉⣠⣴⣾⣿⡟⢻⣿⣷⣿⣿⠟⠁⢀⣤⠞⠋
⠸⣆⠀⠀⠀⠀⠀⠈⢳⡄⣾⣿⣿⣿⣯⣤⣴⣫⠶⣋⡵⣛⡵⢛⡵⢛⣧⡀⠀⢀⣠⠶⠛⠁⣠⣴⣿⣿⠟⣻⣿⣷⣾⣿⠟⠋⢁⣤⠾⠋
⠀⠙⣦⡀⠀⠀⠀⠀⠀⠙⣮⡉⣿⣿⣿⣿⣏⣥⢞⣽⠞⣩⠞⣫⡴⠋⢁⣽⡾⠋⠁⣠⣴⣿⣿⠿⣿⣿⣴⣿⣧⠾⠋⢁⣠⠶⠋⠁
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⢳⣜⠛⣿⣿⣿⣿⣫⡴⢛⡵⠞⠁⣠⠶⠋⠉⡇⠀⢾⣿⣷⣾⣏⣸⣿⣽⡿⠟⢁⣠⠶⠋⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠙⣾⡿⢿⣿⣿⣿⡾⠋⣀⡴⠛⠁⠀⠀⠀⡇⠀⢸⡇⣾⡟⣿⣿⠟⢉⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠈⠳⣌⠿⠿⢟⣵⠞⠉⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⠟⢉⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠙⢷⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⠋⣡⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢸⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣆⡴⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣿
⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣾⣿⡟
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⣾⣿⡿⠃
⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⣿⣿⡟⠁
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣜⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣴⣻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⡝⠁⠀⠀⠀⠀⣤⠒⠒⠒⠒⠒⠒⢢⠒⡒⢒⡒⢒⢦
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢫⠏⠀⠀⠀⠀⣀⣎⣼⣿⣿⣿⣿⣟⣡⣿⣿⣿⣿⣿⠋⢸
⠀⠛⠒⠿⠿⠿⠿⠾⠿⠷⠶⠾⠷⠶⠶⠒⠛⠁⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⢻⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⣿⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣾⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⣩⠋⠙⠉⠋⠛⠙⠉⠛⠙⠋⠉⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠌⠤⢤⣤⣤⠤⠤⠤⢤⣤⣤⣤⡀⠠⠷⠁

⠀⠀⠈⠐⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠀⠀⡤⠊⠀⠀⠠⠤⡐⠀⠾⠓⠒⠒⠈⠀⠀⢀
⠁⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈
⢤⠖⠀⠀⠀⢠⣀⠄⠀⠀⠀⠌⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⣄⠀⠀⠀⢶⠄
⠀⠀⠀⠀⠀⠋⠀⠀⠔⠛⠀⠀⠀⠀⠀⠢⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠱⡁⠀⠀⠈⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⡄⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢆⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⡈⠄⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⡎⠂⠀⠀⠀⠀⠀⠀⢄⡇⡀
⠀⠀⠀⢆⠀⠁⣦⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⠇⠀⠀⠀⡇⠀⢀⠀⠈⠃⡃⠀⠀⠀⢀
⣄⠀⠐⢾⡔⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢣⠃⠀⡠⠂⠁⠀⠀⠀⠀⠀⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⢀⡴⠀⠀⢄⡀⠀⠈⠀⣠⡆
⠀⠀⠘⠀⠘⠀⠀⠀⠀⣠⠀⠀⠀⠇⠀⠀⠀⢀⠀⠊⠙⠀⠀⡠⢺⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡊⠔⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠝⠋⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⠊⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠞⠋⠀⠀⠰⠀⠀⠀⠘⠄
⠐⠠⢀⣀⠀⠀⠀⠈⠉⠉⠁⠌⢠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠀⠀⢀⡀⠀⠀⡀⠀⠀⡀⠈⠚
⠀⠀⠀⠀⠀⠠⠄⡐⢀⣀⡀⠀⠀⠑⠀⠀⢠⣧⠀⠀⠀⠀⠀⠐⠢⢤⣤⠀⡠⠋⠀⡠⠊⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠑⠹⠯⡁⠀⠀⠀⠀⠀⠈⠃⢀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠈⣢⣀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠗⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⡀
⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣐⣛⣛⣉⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢾⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣼⣇⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⣸⠀⠸⣧⣸⣦⡀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣠⠀⣿⣿⣿⡇⢰⡀⣷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣼⣿⢰⡄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⣷⣾⡟⣼⠁⣿⣿⣿⡧⣸⣴⠀⢀⣼⣿⠃⢠⡆⣿⣤⣿⣿⣿⣀⣾⡷⣸⣷⠀⣦⠀⠀⢀⠀⣼⣿⣿⣿⣿⣎⣿⣾⣿⣦⠄
⠀⠀⠀⠀⣄⠀⣴⣿⡇⣿⣿⣿⣿⣿⣤⣿⡿⣹⣿⣿⣿⣷⣿⣿⣿⣴⣿⣃⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣄⢷⣌⣷⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣀⣦
⠀⠀⠀⠰⣿⣿⣿⢻⣷⣟⠹⡙⢿⣿⣿⣿⠰⣿⣿⠟⣿⣧⣿⣿⠙⢿⢻⣿⣿⡏⣿⠃⢿⠙⣿⡟⠠⣿⣿⢹⣿⡿⣿⣿⠻⣿⣿⠐⣿⣿⡿⠃⠟⢛⣿⣷⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠉⠁⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣠⣶⠃⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⢰⣿⡇⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⣿⠀⢠⣿⠀⠀⠈⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⡏⠀⠀⢸⣿⣇⢸⣿⡄⠀⠀⠘⣧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢸⣿⡇⠀⠀⢸⣿⣿⡆⠻⣿⣦⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣤⣾⡿⣿⡇⠀⢹⣿⣧⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠸⣿⣷⡙⠛⠛⣵⡿⠁⠀⠀⣿⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢿⣿⢀⣾⡟⠁⠀⠀⢀⣿⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣾⣿⠀⠀⠀⠀⣼⡿⢹⣿⠏⠀⠀⢰⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣦⣴⣏⠀⠀⢀⣿⣿⣿⣤⣀⣤⠞⡉⢀⣿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠾⣿⣿⣦⣤⣾⣿⢻⣿⣿⣿⣿⠀⡇⢸⣿⡿⣶⣾⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡛⢿⣿⠀⠻⣿⣿⣿⠀⡷⠄⢀⣴⡿⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⣧⣀⠁⠰⣆⠙⠛⠋⢰⠁⣴⣿⣿⡀⢀⣠⡿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⣠⢀⣿⣿⣷⣦⡈⠻⣶⣶⣿⠀⠻⣿⠟⣿⡿⠋⠀⠀⠀⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣧⠀⢹⣿⣿⠷⠖⠀⣰⣿⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡙⠛⠋⠁⢸⣿⡏⣴⢿⡟⠁⢀⣴⣾⣿⣿⣀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄⠀⠘⠋⢸⣿⠘⠀⠀⣿⣿⣿⠟⠛⠛⠋⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠘⡆⠀⣿⡀⠀⠀⠈⠉⣡⠖⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣰⣿⠇⢀⣵⠀⣽⠹⣦⣤⣤⣾⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡟⠀⣾⣿⢾⣿⡀⢻⣿⠏⣿⡿⢸⣿⣧⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠄⠘⠿⠄⢉⣤⠆⠁⣰⠟⠡⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠁⠀⠀⠀⠀⠉⠙⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢠⡾⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠛⠛⠿⠶⣦⣄⡀
⢿⡇⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠛⠁⠀⠀⣿⠀⠀⠀⠀⢀⣤⡶⠿⢶⣄⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣄
⠸⣧⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⣀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣾⠋⠀⠀⠈⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧
⠀⢻⣧⠀⠀⠀⠀⠈⠻⣿⣟⡛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠇⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢠⣿⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⢸⣿⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢻⣧⠀⠸⣿⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣀⣀⡀⠀⢀⣀⣤⣴⠾⠋⠀⢠⣿⣆⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠙⠛⠛⠛⠛⠋⠉⠀⢀⣠⣶⠟⠉⣿⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠐⠾⠛⠋⠀⠀⠀⣸⡗⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⡀⠀⠀⣀⣀⣤⣴⠾⠛⠁⠀⠀⢀⣾⠟⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣇⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⢀⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠹⣷⣄⠀⠀⠀⠈⠙⠛⠿⠶⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠈⠙⠷⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣀⣀⣤⣾⠋

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢂⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠠⠂
⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠀⠀⠄⠀⠀⠀⠀⠘⡀⠀⠀⠄⠀⠀⠀⠀⠆⢀⠠⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⠀⠒⠂⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⡄⠂⡄⠀⠀⠂⠀⠄⢦⡀⡀⠄⠄
⠀⠀⠀⠡⡳⠀⠀⠠⠐⠩⠀⠀⠀⠀⠀⢃⠀⡀⣀⡈⠎⠀⠀⠀⠈⠒⠜⠄⠤⠊⠀⠀⠀⠐⠆⢠⠃⠠⠂⠠


⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⠀⠁⠸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⠄⠐⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠩⠤⠊⠄⠈⠀⠀⠀⠈⠂⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠔⠤⠀⠂⠁⠂⠀⠀⠀⠀⠀⠂⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢋⡏⡄⠀⠔⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡰⠁⠀⠐⠀⢠⠇⠀⡡⢢⡄⢀⡴⢊⠭⢍⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡴⢡⠀⡇⡆⡆⡿⠀⠸⣇⠀⠀⠘⠀⣇⣶⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣇⠰⡀⢧⢸⠸⣿⡀⢆⠉⠳⣄⠐⡄⠻⣝⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠁⠈⠢⣳⡈⢆⢣⠻⣿⣌⢧⢀⠘⣷⠸⡄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠪⡅⠀⠀⠉⡡⡄⢱⣌⢿⣾⣷⣧⠹⣿⠘⣾⣦⣿⠇⡇⣈⠀⢸⡆⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣺⢩⣛⢦⣿⣿⣿⣶⣿⢀⣿⡿⣫⠞⠁⡡⢀⣼⠃⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠠⡸⠖⠀⢸⠂⠹⣿⣿⣿⣇⣾⣯⡟⢁⣴⣿⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⢀⠀⠀⢤⠀⢿⡄⢆⠹⣯⣿⣿⣿⣿⣷⣿⣴⣆⡀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣋⠍⢹⠒⡟⣰⠺⢿⣶⣷⣿⣿⠟⠛⠋⠉⠉⠁⠱⠉⠃⠙⠒⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠿⠃⣼⠀⠀⠁⠐⣠⠼⠛⠁⠁⠀⠠⠀⠀⠀⠰⢆⠀⡀⠀⣼⢡⢺⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⠶⠶⣧⠤⢤⣞⣭⣀⡀⠀⠀⠀⢀⡴⣋⠀⢤⠶⢦⠶⢾⣧⣷⡘⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡗⠀⣤⡤⡀⢀⢸⡀⢀⢮⣹⡆⠀⣨⣾⠛⢿⣃⡏⠠⠄⠀⣼⣿⡍⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢉⡇⢀⡛⡇⠀⣀⣸⣧⣐⣊⣿⢷⣿⡁⠠⠀⠘⣿⡷⢦⠤⠼⠋⢙⡃⢁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠲⠶⢒⣻⠿⢛⠅⠈⢛⢽⠞⠻⢿⣴⣶⠟⠃⠐⠀⠁⠀⠃⠀⠀⠴⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⢠⡀⠀⠀⢀⣴⢟⠁⠀⠐⠂⠀⠀⠀⠠⠛⢾⠇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⢯⠀⠀⠀⠈⠀⠀⠀⢹⡏⠀⠀⠈⠁⠀⠐⠉⠀⠀⢀⡅⠀⠀⣀⣼⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣶⡀⠀⠀⣠⠃⠀⠀⠀⠂⠀⠒⢋⠀⠀⠀⠀⠲⠤⠆⠀⠀⠀⡁⠀⠀⢸⣟⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣶⣿⣿⣿⣇⠀⠀⠀⠀⠀⠘⣋⢶⣆⡀⠀⠀⠀⠀⠀⠀⠙⣤⣾⡟⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣿⣿⡿⣿⣿⣷⣿⡶⠀⠐⢶⣿⣾⣏⠙⠳⠀⢾⠶⠀⣠⣤⣼⣿⡋⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢿⣿⣷⣾⣿⣿⣿⣃⣀⠀⠘⠟⠿⠇⠀⠶⠀⠀⠀⠀⣨⣿⣿⣿⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⠿⢛⣯⣭⣭⣽⣿⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣍⣛⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣉⣉⣁⢀⣿⣿⡿⠛⠿⠿⠻⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠶⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣼⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⣤⣴⡿⢛⣽⣿⣇⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⡄⠀⡀⠀⠀⢀⣄⠀⠀⢺⣴⡄⣿⠿⠛⢀⣴⣾⣿⡿⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⡆⠛⠺⠗⠛⠻⠟⠀⢘⣋⣡⣤⣶⣾⣿⣿⣿⣿⣧⣾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⡏⢙⠿⣿⣿⡿⢿⠋⠁⠀⣼⣿⠀⠄⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⠿⣷⣿⣧⣿⣶⢿⠁⢰⠀⠀⣄⢰⣿⣿⣷⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠉⠻⡇⠀⠀⢹⣿⡟⠁⠀⠀⠀⠘⣄⡀⣿⣾⣿⣿⡋⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢁⠀⢀⡀⢿⣿⠀⣤⣀⠀⢰⣿⣿⣿⣿⣿⡟⠁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠿⠛⠺⢿⡇⠈⠁⠀⠘⣿⣿⣿⣿⠏⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢡⡴⠀⠀⠀⠀⢿⣿⣿⡏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢠⣴⡇⢸⣿⠀⢈⣿⠀⠈⢿⣿⢧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠘⠛⠃⡸⢏⠀⠉⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣾⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⣼⣿⣇⣸⡿⢀⣾⡷⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣇⠀⠀⠙⣷⣌⠁⠀⢰⡀⠀⠀⣀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⢿⣤⣤⠞⢿⡏⠓⣶⣾⣧⢠⣌⣙⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢓⣘⣛⠓⢠⣾⣷⣬⣿⣯⣭⣬⣥⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⢿⣸⠿⠿⢿⡿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⢀⣀⠀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⡀⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣷⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⢠⣿⠁⠀⠀⢀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⢰⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⢀⣾⠃⡀⠀⣴⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡏⠀⠀⢠⡇⠀⠀⠀⣰⣿⣿⠏⠀⢠⠀⣼⡟⢀⠃⣼⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⣠⣿⠀⠀⢀⣼⣿⣿⡟⠀⠀⡄⢰⣿⡇⢸⠠⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⣿⣿⣿⠀⢀⣼⣿⠃⠀⣴⣿⣿⣿⡟⠀⠀⢰⡇⢸⣿⠇⣾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⣿⣿⣿⡇⢠⣾⣿⠏⢠⣾⣿⣿⣿⠏⠀⠀⠀⣾⠃⢸⠟⣰⢇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢰⠃⠀⠀⠀⣰⣿⠟⣿⣿⣇⡞⣿⡟⣴⣿⣿⣿⡿⠁⣰⢣⠆⣼⠟⠀⠈⣰⢏⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢺⠀⠀⠀⢠⣿⠃⠀⣿⠟⣿⢀⣿⣿⣿⠋⣼⡟⠀⢰⣷⣿⣾⠏⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⠀⣻⡄⠀⠀⣼⡇⠀⠀⡇⠀⡏⢸⣿⡿⠃⣼⣿⢡⢇⣿⣿⣿⠏⢀⣾⣿⠟⠁⠀⣠⡾⠀⠀⠀⢸⡇
⠀⠀⢸⠀⢿⣿⠀⠀⣿⠃⢰⣴⠇⠀⠀⢸⣿⠁⢰⣿⡏⣾⢸⣿⣿⣿⢠⣿⡿⠋⢀⣴⣿⡿⠃⠀⠀⠀⢸⡇
⠀⠀⢸⡄⠈⢿⡇⠀⣿⠀⣼⣿⠀⠀⠀⣼⡇⠀⣼⣿⢧⣿⢸⣿⣿⣿⣾⣿⠃⢠⣿⡫⠋⣠⡜⣡⠆⠀⢸⡇
⠀⠀⢸⣇⢠⣞⣿⠀⠁⢀⣿⡟⠀⠀⠀⣿⠀⠀⣿⢻⢸⣿⣸⣿⣿⣿⣿⡇⢠⡿⠁⣤⣾⡿⠛⠁⠀⠀⢸⡇
⠀⠀⠸⣿⣆⣿⣿⠀⠀⣸⣿⠃⠀⠀⠀⠉⠀⢀⣯⡏⣿⣿⢹⣿⡇⣿⣿⡇⢼⠁⣾⡿⡋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⣿⣿⣿⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⢱⣿⡟⣼⣿⡇⣿⣿⡇⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⣿⣿⢠⣾⠏⠀⠀⡆⠀⠀⠀⣰⣿⢇⠏⡿⣹⣿⣿⣾⣿⡿⠁⣼⡿⠋⠀⠀⠀⠀⣠⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣿⡇⣼⠏⢠⠃⣴⡇⠀⠀⢠⣿⡏⢀⠀⣽⢿⣿⣿⠿⠋⡀⠞⠉⠀⠀⢀⣀⣴⡾⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠋⣼⠏⢠⠇⣰⣿⠃⠀⢀⣾⠏⢀⣾⡼⣿⣿⠟⠁⢀⡠⠔⠀⣠⣴⠾⠛⢡⡞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡏⢀⣾⣴⣿⠏⠀⢀⣾⠃⠀⣼⡇⣿⡿⠁⢀⣴⠋⠀⣠⡾⠋⠀⣰⣾⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡿⠀⣸⣿⣿⠏⠀⠀⡾⠁⠀⣰⣿⣧⢹⠇⢠⣿⡏⠀⠞⠁⠀⣠⣾⣿⣿⠃⣸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠘⠁⠀⣰⢿⣿⡏⣻⢠⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⠏⢀⢻⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡴⢋⣾⣿⢳⣿⣾⣿⠃⠀⠀⢀⣾⡿⣭⣿⠏⠀⠎⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡜⣡⣿⣿⣢⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⡿⠃⠀⠜⢀⡾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢈⣼⣿⣿⣿⠿⠋⠁⠀⠀⠀⣰⣿⣿⠟⠋⠀⣠⣎⣤⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠀⢀⠞⣵⠿⠋⠁⠀⣀⡀⠀⣠⠾⠟⠋⢁⣀⣤⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣶⠀⠀⢂⠞⠁⠀⣠⣶⠿⠋⠛⢋⣠⣤⣶⣾⣿⣿⣿⣿⣿⣻⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⢀⠀⣠⠞⢉⠀⠀⡠⢚⣩⣭⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⡿⠃⠀⢀⡴⠁⠔⣡⡖⠁⠀⢀⣴⡿⠿⠿⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣿⠋⢀⣤⣶⠟⠁⣠⣾⠋⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠜⢁⣴⡿⠟⠁⠠⠾⠋⠁⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠄⠀⠠⠤⠠⠄⠼⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠄⠀⠀⠘⢌⠢⠀⠀⠰⠀⠠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠊⠀⠘⡠⠀⠀⠀⠠⠑⠀⠀⠀⠐⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠂⠀⠀⠀⠀⠀⣄⠀⠀⡰⠁⠀⢹⠀⠀⠀⣀⠀⠈⠀⠀⠀⠄⠁⠄⠘⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠐⢀⠁⠳⠉⢦⣠⠁⠀⠀⢸⠀⠲⢀⡶⢱⠀⢀⠀⠸⠎⠀⠀⡄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⡉⠀⠀⠹⡀⠀⠀⠸⠀⠀⠰⠁⠀⡇⠈⠴⠃⠐⡆⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢾⠣⣄⣠⠀⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠀⡇⠂⠀⠀⢠⠃⠀⠀⣀⢀⡀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠀⢇⠀⠀⢠⢃⣄⣀⣤⡔⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡞⢈⠼⠊⢰⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠙⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠀⠀⠀⢀⠔⢠⠀⠀⠱⠁⠀⠰⠁⡾⠁⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣄⣅⣀⠀⠀⠀⠀⠀⠀⠃⠄⠂⠖⠄⠀⡀⠈⠠⠄⠂⠀⠀⠠⠴⠋⠉
⠀⠀⠀⠀⠀⠀⢐⡖⣻⣉⣉⠉⣩⣽⠶⣀⠀⠀⣀⣠⠤⢤⣀⡄⡦⠀⡠⢦⠐⠠⣤⢄⠂⠠⠾⠿⢷⣿⡂⡀⢰
⠀⠀⠀⠀⠀⠀⠹⠷⠼⠷⠶⠚⢧⡇⢠⣾⣶⣛⡻⣄⣵⢠⣼⠃⣓⢲⠀⢘⣧⡂⠚⢺⣇⣀⡀⡶⡞⣮⣭⠴⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠙⠛⠛⠁⠀⠙⠶⠖⠛⠉⠙⠳⠚⠋⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠉⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠀⠘⢃⣴⣶⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠻⣿⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠤⠀⢧⠀⠀⠀⠀⠀⠀⢹⠃⣸⠇⠤⠤⠀⠀⢀⡀
⠀⠀⠀⢀⡠⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⢀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢤⡀
⠀⠀⠀⢿⣀⣀⣀⣤⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠙⠛⠶⢶⣤⠤⣤⣀⣀⣀⡀⠀⠀⢀⣿
⠀⠀⠀⠀⢈⠙⠿⠿⢥⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢒⣬⣍⡽⠿⣿⣿⡉⢠
⠀⠀⢸⠀⢸⢰⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠓⠒⠒⠒⠒⠒⠛⠛⠉⠉⠉⠉⠉⠀⠸⠀⠀⢻⣿⡇⢠
⠀⠀⠘⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⡞⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⣼⣿⡇⢀⡆
⠀⠘⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡾⠃
⠀⠀⠀⠉⣿⣿⣶⣶⣤⣤⣄⣀⣀⣀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⣀⣀⣤⣤⣴⣶⠿⣟⣿⣿⠁
⠀⠀⠀⡰⠛⢿⣿⣶⣾⣭⣭⣽⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣫⣭⣭⣵⣶⣾⣿⡿⠟⠉⢂
⠀⠀⠰⠁⢀⡄⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⢠⡀⠀⠀⣆
⠀⢀⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣇⠀⢳⣄⠀⢸⡄
⠀⡘⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠀⠀⠘⣿⡄⠀⢻⣆⠀⣷
⠀⠇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢹⣇⠀⢸⣿⠀⢹
⢰⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⡄⠸
⠸⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⡆⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿⡇
⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⡇
⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡇
⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⡇⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡇
⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⢠
⢸⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⡄⠀⠸⣇⠀⠀⠀⡀⠀⠤⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠂⠀⠤⠤⠀⣀⠀⠀⠀⢀⣿⠃⡾
⠀⢡⠀⠀⠙⢆⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⡏⣸⠃
⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⢋⣼⠏
⠀⠀⠀⠈⠻⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠲⠭⣉⡛⠛⠳⠶⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠶⠶⠒⢚⡋⠭⠓⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣒⣒⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣒⣒⣂⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⡿⠋⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠻⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⣤⣤⣤⡤⣤⢤⡴⡶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣶⣶⣦⣦⣤⣀⡀
⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠻⠿⠿⠯⠿⠽⢿⣧⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠘⠿⠿⠿⠽⠮⠧⣿⣽⡿⡿⠿⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣄⣀
⠀⠀⠀⣤⣤⣤⣶⣶⣶⣶⣶⣆⣶⣶⣿⣿⣿⣿⣿⣿⣿⢸⣻⣿⣿⢻⡟⣿⢹⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢿⣿⣿⢸⡇⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢻⣿⣿⢸⡇⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⢸⣿⣿⢸⡇⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣻⣿⣟⣿⣿⣸⡿⣿⣿⢿⣿⣿⣿⣿⣿⢸⣸⡿⣿⢸⡇⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠛⠛⠛⠛⠓⠛⠘⠺⠷⠿⠾⠷⠿⠮⣿⣽⡿⡿⠿⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢰⣿⡇
⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣸⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⣿⠄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⣿⡿⠁⠀⠀⣸⣄⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣿⣿⡟⠇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣠⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢠⡀⠹⣿⣿⣾⣿⣆⠀⠀⠀⢀⠀⠀⠀⣼⣇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⠏⠀⠀⠀⣄
⠀⠀⢀⡀⠀⠀⣸⡗⠀⠙⢿⣿⣿⣿⡆⠀⠀⠘⣧⠀⢸⣿⣿⣶⣤⣴⣿⣿⣿⠀⠀⢻⣦⡀⠀⠀⠙⢿⣿⣿⣿⠟⢰⣿⣿⣿⣿⣿⣆⡀⠀⣰⣿
⠀⠀⠀⢻⣷⣼⡟⠀⢸⣆⣨⣿⣿⣿⡟⠀⢀⣼⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⣿⣿⡆⠀⠀⠀⠙⠿⢣⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⡆⠀⠀⣿⡿⠀⠀⠈⢿⣿⣿⣿⣿⢃⣴⣿⡟⠁⢀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣾⣿⣶⣾⣿⡇⠈⣷⡄⠀⠛⢿⡿⢡⣿⣿⣿⣧⣤⣼⡧⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠙⣶⡘⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀⢸⣇⢿⣿⣿⣿⣿⣿⢃⣶⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠈⠹⣿⣿⣿⣿⣿⠏⠀⠀⢸⣯⣿⣿⣿⣿⣿⡿⠁⣾⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣤⣾⣿⣿⣿⠏⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠙⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⣷⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠉⠉⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⣄⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣤
⠀⠀⠀⠀⠀⠀⣾⣥⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣦⣌⣃
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⠉⠉⣿⡏⠉⢹⣿⠉⢹⣿⡏⠉⣿⣯⠉⢹⣿⠉⢹⣿⠉⠉⣿⡏⠉⢹⣿⠉⢹⣿⡍⠉⣿⡏⠉⢹⣿⠉⢹⣿⠉⠉⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⣇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⡇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⣏⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⡇⠀⢸⣿⠀⢸⣿⠀⠘⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⡇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⡇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⡇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⣿⡇⠀⢸⣿⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⣶⣶⣿⣷⣶⣾⣿⣶⣾⣿⣷⣶⣿⣿⣶⣾⣿⣶⣾⣿⣶⣶⣿⣷⣶⣾⣿⣶⣾⣿⣷⣶⣿⣷⣶⣾⣿⣶⣾⣿⣶⣶⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡴⠲⡄
⠀⠀⠀⢰⣊⣉⡉⡦⠀⠀⠀⠀⠀⢸⠓⠒⠒⠒⡻⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⠀⠀⠀⠀⠀⠀⢠⠛⠙⢦
⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⢤⡤⠔⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⢨
⠀⠀⠀⢸⣀⡀⣀⡇⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡍⠉⠉⢸
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠲⠖⠋⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠐⠤⠤⠞
⠀⠀⠀⠀⣤⣒⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢸⠉⣏⠀⠀⠀⠀⠀⠀⡴⣾⣭⣛⠽⢛⡆⠀⠀⠀⣴⣒⣉⣉⣑⣲⡀⠀⠀⠀⠀⢠⣖⣦⣶⡶⣄
⠀⠀⠀⢠⠁⠀⠀⣆⠀⠀⠀⠀⠀⡇⠒⠚⡄⠂⠈⡇⠀⠀⠀⠘⡄⠀⠀⠀⢀⠯⡆⠀⠀⠀⡞⢺⡖⠓⠉⢽
⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⣀⠀⡇⠠⢐⠇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠟⠁⠀⠀⠀⡇⢸⠇⣀⣤⠼
⠀⠀⠀⢠⣀⠀⣀⡼⠀⠀⠀⠀⠀⠛⠶⠽⠧⠊⠁⠀⠀⠀⠀⠀⠓⠤⠤⠄⠊⠀⠀⠀⠀⠀⠈⠚⠋⠁
⠀⠀⠀⠀⠈⣉⠉
⠀⠀⠀⢀⡸⠷⠿⣀⠀⠀⠀⠀⠀⢰⢍⡈⢉⣉⢷⠀⠀⠀⠀⢀⡤⠤⠤⠤⣄⠀⠀⠀⠀⢀⣴⣖⠒⠒⢒⣶⣄
⠀⠀⢰⠁⠀⠀⠀⠈⢷⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠙⠒⠒⠒⢻⠟⡆⠀⠀⠀⢻⠿⠿⠿⠿⠛⡇
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢣⠀⠀⠰⠁⠀⠀⠀⠀⡼⢦⣀⣀⣠⠾⡏⠀⠀⠀⠀⠸⡈⠐⠀⠀⣼⠃
⠀⠀⠸⡄⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⡀⢀⡄⠀⠀⠀⠀⠀⠙⠒⠲⠷⠒⠚⠁⠀⠀⠀⠀⠀⠳⢤⣀⡠⠇
⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⢀⢤⣶⡶⠶⣶⢢⡀⠀⠀⠀⠀⠀⢰⣊⡳⠀⠀⠀⠀⠀⠀⠀⠀⢺⢿⠂
⠀⡾⠯⣤⣄⣀⣀⣈⣸⢷⠀⠀⠘⡿⠭⠭⠭⠽⡟⠁⠀⠀⠀⠀⢀⠎⠉⠙⡄⠀⠀⠀⠀⠀⠀⠀⠸⠸
⠀⢻⠲⠤⣤⣤⡥⠤⠴⡺⠀⠀⠀⢰⠀⡖⠉⠑⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣐⠁⠀⢱
⠀⠈⢭⣐⢒⠒⢒⢘⣩⠃⠀⠀⠀⠸⡀⠓⠤⢰⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠊⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⠑⠦⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⢠
⠀⠐⠈⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣦⣥⣽⣥⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡤⠴⠶⠶⠶⠶⠶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠎⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣰⠟⠛⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣿⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠙⠛⠶⢶⣤⣤⣤⣤⣶⠶⠛⠋⠀⠄⠘⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⣀⣀⣠⣶⣦⣀⣀⠀⠀⠀⠂⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠫⢌⠛⢤⠞⡉⠝⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠋⠀⠀⣠⠤⣿⣴⣽⡧⢦⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠀⣌⠛⣉⠔⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢾⡙⠻⣿⡾⠛⡻⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣤⣶⣊⡴⣆⣤⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣙⠲⢦⣬⣑⠆⢘⣀⣀⣀⣄⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣝⠉⠁⠐⠛⠛⠛⠓⠀⢙⡫⡍⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢶⠶⠶⣶⠶⢶⣶⠶⠶⡶⠶⢶⠶⢶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣿⠀⢸⣿⠀⢸⡇⠀⣿⠀⢸⣿⢸⡟⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣿⠀⢸⣿⠀⢸⡇⠀⣿⠀⢸⣿⢸⡇⣿⡏⣿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣿⠀⢸⣿⠀⠸⡇⠀⣿⠀⢸⣿⢸⡇⣿⡇⣿⢸⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣿⠀⢸⣿⠀⢀⡀⠀⣸⠀⢸⣿⢸⡇⣿⡇⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⣤⣿⡤⢸⣿⠀⣼⣧⣤⣻⠀⢸⣿⢸⡇⣿⡇⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣿⠀⢸⣿⠀⢸⡇⠀⣿⠀⢸⣿⢸⡇⣿⡇⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣿⠀⢸⣿⠀⠸⡇⠀⢿⠀⢸⣿⢸⡇⣿⡇⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡁⢨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⣿⣧⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡟⢿⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣏⣸⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣅⠀⠈⠉⠲⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠛⠻⠿⢿⣿⣿⣇⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠉⡿⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠐⣧⣧⠤⠶⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣧⡀⠀⠀⠀⣿⠿⡄⡃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣄⣠⠀⣄⢀⠀⠀⠸⣿⣿⠁⠀⠀⣸⠯⢾⠗⣷⣤⡤⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣿⣿⣀⣀⣴⣿⠀⠝⠀⠀⠀⠀⢈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⡿⣿⣿⠇⢸⠃⠀⢀⣰⠃⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣆⣈⣠⣴⣸⣿⢠⣀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⢿⣯⡿⠟⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠻⠿⠿⠿⠛⠻⢿⣿⣿⣿⡧⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⣀⣠⣄⣠⣤⣶⣶⣄⢹⣤⣿⡇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠞⠋⠁

⠀⠀⠀⠀⠀⢀⣴⡞
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄
⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⢸⣿⡟⣧⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⠀⠀⢀⠀⢸⡇⠀⢸⠀⢱⢰⣿⣇
⠀⠀⠀⠀⠀⡜⠉⢠⣸⣇⠀⢀⣼⣇⠀⠀⢀⣿⣿⡀⠀⡘⢦⣈⣿⡄⢸⣧⣾⣿⣯⠻⡄⠀⠀⢱
⠀⠀⠀⠀⠈⠀⠀⠸⣿⠙⠳⡾⠉⣿⣦⣴⠟⢻⡿⢻⣦⡇⢸⡟⢹⢇⡞⢿⣯⡾⠋⠀⢿⣾⠀⠚⠀⢠⡀
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠰⡀⢻⡇⠉⠀⠘⠇⠀⠀⠁⡞⠀⠀⠊⠀⠊⠀⠀⠀⠀⠈⠟⠁⠀⠀⢼⠁
⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡟⡆⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣦⢸⢸⠿⡆⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢰⢿⠇⡇⣴⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣿⠀⡼⠹⠘⢸⠀⣷⠀⠀⠀⠀⣶⡰⣭⡆⠀⠀⠸⠈⡇⣇⠏⢣⠀⣿⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⣡⣾⢸⣧⣙⣡⣧⠸⣤⣿⠀⠀⠀⣰⣿⣷⣜⠃⠀⠀⣷⡆⡇⣼⣌⣋⣼⠉⣧⣜⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣼⡟⣷⢧⡄⡇⢹⣥⣨⢣⣿⣾⠀⢀⢴⣿⡏⣿⢻⡦⡀⠀⣷⣿⡜⣅⣬⠋⢸⢰⡹⡿⢿⣧⢿⡇
⠀⠀⠀⠀⢠⡄⠀⢸⣦⣹⣷⠏⣿⠀⣿⣷⢻⡏⣿⠇⡿⢀⠎⣼⢹⡇⣿⢼⢃⣵⡀⢿⠸⣳⢹⡏⣾⣿⠘⣷⠻⣾⣿⢬⡇⠀⢠⡄
⠀⠀⠀⢠⡿⠀⠀⢸⣟⣹⡏⢆⡏⢸⡏⡏⡌⠑⡎⡀⡇⢸⡟⣿⠘⣇⢹⣧⣸⣿⡇⢸⢀⣿⠸⢡⣿⢻⡆⢣⢁⢹⣏⣾⣇⠀⠀⢿⡄
⠀⠀⢠⠏⣧⠀⣠⠇⡻⣿⣱⡟⣾⢸⡇⢣⢻⣄⠁⡇⢿⡜⣧⡇⣶⢻⡀⣿⣿⣼⣣⡟⣼⠈⣰⡟⡏⢸⡇⣧⢿⡆⣿⣿⠹⣄⠀⡸⢹⡄
⠀⠀⢸⢠⡘⣿⣡⣞⢠⡿⠿⢹⢿⢸⡇⣼⣆⠛⠆⢿⢸⣙⣿⢡⢿⣾⡇⢹⡇⣿⠉⡇⡿⠸⠋⣰⢧⢸⡇⣿⠘⠃⢿⡈⣣⣎⣿⢃⡌⡇
⠀⢠⣼⢼⡇⣹⡶⡽⣾⣷⣇⢸⡿⢸⡇⡟⡿⡜⡟⡌⣼⣿⢃⡿⢸⢸⣇⣼⡇⣏⠿⣏⢃⢿⢱⢿⢸⢸⡆⢿⠀⣼⣾⣇⢯⢶⣏⣼⢱⣧⡄
⢀⡾⢻⡏⣱⠏⣀⡰⣻⢿⡟⣧⢿⣸⠃⠀⠁⡇⡷⣱⢋⣿⣏⣷⡼⢸⡿⣿⢱⢟⣄⠹⣎⢾⢸⠈⢀⠘⡇⠿⣾⢹⡟⣿⣆⣀⠻⣟⢸⡟⣿
⣼⡷⣾⡿⣱⢇⣿⠾⣇⣾⠣⢻⣄⡿⢸⣧⢠⠃⣼⠃⢸⣿⡜⠘⢁⡾⢁⡏⡿⣾⠙⣦⡹⣦⢸⣠⣼⡆⣿⢰⡟⠸⣧⣼⠧⣿⡼⣞⢿⣗⢾
⡏⢻⣿⢣⢻⡘⣯⠘⡿⢣⡶⣸⣿⠇⣦⣿⢸⡾⡏⣶⠸⢀⡖⣸⢹⣧⣾⡗⣧⢣⡆⣼⢣⣿⢿⡇⣿⣰⠸⣿⣇⢶⡘⢯⠃⣽⢣⡟⡜⣿⡞
⢷⡟⢸⡸⠞⠄⣩⢞⣡⡞⣰⠿⡏⢸⣇⢻⠘⠀⡇⠙⠘⠏⡟⢉⠘⠛⡏⢣⠉⠘⠃⠛⠈⣿⠀⠁⡇⢸⡇⢻⠻⣦⠷⣌⠳⡍⢠⠳⢷⡇⣷
⠈⢿⡈⢧⠻⣷⣯⣞⣠⡾⢃⢸⠁⢸⡘⡏⣴⡄⣿⠀⡆⠀⢰⠞⠀⠀⠀⠈⠟⣧⠐⠑⠀⣿⠀⣆⢱⢙⡇⠘⡇⡘⣷⣌⢳⣽⣦⢟⡞⢀⡿
⠀⢬⣳⡌⢳⣿⢋⡞⣿⢣⣿⢸⣸⣇⠻⢠⡏⠀⡿⡄⠀⢸⡄⢶⠀⠀⠀⠳⠀⢹⡄⠎⣸⢿⠀⢹⡌⠟⣼⣇⡇⣿⡘⣟⢿⡙⣷⡞⢠⣞⡤
⠀⢸⣿⠇⢾⡻⣟⠠⣿⢸⣻⠸⣿⠘⡆⢸⢰⢰⡇⢰⣸⡿⡇⣾⢷⡆⢾⡆⢸⠂⣷⢰⡇⠸⠀⣌⡇⢸⠁⣿⢣⣟⡇⣿⢤⣿⢿⡁⣸⣿⡇
⠀⢸⡏⢧⡾⡁⣧⠃⣿⣸⢻⡇⢡⡄⣷⢸⢃⡟⢰⢸⡿⣦⢃⡟⠸⣳⢢⣿⠸⠁⡘⣿⢳⡄⢷⡘⡇⣿⢰⡅⢸⣿⡇⣿⠘⣾⢈⢧⡽⢹⡇
⠀⢸⣷⡜⢱⢃⣿⢷⡟⣿⡟⡴⢛⠃⣿⣴⡟⣠⢿⢸⡇⡿⣸⣇⢠⣏⡎⣿⣴⢸⡇⣿⡞⢳⣌⢻⣦⣿⠸⡛⢆⢻⡇⢿⡾⣿⡸⡎⣡⣾⡇
⠀⢸⡟⣸⠀⣼⣿⢀⡁⣿⢸⢱⡏⢸⢫⢏⣾⡟⡘⣼⠉⣵⡿⡏⣾⢏⡗⢹⡅⣼⠇⡿⢧⠎⣿⣦⡹⡝⡆⢹⡞⡇⣿⢈⡀⡿⣧⠀⣇⢻⡇
⠀⠀⢧⠻⢸⠇⣿⢘⢣⠈⣤⣾⠃⡾⠃⣎⢸⢱⢣⡇⣼⠹⡇⢳⡿⢠⢗⡟⡼⠁⣸⣧⠸⡜⡜⢂⢳⠘⣷⠘⣧⣥⢁⡞⡃⣿⢸⡆⢏⡾
⠀⠀⠈⢷⣼⡆⣿⠸⡾⣇⢿⡘⣸⢃⠸⡹⠘⣿⢸⡿⣏⣆⢳⡀⣷⠺⣬⢡⣽⣼⠋⣿⢿⡇⣿⠸⢾⠀⡘⣇⢇⡟⣸⣣⢇⣿⢸⣧⡾⠁
⠀⠀⠀⠀⠉⠃⠈⢧⡹⣿⣮⢡⡏⣷⡀⣧⡄⢻⠘⠁⣿⢙⣦⠳⣌⠷⠏⢣⡿⣱⡏⣿⠘⢠⡇⢠⡼⡄⣧⢻⡈⡿⣿⢋⡾⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠐⣦⡻⣍⣮⣾⡇⢿⣿⠘⠇⢸⠀⣷⠸⡎⣿⡆⠹⣷⣴⠏⢐⣯⣹⠇⣧⢸⡇⡞⠃⣿⡹⢸⣧⣽⣱⠟
⠀⠀⠀⠀⠀⠀⠀⢫⣳⡹⡏⢿⠀⡟⢸⣠⢰⣾⢰⡏⡄⣿⠘⣷⠀⢻⡏⢺⣘⠃⣿⢰⢹⠸⣇⠇⣄⠃⣷⢸⡇⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣥⢻⣾⡄⡇⣾⡿⡄⢿⡘⣧⡇⣿⠃⡁⣾⢸⣷⠸⡏⢸⣿⢾⣸⢰⡟⢸⣿⢰⣶⢸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣏⣣⢷⠉⡇⢷⢰⣄⠻⠇⣿⢠⣿⡏⣾⠛⣧⢻⢹⣿⠸⠏⣨⡄⣷⢸⢀⡏⣜⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠳⣏⢸⠸⣘⢧⠀⣿⠸⢹⣧⡟⢀⢹⣦⡟⣿⠀⠾⢹⠀⡇⣸⠟⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡻⠄⢿⡀⡆⣿⡼⣼⢹⡇⡿⢸⣿⡕⡿⢸⣠⡟⠰⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣍⣃⠘⣷⡉⢾⡕⣣⡟⣩⣼⢃⣉⣭⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣾⣷⡟⢞⣵⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡶⠶⠾⠛⠛⢋⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠾⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢠⡞⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠟⠛⠋⠉⠀⠀⠀⠀⢀⣤⣄⣤⣀⣤⣴⠒⢶⡟⠻⣏⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠀⠀⣀⡀⢀⣠⣤⣤⣄⡾⠛⣿⠀⠘⣧⠙⣧⣿⡀⠘⣷⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⣤⣤⣾⠉⠙⣿⡅⠈⢻⡀⢻⡀⠸⣆⠀⣿⣀⣿⣼⣷⠾⡟⣟⢫⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠸⣇⠈⣿⡀⠀⢹⣷⡀⣸⣇⣼⠻⡾⡋⡟⢻⣏⢣⢻⢸⡇⢳⢹⡸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢿⡀⣸⣷⣤⣾⡛⣻⠻⠘⣸⡄⢷⢡⠸⡀⢿⡘⣜⣇⣷⣼⣶⣷⠷⠾⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣟⢻⣿⣷⠸⣇⢹⡆⢃⣁⣷⣼⣾⡶⠿⠿⠛⠛⠋⠉⠉⠀⠀⣀⣀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⡈⣟⣿⣦⣿⠾⠟⠛⠉⠉⠁⠀⠀⢀⣀⣠⣤⠴⠖⠚⠛⠉⠉⠉⣇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣸⣟⠋⠉⠀⠀⠀⣀⣀⣤⠴⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣸⠋⢻⡀⠀⢰⡞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⢏⠠⢸⣇⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠆⠁⠀⢹⡄⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⡜⣧⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢤⡀⡀⢹⡄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠁⠀⣷⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⣄⡀⠸⡆⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠁⠀⣿⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⡀⠘⠸⣇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢁⠀⢻⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠤⣎⠀⠈⣧⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠖⠛⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠈⠈⠀⢹⡀⠈⣧⠀⠀⠀⠀⢀⣀⣠⠤⠶⠒⠛⠉⢉⣁⣠⣤⣴⠶⣾⠟⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠁⠸⣇⠀⠹⠶⠖⠛⠋⢉⣁⣠⣤⣴⡶⠾⠛⠛⠉⠁⠀⠀⠀⢹⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⢸⣄⣠⣤⡶⠶⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠄⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠎⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣰⡏⠀⠀⣀⣠⣤⣶⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠾⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣠⣭⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣯⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠊⣰⣏⣀⣀⣀⠀⣀⣠⡀⠀⢠⠎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢠⣿⣿⣿⣶⣿⣿⣷⣼⣿⡿⠃⢬⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣥⣤⣯⣿⣿⣿⣭⣿⣿⣿⣭⣿⣥⣬⣭⣿⣯⣽⣿⣿⣤⣽⣽⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣼⠟⣿⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢠⣾⣿⡄⣿⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⣰⡟⣱⣟⡇⣿⡇⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠂⢀⣼⠟⣰⣿⣿⣷⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠔⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠄⠀⠘⠀⠀⠀⠀⠀⠀⡐⠁⣠⣿⢋⣼⣿⣿⢿⠏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠠⠂⠀⠀⠠⠀⠌⠠⠅⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣡⡾⢼⡿⣻⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣴⠟⣴⣿⣿⣿⡿⣳⠏
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⡾⢣⣾⣿⣯⣿⡟⡽⠃
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣠⡟⣵⣿⣿⣿⣽⣿⡞⠁
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣰⢏⣾⣿⣿⢿⡿⣵⠋
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⢀⡼⢣⣿⣿⡿⣫⢟⡾⠁
⠀⠀⣾⠛⠛⠛⠛⠛⢛⡟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢛⠛⣻⣥⣿⣿⣺⡽⢣⡞
⠀⠀⣿⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣡⠏
⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⢟⡴⠋
⠀⠀⣿⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣣⠞⠁
⠀⠀⢻⡔⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣹⣷⠋
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⢸⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⡰⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⣼⡿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⣼⡿⠁⡀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⣠⡞⠀⠀⠀⠀⣴⣿⣿⡿⠁⠀⢀⠀⣸⣿⡃⣸⠁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⢀⣴⣿⠁⠀⠀⣠⣾⣿⣿⡿⠁⠀⢠⡄⢰⣿⣿⠃⡿⢰⠋
⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠁⠀⣰⣿⣿⠃⠀⣠⣾⣿⣿⣿⡿⢁⠀⠀⣸⡇⢸⣿⡿⢀⠳
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⢀⣼⣿⣿⠃⣠⣾⣿⣿⣿⣿⠋⢀⠂⠀⢠⣿⠁⢨⡿⢁⡿⣈
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣼⣿⡿⠙⡿⣿⠇⣼⢡⣿⢇⣴⣿⡿⣿⣿⠟⠁⣰⢋⡞⣠⣾⠋⠀⠈⣠⣟⣼⠃
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢰⣿⠏⠀⢰⡇⣿⢸⠇⣸⣿⣾⣿⠋⣴⣿⠏⡌⣰⣧⣞⣴⡿⠁⠀⣠⣾⣿⡞⠁
⠀⠀⠀⡄⠀⣿⣆⠀⠀⠀⣿⡏⠀⠀⢸⠃⠋⢸⠀⣿⣿⣿⠁⣸⣿⡏⣼⢠⣿⣿⣿⡿⠁⣠⣾⣿⡟⠉⠀⢀⣤⡾
⠀⠀⠀⡇⠀⢿⣿⡄⠀⠀⣿⠁⢠⡇⣾⠀⠀⠈⢀⣿⡟⠀⢠⣿⣿⢰⡏⢸⣿⣿⣿⠁⣼⣿⡿⠋⢀⣠⣶⣿⡟⠁⠠
⠀⠀⠀⣧⠀⡘⣿⣷⠀⠀⡿⠀⣸⣿⡟⠀⠀⠀⢸⣿⠀⠀⣼⣿⡇⣾⣇⢸⣿⣿⡿⣸⣿⡿⠁⢀⣾⣿⠟⠋⣠⠶⣡⠔⠁
⠀⠀⠀⣿⡄⠱⣍⢿⡆⠀⡇⢀⣿⣿⠇⠀⠀⠀⢸⡇⠀⠀⣿⢩⢣⣿⣿⢸⣿⡟⣿⣿⣿⠁⢠⡿⠋⣠⣴⣿⡷⠞⠁
⠀⠀⠀⢿⣿⡄⣿⣿⣷⠀⠃⣸⣿⡿⠀⡀⠀⠀⢸⠀⠀⢸⡇⡾⣸⣿⣿⢸⣿⣧⢹⣿⣿⠀⣿⢀⣼⣿⢟⠁
⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⢠⣿⡿⠁⠀⡇⠀⠀⠸⠀⢸⣼⣸⢣⣿⣿⢇⣾⣿⡇⣼⣿⣿⠀⠁⣼⣿⡿⠋
⠀⠀⠀⠀⠘⣿⢿⣿⡇⢠⣿⡿⠁⠀⢀⡇⠀⠀⠀⠀⣾⣿⢇⡞⢹⢟⣾⣿⣟⣴⣿⣿⠟⢀⣴⣿⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠸⣿⣿⢁⣿⠟⢀⡞⢀⣼⡇⠀⠀⠀⣸⣿⠏⢌⢀⢫⣾⣿⣿⣿⢿⠿⠋⣠⠞⠋⠀⠀⠀⠀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠘⠇⣼⠏⢀⡞⢀⣼⣿⠁⠀⠀⣰⣿⠋⠀⣼⣠⣟⣵⣿⠿⠋⠀⢀⣈⠄⠀⢀⣤⣴⠾⠛⣩⡟
⠀⠀⠀⠀⠀⠀⡈⣾⠃⢀⡾⣠⣾⣿⠇⠀⠀⣼⡿⠁⠀⣼⡇⣏⣿⡟⠁⢀⣠⠞⠋⠀⣠⡾⠋⠉⢠⣴⣾⡟
⠀⠀⠀⠀⠀⠠⢰⠏⠀⣼⣿⣿⣿⠋⠀⠀⣼⠟⠀⠀⣼⣿⣇⢻⡟⠀⣠⣾⡏⠀⣠⠞⠉⠀⣠⣶⣿⣿⡿⠀⣾
⠀⠀⠀⠀⠀⠀⡎⠀⢠⣿⣿⡟⠁⠀⠀⢼⠋⠀⠀⣰⣿⡿⢻⢸⠃⣰⣿⡿⠀⠀⠀⠀⢀⣼⣿⣟⣾⡿⠁⢠⢻
⠀⠀⠀⠀⠀⢠⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⢀⡼⠋⣾⡷⣄⣾⣰⣿⣿⠃⠀⠀⠀⢠⣿⣿⢻⣾⡿⠁⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢠⠏⣰⣿⡟⢇⣿⣿⣿⡿⠁⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⢠⠊⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠰⢣⣾⣿⣿⣿⣾⡿⠟⠋⠀⠀⠀⢀⣾⣿⣽⡿⠛⠁⢀⡴⠁⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣶⠀⠀⠀⡴⢋⣽⣿⠿⠋⠁⠀⠀⠀⠀⢀⣴⣿⠿⠟⠉⢀⣠⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠠⢸⣇⢰⡇⠀⢀⠞⣰⠟⠋⠀⢀⣀⣤⡶⣮⡤⠞⠋⣁⣀⣤⣴⣾⣿⣿⣿⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡟⠀⠀⢀⠞⠁⠀⣠⡶⠟⠋⠁⠀⣀⡴⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⢠⣡⣿⡟⠀⠐⠀⣠⠄⢀⡾⢋⡠⠆⠀⠀⢊⣥⣶⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⢀⣠⡾⠁⠐⣡⡴⠋⠀⠀⣠⣴⠿⠛⠋⠉⠉⠁
⠀⠀⠀⠀⢰⡿⠋⢀⣤⣶⣿⠋⠀⣠⣾⠟⠁⠀⣠⠾⠋⠁
⠀⠀⠀⢀⠟⢠⣾⠿⠟⠋⠀⠐⠚⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠉⠁⠀⣀⣄
⢀⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠋
⠙⠛⠛⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠐⢾⠛⠒⠂⠀⠐⠒⠙⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⣀⣈⣀⣀⡌⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠘⠁⠀⠀⠄
⠸⣶⣶⣶⢶⣶⣶⡟⠀⠀⠀⠀⠸⣿⣿⡇⣿⣿⣿⠁⠀⠀⠀⠀⢰⣶⣤⣤⣤⣤⣤⣶⠆⠀⠀⠀⠑⣶⣤⣤⣤⣤⣤⣤⡶
⠀⠹⢿⣿⣺⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣧⣿⡿⠋⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠿⠿⠿⠿⠷⠆⠀⠀⠀⠀⠀⠸⣶⣶⣤⣶⣶⢅⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⡶
⣤⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣤⣤⣴⣾⠀⠀⠀⠀⠀⠨⣶⣶⣦⣶⣶⣶
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣉⣉⠉⣉⣉⣡⡇⠀⠀⠀⢐⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠯⠭⠭⠿⠿⠁⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡄
⣠⣴⣿⣿⠀⠀⠀⠀⠀⣠⡀⠀⡄⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⣀⠉⣀⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⣀⣀⣀
⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠀⠸⠇⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⠀⠀⠀⠀⡇⣸⠃⢀⠀⢠⡈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠁⠀⠀⣰⠃⠀⠀⡞⠀⢈⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠸⠁⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠚⢸⡆⠀⢰⡆⠀⠀⣠⠀⢀⡀⠀⠈⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣞⠀⠀⠀⢻⠀⠀⢸⠆⠈⠁⠀⠀⣇⡔⢰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠋⢠⠀⡄⠀⠀⠀⣀⠀⠀⠀⣼⠀⢉⠀⢸⣧⡇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣼⠃⠀⡠⠀⢀⡀⠆⠀⢀⢉⠀⠀⣧⠁⠀⠈⠀⠏⠏⠀⢰⠀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣇⠐⡇⠀⠀⠁⠀⢸⣷⠀⠃⣾⠀⠀⠂⠈⠀⠀⠁⠆⠀⠀⠀⢸⢃⡁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⣸⡆⣦⢠⠀⠀⢠⠀⠁⠀⠀⠛⠀⢀⢰⠀⡄⠀⣆⠀⠁⠀⠀⠀⠸⠃⢸⠆⠋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⡀⣠⠠⠈⡃⠞⠀⠀⠁⡀⠀⠀⢀⠀⢀⠸⠆⣆⠁⡞⠑⠀⡀⣤⠀⠀⣠⠀⠊⠀⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⢳⣉⠀⠁⠑⠰⠓⣠⣷⠇⠀⠰⡌⠂⠰⠀⡄⠉⠀⠇⠐⠾⠁⣸⠀⣶⠀⣶⠄⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠋⡤⠛⠐⣷⡀⠜⠿⣿⡏⣻⣧⡐⠁⢀⠀⢸⡇⠀⠀⠀⠀⢠⣘⡧⣶⠛⣷⣤⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠘⠄⢀⡀⠀⣄⣀⠈⡄⠙⠉⣿⡄⠉⠀⡠⢁⡯⠀⢠⠀⣸⡟⠁⠇⠞⠟⢁⣀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡄⠙⣻⡆⢨⢉⣠⣿⣾⣏⢀⣔⢃⠈⣁⠀⡀⠇⠋⠛⠀⠀⠀⠋⠅⢡⣴⡀⢸⡇⢦⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠓⠉⠙⠛⢾⡌⠙⣿⠇⠁⠈⠛⠙⢷⢈⣤⣅⠀⣄⢂⣼⣷⡀⢰⣾⠆⡿⠃⣾⣿⠆⠘⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⢠⣿⡿⠂⠉⠁⠒⢶⡄⠀⣤⣼⡿⠋⠁⠀⠈⠙⢿⣿⡆⠀⠀⠆⠉⠘⠳⣦⣤⡠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⡒⣶⠞⣻⣟⡀⢀⠀⣀⣼⣮⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⣸⣆⠀⠀⢀⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠯⢉⣛⣛⡽⣿⣦⢠⠞⣿⣿⠦⡀⢀⡀⣰⡿⠿⣿⣿⣯⣾⣽⣷⣀⣫⣾⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠦⠤⠔⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⢠⣾⣿⡟⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢿⠀⢠⣿⣿⡿⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⠏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠈⣿⡀⣸⠏⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣦⠀⠀⠸⡇⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠹⣷⡈⢿⡆⠈⣿⣿⣿⣿⣿⣷⡀⠀⠁⠀⠀⠀⠀⣾⠏⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠈⠁⠈⠿⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⣤⠀⠀⠏⠀⠀⠀⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⣾⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠙⣿⣦⠀⠸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣧⠀⠀⠀⠸⢏⣼⠟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠀⢻⣿⡇⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡆⠀⢀⣀⠸⠁⠀⠀⣠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣧⠀⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠁⢰⣿⠃⠀⢠⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣤⠀⠀⣿⣿⣿⣷⣿⣿⣿⠀⠀⠹⣿⣿⣿⣧⣠⣾⣿⡄⢰⠟⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡙⣷⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⠀⠴⢋⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠘⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠒⠫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⢹⣿⣿⡇⠙⢿⣿⡀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠉⠙⠀⠀⠀⢀⣿⡿⠃⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠘⠋⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣦⣀⠀⠀⠀⠀⣀⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠒⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠐⠂⠂⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⡀⠁⡘⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠐


⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣇⠀⠀⠀⠐⠊⠙
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢸⡁⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⠀⠀⠀⠀⠀⢁⠀⠑⠈
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡓⣇⢀⡴⠒⢁⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣁⠄⠈



⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠖⠦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⠡⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠉⢤⡀⠒⠈⢂⡇⠀⠄⡠⠇
⠀⠈⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠄⠂⢹⠀⠀⡈⡅⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠉⠀⢀⣸⠷⠚⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⣱⣿⢻⣧⠀⠀⠀⠀⠀⠀⢀⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿
⠀⠀⠀⠀⢸⣿⠘⣿⡀⠀⠀⠀⠀⠀⠐⢠⠘⣏⢻⢿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢁⣾⡏⠀⠀⠀⠀⠀⠰⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣦
⠀⠀⠀⢰⣼⠀⣤⠛⡇⠀⠀⠀⠀⠀⢰⢿⣿⢃⡀⢸⣶⠂⠀⠀⠀⠀⠀⢸⡇⠋⢀⢸⣯⣔⡇⠀⠀⠀⠀⣾⠉⡀⢸⡇⠀⠀⠀⠠⢠⣠⡾⣿⢠⡌⠹⣷⡀
⠀⠀⠀⢈⣏⠃⡝⣆⣷⠀⠀⠀⠀⠸⣿⠆⢁⡞⣇⠀⢛⡟⠀⠀⠀⠀⠀⠘⢿⣄⢼⣦⣿⡟⠀⠀⠀⠀⠀⣾⠄⣧⣈⡇⠀⠀⢀⣴⠛⢛⣄⣁⣼⣧⡆⡹⠁
⠀⠀⠀⠀⠉⠛⠓⠋⠁⠀⠀⠀⠀⠀⠈⠑⠂⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠃⠀⠀⠀⠀⠀⠈⠑⠛⠉⠀⠀⠀⠘⠛⠀⠛⠈⠉⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⢃⣴⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠴⠀⠠⠀⠀⠀⠀⣶⣄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠄⢀
⠀⠀⠀⠀⢾⡟⢿⣷⡈⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠀⣠⣴⣶⡿⠀⠀⠀⠀⠀⠀⢿⣿⠿⣷⣆⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣴⠁⠀⢀
⠀⠀⠀⠀⣸⡇⠀⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣆⡿⠋⣜⣥⣠⡌⠀⠀⠀⠀⠀⠻⣆⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢃⡴⣿⠠⣶⡿⢻⣿⢀⡴⣯
⠀⠀⠀⠀⠿⣟⢸⡄⡿⠀⠀⠀⠀⢀⣠⣰⣶⠏⣸⠟⢡⠜⠙⠻⢛⡵⠆⠀⠀⠀⠰⡶⣿⢠⡄⢸⣦⠀⠀⠀⣠⣴⣤⡴⠋⣸⡿⢚⡉⣠⢿⡿⠋⣴⠏⡀
⠀⠀⠀⠀⠀⠈⢿⠟⠁⠀⠀⠀⠠⣿⢋⣛⣡⢦⣠⠶⠻⠴⣦⣤⣀⡶⣧⡀⠀⠀⠀⢹⡦⣾⡁⡸⠃⠀⠀⠞⡋⡉⣩⣞⣀⡀⢴⣋⣈⣥⣠⡄⠚⠟⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣟⣻⣦⢀⠀⠀⠀⠀⠀⣀⠂⢀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠃⠘⣿⡏⢻⣆⠀⠀⠀⢸⣿⣷⣄⡠⢀⣁⣀⣒⣠⣤⣌⡁⠀⠀⠀⠀⠀⠀⣿⡿⣦⣤⣇⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣧⠈⡀
⠀⠀⠀⣠⣦⡿⡁⠘⠉⡇⠀⠀⠀⠻⢿⣟⠛⠟⠛⠻⠿⣿⣿⠛⣿⣤⡀⠀⠀⠀⠀⠹⣧⠈⠙⠻⡿⣦⣷⣦⠀⣂⠀⠀⠀⠀⠀⠀⠀⣿⠻⣷⣿⣦⣤
⠀⠀⢼⡏⠋⣼⣧⠄⠸⣿⠀⠀⠀⠀⠐⠉⢻⣷⣶⡄⢲⣤⣈⡆⠙⢿⠇⠀⠀⠀⢀⢳⣾⢷⡄⠰⣤⣤⣍⣉⠻⡟⢷⠀⠀⠀⠀⢀⣗⠾⣆⢉⡛⠈⠻⠿⣆
⠀⠀⠈⠻⠦⠿⠸⠧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠧⠼⠷⠽⠷⠦⠟⠀⠀⠀⠰⠾⠿⠋⠁⠠⠶⠾⠧⠉⠉⠳⠤⠌⠦⠀⠀⠀⠸⠿⠆⠠⠾⠟⠓⠛⠲⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤⣤⣤⣀⢀⠠⠤⠄⠐⠒⠩⠉⠹⠷⠄⠄⣀⣀⣠⠤⠤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⢠⡔⣒⠥⠾⠿⠟⠛⠛⠂⠀⠀⠀⠀⠀⠔⣈⣽⣾⣷⣤⣀⣀⣠⠤⠤⠶⣚⣿⢿⣄
⠀⠀⠀⠀⣠⣴⡛⡣⠭⠅⣈⣉⡉⣉⠒⠒⠂⠠⠤⠄⣈⡒⣿⡿⠟⠟⣛⣫⣩⡴⢴⣺⡿⠽⠚⣻⡹
⠀⠀⠀⠀⢧⣂⠈⠉⠉⠉⠁⠐⠒⠩⠥⠍⣉⣑⣒⠠⠤⠶⠶⢖⣛⣻⢭⢭⠲⠚⢩⣡⡤⣶⣿⣿⠁
⠀⠀⠀⠀⠀⡯⡯⠫⠛⠖⡲⢂⠤⢤⢀⣀⣀⠀⠀⠈⠉⡉⠉⡍⡀⣄⣬⣴⣾⣾⢿⢗⣏⣿⣻⡇
⠀⠀⠀⠀⠀⡧⠁⡢⢂⠀⠀⠈⠀⠁⠈⠐⠡⠋⠝⡹⢛⢛⣿⠻⣿⡯⡊⣞⣽⡿⣡⣾⣿⣿⢿⠇
⠀⠀⠀⠀⠀⠙⡃⠔⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⢔⢕⣷⣿⡾⢋⢾⠟⡡⡇
⠀⠀⠀⠀⠀⠀⠀⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣼⠗⡡⡫⢻⢟⠔⡵⣣⢊⢞⡇
⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡡⣪⢎⠔⡵⡣⢊⢜⡕⡡⢫⡇
⠀⠀⠀⠀⠀⢨⠀⠁⠀⣠⠤⠶⠐⠃⠱⡄⠀⠀⠀⠀⠀⠀⣟⠔⡵⡡⢮⢞⡔⡡⡪⢎⠔⣱
⠀⠀⠀⠀⠀⢸⠀⢀⡞⢁⡴⠒⠈⣧⠖⢡⡪⢝⣦⠀⠀⠀⡧⣪⢞⠕⡵⢋⢞⢔⠕⡡⢊⢾
⠀⠀⠀⠀⠀⢸⠀⣼⣃⡎⠀⠀⠀⠀⠀⠀⠙⡆⠘⡆⠀⠀⣟⠕⡡⢊⢔⡵⡡⢋⠜⡱⢣⢾
⠀⠀⠀⠀⠀⢸⠀⠈⢠⢤⣀⣀⠀⠀⠀⠀⠀⡅⠀⡇⠀⠀⣯⢎⠔⡡⢊⢞⡴⢡⢎⠔⡵⢫
⠀⠀⠀⠀⠀⢀⠀⢠⡃⠀⠰⡋⠀⠀⠀⠰⠟⠃⣰⠃⠀⠀⡷⢡⢎⡔⣡⠊⡴⡡⣪⠊⡴⡋
⠀⠀⠀⠀⠀⣾⠀⠈⠉⢦⠀⣈⠙⢋⢃⠸⣀⣀⣳⡀⠀⠀⡟⡡⢊⡔⡡⢪⢎⡔⣡⢊⢜⡅⢀⠔⠊⢒⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠒⠼⠽⠾⠂⠀⠀⠀⠀⠀⠀⡟⡠⢋⡞⠙⡁⢋⣜⡕⡋⣫⣿⡏⢀⣾⣿⣿⣷
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡰⣩⢏⣜⡁⣩⢋⢜⡁⣡⣛⡁⣈⣉⣻⣿⣿⣀⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠜⡡⣫⢊⠜⡡⡣⢊⢞⣽⣿⠍⢾⣽⣩⣛⣟⣉⣐⣂⣀⣀⡀
⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣃⢜⠔⡡⢫⣮⠞⡕⡱⡹⣡⡿⣻⡼⣮⣿⣛⢒⣒⣒⣒⣐⣂⣒⣂
⠀⠀⠀⠀⠀⢥⡂⠠⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢸⡡⢊⢞⡔⡕⣡⢊⢎⠜⣔⣿⣓⣘⢛⣐⣒⣒⣒⣒⣒⡒⠒⠒
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⢥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡵⡣⢪⠊⡔⡱⣣⣥⢾⠛⠒⠒⠒⢒⣂⣀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⢄⣐⠠⠖⢸⢎⠔⣡⣪⠾⠟⠩⠭⠭⠉⠠⠤⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠡⠭⠭⠭⠩⣭⣤⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⢀⡀⠀⡏⠈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠙⡆⠀⠀⠀⢨⢷⡀⠰⢈⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡟⠀⠀⠀⠘⣕⠙⠢⡀⠁⠪⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⠀⠀⠀⠀⠀⠱⡨⢀⠌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠊⣣⠀⣴⠀⠀⠀⠀⠘⢤⡀⠀⠂⠑⣄⠀⠀⠀⠀⢠⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠌⡠⢁⡿⢸⠹⣄⢀⡀⠀⠀⠀⠙⣬⠀⢑⠙⡆⠀⠀⡔⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠅⡢⠰⡇⠀⠑⠬⣢⠹⡳⣄⠀⠀⠸⡆⠐⠂⢻⠀⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠐⠁⣄⠀⢷⡀⠀⠀⠈⠀⠇⡈⣷⠀⡸⠂⡷⠀⢹⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⢸⡨⠀⡄⠣⡀⠑⢄⠀⢀⡬⠁⠀⣾⡼⠃⡐⢨⠀⡵⠀⢸⡇⣷
⠀⠀⠀⠀⠀⠘⡈⠑⢢⡀⠀⠀⢣⠀⠸⡄⠈⠢⡈⢳⡞⢀⠞⠀⠍⢀⠞⠀⠎⢰⠃⠀⡸⡁⢸⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠀⣙⡀⠀⠀⠑⣄⠐⡄⠀⠐⡀⢠⠃⠀⣇⠔⠁⠀⠌⢠⠏⣠⡞⠃⠀⠀⡇⠀⠀⠀⣠⢷
⠀⠀⠀⠀⠀⠀⠈⠹⢖⠤⣁⠀⠀⠀⠘⡆⠈⠀⠀⢱⡎⠀⠀⠋⠀⠀⡜⢠⠿⠚⠉⣀⡤⠀⢰⠇⠀⢀⠌⣁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⢦⠀⠀⢿⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠁⢀⠄⠒⠉⡔⠀⢰⠏⠀⠀⠧⢊⡁⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢦⡀⢳⢀⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⢀⠎⢀⡴⠃⢀⡀⢀⡔⢉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡀⠰⡈⡟⢁⠌⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⡞⢡⣴⡏⢰⠋⠈⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠃⠀⠆⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠀⣷⠟⠲⣇⡟⡰⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡀⠇⠸⠀⠟⠱⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⠀⠱⣄⡃⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠔⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠈⠀⠀⠤⠀⠀⢀⣀⣀⣀⡀⠀⠔⠈⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠈⠀⠈⠐⠠⠐⠨⠀⡂⠄⡀⠄⠠⠀⡠⢀⠐⠀⠠⢐⠀⢀⠀⡠⢔⠢⢀⡐⢀⠔⡢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⢀⠀⡀⣀⡀⠀⣀⠀⢀
⠀⠀⢀⡗⠀⠀⠀⠀⢀⣴⣿⣾⡆⠀⣼⣿⣷⠀⠀⠀⢀⡀⡀⠀⠀⠐⣾⣿⣷⣿⣿⣿⣾⣿⣿⡿⢀⣉⡇⣨⡽⢃⡀
⠀⢠⣾⣿⣴⡄⠀⠀⣸⣿⣿⣿⣿⡀⠁⠹⠏⠀⡐⡜⣨⡵⣿⣷⣤⡠⠙⠛⠛⡛⠛⣛⣛⣛⣋⠀⡸⣿⣜⡿⢳⡞⣖
⢀⣿⡟⠉⢿⣷⠀⠀⠻⣿⠿⢿⣿⠇⠀⠀⣠⣾⣷⣿⣿⣿⣿⣯⣿⣿⣦⣄⢼⣿⣙⢺⣟⡋⠛⠁⠘⠿⠻⣷⣾⠟⠋
⠀⠛⠄⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣼⣿⣿⠋⠀⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⢰⣠⢼⡰⢠⢠⣀⠀⠀⠁⠊⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠀⠢⠀⠀⠰⡄⣠
⠀⠀⠐⡄⠀⢰⣺⣽⣿⣿⣾⣽⣼⣿⣿⠀⠀⠀⠀⠀⣠⣦⣤⣄⠀⠀⠀⠀⠐⠀⠈⠼⠷⠁⣨⢷⣿⢕⠗⠀⢀⣘⡆
⠀⡄⣾⢁⠀⠈⠙⠛⡉⠉⠉⠙⠛⠛⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⠀⠀⣠⢞⢢⡇⠀⠀⢀⣈⣿⣿⣏⣀⠰⣞⣮⣷
⠈⢷⣙⣾⣦⠀⠀⣴⣻⣶⡄⠀⢀⡺⣿⣎⠠⣄⠈⠻⣿⣿⣿⣿⡿⢠⣶⣿⡏⣿⣷⡇⠀⠉⠉⠉⠉⠉⠉⠀⣤⣿⠇
⢀⣾⣼⣿⡏⠀⣼⡏⠹⢿⣿⠀⢢⣹⣯⣟⣵⡝⡄⠠⣿⡿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣗⠀⠀⢀⣠⡄⠀⠀⠈⠻⣧⣄
⠀⠛⠿⠟⠃⠀⠈⠓⠀⠘⠃⠀⠘⠛⠛⠛⠛⠛⠋⠀⠘⠇⠸⠋⠀⠰⠿⠿⠿⠿⠿⠟⠀⠶⠿⠿⠿⠷⠶⠀⠶⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⡶⠾⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣾⠿⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⢠⡞⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠾⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀⣀⣤⣴⠶⢶⣴⡟⠻⣷⡋⠀⠀⠀⣿
⠀⠀⠀⢀⣴⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢦⣿⠉⠙⢿⡉⠻⣿⠘⣿⠀⠀⢿⣇⠀⢸⣇⠀⠀⠀⢻⡄
⠀⠀⢀⣾⣷⠀⠀⠀⢀⣤⣤⣄⣀⣴⠖⠶⣶⠖⢶⣿⠀⠀⣿⠀⠀⠘⣷⠀⢹⡇⢻⣆⠀⢸⣿⣄⣸⡿⠀⠀⠀⢸⡇
⠀⠀⣼⡏⢹⣧⣶⣤⣾⠃⠀⠙⣿⣿⠀⠀⠹⣧⠀⢹⡄⠀⢹⣧⡀⢀⣿⣂⣸⣿⣤⣿⡿⢿⡇⢹⡋⣧⠀⠀⠀⠘⣿
⠀⢰⡟⠀⠈⣿⠀⠘⣿⡆⠀⠀⢸⣿⣆⠀⠀⣿⠀⣸⡿⣦⣴⡟⢿⠛⢏⡟⠹⡆⢿⠘⣿⠀⣷⠘⣇⢸⡄⠀⠀⠀⣿
⢀⣿⠁⠀⠀⢹⡆⠀⠸⣷⡀⠀⢸⣯⣽⣶⣴⡟⢿⢻⡄⠸⡆⢷⠘⡆⢸⣿⡄⢷⠸⡆⢻⡇⠸⡄⢹⣈⣧⣀⣤⣤⣼⣧⡀
⣼⠃⠀⠀⠀⠘⣿⣀⣸⢿⢻⡾⢿⡷⠀⢿⡄⢷⢸⡌⣧⠀⢿⠸⡆⢱⠀⢻⣧⣨⣦⣷⣼⣿⣾⣿⠿⠿⠛⠛⠋⠉⠉⣿⡇
⣿⡄⠀⠀⠀⠀⢹⡏⢹⡌⡿⣷⠈⣿⡆⢸⣧⠸⡇⢱⣸⣆⣸⣧⣿⣾⡿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⣧
⢸⣇⠀⠀⠀⠀⠀⣿⠀⢧⢻⢿⡇⢹⣿⣤⣿⣶⡿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠛⠛⠋⢻⡄⠀⠘⣿
⠘⣿⡄⠀⠀⠀⠀⢸⣇⣼⣶⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣿⡇
⠀⢻⣧⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧
⠀⠈⣿⡄⠀⢀⣾⠋⢹⡇⠀⠀⠀⣶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠘⣿
⠀⠀⢹⣇⢀⣾⠏⠀⠘⣿⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⡇
⠀⠀⠈⣿⣾⠋⠀⠈⠀⢿⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢹⣧
⠀⠀⠀⢻⣿⠀⠇⠀⠀⠀⢿⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿
⠀⠀⠀⠘⣿⡾⠀⠀⠀⠀⡾⣧⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⡆
⠀⠀⠀⠀⢹⣧⠀⠀⠀⢸⠀⢿⡄⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢹⣧
⠀⠀⠀⠀⠈⣿⣍⠐⠆⢄⠀⠸⣧⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡀
⠀⠀⠀⠀⠀⢹⣿⡦⠀⠈⠀⠀⣿⡆⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⣀⠀⡄⠐⠈⣷⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡄⢹⠓⠆⠀⢹⡆⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣧⡈⠀⠀⠀⠘⣿⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢀⠀⠈⠃⢸⡇⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠁⠂⠀⠀⢿⡀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠐⠀⠀⢸⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣧⡘⠀⠀⢻⡄⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠖⠛⠛⠉⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣠⠁⠘⢠⠘⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠾⠛⠋⠉⠉⠀⢀⣀⣠⣤⣶⣶⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢀⠁⡆⠀⣿⡄⠀⠸⣇⣀⣀⣤⣤⠶⠶⠞⠛⠋⠉⠀⢀⣀⣠⣤⣴⣶⠾⠿⠛⠛⠉⠉⠉⢻⡄⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠸⠀⢸⠘⣧⠀⠀⠙⠉⠉⠀⢀⣀⣠⣤⣶⣶⡾⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⡈⠀⢿⣄⣀⣤⣴⣶⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡾⠀⠀⠀⢸⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⡇⠆⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠸⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⣼⡇⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣠⣤⣶⣿⡿⠿⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣟⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣍⡉⢉⣩⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⡉⠉⢨⢨⠉⡏⣯⡽⠛⠛⣏⣯⣭⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠾⠿⠛⠿⠾⠿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢴⣒⠟⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄⠤⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠖⠒⣊⣉⠵⠋⠉⣀⠴⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⢀⡀⠈⠉⠁⠒⠠⢍⡉⠓⠒⠢⠤⠤⢤⡴⠾⠿⠿⠿⠯⠥⠖⠒⠒⠛⠉⡉⠀⠤⣶⠉⠇⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠒⠬⠁⠒⠂⠀⠀⠛⠁⡲⠋⠉⠉⠉⣷⠤⠤⢴⡖⢲⡚⠋⠙⠏⠹⠉⠁⠀⠀⠘⠂⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠰⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠃⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠀⠀⡀⠀⡇⡀⠀⢃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⢉⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠓⠤⠀⠀⠀⠀⠀⢵⣟⢤⡈⠈⠁⣀⠶⡆⡄⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠄⡀⠀⠀⠀⠀⠀⠀⠈⢳⡤⠖⢁⠴⠊⠉⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⣿⣶⠤⠀⠀⠠⠔⢈⠀⠺⡉⢀⠀⠀⠀⠀⠀⣽⠶⠀⠒⠲⢾⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢠⠁⠙⡄⠰⢏⡠⠂⣿⠀⠀⡙⢀⣑⡄⢀⣠⠞⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⡘⠀⢀⡇⠀⠈⠇⢀⣇⠑⠂⡇⠀⠏⠁⠀⡝⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⡇⠀⠐⠀⢸⠀⠘⠢⡇⢀⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠧⣾⣛⡒⢋⣉⣁⣠⠥⠤⠴⠊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⣰⡇⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣸⠀⠈⠉⠀⡀⠈⠉⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⣸⠉⠁⠀⠐⠂⢀⣀⡐⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣿⡦⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡟⠀⠀⠀⠀⠠⢤⡿⠀⠀⠀⠠⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡿⠃
⣧⠀⠀⠀⣀⣠⡼⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠷⠄⠀⠀⢘
⢸⡄⠀⠈⢁⣴⡇⠀⠀⢀⠆⠀⠀⠒⠀⠛⠋⠉⠉⠉⠒⠒⠒⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⠤
⠀⢳⡀⠀⡴⢾⡄⠀⣠⡞⠀⢀⡠⠴⠖⠚⠛⠛⠛⠓⠲⠦⠤⢤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⡄
⠀⠀⠳⣄⡀⠸⢷⣾⣿⡁⠀⠀⠀⢀⣠⣤⠶⡞⢿⡟⢿⡷⣶⡶⢤⣀⠀⠑⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠁⠀⢠
⠀⠀⠀⠀⣿⠉⠉⢺⣿⡿⠖⠛⢿⠁⢹⠃⠈⠁⣼⠁⢾⠀⣼⠀⠀⢿⠿⣷⣶⣄⡀⠀⠀⢀⣴⠟⠄⠀⠀⠀⡾
⠀⠀⠀⠀⣿⠀⠀⠉⠈⠃⠀⠀⠈⠀⣿⠀⠀⠀⠛⠀⠀⠀⠛⠀⠀⣿⠐⠏⠀⣿⣿⠉⠛⠛⢇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠈⣿⣄⠀⠀⠈⠂⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡁⠒⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠤⠤⣤⣤⣀⣀⣀⣤⣤⣤⣤⠶⠾⠋⠁⠀⠀⠀⠀⠉⠙⠛⠲⠶⠤⠤⠤⠤⠤⠶⠖⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢰⡿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢠⢿⡅⠀⠀⠀⠀⠀⠀⠀⠸⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢻⡆⠀⣼⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇⠀⣿⠈⣇⠀⠀⠀⠀⠀⢀⠀⠀⢹⡄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣇
⠀⠀⠀⠀⠀⠀⣸⠃⢸⠃⠀⢹⠀⢻⡀⠀⠀⠀⠀⠀⢷⡄⠀⢳⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⠇⠀⣼⠀⠀⢸⡇⠈⣇⠀⠀⠀⠀⠀⢸⢹⡀⠈⢧⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⡞⠀⠀⡏⠀⠀⠀⣇⠀⢸⡄⠀⠀⠀⠀⣸⠈⡇⠀⠈⣇⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡿⡇⠀⢸⡆⢻⡀⠀⠀⠀⠀⠀⠀⠀⡏⣿
⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⠀⣿⠀⠀⣧⠀⠀⠀⠀⡟⠀⡇⠀⠀⠸⡆⠀⠀⢻⡀⠀⠀⠀⠀⢰⠃⡇⠀⠀⣧⠈⣇⠀⠀⠀⠀⠀⠀⠀⣇⢹⡀
⠀⠀⠀⠀⢸⠀⠀⠀⢿⣤⣀⣴⠏⠀⠀⢻⢠⣇⠀⠀⡇⠀⣇⠀⠀⢀⣇⠀⠀⠀⣇⠀⠀⠀⠀⡟⠀⡇⠀⠀⢹⠀⢸⡄⠀⠀⠀⠀⠀⠀⢻⠈⣧
⠀⠀⣄⠀⢸⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⡿⢸⢹⡄⠀⣧⠀⢻⡶⠶⠾⡟⠀⠀⠀⡿⢠⡄⠀⢠⡇⠀⣧⣀⣀⡼⠀⠀⣧⠀⠀⡄⠀⠀⠀⠈⣇⠘⢧
⠀⠀⣿⡇⠸⡆⠀⢀⠀⠀⠀⠀⠀⠀⢠⠇⠘⡆⢳⡀⠸⡄⠀⠙⠀⠀⠁⠀⠀⢀⡇⣿⡇⠀⠸⡇⠀⠘⠿⠋⠁⠀⠀⢹⠀⠀⢹⣆⠀⠀⠀⢹⡄⠈⢳⡄
⠀⢀⡇⣿⠀⢧⠀⡾⣆⠀⠀⠀⠀⢀⡟⠀⠀⢻⡈⢧⠀⠹⣆⠀⠀⠀⠀⠀⠀⣼⠁⢷⢻⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡟⡆⠀⠀⠀⢷⠀⠀⠙⣆
⡀⣸⠁⣿⠀⠸⣶⡇⢸⡄⠀⠀⠀⡾⠀⠀⠀⠀⣧⠈⣧⠀⠘⢦⠀⠀⠀⠀⢠⡇⠀⢸⡆⢳⠀⢹⡀⠀⠀⠀⣶⠷⣦⣸⠀⠀⠸⡇⢹⡀⠀⠀⠘⡆⠀⠀⠘
⡇⡏⠀⣿⠀⠀⢹⡇⠀⡇⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⢸⡄⠀⠈⢷⡀⠀⠀⢸⣇⠀⣸⡇⠘⡇⠀⢷⠀⠀⠀⡇⠀⢸⡏⠀⠀⢸⡇⠈⡇⠀⠀⠀⢻
⣿⣷⠀⢻⣄⠀⠀⠀⢠⡇⠀⠀⠸⣧⠀⠀⠀⣠⣿⠀⢘⡇⠀⠀⠘⣧⠀⠀⠘⣿⠛⠉⠁⢠⡇⠀⠈⢧⠀⠀⢳⡀⠀⠀⠀⠀⣸⠁⠀⣷⠀⠀⠀⢸⡆
⠹⣿⡄⠀⠻⣷⠶⠾⣿⠃⠀⠀⠀⢻⣿⠷⠿⠻⠃⠀⢸⠃⠀⠀⠀⣿⡆⠀⠀⠈⠀⠀⠀⢸⠁⠀⠀⠈⢧⠀⠀⠻⢶⣤⣴⡾⠋⠀⠀⡟⠀⠀⠀⣸⡇
⠀⠈⠻⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣦⣤⣤⣴⣿⡇⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⣀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⣼⠻⠿⠛⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠈⠉⠉⠉⣿⠃⠀⠀⠀⠀⠀⠀⠘⢿⠛⠛⠋⠁⠈⣇⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠶⢒⣒⠋⢲⡤⠶⠒⠚⠉⠉⠀⠀⢸⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠴⠒⢚⣉⣭⠥⠴⠒⠒⠚⠉⠉⠈⠳⡄⠙⢦⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⣀⡤⠴⠖⠛⣉⣩⠤⠴⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠈⠳⣄⢤⣤⣶⣺⣯⡼⢦⠀⠀⠹⣄
⠀⠀⠀⠀⠀⣿⣄⠀⢮⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠘⢷⣿⣿⣿⣿⣿⡿⠧⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣷⠈⢧⡀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠹⣽⣿⣿⡿⣗⣉⣽⣧⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢻⠀⠈⢳⡀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡧⠀⠈⢻⣿⣿⣿⣯⣤⣶⠧⠴⠒⠛⢛
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⣆⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠶⠒⢛⣉⣉⡥⠤⠶⠒⣛⡉⣿⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣧⠀⠱⣄⣤⠤⠴⠒⠊⣉⣉⣠⣤⠤⠶⠖⣚⣋⣭⡥⠴⠒⠚⠋⠉⠉⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡤⠖⠒⠉⠉⠉⢳⣄⡤⠤⠖⣒⣛⣉⣭⠥⠶⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⡇⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⣀⣀⡤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣐⣠⠿⣸⠖⠚⠋⠉⠁⠀⠀⣀⠐⠉⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⣶⣚⣿⠭⠟⠒⠋⠉⠀⠀⠀⣀⠤⣒⣭⡼⠦⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡿⡇⠀⠀⠀⠀⢀⣀⣠⡤⢶⣒⣛⠭⠽⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⢀⡠⢖⣪⣷⣿⣛⣁⠤⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡇⠧⠤⢴⣚⣿⠽⠷⠚⠋⢉⣁⣤⣤⠤⠤⠤⠤⠶⠶⠶⠶⠒⠒⢚⣋⣉⣉⣉⣩⣥⣤⣤⣀⡀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠉⠉⠀⠀⠀⠀⠀⠀⠻⣤⣤⣶⣶⣲⣶⣶⣶⠶⣿⣿⡿⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⣯⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⠋



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

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⢌⣀⣈⣒⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣽⣿⣿⡿⣿⣿⣿⣿⣯⣥⣌⣩⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⢙⣿⡿⣛⡿⣿⣿⢿⣿⣿⣧⣤⣶⣿⡮⠁⣼⣿⣿⣿⣿⣷⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⡟⢯⣷⠸⣿⣟⢿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣾⠟⣁⡀⠚⢿⣿⠿⢿⡿⠿⠷⠶⠶⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣷⣄⡌⣿⣿⣿⡈⣟⡛⠿⢿⣿⣿⣿⣽⣿⣿⣿⣯⣥⣤⣀⠤⠀⠀⠀⠰⠀⠀⠀⠈⠷⣧⣧⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣟⣿⣿⣷⣿⣽⢿⠿⠅⣿⣷⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠨⠓⠀⠎⢈⢲⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⠛⢯⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣟⣂⠀⠀⣀⠀⠦⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣤⠀⠀⣿⣿⣿⣿⣷⡄⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢱⣿⣿⣿⣽⡿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣆⢰⣿⣿⣿⣿⣿⣷⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⣼⣿⣿⠻⢿⣿⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣵⡎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠁⢹⣿⣿⣷⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣛⡛⠻⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡹⢿⣿⣿⣿⣿⣿⣿⣿⡿⢟⢡⣬⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣶⣋⣭⣥⣀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠐⣻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣷⡆⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣤⣽⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠐⣱⣿⣿⣿⡿⠂⠀⠀⠀⢀⣠⣤⣤⣴⣦⣤⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⢀⣴⣿⢟⣿⣿⣿⣿⣟⠂⠀⠀⠀⣿⣿⣿⣿⣿⣝⡛⢿⣿⣦⠀⢸⣿⣿⡿⠴⢤⡀
⠀⠀⠀⠀⢀⡸⢿⣿⣿⣧⠀⢠⣿⣿⣩⣥⠭⣿⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠆⢰⣿⣿⣠⠚⣨
⠀⠀⠀⠀⢸⠁⣤⢹⣿⣿⡀⣸⢿⣿⣿⣿⣷⣶⣿⣿⣿⣷⠀⠀⢸⣿⣿⡴⢿⣿⣿⠛⣻⡉⠉⢩⣁⣿⣿⠃⠀⡇
⠀⠀⠀⠀⠀⢳⠀⢿⣿⡿⠁⠀⠀⠀⠹⣯⣩⣽⡷⠒⠋⠁⠀⠀⠀⣿⣿⣷⣾⣭⣷⡶⠏⠀⠀⠘⣿⣿⣿⡇⢐⡇
⠀⠀⠀⠀⠀⠘⡆⣾⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⢸⠁
⠀⠀⠀⠀⠀⠀⡇⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⣼⣽⣿⡧⠋
⠀⠀⠀⠀⠀⠀⢻⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡟⠉⢦⣀⠀⢀⣼⣟⢘⡿⢿⣷⣄⠀⠀⠀⣠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆⠀⠀⠀⠀⣠⡴⠞⠉⠀⠙⠛⠷⢬⣽⣿⡿⠛⠋⠁⠀⠈⠛⣦⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠐⣿⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⣾⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢧⡀⢋⢰⢧⣶⡶⠶⠿⣿⣷⣶⣶⣾⣿⠿⠟⠶⣶⣷⡟⣾⢿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠻⣷⣦⣠⣁⡉⠙⠁⢀⣀⣤⣶⣿⡏⠋⣰⣯⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠉⠘⣿⣿⢿⣿⣿⣿⣿⣿⢋⠏⠀⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠘⠧⡀⠈⢉⣩⡿⠇⠈⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠓⠶⣤⣤⣤⣴⣦⡄⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⢶⠶⠾⠿⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⣤⣾⠟⠋⠉⠉⠈⠉⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠉⠉⠉⠙⠻⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟
⠀⠀⠀⠀⢠⣾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⢈⣿⣷⣄⣠⣾⠿⠿⣷⣦⡀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠁
⠀⠀⢀⣴⡿⠋⠈⠻⣿⣅⠀⠀⠈⠻⣿⣤⡀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⢀⣤⣾⠟⠋
⠀⠀⠘⢿⣦⡀⠀⠀⠈⠻⣷⣄⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠙⢿⣷⣄⣠⣶⠿⠋⠁
⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⠻⣷⣄⠀⠀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⠻⣷⣄⠀⢀⣾⡏⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⠻⣷⣿⠋⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⣻⣷⣄⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣠⣾⠟⠉⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠒⢄⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⡁⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢡⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠉⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠚⠻⣤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠰⡀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⡰⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠄⢀⠊⠁⠀⠀⠈⠐⢄⣀⠀⠀⠀⠈⢣⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢠⠏⠀⠀⠀⢀⣀⠔⠈⠀⠀⠀⠉⠃⠐⠄
⠀⠀⠀⠘⠀⠀⢆⠀⠀⡿⠓⠀⠀⠃⠀⠙⠃⠀⠘⠭⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠏⠞⠀⠀⠊⠁⠀⠁⠀⠐⠾⡇⠀⢀⠎⠀⠈
⠀⠀⠀⠰⠀⠀⠈⠒⢾⠀⠀⠒⣂⣀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⢀⣀⡒⠂⠀⢸⠖⠈⠀⠀⠰
⠀⠀⠀⠀⠑⢀⠀⠀⠘⠋⠁⠈⠁⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠉⠛⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣡⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⠤⠤⠒⠠⢤⣀
⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠁⠂⠀⢀⠠⢾⠁⠉⠁⠢⡀⠀⠀⠀⡠⠃⠀⡾⠋⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠛⠀⠈⠿⠁⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠈⣀⠤⠠⠠⣀⢀⣠⢀⠃⠀⡼⠀⠀⠀⠀⠘⡆⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢸⠀⠀⠀⣀⠟⠉⠻⣈⠀⠀⠁⠀⠀⠀⠀⢀⡇⠀⢸⠀⠈⢿⣆⢀⡶⠀⠀⠀⠀⡈⠀⠀⠀⠈⡒⠒⠂⠠⠀⠀⠐⠒⠠⡀⠀⡜
⠀⠀⠀⠀⠈⠃⠀⠈⠛⠀⠰⡄⢦⠃⠀⠀⠀⠀⠀⢠⠎⠀⠀⠻⣄⠀⢀⡏⠹⣄⣀⡀⠄⠊⠀⠀⠀⢠⠊⠀⠀⠀⣰⠮⠶⢤⣤⣴⡭⠍
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢠⠃⠈⠀⠀⢷⡶⠚⠁⠀⠀⠀⠀⠈⠙⠋⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⡆⠀⠀⡰⠒⢣⠀⠀⠒⠄⠀⠘⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠉⠢⠄⠠⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠀⠀⠘⠿⠂⠀⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⢦⣀⡀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⢄⠈⢷⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣸⠃⢻⡀⠀⢀⣾⠀⠀⢹⡆⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣸⠇⢰⠀⠀⠀⠀⠀⢠⠈⢻⡄⠀⡄⠀⢠⣿⠀⠈⣷⠀⢸⣷⡀⠀⣸⡇⠀⠀⠀⢠⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢀⠀⣿⠀⠘⣧⠀⠀⠀⠀⣼⠀⠀⣿⠀⣧⠀⠀⢿⣷⣶⡿⠀⠸⣿⣿⣿⡿⠀⣧⠀⠀⣾⠀⠀⠀⣿⠀⣇
⠀⠀⠀⠀⢸⡄⠀⢀⣾⠇⢾⡀⣿⡄⠀⢸⣧⠀⠀⣼⣇⠀⣠⡿⠀⢻⣧⠀⠀⢻⣿⡇⠀⠀⢻⣿⣿⠁⠀⠹⣆⠀⢹⣷⣤⣶⡏⠀⢹⡄
⠀⠀⢀⣴⡿⠁⠀⣾⡟⠀⠈⣷⠘⣿⣷⣿⠇⡄⠀⢿⣿⣿⣿⠁⠀⢀⣿⠀⠀⠈⣿⣧⡀⢀⣾⣿⣿⡀⠀⠀⣹⡆⠀⢻⣿⣿⡀⠀⢀⣿
⠀⢀⣾⡟⠀⠀⠀⣿⣷⣄⣤⡟⠀⠘⣿⡟⠀⢳⡀⠈⢻⣿⣿⣄⣠⣼⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⡿⠁⠀⠀⣿⣿⣷⣶⣿⣿
⠀⠈⣿⣇⠀⠀⢀⣿⣿⣿⣟⠀⠀⢀⣿⡀⠀⢘⣇⠀⠀⢹⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⣿⣦⣤⣾⣿⣷⣶⣾⣿⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠶⢻⣟⣛⣛⣟⣛⣛⡷⠶⠶⣦⣤⣄⡀
⠀⠀⠀⠀⣠⡶⢛⣯⠴⠞⢛⣋⣩⣭⣭⡭⠭⢭⣭⣭⣟⣛⣓⠶⢭⣟⡻⢶⣄⡀
⠀⠀⠀⢰⡟⣷⢋⣴⠞⠛⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠙⠛⠺⣿⣷⣌⠿⣆
⠀⠀⢠⡿⣷⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣈⣙⣧⢹⣇
⠀⠀⣾⠁⠘⣶⣟⠷⣄⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣠⠏⡿⢿⡄
⠀⢰⣿⣄⣰⡇⠉⠓⣾⣏⣛⠳⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⢚⣻⣶⠚⠁⠘⣧
⠀⣾⠃⠈⠙⠳⢦⣼⣇⠀⠉⢹⠛⠒⠶⢾⡷⠤⠬⢽⡧⠤⠾⡷⠒⢻⡏⠁⣸⣦⠴⠚⢻⡄
⢠⡟⠳⢦⣀⠀⠀⠀⠉⠙⠓⠻⠶⠤⢤⣼⣧⣤⣀⣸⣧⣤⣤⠿⠶⠞⠛⠋⠉⠀⠀⠀⣸⣇
⢸⡇⠀⢀⡏⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠋⠁⣿
⣼⣇⠀⢸⡇⠀⠀⡇⠈⠉⢙⡟⠒⠶⠶⢤⡤⠤⠤⢤⡤⠤⠴⢶⠖⠚⢻⡏⠁⠀⣇⠀⠀⢿⡄
⣿⠉⠳⣼⣇⠀⢠⡇⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⢀⣠⣼⡇
⣿⣦⡀⠀⠉⠓⠾⣧⣀⣀⢸⡇⠀⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⢀⣇⣠⡴⠞⠋⠁⢸⡇
⣿⠀⠙⢶⣄⡀⠀⠀⠈⠉⠙⠓⠲⠶⠦⠿⢤⣤⣤⣤⠷⠤⠶⠾⠓⠛⠉⠉⠀⠀⠀⣀⡴⢻⡇
⣿⠀⠀⢸⠀⠉⢳⡦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢻⡏⠀⢸⡇
⣿⠀⠀⢸⠀⠀⢸⡇⠀⠉⢹⡟⠒⠶⠶⣦⠤⠤⠤⢤⣤⠶⠶⢶⡖⠛⠉⣿⠀⠀⢸⡇⠀⢸⡇
⣿⠳⣄⣸⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇
⢿⡄⠈⠙⠳⢤⣸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⡿⠀⢀⣸⡧⠞⢻⡇
⠈⣿⠲⣄⡀⠀⠈⠉⠓⠲⠾⣧⣄⣀⣀⣏⣀⠀⠀⢀⣇⣀⣀⣸⣇⣤⠴⠿⠚⠉⠁⠀⢀⣼⠇
⠀⢿⡀⠨⡟⠲⣤⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣠⣴⠞⢉⣟⣤⣤⡄
⠀⢸⡇⠀⣧⠀⠀⣏⠙⠓⢶⡤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⢶⡟⠉⠀⡏⠀⢸⢿⣿⢸⣯⣿⣿⣿⣿
⠀⠀⣿⠀⢿⠀⠀⣿⠀⠀⠈⡇⠀⠀⠀⡏⠉⠉⠉⢹⡏⠀⠀⣿⠀⠀⢸⠇⠀⢸⣧⠶⢿⠛⠛⠚⠛⣷
⠀⠀⢹⡟⠾⣄⡀⢹⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⣼⠀⠀⣼⠛⢶⣾⣀⣀⣀⣀⣿
⠀⠀⠈⣿⠀⠀⠉⠛⠳⠦⣤⣿⣀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢀⡇⠀⢀⣿⣠⡤⠿⠚⣿⠏⠉⢿⣼⠟⠁
⠀⠀⠀⠸⣿⢶⣤⣀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠒⠒⠛⠒⠚⠛⠛⠋⠉⠉⠀⠀⣀⣠⡟⠀⠀⠈⠉
⠀⠀⠀⠀⠙⢿⣇⠉⢻⠓⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⣴⠶⢚⡟⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠷⣼⣇⠀⠘⡇⠀⠀⢿⠉⠉⢉⡏⠉⠉⡟⠀⢀⡏⠀⣼⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢷⣤⣤⣼⣄⣀⣸⣇⣀⣸⣧⣤⣼⠷⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠚⠛⠲⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠚⠀⠀⠀⠀⠀⠀⣀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠖⠒⠲⢦⡀⠀⠀⠀⣀⡤⠤⣄⡀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢰⡏⢠⣞⣻⢮⠇⠀⢀⡞⣡⣤⡀⠈⢳⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡟⠀⠀⠘⣇⡀⢉⣡⠏⠀⠀⢸⡰⣿⣼⡿⠀⢸⡇⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡿⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠳⣌⣉⣀⡴⠋⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡇⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣀⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣼⡇⠀⠀⣠⡄⠀⠀⠉⠛⠛⠋⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣿⠗⣦⢤⣀⣀⣀⣠⣤⢶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⣧⡛⢦⣠⡁⠀⣇⣠⢜⡿⠁⠀⠀⠀⠀⠀⢠⣤⣤⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠧⣄⠀⠁⢸⣀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⢌⣙⣛⡋⠁⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⣀⣀⣠⣤⠶⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣟⠛⠉⠉⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡟⠻⣦⣄⣀⣀⣠⣴⠟⢙⡷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠈⠳⠶⣤⣌⣉⣉⣉⣠⡴⠛⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⠀⣀⣀⣠⠤⠴⠖⠒⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⢒⣚⣉⣩⠥⠴⠒⢆⠈⠻⡁⠀⠀⠀⠀⠀⠀⡿⠀⠱⣄
⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⢚⣋⣉⠥⠴⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠳⡄⠙⣆⠀⢀⣀⣠⣴⣛⣆⠀⠈⠳⡀
⠀⠀⠀⠀⢰⣏⠁⢠⠖⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⠻⣽⣿⣿⣿⣯⡴⣧⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠸⡏⢳⡀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠙⢿⣻⣿⣯⣿⠂⢘⣦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⡇⠀⠹⣄⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣆⠀⠹⣿⣿⣿⣿⣍⣤⣧⡤⠴⠖⠓
⠀⠀⠀⠀⠀⡇⠀⠀⠘⢦⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠴⠖⠒⢚⣉⣩⠥⠴⠖⢒⠻⡟⠛⠋⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢳⡀⠙⢄⡤⠤⠴⠒⠋⣉⣉⣠⡤⠴⠶⢒⣛⣩⡭⠴⠶⠚⠋⠉⢹⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢷⠖⠒⠋⠉⠁⠻⣤⠤⠶⢒⣚⣋⣩⠥⠶⠖⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⣧⡞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠘⡇⡇⠀⠀⣀⣀⡤⠤⠶⠒⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣠⣤⠴⣋⡗⠋⠉⠁⠀⠀⢀⢠⠊⠁⣀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⢖⣚⡿⠽⠟⠛⠋⠉⠁⠀⠀⢀⡤⣖⣯⡵⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣿⡧⠀⠀⣀⣠⡤⢴⣖⣻⠭⠿⠒⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠽⠾⠿⠯⠤⠚⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣓⡯⠭⠷⠒⠋⠉⠀⢠⡖⠒⠒⢒⣛⣋⣉⣉⣉⣉⣩⣭⣤⣤⣴⡶⠶⠶⠶⠶⠛⠢⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⣛⡿⣛⣽⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠈⠈

⣿⠋⠉⠋⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⢫⣍⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡟⠃⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⣿⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⣀⡾⣿⣿⡟⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⣿⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣤⣾⣿⣦⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢰⣿⣿⡄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⢰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣿⡥⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡼⠆⠀⢷⣤⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣀⣴⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⠛⡛⣟⢙⠛⣻⢛⣿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉⠉⠛⠛⠛⠛⠛⠛⠋⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠁⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠙⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⣀⣤⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⠛⠳⠦⣄⡀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣴⠟⠁⠀⢀⡴⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣤⡀⠈⠻⣦⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣰⠟⠁⠀⢀⡴⠿⠀⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠷⣤⡀⠈⢳⡄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⡼⠃⠀⠀⠀⢿⡀⠀⣀⡀⠻⣄⢀⣀⣤⣤⣤⣤⣀⣀⢀⣠⠟⢀⡀⠀⢨⡗⠀⠀⠹⣆⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⡾⠁⠀⠀⠀⠀⠈⠙⠛⠉⢙⡶⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⠳⣾⡋⠙⠛⠋⠀⠀⠀⠀⠹⣆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡋⠉⠉⠛⢦⠀⣰⠟⠉⠉⠙⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⢷⣄⣀⣀⡼⠛⢧⣀⣀⣠⡼⠃⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣹⠶⣄⠈⠉⠉⠀⠶⠄⠈⠉⠉⣤⠾⣏⣠⣤⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠉⢁⣶⢻⡇⠀⣀⠀⢀⠀⢀⠀⢸⡟⣦⠈⠁⠀⢙⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⢹⠆⠛⠚⠛⠶⠟⠓⠛⠳⠛⢸⡇⠀⢀⡶⠚⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠈⠉⠛⠲⠦⣤⣤⣤⣀⣤⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⣤⣤⣄⣠⣤⣤⠴⠶⠛⢹⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠙⠓⠶⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠴⠖⠛⠉⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠓⠒⠒⠶⠶⠒⠒⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠒⠒⠶⠶⠶⠶⠶⠶⠶⠖⠒⠚⠛⠛⠋⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⠛⣛⣛⣛⣙⣛⣟⣛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣉⣿⣿⣿⣿⣛⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡁⠀⢣⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⣀⡈⣰⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡼⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣿⣬⣟⣲⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡽⣿⣿⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣿⣿⣇⣿⡇⢠⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣿⣾⣿⡎⠛⢣⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠈⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠇⠀⣼⣿⠀⠀⠀⢤⣹⣿⣿⣿⣿⣿⣿⣿⣛⠞⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⡄⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⡿⢂⣠⣽⣿⣿⣿⣶⣿⣿⣿⣿⣟⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⢷⡉⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠛⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣤⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠉⠙⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⣠⣿⠀⠀⠀⠀⣸⣧⠀⠀⠀⢀⣰⣷⡄⠀⠀⢀⢰⣧⠀⠀⠀⠀⣾⣧⠀⠀⠀⢀⣼⣧⡄⠀⠀⠀⣾⣄
⠀⠀⢾⠿⡹⣿⠀⠀⢰⣿⣿⣿⠀⠀⢾⠟⠙⣿⠀⠀⢸⣾⣯⡤⠀⠀⢰⣿⣿⡆⠀⠀⣽⡟⠻⣷⠀⠀⣿⠏⢻⣶
⠀⠀⠈⠘⠃⠉⠀⠀⠈⠛⠛⠁⠀⠀⠈⠃⠐⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠐⠃⠉⠀⠀⠈⠀⠀⠁
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣄
⠀⠀⣰⣾⣿⣧⠀⠀⠀⣼⣿⠀⠀⠀⢀⣿⣷⡀⠀⠀⢀⣿⢷⣄⠀⠀⢠⣾⢿⡆⠀⠀⣠⣿⣿⡀⠀⠀⢠⣶⢿⡄
⠀⠀⠻⢿⡽⠟⠀⠀⠰⣿⣿⠟⠀⠀⠸⡏⠘⠟⠀⠀⠸⣧⣆⠟⠀⠀⢸⡇⠸⡇⠀⠀⢻⣿⣿⡟⠀⠀⢸⣣⡌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣰⣷⡄⠀⠀⠀⢰⣦⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⢰⣶⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣾⣆
⠀⠀⢴⠟⡈⣷⠀⠀⢨⡟⢹⣶⠀⠀⢰⡿⠹⣧⠀⠀⢀⣿⢿⣀⠀⠀⣰⣿⣿⡇⠀⠀⣤⣿⢿⣧⠀⠀⣤⡟⢹⣧
⠀⠀⠘⠚⠃⠋⠀⠀⠘⠃⠀⠋⠀⠀⠘⠃⠀⠃⠀⠀⠈⠃⠈⠃⠀⠀⠘⠻⠿⠋⠀⠀⠙⠋⠈⠋⠀⠀⠙⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠟⠛⠛⠻⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣤⠶⠶⠶⠶⠦⠄⠀⠀⠀⠀⠀⠀⠠⠴⠶⠿⠶⠶⣄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠟⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠿⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠾⠛⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠛⠷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣏⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠉⠉⠀⠀⣀⣀⣄⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢻⣯⣭⣭⣛⣛⣛⣿⣿⣭⣽⡟⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠈⠻⣄⣀⠀⠉⠀⠀⣀⣠⠞⠁⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠻⠶⠶⠶⠶⠟⠛⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠾⠿⣿⣤⣄⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣽⠿⠷⢶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⢶⣤⣀⡀
⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠈⠙⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠠⣁⠔⠒⠒⠂⠤⣈⡉⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢨⠎⠻⠀⢀⠤⠀⠀⠀⢈⠝⠀⠀⠌⢉⠓⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠁⠐⣾⣶⡀⠀⠒⢶⡒⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠄⢉⠓⠲⢤⡀
⠀⠀⠀⠀⢀⠞⢁⠂⠀⠀⠀⠀⠉⠙⢦⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣠⡟⠓⣦⣤⣀⡀
⠀⠀⠀⠀⣿⣲⣤⣌⡀⠀⠄⠀⠀⠀⠀⠉⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⡓⠬⣱⣿⣿⢯⣿
⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣶⣤⣌⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣼⡓⠬⣱⣿⣿⣷⠋⣸
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⢤⣈⡐⠀⠀⠀⠀⠀⠀⠀⡐⢀⣞⡓⢭⡽⣿⡿⡽⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠄⣠⣟⡲⢬⣿⣽⣟⡟⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠿⢦⣭⣿⣿⢿⠞⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠯⣿⣿⣿⣿⣿⡅⠀⠀⠉⠑⡇⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢽⠷⠦⣄⡀⠀⣇⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡰⠀⠀⠦⢔⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠞⠀⠘⡄
⠀⢰⠃⠀⠀⢀⡾⢡⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡘⣧⠀⠀⠀⢹⡄
⠀⣿⠀⠀⠀⠘⠷⢋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠳⠟⠀⠀⠀⢸⡇
⠀⠹⣆⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢀⡼⠁
⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⣠⣾⠟⠋⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⣼⡿⠋⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠰⡜⢿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣿⣇⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠘⡎⢿⣧⠀⠀⠀⠀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿⡀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⣠⣾⠟⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠻⣷⣾⣟⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⡿⠀⣼⡿⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠈⣿⡆⠀⠉⠛⠛⠉⠀⠀⠀⢀⡼⡇⠀⠀⠀⢰⣿⠇⣼⡿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢠⠏⠀⢧⠀⠀⠀⢸⣿⣾⡿⠁⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⡀⠀⠀⠹⣿⡀⠀⠀⠀⠀⣰⠃⠀⠀⠘⡆⠀⠀⠀⠉⠉⠀⠀⠀⡄⣼⠃⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣷⡀⠀⠀⣿⡇⠀⠀⠀⢰⠇⠀⠀⠀⠀⢹⡀⠀⠀⢀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢀⠞⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡘⢿⣦⣤⣿⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣇⠀⢀⠜⠀⠀⠀⠀⢣⡹⢦⣤⠴⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⠓⠮⠭⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠉⢂⠱⡀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠑⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⢿⠢⠞⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠐⢂⡿
⠀⢶⣶⡌⢷⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⡾⣡⣾⠗
⠀⠀⣿⠙⠟⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠛⢋⡿
⠀⠀⠘⢧⣀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣠⡾⠁
⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⣞⣿⣻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⢰⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⡦⣄⡀⠀⠀⠀⠀⠀⠸⢏⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣄⠀⠀⠀⢀⡞⠁⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣽⠁
⠀⠀⠀⠀⠀⢠⣶⣲⣖⣶⢦⣄⠉⠛⠿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢸⠋⠛⠻⡅
⠀⠀⣠⣤⣤⣿⣿⣛⢿⣻⣿⣿⣽⣦⣄⠀⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⡟⠀⠀⠀⢳⡀
⢀⣾⣿⣿⣾⣿⣾⣝⣿⣿⣿⣿⣿⣿⢿⣷⡄⠀⠀⠀⠙⠻⢿⣿⣿⠿⠟⠀⠀⠀⡜⠁⠀⠀⠀⠈⢷⡀
⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣶⣶⣄⠀⠀⠀⠀⠀⢀⠎⠄⠀⠀⠀⠀⠀⠀⠳⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡄⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠘⢯⣻⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠙⠻⠿⠯⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⠞⠁⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠸⠀⢀⣠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠂⡀⠀⠀⡒⠠⢄⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠂⠂⠀⠠⠤⠄⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢐⡅
⠀⠀⠀⠀⠀⡀⠀⠀⠢⠀⢀⡀⠌⠀⠂⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⢀⡌⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠐⠒⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠁⠀⠀⠈⠒⠤⠄⣀⡀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠄⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⢰
⠀⠀⠑⠠⢀⠀⠀⠀⠀⢀⡀⠄⠊⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⣀⡀⠀⠁⠀⠀⠀⢐⠁⠐⠈⠉⠀⠉⠁⠄⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠂⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⡰⢀⠋⠀⠀⠁⠀⠀⠈⠁⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⢀⠁⠀⠀⠁⠓⢴⡂⠀⠀⠀⠀⠀⡀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⠂⠀⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠧⣤⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣀⡀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠂⠉
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠠⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⢠⡀⠀⠤⠤⠄⠐⣨⡰⠀⠀⠀⠈⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⢨⠀⠒⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⢈⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢀⠠⠑⠉
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠠⠁⠀⠀⠀⠀⠀⠀⠂⠠⠤⠤⠐⠀⡠⠔⠚⠉⠉⠉⠁⠓⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠄⡊⠁⠀⠀⠀⠀⠀⠀⠀⡠⣐⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⢀⣁⠂⠀⠀⠍⠈⠒⢄⡠⡀⠀⠠⠀⠕⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡀⠀⠀⠀⠀⠂⠈⠀⠀⢀⠄⠈⠀⠀⠀⠈⠐⠠⢀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠠⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⡰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢄⠀⠀⡆⠀⠀⠀⠀⠀⢀⡄⠀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠⠀⠀⠀⠀⣠⡤⢠⠀⠃⠀⣘⠀⢀⡏⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠘⠦⠂⢠⠊⡜⠁⠈⠀⢡⡀⠘⠦⠚⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⢧⣀⠄⠀⠀⡇⠀⢀⢻⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⣆⠀⠀⠇⠀⠀⠀⠀⢰⡥⠔⠁⡸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠸⡌⠂⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⢃⠀⠈⠓⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣫⠟⣠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢀⡈⠑⠊⠿⣶⣲⠤⠶⠦⠖⠶⠭⠐⠋⠁⠀⠒⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠐⠂⠀⠀⠩⣒⡦⢤⣀⡈⢉⠥⠔⠒⠈⠁⠀⡀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⣄⣀⠀⢀⠀⠀⠐⠊⠉⠀⠀⠀⠠⠂⢀⡀⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠚⣁⠄⠀⢀⡀⠠⠀⠀⡀⠀⠒⠊⠉⣁⠒⠄⠤⢀⡀
⠀⠀⠀⠀⠀⠀⡰⢚⡑⢂⠀⠀⠀⠀⣀⡠⠤⠒⠊⠁⠤⢀⣐⡦⠤⢀⠀⠀⠀⢀⠞⠩⣁⢀
⠀⠀⠀⠀⠀⠀⠳⣎⣁⣾⠠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⣄⡀⢸⠀⠄⣰⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⠛⠉⠀⠀⠀⢠⡐⣯⣦⠀⠀⠀⢠⡾⠋⠁⠀⡀⠀⠀⠀⠈⠉⠉⢻⣿⡿⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⠀⠀⠀⠀⣨⣓⡶⣶⣷⣿⣿⣇⠀⢠⡟⠀⢰⣸⣀⡇⠀⠀⠀⠀⠸⣀⢂⣓⣃⣈⣭⠙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢁⠀⠠⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠛⠙⠛⠻⣿⣧⢴⣝⣿⣿⣿⠀⣀⠀⠀⠀⠽⠈⠹⡿⡟⣿⣴⡠⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⢡⣍⠰⠁⠁⠀⠀⠰⠶⢿⣻⣿⣿⠉⠁⠀⠀⠀⠘⣿⣿⣿⣷⣿⣯⣾⣇⠀⠀⠰⣄⠠⡀⠖⠩⠚⡯⠕⠒⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⡼⣿⠸⢄⠀⠀⠀⠀⠀⣿⡿⢿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡟⢿⣄⠀⠐⠹⡄⠐⡀⣠⡚⠈⠢⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣶⣄⣧⢸⠋⡡⠊⣠⠞⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣳⠗⠦⢤⣀⣘⠶⣦⠵⠖⠚⠃⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣟⠿⣆⣞⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡯⠓⠆⠀⠀⢉⡉⠑⠶⠒⠒⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣮⠟⡋⡠⠼⠗⠁⠀⡢⡦⢛⠆⠀⠀⠀⠀⢠⣼⡏⠻⣿⣯⣏⣿⡷⠀⠀⣠⣌⡉⣶⡔⠺⠒⠄⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡌⣳⢷⣆⠀⡋⡴⣐⠉⠁⠀⠀⠀⠘⣦⣴⠟⠀⠀⢿⣿⣿⣿⣷⠀⢢⣿⡿⢒⢘⠶⣧⣶⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣌⣍⣻⣆⢑⡌⢫⡀⡄⠂⠀⢠⣴⡿⠋⠀⠀⠀⣴⣿⣟⣿⣿⣿⣦⣤⣄⣀⣀⣠⣩⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡿⣞⠺⣅⡀⢷⣄⣴⣾⣿⣿⣿⡄⠀⠀⠀⣿⣿⠋⠙⠛⠛⠻⠿⠿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣿⣷⡿⠟⣛⡿⠿⠻⣿⣿⡄⠀⢸⣿⣿⣷⣶⠶⠶⠤⠼⣃⣀⡀⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠧⠖⠊⠉⠀⣀⣽⠴⠿⢿⣧⠀⣿⣿⡇⠀⠉⠙⢓⡒⠂⠬⠽⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣈⣧⡴⠖⣉⣡⣤⣴⠾⠟⠛⠀⢿⣿⣿⡛⠓⠶⠮⣥⣄⣀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣶⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠁⠈⠉⠉⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢛⢾⡇⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⡶⢓
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⢿⡇⠀⠉⢀⣙⣻⡶⣦⡀⠀⠀⢀⣤⠶⣟⣋⣉⠉⠀⠀⢸⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡇⠀⠐⠋⠹⠟⠙⠄⢠⡄⠀⡤⠀⠎⠛⠏⠈⠁⠀⠀⣸⡁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢆⣸⠀⠀⢀⢰⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣀⣬⣽⣶⢲⣭⣤⣤⣀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢰⠛⠋⠉⠉⣀⢊⡉⠉⠉⠛⠻⠆⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢀⡀⠀⠲⠤⢤⣥⡤⠔⠁⠀⢡⣆⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠺⣿⣄⡀⠀⠉⣉⡁⠀⢀⣴⣾⠟⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣾⣿⠀⠀⠙⢿⣿⣷⣾⣿⣿⣶⣿⡿⠋⠀⠀⠀⠙⢻⠒⢤⣄⡀
⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠸⣈⠉⠀⠀⠀⠀⠈⠉⠛⠛⠃⠀⠉⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠉⠛⠶
⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠖⠋⠁⠀⡴⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠋⠉⠀⠀⠀⠀⣠⠞⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠄⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣇⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⠇⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⢻⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠈⣿⣏⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢸⠀⠀⠀⠀⠀⡼⠛⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⡇⠀⠁⢸⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⡇⠀⡆⢸⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⢸⠀⠀⠀⠀⠀⡇⢸⡇⢸⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⠀⡇⢸⡇⢸⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢠⡇⢸⡇⢸⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⡇⢸⡇⢸⠀⠀⠀⠀⠰⡟⢻⣿⡇⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⡇⢸⡇⢸⠀⠀⠀⠀⢰⡅⢸⢹⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⡇⢸⡇⢸⠀⠀⠀⠀⢸⡇⢸⢸⡇⠀⠀⠀⠀⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⡇⠸⡇⢸⠀⠀⠀⠀⢸⠀⢸⢸⡇⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⠀⠀⠀⢸⠀⢸⢸⡇⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⡇⢰⡇⢸⠀⠀⠀⠀⢸⠀⢸⢸⡇⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⠃⠀⡇⢸⠀⠀⠀⠀⢸⠀⢸⢸⡇⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⢸⢸⡇⠀⠀⠀⢸⠁⢹⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⢸⠀⠃⠀⠀⠀⢸⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠸⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢠⡄⠀⠀⠀⠀⠀⠂⠒⢢⡄⠀⠀⠀⠀⠘⠐⠒⡤⠀⠀⠀⠀⠘⠐⠒⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠙⠀⠀⠀⠀⠚⠋⠙⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣶⡄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⠉⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠁⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣁⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⣿⣿⠤⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠻⡋⠉⠉⠉⠙⢛⠿⣿⣦⠀⢀⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠈⠐⠂⠒⠊⠁⠀⠈⣻⣶⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣀⣀⣀⣠⣤⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣤⡀⢬⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠘⣿⣿⣦⠹⣧⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢠⠀⠀⠀⠀⢹⣿⣿⡇⠟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠲⣦⡀⠀⠀⠀⠘⢠⣿⣇⠀⠀⠀⢸⣿⣿⡇⠀⡀⠀⠀⠸⣷⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠉⠹⠀⠀⠀⠀⣼⣿⣿⣶⣶⣶⣿⣿⣿⠀⠀⢻⣷⣄⠀⠈⢿⡿⠄⢹⣧
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢸⣿⡇⠀⠀⠀⠀⢳⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣠⣼⣿⣿⣇⠀⠘⡏⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣾⣿⠃⠘⣶⣄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢰⡄⣰⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣦⣴⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⢃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⡇⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠃⠀⢀⣾⠟⠁⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣌⠀⠀⣠⣿⡏⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣠⡾⢻⡿⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣠⣾⡿⣡⣿⣧⠆⠘⠁⢸⣿⠀⡀⠀⠷⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢻⣟⣵⣿⣿⣇⠀⠀⣠⡾⣿⣿⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⣿⣿⣿⣿⣦⡾⣽⣿⣹⣟⡽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢯⣾⡟⣱⣿⣿⣿⣿⡟⣱⣿⣿⣧⠻⣯⠜⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣿⢣⣿⡟⣼⢿⣯⣿⣿⣽⢣⠟⣿⣿⠹⣦⠙⢧⣀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡀⡆⢸⣿⠃⣿⡿⢰⡟⣿⡿⣿⣿⣏⠸⡆⢸⣿⣿⣿⡄⡈⢻⣧⡈⢀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣼⣿⡇⣿⢱⣿⣿⣿⣿⡿⠇⠁⣾⡈⣿⣿⣧⢰⡀⢻⣿⣆⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣿⣿⣿⠀⢿⣰⡇⡿⠁⣘⡇⢀⣼⣿⠇⢸⢻⣿⢸⡇⠀⣿⣿⠆⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢿⡄⢿⢻⣿⣿⣀⢸⣿⡇⠀⠀⣧⣷⠘⢿⡏⠀⣿⢹⡏⣸⣇⡀⢸⣿⡇⠀⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣷⣿⣿⣿⣌⣿⣧⣀⠀⠻⣿⠆⠈⣿⣰⢃⡾⢡⣿⣿⡁⢸⣿⡇⠀⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠻⣿⡿⠿⣧⣼⡻⣷⡄⠉⠀⣰⣯⣿⡾⢡⡞⢹⣿⠁⣸⣿⡇⠀⣿⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣙⠓⠪⡿⠶⣿⣷⣿⣮⡻⣦⣬⣿⡿⠋⠐⠋⠀⣼⠟⣰⣿⡿⠁⢠⡏⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢽⣿⣿⣿⣷⣬⣉⠉⠁⠀⠀⠴⠊⣅⣼⣿⠋⠀⢠⠟⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⡏⢿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⡿⠟⢀⡴⢋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣽⣿⣿⣿⣿⣿⣿⠿⢛⣋⣤⠶⠋⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣷⠶⣿⠶⠶⠊

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡟⠿⡿⠛⡿⠿⣿⠿⢿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣷⣾⣶⣷⣶⣿⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠂⠄⢀⣀⠀⠀⣀⡀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣀⣀⣀⣴⣦⣤⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡟⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠑⠦⣄⠈⢷⡈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠻⣎⠻⣦⣀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠸⣆⠀⠀⠀⠀⢹⡆⠈⠙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⣦⠀⠹⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⠁⠀⠘⣇⠀⠘⢷⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢻⡀⢀⡾⠃⠀⠀⠀⠘⢧⣀⣸⡇⠀⢻⣆⠀⠀⠀⠀⠀⠀⣇⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡼⠃⣾⡁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢦⡀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⠀⠀⠈⠻⢦⣄⣀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠻⢦⣄⡀⠀⠀⠹⣦⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣆⠀⠀⠀⠀⠈⠉⢿⠓⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡟⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣰⠃⠀⠀⠀⠀⣠⠞⣿⠇⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⡆⠀⢀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⠞⠁⠀⢀⣤⠶⠛⠁⣠⡟⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⡇⢰⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠐⣇⠀⣤⠞⠋⠀⠀⠀⠰⠋⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡇⢸⡇⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠈⣻⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⡀⠀⡼⠃⠀⢀⡼⠁⢸⡇⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢰⡄⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⡇⠊⠀⠀⠀⣾⠤⠒⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⡄⠀⠰⠾⣡⣶⣟⣯⣤⣦⠀⠀⠈⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣦⡀⠀⠀⠀⠀⢀⡀⠛⢿⣷⣦⣤⣠⣴⣿⣷⣿⣿⠟⠃⠀⠄⠀⠂⠈⢁⣤⣶⡆⠀⠀⠀⠀⠀⣠⡟⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠶⠾⠿⣶⣤⡀⠰⠄⠠⣤⠄⢈⣩⣥⣾⠿⠿⠛⢉⣤⣄⡀⠰⣶⡿⠿⣿⣛⡉⣁⡀⠀⠀⢀⣤⡾⠛⠛⠛⠛⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⠛⠛⠋⠉⠉⢛⣿⣶⣦⣤⣤⣿⡿⠛⠿⣶⣶⣦⣬⣽⣿⣿⣿⣶⣄⠈⠙⠛⣛⣩⣷⡶⢾⣿⣿⣿⣿⡿⠛⠛⠛⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⢧⣤⣤⣤⠶⠶⠒⠛⠛⠋⠉⠀⠀⠀⠀⠀⣀⡤⣶⣿⣿⣿⡿⠿⢿⣿⣿⣛⣛⣻⣛⡻⠿⢿⣶⡶⠶⠦⠬⣭⡉⠛⠛⠓⠶⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠋⢁⣚⡯⠟⠛⠉⣀⣤⣄⠀⠀⠀⠉⠁⠉⠉⠙⠒⠲⠤⣀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⣻⣅⣤⡤⠶⠚⠋⠉⠁⠀⠰⠞⠛⣁⣀⣘⠛⠉⠙⠛⠒⠒⢦⣤⣤⣤⡶⠒⠾⢭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⠉
⣀⡀⣀⣀⡀⣀⣀⣀⢀⣀⢀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠
⠤⠀⠀⠀⠀⠀⠀⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠠⠄⠠⠀⠄⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠁⡀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠬⢠⣀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢰⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠒⠋⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⢿⣿⣿⡗⠶⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠈⠁⠐⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡯⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠒⠂⡀
⠀⠀⠀⠀⠀⠘⠿⠾⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠈⢂⣴⣶⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣐⣠⠄⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣾⠿⠿⠾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⣿⣾⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠘⢤⡀⠀⠑⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠛⠙⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠈⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠠
⠀⠀⠀⠀⠀⠀⠀⠁⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣶⣦⠀⢰⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠶⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⠶⢶⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠛⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣠⡾⠟⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢀⣴⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣠⣾⣧⣤⣀⣤⣀⢸
⢀⣴⣿⢻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠃⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣴⡿⢿⣿⣿⣿⣉⣿⣾
⣾⣇⣧⣇⢏⠏⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣤⣀⣀⠈⠉⠀⠈⠀⠈⠁⠈⠑⠒⠂⠠⠤⠶⠆⠀⠟⠀⣼⣿⣷⣿⣿⣿⣿⣿⣿⢸
⣿⡏⡿⣿⡼⡜⡌⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⠿⢷⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⣼⠏⠉⣿⠛⠛⣿⠟⢻⣿⢸
⠘⢿⣧⣿⣿⣷⣥⣌⡝⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡟⠀⢸⡏⠈⣿⠉⠙⣿⠛⠛⣿⠟⢿⣿⠀⠀⣿⠀⠀⣿⠀⢸⡇⣿
⠀⠈⠛⠿⣿⣿⡿⠿⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠁⢰⡏⠀⢀⡇⠀⠀⡟⠀⢸⣿⠀⢀⣿⠀⢸⡏⠀⣾⠇⣿
⠀⠀⠀⠀⠈⢻⣆⢰⣄⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠃⠀⣾⠀⢸⡇⠀⢸⡇⠀⢰⡇⠀⢸⡇⠀⢸⡏⠀⢸⡇⠀⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠹⣧⡙⢷⣄⠀⠈⢳⡀⠀⠀⠀⠀⠀⣾⡇⠀⣸⠀⠀⡏⠀⣸⠁⠀⢸⠁⠀⢸⡇⠀⢸⡇⠀⣼⠇⠀⣿⠁⢰⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠹⣧⡀⠀⠙⢦⠀⠀⠀⠀⣿⠁⠀⣿⠀⢰⡇⠀⣿⠀⠀⣿⠀⠀⣾⠁⠀⣾⡇⠀⣿⠀⢀⣿⠀⢸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⢻⣦⠀⠈⠳⣄⣠⣾⣿⣶⣶⣿⣤⣾⣧⣴⣿⣄⣠⣿⣄⣠⣿⣀⣀⣿⠀⢸⡟⠀⢸⣿⠀⣼⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢷⣄⠀⠙⢿⣃⣀⣀⣀⡀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⢶⣿⣷⣤⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠻⣦⡀⠈⠻⣿⡽⡻⡻⡻⡿⡿⡿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠉⢻⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠈⢻⣄⠀⠈⢻⣷⡷⣼⣾⣼⣼⣧⣏⣾⣏⣾⣣⣏⡾⣹⡹⣹⢫⢏⡟⣿⠁⢸⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢧⡀⠀⠙⢦⡀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠿⠿⠛⠀⢸⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⢻⣆⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⣷⡀⠀⠘⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⢻⣦⠀⠈⢻⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠙⣷⡄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣦⡀⣹⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣁⡀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣡⣴⣾⣿⣛⢿⣿⡿⢿⣦⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣡⣴⣾⣿⣿⣿⠿⠛⠛⠿⠿⠿⣷⣝⢿⣦⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢋⣡⣴⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡻⣷⣄⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⣁⣴⡾⢟⣻⣿⡿⠟⠋⠁⢰⡶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⣻⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠁⠀⠀⠀⣿⡇⢰⣿⠋⠁⠀⠀⠀⠀⠈⡇⠀⢹⣤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠹⣿⣟⢿⣦⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠙⣦⠀⠀⠀⣠⣿⠇⣼⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⠃⠀⠙⣆⠀⠀⠀⠀⠀⢀⣾⡟⢀⣿⡟⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⢀⣠⠶⣛⣷⡄⠀⢿⣯⡸⣿⣅⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⠀⠀⠀⠀⢻
⠀⠀⣀⡴⠞⠋⢁⠀⢀⣤⣞⣯⠶⠛⠉⣀⡽⣦⡀⠹⣿⣜⢿⣦⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣠⣴⡿⣃⣼⣿⠇⣀⣤⠞⠋
⣴⣟⠉⠀⣴⣿⣿⣿⡿⠛⠉⠀⣠⡴⣚⣭⠞⠋⣳⣄⠈⢿⣷⡹⣿⣄⠀⠀⠀⠀⠀⠙⠓⠶⠞⠋⠁⠀⢀⣤⣶⡿⣻⣽⡾⠟⣩⡴⠞⢉⣤
⣿⠈⢷⡘⣿⣿⣿⣿⣧⣤⣴⣻⡵⠞⢉⣠⡴⣻⡽⠛⣦⡀⠙⣿⣮⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⢟⣫⣷⠿⢛⣡⠶⠛⢁⣤⣾⣿⣿
⣿⠀⠀⠹⣎⠉⣽⣿⣿⣿⣿⢁⣠⠶⣫⡵⠛⣡⡴⠞⣩⡷⣄⠈⠻⣷⣝⠿⠟⠿⣷⣦⣠⣴⡿⣛⣭⣾⠟⢋⣥⠞⠋⣁⣴⣾⣿⡟⣻⣿⣿⠀⢸
⣿⠀⠀⠀⠈⢷⡜⠛⢻⣿⣿⣿⡷⠛⣡⡴⠛⣡⡴⠛⣡⡴⢛⣦⡀⠙⢿⣶⣶⣶⣤⣙⣫⣵⡾⠟⣉⡴⠞⠉⣀⣴⣾⣿⡟⢻⣿⣷⣿⡿⠛⠀⣸
⣿⠀⠀⠀⠀⠀⠙⣦⠻⠿⢿⣿⣷⣾⣧⡴⠛⢁⡴⢞⣡⠶⣫⠟⣳⣄⠀⠀⠀⠈⠙⠻⢛⣡⡴⠛⠉⣠⣴⣾⣿⡟⢻⣿⣷⣿⣷⠟⠁⢀⣤⠞⠋
⠸⣆⠀⠀⠀⠀⠀⠈⢳⡄⣾⣿⣿⣿⣯⣤⣶⣫⠶⣋⡵⣋⡵⢛⡵⢛⣧⡀⠀⢀⣠⠶⠋⠁⣠⣴⢿⣿⠏⣻⣿⣷⣾⣿⠟⠋⢀⣤⠞⠋
⠀⠙⢦⡀⠀⠀⠀⠀⠀⠙⣮⡉⣿⣿⣿⣿⣏⣥⢞⣽⠞⣩⠞⣫⡴⠋⢁⣽⡞⠋⠀⣠⣴⣿⣿⠿⣿⣿⣴⣿⣧⠾⠋⢁⣠⠞⠋
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠳⣜⠛⣿⣿⣿⣿⣫⡴⣛⡵⠞⠁⣠⠶⠋⠀⡇⠀⢾⣿⣷⣿⣏⣸⣿⣽⡿⠛⢁⣠⠶⠋⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠙⢾⡿⢿⣿⣿⣿⡾⠋⣠⡴⠛⠁⠀⠀⠀⡇⠀⢸⡇⣾⡟⣿⡿⠛⢁⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠈⠻⣌⠿⠿⢟⡵⠞⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⠟⢉⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠘⢷⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⠋⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢸⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢸⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣆⡴⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⣀⣀⡀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠔⠊⠉⠀⠀⠈⠙⠢⣄⠀⠙⠲⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣉⠙⠒⠒⠒⠒⠒⠒⠒⠢⢄⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⠭⢳⡶⢲⢲⡞⣷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢳
⠀⠀⠀⠀⠀⠀⢠⠇⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡱⠋⠀⢰⢳⡇⢸⢸⡇⣿⠈⡏⡆⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⣸⣸⣇⣿⣸⣇⣿⣠⣇⣇⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢧⡀⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⣞⠀⠀⠀⠀⠀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣭⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣳⣄⡀⠀⠀⠀⠀⢱⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠈⢧⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢣⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⢳⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢧⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⢧⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢳⠧⠤⠤⢤⡤⠤⠤⠤⠤⠤⣤⡤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⢤⡤⠤⠤⠬⡿⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⠀⠀⡴⠋⠘⢦⡀⠀⢀⡜⠁⠙⣄⠀⠀⣠⠞⠈⠳⡄⠀⠀⡰⠋⠈⢦⡀⠀⣧⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣠⣞⣀⣀⣀⣀⣱⣴⣋⣀⣀⣀⣈⣳⣴⣁⣀⣀⣀⣘⣦⣞⣁⣀⣀⣀⣙⣄⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉

⣿⠟⠉⠛⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢋⡍⠛⠉⠛
⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠀⠀⣠⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣷⣾⣿⣿⣿⣆
⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙
⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠿⢿⣿⣿⠛⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠⣴⣶⣤⣾⣿⣿⣦
⠀⠀⠀⠀⠠⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠉⢻⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠟⢹⣿⣿⡟⠛⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣀⣤⣶⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢀⠈⠈⠁⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⣤⠰⠛⠆⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡿⢻⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⢻⣿⣿⠉⠛⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣠⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠛⠛⠛⠛⠳⠶⢾⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣠⣄⠀⠀⠀⠀⠀⠀⣡⡶⠿⢷⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣴⣤⡀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠹⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⣿⡇⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⡄⣸⣿⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣿⠀⠀⠀⠀⣿⡀⠀⠀⣸⣿⠃⣠⣾⣿⣿⣽⣻⣿⣿⣿⣶⣜⠁⠀⣠⣆⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣸⡿⣀⣀⣀⡀⠹⣷⣤⣼⣿⣷⣾⣿⣿⣿⣿⣿⣻⢿⢿⣿⣿⣿⣿⣦⡛⠉⠓⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⡿⣿⣭⣡⣤⣤⣭⣤⣬⣿⣿⣿⡿⢿⡋⠁⠀⠀⢸⡟⠻⣶⣼⣿⣿⣿⣿⣿⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⠋⠁⠀⠈⣇⠀⢀⣀⣼⣧⣤⣿⣿⣿⣿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⣾⣿⣿⣿⡏⠉⠀⠀⣸⣅⣠⣤⣴⣿⣿⣿⡿⠟⠛⠻⣿⣿⣿⣿⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣤⣴⣿⣿⣿⣿⣿⣾⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⠈⣿⠿⠿⠿⣿⠟⠛⠉⠉⢿⣁⣴⣿⡟⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⠻⣶⣶⣤⣄⣿⡀⠀⠀⢿⡀⠀⠀⢀⣼⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⠁⠀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣶⣷⣾⢿⣿⣿⣿⣷⣾⣷⣶⣾⣿⠿⣋⣴⣿⣿⣿⣿⣿⣿⣿⣧⣿⠶⣫⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣫⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣻⢿⢿⣿⣿⣿⣿⣿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣭⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢯⣿⣿⣿⡍⠉⠉

⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⢰⣿⣆⠀⠀⠀⠀⠀⠀⠐⣿⣷⠀⠀⠀⠀⠀⠀⡀⣸⣷⠀⠀⠀⠀⠀⣴⣿⣦⡀
⠀⠀⣴⣿⠏⠙⣿⣤⠀⠀⣰⣼⡿⣿⣷⢀⠀⠀⠀⠐⣿⣿⣿⣤⡄⠀⠀⠀⢰⣿⡿⣿⣤⠀⠀⠀⡀⣿⣿⢻⣿⡄
⠀⠀⢿⡏⣰⣆⢹⡿⠀⠀⢻⡿⠃⠘⢿⣿⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠐⣾⠟⣀⠘⣿⠀⠀⠀⣿⡟⠁⠀⢿⡿
⠀⠀⠈⠓⠝⠋⠊⠀⠀⠀⠈⠓⠀⠀⠘⠁⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠙⠐⠿⠂⠋⠀⠀⠀⠘⠃⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⢀⢸⣧⡀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⢠⣼⣿⣿⣿⣇⠀⠀⠀⢀⣼⣿⣿⣧⠀⠀⠀⢠⣄⣿⣿⣿⣀⡀⠀⠀⠀⣼⣿⣿⡄⡀
⠀⠀⢀⣿⣿⣿⣴⡆⠀⠀⣸⣿⠟⠙⢿⣿⠆⠀⠀⣾⣿⠏⢹⣿⡇⠀⠀⢺⣿⡿⠃⢻⣿⡇⠀⠀⣶⣿⣿⣿⣿⣧
⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠘⢿⢠⣷⠈⠟⠀⠀⠀⢻⡇⠀⠘⢻⠃⠀⠀⠘⠯⢀⣶⡄⠿⠁⠀⠀⠻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⣀⣼⣿⣦⠀⠀⠀⠀⡀⢿⣧⣷⠀⠀⠀⠀⠀⣾⣿⣧⣄⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⣾⣿⡇⢀
⠀⠀⣼⣿⡿⠻⣿⣇⠀⠀⢸⣷⣼⣿⠃⣠⠀⠀⠀⣄⣿⠏⠻⣿⡆⠀⠀⠀⢸⣿⣿⡟⢀⠀⠀⠀⣄⣿⣿⣿⣿⣀
⠀⠀⢻⣏⠀⠀⢹⡟⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⢿⡏⣠⣆⢹⠇⠀⠀⠐⣾⡿⠈⣿⣿⠀⠀⠀⣿⣿⡏⠘⣿⡿
⠀⠀⠀⠉⠒⠒⠋⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠈⠁⠀⠘⠁⠀⠀⠀⠈⠃⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⣉⣩⣭⣽⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⣡⣤⠶⠖⠒⠒⠀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠭⠭⠭⠭⠭⠭⣭⠽⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⠀⠀⠀⠈⣁⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿