Explorer Emojis & Symbols

⠀⠀⠰⣿⢀⣄⠀⠀⠀⠀⠀⠈⣉⣹⣶⣦⣷⣤⣶⣿⡟⠉⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣤⣄⣠⣴⣤⣤⣤⡀⣀⢀⡶⠛
⠀⠀⠀⠈⠉⠙⢷⣄⠠⠀⠀⠀⠀⢀⠛⠋⠁⠀⠀⠀⢀⣀⡤⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠻⠛⠛⠛⠛⠋⠛⢻⣿⠛⠛⠻⡿⠃
⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⡂⠰⠦⣈⡿⢧⠄⠀⠀⠀⠀⠉⠓⠦⢀⠀⠐⠛⠠⡄⠀⠀⠀⠀⠀⠠⣄⠀⠀⠒⠐⣤⢄⣠⣤⠀⡄⠀⡐⠸⣿⡄⣠⠤⠟⠀⠘⢇⡀
⠀⠀⢀⢀⣀⡖⠖⣀⢀⠀⠠⡐⢶⣄⢀⡾⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠳⠶⣔⠃⠀⠀⠀⠀⠰⠗⠈⣷⠀⠀⠸⠀⠀⢉⠇⠈⠁⢠⠇⢤⣿⣧⡏⢀⠀⣠⡄⠀⠛
⠀⠀⣠⠔⠁⠀⣸⡇⠈⣷⠂⠉⢀⡈⠛⢤⣅⣨⠀⢠⡀⢰⠀⢀⠄⠀⠀⡀⠈⢳⠀⠀⠀⠀⢀⣴⣿⣼⢧⠀⠚⠤⠰⠦⡀⠀⠀⣾⠰⣆⣰⢉⢀⡉⣠⠈⠳⣄⠤
⠀⠞⠁⠀⢠⣼⣿⡇⠀⢻⠀⡀⠈⢃⣦⠀⠙⠃⠀⣾⣿⡀⠀⠛⠀⠀⢠⡇⠀⢸⢀⣀⡈⠛⠛⡿⢻⡁⡌⣀⠀⠀⠀⠐⣱⠀⢺⡇⢀⢡⡏⢻⠀⠁⠙⠃⠀⠀⠤
⠀⢀⠀⠚⠈⡿⢿⡁⠀⠈⠀⠁⠀⠈⠆⢰⡄⠀⠈⡜⠻⠀⢀⡈⠀⠠⠊⠀⢰⣾⣿⣿⣏⡆⠀⠘⠈⣟⣠⡟⠐⡆⠠⡤⠟⠀⡿⢀⡈⣧⠀⠀⠰⢦⠀⠟⠙⠓⠖
⠀⢻⣿⠀⢰⠀⣼⠋⠀⢀⠀⠠⠀⠀⠈⣿⣦⣤⠀⢰⡄⠀⠋⠙⠒⠇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠘⡇⠉⠿⠟⢻⠶⡷⠀⢰⠀⣶⠞⠆⠀⠙⠳⣎⠙⠆⢰⠘⣥
⠀⢸⣿⡆⢸⠆⠙⠃⠀⠂⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⠀⠀⠐⠉⣀⡈⠛⠿⠛⠿⠟⠓⢀⣤⠴⡷⠦⣀⠀⢠⡦⠀⢨⣤⣤⣥⠀⠀⠐⠲⠿⠊⠁⠀⠀⠈
⠀⠘⣿⡇⢸⠁⣠⡀⠀⣀⡀⢠⣄⢀⣤⣆⠀⢀⣀⡀⠚⠓⠠⡄⠲⣤⡋⣡⣤⠄⠀⠀⠀⢀⣀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢿⣿⡄⠀⢸⡄⢠⡭⠷⢶⣿⣿⣿⣷⠄⠠⠤⡉⢉⠐⠶⡌⠃⠙⠁⠛⣋⢀⠀⠤⣄⠉⠉⠁⠀⠀⠀⠀⢀⠄⠀⠢⡰⠄⠀⠀⠶⠀⠈⠁⠀⠀⠀⢠⠶⡴⠶
⠀⣿⣾⡷⠛⢠⣿⣡⠀⠒⠐⠂⠀⠶⠠⣤⢄⠀⠀⠸⠀⡈⣯⣄⠀⠀⢠⣌⡉⢶⡜⢶⡌⠀⠀⠈⠀⣀⣀⣘⡴⠄⠀⠂⠐⠲⠖⠉⠁⠠⠌⠉⠀⠀⠰⠏⠀⠀⠐
⠀⣿⣿⣷⡀⠸⠋⣀⡀⢠⣤⣄⢤⣄⣀⡀⠀⠉⠐⠓⠤⠀⣀⠀⠀⠀⠀⠈⠀⠀⠽⠀⠀⣀⣀⣀⠲⣿⠛⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠀⠭⢐
⠀⣿⣿⠉⠀⢶⡈⠉⠈⠁⠀⠁⠀⠉⠉⠀⢀⡀⠠⠤⠐⠈⠉⠀⠀⠀⠀⠀⠀⠛⠀⠤⣀⡉⠉⠙⠀⠈⠃⠀⠦⢀⣀⣀⡀⠀⢠⠖⠀⠀⡀⠀⣠⠄
⠀⢻⡏⠀⣸⠒⠛⢤⠀⠀⠀⠀⠀⠀⡠⠶⠨⠦⢀⠀⠀⠀⠀⢨⣨⣥⣤⣄⣷⠀⠀⠠⠙⣿⡇⠀⢀⣠⣄⢀⣀⠀⠀⠀⠈⠁⠘⠃⠲⠿⠀⢈⠃⢀⢠⠠⠘⣰⠆
⠀⠟⠙⠀⠹⠞⠁⠐⣿⡷⠖⣠⢤⠀⠀⠂⣀⣀⣀⣀⣉⣀⠰⠄⠀⠀⠀⠀⠀⠖⠀⠀⠉⠀⠀⠀⠈⠉⠋⠀⠛⠂⠀⢔⠒⠀⣀⣐⣦⣠⠀⠈⠉⠉⠓⠒⠂⠘⠂
⠀⣤⠄⢀⣀⣀⣀⠈⠠⠀⠒⠙⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠰⠀⠀⠠⠒⠳⠦⣀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⢁⠀⠀⢀⠤⠄⠀⣠⣤⣂⠝
⠈⣿⠾⢛⣛⣉⣉⣀⡀⠀⠀⠀⠀⠀⢠⣄⣀⣙⢦⡌⠀⠀⠀⠀⠀⠀⠀⠾⠿⠶⢶⡖⢶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢀⡀⢀⣉⣛⣉⡾
⠀⣿⣿⣿⣿⠿⠃⠶⢤⣄⠀⠈⠉⠑⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠦⡤⠀⠀⠀⠀⠙⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠄⠈⢁⣠⣤⣤⣤⣬
⠀⢿⠿⠏⠀⠀⠀⠀⠀⠈⠑⢦⣄⠘⠃⠚⠃⠀⠀⠀⠐⡀⠃⠀⠤⠀⠀⠀⠰⣆⣀⣀⣤⠴⠦⢈⡆⠀⠀⠀⠀⠀⠂⠠⣤⡄⠀⠀⡀⠂⣤⣶⣾⣿⣿⣿⠟⠻⠿
⠀⢈⠀⠀⠤⠤⠀⠠⠤⠴⠶⠄⠤⠂⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠁⢴⣀⠄⢸⡏⣀⣈⣶⡞⠛⠠⠾⣆⣿⡀⠸⠧⣼⡿⠛⠛⠉⣁⢔⣀⣀⡀
⠀⠈⠁⠀⠀⠀⣀⠀⠀⢀⠀⠐⠂⠉⠁⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⣄⣈⡀⣡⠀⡀⠀⣁⡀⠴⢟⠉⠋⠀⠷⠗⢚⣂⡄⠀⠈⣷⢠⠀⣀⣀⡀⠰⠇⡈⠁⠄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢤⣤⡟⠓⠜⠛⣤⣀⣀⠀⠉⠉⠀⠰⣶⠀⠀⠀⣉⣤⡴⠋⠀⠀⠛⠛⠻⠿⢿⣷⣶⣶⣆
⠀⠀⣠⡊⠀⠀⠀⠀⠀⠀⠴⠁⠻⠏⠙⠳⠾⠒⠦⢠⡏⠀⠘⠣⠤⢄⣀⡀⠉⢸⠗⢛⠉⠉⠹⢶⡆⠀⠀⠀⢀⠤⠒⠊⡁⠂⠁⠀⠀⠀⠀⠘⠛⡷⠾⠿⣿⣿⣿
⠀⢀⣩⡿⠳⢤⣄⠀⠀⠀⠀⠀⣀⣀⣀⠰⠰⠄⠀⢸⡇⠀⡀⠀⠀⠀⠹⠁⠚⠛⠁⠈⠀⠀⣀⡀⠦⠄⢊⠀⠤⠐⠊⠀⠀⠂⠀⠀⡀⢤⠴⠞⠈⠁⢠⣦⣤⣼⣟
⠀⢀⡱⢶⣉⠙⠻⠂⢤⣤⣄⠀⠉⠁⠉⠃⠀⠀⠀⠀⠛⢶⣬⣙⠓⠀⠀⠀⠀⠰⠒⢈⣉⣉⠠⣤⣈⣁⣀⣤⡤⠄⠈⠉⠉⠙⠙⠛⠻⢶⡄⠂⠀⠀⣠⣴⣶⣿⣿
⠀⠈⠍⠉⠀⠀⢀⡀⠀⠀⠹⢤⡤⠐⠶⠀⠀⠀⠀⠐⠂⠁⠈⠛⠛⠛⠋⠋⠉⠉⠙⠁⠄⠋⠉⠉⠀⠀⠀⣄⠀⠀⠀⠉⠉⠘⠃⠦⡄⣀⠓⣀⣀⠀⡀⠚⠛⠻⠟
⠀⠀⠂⠀⠀⠀⠾⠷⠷⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠐⠀⠀⠠⠀⠀⢸⡇⠔⠉⠀⠀⠀⠀⢸⣆⠀⠀⢻⣿⣷⡞⠀⡀⠀⡄⠀⠀⠈⠘⠫⣏⣿⡟⠥⣐⠢
⠀⠀⠶⠈⢁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠆⣠⡀⠠⠆⠀⠀⠀⠀⠠⡞⠁⠀⠀⢰⡆⠀⠀⠘⣁⣠⠄⠀⠹⣏⢻⡄⠀⠈⣁⣀⠀⡇⠀⠀⠈⣿⠙⢦⡈⠻⡆
⠀⢀⠰⠉⠉⠉⠛⠛⠋⠢⣀⠀⠀⠀⢀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡀⠀⠙⠀⠀⠈⠛⠻⠆⠀⡀⢳⡄⢹⠀⠐⢉⠛⢡⡾⢢⡄⡀⣿⣆⠈⣿⢆
⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠈⠻⠉⠶⢈⡷⠀⠀⠀⢀⡈⠂⠀⠀⠉⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠈⢻⡀⢷⡀⠘⠀⢰⢁⠞⠀⡀⣿⣿⠇⡸
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⣤⠀⢠⣴⡛⠀⠀⠀⢀⡈⠤⠈⠀⣤⠐⢤⠀⠠⡌⢧⠘⣧⠄⠀⠘⠁⠀⠀⡟⢃⣸⠀⠙⠁
⠀⠀⠀⠀⠰⠶⣶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡌⠉⠁⠀⣤⠛⠉⠀⢰⡶⠛⣋⠀⠀⣀⠀⠋⠘⡄⠙⣦⠘⠇⢀⠀⠀⡇⠘⣿⢀⣀⡀
⠀⠴⠂⡀⠀⠀⠀⠀⠀⣉⠉⡁⠲⠦⢤⣴⡦⠴⢦⣄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⢠⡴⠛⠉⠀⠀⠀⠒⠀⠀⠀⠀⠀⣿⠀⢿⡄⠀⠐⠦⣤⡇⠀⢟⠁⠘⠒
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠛⠻⢿⣿⣦⡀⢸⡆⢸⡇⠀⡈⣷⢈⠁⠀⠸⢫⣤⣀
⠀⠀⠀⠀⢀⡀⠀⢀⣀⣤⣤⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠒⠂⠁⠀⠠⠄⠲⠾⠟⠛⠛⠿⠿⠿⠿⠿⢿⣶⣌⣿⣷⣸⡇⠀⣷⠀⠄⠀⣀⡀⠐⠛⢣⣈⢏
⠛⠛⠛⠛⠀⠁⠀⠀⠈⠉⠀⠀⠀⠀⠖⣁⠀⠀⣤⣀⠀⠀⠀⠠⣤⡴⠆⠑⠀⢶⠶⣶⣶⣶⣶⣤⣀⣀⣠⣶⣾⣿⣿⣿⣿⡇⠀⢻⣀⡤⠀⠤⠄⣀⠐⢦⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡻⠀⠀⠀⠀⠀⠀⢀⣀⠐⠃⠀⠀⠀⠀⢠⣤⣬⣍⣛⣛⡻⠿⣿⣿⣯⣼⣯⣿⣿⡇⠀⠘⣯⣡⡖⠰⣂⡈⠋
⢀⡔⠂⠀⠘⠋⠀⠀⠀⣀⠀⠠⡚⠡⠘⠃⠀⠀⠀⢀⡤⠲⠹⠅⠀⣤⣤⡄⠄⣰⢆⣀⣩⡍⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠈⡉⠉⠒⠛⢃⣠⣤⣿⣦
⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠙⠋⠉⠁⠀⠉⠈⠉⠋⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠘⠃⠀⠛⠛⠻⠿⠿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡉⣡⣉⢿⡁⣸⠉⣉⣭⠋⣸⣽⠋⣅⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡒⡂⠀⠀⠀⠣⠝⠀⠀⠀⢔⢲⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⢄⠀⠈⠀⢀⢤⠀⠀⠀⠀⠤⡀⠀⠁⠀⠤⡄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⢸⡆
⠀⠀⠀⢀⣠⡼⠶⠚⢛⣉⣉⣭⣤⡤⠤⢶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠤⠤⣤⣭⣉⣉⡛⠓⠶⢧⣄⡀
⢀⣴⠞⣉⡤⢶⡞⠋⣉⠭⠋⠁⠀⣀⣀⠈⢆⠀⠀⢀⡀⠀⠀⠀⡇⣀⣀⠀⠉⠙⠬⣉⠙⢳⡶⢤⣍⡳⢦
⣾⡵⠋⠁⠀⢸⠗⠊⠀⠀⠀⠀⠋⠁⠀⠀⠀⠑⠦⡼⠉⠓⠲⠦⠇⠀⠈⠙⠀⠀⠀⠈⠑⠺⡇⠀⠈⠙⢮
⠹⣦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴
⠀⠈⠙⠶⢤⣼⠀⠀⠀⠀⠀⠸⠋⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠙⠇⠀⠀⠀⠀⠀⣧⡴⠖⠋
⠀⠀⠀⢀⡞⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢳
⠀⠀⠀⠀⠳⣄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⠞
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠘⠉⠓⠢⠤⠤⠔⠚⠉⠑⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏
⠀⠀⠀⠀⠀⡠⠒⠁⡇⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⢸⠈⠒⢄
⢀⡠⠤⠐⠊⠀⣠⠤⣇⢀⡄⠀⠙⢙⠓⠶⠤⠤⣤⣤⣤⡤⠤⠤⠶⠚⡉⠋⠀⢀⡀⣸⠤⣄⠀⠑⠂⠤⢄
⢎⣀⠔⣦⠀⢀⡵⠀⠀⢨⠃⠀⠀⠈⠙⠲⢤⠔⠃⡀⠀⠑⠦⠤⠖⠋⠀⠀⠀⠘⡅⠀⠀⢮⡀⠀⣴⠢⣀
⠀⠑⠊⠘⢤⣈⠆⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠰⣅⡤⠃⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠉⠁⠐⠒⠂⠀⠠⠀⠀⠄⠀⠐⠒⠂⠈⠉⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⠒⠒⠒⠶⠃⠘⠖⠒⠒⠒⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠤⠤⠤⡦⠀⠀⢤⠤⠤⠤⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⠀⠉⠓⠢⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣶⣶⣶⣟⣻⣿⣿⣏⣀⣀⣹⣿⣿⣟⣿⣶⣶⣶⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡉⣉⣶⢤⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠁⠀⠀⠂⠀⠀⠉⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢠⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⢠⠴⠒⠶⠊⠉⠉⠀⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⠒⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣏⠙⠋⠓⠒⠶⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣀⣠⣤⣴⣶⠾⠛⠋⠉⠙⠿⠿⠟⠛⠛⠉⠙⢿⣿⣦⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⣴⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡦⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠞⠉⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣁⠀⠀⢀⣈⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⣹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠘⠁⠀⠀⠈⠇⠀⠀⠀⠀⠠⢁⠈⠀⠐⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⣀⡄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⣄⣀⠀⠠⠒⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠀⠙⠅⠒⠢⠤⠤⠤⢤⣤⡐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡐⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⢦⠀⠀⠀⠀⢤⠀⠀⢀⡄⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠶⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠈⠳⠀⡀⠂⠀⠀⠀⠁⠈⠣⡬⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠈⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⢀⠔⠁⠀⠀⠀⠈⠁⠀⠀⠘⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⠆⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣵⡀⠀⠀⠁⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⠀⠈⢂⠀⠀⠇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠊⠙⢷⣄⠀⢀⠃⠀⢀⣉⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠈⡁⣯⠀⠀⠀⠀⠀⣠⣄⠀⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⣀⠠⠌⠀⠀⠀⡿⢛⠃⠀⠀⠀⡁⠘⠀⢀⠃⠀⠀⠀⠀⠀⢁⠀⠃⢀⠁⢰⡀⠀⢀⡤⠞⠁⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠞⠀⡆⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⡗⠉⠁⠀⠀⠀⠀⠀⠘
⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠑⠀⢀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠓⠀⣀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡀⠾⠀⠀⠀⠀⠈⠑⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⡔⠁
⠀⠀⠀⠀⠐⢄⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠀⠀⠀⠀⠀⡇⠈⠉⠉⠛⠛⠛⠒⠒⠒⠒⠓⠒⠚⠋⠉⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⡀⠀⠉⠐⠂⠠⠤⠤⠠⠤⠴⢧⠀⠀⠐⠒⠂⠉⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠋⠒⢶⡀⠀⢎⠙⠋⠉⠉⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡈⠐⠒⠀⠁⠃⠰⠈⡀⠀⠀⠀⣐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠑⠴⠤⠆⠘⠁⢸⠀⠁⠈⠙⣀⠈⠁⢠⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠒⠂⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠇⠤⠀⠀⠀⣀⣀⡀⣀⣀⠀⠠⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣦⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠤⠤⠤⠀⣑⡉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⠀⠀⠀⢀⢀⣒⠒⠀⠩⠝⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡈⢀⡆⠀⠀⠀⠄⠒⠤⢌⡹⠐⠠⣀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⡟⠁⢀⠀⠀⠀⠐⠆⠀⠀⢣⡠⠦⡀⠑⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠈⠀⠀⠀⠀⢀⡀⠀⠈⢑⣁⠆⡠⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣄⠀⠀⠰⡶⠂⠉⡇⠀⣤⣡⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠢⠤⣬⠭⠍⢀⣠⠖⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⡤⡔⣤⡠⡗⠋⢹⠿⠡⠤⢄⠀⠀⡿⡐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢮⠠⡢⣱⣱⠟⠈⠙⠉⠓⠬⠼⣁⠁⠢⡑⡜⠀⠘⡁⠂⢭⡲⡤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⢻⣦⢱⡞⠓⣬⡙⡉⢆⠀⠀⠀⠀⠀⢀⢴⣟⠀⠀⠀⣎⡱⠁⢰⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⡹⠞⣧⣀⢹⣧⠛⢾⣉⣉⣹⠟⠉⠁⢁⠶⠬⣔⣒⡤⣷⣀⠇⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⡲⠾⠌⢉⠉⠑⠢⣄⠀⠀⠀⠀⠁⠈⢹⠒⢇⣂⣀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠤⠤⣀⠀⠑⢄⠀⠀⠑⢦⣀⣀⣀⣀⡸⠶⢶⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⠊⢐⠢⡀⠱⡀⠈⣆⠀⠀⠀⣀⠤⠋⢄⠐⠁⣸⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣀⠀⠉⢦⢄⣊⢠⣀⠌⢼⡴⠧⠊⠑⢌⠉⠀⠀⠀⡁⢸⣴⢾⣟⠓⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠳⢦⡈⠛⣧⡰⠃⡠⠊⠀⠀⠀⠀⠀⠉⠒⠒⠒⠧⠊⢹⠟⠉⠻⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠈⠻⣄⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⣉⣈⡳⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⡉⣩⡦⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⡁⠋⠀⠀⠀⠰⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠠⠁⠀⠀⢀⣀⣀⠨⠤⠤⢀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠤⠘⠂⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠉⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣀⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠚⠛⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣠⠴⠖⠛⠉⠩⡄⠀⠀⠀⠀⠀⠀⠀⢀⠲⣦⣀⠀⣦⡈⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠀⠉⡽⠟⠾⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⠁⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⣠⡄⠀⢀⣠⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣦⠖⠛⠈⠓⠦⡀⠐⠟⠀⠀⠠⠀⠙⠁⠐⠀⠛⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠐⢲⣄⠀⠉⠀⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣀⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣦⣄⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢿⣿⡿⠿⣿⡿⠿⢇⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠁⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠫⡓⠢⠤⣈⡀⠀⠀⠀⠀⠀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣀⠀⠈⠀⠀⠀⠀⠀⠀⠜⢡⠶⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡤⠚⠁⠀⠀⠀⠀⠈⢆⠀⠀⠠⠚⢃⠀⢠⠚⢧⡀⢸⠀⠀⠀⠀⠀⠀⠀⠔⠃⠘⠢⡀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠸⡀⠀⠀⠀⢀⠄⠀⠀⠀⠓⢴⠂⠀⠘⡤⠁⠀⠀⠙⣎⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡃⢀⡶⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠑⢄⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⡠⠀⠀⠑⠦⠐⠁⢪⡄⠤⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⠀⡜⠁⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⡆⠰⠀⠉⠀⠀⠀⡀⠀⠀⠀⠸⣅⢀⠔
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠱⣄⠀⠀⠀⠀⠌⠀⠀⠙⣷⣄⡜⠀⠀⠰⠤⢴⣄⠀⡸⠀⠀⠀⠀⠀⢣⠀⢀⡦⠠⠄⢡⠀⠀⠀⠀⢀⡼⠉⠢⠤⠤⠔⠉
⠀⠀⠀⠀⠐⠀⡀⠃⠀⠀⠀⠈⠣⡄⠐⠚⠀⠀⠀⣠⠏⢰⠁⠀⠀⠀⠡⢀⣀⠠⠃⠀⠀⠀⠀⠀⠀⠠⠀⡅⠄⠀⢸⣀⣀⡤⠚⠁
⡀⠀⠀⠁⠀⠀⠀⠀⠰⠀⢀⠔⠈⠠⠀⠀⠀⠀⠘⢃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠈⠀⢀⡄⠀⠂⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠂⠐⠀⠀⠀⠈⡄⠀⠈⠛⢄⣰⠀⠀⢸⠉⠂⠴⠹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠙⢄⠀⠀⠀⠃⡄⠐⠀⠀⠀⠱⠒⠋⠀⠀⠀⠀⠃⠈⡿⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠁⠀⣀⣀⡠⠤⠲⡏
⠀⠀⠀⠀⠀⠳⠀⡠⠀⠀⠀⠀⠀⠀⡈⠋⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⠀⠀⠘⠀⠀⠀⢀⠌⠀⠀⠀⢰⠀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢇⡀⠀⠀⢀⠌⠀⠀⠀⠀⠸⡀⠀⠀⠁⠐⠂⠠⠤⠤⠤⠀⠛⡆⠒⠒⠂⠀⠉⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣤⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡛⠛⠛⢻⢦⠀⠸⡉⠛⠉⠈⠩⢇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡌⠁⠘⠂⠈⣰⠀⢦⣈⠀⡀⠀⢠⣜⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠀⠛⠀⠆⠂⠓⠉⠰⣇⠀⠉⠉⠏⡀⠀⠀⡢⠄⠒⠒⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠐⠢⡁⢀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠁⠎⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⠰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠠⠤⠤⠤⠤⠤⠀⠀⠂⠀⠒⠂⡀⣉⠀⠀⠃⠂⠠⠤⠀⠀⠄⠀⠤⠄⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡬⠩⠖⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⢀⣀⣀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⡠⠀⠂⠒⠉⠁⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠂⠀⠀⢀⣀⣀⡀⠀⠤⠤⣤⣤⣬⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠤⠄⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠈⠁⠐⠨⠄⠠⡂⢄⠠⢄⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⡄⠀⠀⠀⣀⣀⠠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⣍⠛⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡁⠀⠀⢀⠄⠀⠑⢤⠈⠀⢠⠂⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⠄⡤⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠠⠀⠈⠂⠂⠀⠤
⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠘⠻⣦⡀⡰⠀⢀⠁⠀⠀⡀⠀⠀⠀⠄⠀⢀⢀⢫⠀⠀⠀⢀⡴⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠂⠀⠀⡼⢫⠁⠀⠀⢀⠁⠠⠁⠀⠀⠀⠈⠀⠁⠀⠈⣄⣠⠔⠋
⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⠀⢀⠀⠀⠀⠀⠀⠄⢀⠀⠀⡓⢀⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⢄⠀⠀⠀⠀⠀⢉⢠⡜⠁⠀⠀⠊⢙⠂⠂⠤⠄⣀⣀⠀⠀⡀⣀⣀⠤⠔⡖⠁
⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠸⠀⠁⠂⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣀⣀⡀⠀⢸⣤⡠⡤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠣⠀⠀⠛⠄⡐⠈⠐⠐⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⠢⠀⠤⠘⠠⢉⠁⠒⢂⠙⠂⠤⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠂⡠⠐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠃⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠐⠂⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡛⠛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠆⠀⠀⣀⣠⣀⣀⣀⠀⠀⠀⠉⠛⠛⠻⢧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠂⠀⣤⡀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⢀⡀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠾⠿⠛⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢉⣤⠖⠂⣀⡀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠶⠖⠀⠀⠀⠲⣶⡶⠈⠀⠀⠀⠀⠉⠲⣆⡈⠙⠶⢀⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠎⠉⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠄⠀⠈⠙⠦⣀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠁⠀⢀⣀⠀⠀⠀⢀⡀⠴⠶⠟⠀⠀⠀⣀⡀⢶⣦⡀⠀⢀⠀⠿⣆⡀⠀⠀⠈⠁⠀⠘⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⣠⠶⠈⠉⠁⣀⠀⠀⠈⠉⠐⢶⣦⠉⠁⠀⣶⠀⢀⡠⠶⠉⠠⣄⡈⠉⠙⠛⠷⣦⠈⠣⡄⠀⠀⠀⠀⠐⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣰⠞⠁⣠⣴⠆⠈⠉⠙⠿⠟⠀⢦⡀⠉⠢⣀⠀⢰⡶⠈⠁⣠⣾⠷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⢆⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⢀⡘⠁⢠⡄⠛⠃⢠⡞⠉⢀⣀⣀⣀⠀⠈⠣⡀⠉⣠⠀⠀⠀⠀⠛⠃⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⢃⡀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡆⠀⠸⠃⢀⡄⠀⠀⠀⠈⠁⣰⡿⠿⡿⠿⣷⣦⠀⠋⠀⠸⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠸⢿⡀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⠀⢸⡇⠀⠀⠀⠀⠀⠏⠀⡄⠀⣀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⠟⠋⠉⠁⠀⣰⣶⡟⢿⡆⠀⠀⠠⡇⠀⢃⡀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠈⠁⠀⠀⠆⠀⠀⠀⠠⣇⡈⠻⣿⡿⢿⡟⠀⢀⠀⠀⠀⠀⠠⠀⠀⢠⠈⢿⣿⣿⠇⣸⡇⠀⠀⠀⠃⠀⠈⢳⡤⠙⠋⠙⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠘⢷⣆⠀⠀⠀⠀⠀⠘⠛⠀⠈⠁⡦⠁⢀⠛⠀⠀⠀⠀⠀⠀⠀⠛⠓⢠⣤⣄⠘⠋⠀⠀⠀⡀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡜⠛⠻⣷⣄⠀⠙⠃⣤⣄⠀⠀⠀⣠⣤⠐⠁⢀⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡄⠋⠀⣀⡞⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢀⡈⠀⠈⠙⠂⣀⣀⠙⠿⣦⣤⠀⠀⠀⠀⠀⠀⢀⣤⡴⠛⠋⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⡦⠀⠀⢀⠟
⠀⠀⠀⠀⣠⠞⠛⠓⠀⣀⠀⠀⠀⠙⢣⡀⠘⠏⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟
⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠻⢧⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠐⢶⣿⣇⣀⣴⠃⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢀⣤⡴⠟⠁
⠀⠀⠀⢾⠁⠀⠀⠒⢀⣤⡀⠀⠀⢠⡶⠂⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⢀⣀⡀⠘⠒⠀⠀⢀⡹⠃⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠈⠁
⠀⠀⠀⠀⣶⣄⡀⠀⠀⠉⠳⢄⣀⣼⣧⣤⠾⠉⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠛⠁⢀⣀⡀⠶⠋⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⣀⡀⠀⢀⠿⣦⣀⠈⠉⠹⠏⠁⠀⠀⠀⠀⠉⠷⢦⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⣀⡴⠾⠋⠁
⠀⠀⠀⠀⣉⠓⠲⠿⠆⠉⠉⣀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠶⢄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠶⠊⠉⠁
⠀⠀⠀⢀⡟⠀⠀⠀⢀⡶⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠰⣶⠖⠒⠀⠈⠉⢻⠃
⠀⠀⠀⡿⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡶⠆⣀⠀⠀⠀⠀⠀⢿⣶⡶⠦⣀⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠶⢶⣾⠋⠀⠿⣀⠀⠿⠀⠀⠀⠀⡀⠼⠋⠀⠀⠉⣿⣇⡀
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠀⠉⠉⠀⠀⠀⠛⡀⠀⠀⠙⠿⠀⠀⢠⡴⠈⠉⠀⠀⠀⢠⠚⠁⠉⠉⠲⣦
⡀⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣴⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠙⠷⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⠙⢃⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣧⠀⣾⠀⠀⠀⠀⠺⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠙⣶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯
⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠙⠳⣤
⠀⠈⢷⣤⡀⢀⣤⣤⣤⡤⠔⠛⠛⠛⠻⠇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠘⠛⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡘⠋⠀⠀⢀⡴⠛⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡼⠟⢀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡴⠋⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⡀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⢠⠞⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡀⠀⠀⣀⣀⠀⠀⠲⠶⠖⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠲⠶⣶⣄⠘⢧⡴⠃⠀⠀⠀⠀⢀⣀⠴⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⡊⠉⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⣏⠰⠎⠀⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⠶⠶⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠰⢿⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠶⠂⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣶⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠺⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⢠⣿⠀⠀⣶⠀⠀⢀⣀⣠⣄⡀⠲⠾⢿⡟⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠈⠉⠁⠀⠙⡿⠁⠀⠁⠀⠀⠛⠓⠺⣿⠁⠉⠁⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⣇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⣿⠟⠁⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢾⣿⡷⢠⣤⠄⠈⠋⠀⣿⠀⠀⠀⠀⠀⠀⣤⡴⣿⡦⠤⠄⠀⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠀⠀⢀⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢈⣤⠀⠀⠀⠀⢀⠘⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢘⣇⠀⢀⣤⡞⠛⢷⠀⠀⠀⠀⣴⠛⠛⠁⠀⠀⠀⠛⢃⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠻⠧⣤⣤⠀⠀⣘⠛⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣠⣼⣇⡀⢠⣄⣴⣶⡀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠉⠉⠉⠙⠓⠀⡀⠀⠀⠻⠿⠶⠟⠛⠛⠻⣿⠁⠀⠀⠀⠀⠀⠉⣿⠟⠓⠀⠀⠀⠀⠙⠃⢀⠙⠋⠁⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⣿⡄⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠒⠾⠛⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⣀⣀⣀⣤⣄⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠲⠖⠘⠛
⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠴⠶⠶⠖⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣀⡀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠆⠀⣀⣀⣀⣀⣀⣀⣀⡀⠰⠶⠖⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠾⠿⠛⠛⠛⠿⠿⠷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠋⠁⠀⠀⠠⠐⠆⠀⠀⠀⠠⠒⠄⠀⠉⠛⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠛⡙⠂⠀⠀⠀⠈⠀⠀⢀⡀⠀⠉⠀⢀⠔⢆⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠤⠀⠀⠀⠀⠔⣱⠆⠀⠀⠀⠠⠄⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠐⠀⢀⣀⣠⣤⣤⣤⣶⣶⡾⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⣿⣿⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣀⣤⣴⡶⠾⠿⠛⠛⠉⠉⠁⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡈⠉⠛⠻⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⠛⠋⠉⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣶⣤⣄⡉⠻⣷⡄
⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⠉⠻⢷⣌⣿⡆
⠀⠀⢀⣴⡿⠋⣁⣤⡶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⡿
⠀⢠⡿⠋⣠⡾⠟⠉⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣾⠇
⢠⣿⣡⡾⠋⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⣴⡿⠃
⠸⣿⡟⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠂⢻⣏⣀
⠀⠹⣷⣄⡴⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⠀⣼⣿⣿⣧
⠀⣴⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⠉⠈⠙⠻⠟⠁
⠀⠘⠛⠛⠛⠓⢛⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢶⡀⠀⠀⠀⠀⣿⣿⣿⠟⠓⣄
⠀⠀⠀⠀⢠⡶⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣼⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⡟⠀⠀⠀⠀⣿⣿⣇⣀⣾⣿⣷
⠀⠀⢀⣶⣿⣧⠀⢹⡟⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢧⣀⣠⠇⠀⠀⠀⢀⣿⠃⠀⠙⠿⣿⡿
⠀⠀⠸⣿⣿⡿⠛⠋⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣤⡀⠀⠀⠀⠀⣠⣤⠞⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠔⠛⠉⠛⠛⠿⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠿⠟⠻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⡁⠀⠀⠀⠀⠀⠀⣤⣤⣀⡀⠉⠉⠉⢉⣹⡟⠉⠹⣧⣤⣤⣾⡇⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠁⠀⣸⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠐⠀⠐⠀⠀⠀⢻⣧⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⢀⣴⠞⠛⠉⠙⠛⢺⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡁⠉⢷⡄
⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⡾⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⢀⡀⠀⡀⠀⠘⠀⠀⠀⢹⣟⠱⣦⠀⠙⠶⣄⣀
⠀⠀⠀⠀⣀⡶⠏⢀⣴⠟⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠠⠤⠠⠀⠤⠀⠤⠤⡄⠀⠀⢀⡄⠀⠀⠀⠈⣿⡄⠈⠛⠶⣄⣀⠉⠛⠦⣄⣀
⠀⠀⣴⠟⠉⠀⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⠉⠳⢦⣤⡈⠉⠉⠛⠷⣦⡀
⣀⡼⠃⠀⡀⠀⢠⡷⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠂⠀⠀⠀⡁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠛⠶
⢻⣤⣤⡾⠻⣦⣼⠃⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⢠⡾⠿⣶⣶
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠂⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠙⠛⠻⢿⣆⠀⠀⠀⠀⠀⢀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣧⣀⡀⠂⠠⠈⣿⣧⠀⠀⠠⠀⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠻⢷⣦⣤⣽⣿⣷⣀⣠⣾⠟⠋⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⢀⣼⠏⠙⠋⠀⠻⠟⠛⢷⡄⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠉⠁⢀⣀⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⡋⠉⠁⢀⣠⣴⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣶⣾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣶⣤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡉⣽⠦⠤⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠡⠁⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⣁⣠⡤⠤⠐⠒⠒⠒⠒⠺⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠴⠊⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠉⠀⢀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣖⠒⠒⠒⠺⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⣴⣶⡾⠛⠋⠉⠉⠛⠛⠛⠋⠉⠈⠻⣿⣆⠀⢀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣁⠴⠚⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⢙⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠠⣏⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣀⠙⠋⠀⠀⠀⠀⠀⠀⠘⠃⠈⠈⠋⠀⠀⠘⡆⠀⠀⠀⠀⡀⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⣷⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠁⠀⠀⠀⠀⣅⡀⠀⠀⣀⡀⠀⠀⠀⠀⣾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠹⠿⠿⠆⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣄⠀⠀⠀⠀⠈⠁⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠃⠈⠈⠑⠒⠢⠶⠶⢖⣋⠁⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠊⠀⠀⠀⠘⢄⠀⢀⠐⡆⠀⠔⢤⠀⡀⠀⠀⠀⠀⢀⠄⠐⠛⢟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢁⠀⠀⠀⡀⠀⠀⠈⠰⠆⠀⠰⠄⠀⠀⠹⡅⠀⠀⠀⠀⠆⠀⣄⣀⣨⠁⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠑⢤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⡀⠄⠣⠀⠈⠉⠁⠀⢈
⠀⠀⠀⠀⠠⠂⠀⠘⢆⠀⠀⠀⢀⠈⠙⢷⣄⢀⠂⠀⣨⢉⡀⠀⡒⠀⠀⠀⠐⡂⢀⡈⡩⢹⠀⠀⠀⠀⡤⡄⠀⢠⢀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠳⣄⠀⠂⠀⠀⣠⠋⠏⠀⠀⠀⠀⡃⠀⠁⠀⠀⠀⠀⠁⠈⠥⠀⢸⣀⣠⠔⠋
⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠈⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⡀⠃
⠈⢄⠀⠀⠀⠀⠀⠀⠅⡄⠀⠀⠀⡴⢄⡁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⡀⠀⠀⠀⡀⠀⠈⠉⢠⠴⠓⠀⠉⠘⠑⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡔⠂
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢁⠀⠉⠉⠉⠛⠛⠓⠒⠚⠚⠉⠉⠀⢠⠃
⠀⠀⠀⠀⠀⠈⠒⠄⢀⠀⠁⠀⠀⡐⠀⠀⠀⡆⠘⠄⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠠⠞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠳⢄⠀⠀⠈⠁⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠒⠲⢦⡀⠸⠳⢖⠚⢻⣋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠁⠐⠂⢉⡃⡦⢁⣀⢀⠠⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠈⠁⠁⠉⠉⠀⠣⡀⠀⠉⠄⡀⡑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠈⠢⡔⠈⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣟⢿⠿⢿⣿⠻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣿⠁⠩⡿⠸⣷⢺⢴⣎⢿⡣⢶⡜⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⣻⡷⡇⣨⣤⣥⡖⣿⣷⣷⣿⣧⣵⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣬⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⣭⣭⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⣻⣿⣿⣯⣿⣿⠿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣯⣿⣿⡿⢿⣿⣿⣿⡏⠁⠀⠀⠀⡠⠤⣤⣽⣿⣿⣿⢧⡤⠤⠴⠶⠒⠒⠿⠛⠅⠀⠀⠘⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡏⠀⢢⠈⢻⣿⡄⠀⠀⠀⠀⠠⢾⠉⠙⣽⡅⠠⠠⠀⠀⠀⠀⠀⠂⠒⣋⡑⠀⠂⠒⣟⡓
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⣿⣿⣿⣿⣷⡀⢘⡄⠀⠻⡷⠀⠀⠀⠀⠀⠈⠿⠟⠿⡷⠶⠶⣤⣤⣤⣤⣤⣶⣾⣰⣇⣀⣀⣀⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣌⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣈⠄⠀⢸⡇⠀⠈⠉⠉⠛⠛⠛⠛⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⡀⢀⡴⠋⠉⢋⣹⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠒⠶⠖⣻⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣧⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⣿⣖⠤⢀⡀⠀⠀⠀⠀⢰⣤⣀⡀⣀⡠⠞⠀⠀⢀⠴⢺⣷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⡉⠛⣻⢦⣌⡁⠒⠒⠲⠼⢿⣤⠀⠀⠀⠀⠀⢠⠃⢹⣹⠏⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣿⣿⣷⣾⣿⣷⣦⡀⣠⠞⠍⣇⠀⠀⠀⠀⣨⠀⠞⠁⢀⡿⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣯⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⢷⣄⡀⢺⡄⠀⠀⠀⡏⠀⠀⠀⣿⠁⠨⣺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⡿⡍⠿⡀⢈⢧⠣⢭⣛⣿⣿⡆⣿⡷⣦⣷⡀⢀⡀⣇⠀⠀⠀⠙⠖⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡛⣴⣷⡘⣿⣆⠃⡣⣙⢻⣿⣿⣿⡮⣹⣼⣿⣷⣿⣟⠻⣄⠀⠴⡆⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⡒⢌⢻⣿⡞⢿⣳⡔⣹⣿⢿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣷⣌⡟⡿⢷⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡛⡹⣭⣻⣩⣿⣎⡿⣽⣞⡹⣶⣿⣿⠏⣽⡿⣿⠿⣿⣿⣫⣻⣿⣷⣶⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡼⣏⣿⠿⣤⡹⣿⣿⣳⡬⢿⣿⣿⡿⣿⣿⣼⣿⡷⢝⢾⣿⡿⢳⡿⣛⡞⡿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣁⢻⡽⣌⢿⣿⣣⡺⠳⣿⣿⡽⣿⣻⡻⣾⣫⣷⣽⣷⣯⣟⡟⡼⠁
⠀⠀⠀⠀⠀⠀⣿⢾⣿⢯⢿⢞⢾⢿⣻⣽⣇⢻⣿⡷⡴⡹⣽⣿⣿⣷⣷⠣⠛⣯⣻⣿⣜⣏⡜⠁
⠀⠀⠀⠀⠀⠀⣿⡽⣿⡿⢻⣧⢳⡠⣻⡻⣯⣷⣿⣿⣟⣵⡜⣿⣿⣿⣧⡆⡀⠙⢻⣿⣟⠞
⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠁⠁⠁⠈⠁⠁⠋⠛⠙⠛⠋⠉⠈⠈⠛⠛⠉⠙⠘⠂⠙⠋

⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⢀⠆⠠⢤⠀⠠⠄⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢄⣀⡄⠀⠤⡀
⠀⠀⢀⡀⢂⡈⠁⠀⠀⠀⠈⠉⠉⣉⣁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠎⠀⠀⠀⠀⢀⠀⠑⠈⢊⠂⠀⢂
⠀⠀⠋⠀⢸⠠⢯⡟⢴⢂⢤⡭⠟⠦⠈⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⡎⠄⠀⠀⠉⠉⠉⠀⠀⠸⣄⠀⠀⠆⠃⠀⡆
⠀⠀⢠⠀⠀⣾⠘⠿⠃⠀⠀⠶⠖⢠⡄⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠄⣲⡲⡀⢤⡐⣦⡜⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠑⠤⣟⡃⠀⣀⠀⠀⠀⢀⣟⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣯⠈⠘⢷⡷⣶⠈⢶⠃⡀⢠
⠀⠀⠀⠀⠀⠈⠙⢃⣄⠀⣠⠴⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠄⠀⠀⠀⠀⠀⠁⠀⢀⠺⠀⠁
⠀⠀⠀⠀⣰⠂⠐⠀⢇⠀⠚⠀⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠂⡀⠲⠖⢛⡠⠞⠃⠀⠀⠠
⠀⠀⠀⠔⠈⠀⠘⢆⡠⠳⡱⢧⣀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⡒⢈⠏⠀⠄⡀⡔⠉⠉⠂⢹⠟⣇⡀⢠⠀⠁⠀⡄
⠀⠀⢈⠀⠀⠀⢰⠀⢱⣀⣽⡧⠌⠆⠀⠀⡈⠢⡀⠀⠀⠀⢠⠌⠸⠀⣠⣤⠖⠂⠄⣈⡄⢂⠀⡁⠉⢈⡀⠱⠄⡀
⠀⠠⠀⢀⣢⡄⠀⡰⠋⠈⢸⣧⠀⠀⡔⠘⠄⠀⠈⡆⠀⢠⣵⠂⠀⠈⢏⢸⡒⠐⢄⠘⠀⠀⠃⠀⠀⠀⢳⠀⠐⠁
⠀⢃⠀⠀⠀⢹⢽⠀⢀⠶⢥⡜⠉⠀⠂⠐⠀⠀⡰⠀⠀⠀⠙⠳⠾⠀⠈⠁⡝⠛⠛⠀⠀⠀⠀⠆⢀⣐⠁⠢
⠀⠀⠀⠀⡅⠀⡍⠀⠀⠀⠀⠂⢀⡄⠙⠂⢀⠔⠀⠀⠀⠀⠀⠀⠐⢀⡀⢰⠁⠀⠀⠀⠀⠀⠰⠀⡠⠁⠀⡀
⠀⠀⠀⠀⢃⠀⢙⡒⠂⠐⡂⢻⠈⠗⠖⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠀⠀⡗⢀⠔⠊
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡘⡄⠀⠐⡊⠄⠀⠹⡆
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠒⠐⠁
⠀⠀⠀⠀⢠⠀⠀⠈⠀⠀⡐⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠠⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⣅⣀⡀⠁⠀⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠢⠠⠀⣀⢀⠄⣀⡀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠤⠤⡁⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀
⠀⠀⠘⡖⠔⠂⠀⠀⠀⠠⠄⢠⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢰⠀⠀⠀⠀⢸⠆⠘⣦
⠀⠠⠐⠉⢐⠀⠀⠀⠀⢠⡀⢀⣀⢉⣉⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠧⡔⠈⠃⠀⠀⠀⠈⠓⠠⠓⢕⠄⡀
⢠⡥⠀⢄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⢀⡠⠢⠃⠀⠀⠀⠀⠀⠢⡀⠂⡠⠚
⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣊⣉⣵⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠁⠁⠀⠀⠀⠢⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⠀⣀⡠⠤⠐⠒⠒⠒⠒⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⠔⠂⠉⠠⠁⠀⠀⠀⠀⠀⠀⠠⠠⠠⠦⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⠀⠀⠀⠃⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣴⡤⠶⠤⠤⠥⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣤⣤⣶⠾⠛⠛⠛⠿⠿⠿⠛⠛⠙⢿⣿⣄⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡤⢶⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠚⠁⠀⠸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠴⢿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠙⠿⠟⠀⠀⠀⠀⠀⠀⢴⠆⠐⠐⠷⠀⠀⠱⡄⠀⠀⠀⢀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⠂⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠉⠀⠀⠀⠀⢹⠀⠀⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣤⠀⠒⠀⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠉⠉⠠⠄⠀⠀⠀⢀⡜⠁⠀⠀⠁⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠶⠠⣀⡀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠘⡀⠄⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠈⣄⠀⠀⠀⢀⠉⠉⢙⡱⠒⠀⠀⠀⠀⠀⠀⠀⠐⣠⣤⣌⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠖⠁⠀⠀⠀⠀⠈⢆⠀⡐⠀⠇⠠⠊⠱⣄⠄⠀⠀⠀⠀⠠⠂⠀⠉⠁⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠀⠀⡠⠂⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠈⢆⠀⠀⠀⠀⡁⠀⠲⡤⠤⠂⢀⡆
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠁⠞⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣆⠀⠄⠂⠑⠀⠀⠂⠈⠈⠠⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠑⡀⠀⠀⠀⠄⠀⠙⢷⣤⡌⠀⠀⠦⢠⠀⢠⠁⠀⠀⠀⠈⡄⢀⡄⠔⢰⠀⠀⠀⢀⡔⢲⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢⠀⠈⠀⠀⣠⠎⡰⠀⠀⠀⠐⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡦⠤⠚⠁⠀⠘
⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⢈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠐
⠈⠢⠀⡀⠀⠀⢀⠀⠰⠤⠀⠀⠀⢐⠣⢀⡸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢄⠀⠀⠘⢀⠀⠀⠉⠱⠤⠞⠀⠀⠀⠉⠰⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⡖⠁
⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⡀⠀⠈⠉⠉⠉⠛⠛⠙⠉⠉⠉⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠑⠄⢀⡀⠁⠀⠀⢀⠃⠀⠀⢠⠀⠳⠄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠀⠈⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⠶⣆⠀⢸⠲⠖⠚⢻⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡌⠁⠒⠐⠹⠠⡄⡃⠀⠀⢠⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠒⠒⠒⠊⠰⢁⡈⠉⠉⠄⡁⠐⠒⠒⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠈⠉⠁⣄⠄⠊⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠤⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣄⣀⣄⣄⣀⣰⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⠙⣡⡁⠀⢀⣉⠂⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠛⠛⠛⠛⠛⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⡆⠉⠁⠀⠈⠉⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡤⢖⠂⠀⠐⣉⣒⠹⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠙⠟⠀⠈⠛⠉⢸⠄
⠀⠀⠀⠀⠀⠀⢠⠀⠏⠈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠠⠄⠀⠊⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⠂⢀⠀⡀⠓⠰⠀⣦⠄
⠀⠀⠠⡀⠀⠀⠀⠈⢣⣀⣷⣣⠀⠀⠀⠘⠂⠀⠀⠀⠨⠐⢅⢀⣀⡴⠁⢄⣀⠐⠁⠈⠐
⠀⠀⠀⠈⢢⣀⠔⠰⠋⠉⢻⡅⠀⠃⠐⠀⠀⠀⠀⢠⠴⠀⠀⢿⡰⡒⠀⠀⠛⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢈⣿⠀⠠⠢⠤⠋⠀⢀⠀⠀⠀⠀⠀⠀⠑⠛⠃⠀⠁⡝⠓⠋⠀⠀⠀⠐⠀⣤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠡⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠨⠍⠡⠸⠀⠁⠘⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠼⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠃⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠀⠄⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣉⡐⠂⠀⠀⠀⠐⠀⢔⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⣒⠀⠀⠀⠀⡃⠘⡦
⠀⠀⠀⠿⠃⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠋⢀⠀⠀⠀⠀⠁⠐⣿⣦⠢⡀
⠀⠰⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠉⠤⠖⠈⠀⠀⠀⠀⠀⠐⢬⡁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⠞⠡⠈⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠲⢾⣷⠒⢿⣥⠤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣻⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⣩⠟⠁⠀⠈⠑⢬⡹⣍⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⡜⢁⡾⠃⠀⠀⠀⠀⠀⠀⠑⡌⢦⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⢀⡞⢠⡟⠀⠀⠀⠀⠀⢀⣀⣀⣤⣼⡾⠷⠶⠶⠶⠶⠶⠿⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣞⣠⣿⣠⡤⠶⠖⠚⠉⠉⠁⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣉⣙⣛⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡤⠶⠛⠋⠉⠉⠀⣀⣀⣤⣤⠴⠶⠖⠛⠛⠉⠉⠉⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⢀⣀⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣴⣤⠀⣀⣿⡿⠟⢻⣿⢿⣿⣿⣿⠿⣿⡷⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⣀⣤⣴⣾⣯⣥⣀⠀⠀⢶⣒⠂⠀⠀⠀⠀⠀⠀⣴⠋⣿⡶⠔⢛⣭⣍⠁⢁⡿⠁⣰⠋⠙⣷⠊⣳⠀⡴⠋⠙⣷⢂⣳
⠀⠀⠀⠀⠀⠀⢠⣾⣁⣤⣶⢿⣿⣿⣿⠛⠁⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⢰⣿⣴⠾⡿⠁⢸⡇⣸⠃⢠⡇⢀⣄⢸⡄⢼⢠⠃⣠⡄⢸⡄⢸⡆
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢺⡛⠃⠀⠀⢠⡟⠀⣿⠀⢸⡇⠸⠏⢸⠃⢺⢸⠀⠻⠃⢸⠃⢹
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⢷⡄⠀⠀⠘⣧⡀⢻⡀⠈⢧⠀⢀⣼⢠⡏⠘⣇⠀⢀⡾⣂⡿
⠀⠀⠀⠀⠘⠻⠿⠿⢿⠿⠿⠿⠿⠟⠛⠛⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⣠⠾⠖⢻⡟⠛⢷⣤⣬⣿⠿⠷⠛⢦⣄⣹⣿⣿⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⡾⠋⠀⠶⠋⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⣤⠞⢁⡴⠋⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣴⠟⢁⣠⣾⠁⠀⠀⣶⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⣶⣆⢸⣿⠙⢷⣴⠟⠉⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠙⠶⢻⡟⠁⠈⢻⣄⣸⣿⣶⣤⣄⣀⣀⣀⣤⣴⠛⠓⢶⠞⢻⣷⣤⣤⣤⣤⣤⠶⢶⣤⠖⠋⢉⣠⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⢻⡟⠀⠀⠀⠀⢻⡿⢹⠿⣄⠉⠉⢉⣉⣽⢿⡇⠀⣿⠀⣾⠤⣤⣤⣤⣤⣤⣶⣾⠇⢈⠻⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⠀⠀⠀⠀⢸⡇⠀⠀⢀⣴⢿⠄⣾⠀⠹⡟⠛⠋⢩⡇⢸⡇⢠⡇⠀⣿⠀⠀⠀⠀⠀⠈⢀⡤⠀⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣿⠀⠀⠀⠀⠸⣷⡴⠾⠋⠁⣿⢠⡟⠀⠀⣿⠀⢠⡞⣷⠸⣇⠸⠷⠞⠛⠀⠀⠀⠀⠀⠀⠈⠛⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⢀⣼⠇⣼⠃⠀⠀⢸⣧⠟⠀⠸⣆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⣴⠞⣡⡾⠁⠀⠀⠀⢸⡏⣀⠀⠀⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠛⢰⣿⣋⣤⠞⠋⠀⠀⠀⠀⠀⢸⡆⣉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠛⢀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢨⡄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠚⠛⢻⣏⢀⣀⠀⢀⣀⣀⣀⣀⣀⢀⣿⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣦⠴⠞⠛⠉⠉⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⣿⢯⣭⠿⠈⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘⠃⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣼⣇⣀⣀⣀⣀⣀⣠⣄⣤⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠿⠿⠛⠛⠋⠋⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠉⠀⠀⣀⡀⠰⠄⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⠀⣀⢸⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠤⠄⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠟⠃
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣤⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠃⠀⠀⠀⠠⠀⡀⠀⠠⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣴⣤⣤⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⢉⡉⠉⠣⠘⠻⠀⠉⠃⠐⠒⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⠊⠐⠀⢀⠀⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⣠⠇⠀⢀⡄⠀⠀⠀⠀⢄⠀⠀⠁⠐⠀⢀⠀⢀⠔⠁⠠⡆⠀⠀⠀⠀⠀⢠⠈⠰
⠀⠀⠀⠀⢠⡏⠀⢠⡟⠀⠀⠀⡀⠀⠀⠑⠠⡀⠀⠠⠈⠁⠁⢀⠈⠀⡇
⠀⠀⠀⠀⢸⠁⠀⡾⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠘⠃⠀⠀⠀⡏⠀⠀⠁⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⡟⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠈⠁⠀⡰⠄
⠀⠀⠀⠀⢸⠀⠻⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠀⠤⠐⢈⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣿⠀⠀⣸⠿⠶⢦⡄⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢈
⠀⠀⠀⣰⠏⠀⠀⡗⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⣇⠀⠀⠀⠗⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠁
⠀⠀⠀⠈⢙⠇⠀⠀⠀⠀⢠⣶⣤⣤⣤⣤⣤⣤⠀⠀⠀⢰⣶⣶⠏⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡔⠀⠀⢀⣴⠂⢢⠀⠉⢹⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢧⢀⣴⠏⡋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣀⣀⣀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠓⠀⢠⣄⣀⣀⣀⠀⠀⠀⠀⢀⠀⠛⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠉⠙⡟⠁⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠆⠀⠀⠀⢠⠉⠀⠘⢁
⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⡄⠀⠀⠸⡀⠐⠂⠀⠂⡿
⠀⠀⠀⠀⠀⠀⠀⣧⠘⠒⠒⠁⢀⠃⠀⠀⠇⠀⠀⠀⠸⠆⡀
⠀⠀⠀⠀⡀⠀⠀⠘⠂⠀⠀⠀⡌⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠠⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠇⠀⠀⠀⠉⠁⠒⠖⠒⠠⠤⡤⠤⢤⠰⠆
⣭⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣽⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢯⣿⣿⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠭⣿⣿⢚⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢻⣿⣿⣿⣭⡤⣂
⠀⠀⠀⠀⠀⠀⠀⠀⣓⣾⣿⣿⣋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠄⠀⠀⠀⠀⠀⠀⠀⡈⠻⢶⣿⣿⣿⣷⣾⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠲⢬⣿⣿⣿⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⠀⣀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⢀⠓⣶⣿⣿⣷⣶⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢠⠀⠀⢶⣼⣹⣿⣿⣿⣿⣯⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⣿⡿⠿⠋⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠾⣦⣌⣿⣿⣿⣿⣿⣿⣯⣤⣶⠖
⠀⠀⠀⠀⠀⠀⠺⢾⣿⣿⣿⣿⡚⠋⠁⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣾⣷⣞⣌⣻⣿⣿⣿⣿⣿⣿⣿⣥⣴⣾⠄
⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣛⣡⣤⣠⣄⣀⡀⣀⣀⣀⣠⠰⠂⠀⠀⠀⣉⣻⣿⣋⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠧⠶⢯⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⢠⣄⣘⣿⣿⣿⣿⣿⣿⡷⠛⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤
⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⣿⣿⣿⣿⠛⠋⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢌⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃
⠀⠀⠀⠀⠘⠚⠛⠛⢿⣿⣿⣿⣿⣿⢿⡾⠈⠁⠀⢸⡆⢰⡆⠀⡄⠀⣧⠘⣧⠠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡛⠶⢢⡂⢀⣀⢸⣇⣼⡇⣀⡏⠀⣷⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣾⣇⣸⣿⣿⣿⣾⣿⣿⣿⣶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁

⣠⣄⣤⣀⣠⣀⣄⣀⣠⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣄⣠
⣿⣇⣺⣯⣍⣭⣭⣍⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⢸⡇⢰⢸⠀⠀⣇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⡀⡀⢀⢀⣯⣿⣀⠀⠀⠀⠀⢠⡄⣹⠀⡀⠀⠀⣼⣿⣧⣾⣿⣶⡎⣿⣰⣷⣿⡷⢶⣦
⠀⠀⠀⠀⠀⠀⢀⠀⣼⣷⣾⡇⠣⢺⣿⣿⣿⡇⠀⠀⢨⣶⣾⣷⣿⣿⠃⠀⠑⠛⣿⠃⠈⢻⡿⠥⣏⡷⢻⣾⣡⡎⣻⡆
⠀⠀⠀⠀⠀⢀⣶⣶⣿⣿⣿⢷⡞⠐⢿⣿⠏⠃⡀⠀⠈⠋⢻⣿⣿⠛⠂⠤⠦⢴⡇⠀⠀⠀⢿⣩⡿⡗⢺⣋⣾⣷⣨⡇
⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣟⣼⠁⠀⢸⡟⠄⢸⡵⣶⡎⠀⠈⣿⣇⡀⠀⠈⢀⠬⠃⠈⠉⠒⠠⣿⡀⣿⣾⠛⣷⡉⡦⠄
⠀⠀⣼⣿⣿⠿⠿⠛⠛⠉⠉⠃⠀⠀⢸⠃⠤⠈⢷⡿⠁⠤⠐⠙⠛⠛⠛⠿⢾⠀⠀⣤⢴⣤⣤⣏⣿⣷⡟⠀⢿⡙⠫⡀
⠀⠀⠋⠁⠠⡆⠀⠀⠐⢺⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠠⠀⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠒⠀⠩⡥⠚⣽⡦⢘⣲⣄⣀⣐⣢⡄⠀⠀⠀⠀⠠⣤⣀⣄⠐⢢⡄⠁⠲⡆⠈
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠒⠶⠶⠶⠄⠲⠚⠃⠀⠼⠖⠒⠛
⠀⠁⠀⠀⠒⠿⠿⣥⣶⣤⣿⣆⡀⢰⣦⠀⡀⣰⡄⡄⠀⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣦⣽⣿⣯⣿⣼⣿⣯⣿⣧⣷⡄⠀⠀⡀⠀⠀⣸⣶⡄⠀⣦⠀⠀⠀⢢⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢹⢿⣶⢶⣿⣿⣻⡃⠀⠆⣿⣷⣶⣿⣿⣇⠘⢻⡇⠀⠐⣘⣧⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⢐⡒⠈⠃⠈⠁⣥⠛⢛⡻⣿⣿⣚⣉⣁⠤⠠⣿⣿⠗⠛⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠉⠈⠛⠉⠉⠝⢿⣦⠤⢴⣾⡿⢿⠭⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣲⣱⣦⣄⣀⢀⠀⠀⡀⢀⣀⣴⣦⡀⠀⠀⢠⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⡉⢛⠙⠉⣴⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠌⠛⣿⣛⠟⠛⢭⡈⠟⢆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠍⠉⠓⠦⢥⣅⡔⡚⣷⣶⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⢆⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠁⠀⠀⠎⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⢠⣤⣀⣠⢀⣀⡈⠄⠈⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠅⠾⠋⠁⠚⠷⠸⠙⠃⠀⢀⣀⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⡎⠀⠀⡀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠈⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠸⢃⡀⣰⠃⠀⠀⠀⠠⠈⠀⠌⠁⠠⠀⠀⠐⡀⠐⢛⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠀⢉⡳⠎⠀⠂⠀⡠⠄⠤⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢥⠐⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢇⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠤⠀⠀⠀⠀⠀⢀⡠⢀⣡⠖⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⢆⠀⠀⠀⠀⡤⣉⡉⠁⠸⠀⠀⠀⠀⠇⢠⣀⡀⠤⢀⡠⢀⡄⠄⠀⠀⢐⣂⠐⢆⠀⠀⠀⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠑⠢⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠠⣍⠺⠀⡁⠈⠉⠏⠉⠀⠂⠄⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠊⠀⠀⠀⠀⣀⠔⠀⠉⠩⢥⠖⠃⠀⠀⠐⠢⡀⠀⠆⠀⡆⣀⠦⠀⠠⠀⠠⠤⢤⠀⠀⠠⠤⠃⠀⠀⠀⠀⠀⠔⠀⢚⠖⠈
⠀⠀⡐⠞⠖⠃⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠤⠀⠀⠈⠰⡄⡀⠉⡐⢀⠂⠁⠀⠀⢁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣀⡔⠁
⠀⠀⠸⡆⠀⠀⢐⠀⡀⠠⣀⠀⠀⠀⢠⣦⣰⡧⢀⣠⣀⠀⡌⣀⠀⠀⠄⡀⠁⠈⠅⠄⠀⠄⡀⠄⢈⠆⠰⢀⠒⠰⠂⠀⡀⠀⠐⠂⣁⣤⠞⠈⠑
⠀⠀⠀⠈⠀⠀⠀⠀⠣⠀⠀⠐⠛⠀⠊⠁⠤⠀⠋⠑⣏⠀⠀⣀⡁⢀⣀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠉⡀⡀⠀⠀⢀⡄⣀⠠⡖⠒⠀⠈⠀⠀⠉⠐
⠀⠀⠀⢠⠀⡠⡤⣠⠀⠀⠀⠀⠀⠠⠚⠂⠀⣀⠀⠮⠀⠄⠀⠀⠂⠄⣉⡟⠻⠯⢉⠝⠋⠉⠉⠉⠉⠉⠉⠉⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⣀⡀⠋⠑⢢⡀⠀⠠⠤⠇⠀⠀⢰⡉⠐⠄⢀⠀⠀⠀⠀⡨⠛⠛⠐
⠀⠀⠀⢆⡀⠀⠀⠀⠀⠈⠰⠁⢀⢠⣤⢆⢄⡁⠈⠚⠄⣀⠁⠀⠀⠀⠀⠐⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢋⠀⠀⠀⠀⡄⠀⠀⠂⠋⣗⡡⠿⠀⣻⣳⣄⣠⠃⠀⠀⣀⡀⣀
⠀⠀⠀⠊⢀⠀⠀⢀⣑⠦⠀⠀⠈⠁⠀⢼⠀⠋⡩⠋⠈⠐⠠⠀⢌⣀⠈⠁
⠀⠀⠀⠀⠈⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⢒⠀⡘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⡎⠱⠄⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⡾⣻⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠙⠀⠀⠘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠤⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠀⠀⠀⠀⠀⣠⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⣿⣃⡀⠀⠀⠀⢀⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠈⠀⠀⡀⢀⠀⠀⠀⠀⠈⠛⠻⠿⠇⠀⠀⣠⣿⣿⡿⠟⠀⠀⠀⡖⠀⠀⢀⠀⠀⠀⠀⠈⡇⡂
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠁⢠⡀⠀⠀⠀⠂
⠾⠿⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠙⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⡁⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣄⣀⠀⠀⠀⠀⢠⠆⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣭⣽⣿⣽⣷⣄⠀⣀⣤⣀
⠀⠀⠂⠀⠀⠀⠁⠈⢀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠀⢠⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠘⣻⡻⣿⣿⣿⣿⣿⣿⣿⣿⢶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠿⠖⠾⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠛⠷⣴⣤⣿⣿⣿⣿⣷⣶⣒⠛⠠⠈⠉⠹⣛⣩⣽⣽⣿⣿⣧⣯⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠠⠀⠙⠚⠀⡠⢸⣿⣿⣿⣟⡶⠾⣿⣟⡁⠈⠰⠀⠀⠈⠁⠈⠛⢷⣿⣿⣕⣄⡀⠀⣀⣿⣿⣷⣤⣶⣦⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡠⠀⠀⡀⠀⠈⠀⠀⠙⠛⣻⣿⣿⣿⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣦
⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠂⠃⠀⠀⠀⠀⠀⠀⣽⣿⣿⣟⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠿⢿⣿⣿⣿⣿
⠀⠀⠰⠀⠒⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⣿⣿⣿⣟⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠘⣸⠁⠀⢸⠧⡄⠀⠀⠀⢀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠄⠀⠈⠀⢶⣇⡀⠀⠀⠀⠀⠀⠀⠀⠋⢻⡷⠆⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠠⢿⡵⠀⠀⠀⠀⠀⢋⣠⣄⣴⣤⣤⣤⣦⣴⡀⠀⢡⡄⠈⢉⠋⠀⠀⢠
⠀⠀⠀⠚⠿⣤⣤⡀⠀⠰⣾⣰⠼⠄⠘⠛⠃⠀⢈⣁⠀⠁⠉⠛⠀⠼⠅⠀⠸⢶⣰⠄⠘⠀⣸⠖
⠀⠀⠀⠀⠀⠘⠿⢷⣶⣄⠀⠉⠁⠀⣀⣀⣴⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣤⡤⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⡿⠿⠟⠁⡀⢿⣧⡀⠚⠛⢿⣛⡛⠛⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠐⠚⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣰⡿⠋⠀⣚⣻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⣉⣼⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢹⣞⣿⠃⠀⣾⠿⠇⠈⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡬⠡⠿⣿⣿⣾⣇⢀⣀⣤⢀⡄⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⠄⠀⠀⠀⠡⣙⣀⠘⠃⢈⠀⠀⠂⠘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡏⠞⢁⣀⣘⣿⣿⣿⣼⣿⢿⡨⠟⠋⠁⠀⠀⠀⠀⠀⢀⣔⣩⠷⡿⠛⠀⠀⠀⣤⢞⡟⠉⠹⣧⠰⢆⠙⠒⠈⠀⠺⠆⠉⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡡⡾⢋⢡⣤⣮⣿⣿⣿⣿⣾⣿⣁⠠⣄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣎⡀⠀⠀⠀⢈⠇⠀⢠⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣾⢈⣯⠿⣿⣿⣿⢿⣿⡎⠛⠂⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠘⠀⠀⢸⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⡘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣷⠟⠁⠈⣿⠃⣽⠀⠰⣻⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠃⢰⠈⠀⠀⢹⡀⠀⠀⠖⢀⠀⠀⣿⢢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠃⠀⠀⢴⣿⡴⠁⠀⠀⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢘⡄⠀⠀⠈⣇⠀⠀⠀⠁⠀⢸⠃⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢹⡄⠀⡁⠀⠁⡞⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠕⠀⢰⠇⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⣾
⠀⠀⠀⠀⢀⠀⠀⣬⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡇
⠀⠀⠀⠀⢸⠆⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣈⣓⣦⣀⢀⣀⣠⣿⡀⢠⡇
⠀⠀⠀⣐⣽⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣴⣇⠨⢠⣋⡼⢛⡉⠭⠜⣼⡇⢸⡇
⠀⠀⠽⣿⢿⣿⢛⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⡥⣪⢤⡄⡤⠉⣡⡴⠋⠄⠶⡶⣿⣟⣸⣷⡆
⠠⢤⣤⣶⢿⣿⣾⣷⣶⠂⢤⡀⣀⣠⣦⣀⠀⠀⠀⠀⢠⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⢟⡉⠄⣉⡚⡤⠐⢊⣉⣠⣌⢁⣤⢯⣿⣿⣿⣿⣿⠧⠶⠂
⠀⢠⣾⣿⣼⡿⢯⣿⣏⣋⣈⣿⣷⣿⣿⣿⣿⣤⠀⢀⡟⠙⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣿⣿⣥⣤⣶⣶⣬⣭⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠠⣿⣿⣿⠃⠀⠀⠈⠱⣶⣾⣿⣿⡿⢛⡿⠇⠀⢸⠃⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢉⢛⣻⣟⣻⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣈⡛⢻⣷⡀⠻⢦
⠀⣀⣹⠿⢃⣤⠶⣶⣶⣶⣿⢿⡉⠩⠭⡗⠀⠀⠀⠨⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠌⣉⣻⣿⣿⣿⡿⢿⣷⡛⠋⣿⣟⠿⢭⣿⣿⡿⠏⠹⣷⠀⠀⢢
⠀⣩⠖⠁⠀⠀⠀⠡⠶⠏⢁⡴⠄⠀⠀⢀⠤⠀⠀⢰⠀⡆⢸⠄⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣠⠶⠿⠿⣒⠶⣷⢤⣤⢌⡛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠚⠷
⠹⠃⠀⠀⠀⠀⠀⠀⠤⠀⠀⣉⡐⠖⣂⣤⣀⣀⠀⢸⣤⣧⢸⠀⠐⠐⠒⠒⠚⠋⠉⠿⠖⠲⠶⣶⡿⠟⣿⣗⣀⡳⠫⠽⠟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠣⠳⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠞⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠘⡓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠑⠠⡀⠀⠀⣠⠔⢪⠃⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠈⢇⠀⠑⠀⠀⠀⠀⠘⠒⠦⡁⠀⠀⠀⠀⠐⠀⣄⠀⢄⡠⠊⠁⢠⠂⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀
⠀⠀⠀⣀⠄⠊⠀⠀⠀⢨⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⡡⠄⠚⠉⠀⠀⠀⠁⠀⢀⠀⠘⣄⠀⠀⠀⠀⠈⠉⠉
⡠⠴⠊⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠄⠋⠉⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⡀⠀⠈⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠉⠂
⠀⠀⠀⠀⠸⡄⢁⠀⢠⢦⠤⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣀⡄⠀⡄⣰⠃⠀⢅⣘⢘⠂⠄⠀⠀⠀⢀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠛⠡⠎⢥⠀⢨⡤⠔⠀⠰⠤⠀⢀⣌⣀⣘⣃⡐⣲⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠁⢁⢀
⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠤⠀⠠
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠴⠀⠀⢠⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⡄
⠀⠀⠀⠰⠁⠀⠀⠛⠟⢫⠁⠀⠀⠀⠀⠀⠀⠀⣤⣿⠂⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠃⠠⣆⠉⠋⢩⡀⣀⠈⢂
⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⡧⠂⠀⢫⠁⠬⢴⣏⡠⢰⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠈⢥⠀⠀⠁⠉⠀⠾⢤⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠉⢩⠁⠴⢻⠓⡀⠀⠀⠀⠀⠀⠀⠘⠀⠉
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠗⠷⢽⣿⠤⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⡄⠀⠀⠉
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠈⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡉⠉⠀⠀⠒⠒⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠒
⠀⢀⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⡀⠈⠁⠀⠤⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠓⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡀
⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠢⠤⣤⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠒⠀⠀⠀⠀⠀⠀⠁
⢤⢶⣆⠀⠀⠀⠀⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠘⠒⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠄⠠
⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠢⠀⢨⡦⠀⢀⠄⠢⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠢⠀⠀⣠⡶⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣢⡄⢲⣄⣦⠀⣤⠎⢀⡲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡘⣢⡔⠵⡄⢁⡔⢨⡡⠂⢑⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠍⠶⠬⠑⠄⣀⡡⢊⠴⢂⡉⠉⡥⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢀⡉⠲⡐⢈⢁⠋⢠⠶⠅⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠂⠐⠶⢂⠈⢣⡿⡿⣭⢀⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣆⢀⣩⠀⣙⢾⣿⠥⢄⠛⠛⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢷⠖⢉⣠⢺⡇⡛⣄⢢⡉⠣⡁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡤⢊⡰⠋⣽⡅⠠⣌⠫⡀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡠⠪⠉⠍⢁⡆⢸⣄⡓⠠⢔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠂⠘⡤⡏⢠⡑⢻⠸⢶⠈⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⠃⢿⡼⢿⣄⢸⢹⡷⢂⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⡘⠱⡎⠄⠇⠱⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢨⡁⣙⠲⣹⣯⣍⠰⡍⠶⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠃⡇⢘⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⡺⢾⠝⢓⣽⢿⠿⣥⡞⢇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠧⡸⢻⠟⡎⣷⢳⠙⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⠀⡔⡡⠾⠛⣤⠎⠏⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠎⢰⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠇⠀⡗⠚⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠓⠂⠠⢄⣀
⠀⠀⠀⠀⢀⠀⡠⠋⠉⠀⠀⠀⠓⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡸⠀⠀⠀⣀⡨⠉⢰
⠀⠀⠀⠐⠁⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠅⠐⢀⠩⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢔⡀⠉⠒⠠⢀⡀
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡻⠀⠀⠀⠈⠁⠂⢄
⠀⠀⠀⠀⠈⠒⠤⠀⣀⣀⠀⠈⠂⠀⠀⠉⠀⠈⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠰⠶⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠹⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠰⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡎⠀⠀⠀⠀⠀⢀⡴⠂⠉⠀⠀⠈⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢢⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠧⡀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠒⠈⠉⠁⠀⠀⢀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉⠉


⣶⣶⣶⣶⣶⡷⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣶⣿⢷⡶⢶⠿⠿⣶⣾⡿⠿⠿⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡔⢢⢀
⠀⠀⠀⠀⠀⢀⡀⢀⡂⠀⠀⡼⠀⠨⠛⢯⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣉⣷⡄⠀⣤⠴⣶⣾⣤⡀⠀⢀⢾⣿⣧⣤⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠰⠉⠩⠍⠉⠹⠎⠹⠄⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠻⠋⠀⠈⠉⠉⠻⠀⠀⠀⠛⠻⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⣀⣀⣠⣀⣠⣤⠀⠀⢰⣺⣇⠀⠀⢸⣇⠀⠶⠏⠉⠁⢀⡀⠀⠀⠀⠀⡠⣾⠃⢀⣾⡧⠄⠀⠉⠀⠀⢤⠤⡀
⠀⠀⠀⡿⠯⠭⠿⠿⠿⠾⢷⠄⢀⡤⣿⠃⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⠀⢀⡄⠀⢉⡿⡇⠈⣹⣧⠀⠸⡻⣄⠀⠭⠭⠇
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠀⠈⠈⠛⠀⠀⠘⠛⠀⠀⠒⠀⠀⠙⠋⠀⠑⠠⠆⠚⠀⠃⠐⠃⠙⠀⠐⠓⠂⠀⠛⠛⠁
⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠸⠿⠿⠟⠀⢸⣼⡇⣽⣭⣿⠀⠠⢶⣷⣶⠷⡍⡿⠃⢀⣀⣤⣶⣶⡿⠿⠗⠀⠀⢰⣛⣛⣳⠀⠀⣠⠟⣐⣧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁⠀⠀⠀⢿⣿⡾⠃⠀⢠⡾⢿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠑⠀⠀⠙⠁
⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠑⣤⣄⡀⠺⢿⣿⠏⠙⠉⠁⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢀⠀⣠⣇⣀
⠀⠀⠀⠘⣷⣤⠾⠁⠀⢰⣿⣶⣷⣿⠄⠀⠀⠀⠀⢹⠟⠁⠀⠘⠟⠀⠀⠀⠰⣶⠄⠀⠀⠀⣼⠓⢻⡆⠀⠈⠉⠹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⢤⣴⣶⣶⣶⠀⠀⡠⠀⠐⢀⠀⠀⣤⡄⠀⢀⣠⣀⡀⢀⣤⡀⠀⣶⡀⠀⣀⣤⡀⠀⢰⣖⣒⣶⠀⠀⢠⣤
⠀⠀⠀⠘⠿⠛⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠉⠉⠁⠀⠿⠀⠀⠛⠁⠀⠉⠉⠁⠀⠸⠿⠿⠿⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀
⠀⠀⠀⡞⠉⢓⠀⢀⣀⣀⣐⠀⠀⠀⢰⠫⣭⢳⠀⠀⣋⣋⣃⠀⠀⠟⠛⠏⠀⢰⣷⠀⠀⠀⠀⠁⡀⠀⠀⣷⡶⢿⣿⡆
⠀⠀⠀⠙⠶⠋⠀⠈⠉⠙⣿⣷⡷⠀⠘⢄⣡⠜⠀⠀⡻⠿⡏⠀⠀⡅⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠿⡿⠿⠿⡇
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠙⠁⠀⠀⠘⠟⠀⢀⡀⠀⠀⢀⠀⢀⠀⣠⡟⢀⠾⢧⠀⠸⣥⠀⠀⠶⠀⠀⣀⠀⢀⠈⠋
⠀⠀⠀⠀⣶⣶⣀⣰⣶⠀⠉⣷⠀⢘⢉⠀⢸⠘⠿⣷⠀⣷⡇⠀⣬⣽⠀⠀⣠⣦⠀⢨⣿⣧⠀⣠⣦⠀⠀⢀⣖⣲⣲⣀
⠀⠀⠀⠀⠯⠿⠀⠹⠽⠀⠀⠇⠀⠺⠶⠇⠿⠀⠀⠿⠀⠻⠇⠀⠿⠹⠀⠰⠿⠿⠇⠸⠿⠿⠀⠿⠿⠗⠀⠛⠿⠛⠿⠛⠃

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

⠀⠀⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣮⣶⣿⠋⠀⠈⠉⠁
⠀⠀⠀⢹⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣄⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠈⠁⠉⠙⢷⣦⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣷⣶⣿⣿⣿⣷⣶⣀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠿⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⠿⠛⣭⣭⣭⣍⠛⠻⣿⣿⣇⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠉⠁⠀⠀⠈⠛⠹⢿⣿⣦⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣠⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣶⣾⣿⡶⠆⠀⠀⠀⢼⣿⣿⠟⠉⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠛⠛⠻⢿⣧⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣦⣄⣀⠀⠀⢀⡀⠉⠛⣿⣿⠀⠀⠀⠀⠀⠀⠉⣯⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢉⣷⣦⣤⣄⣀⣤⠛⠉⠉⠻⡟⠛⠉⠉⢿⣿⣷⠀⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣤⣀⣀⣴⣾⡿⠿⠋⠀⠉⠛⠙⠇⠀⠀⠀⢠⣄⠀⠀⠀⣸⡟⠁⠀⠀⠰
⣿⣶⣤⣤⣤⣶⣿⣿⣫⠶⣙⡿⠛⠋⠛⢦⣤⣤⣴⣄⣀⣀⠀⠀⣴⣶⣿⣿⣦⣤⣿⣏⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣖⣯⣷⡄⠀⢀⡀⠈⢉⠹⠿⠿⠿⢿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣷⣶⣤⣤⣄⣤⣾⣿⣿⠋⠀⠀⠀⠀⣤⡄
⣿⣯⣿⣿⣦⣿⣿⢻⣿⣿⣿⣿⣿⠆⢸⣿⣆⠀⠀⠄⠠⠱⠤⡤⢿⣿⣿⣿⣿⠿⠟⠉⠹⣿⡟⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣨⣿⣦
⡉⠿⣾⣿⡈⣿⣿⠏⣛⣿⣿⣿⣇⣠⣯⣿⣿⣦⠦⠭⠤⣌⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣠⣄⣀⠁⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣦⠈⠿⠁⠾⠋⣴⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣀⣒⣒⠐⠰⢿⣿⣿⣿⣿⣿⣷⠀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠀⢀⣤⣤⣤⣭⣷⣿⣿⣿⣿⣯⣿⣿⣿⣿⣶⠒⠲⠧⠬⠭⠭⣽⣽⣿⡻⣧⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠙⠉⠉⠻
⠻⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠥⣏⣻⡓⠒⡺⢺⣾⣣⣌⡹⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⠛⠿⣿⠇
⡀⠀⠀⠙⣿⣿⢫⣿⣿⢋⣿⣿⣿⣿⣛⣁⣛⣟⣛⣿⣷⠾⡟⣉⣛⡙⢯⠤⢴⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⡀⠀⣀⣀
⣿⣦⠀⠀⠹⠏⣠⣤⣴⣿⣿⣿⣿⣿⣽⣼⠛⡛⠟⣉⣙⣛⠟⢻⣝⢠⢾⠾⣇⣚⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣀⣀⣠⣶
⣿⡏⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣶⣶⡿⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠉⠀⠀⢀⣾⣿⣿⣿⣿⡿⠿⣿⣟⣿⣿⣿⣿⣿⣶⣿⣿⢻⣷⠙⠧⢉⣿⠶⣦⣾⣿⣿⠮⡌⠻⢿⣿⢠⣿⣿⣿⣿⣿⣿⡭⢿⣹⠌⠛⢻⡶⠤⢀
⠀⠀⠀⣿⣿⡿⣻⡿⢿⣟⡻⣮⣭⣴⣿⣿⣿⣿⣿⣿⣿⣼⣿⣆⢻⡻⣆⣠⣿⣿⣿⣯⣸⡏⠀⣽⣿⣿⣿⣷⠿⣿⣿⣿⣷⣍⠅⠀⣰⣿⣏⠀⢰
⠀⠀⠀⣹⣿⠛⠋⠐⠀⠻⢄⣾⣿⣿⣿⡿⣵⣠⣿⣿⣿⣿⣬⣿⣾⣷⣿⠟⣿⣟⣿⡏⠉⣁⡀⣼⣉⣏⣭⠷⣾⣿⣿⣿⣿⣧⠀⠸⠄⢿⡃⠰⠮
⠀⣤⠤⢌⠣⢥⡾⣠⡄⡼⠛⠻⣿⣿⣟⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣭⡷⠾⠿⠛⠉⠀⠀⠀⠛⢶⣽⡿⣿⠿⠿⠟⢿⣿⣿⣿⣿⡆⠒⠈⠀⠀⣀
⣶⡟⠉⣿⣿⣿⠒⠇⠀⠀⠀⠈⠿⠟⢺⣿⣿⣿⣿⢿⠟⠋⠉⠉⠁⠀⠀⢀⠀⠈⠋⠁⠉⠀⠀⣤⣘⡓⠛⠆⠀⣢⣼⣿⣿⣿⣉⣥⣄⣁⡀
⣿⠛⠛⠛⠻⠻⢀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠉⠛⠋⠀⠀⠀⣠⣾⡿⠟⠚⠋⠘⠉⠙⠢⡔⠀⠀⠀⠈⢉⣴⣷⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠿⠖⠟
⣻⠅⡀⠀⠀⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣨⠟⠁⠀⠀⠀⠀⢀⣴⣾⣿⣦⣶⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣰⣷⣄
⠀⣈⠛⢁⠁⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⣠⠼⠿⠶⢶⣦⣖⣠⣾⣿⡿⣿⡆⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣻⡏⠁⠈⠉⠙⠛
⢴⣤⣿⡿⣷⣝⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠘⡣⠂⠀⠞⡉⢙⣹⣿⣷⠿⢷⣾⣿⣿⣿⣿⢿⣟⣿⠟⢟⠚⢡⢻⡿⢇⠈⣉⠚⠯⠄
⠀⠝⣿⣷⣋⠀⠀⠀⠀⠂⠀⠀⣠⣴⣾⣷⣄⠀⠀⢰⣠⣿⣴⣦⣶⣶⣿⣿⣿⣿⣟⠟⣲⣿⠿⣿⣿⣿⢃⣝⠉⠀⠈⠿⣿⣤⣄⣻⡶⢾⠳⣷⣾
⠸⠂⠙⢛⣥⢷⡤⡀⠀⠀⠀⠈⠻⣿⣿⡿⠋⠐⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⠟⠉⠀⠘⠡⠀⠸⣿⣷⣤⣾⡆⠻⠧⣘⡃⢀⠀⣸⡷⢿
⢀⡻⠋⣈⣅⣨⠓⠆⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣹⣿⣿⣿⣿⣿⣯⠉⠉⠀⠀⠀⢀⡆⠀⡴⠂⡄⢀⣿⠋⠈⣴⡀⠀⢨⣏⣀⠛⣦⡼⢻⡟
⠈⠑⠂⠬⣹⢉⠐⠞⠠⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠈⠙⢛⣾⣟⠙⢶⡹⣟⢿⣦⡿⣇⣼⣿
⠀⠀⠀⠀⡟⠀⠉⠓⢠⣤⠈⣠⠀⠀⠘⠀⠾⠇⠀⠻⣷⠀⠀⠉⠀⠀⠀⣠⠀⠁⠐⣄⠘⠀⠀⢀⡀⢸⣆⣠⣿⣿⣿⣿⠘⢿⡿⠟⢉⣷⣿⢷⣻
⠘⠀⠀⢸⡷⠀⠀⣿⠀⠀⠐⠉⡀⠀⠀⠀⡤⣰⠦⣦⡀⠀⠀⠀⠒⠶⢁⠀⢃⠑⣤⡯⠁⠀⢠⣿⣿⢿⠮⣿⠷⣿⡟⢿⡟⣰⢿⣷⣾⣷⣦⣜⡛
⠀⠀⠀⠈⣿⠀⠀⠠⢈⠁⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⢠⣴⣦⡄⠐⠀⠀⠀⢠⣤⡝⠀⠀⣀⣠⣾⣿⢾⡷⣙⣻⣿⢿⠺⢿⣷⢿⣿⣿⣿⡄⣹⣟
⠀⠀⠀⠀⣿⢀⠠⣾⠘⠟⠁⡀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠇⠀⠀⠀⣸⢹⣿⣥⣤⣴⠛⣿⣿⣿⣾⣿⢿⣾⣷⣼⢆⣬⣿⣶⣬⣏⣻⡷⢿⡘
⠀⠀⠀⠀⠁⢠⡄⢀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣴⣾⣷⣁⢼⣿⣼⣿⣿⣋⣜⣽⡟⢿⡓⢻⢿⣿⣿⣇⡔⣨⡤⠙⣿⣷⣼⣟

⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣀⣀⡀⢴⣀⣠⣷⣾⣯⣤⡴⢀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⢛⣿⣿⢿⣿⣿⣟⣾⣷⣠⢦⣬⣽⣏⣴⣶⣶⡀
⠀⠀⠀⠀⠈⠛⢩⣿⣟⣿⣿⣿⣿⢿⣿⣿⣿⣿⣾⣯⣦⣽⣿⣏⠉⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⢠⣬⣽⣿⡿⢿⣿⣁⡀⣹⣿⣿⠋⢿⣿⣿⡿⠻⢿⡟⣷⣴⣲⣿⣧⠆
⠀⠀⠀⠀⠀⠀⣿⣿⠿⠃⠀⠀⠹⣷⣿⡟⠀⠀⠀⠀⠹⣧⣤⡿⣽⣿⣟⡉⠱
⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠘⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠿⠻⠿⣿⣿⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠠⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⣄⠑⢧⡄⠀⠀⠀⠀⢀⢉⣀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠈⠃⢆⢳⡄⡠⢨⣀⡬⡉⠉⠀⠁
⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⡄⠀⡄⠍⣑⢪⣾⡸⠾⠛⠑⠒⠛⠂
⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠴⡀⠀⠀⠀⠀⠀⠀⠁⡠⣴⡾⢿⣯⣿⣿⡧⡴⠖⠒⢏⢲⡢⡲⡠⡄⡀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡘⠻⠛⠟⣻⡏⣽⢁⣧⢀⠀⠀⠁⠀⠀⠈⠈⠃⠠⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⠋⠠⡢⠊⠨⠇⡏⢸⠁⢋⠀⠘⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡱⠁⠀⠀⢶⣇⠏⠀⠀⠘⠆⢠⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠄⠀⠀⠀⠀⠀⡤⠤⠠⠤⠤⢤⣄⣤⣤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠒⠀⢰⡆⣸⡤⠄⠀⣀⡀⠀⠂⠋⠉⢉⣛⣉⣉⣉⣉⡩⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠁⠁⠉⠁⠈⠀⠀⠀⠁⠒⠢⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠶⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⢹⠗
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠒⠀⠀⠂⠀⠤⠤⠤⠄⠶⠒⠚⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⡉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠉⠒⠠⣍⡒⠤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣠⣤⣄⡀⠀⠈⠑⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⠀⣤⢄⣀⡜⠁⢀⡴⠋⠁⠀⠈⠙⢦⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢥⡇⠀⠀⠀⠀⡔⠉⢆⠀⠀⠀⠀⠀⡠⡫⠊⠁⠀⢀⡠⢘⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⣠⠔⣉⠉⠭⣉⣿⡲⡀⠀⠀⠹⠛⡟⠀⠀⠀⠀⠀⣿⠁⠀⢀⠔⠁⢀⠎⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⡾⠁⠀⡌
⠀⠀⠀⠀⢰⠁⡜⠁⠀⠀⠈⡇⢸⠄⠀⡠⠃⡰⠁⠀⠀⠀⠀⠀⣾⠀⢠⠋⠀⠀⡌⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠋⠀⠀⡰
⠀⠀⠀⠀⣾⠟⢆⠀⠀⠀⡠⠃⡜⢆⠉⣀⠔⠁⠀⠀⠀⠀⠀⠸⡆⢠⠃⠀⠀⡸⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⢠⢻
⠀⠀⠀⠀⠑⠵⢤⣉⣉⣉⣤⠮⠒⡌⢷⠻⠼⢖⢄⠀⠀⠀⣠⡳⢁⠏⠀⠀⢀⣇⡀⠀⠀⠀⡰⠁⠰⠁⠀⠀⠀⠀⢀⠇⠈⡄
⠀⠀⠀⠀⠀⢀⠊⢣⠰⡊⣴⣺⢷⢘⡌⢆⠀⠀⠑⠵⣖⠮⠛⠁⡘⠀⠀⣠⠋⠑⡦⠕⡒⡴⠁⣰⠁⠀⠀⠀⠀⢠⠋⠀⢠⠁
⠀⠀⠀⠀⠀⢸⠀⡇⠱⡑⣌⠥⠒⢉⣠⡼⣆⠀⠀⠀⠀⠀⠀⠀⢧⠀⢠⠁⠀⠀⠉⠒⠼⠁⢠⠏⣒⠤⣀⠀⡰⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠘⡄⠸⡄⢱⣠⣴⠾⠛⠉⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢰⠁⢠⠋⠒⠢⠟⠒⠿⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠘⣶⠛⢹⠻⡀⠀⠀⠷⣄⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⢠⠃⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠱⡀⠀⠸⡈⠢⡀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢃⠀⠘⣦⠜⠓⢒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠜⢆⠀⠈⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡡⠔⠊

⠀⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠄⠀⢤⠃⠼⠛⣡⠁
⠀⣤⠀⠀⠠⠀⢨⠴⠀⢀⡠⡔⠂
⠀⠀⠐⠇⠀⠀⣞⠀⠀⣄⣁⠀⢤⡀⠀⠀⢤⠄
⠀⠀⠀⠀⠄⠛⠿⡀⠀⠭⠉⠙⠆⠰⣚⠴⠓⣀⠆
⠈⠉⠀⠀⠀⠀⠅⠈⠀⠠⢦⠴⠂⣰⡇⠰⣆⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠋⠰⠀⠘⠁⠁⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡺⡏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠄⠀⠀⠀⠀⠀⣤⣴⡂
⠀⠘⠧⠀⠀⠀⠅⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠇⠀⠉⠐⠃⡂⠤⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⢂
⠀⠀⠀⡀⠀⠀⡇⠀⠂⠠⢀⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠒⠂⠠⠔⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠚⠃⠂⠀⠀⠬⠤⠀⠀⠀⠐⠠⠀⠀⠂
⠀⠀⠀⠉⣤⡀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⠀⠤⠤⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣀⠀⡉⠀⠁⠀⠀⠀⢠⠀⠁⠀⠠⣾⣀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠉⠁⢀⠀⣄⣄⠀⣀⣈⣧⣀
⠁⠀⠀⡀⠀⢀⣠⣷⣷⣾⠵⠟⠛⠓⠐⠛⠉⠉⠁⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠷⠶⠦⠴⠶⠿⠿⠛⠛⠛⠛⠛
⠛⠻⠻⠿⠋⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣄⣀⣤⣤⢠⣤⠀⣠⣄⣠⣀⣤⢤⣤⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣇⠄⠀⠀⠀⠀⠟⠟⠛⠳⠸⠚⠀⠳⠐⠛⠛⠆⠘⠘⠟⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⡀⠀⠀⠸⡆⢸⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠨⠀⠀⠀⠛⠿⠘⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⡄⢿
⠀⠀⠀⠀⠇⢀⠰⠀⠀⢀⣸⠋⠀⠀⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠆⠀⢐⠀⣾⠀⢾⡇⠀⡇⢲
⠀⠀⠀⠀⠛⠈⠓⢀⠚⠻⠿⠀⠀⠀⠄⠀⠂⠉⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠘⠓⣾⡿⢹⢲⣼⡇⡆⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠁⠸⣼⢸⣿⢹⣡⡇⡄
⠀⠀⠀⠀⠀⠀⣀⡀⣀⡲⡄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣀⠠⠴⠀⠄⠠⠐⠚⠂⠀⠀⠀⢀⡀⠀⢀⡹⠎⠋⠈⠋⠾⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠁⠈⠉⠉⠠⠀⠖⢊⠁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠦⠄⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢋⣠⣤⣤⣀⠀⠀⠰⠶⠂⠀⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠶⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠠⡀⣤
⠀⠀⣀⠀⠤⠀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢀⣤⠀⠤⠆⠀⡀
⠒⠁⠀⠀⠑⠈⠩⠖⣨⡥⠔⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠁
⠀⠐⠊⠁⠈⠔⣊⠁⢀⡴⠖⢉⡤⠄
⠆⡠⠖⣂⡴⠏⣡⡶⠋⣴⠞⢉⣴⠞⢋⣡⣤⠀⢀
⣉⡤⠞⠡⠤⠖⠋⢀⡼⠁⠔⢉⡴⠚⣉⠥⠴⠚⢋⣠⠴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠟⠀⣁⠖⢚⣩⠶⠋⣠⠶⠒⣩⡤⠶⠒⣠⡤⠞⠛⢁⣴⡶⠗⠋⣉⣤⠔⢂⠄⢀⣀
⣤⡿⢃⡴⢋⣤⠶⠛⣉⡤⠞⣋⠴⣠⡿⢋⣤⠾⠁⣸⠟⠀⣠⡾⠋⣠⡾⠋⣠⡾⠋⢠⣼⠃⣠⡤⢀⣠
⣩⣔⣯⣖⣋⣴⣞⣉⣡⣶⣛⣡⣞⣩⣴⣛⣈⣴⣚⣁⣤⣞⣉⣤⣾⣋⣠⣾⣛⣡⣴⣛⣀⣜⣡⣾⣋⣡⣤⣟⣃⣤⣶⣂⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⢀⡤⠂⠊
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢀⡴⠋⠀⠀⣠⠆
⠀⠀⠻⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠚⠉⠀⠀⠀⠈⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣤⣤⣀⠀⢳⡀⠀⠀⠀⠀⢢⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠗
⠀⠀⠈⠳⣬⠙⠄⠀⠀⢠⣮⡿⠙⠢⡄⠀⠀⠀⠀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠻⣦⠀⢀⣺⣿⡇⠀⠀⡄⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⠀⣸⣿⣄⠀⢀⠀⠀⠀⠘⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⡼⠁⢀⡄⠀⣀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡗⠀⠀⣴⣾⣄⠀⠀⠉⠻⢆⡀⠳⠆⠀⠀⠀⠠⣾⢿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⠀⣉⡴⠂⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢰⣿⠋⠁⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠁⣠⠖⠀⠀⠀⠀⠉⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⢰⠞⠁⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠼⠅⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣟
⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⣼⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡘⠛⠛⠷⠶⠶⣤⣤⣀⡀
⠀⠀⠀⣼⡿⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡀⢉⠉⠛⠻⠶⣦⣦⣤⣦⣤⣠⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣼⣿⣿⡿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢀⣼⣿⣿⣿⡿⠤⠸⠘⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠛⠛⠻⠿⢿⣶⣶⣤⣤⣀⡀
⠛⠛⠛⢿⠿⡁⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣷⣶⣧⣤
⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠻
⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠈⠉⠊⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡏⠀⣴⠖⠈⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠹⣄⣷⠗⢋⣈⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡗⣒⣤⡈⠷⡸⣁⠟⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠋⠈⢀⣁⠵⠦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣥⣄⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠉⣍⡙⣿⣯⣉⢉⣍⣉⠛⠿⠿⠿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⠀⠈⣰⣿⠛⢻⡴⣶⠏⢀⡀⠠⠿⠛⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣠⣀⣀⣀⡉⣘⣷⡀⢊⣿⣿⣤⣴⠦⣤⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣬⣿⣭⣭⣿⣿⣿⣿⣦⣾⣵⣿⣷⣯⣼⣿⣿⣿⣵⣧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⣤⣉⡈⢻⢈⡇⢀⣉⢯⢈⣿⠏⢀⣭⣿⠀⣨⢹⢀⣄⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢨⣯⢿⠟⡗⠿⡗⣬⣵⣿⣾⠿⣴⠿⠋⣼⢨⠳⢿⢨⣯⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣴⣤⣿⣦⣧⣾⣿⣷⣤⣴⣷⣤⣴⣿⣼⣷⣼⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⡠⡔⣒⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡄
⠀⠀⠀⠀⢀⡀⠤⢠⠀⣀⠀⠀⠀⠀⠰⠒⣟⣏⢭⠅⠃⠀⠀⣀⣀⣤⡤⠚⡁⠀⠊⠁⢲⠶⢤⠤⣄⡀⠀⠀⢍⠀⠈⠁
⠀⠀⠀⢸⢼⢶⢺⠻⠉⠉⣀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢠⡖⠋⠹⣉⡀⠀⠀⠁⢀⢀⣀⣉⠤⠖⠁⣢⣿⠂
⠀⠀⠀⠘⠚⠉⠈⠀⠀⠀⠘⢻⡿⢛⠁⠀⠀⠀⠀⠀⠈⠻⠶⣤⣄⣈⣉⣉⣉⣉⣡⣤⣤⣤⡶⠾⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠉⠉⡤⠑⠀⠀⢀⡀⡔⢢⠀⠻⠇⠀⠀⠀⠀⠀⠠⠀⠀⡤⡤⣦⢤⢤⡀⠺⠿⣀⡖⡗⢟⢻⢹
⠀⠀⢀⡠⡤⠴⠀⠀⠀⡈⠀⠀⠀⠁⠀⠀⠀⠀⠛⠀⠈⠐⠃⣠⠀⣇⢀⢹⣸⡼⠋⠀⠀⠏⠳⠶⠼⠼⠼⠄
⠀⠀⠀⠀⠀⠀⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠼⠀⠠⡻⠀⠈⠉⠉⠀⠀⠀⢐⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠄⠖⠈⠀⠀⠀⢀⢀⢀⠀⠀⣈⠋⠀⠀⠀⠸⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠈⠑⢦⣀⣷⣦⡀⠀⠀⢠⣶⣶⣄⣴⠀⢰⣷⠿⣿⡟⠃⣿⣯⣩⢹⣿⣹⠗⠀⠀⠀⠈⡗⠒⠂⢰⠂⠐
⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠉⠃⠀⠀⣿⣁⣿⡏⢿⣧⣼⠏⠀⣿⡇⠸⣯⣤⣤⠸⡿⠙⠷⠀⠀⠠⠀⢹⠋⠓⣾
⠀⠀⠀⠀⠀⠈⠃⠀⠀⡤⠶⣄⠀⠀⠘⠛⢉⣤⣄⢩⡤⢤⡀⣴⢶⡀⢀⣴⢶⢠⡶⠖⡂⠈⠀⠀⠀⠾⠲⠶⠻⣤⠐⠐⠂
⠀⠀⠰⠖⠶⢶⡀⠀⠀⢧⣀⠞⠀⠀⠀⠀⠘⠳⢦⣼⣷⠾⠛⣿⠾⣧⢼⣇⣠⠜⣟⢋⣁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠻⠷⠖⠀⠀⠀⠀⠀⢈⠠⠂⠈⠛⠛⠁⠛⠀⠀⠉⠀⠀⠀⠉⠁⠀⠛⠉⠉⠀⠀⠀⠀⠉⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⡰⠀⣔⢈⠀⢀⡀
⠀⠀⠀⠀⠉⠰⠶⠀⠀⠀⠀⢀⠖⠉⢠⠄⡀⠀⢁⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢦⣀⡀⠉⠀⠄⠈⢁⣸⠔⠋⠀⠀⠇⠀⠀⠄⠀⠈⡷⠀⠀⢀⠀⠄
⠀⠀⠀⠃⣄⡀⢀⡀⠤⢾⡁⠀⠀⠀⠉⠋⠍⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠁⠀⠀⣌⠀⠄⠀⢀⡀⠀⢠⠐⢀
⠀⠀⠀⣴⠇⠀⠈⠇⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠋⠀⠐⢦⣤⠄⠒⠾⠛

⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠐⠂⠐⠂⠔⠀⠂⠀⠒⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣿⣿⣿⠀⠀⢸⣿⣿⣿⡤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⠤⡤⣤⡤⢤⠤⡄⡄⠀⢀⣀
⠿⡿⠁⠀⠀⠈⠉⠁⠉⠀⣿⠀⣯⣿⣿⣾⣷⡆⠚⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣫⠄⠡⠀⠌⠰⠤⢾⠶⠲⠃⠛⠀⠘⠑
⣒⠛⠺⡷⠟⢓⣛⡖⢛⣉⠟⠀⠣⠤⠤⢤⠀⠀⠀⢤⠄⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀⠀⠄⠀⠀⠀⢀⠘⠐⠐⠃⠐
⣽⣿⣿⣧⣀⣾⡿⠿⣿⣿⣶⣶⣶⣶⣶⣆⣀⣀⣀⣀⡀⣀⠃⡂⠀⠀⢀⢀⠀⡛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠨⠝⢡⣿⣿⣿⣿⣽⣿⣀⠀⢰⡆⣴⣴⣤⡀
⠉⠍⠻⠀⠠⡈⠙⠉⣉⣉⠉⠉⣉⣉⣉⣙⡉⡋⡙⢻⣏⣏⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠞⠨⢡⠁⠀⠂⠂⢰⡆⠀⠘⣿⣿⣿⣻⣿⣿⡁⠀⠀⠁⠛⠙⠋
⠠⠀⡀⣦⠰⠇⢀⠀⣀⡉⢀⣶⣮⠈⠁⢈⠁⣡⠉⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠂⠀⠀⠀⠀⠀⠸⠀⠀⠓⠛⠛⠛⠛⠛⠋⠁⠀⠀⠐⠂
⠀⠀⠨⠃⠤⠤⠶⠸⠯⠥⠨⡭⠥⠠⠄⠿⠿⠏⠠⠤⠄⠤⠠⠤⠤⠠⠄⢤⠀⣸⣇⠀⡀⢀⠀⠂⢰⠖⠃⠚⠀⠀⠀⢤⠀⠾⢺⠀⣓⣀⣒⣀⣃⣐⣘⣃⣙
⣿⡆⠀⠀⠀⠀⠒⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈⠀⠤⠬⠄⠀⢸⣿⣿⣿⠀⢸⡇⠐⠀⡄⢘⢸⣿⡟⣿⣿⣿⡍⣿⣻⢛
⣿⡇⠀⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠆⢸⡇⠀⠀⢁⢸⢸⣿⣿⣿⣿⣿⣦⣿⣿⢹
⠏⢁⣠⡿⡿⠀⠀⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀⠉⠫⢍⣙⢠⡀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠘⠘⠛⠧⠛⠛⠛⠛⠛⠛⠛
⠀⣭⣭⣼⢇⠀⡂⠓⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠁⠀⠏⠀⠄⠘⢆⣤⠶⡆⣴⠒⠗⢰⣿⣷⠤⡦⣤⣤⣤⣤⡄⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠂⠈⠍⠅⠘⠀⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠒⠁⢩⡴⠤⠦⡆⡶⠄⠾⡕⠢⠶⠐⠀⠂
⣤⣠⡤⠧⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⣀⣠⣴⡄⠀⠀⣤⣴⣶⣦⠀⠇⣤⣀⡈⣥⢠⠘⡇⠀⠐⠀⣿⣻⣿⠃⡀⣄⣀⡀⣤⣴⠂⠉⠀⠀⠀⠀⡀⠀⠁⠈
⡿⣉⢁⣒⡂⣀⢰⡀⣀⢀⠀⡀⢠⠐⣈⢊⠻⡃⠀⠀⣿⣿⣿⡇⠈⠳⣿⣾⣗⣿⣿⠉⡇⠀⠀⠀⣿⢯⡴⠀⠔⡁⠂⡜⠸⣿⠈⡉⡉⠉⠉⠉⣉⣉⣉⣁⣉
⡏⠁⠈⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡀⠉⠀⠀⠀⣿⣿⣿⡗⠀⣀⣿⣾⡿⣿⣿⣺⡇⠀⠰⢠⣿⡟⠀⠁⠀⠁⠈⠁⢀⡶⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿
⡇⠀⠀⣿⣿⣿⠉⠉⠉⠙⣿⠙⡗⠙⣿⡇⠀⠀⠀⠀⡿⠿⠿⠗⢸⣿⣤⢠⡌⠸⣿⣿⡇⠐⢀⣽⣿⣇⠀⢸⣿⡿⣿⢿⣿⣟⠀⠁⠀⢈⠀⠀⣟⣿⣿⣿⡿
⠃⢠⣤⣿⣿⣿⣀⣀⣀⣀⡟⡟⢇⣠⣿⡇⠀⠀⠀⠀⠀⢀⣘⣇⣻⡦⠷⠒⠓⠰⣇⣼⡷⢾⠢⠀⢸⡿⠀⢸⣿⣿⣿⣿⣿⠯⠀⠄⠀⠠⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠁⣿⣿⣿⣤⣄⣤⣤⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠟⠛⠛⠀⠛⠀⠭⠀⠀⠀⠘⠉⢸⡇⠀⢹⣿⣿⣿⣿⣿⡗⠀⠂⠀⢰⡶⢿⣍⣿⣿⣿⣿
⡀⡈⠀⠙⠛⢉⣉⣉⣉⣉⣙⣁⣀⣉⣋⣉⠂⠀⠀⠀⠀⠀⠀⠀⢸⠠⠄⢀⣀⢠⡆⢰⣤⣰⣤⣤⣾⡇⠀⠈⠉⠉⠉⠄⠛⢻⠀⠁⠀⠈⠉⠙⠟⠛⠛⠟⠿
⠤⣤⢤⣶⣶⢸⣏⣿⣿⣟⣽⣟⣋⣻⡏⣿⣾⣷⡆⠀⠀⠀⠀⠀⢸⡀⢘⣟⡏⠀⠁⢸⣿⣿⣿⣿⣿⡷⠐⣲⣶⠿⣶⡟⢻⣿⡶⢶⡖⠒⣶⡖⢲⣶⠒⣶⡆
⣠⣤⣴⣿⣿⠈⠟⠛⠛⠛⠛⠛⠛⠻⠛⠿⢙⣿⡇⠀⠀⡀⣦⡀⢸⠀⠻⡿⡟⠀⠀⢺⣟⣟⠉⠻⠋⡇⠀⠹⣿⣼⣿⣦⣼⣿⣦⣾⣧⣤⣿⣧⣿⣷⣤⣿⡇
⠀⠀⠀⠉⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⣅⠉⠀⢸⠀⠸⣗⣧⠀⢀⠘⠉⠁⠀⣀⣀⠉⢰⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠐⡦
⡀⠀⠀⠀⣇⣀⡀⣀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢰⡄⠀⠀⣿⢠⣤⣤⣷⣾⠋⢁⠔⠁⠀⠃⠀⠐⣿⣽⡇⢸⢿⢿⠿⢿⣾⡿⠷⠿⠷⣾⣿⠀⠀⠀⠀⠀⠀⣌
⠄⢶⣾⡦⣯⠁⠅⠀⠀⠀⢰⠄⠀⠄⠀⠤⢤⣼⡇⠀⠀⣿⢸⣧⣽⠏⢂⠆⠉⠀⠄⠀⠀⠀⠀⣧⣼⡇⠀⠰⠾⠿⠴⡴⠿⠄⠀⠂⠀⠟⠀⠀⠀⠀⠀⠀⢀
⠀⢺⣿⡷⣿⠛⡃⠀⠀⢠⣤⣴⣆⣶⣶⡀⢸⣿⡇⠀⠘⣿⢨⣿⣿⡖⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡇⠀⢠⠀⠀⣤⠄⠀⠀⠀⠶⡿⠟⠀⠀⠀⠀⠀⠀⠨
⠀⢼⣿⣿⡷⠶⠆⠀⠀⠀⣤⣈⣀⣩⣬⡀⣼⣟⡇⠀⠠⣿⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⢛⣻⡇⢀⠘⠀⠀⢼⣿⣷⠀⠀⠸⡷⣾⠀⡀⡀⡀⡆⠀⢠
⠀⢸⣿⣿⡟⠛⠃⠀⠀⠺⢀⣀⣀⣠⣤⣄⣘⠀⠀⠀⠀⠘⢻⣧⣾⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⣶⣿⣾⣿⣴⣿⠒⠂⠐⢒⣗⠀⠀⠠⠤⠄⠀⠘
⡀⠉⠙⠉⡇⠀⠀⠀⠀⢀⠉⡉⠉⢙⡉⠉⢹⢠⡄⢠⡈⠁⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⣿⠁⢸⣶⣿⣶⣶⣾⣿⡀⠙⠒⠉⡉⠀⣀⣀⣀⣀⣀⡀
⠀⠒⠐⠐⡇⠀⠀⠀⠀⠛⠛⠋⠛⠻⠛⠛⠛⠀⣼⢿⣿⣿⣿⣿⣧⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⠈⠋⠀⢸⣶⣶⣶⣾⣿⣷⣄⠀⠀⢸⡇⠀⣿⡏⢻⠏⢹⣇
⠄⠀⠀⠐⡇⠀⠀⢀⠀⠀⡀⠀⢀⢠⡀⢈⠀⠀⢘⣿⡏⡉⠉⢉⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠔⠇⢸⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⠗⠘⠿⠿⠿⠷⠾⠟
⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⠀⠀⠀⠃⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠈⣀⠀⠀⠀⠀⡆⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠀⠃⠀⠁⠀⠀⠈⠀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⢀⠀⡏⢱⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡜⣙⣎⣸⢇⡇⡀⠀⡀⠆⠀⡔⠀⠀⠀⠈⠁⠀⠀⠀⣀⡀⠀⢠⠖⢦⠀⠀⣀
⠀⠀⠀⡋⣇⡇⢸⣠⢏⡇⠀⠀⠀⢠⠶⡄⠀⡇⠉⢩⠦⢊⡀⠘⢆⣡⡠⠜⣀⡔⠉⢀⠀⠠⠖⠀⠀⠁⢀⠀⢸⠁⠸⠀⡰⠁⠱⡀⠀⠠⡄⢀⡀
⠀⠀⠀⠳⠤⡄⢰⠓⠊⠉⠉⠉⠓⢣⠀⡧⢬⡇⠦⣿⢰⢳⠈⢲⠊⠀⠀⠀⠀⠙⣏⠥⠒⠐⠒⡀⠀⠉⠁⠀⠸⡄⠆⢧⣇⠀⠀⡇⠀⡦⡇⢨⠀⢶
⠀⠀⠀⠀⠀⠁⢸⡄⠀⠀⠀⠀⠀⠈⣆⠉⠒⠁⠀⠻⠞⣸⢅⣣⠀⢀⣀⠤⠔⠒⠛⠂⠉⠉⠀⠀⣫⡄⡿⠋⠉⠙⠤⢠⡀⠀⠀⠳⢖⣡⢇⣈⡄⡇
⠀⠀⠀⠀⠀⠘⠈⠀⠀⠰⠷⠆⠀⠀⠈⠑⠒⡄⠈⡴⠉⠁⠀⠉⠙⠢⠤⢀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠈⠁⠠⠄⠀⠀⠀⢣⠀⠀⡏⠉⠀⠀⠈⠇⠀⢤⣀
⠀⠀⢀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⣶⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠁⠒⠀⠒⠀⢠⠀⢠⢸⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⠈⠁
⠘⠚⠚⠓⠚⠒⠒⠛⠛⠛⠚⠒⠒⠚⠒⠚⠒⠓⠂⠛⠓⠚⠚⠚⠓⠓⠚⠓⠒⠚⠒⠒⠓⠒⠒⠒⠒⠒⠚⠒⠛⠚⠒⠓⠛⠒⠒⠓⠛⠒⠒⠒⠒⠚⠛⠚⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢲⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠰⠀⠇⠀⠤⠄⠀⠀⠀⠀⠀⠤⠄⠇⠰⠇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠆⠤⠀⣼⠁⠀⠀⢹⣄⠀⠤⠶⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠀⡠⢴⣶⢶⣤⡀⢠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠴⠀⠘⡄⣠⠟⠋⠛⢆⡁⡜⠀⠰⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⣠⠀⠀⠀⠘⠈⠂⠀⢀⡀⡤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠈⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠐⠒⠂⠲⠀⣶⡇⠀⠀⠀⠀⠀⠀⣷⡆⠰⠂⠂⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣾⣿⡷⠒⠒⠒⠒⠒⠲⣿⣿⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⠿⠁⡇⠀⠀⠀⠀⠀⠀⡏⠹⢯⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠲⡞⢣⣤⣧⠤⠤⠤⠤⠤⢤⡧⣤⢛⡶⠒⠒⠒⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠾⠉⡈⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⡘⡹⢦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⢞⡥⠊⠀⠇⢠⣇⣀⣀⣀⣀⣀⣀⣇⠀⢇⠈⠢⣙⢦⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠴⠋⢀⠔⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠈⠑⢄⡈⠓⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⠔⠉⣠⠜⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⢤⡀⠓⢌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠔⠁⠀⢸⠀⠀⠀⠀⠀⠀⢈⡯⠉⠉⣩⡉⠉⠨⣏⠀⠀⠀⠀⠀⠀⢨⠀⠀⠑⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⡎⠀⠀⠀⣿⡇⠀⠀⠈⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⡇⠀⠀⠀⣿⡇⠀⠀⠀⡟⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠲⡞⠀⠤⠄⠀⣀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⡇⢀⣀⠀⠤⠄⠐⠶⠂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⠬⡇⢹⡶⠶⢾⣿⡗⡶⠶⢺⠃⡻⠤⠤⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢁⠀⣿⣿⣿⡿⢿⣿⣿⣿⠀⠃⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣰⣄⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣴⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠄⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠄⠤⠤

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

⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠛⢠⣶⠔⠿⣿⣷⣶⡾⠛⢻⣿⡿⢻⣿⣿⣿⡇⢰⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣦⣠⣴⡄⠀⠀⠉⠀⠀⠘⢿⠇⠼⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠘⢋⣉⡀⠀⢀⡀⠀⢠⠀⠲⢄⡈⠛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣰⣶⡀⡏⢴⠲⠀⡗⠂⠀⠀⠀⠀⠀⠀⢀⣉⣠⣴⣞⡛⣠⠁⢤⣬⣆⠟⠋⠀⠀⢻⣷⣸⣿⣻⣟⠿⢿⣉
⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀⠈⠈⠁⠉⣮⠀⢀⠁⣀⢀⠀⣀⡀⠀⢀⡛⠋⠘⠉⢩⠟⠁⠶⠀⠀⢿⣴⡄⠆⠀⠈⣉⣿⢿⣥⣿⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠸⠽⠇⣿⠸⠿⠇⢿⢼⠀⡿⠁⠀⠈⠁⠀⠀⠀⠘⣇⠀⢀⣷⣼⡘⠟⢳⠖⣠⣄⠟⠁⠀⢻⣍⣁⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⣀⠿⠿⣿⢣⣄⡿⣶⣛⣋⠁⠀⣀⣚⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣯⣴⡏⠉⢻⡧⣿⠀⠛⠀⣀⣨⣿⣷⠠⣿⣿⣇⣤⣿⣿

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡞⠙⡀⢸⡇⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣴⡇⠀⠀⠀⠜⠀⠀⢿⡀⢀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⡴⠛⠀⣠⠂⠠⠀⠠⡀⠀⠀⡇⠘⠃⠈⢧⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢀⠤⠚⠉⠀⠀⠀⠐⠃⢠⠆⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⡸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⣼⠀⠈⠉⠉⠁⠀⠀⠐⠤⠤⠀⠄⠀⠈⠁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣸⣿⣇⢠⣿⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⣷⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣸⡀⢀⣿⣿⣿⠼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢼⣿⣿⡀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠿⠷⠖⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡯⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠙⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠋⠉⠙⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⡀⠀⠀⠀⡠⡀⠀⠀⢀⡀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⡀⠘⠀⠀⢀⠀⠁⠀⢀⠀⠑⠊⠀⡀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⠜⠀⠀⠀⠠⠂⠀⠀⠢⠀⠀⠀⠘⠜⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠗⣚⣉⣭⣭⠤⠶⠶⡶⠖⠒⠒⠒⠒⠶⡶⠶⠤⢤⣭⣍⣙⣓⠻⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡩⠖⢺⡋⡩⠔⠀⠀⣠⠤⠄⠑⢄⠀⢠⢄⡀⠀⡇⠤⢤⡀⠀⠓⠬⡉⣿⠓⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡋⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠈⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣸⠀⠀⠀⠀⠰⠖⠒⠒⠄⠀⠀⠀⠀⠀⠀⠔⠒⠒⠶⠀⠀⠀⠀⢸⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⣰⣀⠀⠀⠀⢀⣰⣀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠋⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠐⠁⡀⢴⣀⣀⠀⠹⡓⠶⠤⣤⣄⣄⣀⡤⡤⠴⠖⡻⠁⢀⣀⣲⠄⡀⠑⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠪⡠⢰⠆⢀⠕⠀⠀⡇⠀⠀⠈⠑⠠⠔⠁⠀⠈⠑⠤⠐⠁⠀⠀⠀⡅⠀⠐⢅⠀⢶⠤⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠤⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠔⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠀⠐⠒
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠃


⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘
⠐⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⢀⣀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠘⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠚
⣥⣤⣴⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⡇
⠀⠀⠀⢘⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⡋⠀⠀⠀⠀⠀⠀⢠⠆
⣀⠀⠀⢸⣖⠀⠀⠀⠀⣠⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⠀⠈⠻⠻⣿⣦⣤⠀⠐⢤⡀⠈⠀⡀⢰⡄
⠿⠆⢨⢊⣵⠄⠀⠁⠐⡫⠀⠀⢐⠀⠀⠀⠀⠀⠤⠄⣈⠉⠉⠀⠐⠠⠀⠈⠋⢻⡆⠀⠀⠉⣶⠀⠋⠿⠃⠀⠈⠁
⠰⠀⠠⠋⣙⠧⣀⢰⠊⢠⡀⢀⡎⠱⠲⠦⢀⡄⠲⢤⠍⠀⡒⠄⠄⠀⡀⠀⠀⢈⠙⠆⢀⠆⠽⠂⣰⠄⠤⠀⡀⠰⠆
⠇⡀⢀⣾⢁⠢⡀⣲⣿⣾⠲⣿⣟⣼⣧⣦⣼⣷⣶⡆⣶⣰⡇⣸⣤⣦⣟⣴⣶⣿⣴⣴⣿⣠⣴⣠⣿⣿⣠⣼⣇⡄⣀
⡟⡟⢁⠜⢁⡤⠸⠿⢿⣿⠊⠿⠟⠛⠻⠿⠿⠷⣿⠿⠗⠿⠤⠟⠟⠻⠿⠿⠿⠿⠿⠻⣿⠿⠿⢿⣿⣿⣿⢿⠿⠾⠿
⠀⣴⣋⡼⠛⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠛⠿⠛⠛⠋
⠄⠈⣹⠴⠺⣿⡭⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋⠉⠉⠁⠀⠀⠀⠉⠉⠉⠙⠋⠙⠛⠛⠿⠿⣷⣶⣦⣄⡀⠀⠀⡄⠒
⠤⠜⠃⣤⢠⣯⣗⢆⠀⠀⠐⠀⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠟⢀⠀⠐
⠀⠠⠞⠁⣸⣟⢯⠁⠀⠀⠀⠀⠀⠉⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣯⡀⠀⠀⣀
⠁⠀⣼⡟⠹⣿⣧⠧⡄⠀⠀⠀⠀⠀⠀⠘⢠⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣋⠀⠀⠀⠈⠉
⣀⠀⠉⠀⠀⠈⠉⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⣀⣼⣷⠶⣄⡀
⠀⠀⠀⠀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⣡⠞⠁⠈⠢⡀
⠀⠀⠘⠤⡐⢂⣉⡵⠦⡀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⣀⣠⣴⣾⡿⠛⠉⡠⠞⠁⠀⠀⠀⣠⣾⣆
⠀⠀⠀⠀⠈⠢⢔⢛⣮⣍⠶⣄⠀⠀⣜⣀⣠⣤⣶⣾⣿⠿⠛⠋⠁⠀⠔⠉⠀⠀⠀⠀⣠⣾⡿⢋⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠑⢶⣊⣻⣭⠵⠴⡿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢋⣴⠟⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠄⠒⠒⠻⣦⣀⡛⣮⠡⠤⣐⢖⠐⡤⢄⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⣁⠴⠋⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠟⡆⠘⢸⡟⣿⠃⡇⢸⠚⣀⣤⣴⣾⣿⠿⠛⣁⠤⠊⠁⠀⠀⠀⣠⡾⢋⡅
⠀⠀⠀⠀⢀⡇⠤⣐⣒⠆⠀⢰⣿⢀⣧⣥⣬⠷⢷⡛⣿⡿⠛⠋⣁⠤⠐⠈⠀⠀⠀⠀⢀⡠⠞⢁⣴⣿⠃
⠀⠐⠽⠋⠉⠹⡀⠀⠀⠀⠀⡼⠉⠀⢛⣉⣁⣲⣉⠽⠦⣚⣺⡉⠀⠀⠀⠀⠀⢀⡠⠒⠉⢀⣴⠟⠋⠎
⠀⠀⠀⠀⠀⠀⠈⠢⠄⠤⠚⠁⠀⠀⠀⠻⡉⠁⠈⠒⢍⡩⠷⣜⣓⣄⡠⠔⠊⠁⣀⣤⠾⠋⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣤⠴⠷⠮⡩⠕⠐⠛⢢⠶⠛⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣤⣶⠾⠒⠋⠉⠀⠀⢀⡀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣄⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⣀⣀⠀⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠳⣿⠀⠄⢀⠈⣿⣷⣰⠀⠀⠀⡄⡷⠋⠠⠇⠀⠀⠀⠀⠠⣀⠁⢸⡆⠀⡏⡼⠀⡘⠀⡇⢺⡇⣴
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰⡀⢠⡸⡇⡘⠁⠀⣿⠊⠁⠀⡴⠒⠃⡧⠀⠀⠀⣠⠆⠀⠀⠀⠸⡄⢸⣧⡮⠁⠖⠀⠘⠀⡇⠰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠼⠃⣿⠁⠀⠀⣿⣿⠀⠆⠀⠀⢠⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠑⢾⡛⠃⠂⠀⠀⠀⠓⣧⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡀⠸⡆⠀⠀⣿⣿⠀⢀⡀⠀⠘⠒⠛⠉⢅⡀⠀⠀⠀⠀⠀⠀⢸⣏⡘⠀⠈⠀⠀⠀⣿⡁⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣧⡀⠀⣿⣟⠐⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡏⠓⠀⠀⠀⠀⠁⠸⣅⣤⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣇⢸⠀⡀⠄⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣿⣿⡶⣄⡴⠒⠒⠒⠀⠀⠀⠀⠀⠀⣀⣴⣦⣄⠀⡟⢸⣼⣤⣤⡀⠀⢰⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢷⡿⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⣻⣿⣿⣿⣿⡿⡼⠀⠛⠻⠷⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠃⢀⠀⠀⠀⣠⣴⡾⠋⠉⠛⢿⣿⣿⣿⣿⣷⡅⣷⣤⡀⠀⠀⢾⡄⡤⠞⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⢈⣿⣿⡇⠀⢁⣠⣴⠟⢛⠁⠀⠀⠀⠀⠀⢙⠻⢿⣿⣿⣠⢻⣿⣿⣶⣄⢸⣇⡗⢦⡾⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⠀⡙⠛⣿⣿⢣⡶⠛⠉⠰⠊⠐⠀⢀⠄⢀⠀⠠⠜⡆⢄⠙⣿⡉⢸⣿⣿⣿⣿⣿⣷⣇⠈⠿⢿⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠌⠀⣀⡴⣿⡿⠓⡇⠀⠀⣸⠞⠛⠚⠛⠒⠋⠀⠀⠁⡟⢸⠶⣿⣧⡼⢿⣿⣿⣿⣿⡿⣿⠷⢀⣡⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢉⣠⡶⠛⠉⣠⣿⣿⠀⠇⣠⠞⢯⡠⠐⠉⠆⡇⠀⠀⠀⠀⠀⠘⣷⣿⡏⠇⠀⠙⣿⣿⢻⣷⢶⣞⣻⣧⢠⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠋⠁⢀⡀⢠⠀⢹⣶⠦⡾⢻⠀⠀⠉⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⣠⣾⣿⣿⣾⡏⢸⡏⡿⢿⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠡⣾⠀⣀⢀⡷⠾⢄⢸⣿⠀⢉⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣿⠇⣼⣿⣿⣿⣿⢸⡇⢸⡿⡇⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⡅⠀⠈⢻⣿⡃⢈⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡄⣿⣿⣿⣿⣿⢸⡏⢨⡇⢇⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡃⠀⠃⠀⠀⢸⣿⠃⠬⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣾⣿⢁⢸⣿⣿⣿⣿⢸⡇⢸⡇⣸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⡀⠀⠀⢸⣿⣀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠁⠇⠀⣠⣾⣿⣿⢼⡸⢿⣽⣿⣿⢸⡇⠈⡇⣿⡅⣇⡄
⠀⠀⢠⣠⡠⠖⣶⣤⣿⣿⣯⠀⠀⠀⠃⢀⠀⢸⢿⣿⢨⠀⠁⠀⠀⠀⠀⠁⠀⠀⠂⢀⣾⣿⣿⣿⣿⢛⠁⣾⣿⣿⡿⣸⡇⣠⠃⣿⡇⣟⡆
⠀⢰⣿⣿⣿⡷⣿⡟⣻⣿⡧⠀⠀⠀⠀⠈⠀⣸⣿⣿⣚⠀⠀⠀⠀⠀⠀⡄⠀⠀⣴⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⡇⣿⠷⣿⡄⣿⠁⣏⡇
⠀⢸⣿⣿⣿⢣⡟⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⣿⣿⡿⢹⠀⠀⠀⠀⠀⠀⠏⠠⣾⣿⣿⣿⣿⣿⣿⣿⣇⡞⣿⣿⣿⡇⢻⡆⣿⣷⣿⣶⣿⠁
⠀⠈⠛⠉⠉⠉⠋⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⢹⡏⢿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣦⠀⠀⡀⣿⣷⠐⠀⠘⠈⠀⣸⡿⠇⠁⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠿⠀⠈⠀⠀⢐⣿
⠀⠠⠀⢠⣄⣤⡤⠄⠀⠤⢤⠤⢤⡤⠶⠴⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠦⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⢿⣍⣹⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠟⠛⠉⣉⡠⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠏⠛⠁⠀⣄⣠⢦⣿⣿⣿⣿⣿⣿⡿⠿⢛⡛⢿⣿⣿⣿⣿⠋⠄⠠⠀⠀⢽⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠊⢴⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡆⡛⢿⢣⣻⣿⡏⠃⠀⠀⣤⠀⠁⢸⣿⡇
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠿⢿⣿⣷⡤⡺⣿⣿⣿⣿⠟⠋⠉⠈⠀⢀⠀⠀⠃⢧⠈⡆⢿⣿⣇⣄⠀⠃⢬⢀⣸⣾⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣥⣠⣼⣿⣿⣿⡿⠿⠛⠉⡏⢰⠻⣦⠀⠀⢸⡀⠀⠰⡘⣤⣇⢸⣾⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⣏⣽⣿⡟⢻⠟⠋⣹⣀⢀⣀⣬⣧⣺⡟⢿⣴⣴⣾⣷⣤⣤⣵⣌⣛⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⢻⣿⣿⣷⣾⣷⣿⣿⣿⣤⣿⣿⣿⠟⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢭⡏⠈⠢⠤⠌⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⢿⡿⠫⢉⡉⠉⠉⠛⣿⣿⢿⣿⣿⣿⣿⡏⣾⣛⡄⢉⣿⣿⣦⢹⣟⢇⣠⣿⣯⠂⠀⠀⠀⠀⠀⠄⢻⠿⣿⣿
⠀⠀⠀⠘⠟⣽⢛⠀⠉⠉⠀⠂⠀⠀⠀⠻⣷⡽⣿⣯⣿⣿⣿⣿⣇⡸⣿⣿⣿⣷⣬⡠⣉⡛⠿⢦⣥⣀⣀⠀⠀⠀⠘⠓⠈⠙
⠀⠀⠀⠀⣼⣛⣊⠈⠁⠀⠀⠀⠀⢠⡾⠟⣉⢹⣽⣿⡟⢯⠍⠛⠾⠦⠉⣽⠛⣿⣿⣿⣿⣿⡗⠲⣤⣍⣙⠛⠻⠿⠁⠀⠘⠀⣱⡄
⠀⠀⠀⠀⣿⣅⣈⣀⣤⣤⣤⢶⣦⠟⠁⠀⠁⣼⣿⣿⣿⢋⣶⣶⣶⣶⣶⣶⠐⣿⣿⣤⣿⣿⣿⣤⣀⣠⣍⣝⣿⣿⣶⣤⣶⠛⠓⠚
⠀⠀⠀⠀⣿⡆⠛⠮⠙⠷⠟⠋⠁⠀⠀⠈⣠⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⡅⢿⣿⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣉⣿⣿⠏
⠀⠀⠀⠀⠘⢧⣤⡀⣀⠀⠀⠀⠐⢀⣤⢾⣵⣿⣿⣿⢃⣷⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⡿⡃⠀⠁⣉⣾⣿⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠳⠶⢆⣀⣂⣈⡿⢞⣿⣿⣿⣿⡏⡸⣻⣿⣻⣿⣿⣿⣿⣿⡆⣿⣿⣟⣫⣅⣤⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⣶⣿⡿⣾⣿⣿⡿⢃⣷⣿⣿⣿⡟⣿⣿⣿⣿⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡷⣻⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣷⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢹⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠉⠉⠉⢧⡙⠉⠉⠉⠉⠈⢿

⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠇⠠⠠⠶⠴⠴⠲⠠⢴
⠀⠀⠀⠀⣶⡆⠐⣶⣆⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⣿⡇⠰⣿⣃⠀⣿⡇⠰⡆⢻⡆⢠⣾⠰⡦⢸⣿⣀⡀
⠀⠀⠀⠀⣿⡇⠀⣸⣿⠂⣿⡇⢰⣏⢸⡇⢸⣿⠀⠀⢸⣿⠹⣷⡀
⠀⠀⠀⢀⣉⡁⠈⣍⡁⢀⣍⡁⠈⠁⠈⣥⡀⠈⠉⠀⣤⡉⠀⢈⠉
⠀⠀⠀⣘⢿⡰⠆⠹⣿⡄⢻⣷⡀⠀⠀⣿⡇⢶⡄⠀⣾⡄⢰⣿⢀⣴⠀⣦⡀⠀⠠⠄⠤⠀⠀⠄
⠀⠀⠀⠙⠊⠛⠁⠀⠹⣿⠄⢻⣷⠀⠀⣿⡇⢸⣿⠀⣿⡇⢸⣿⢸⣿⠀⠉⠁⠀⠒⠐⠀⠀⠒⠐⠂⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠃⠀⠀⠛⠁⠙⠋⠀⠛⠁⠈⠛⠀⠙⠓⠀⠀⠀⠀⠀⠈⠁⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⣿⠀⠀⠀⣼⡿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⣿⡄⠀⠀⣾⣏⠀⠀⣸⣿⡇⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⣧⣸⣿⡄⢠⣿⣿⣠⣤⣿⣿⣆⣠⣾⣿⠁⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣦⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⢤⡿⠿⢻⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣀⣠⣸⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠖⠀⠛⣿⠗⣋⣵⣿⣟⣷⠆⠟⠚⠁⠊⠙⢿⣿⣷⣏⣠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠰⣾⡋⣤⣙⣯⣵⡞⠁⠀⠀⡐⠂⠀⠀⠀⠙⠛⠿⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢈⣼⣷⣭⣿⣿⣟⣵⣾⣿⣿⣿⣳⡶⣦⣄⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠶⢾⡿⣛⣿⣿⣿⣿⣿⣿⣿⠿⣡⣾⡁⠀⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⣬⣅⢻⣿⣿⣍⣤⣀⣠⣭⣿⡿⣣⣽⡇
⠀⠀⠀⠀⠀⠚⠋⠉⠉⣻⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠁⢸⣿⣿⣿⣿⢿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣸⣿⡿⢻⣿⢻⣿⣿⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠛⠛⠀⢠⣟⣸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢄⠀⢀⣠⣿⣷⣿⣟⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢿⣿⣿⠗⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠃

⠀⠀⠀⠄⠄⡀⠀⠀⡠⠀⠀⢀
⠀⠀⠀⠀⠄⠀⢆⠊⠼⠛⢩⠇
⠀⣤⠀⠀⠠⠀⢨⠴⠀⠀⣀⡄⠒
⠀⠀⠐⠃⠀⠂⣴⠀⠀⣄⣀⠀⢤⡀⠀⢀⢠⡄
⠀⠀⠀⠀⠄⠋⢿⡡⠀⠽⠉⠉⠂⠐⣔⠡⠛⣀⠆
⠈⠉⠀⠀⠀⠀⠀⠘⠂⢠⢤⡤⠂⣠⡏⠐⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠛⠱⠀⠘⠁⠁⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⢐⠀⠀⠀⠁⠀⠀⠀⠀⢠⡼⢏⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠦⠀⠀⠀⠀⠀⣠⣴⣗
⠀⠘⠧⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠟⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠏⠀⠉⠐⠃⡒⠰⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⢛⡁⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠖⠒⠀⠤⠂⠀⠀⠀⠐⠂
⠀⠁⠀⠀⠀⠒⠃⠀⠀⠀⠨⠥⠄⠀⠀⠀⠠⠀⠁⠒
⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⠤⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢀⠀⠀⣀⠀⡉⢀⠁⠀⠀⠀⢀⠀⠁⠀⠐⣾⣣⣠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⡀⠈⠉⢀⡀⢠⣠⣀⢀⣀⣹⣄
⠋⠡⠀⣀⠀⢀⣠⣷⣿⡾⠵⠞⠛⠓⠒⠛⠉⠉⠉⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠶⠶⠤⠶⠾⠻⠿⠛⠛⠛⠟
⠛⠛⠛⠛⠋⠉⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠀⠈⠉⠉⠉⠉⠁⠂⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣀⡀⠠⠤⢀⣀⣀⡀⠤⠄⢀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠉⠙⢿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡟⣿⣿⣧⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣄⠙⢿⣿⣯⣝⣛⠛⠻⠿⠿⠟⠛⣛⣫⣽⣿⡿⠋⢠⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣉⠀⣿⣷⣄⠉⠛⠿⠿⣿⢿⣿⣿⡿⢿⠿⠻⠛⢉⣠⣶⣷⠀⣈⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣿⣿⣿⣿⣿⣸⠑⠒⠂⠤⠄⠄⠀⠤⠤⠐⠒⠋⣿⣿⣿⣿⣿⣿⢿⡋
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⢷⠸⠘⡀⠈⠙⠻⢧⣖⣀⣀⣀⣀⣀⣀⣀⣀⣲⡬⠟⠋⠁⢀⠃⠆⡾⠉⠒⠢⢄⡀
⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⢸⠀⡆⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⢠⠀⡇⠀⠀⠀⠀⠙⢢⡀
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢠⡇⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡔⠉⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠉⢢⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢠⡀⣠⣶⠶⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢐⡖⢾⣿⣿⣫⣤⣤⣤⣀⡈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⠿⣿⣟⠶⠲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠐⣡⣾⠼⡈⢳⣄⠈⠙⠆⠳⡅⠀⠀⠀⣠⣼⣶⠒⠲⠦⠀⠀⣦⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣡⣠⢹⣿⡀⠛⢾⡿⢦⠀⠀⠀⠰⠀⠀⣩⢖⢻⣷⣄⠙⠦⣴⢞⣻⢿⠛⡛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠈⢿⠃⠀⠀⢿⡊⠀⠂⠀⠀⠀⢰⠛⠀⠀⡿⣽⣧⠀⠏⠸⠏⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⢦⡄⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠒⠒⢒⠂⢒⣒⣀⡀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⡀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠶⠷⠶⢷⠷⠶⠾⠷⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠉⠲⣄⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣔⠀⠀⠈⠙⢦⣄⠀⢈⡷⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢄⠉⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠳⣷⡀⠀⠀⠻⡡⣌⡁⠻⣆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠤⢴⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠦⢤⡵⠦⡙⣿⣤⢴⣎⣀⠚⠓⡦⣭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣠⠴⠾⠛⠉⢀⠄⠠⠞⠓⠲⢵⣬⣣⣿⠚⠻⢴⣿⣇⠨⠟⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣛⠂⠀⡰⠀⢠⡿⠋⡄⡆⣠⣿⣄⣻⣿⣮⣷⣄⣠⠀⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏⣉⣴⢏⣴⣿⠅⣘⡁⡥⣸⣷⡿⣿⠿⠻⠿⠿⠿⠿⠟⢿⣶⡈⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠟⠛⠿⠿⡷⠾⠿⠷⠿⠋⠋⠀⠀⠀⠀⠀⠀⠑⠀⠐⠁⠀⠉

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

⠀⠀⠀⠀⠀⠀⠃⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⡀⠄⠀⠐⠄⢀⠀⢀⠀⡆⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠐⠀⠀⠀⠀⠄⠀⠀⠲⢂⠁⠇⠀⠀⠀⣸⡿⠃⠈⠀⠀⠀⠀⠀⢴⣧⠇⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⢠⠀⠙⢂⡀⠀⠰⠋⢁⡔⠡⠀⠈⠀⠀⠀⠈⡞⣀⠀⠀⠀⡄⠀⠛⢡⣀⣀⣠⣤⠶⠶⠦⠤⠤⠤⠞⠉⠈⢀⣶
⠀⠰⠀⠀⣁⡗⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢶⢀⡀⣬⣯⡄⢀⠀⡀⣠⠀⠠⠀⠀⠀⠀⠀⢰⠌⠀⠀⠀⠀⠀⣴⣾⣭⠿⠚⠛⠚⠛⠓⠂⢄⠴⣤⢠⣍⣉⣉
⠀⠀⠀⠀⢈⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠙⢿⣿⢶⣤⣴⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡗⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⢺⣇⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⢀⠀⠈⠿⣭⣟⣻⡾⣷⣦⠘⠤⠶⢖⢀⠛⠁⠂⠀⠀⠀⠀⣇⢸⠀⠀⡃⠰⠀⣀⡀⠀⠀⡀
⠀⠀⡀⠀⢈⣧⠀⠀⢸⢸⡱⠀⠸⢀⣿⣴⠤⣾⠁⠰⠀⠀⠴⠀⠀⢤⠙⡚⠰⠒⠉⠁⡤⠀⡀⠀⠀⠈⠉⠒⠦⢀⡀⠀⠀⠛⢀⡄⠈⠀⢠⡄⢹⢩⢾⢤
⠀⠐⡏⠀⠛⡷⠄⠀⢻⠆⢿⠶⣼⣬⠿⠛⠀⢨⠄⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⡡⠄⠐⠂⠀⠀⠀⠀⠙⠀⣄⠀⠀⠀⠈⠀⠀⠐⡀⣘⠃⢆⠀⢀
⠀⠀⣧⢀⡰⡧⠄⠀⣽⠁⠀⠀⢾⡀⢀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠄
⠀⠈⠉⠀⠰⠀⠀⠁⠀⠐⠆⠀⡸⡀⠀⠀⠐⠚⠄⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠔⣟⣋⣛⣛⣛⣛⣛⣛⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠑⢄⠀⠙⢆⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡑⢌⠠⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣁⣈⣈⠀⠐⢁⠈⠀⢁⡤⠞⢉⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠈⢂⠔⢁⠔⠉⠀⠀⠀⠈⠐⢄⠠⡒⠁⠐⢊⠁⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠅⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠛⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀⠀⠀⠈⠀⢣⣀⣤⡤⣄⣀⡤⠴⠚⠉⠁⠉⠉⠀⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠒⠪⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠚⢉⠰⠲⠁⠀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⠻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢬⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢴⣐⣋⡂⠆⠀⠀⠐⢉⠀⠞⠁⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠶⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠭⠍⠭⠟⠉⠉⢰⡄⠒⠐⠒⠒⠊⠉⠐⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠺⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⣿⣿⠿⠿⠿⠿⠀⠺⠿⠿⠿⢿⣿⡆⠀⢀⣿⣿⡆⠀⠀⢸⣿⡿⠿⠿⠿⠇⠀⠿⠿⠿⠿⠿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣧⠀⠀⢸⣿⡇
⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡟⢸⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢸⣿⡿⠿⠿⠿⠃⢰⣿⡇⠘⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⣼⣿⢠⡆⣿⣿⠀⢸⣿⡇
⠀⠀⠀⣶⣶⣶⣶⣿⡿⠀⢸⣿⠀⠀⠀⠀⠀⣿⡿⢸⡇⢸⣿⡄⢸⣿⣷⣶⣶⣶⡆⠀⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛
⠀⠀⠀⠀⠀⢀⣤⣄⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⣤⠀⢰⣶⣾⣿⣿⣷⣤⣄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧⣤⣀⣠⣄⠀⢀⣤⣀⣸⣷⠀⢀⣀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⣶⠶⠆⢲⣄⡶⢰⡶⢶⡆⣶⠀⠀⡶⠶⣦⢰⡶⢶⡄⢰⣶⠀⠶⣶⠖⢰⠀⣶⠶⣶⢰⣆⢰⡆
⠀⠀⠀⣿⣦⠄⠈⣿⡃⢸⣇⣸⡇⣿⠀⠀⡇⠀⣿⢸⣧⡼⠇⣼⢻⡄⠀⣿⠀⣼⠀⣿⠀⣿⢸⣿⣼⡇
⠀⠀⠀⣿⣀⡀⣸⠟⣧⢸⡏⠀⠀⣿⣀⡘⣧⣀⣿⢸⡏⢷⢀⡿⠺⣇⠀⣿⠀⣿⠀⣿⣀⣿⢸⡇⢿⡇
⠀⠀⠀⠈⠉⠁⠉⠀⠈⠈⠁⠀⠀⠉⠉⠀⠉⠉⠁⠈⠀⠈⠀⠁⠀⠁⠀⠁⠀⠈⠀⠉⠉⠁⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⢠⣄⢠⣄⢀⡄⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢸⣿⠸⣏⢸⡇⢰⢸⡆⣾⠰⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠂
⠀⠀⠀⠀⠀⠘⡋⠐⢋⡘⢃⠙⠘⠃⡙⠂⠈⣋⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⣿⣿⣅⠠⡄⢣⠀⠀⠀⠀⠀⢃⠀⠀⡀
⠀⠀⠀⠀⠀⣼⢷⡄⠈⢷⡘⣷⡀⢸⡇⢰⡄⣿⢰⡇⣰⠂⡦⠀⠀⠀⠀⠀⠀⣸⡟⢿⣽⣿⣥⠀⢀⠘⡘⣾⠄⠀⠀⡀⠀⣸⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠃⠈⠃⠘⠃⠛⠈⠃⠙⠃⠀⠀⠀⠀⠀⠀⢀⡿⣷⣿⣿⣿⣿⠗⢈⣿⣷⣿⣾⣷⣄⣤⠰⣿⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣿⣿⣄⣭⡏⠉⡿⢣⣮⠉⠉⠛⠛⠛⠉⠁⠀⠈⣟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠯⠿⠟⢁⠴⠃⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣦⡤⠀⠀⠀⠰⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣽⠗⠀⠒⠲⣄⠀⠧⣈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠰⠶⠶⣿⠀⠀⠈⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠙⣻⣿⣿⣿⣵⠇⢸⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣡⠀⠀⠀⠀⠁⠀⠀⠀⣿⢋⡁⠀⠀⠀⢆⢸⣷⣒⠐⢛⠓⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠘⠿⠶⠃⠀⠀⠀⠀⠀⢩⣿⡇⠀⣧⠀⠨⣶⣿⣿⣶⣾⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣶⣶⡶⠾⠇⠀⠀⠀⠀⣿⣿⣷⡠⢻⡆⢀⣋⠻⣿⣿⣟⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠼⠯⠻⠾⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⣤⣤⣤⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣯⣈⣈⣻⣆⠀⠉⠀⢀⣾⣿⠿⢿⠆⠀⠀⠁⠀⢸⠃⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⢿⠁⠁⠸⣟⡀⠀⠉⣿⡇⣤⢸⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⢻⣷⣦⠀⢹⣧⣹⣾⡇⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢿⠌⣹⡄⣿⢹⠉⣶⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⡀⠀⠀⣿⠿⠏⢸⠀⣿⠀⠀⠀⠀⠀⣀⠀⢠⣄⢀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣀⣶⠀⡏⣿⡿⠭⠓⢷⣤⣀⣿⡀⠀⠀⠠⠄⠭⡥⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠶⠋⠀⠄⠀⠀⠈⠻⠿⢿⠇⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢤⡦⢠⡄⣤⠀⣀⢀⢀⡀
⠀⠀⠀⠀⠀⠟⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠸⠷⠸⠇⠿⠀⡷⢸⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠠⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⣀⡀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⣀⠀⠀⠀⠠⡴⢦⡚⠟⠛⠛⠋⠁⠉⠙⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⢉⢄⠠⠄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠦⣼⢇⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣯⡇⠁⢘⡆⢀⠀⠀⡬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⠉⠉⠋⠙⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠆⠦⠼⢀⠀⢠⠃⠐⠃⠀⠀⠰⠃⠀⡠⠈⢠⠰⢿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡜⠂⡌⡿⠃⠀⠀⠀⠀⡀⠀⠐⠈⠃⠀⠘⢸⡄⢣⢹⡀⠀⡤⠂⡄
⠀⠀⠀⠙⠀⠀⠀⠀⢠⡄⠉⠙⠛⠋⠁⢀⣀⡠⠄⡎⡠⣶⡄⠀⠀⡇⠀⠀⡎⢧⣸⠀⡇⠀⠉⠈⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⠉⠀⠀⣰⣆⢷⠶⣧⣣⣪⣼⣄⢀⣘⣌⠻⡦⠇⠀⢀⠀⠀⡀⠆⡀⣀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡶⠿⠷⣿⠿⠿⠿⠾⣇⢽⠟⠛⣿⠋⠛⠟⠟⠛⠈⠀⢠⢿⠉⠁⠀⠈⠁⠘⡁⠈⠀⣸⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠉⠉⠀⠀⠀⠀⠀⣄⠀⠀⠈⠓⠓⡟⠃⠀⠀⠀⠀⢀⣠⣴⣿⣭⡀⠀⢀⣠⡴⢟⣣⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢹⢿⠁⠀⢀⣄⣰⡗⠆⠀⠀⠀⣴⣿⣿⡁⠹⠿⢗⣛⡩⡵⠞⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣤⡀⠀⠀⠀⠀⠀⠈⠛⢹⡇⣁⠀⠀⠀⠈⠀⠉⠉⠁⠈⠛⠳⠦⠒⠊
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠠⠆⠀⠀⠀⠁⠛⣷⡄⠀⠀⢀⣀⠶⢿⣛⡈⠓⣄⡀⠀⠀⢀⣤⠐⠀⠄
⠀⠀⠀⠀⠀⠀⢰⣏⠤⠀⠀⠀⠀⠀⣾⠋⠤⢻⠀⠀⠘⢭⣕⢠⣬⡍⢘⡏⠀⠀⢀⣿⠃⠀⡀
⠀⠀⠀⠀⠀⠀⣿⡏⠒⢀⣀⣤⣶⡶⠃⠀⠀⣺⡇⠀⠀⣸⡏⢸⣿⠇⠈⣿⠀⠀⠀⢿⣧⣠⡁⢀
⠀⠀⠀⠀⠀⠀⢸⣏⡾⠯⠛⠟⠋⠀⡀⢐⣴⡿⠀⠀⠄⡿⠁⢸⣿⠀⠀⢹⡄⠀⠀⢠⡙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⡀⢴⣮⠟⠀⠀⠀⣸⠇⠀⢸⣿⣀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠘⠚⠉⠀⠀⠀⠀⢀⢃⣠⠞⠁⣿⠈⠳⣄⢸⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣮⠿⠁⠀⠀⠀⠀⠀⠈⠹⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠀⠀⠀⠀⣼⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠇⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⡀⢀⣀⢠⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡀⢘⠂
⠀⠈⠄⠠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⢀⡀⣀⠠⠤⠐⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣔⣀⡂⠐⢀⠰⠤⠀⠚⠛⠉⢁⡐⠢⡀⠀⠀⣀⠤⠶⠒⠙⠒⠚⠓⠒⠲⠤⠤⣀
⠀⠀⠄⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⢁⣀⠥⠤⣁⣉⡐⠒⠲⠀⡀⣀⡠⠤⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⣀⡠⠤⠤⠑⠐⠒⠚⠃⠀⠈⠑⣄⣀⠤⠉⠁⠠⢍⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣀⡀⠠⠄⠀⣀⡀⡀⣈⡆⡴⠒⠉⠉⠀⢄⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⢦⡠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠀⠀⠀⠀⠀⠀⠀⡆⠉⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⡿⡖⣄⠀⠀⠀⠀⠀⣇⢀⣀⠀⠀⡄⡾⠀⠀⠀⠀⠀⠀⢀⣀⣠⠒⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⣿⣆⠀⠀⡀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⣄⢠⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡄⡀⠆⠘⠀⣳⠀⠀⠀⠃⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠈⠁⠀⣄⣃⣌⠏⣄⡈⠀⠱⠄
⠀⠀⠀⣰⡀⢸⡽⠁⠈⠦⣽⡉⣿⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠐⠶⠇⠐⡆⠸⣄⣸⠖⠃⣦⠇
⠀⠀⠀⠙⣍⡩⢀⡀⢤⠀⣼⠀⡆⠀⢀⡀⠀⠀⠀⢀⢤⡀⠀⠀⡤⣤⡄⠀⠀⠀⠀⣟⣉⣀⣇⠖⠈⠩⣤⣴⠁
⠀⠀⠀⠀⠀⣇⠀⠈⡿⢺⣼⣿⠗⠀⠈⠁⠀⠀⠁⠋⠉⢳⠄⠁⣄⠀⡇⡰⠙⠂⠀⠈
⠀⠀⠀⠀⠐⠚⠛⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠼⠤⠤⠞⠚⠁⠓⠀⠒⠃

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣅⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⡾⠼⣿⠷⠀⠀⠀⠀⠀⠀⠀⡜⡽⢶⣾⡵⡄⠀⠀⠀⠀⠀⠞⣿⣟⣻⡟⢢⠀⠀⠀⠀⠀⠀⢠⡾⣷⡀
⠀⠀⠀⢰⡷⣤⢾⡇⠀⠀⠀⠀⢠⣶⣾⣿⠿⠛⣴⣶⡄⠀⠀⠀⠀⢇⢿⣦⣼⢇⠇⠀⠀⠀⠀⠀⠲⣧⣿⣯⣷⠞⠀⠀⠀⠀⠀⠴⠟⠁⠈⠻⠦
⠀⠀⠀⠸⠷⠯⠾⠇⠀⠀⠀⠀⠀⠉⠉⠋⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠓⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⢧⡄⠀⠀⠀⠀⠀⣠⡞⣿⣆
⠀⠀⠀⣠⣾⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⢀⣼⠏⠀⠙⠁⠀⠀⠀⠀⣠⣀⣀⠀⢸⠀⠀⡂⠀⠀⠀⠀⠀⣿⠈⢁⡿
⠀⠀⠀⠹⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠿⠋⠁⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⡦⠁⠀⠀⠀⠀⣐⣾⣶⣼⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠦⣄
⠀⠀⡤⠚⢿⣶⡿⠓⢦⠀⠀⠀⠀⠉⠹⣭⣴⣬⡽⠉⠀⠀⠀⠀⠀⣀⣴⣾⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡳⠉⠀⠀⠀⠀⠀⠀⠰⡇⠉⠁⢸
⠀⠀⠳⠤⠞⠉⠳⠤⠞⠀⠀⠀⠀⠀⠀⠂⠉⠉⠄⠀⠀⠀⠀⠀⠀⣇⠖⣹⣽⣯⡇⠀⠀⠀⠀⠀⢀⣾⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠭⠽⠿⠃⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⠃⠀⠀⠀⠀⠀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⡔⠊⠙⠑⠤⡀
⠀⠀⠀⠀⠃⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢡⣦⣀⣠⣴⣯
⠀⠀⠀⠀⢢⣤⣴⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣺⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⣼⣿⣿⠏⠀⡀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷
⠀⢠⣯⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⠇⠀⣼⣿⡋⠀⠀⢠⡀⠐⠀⠀⢀⠉⠀⠀⠀⠀⣴⣯⠀⠀⠀⠀⣀⠀⠈⠙⠋⠉
⣇⢸⡏⠄⢀⠀⠀⢴⣾⠛⠀⠀⠀⠀⢠⡄⠀⢴⣿⡿⠁⠀⣠⣾⡿⠃⠀⠀⠈⡏⠀⠀⠀⠀⠉⢋⣤⠀⢀⡿⠋⠀⢀⣴⣾⣿⣶⠈⠓
⠷⣿⡧⠀⣾⠃⢠⠆⠿⣷⡀⠈⠁⠀⠈⠀⠀⠺⣿⣣⣴⣿⣿⣿⣦⣄⡀⠀⠀⠛⠀⠀⠀⠀⠀⠼⠿⠀⠈⠁⣠⠖⠋⠈⠈⠁⡄⠀⠀⡉
⢷⣿⡇⠉⣿⡊⣿⠠⣤⠸⣿⣗⠀⠀⠂⢀⠀⠀⠉⠙⠿⠛⠻⠟⠛⠿⣿⠀⢠⠄⠀⢀⣴⡇⠀⠀⠀⢀⡀⠀⠀⢀⡗⢀⡀⣸⡇⣠⢰⡧⣾
⣿⣿⣷⠠⣸⣿⣿⡐⣿⠀⠀⢀⣜⣧⣀⢆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⠀⠘⠃⠀⢀⢠⢀⢸⣠⡇⣼⣷⣾⣴⣸⡷⢼⣷⣿⣷⣿⣾⡧⣿
⠙⠛⠛⠛⠻⢿⣿⣦⣿⣴⣾⢾⡷⣮⡷⠞⠀⠀⠿⠇⠀⠘⠟⠿⠿⢶⠾⢾⡶⠀⠀⠺⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⡍⣼⣿⣿⡿⣿⠿⠷⠛
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠉⣀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠂⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠘⠛⠟⠉⠀⣀⣀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠒⠶⠦⠀⠀⠈⠁⠒⠒⠲⠾⠿⠿⠶⠶⢶⣤⣤⣤⣤⣤⠴⠶⠾⢿⣿⣿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠘⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠛⠂⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⢿⣧⣤⣀⣁⣠⣿⣯⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠂
⠀⠀⠀⠁⠀⢠⣦⡌⠙⣿⣷⣴⣾⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡦⠞⠓⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣠⣤⣤
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠉⠛⠿⣿⣗⣂⠉⠉⠀⠈⠙⠿⠿⠿⠷⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⣰⠃⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢀⣴⠖⢣⣼⢋⣠⠟⣰⡇⣠⣄⠀⠠⣄⡀⣶⣤⡀⠀⠀⠀⠀⠀⠀⢸
⣀⣤⠀⣤⣤⣤⣤⣤⣶⣿⣿⣿⣾⣿⣧⣾⣿⣵⠟⢃⣶⡿⣹⣏⣉⡠⡴⣠⢥⣭⣿⣿⣷⠶⣶⣶⠶⣦⢸
⠘⠛⠛⠒⠒⠲⠚⠇⠬⠭⠭⠯⠍⠿⢉⣙⣓⣛⢒⣛⣿⣅⡠⠤⠽⠿⠿⠿⠛⠛⠻⠛⠛⠛⠛⠉⠉⠉⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠴⡔⣄⠄⠀⡀⠀⠆⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠐⠅⢤⡈⠊⠂⠈⢌⢆⢀⠱⡘⢀⠀⠀⠀⠇⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠬⢀⠑⠢⣀⠀⠑⣤⠠⠤⠤⠦⠒⡲⣿⡄⣤⠀⠀⠀⠀⡄⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠭⣑⡒⠤⢄⣭⣿⣈⣽⣍⣡⣤⢶⣖⣺⣿⣻⣿⣿⢹⡾⣶⡇⣰⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⣀⡀⠩⠁⠐⠀⠍⠛⠫⠛⠉⢋⠙⠉⠛⠙⠍⠉⢣⣿⣿⣿⢠⡏⠀⠀⠀⡠⣪⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠉⠐⠒⠤⠄⢀⡀⠐⢂⠤⢄⣀⠀⠠⣤⣤⣜⣿⣿⣿⣾⡄⢀⣠⡿⣛⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⠉⢁⡀⠀⠒⠂⠠⠄⠀⠀⠀⢈⡙⠒⢞⠴⣶⣿⣗⢊⡿⡋⠊⠁⠈⠻⣿⣿⣷⡞⠅⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠬⠄⠀⢀⣴⣰⣶⡒⠒⠒⠚⠛⠻⢿⣿⡟⠀⠀⠀⠀⠐⢸⣿⣿⣯⡉⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢀⡀⢀⠀⢀⣠⣤⠬⣑⣣⣶⣤⣭⣒⡁⠑⣮⢍⣭⣿⣷⡣⡀⠀⢀⣠⣷⢛⣯⣿⠟⣁⣄⣮⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠊⣉⠰⣚⡶⠛⠺⣯⠟⠻⠿⠟⢩⡿⢶⣭⠿⣾⣿⠟⣿⣶⣿⣷⣿⣯⣿⡶⣾⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⣡⣾⣿⣿⣝⣥⣦⣲⣤⣶⣴⣆⣦⣽⣶⣶⣤⣄⡻⢏⣻⣴⠕⣿⣿⣿⣿⣿⣿⣍⢢⡪⡪⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠿⣛⠽⠛⠟⡻⠟⠛⠛⡩⢟⠻⣻⣿⣻⣿⣿⠿⢛⡟⡱⠋⠀⠈⣟⡿⡹⡟⢮⡿⢧⡘⠊⠪⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠩⠂⠬⡒⢕⠪⣐⠌⠀⢂⢵⣿⠿⣻⢵⠟⢁⢤⢎⠂⠡⠀⠀⠀⠙⢻⠀⠈⠈⠵⣜⢵⡔⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⡒⠅⢢⡐⢝⣡⡤⣨⣾⣿⡧⡊⡤⡡⣢⣪⢫⢆⢠⢂⢃⢂⢀⡄⡄⠠⢂⠀⠀⢌⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⣺⣵⣾⣿⣿⣾⣿⣿⣿⣿⡿⢾⡷⢷⣧⣯⣎⣞⣾⣼⣾⣾⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠾⠦⠄⠀⠛⢉⠉⣙⢛⠿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣟⣉⣉⡤⠀⠀⠒⡨⠑⣩⠐⡻⠻⢷⣮⣟⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢻⣿⣧⣬⣉⣛⠷⠔⣂⠍⢐⠩⠒⠍⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠈⠑⠂⠉⠀⠉

⣤⣶⣶⣴⣶⡶⠶⠶⠶⠶⠒⠒⢂⣀⡒⣒⣂⣰⣆⣠⣤⣤⡤⠤⠤⠤⠤⠤⣤⣄⡠⠄⠀⢀⣀⣀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣷⣶⣤⣤⣤⣤⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣯⣽⠛⠛⠻⠏⠡⢤⡤⠄⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣤⣤⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢤⢢⡰⡀⣠⢠⡄⣄⢠⡀⡀⣄⣀⣀⣀⣀⣀⣤⣀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣏⣎⣷⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠢⣄
⠀⠀⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠴⣿⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠙⡗⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤
⠀⠀⠀⠀⠰⠝⣒⣶⢠⡿⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠐⠄⠀⠀⠈
⠀⠀⠀⠀⠈⢁⢻⠿⣿⣟⣿⣗⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠚⠀⠀⠈⡷⠀⠀⠀⠈⠰⡐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⢓⡀⠂⠀⠀⠐⠛⠛⠛⠒⢚⠂⠒⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠉⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠈⠙⠒⠲⠶⠶⢶⣤⡤⠤⠴⠂⠀⠀⠀⢉⠒⠤⠄
⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣖⣉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣁⡀⠀⠀⠠⠤⢤⣄⣀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠂
⠀⠀⠘⠛⠟⠛⠏⠋⠉⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⣀⡀⠀⠈⠉⠙⠛⠛⠛⠛⠓⠲⠿⠟⠉
⠀⠀⠀⠀⢤⠀⡤⠄⠀⠘⠂⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠙⠟⠃⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠉⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣴⣟⡁⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠀⠙⣿⣿⣿⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⡟⢉⣝⣿⣿⠙⠂⣄⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡤⠞⠋⢹⣿⡟⣦⡌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⡁⢈⡼⢇⣻⣟⢧⡄⠘⢿⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣰⠟⠀⣰⡶⢀⢻⡏⠙⢇⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠐⢊⣷⢀⣿⣏⢣⠙⣦⡀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣴⢿⠇⢨⣿⠃⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢸⡟⠈⡎⠛⠷⡌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠀⠠⠂⠛⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁⠀⠀⠀⢀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡁⠀⠀⣤⠴⠒⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡀⠀⡀⠀⠀⠼⢷⠔⠁⠀⠀⠀⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣠⠉⠀⠀⠠⠙⠂⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠇⣿⣏⠀⠰⣆⣴⣶⠶⠀⠄⠼⢇⡀⠀⠠⡄⠀⣧⡧⠀⠂⢴⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⣧⣴⡶⠉⢀⣀⠹⡟⠋⡁⠀⠀⠀⠠⢨⣧⠀⣰⡷⠿⣿⠟⠀⢶⣬⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢦⣄⢴⣦⣄⣼⣧⣤⣤⣠⣦⣾⣿⣿⣟⣀⣤⣧⡤⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⠔⢊⣠⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠒⠒⠺⠳⠦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣀⣀⣀⢀⣀⣀⣤⠴⢋⣁⣋⣈⣉⡙⠋⠁⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠀⠙⠚⠛⠉⠀⣠⣀⡉⠩⠤⠤⠚⠷⠤⠍⣉⠉⠉⠉⠁⢒⣪⣭⣩⠩⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢩⣭⣥⡤⠖⠠⣬⡍⠁⠀⠈⠉⠁⠒⠒⠂



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡸⢴⢔⡁⠃
⠀⠂⣄⠀⣧⡐⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⡀⠀⡀⠀⠍⢰⢟⠈⠉⠂
⠀⠐⠡⠈⡿⠙⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠻⠛⠀⠀⠤⠠⠀⢿⠴⡄⠀⠀⠂
⠀⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣾⣯⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠁⢰⠀⣠⡄⠌
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢾⣯⠀⠀⠀⠀⠈⠊⠙⠟⠿⠟⠛⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠨⠀⠟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠀⢠⠄
⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡀⠀⠀⠀⠀⠐⠒⠀⠀⠉
⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠄⠀⠀⠀⠒⠂⠀⠠⠄⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠉⣄⠀⠀⠀⠀⢠⣤⠀⠤⠤⠄⠀⠀⡀⠄⠠⠄⠤⠤⣤⣀⡤⠤⠀⣤⠀⣀⣠⣒⠀⠀⠀⠀⢀⣂⠀⠀⠀⠀⠉⠁
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠁⠈⠀⠀⠈⠙⠛⠦⠀⠀⠀⠀⠀⢀⣉⣀⠀⠀⠃⠀⠀⠉⠉⠀⠀⠀⠘⠋⠛⠉⠀⠀⠀⠀⠠⠤
⣤⠒⠀⠈⠉⠑⠂⠀⣀⠀⠀⠈⠉⠉⢀⠀⠀⠀⠀⠀⢤⣤⡄⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⢀⡀⠤⠤
⣀⣀⣉⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀
⣿⣿⢛⡻⢛⡛⢻⡻⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⠱⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⢠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣶⢰⣗⣿⣴⣸⣧⣿⣸⣷⣆⣾⣾⣧⣴⣧⡀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⡇⡀
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⠿⠿⠟⠻⠿⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠛⠀⠀⠉⠀⠀⠀⠀⠀⣤⡀⢹⣏⡤
⠈⠉⠉⠙⠛⠿⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣈⣩⡷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠓⠒⢂⡓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠙⠶⠭⣽⣷⣒
⠀⢀⣶⡄⢀⣸⢁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢨⣽⠆⠸⢾⣉⣻⣟⣡
⠀⢼⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢍⠴⣿⣛⣁⠀⣉⠙⣟⣡
⠀⣿⣿⣿⣿⣿⡿⣧⣦⡤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣾⠡⠀⠭⣽⣿⠧⠖⠀⢨⣿⣿⣥
⠀⠻⠿⠿⣿⡿⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠴⡿⠿⠄⠑⠚⠻⠅⠉⠐⠯⠟⠻⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠒⠒⠒⠒⠒⠒⠂⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣝⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢔⡣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠑⠤⣉⡁⠒⠚⠟⠭⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠒⢉⣁⡴⠋⠒⢤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡠⢤⣀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠲⠤⠐⠒⠒⠒⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠸⢍⡤⠤⠒⠒⢉⣢⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣴⠃
⠀⠀⠀⠀⠀⠀⠑⢶⡊⢉⣳⣤⡬⠗⢭⡭⢑⣒⠂⠴⠤⠤⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠤⠴⣒⣒⡊⠭⠭⠒⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⠤⠛⢶⣈⣙⣦⡀⠀⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠉⠁⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⡉⣉⣷⣤⠜⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠲⢽⣦⡠⠬⠷⣶⣎⣉⣫⡲⢶⡒⠒⣤⣤⡤⠤⠤⠤⠤⠄⠖⠒⠚⠉⠉⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠈⢟⢾⡉⠙⡍⣤⣖⣶⣽⠝⡄⠘⡀⣱⣀⣀⣀⣠⣀⠠⠤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⡎⣷⠀⢸⢹⡏⠻⣆⠀⡇⢀⠋
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⢒⣊⣩⠇⠀⠀⢰⣿⠀⣸⠤⠄⢒⣚⠿⢿⡓⢒⣄
⠀⠀⠀⢸⣹⠞⠓⠊⢫⠉⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠈⠲⣖⠊⣙⣧⣤⠝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⡤⠼⠳⣎⡉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡂⢉⣙⢦⡤⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠤⠛⠒⠉⠉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠉

⠀⠀⠀⠀⢰⡆⠀⣄⢠⣄⣤⣀⢠⡄⣦⢠⣄⣤
⠀⠀⠀⠀⠈⠀⣀⣀⠈⠁⣀⣈⠀⠁⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⢹⣿⣷⡄⠀⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠄
⠀⠀⠀⠀⠀⠐⣿⣿⡇⢀⣼⣿⠟⠁⠀⣿⣿⡇⠀⣶⡀⢲⣶⡄⠀⣤⣶⠀⣴⣦⠀⣿⣿⠀⠀⠒
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠙⣿⣷⣦⠀⣿⣿⡇⠀⢙⣧⢸⣿⡇⢸⣿⣿⠀⠛⠋⠀⣿⣿⣶⣷⡄
⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⢀⣿⣿⡿⠀⣿⣿⡇⢸⣿⣇⢸⣿⡇⠘⣿⣿⡀⠀⠀⠀⣿⣿⡈⢿⣿⣆
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠉⠉⠉⠀⠀⠉⠙⠁⠈⠉⠉⠈⠉⢉⠀⠀⠉⠋⠉⠀⢀⡉⠉⠁⠀⠉⠉⠁
⠀⠀⠀⠀⠀⢰⣆⠷⠀⡀⠙⣿⣷⡄⠘⣿⣷⣄⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠿⠿⠀⠀⣀⣶
⠀⠀⠀⠀⢀⣜⢿⣧⡘⠟⠀⠘⣿⣿⡄⠘⣿⣿⡄⠀⠀⠀⢸⣿⡇⠈⣿⣷⠀⠀⣿⣿⠀⢸⣿⣿⠀⣴⣿⠁⢹⣷⣆⠀⠀⠸⠑⠲⠶⠒⠀⠶⠠⡆
⠀⠀⠀⠀⠘⠿⠆⠙⠿⠃⠀⠀⠹⣿⣿⡄⠸⣿⣿⡆⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⣧⠈⣿⣿⠀⠘⠛⠋⠀⠀⢤⠤⠤⠄⠀⠂⠤⠤⠤⠄⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠻⣿⠃⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⠸⣿⣿⡀⠻⣿⣄⠀⠀⠀⠀⠀⠠⠀⡀⢠⣤⡄⠀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⢨⣿⡆⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⣿⣿⡄⠀⠀⣸⣿⡿⣇⠀⠀⣴⣿⣿⣿⠀⠀⣀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⣿⣿⣆⣹⣿⣿⡄⢀⣿⣿⣿⡫⣤⣶⣾⣿⣿⣿⠤⢀⣿⣿⣿⣟⠀⠀⣠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣸⣿⣳⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⢿⣿⣿⠛⠁⣀⣼⡿⣟⢟⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿⣟⠙⣀⣶⣉⠛⠀⠉⠈⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡿⡿⢻⡿⠟⡛⠿⢿⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣀⣄⢰⡔⢫⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡕⠈⠻⣿⣟⣩⡼⠛⣩⣴⣾⣿⠿⣳⣶⣨⠿⡵⠟⠁⠖⠋⠛⣿⣿⣿⣿⣶⠏⠀⠺⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⣸⡿⠁⠴⢟⣹⡿⠋⣑⣶⠛⠁⠁⠀⠀⢀⡀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣾⣯⣽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⢴⣦⣬⣽⣴⣾⡟⠁⡌⠀⡀⣀⣈⠙⡥⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⡧⢈⣿⣿⣿⣿⢏⣻⣦⣷⣾⣿⡿⢏⡻⣷⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠛⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣟⣷⠾⢋⣽⠟⠛⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⡿⠟⣉⣼⣿⣿⣻⣿⣥⣿⣿⠿⡿⠿⠟⣁⣴⣿⣿⡀⠀⢠⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⢀⡈⢉⡁⢠⡿⢿⣿⣍⢉⣁⠀⠀⠀⠈⣙⢻⣿⣿⠗⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣥⣤⣤⣼⣿⣿⣿⢃⣶⣿⣿⣿⣿⣧⣼⣿⣷⣬⡿⢿⣿⣿⢧⣼⣿⣿⠟⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⢀⣿⣿⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⢸⠃⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⡇⢻⣿⣿⣿⢿⣧⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⢸⣿⣿⣿⡟⠻⣿⣿⢸⣿⣿⡿⢶⡁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⣼⢿⡿⡟⠀⠀⣿⡇⢸⣿⡟⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⡀⠀⢉⡉⠁⠁⠀⣼⣿⢀⣾⣿⡛⠁⢠⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⣱⣿⣷⣴⣿⣿⠁⠐⠬⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢼⣿⣿⣿⣿⣿⡟⠉⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⡿⠹⣇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠏⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉


⢀⡀⣀⢀⣀⣀⣀⣀⣀⣀⡀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀

⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢰⣿⣷⣷⡄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⣀⣤⣤⣤⡀⠀⣀⣤⣤⣤
⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⡟⣽⣿⡏⣧⠀⠀⢠⣤⣤⣿⡟⣿⣿⣿⡿⠃⠀⠀⠻⠿⠾⣿⣿⣤⣟⡿⠶⠿
⠠⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠⡿⢸⣧⠀⠀⠀⠘⠛⠋⠹⠇⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃
⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⣲⣶⣶⣄⠀⠀⠀⠀⠰⣿⣷⢀⣀⣄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣷⣦⡴⣶⡦
⠀⠀⠈⣿⣏⣹⣿⠁⠀⠀⠀⠀⠈⢱⣿⡿⢃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡶⢿⣿⣷⡩⢿⠃
⢀⠄⠀⠿⢿⠿⠿⠂⠠⡀⠀⠀⠀⠪⠉⠰⣿⣷⠆⠀⠀⠘⢿⣿⣿⣿⡿⠏⠀⠀⠀⠸⡟⣧⣜⣻⣿
⠈⠁⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠄
⠀⠀⠀⡀
⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⢠⣾⡂⢀⡀⠉⠉⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣿⡆⠀⠠⣄⡀⢀⣠⣤⣀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⣇⢻⣿⢾⣃⡤⣶⠀⠀⠀⠀⣸⣿⣿⡿⠿⠻⠛⠁⠀⠀⠀⠑⣾⣿⣿⣿⡆
⢀⡾⢿⡿⢷⡆⠀⠀⠀⣿⣿⡿⠉⠀⠉⣾⣿⡷⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠻⠷
⠘⠁⠘⠇⠈⠇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⡀
⢀⣴⣿⡾⣶⣄⠀⠀⠀⠀⢠⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣷⣦⣀
⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⢿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣠⣼⣿⣟⣟⣧⣄⡀⠀⠀⠀⠀⠀⠈⠿⠈⠉⠑
⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⡟⠿⠿⠗⠀⠀⠀⠙⢻⠿⠿⡿⠿⠿⣟⠋⠀⠀⠀⠀⢀⠂⠂⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣤⡎⠤⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢻⣶⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⡀⠀⠀⠇
⠀⠀⢩⠒⣾⠬⡀⠀⠀⡂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢄⣖⢂⠀⠂⠃
⠀⠀⠀⠀⡆⠀⠀⠀⢰⠀⠀⠠⣤⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠀⠀⠀⡆
⠀⠀⠀⠀⠆⠀⠀⠀⠎⠀⠀⠊⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⣀
⠐⠂⠀⣀⠀⡈⠶⠀⢀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣤⣤⡀⠀⢠⣤⣀⣠⣤⣀⣀⣀⣀⠀⣠⣤⣠⡄⠀⣀⡀⠀⣀⡀⠬⠵⠄
⢠⡄⠀⠀⢤⡛⠃⠀⡁⠀⠀⠠⠄⠀⠀⠀⠈⠉⠉⠠⠍⡀⠀⠀⠻⠟⠒⣛⠉⠁⠀⠀⠁⠀⠀⠀⠌⠉⠙
⠀⠀⣀⣄⠀⠁⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠈⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀
⠀⠆⠘⠉⠓⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠁⠀⠀⠈⠭⣤⢤⠶⠆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⠀⣀⣀⠤⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠌⠀⠀⠀⠀⠀⠨⠥⠀⡀⠀⠀⠀⠀⠋⠉⢀⠀⠀⠀⠀⠀⣰⠦⠄
⠐⠇⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉⠏⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠟⢹⠹⢹⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁⡀⣶⡆⣆⡆⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⡇⠈⠀⠈⠀⠀⠋⢻⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⣾⣿⣿⣿⣿⣿⣴⣄⡀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⢻⠀⠀⠀⠠⣶⠄⠀⠈⢸⠙⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣀⡆⣿⣿⣿⣿⠉⣿⣿⣿⡇⣦⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⠀⠘⠀⢠⢶⣿⣷⣄⣀⡀⠸⠀⢹⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⣿⣷⣿⡟⡉⠉⠉⠻⠿⣿⣷⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠋⢸⣿⣇⠈⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣴⡇⠀⢺⣷⣾⣿⣿⣿⡇⢻⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⢈⣿⢹⡇⠀⠀⢀⣠⣸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣷⢀⡆⢹⣿⣿⣿⠟⠇⢸⣿⣿⣿⣿
⠀⠀⠀⠈⠁⠙⠳⣄⡀⢀⣸⣧⣼⣇⣀⡴⠟⠁⠉⠳⠄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣍⠻⣿⣿⠇⠘⠋⠸⠿⢋⣴⣾⣶⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⣿⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⢸⡇⢸⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠈⣿⢸⡇⢰⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⢸⡀⠀⣿⢸⡇⢸⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⡆⣇⠀⣿⢸⣇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢟⣜⢦⣿⢸⢿⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⡰⡀⠘⢮⡪⢗⡏⢸⢸⠋⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⣀⣈⠂⢈⠆⠀⠈⠳⡇⢸⢸⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⢄⡀⠈⡍⠀⠁⣤⠀⠀⠀⣿⠘⢸
⠀⠀⠀⠀⠀⠒⣄⣈⡄⠀⠀⠀⠀⠠⠂⠉⠁⢺⡏⠉⠁⠀⠀⣆⢀⠀⠀⠀⡿⡄⣾⡀
⠀⠀⠀⠀⠓⠚⠋⢀⠂⠀⠀⠐⡄⠫⠤⣄⣀⠬⠀⣠⡼⡿⠿⢛⠛⣃⠤⢄⡇⣇⣿⡇⠀⠀⠀⣀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠉⠁⠉⠛⠚⠛⠁⠀⠀⠀⣰⡤⢀⡨⠞⠁⠀⠈⠙⠤⠤⠶⣧⣹⣿⡇⠀⠀⣨⠴⠒⠧⣄⠀⢈⡆
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠂⠀⠳⢤⣄⣀⣈⡟⠉⠁