Zodiac Emojis & Symbols

⠀⠀⠀⠀⢀⣤⢞⣻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠏⠓⣤⣄⡄⠀⠀⠀⠀⠀⣀⣴⡆⣴⡀⠀⢤⡀⠀⠀⣠⢴⠖⠀⡀⠒⣦⣄
⠀⠀⠀⠠⢊⡅⠛⣟⣦⡙⣿⡄⠀⠀⠀⣈⡣⠰⢶⡚⣿⡀⠀⠀⠀⢀⣴⠟⠍⠂⠈⢻⣷⢦⢹⠀⠀⠙⠞⢠⡆⠰⢰⡟⡏
⠀⠀⠐⣀⣠⣴⡾⠟⣹⣷⢹⡇⠀⠀⢰⡿⠀⠘⠭⠀⡟⠁⠀⠀⠀⢸⣿⠂⣤⠀⠀⠈⢙⣾⢹⠀⠀⠀⢆⣸⣇⢸⡄⣷⠁
⠀⠀⠀⠘⣹⡿⠓⠚⣋⡼⠘⠀⠀⠀⣼⡇⢸⠉⠁⢤⡷⠀⠀⠀⠀⠸⣟⢿⣧⣀⠀⢀⣽⡿⠁⠀⠀⠀⠸⠇⠹⡼⠘⢻
⠀⠀⠀⠀⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠈⠃⠼⠀⠀⠘⠗⠊⠓⠀⠀⠀⠈⠁⠈⠙⠁⠘⠁⠀⠀⠀⠀⠀⠐⠀⠐⠓⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⢄⠤⢬⣯⠤⣀⠄⠀⠀⡀⠘⣿⠍⡄⠀⢀⣀⣀⣀⠀⠀⠀⠀⢰⠤⢤⣔⡿⣥⠴⠂⠀⠀⠀⣰⡿⢸⡇⢿⡆
⠀⠀⠀⠀⠘⠀⢰⡇⠀⠘⠄⠀⠀⢰⣞⣩⠼⠃⠀⢀⠀⢠⣬⠆⠀⠀⠀⠀⡀⠀⠀⠿⡯⠀⠀⠀⠀⠀⢘⢇⣼⡇⡼⢃
⠀⠀⠀⢾⣴⢶⢼⣧⢶⡶⢾⠆⠀⠀⢻⣵⣦⢰⢾⣿⣷⡀⠀⠀⠀⠀⢀⡏⢠⣶⣴⠶⢿⣽⠀⠀⠀⠀⢚⣿⣿⣻⣿⡓
⠀⠀⠀⠀⠉⠁⢸⡟⡀⠉⠁⠀⠀⠀⢀⡟⢹⡆⠀⣹⠀⢸⠀⠀⠀⠀⠈⠀⣎⠙⡇⠀⠙⠀⠀⠀⠀⠀⠁⠁⣿⣿⠈⠈
⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⢦⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠶⡄⠀⠀⢰⣄⠀⣰⣢⠀⣀⡄
⠀⠀⠀⠀⠀⠀⣉⡁⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⢀⡾⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠶⠃⠀⠀⠸⢿⣷⣧⢨⣾⣿⠇
⠀⠀⠀⠀⢶⣖⣀⡀⠀⢐⡢⡀⠀⢠⢶⠖⢭⣉⣉⣤⣶⣿⡧⡀⠀⠀⠒⠇⣴⣄⣠⢰⣾⠃⠀⠀⠀⠀⠀⠈⢸⣤⠋
⠀⠀⠀⢠⠞⠻⢿⡿⢷⣾⠉⠃⠀⠐⠀⠀⠈⠉⢿⠛⣟⢻⠙⠄⠀⠀⣠⠾⣿⠋⣩⣸⡿⠀⠀⠀⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠚⠀⠠⠼⠁⠀⠸⠇⠀⠀⠀⠀⠐⣲⠾⠋⠀⠉⠀⠀⠀⠀⠀⠿⠀⠘⠧⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣠⣴⣖⣆⣦⣶⣵⣵⣰⣲⣢⡄⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣰⡼⠼⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠳⢏⣖⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣲⠾⠋⠁⠀⠀⠀⠀⡀⣴⡀⠀⠀⢰⡆⠀⢀⢀⢰⣦⠄⠀⠀⠀⠈⠙⠷⣆⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⠾⠋⠀⠀⠐⣄⠀⠀⠀⠍⢅⣼⡇⠀⠀⢀⡄⠀⠀⠉⠙⢿⡄⠀⠀⠀⣠⠃⠀⠀⠙⠷⣓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⠟⠁⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⡜⠁⠳⠀⠀⢠⡄⠀⠀⣠⢼⡆⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠻⣕⣄
⠀⠀⠀⠀⠀⠀⢀⣜⡾⠃⠀⠀⠈⠃⣠⡀⠃⠀⠈⢤⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⡤⠁⢠⣿⣇⣈⡀⠀⠀⠀⠘⢷⣣⡀
⠀⠀⠀⠀⠀⠠⣺⠞⠀⠀⠀⠀⠀⠐⢿⠃⠀⠀⠀⠈⢣⠀⠀⠀⠀⢀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠜⠁⠀⠀⣿⣾⣿⠇⠀⠀⠀⠀⠀⢳⣗⠄
⠀⠀⠀⠀⠠⣰⠏⠀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣦⠦⠷⠞⠚⠚⠒⠶⢭⣖⡤⣄⠀⠀⠀⠀⠁⠈⠉⠒⠀⠀⢀⡤⠒⠀⠹⣞⡄
⠀⠀⠀⢀⣳⡏⠀⠀⠀⠀⠀⠐⠂⣄⡀⠀⠀⠀⢀⣔⣯⠟⠋⠀⠀⡒⡄⠀⠀⠰⣦⠀⠀⠙⠳⣯⢤⡀⠀⠀⠀⢀⣠⠐⠂⠁⠀⠀⠀⠀⢹⣖⡀
⠀⠀⠀⣜⡾⠀⠀⢀⠀⣰⣦⣀⠀⠀⠉⠒⠀⡰⣾⠞⠁⡤⠀⠀⠀⢁⡠⡠⡄⣀⡁⠀⠀⠤⠠⠈⠻⡼⣀⠀⠒⠉⠀⠀⡤⠀⠠⢄⠀⠀⠀⢷⣣
⠀⠀⢀⢾⠇⠀⠀⢻⡿⣿⣿⠉⠀⠀⠀⠀⠰⣱⠋⠀⠀⠀⢀⡴⣮⠶⠛⢋⡙⠓⠫⣕⡦⡀⠀⠀⠀⠙⣷⣆⠀⠀⠀⠀⠻⠿⠿⠿⠀⠀⠀⠸⣾⡀
⠀⠀⢸⢽⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⣱⡏⠠⢀⠀⢠⣺⡞⣁⣸⣷⣾⠋⠚⢇⣈⢻⡵⡀⠀⣴⠆⢸⡼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠅
⠀⠀⠨⣿⠀⠀⢀⣀⠀⣀⠀⣀⡀⠀⠀⢘⢺⠁⠀⠀⠀⡐⡿⢀⣽⣿⣛⣧⠘⠃⡈⢋⡀⢿⠅⠀⠀⠀⠐⡯⠇⠀⠀⢀⣀⠀⣀⠀⣀⡀⠀⠀⣯⠁
⠀⠀⢐⣿⠀⠀⠈⠉⠀⠉⠀⠉⠉⠀⠀⢠⢼⡄⠀⡀⠀⠱⣷⠈⣹⣿⣿⡇⠀⠀⠁⣜⠁⣾⡢⢀⠀⠀⠀⣗⡆⠀⠀⠈⢩⡀⢉⡀⠉⠁⠀⠀⣿⡀
⠀⠀⢸⣺⠀⠀⠀⢸⣿⣶⡀⠀⠀⠀⠀⠈⡭⣇⠘⠀⠀⠘⢽⣧⡉⢹⡿⣿⡦⣴⡎⢉⣼⡳⠃⠀⠀⠁⣸⢳⠀⠀⠀⠐⢯⡅⣸⡿⠀⠀⠀⠀⣗⡇
⠀⠀⠈⠿⡆⠀⠀⠠⢼⣿⣿⣾⠃⠀⠀⠀⠰⡹⣄⠀⠀⢀⠈⠻⢟⠶⣤⣌⣡⡤⠔⡫⠗⢁⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠸⢛⡿⢇⠀⠀⠀⢰⣷⠃
⠀⠀⠀⢫⢷⠀⠀⠈⠉⠁⠘⠛⠀⠀⣀⠄⠀⠐⡽⢦⡐⠂⠀⠀⠀⠈⠑⠑⠓⠉⡁⠀⠀⠀⠀⢀⣴⠫⠏⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡝
⠀⠀⠀⠈⠽⣇⠀⠀⠀⠀⠀⠠⠄⠋⠁⠀⠀⠀⠀⠫⡟⣦⣄⠀⠀⠘⠀⠀⠀⠰⠒⠀⠀⣠⡴⡿⠇⠁⠀⠀⠀⠈⠙⠠⠄⠀⠀⠀⠀⠀⣸⠯⠁
⠀⠀⠀⠀⠘⡹⣆⠀⠤⠚⠁⠀⠀⢰⣾⣦⠀⠀⠀⠀⠀⠑⠫⠟⢖⡶⢤⢤⢤⠤⠶⣒⠯⠗⠋⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠈⠓⠤⠀⣰⠯⠃
⠀⠀⠀⠀⠀⠐⠽⢦⠀⠀⠀⠀⠀⢠⠾⡇⠀⠀⠀⢀⡔⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢢⡀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⡴⡯⠂
⠀⠀⠀⠀⠀⠀⠈⢯⢷⡄⠀⠀⠠⠋⠀⠘⠀⠀⢀⠚⠀⠀⢀⡄⠀⠀⠀⢀⡀⠀⠀⣀⣄⠠⠀⠀⠃⡀⠀⠠⠇⠚⠁⠀⠀⠀⢠⡾⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢜⣦⡀⠀⠁⠀⠀⠀⢀⠌⠀⢰⣿⡁⠶⣶⣤⠀⠘⠃⠠⣾⣿⣿⣿⣦⠀⠀⠡⡀⠀⠀⠀⠀⠀⢀⣴⡣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢭⢶⣄⠀⠀⢠⠋⠀⠀⠈⠙⢻⡤⠼⠋⠀⠘⠃⠀⠈⠻⠉⢉⠃⠀⠀⠀⠙⡄⠀⠀⣠⡶⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠹⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⡏⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠽⢳⢶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡴⡎⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠚⠚⠩⠯⠯⠟⠟⠻⠻⠽⠹⠭⠓⠓⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣥⣶⣿⣿⣯⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣽⣿⣿⣶⣬⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠛⠉⢁⡀⠀⣀⠀⠀⠀⠀⢀⠀⢀⠈⠉⠛⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢠⠓⠃⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⡐
⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣿⡿⠋⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠁⠁⠀⠀⠀⠀⣀⣀⠀⠙⢷⣟⣿⣄
⠀⠀⠀⠀⠀⠀⠈⣼⣿⣿⠏⠀⠀⠐⠒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠈⠀⠈⠀⡤⠈⠀⠀⢠⠄⠀⠀⠀⠀⠠⠄⠀⠀⠀⢦⠀⠁⠀⠁⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⡿⠀⠔⠤⠂⠄⠀⠀⠀⠀⣠⠀⠀⠀⣀⠰⣄⣧⣼⣠⠆⣀⠀⠀⠀⢀⠀⠀⠀⠀⠜⡃⠀⠁⠀⢿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⡇⠀⠁⠉⠈⠁⠀⠀⠀⠀⠁⠀⠀⠢⢬⣿⣿⣿⣿⣿⣿⡥⠔⠀⠀⠈⠀⠀⠀⠀⠀⠂⠘⠁⠀⢸⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⢒⣻⣿⣿⣿⣿⣿⣿⣟⡒⠀⠀⠀⠀⠀⠼⠆⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠘⣿⢿⡇⠀⠀⠤⢠⠀⠀⠀⠀⠀⡀⠀⠈⢉⡽⢿⣿⣿⣿⣿⡿⢯⡉⠁⠀⢀⠀⠀⠀⠀⠀⠔⠄⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣯⣷⠀⠀⠀⢀⡀⠀⠀⠀⠀⠑⠀⠀⠀⠰⠋⡼⢹⡏⢧⠙⠆⠀⠀⠀⠊⠀⠀⠀⠀⠾⠠⠁⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠸⣿⢿⡆⠀⠀⠈⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣷⣿⣄⠀⠀⠀⠀⠀⢀⡀⠓⠀⠀⠀⠑⠂⠀⠀⠀⠀⠐⠊⠀⠀⠀⠞⢀⣀⣀⠀⠀⠀⠀⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⢀⡠⠊⠁⠀⠀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠘⠀⠀⢸⢸⢀⡀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣷⣄⠈⠁⠀⠀⠀⠀⠠⡤⡤⠀⠀⠈⠁⠀⢀⠤⠀⠀⠀⠀⠈⠈⠸⠇⣠⣾⣿⣿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠃⠁⣀⠀⠀⠀⠀⠼⠀⠸⠄⠀⠀⠀⢀⣠⣾⣯⣿⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣻⣶⣶⣶⣶⣶⣶⣶⣶⣻⣻⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢹⣏⡀⢀
⠀⠀⣠⠶⠋⠉⠀⠙⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢡⠆⣉⣄⢾⠃⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠙⠛⢃⣤
⠀⠸⣧⣔⡀⢠⠖⢋⡀⠌⢹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡰⠂⠋⠱⢿⡮⣀⣈⢲⡄⡼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⢀⣤⠀⠉⢡⠼⣄⠀⠀⢀⣸⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⢀⠓⠰⢃⠀⣄⠱⡈⡟⠀⠿⢇⣿⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⠀⠀⠀⠀⠀⠀⠀⠁
⠘⢿⡷⠞⠓⠒⠈⢘⢶⣿⠿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢨⡿⠏⣴⠣⠗⣤⠒⢺⣿⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠤⢤⣴⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡞⠹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⠸⠇⠰⣼⢋⡴⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣛⠛⣡⠠⣄⠀⠁⠙⢻
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡐⠴⣶⡀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠙⢿⣭⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠔⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠑⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢷⣶⡾⠶⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠴⠶⠶⠶⣄⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡛⠃⢾⠃⠀⠙⡇⠜⣓⣦
⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⢀⡠⠦⠀⣠⠀⠀⠀⠀⠀⠲⡀⠺⠁⡆⠂⠀⠀⠐⢀⣿⡿⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠰⠶⠲⠶⠖⠶⢆⡙⣁
⠀⠉⠀⠀⠀⠀⠨⣯⠀⣀⠀⠐⠂⠀⠘⠀⠀⠀⠀⠀⠀⠉⠤⢊⣉⣀⣀⡀⡄⠈⠀⢘⢧⣷⠀⠀⠀⠀⠀⠀⠀⢰⡇⣧⢦⡐⠄⣠⠘⣂⡜⣄⢹
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⠦⠁⠀⠀⠋⡸⡿⠹⠃⠀⠀⠀⠀⠀⠀⠀⠉⢡⠾⡱⠶⠙⠶⢋⠶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠑⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠵⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣀
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠂⢬⠤⡉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⣀⠆⢀⠈⢀⢠⠸⣆
⠀⣷⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣘⣧⠐⠀⠠⠜⣌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠏⠀⣄⢀⣤⠈⣃⢙
⠀⠘⢗⠷⡑⠀⠀⢀⣷⡄⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠟⠐⠋⢉⠙⢧⣀⡀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⢈⡭⢤⡃⣤⣭⣆⣉⣩
⠀⠀⠈⠀⢈⡶⠃⠉⠛⢁⣎⠶⣭⡻⠶⣄⠀⠀⠀⠀⠀⢠⡀⠠⠤⡀⢄⠀⠈⠋⠀⠀⠀⠀⠀⢤⡽⠀⠀⠀⠀⠀⠀⠀⠠⢗⣾⣄⠢⣄⢹⣡⢾
⠀⠀⠀⠀⠾⠶⠌⠁⠸⡏⢉⣹⡀⣭⢶⣌⣉⠁⠀⠀⠀⠘⣷⣤⠄⠈⠚⣧⣄⡉⠶⢾⠷⢞⡙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣸⡇⢮⡙⠀⠻
⠀⠀⠀⣠⣀⠀⡠⠤⠞⠀⡈⣉⣥⣖⣈⠉⠉⢃⠀⠀⠀⠀⠈⠓⢒⡒⠓⠁⠈⠙⠦⠤⢤⣾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢀⠃
⠀⠀⠀⠿⠆⠃⠀⠀⣶⠊⠉⠁⠀⠀⢴⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠣
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⡀⠀⠀⠀⠀⣠⠤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢀⡁⠀⠀⠀⠀⢁⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⢛⣟⢛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠐⠇⢋⣀⠲⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⢿⡉⠗⠠⠄⠏⣹⡗⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠧⠈⢯⣄⣿⣾
⠀⠀⠀⠀⠀⠀⣴⣦⠈⣍⢣⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣼⠾⣿⡁⢹⡿⢿⣄⠴⠃⠀⠀⠀⠀⠀⠀⠀⣾⡿⣄⠀⢀⣭⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠠⠼⣷⡿⠤⠀⠉⠻⠃⠀⠀⠀⠀⠀⠀⠠⢘⣿⡉⠛⢫⠀⡦⠈
⠀⠀⠀⠀⠀⠀⠄⠈⣵⠔⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞⠋⠟⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢏⠿⢦⣤⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢻⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⣃⣩⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠱⠈⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣵⠄⡀⠤⠚⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⡆⢰⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣏⣿⣽⣿⠿⢿⣿⣟⡇⢙⣿⣿⠿⠿⣿⣯⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⢿⣿⢿⣛⣯⣽⣶⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣶⣯⣽⣻⠿⣿⡿⣠⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⢟⣄⢰⣿⡟⠋⠁⠙⠛⠉⠛⢿⣿⡇⢸⣿⠁⠈⠉⠉⠉⠙⠻⣿⣿⡆⣠⡻⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣿⡿⣫⣾⣿⣿⣧⢻⣿⣶⣄⠀⡀⠀⠀⠘⢿⡇⣸⣟⣀⣄⠀⠀⠀⠀⠀⣿⠟⣼⣿⠿⢷⣝⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⣵⣿⣿⠿⠿⠓⢾⣷⡹⣿⣶⣶⣷⣶⣿⡿⢣⡇⢸⣷⣾⣿⣿⠀⣼⣷⣶⢎⣾⣿⠅⢿⣿⣿⣿⣮⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣫⣾⣿⣯⠁⠀⠀⠀⢈⡹⣷⡙⣿⣿⣿⣿⣶⣾⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⢃⡾⣫⠅⠀⠀⠙⠿⠿⠟⣷⣝⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣤⣝⠛⢵⣿⣯⣝⠁⠀⠀⠀⣀⢶⣽⣾⣿⡜⠿⢛⣫⣭⣵⣶⡆⠀⠀⠈⠉⠙⠛⠿⢣⣿⣿⣿⣿⣿⠄⠀⢀⣠⢄⣻⣿⡮⠛⣫⣤⡀
⠀⠀⠀⠀⢀⣾⣿⣟⣴⣶⣭⣛⢯⣾⣶⣤⡜⢿⣧⣿⡿⠋⠁⠈⢿⣿⣿⠿⠿⡇⢀⠀⠀⠀⠀⠀⣠⣮⣙⢿⣿⣿⣯⣦⣾⣿⣿⠿⣛⣭⣶⣦⢻⡿⣷⡀
⠀⠀⠀⠀⣼⢻⡿⣼⣿⣿⣿⣿⣷⣮⣝⠻⣿⣾⡿⠋⠀⠀⠀⠀⣈⠱⠖⣛⡉⠥⠬⣍⣛⡳⠆⣀⠿⣿⣿⣷⡝⢿⣿⣿⠟⣫⣵⣾⣿⣿⠿⣿⣯⢿⣿⣧
⠀⠀⠀⢰⣿⣿⢳⣿⣿⣼⣿⡏⣤⣉⣽⢻⣶⡍⡁⠀⠀⠀⣴⢟⡥⠚⠉⠀⠀⠀⠀⠀⠉⢻⣷⣬⠳⣌⠻⣿⡿⠊⢩⣶⣿⣿⠟⠉⠁⢘⣾⣿⣿⡞⣿⣿⡄
⠀⠀⠀⣾⣿⡟⣿⣿⣿⣿⠟⠀⣿⣿⣿⣿⡟⣰⣿⣷⢄⡘⣡⠋⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⣿⣿⣿⣌⢃⡀⠀⠀⠀⢿⡯⣭⠀⠀⠐⠃⠉⠉⠛⣷⢻⣿⣷
⠀⠀⢀⣿⣿⣧⣿⣿⣟⠀⣴⡈⣿⣿⣿⣿⢇⣿⣿⡟⣼⢱⠃⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣿⣿⣿⣿⡌⣧⠀⠀⠀⠸⣿⣿⣿⣧⣶⣶⣶⡔⣤⣿⣼⣿⣿
⠀⠀⠸⠿⠿⠹⠿⠿⠿⠶⠿⠧⠿⠿⠿⠿⠸⠿⠿⠧⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣧⠸⠄⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠇
⠀⠀⢰⣶⣶⣰⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠐⡆⢃⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⢲⣶⣾⡆⣶⣶⣶⠶⣒⣒⠶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⣿⣿⡟⣿⣿⡟⠙⠿⠟⠛⣠⠼⣿⡆⠀⠀⠀⢿⡘⡄⠀⠀⠀⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⢃⡟⣼⣿⣿⢱⣿⠿⠃⠘⠛⢿⣿⣎⢿⣿⢻⣿⣿
⠀⠀⠀⢿⣿⣧⣿⣿⠄⢀⣀⠀⠀⠀⠀⣻⣷⠀⠀⠀⠈⢡⡙⣄⠀⠀⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⡿⢋⡌⠱⢿⣿⠏⣾⡏⠲⠀⠀⠀⠀⠛⢻⣾⡿⣼⣷⡿
⠀⠀⠀⠸⡿⣿⡼⣯⣤⣸⣿⣧⣤⣤⣤⣿⠿⣃⡠⣾⣿⣧⡙⢮⡓⢤⣘⢿⣿⣿⣿⣿⣿⣿⡿⢋⡴⠋⠀⠀⠀⢀⣘⠿⣷⣿⣤⣈⣤⡄⠀⣀⣿⢧⣟⣿⠇
⠀⠀⠀⠀⢻⡾⣷⣻⣿⣿⣿⣿⡿⢟⣫⣴⢿⣿⣷⣜⢿⣿⣿⣶⠉⠰⠦⣭⣉⡛⢛⣉⣭⠴⢆⡉⠀⠀⠀⠀⣠⣾⠛⠻⣦⣝⡻⢿⣿⣿⣾⣿⣟⣾⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣯⠻⠿⣛⣥⣾⡿⡿⠉⠚⢿⢿⣿⣷⣍⡻⠋⠀⠀⠀⠀⠀⠁⢰⣶⣶⣿⣿⣷⠀⢀⣠⡾⢿⣿⣿⠆⠙⠻⢿⣶⣭⣛⠿⢟⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠛⣫⣤⡺⣿⣿⣭⣕⢇⠸⣳⠀⣿⣿⣿⡿⢣⣶⣤⣄⣀⡀⠀⠀⠻⠿⢟⣛⣫⣥⣶⡜⣿⣷⣦⠛⠀⠀⢀⣴⣿⣿⣿⣿⢗⣤⣝⠛⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣝⢿⣿⣿⣞⠆⣹⣯⣔⣹⡿⣡⣿⡿⠿⠋⠉⠛⢿⡇⢸⣿⣿⣿⠻⢿⣿⣿⣌⢿⣿⣆⠀⢠⡿⣿⣿⣿⡿⣫⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡻⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⡀⠀⠀⠈⠃⢠⡇⢸⣿⣿⣿⠃⠈⢹⣿⣿⣎⢻⣿⡧⠀⣱⣿⡿⢫⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢷⣝⢿⣿⣿⡟⣼⣿⣿⡿⠃⠀⠀⠀⣤⣼⡇⢼⣟⣋⠁⠀⣴⣿⣿⣿⣿⣧⢻⣾⣿⡿⣫⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣮⠋⠸⣿⣿⣏⣁⠀⣀⣤⣷⣿⣿⡇⢸⣿⣿⡗⠀⢸⣿⣿⣿⣿⣿⠇⠙⣵⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⣾⣿⣶⣯⣟⣻⠿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣷⣿⠿⣟⣻⣽⣶⣿⣷⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣶⣶⣿⣿⡅⢨⣿⣿⣶⣶⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠇⠸⠟⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠉⢿⠀⠀⢶⣦⣄⣠⣴⣦⠀⢠⡏⠉⠛⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⢠⣶⠀⠀⠘⡆⠀⠻⠍⢿⣿⠫⠟⠀⣸⠁⠀⣿⠄⠀⢈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠠⠶⣦⣼⣿⣾⠈⠀⢻⠀⠠⠀⢸⡅⠀⠀⢀⡏⠀⣴⣿⢿⣷⣤⣤⡆⠈⢻⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠳⣄⠀⠀⣴⢿⣏⠙⠚⠂⠈⣆⢀⣀⣀⣀⣈⠀⡸⠀⠐⣿⣅⢀⣽⡏⠅⢀⡴⠋⠈⠻⣧⡀
⠀⠀⠀⠀⠀⣠⣿⠋⢠⣴⠀⠈⠳⢄⠐⠀⠏⢀⣠⣴⠾⢛⣛⣩⣭⣭⣍⣙⣛⠷⢦⣌⡙⠋⠛⠀⡠⠋⢀⠠⢤⣤⡘⢿⣄
⠀⠀⠀⠀⣰⡿⠁⢰⡾⢇⣸⡟⠀⠀⠑⣤⣴⢟⣩⣶⣻⣿⣿⣿⣿⣙⣿⣯⣿⠿⣶⣬⡻⢦⣤⠊⢤⣄⣠⡶⠞⢻⣇⠈⢿⣆
⠀⠀⠀⢠⣿⠁⠈⢿⢣⣾⡍⠀⠀⢀⣼⢟⣵⣛⣿⡟⣿⣛⣷⡾⠿⠛⠻⠿⢿⣿⣻⣯⣿⣦⡹⣦⡀⠈⣿⣠⠾⠛⠻⠦⠈⣿⡄
⠀⠀⠀⣾⣇⣀⡐⠀⠈⢇⠀⠀⢀⣾⢣⡞⢻⣿⣻⡿⠛⠛⠋⣤⠖⢲⡄⠀⠀⠊⣝⢯⣬⣿⣿⡌⢷⡀⠘⠿⡄⠀⢀⣀⣠⢼⣷
⠀⠀⢰⣿⠀⠉⠉⠋⠒⠢⠤⢄⣾⢇⣿⢿⣿⣿⣿⣄⣠⡄⠀⠀⡀⠆⢹⡄⠀⠀⠀⠫⢿⣯⣽⣿⡜⣷⠤⠒⠒⠉⠉⠁⠀⠀⢿⡆
⠀⠀⢸⡇⢀⡀⠀⣤⣤⡁⠂⢠⣿⢸⣿⢿⡿⠁⠀⠀⠉⠀⠀⠏⠀⣂⡺⠆⣰⢒⠄⠀⠃⣿⠿⣿⣧⢻⡆⠀⣰⡟⣾⣛⣷⣄⢸⡇
⠀⠀⢸⡇⠀⣤⣶⡿⠾⢷⣄⢸⡇⣾⣿⣿⣷⣤⠤⠀⠀⠀⢨⠀⠀⠁⢹⢃⡈⠉⠀⠀⠀⣿⣿⠿⣿⢸⡇⠀⣿⣇⣭⡉⢹⣿⢸⡇
⠀⠀⢸⡇⢰⡿⠿⠓⠈⠀⠀⠘⣿⢸⣤⣿⣿⡇⠀⠀⠀⠀⠈⢆⠀⠀⠤⢿⠠⠀⠀⠀⢤⣿⣿⣶⡟⣼⠃⠀⠉⢿⣭⡿⠸⠋⢸⡇
⠀⠀⠸⣿⠀⠀⢀⣀⣠⠤⠴⠚⢿⡎⣿⣛⣿⠿⣦⣤⠄⡀⠀⠐⠁⠀⠀⡯⠀⠀⠀⢌⣾⣏⣹⣿⢣⡿⠑⠒⠦⠤⣄⣀⣀⠀⣾⠇
⠀⠀⠀⢿⡟⠛⠉⠁⠐⠀⢀⠀⠈⢿⣜⢿⣿⠿⣿⣿⡀⠁⣀⣠⣲⠁⠀⠀⢀⣠⣴⣿⣿⣽⣿⢣⡿⠁⠀⣠⣤⡄⠀⠈⠉⢻⡿
⠀⠀⠀⠘⣿⡀⢤⣤⣿⣥⣼⣧⣀⠈⠻⣮⡻⣾⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠛⣿⣿⣹⣿⢟⣵⠟⢡⣎⣴⠷⢾⣯⡄⠀⢀⣾⠃
⠀⠀⠀⠀⠹⣷⡀⠉⠻⠀⠹⠋⠙⠀⡠⠛⠻⣮⣝⠿⣷⣿⣿⣍⣿⣿⣿⣿⣿⣿⠿⣋⣵⠟⠛⢄⡘⢁⠈⢳⣤⣀⣿⢂⣾⠏
⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠂⣠⠎⠰⢶⡄⠈⠙⠻⢶⣮⣭⣽⣛⣛⣯⣭⣵⡶⢟⡋⠁⠄⠀⠀⠑⢦⡀⠀⢉⠉⢡⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⣤⠞⠁⠄⢀⡼⢋⠀⠀⠀⠀⡎⠈⡉⡉⠉⠉⠀⠰⡠⢾⡻⡄⢠⡀⢳⣔⠀⠙⢦⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠐⣿⣰⡏⢀⡖⠀⣸⠀⣟⣧⡼⠷⣶⣞⡁⣧⠈⠳⢿⣌⣷⣾⠟⠓⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡈⠙⠿⣿⠀⢀⡏⠀⢩⣿⠁⠈⣿⠋⠀⢸⡄⠁⠈⠙⠉⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⣦⣭⣀⣸⠁⠀⠀⠙⠿⠿⠅⠀⠀⠀⣷⣀⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠶⠶⠶⠶⠾⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠠⠤⠴⠄⠂⠒⠲⠀⠠⠶⠤⠄⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠣⡰⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⠀⠀⠀⠀⠀⠀⠓⠈⣷⠉⠅⠀⠀⠀⠀⠀⠀⠀⠐⣤⠊⠘⠃⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡐⢣⣴⠶⠛⣛⠛⠒⢤⡠⠄⣈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢫⣽⣿⣿⣿⣷⡀⠀⠐⠂⠀⢰⡛⣿⡘⢣⠉⡆⠀⡀⠱⠀⠀⠀⠀⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⢈⠳⢄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢩⣼⡟⣃⣾⡿⠟⠅⠈⠀⠀⠀⠀⣼⡟⣻⣷⣾⠟⠓⠪⠁⢾⡃⠀⠀⠀⠀⠀⠁⠪⢀⢄⠀⠀⢀⡴⢋⠟⡮⠳⡄
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡔⣿⠃⠒⢀⣄⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢡⠎⠀⠀⠘⢆
⠀⠀⠀⠀⡴⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⢼⡟⠻⡰⠀⡈⠙⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠄⠀⠀⠀⠈⢧
⠀⠀⠀⡼⠀⠀⠀⠉⠛⢿⠟⠅⠀⠀⠀⠀⣀⣄⠚⢋⢥⣌⣿⡘⣿⠀⠣⢔⣓⣤⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⠀⠀⠀⠀⠀⢣
⠀⠀⡸⠁⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣀⣜⣯⠐⣀⣠⢌⣣⣿⣿⣻⣷⡄⠀⠐⣸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠀⠀⠀⠀⠀⠀⢧
⠀⢠⠇⠘⠢⣄⣀⠀⠠⠀⠀⠀⠀⢰⣿⣅⡤⠦⣤⣹⣿⡿⣻⣫⣾⠫⣽⣿⡿⣛⣽⣿⢿⣿⡕⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣀⣀⠘⡆
⠀⡼⠀⠒⠺⠢⡀⢠⠃⠀⠀⠀⢠⣾⡿⠉⠀⠀⠀⠈⠿⡽⣱⡟⢦⣿⣿⢏⣴⣿⡿⢍⣶⠿⠉⢢⢴⡖⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡇⢁⣀⠇⢛
⢀⡇⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⡟⣽⠀⠀⠀⠀⠀⠀⢸⣇⡟⢬⣿⡿⣱⣿⣿⠯⣹⠞⡗⡀⣪⣾⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠘⠀⠀⢸
⠈⠃⠀⠀⠀⠀⠀⢁⠀⠀⠀⣸⡇⣿⡀⠀⠀⠀⠀⠀⢸⠟⣿⣷⣿⢳⣿⡿⢍⣾⣫⠁⢄⢾⣽⢿⠅⣸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠰⠄⠄⠀⠀⠀⠠⠐⠀⠀⢀⣿⢿⣿⣧⠠⢄⠀⠀⠀⢸⣿⣿⣿⢇⣿⣿⣝⣲⣯⠅⢣⣟⣿⣃⢨⡴⠇⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐
⠈⡀⠀⠀⠀⠀⠀⡈⠀⢠⣾⠙⣸⡟⣿⠀⠀⠀⠀⠀⠸⡇⢿⡾⡿⣿⣟⢢⡟⡔⡈⣞⣾⠋⢃⡾⡄⡟⠭⣰⠏⠁⡗⠂⣴⡄⠀⠀⠀⠀⡀
⠘⡇⡠⠄⠀⠀⠀⢃⠀⠀⣿⣦⣿⠃⡻⡆⠀⢀⠀⠀⠀⣷⣤⡝⢾⣺⣷⣾⣱⠣⢹⢿⡟⢎⢸⣧⣽⣖⣹⠃⠀⡘⠀⣾⣿⠃⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢰
⠀⢼⠒⠤⢤⡤⠄⠘⡀⠀⠀⢸⣿⡀⢳⣷⠀⠀⠳⠀⠀⠸⠦⠿⠷⠽⠿⠿⠟⠛⠛⠉⠁⠀⠀⢤⡤⠋⣱⣄⠀⡧⢰⣿⠏⠀⠀⠀⠀⠀⠆⡀⠬⢍⣓⣶⡘
⠀⠸⣠⠊⠁⠀⠀⠀⠰⠀⠀⢠⢹⣿⣅⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣴⣿⣿⣗⣁⣾⡄⠀⠀⠀⠀⠀⡌⠈⠙⢖⢲⡬⢧⠇
⠀⠀⢫⠉⠁⠀⠀⠀⠀⠑⠄⠘⠻⣿⣿⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠍⠁⠿⣿⣿⡏⠀⠁⠰⠄⠀⢀⢎⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠪⠀⠀⠙⠛⠿⢿⣿⣦⣤⣴⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣀⢄⣪⠟⠏⠐⠀⠐⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠻⠿⠟⠋⠉⣙⠛⠻⠿⠿⠟⠛⠛⠉⠉⠉⠉⡉⠉⠈⠁⣀⡀⢀⠀⠒⡀⠄⠀⠊⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢀⠤⠤⡙⠄⠀⠀⠀⠀⠀⠀⡀⣻⠀⠀⡦⢠⢠⠄⠀⠤⢠⠄⢦⠠⠄⡄⢀⡀⠀⠈⠐⠘⡛⠀⠀⡄⢐⠄⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⢦⠠⠤⠼⡀⢀⡄⠀⠁⠐⡄⡀⢀⠀⠀⠃⠘⢒⠃⠚⠚⠘⠒⠃⠈⠀⠂⠘⠒⠃⠂⠁⢀⡠⡀⠊⠀⠊⠑⢄⠈⠉⢆⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡍⠁⠀⠀⠀⠀⠀⠐⠐⠤⣀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠑⣲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⠀⠒⠀⠀⠴⠔⠲⠤⠄⠉⠓⠀⣩⡤⠤⠄⡀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⢀⠇⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠠⠀⡄⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡠⠾⢀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⣋⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠒⠛⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⡶⠶⡶⢶⣶⣶⠶⣶

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡀⠀⣤⢀⡀
⠀⠀⠀⠀⠌⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⠎⣸⠵⡇⠰⠧⠸⢏⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⡌⠰⠀⡄⠄⠀⠀⠀⠀⠀⢷⡟⣁⡈⣴⠓⠾⠓⠀⣢⠾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠋⠀⢀⢄⡨⠂⠥⡓⢀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢁⡟⠀⢹⠀⠘⠐⡁⡉⠰⣣⠀⠀⠀⠀⠂⠀⠐⢀⣴⡆⠀⡞⠀⠀⠠⠂
⠀⠀⠀⠀⠐⢦⡀⢀⠠⠉⠷⡖⠛⠈⠓⠀⠀⠀⠀⠀⣸⣄⡆⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⡇⢸⡈⡆⠀⠀⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢀⠂⠀⠩⢩⠄⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠋⢻⣧⠐⠀⠀⠣⡁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣷⣄⡀⠘⠇⠀⠤⠁
⠀⠀⠀⠐⠀⠀⠀⡤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠓⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓

⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢴⡠⣄⡀
⠀⠀⠀⣏⠑⠄⠂⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠠⡇⣰⢀⡄⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠤⣤⠀⢐⡛⠁⣷⣔⠋
⠀⠀⠠⣟⣲⣮⢶⣄⣠⡥⠂⠒⠀⠀⠀⠀⠀⢠⠙⣷⣌⣿⡆⢷⠈⠓⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⢾⠠⠄⠁⠈⠓⠀⢦⡄
⠀⠀⠀⢾⣽⣯⡿⣿⡅⢀⣀⠀⠀⠀⠀⠀⠀⢐⠀⠀⢈⡝⠀⠸⠾⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡀⢀⡀⡀⡀⠀⢠⡀
⠀⠀⠀⢻⣿⠟⣿⣿⠓⠈⠙⣹⠀⠠⢸⠀⠀⠀⠀⠀⠘⡇⢄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠉⠁⠀⠀⠀⠰
⠀⠀⠀⡈⠘⠀⠹⠈⠂⠀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠈⢻⡈⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⡅⠤⠸⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⣜⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⢀⠰⢖⠣⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣤⡖⠒⡃⠀⡑⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠓⢀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣄⣀⣀⡰⠀⠀⠀⠀⠀⠋⢥⠀⡁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠴⠈⠁⠣⠀⠀⠀⠀⠀⠠⠚⠀⢸⢁⡀⢀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠣⠤⠿⠫⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣤⣥⠴⠖⠖⢓⣚⣓⣚⠒⠳⠶⢬⣄⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢂⣤⠞⠛⠉⠁⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠳⠶⣮⣏⡳⢦⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⣩⡄⠀⠀⠀⠀⣤⣤⣄⠀⠰⡇⠀⠀⠀⢠⡀⠀⠀⢀⣿⠃⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⢋⡴⠿⠋⢻⡧⡀⠀⠀⢿⡿⠟⠀⠀⣻⠀⠘⠟⠉⠰⠆⠀⣼⠆⠀⠀⠄⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢞⣴⡏⠀⣠⡄⠀⠙⢷⣄⠀⠈⢁⣀⣤⣤⣼⣤⣄⣄⡤⢀⡀⠴⠀⠀⠸⠛⢳⡄⠂⠀⠀⠙⢶⡰⡄
⠀⠀⠀⠀⢀⣄⡼⠁⣾⠟⣠⡞⠃⠁⠀⠀⢀⣽⣓⣼⣿⠛⠋⠉⠉⢹⡍⠉⠉⠛⣾⣧⣶⡄⠀⠀⠉⠃⠀⣠⣴⣿⣎⢛⣀⠄
⠀⠀⠀⠀⠘⡴⠡⣟⣇⡀⠀⠉⠀⠘⢀⡴⣯⠞⠁⠀⠻⣆⠀⠁⠀⢸⠄⠀⠀⢰⠏⠀⠉⠻⣷⣆⢀⣴⡿⠛⠁⠀⠸⣇⠹⡀⠄
⠀⠀⠀⠀⢺⠁⠀⠀⠈⠛⠶⣄⡀⢀⣫⣾⣅⠀⠀⠀⢀⣉⠴⢖⣒⡛⣛⠓⠶⣍⡀⠀⠄⣠⡼⠿⡴⡁⠀⢴⣾⣷⡄⠸⣧⢣⡄
⠀⠀⠀⢀⠇⠀⣀⠀⢖⠀⠀⠈⢹⣿⡟⠀⠉⠛⢦⣴⢟⡵⠚⠈⠁⠁⠀⠈⠑⠦⡝⢦⠚⠁⠀⠀⠹⡝⡎⠛⢿⣻⠏⠀⢹⡎⡒
⠀⠀⠀⣽⠃⠀⠁⡀⠈⠁⠀⢀⣿⡿⠀⢀⠀⠀⡟⣱⠋⠀⠀⠀⢠⣠⡆⠀⠀⠀⠘⢣⢳⡄⠀⢀⣀⣹⣲⠀⠈⠙⠀⠀⣘⣇⣹⢡
⠀⠀⠀⢾⠀⠀⠀⠸⠀⠈⠀⢸⣿⡇⣀⣀⡀⣾⠠⠇⠀⠀⠀⣼⡿⠟⢿⣧⡀⠀⠀⠐⣇⣿⠉⠉⠉⢹⣷⡶⠶⠿⠖⠛⠛⠛⢸⣘⠃
⠀⠀⠀⢯⢀⣀⣀⣀⣀⣄⢠⣌⢹⡛⠋⠉⢁⣿⢘⠁⠀⠀⠀⣻⣇⠀⣀⣿⠯⠁⠀⠀⡆⣿⠀⢄⡘⢸⣿⡇⢀⣀⡀⠀⠀⠀⢘⣣
⠀⠀⠛⢫⠘⢻⠉⠉⠁⠁⠀⢱⢸⡇⠈⠂⠀⣿⡆⢇⠀⠀⠁⠉⣿⡿⢿⠣⠀⠀⠀⢨⢡⡟⢤⣄⡀⢸⣿⠅⣾⠉⠁⠀⠀⠀⢸⠴
⠀⠀⠀⢩⡆⠸⡀⠀⣰⡍⢂⠈⡥⢿⡀⠀⣀⡼⢻⣌⢶⡀⠀⠀⠃⠁⠀⠀⠀⣀⡴⡣⣾⠀⠀⠉⠛⡾⡛⠀⠸⣤⡀⠀⠀⠀⡿⠂
⠀⠀⠀⠈⢷⡸⣧⠀⠲⠞⠀⠀⠘⢈⠷⡟⠋⠀⡀⠙⢦⣌⡓⠦⠤⡄⠦⠶⢒⣫⣺⢾⣅⠠⠄⢀⡾⣱⠧⣤⣀⠀⠀⠀⠀⣼⣗⠁
⠀⠀⠀⠀⠸⣦⢸⣇⠀⠀⠀⢀⣤⡎⠿⣿⣦⡀⠀⠀⠀⣼⠿⠿⠿⠿⠔⡞⠋⠁⠁⠀⠰⢦⡠⠞⡴⢀⣄⠈⠙⠻⢶⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠘⣆⠘⣷⣴⠜⠛⠁⠀⠀⠛⠿⣿⣶⣄⣼⠏⢀⡄⠂⠀⠸⣧⠀⠀⢂⣡⣶⣿⠋⠋⣤⠜⠃⠠⠀⢀⡾⢙⡋⠛⠁
⠀⠀⠀⠀⠀⠀⢯⢧⡌⠀⠀⠀⢰⣿⣾⡆⠀⠉⢛⣹⣟⡳⠶⢤⢤⣤⣬⡽⣶⣾⡿⡿⠛⢬⡀⠀⠉⠀⠖⠁⣠⢏⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⠀⠈⠻⠉⠁⠀⢀⡞⠀⠈⠁⠓⠒⠐⠂⣶⠈⠋⠈⠀⠀⠀⠀⠱⣄⡀⢀⣤⠞⣁⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⣀⠀⠀⠀⢀⡟⠀⠀⢴⠁⠀⡀⠀⠀⢹⠀⢀⠀⣶⣦⠀⠀⠀⠈⠻⠊⣵⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠗⢦⣄⠻⢷⣄⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠉⠀⠀⣀⣠⠐⣯⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠋⢳⠲⡍⢍⣛⡒⢒⠒⣛⢛⣂⢀⣀⠤⡔⠶⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠋⠉⠌⠃⠽⠆⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣶⣒⣶⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣿⣾⣿⣿⣿⣿⣿⣽⡲⢄⡀
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⡢⣄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣧⠻⠿⡿⣿⠿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢆
⠀⠀⣼⣿⣿⣿⣿⣏⣿⣿⣿⡻⣿⡾⢋⣥⠀⠀⠀⠹⠋⠀⠀⢀⣎⡙⠿⣿⢿⣿⣿⣿⣿⣿⣯⣿⣯⢧
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⢋⣥⣶⣿⣿⣿⣻⡶⣬⣋⠀⠀⠀⢻⣽⣿⣿⣿⣿⣿⣧⣼⣏⣇
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡜⡴⢀⣾⣿⣿⡾⠿⠲⡌⠉⠻⣿⣽⣷⡄⣴⡧⠹⣾⣿⣿⣿⣿⣿⣿⣿⡾
⢸⣽⣿⣿⣿⣿⣿⣯⣟⡛⠀⠀⢡⣿⣾⡛⠿⠟⡶⠀⢼⠀⠄⠈⣿⣿⣿⡌⠀⠀⢛⣭⣭⣾⣾⣿⣿⣿⣇
⣿⣿⣿⣿⡿⣿⣿⣧⣼⡇⢀⡀⣿⣟⣿⣿⣤⢤⡇⠈⠱⡀⠁⠀⣸⣿⣟⣿⣠⣤⢸⣿⣿⣿⣿⠿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣻⣿⡏⠁⠀⣻⣿⢿⣮⣥⣀⣰⣄⢀⡃⠀⢠⣿⣾⣿⣿⠘⠓⢹⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⡀⠘⣿⣾⣿⣿⣽⣶⣯⣤⣤⣶⣿⣿⣧⣿⢃⡀⠀⣬⣝⣿⣿⣿⣿⣿⣿⡟
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠋⠓⠈⢿⣿⣿⣿⣿⣽⣿⣾⣿⣟⣿⡿⠃⠛⠷⣰⣿⡿⣿⣿⣿⣿⣿⣿⢿
⠀⢹⣿⣿⣿⣿⣤⣿⣿⣿⡟⣷⠄⠀⠀⣨⠛⢷⣿⣯⣿⢷⡿⢛⣉⡀⠀⠀⣾⣛⣿⣷⣿⣿⣿⣿⣿⣏⡏
⠀⠀⢻⣻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣤⣻⠃⠀⠀⢾⣻⠀⠀⠀⣛⣯⣾⣿⣮⣿⣿⣿⣿⣿⣿⣽⢟⡞
⠀⠀⠀⠹⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⣶⣶⣶⣶⣾⢻⣿⣧⣼⣿⣟⣿⣿⣿⣿⣿⣿⢫⠎
⠀⠀⠀⠀⠈⢮⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡵⠁
⠀⠀⠀⠀⠀⠀⠙⠪⣟⢿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⢿⣻⠽⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠿⠭⠽⣿⣿⣿⣿⠯⠭⠿⠓⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢶⡶⠾⠿⠿⠛⡻⢿⠿⠷⢶⢶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋⠁⢹⡇⠀⠀⢃⣲⡈⢰⡀⠀⠀⠀⣼⠃⠉⠛⣽⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢿⠏⡁⠁⢄⡀⠀⠀⢷⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⡏⠀⠀⢀⠅⠠⠈⠹⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⢠⡈⠡⠄⢠⣀⣠⠼⠷⠖⠒⠒⠛⠓⠓⠒⠲⠾⢧⣄⡀⠀⠈⠁⠸⠈⠀⠈⠻⢳⣄
⠀⠀⠀⠀⠀⣠⣾⠙⠷⣄⠀⠀⠀⠈⣤⠴⠛⠉⠀⢂⠀⠀⠰⠀⠀⢀⡀⠰⠆⠀⠀⠀⠉⠛⠷⣌⠀⠀⠀⢀⣴⠟⠹⡷⣄
⠀⠀⠀⠀⣴⡟⠁⠀⠀⠈⠳⣄⡴⠞⠀⢀⡀⠀⠈⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠐⠈⠑⢦⡴⠋⠀⠀⠀⣈⢿⣦
⠀⠀⢀⣾⢏⠀⡎⡄⠠⠀⣰⠏⢀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠙⢆⡠⣀⠀⣃⡀⡹⣷⡀
⠀⢀⣾⠇⠊⠤⡈⢰⢂⡾⠁⠀⡈⠀⡀⠠⠀⠒⠀⠀⠃⠀⠀⠠⠄⠀⠀⠀⠤⢀⠀⠀⠀⠀⠙⠀⠀⢀⡀⠈⢻⣆⠓⡀⠐⠌⠻⢷⡀
⠀⣼⡏⠀⠈⠁⠢⢁⡞⠀⢀⠀⠀⡈⠃⠀⡀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⡄⠈⠁⣀⠀⢻⡖⠁⠀⠀⠀⠫⣧
⢰⡿⠦⣤⣄⣀⡀⡼⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⡄⠀⠂⢄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢷⣀⣠⠤⠶⠾⣻
⣼⡇⠀⠀⠀⠉⢹⠇⠀⡤⠁⠀⠀⠀⠀⠠⠆⠀⠀⠀⠘⠈⠢⡄⠀⢙⠀⠀⠀⢣⠀⠖⠀⠀⠀⠀⠤⠀⠤⠀⠰⠄⠸⡇⠀⠀⠀⠀⢹
⣿⠁⠀⠀⠄⠀⢸⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠰⠦⢂⠀⢠⠁⠐⢊⠂⠂⠀⠀⡆⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⡖⢀⠢⢘
⣿⡄⢢⣃⠁⢃⣼⠀⠀⠐⠀⠀⠈⠠⠀⠀⠋⠀⠀⠐⠃⠀⠈⢆⠀⠀⣏⠀⠀⢠⠁⠰⠆⠀⠀⠀⠀⠀⠛⠀⠘⠃⠀⣿⠀⠄⠆⠢⢸
⣿⡟⠂⠀⠀⠀⢹⠀⠰⠀⠘⠃⠀⠀⠀⠀⠠⡦⠀⠀⠀⡕⠤⠘⡵⠠⠇⠀⠠⠀⡀⠀⠀⡴⡀⠉⠀⠀⠀⠉⠀⠀⠀⡇⠐⠨⠐⠣⢘
⢻⡇⠀⠀⠀⠀⢸⡆⠀⡀⠀⠀⣠⢂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⡀⠈⠉⠀⠀⡀⠀⠀⠀⠀⠠⠀⠠⠀⢰⣇⣀⠀⠀⠀⣘
⠸⣯⣶⠶⠚⠋⠉⢳⠰⠀⠀⠀⠌⠁⠀⠂⠈⠀⠀⠀⠀⠉⠐⠀⠐⠀⠀⠀⠉⠀⢀⣄⠀⠀⠀⠀⠀⢈⡀⠀⠘⢀⡾⠉⠉⠛⠛⢶⣶
⠀⢻⣆⠀⠀⢄⠀⡈⢧⠀⠠⠀⠀⢠⡄⠀⠂⢤⡄⠀⠀⠀⠁⠀⠀⠀⠐⠱⠂⠀⠀⠀⠀⠀⠀⣀⠰⣾⠁⠙⠁⣾⢣⠤⡄⠀⠀⣼⡿
⠀⠈⢿⡆⠀⡍⡐⠐⠌⢷⡀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠂⠀⠀⠀⣠⠀⢀⣾⢓⠻⣟⠙⠄⡸⡿⠁
⠀⠀⠈⢿⣇⠁⠐⠂⠈⠀⠙⢆⠀⠰⠆⠀⠀⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠁⠀⠀⠲⠀⠀⠈⣠⠋⠀⠋⠀⠨⠀⡼⡽⠁
⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⣠⠞⠓⢦⡀⠀⠀⠃⠀⠉⠀⠀⢀⠀⠀⠀⠦⠀⠀⣀⠀⠀⢀⠀⠀⢀⡤⠚⠻⣦⡀⠀⠀⢀⢜⡟
⠀⠀⠀⠀⠀⠙⢷⣄⣤⡞⠁⠀⠀⠴⠟⠷⢤⣀⡐⠀⠀⠀⠈⠀⠠⡄⠀⠀⠈⠉⠀⠀⣀⣤⣾⠉⣀⡀⠀⠈⠻⣦⣀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢯⣄⡀⠀⠀⢀⢠⢆⡤⠈⠉⢻⡶⠦⠤⢤⣤⣤⡤⠤⠴⢶⡞⠉⠁⠻⢆⢺⡱⡔⢄⢀⢤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⣀⡎⡜⡐⠀⠀⣼⠁⠀⠀⡧⢤⢠⠤⡇⠀⠈⣧⠀⠀⠀⠊⠳⣹⡼⣾⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣅⡀⢠⡏⠀⠀⠀⢁⠀⠰⠀⠁⠀⠀⢹⣆⢀⣠⣴⡮⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⣶⣦⣤⣴⣶⣤⣴⣶⠾⠿⠛⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢀⡰⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣈⣾⣿⣶⡝⢿⣦⠀⠀⠀⠀⠀⢀⣠⣾⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣠⡀⠀⢀⣼⣻⣿⣿⣿⣿⡌⣿⠀⠀⠀⢀⣶⣟⡿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⢀⣾⠟⠉⠉⢻⣿⣦⠀⠀⣴⣿⡿⡉⠉⠙⣿
⠀⠀⠀⠀⢠⣿⠁⣁⣴⣿⣿⣿⣾⣿⣿⣿⠗⠛⠀⠀⠀⢾⣷⣿⡿⠛⠿⠆⠀⠉⠻⣿⣷⣶⣦⠀⠘⣿⣄⢀⣰⡇⢹⣿⣷⣼⣿⣯⠸⣧⡀⣠⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋⠸⣯⠀⠀⠀⠀⠀⠀⣨⣯⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠈⠛⠛⠋⣼⣿⣿⣿⣿⣿⣿⣷⡙⠛⠛⠁
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⢀⣤⣤⣀⠀⠀⠀⣠⣴⣿⣯⣀⠀⠀⠀⣀⠀⢀⣤⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⣻⣿⣿⣿⢋⣿⠃
⠀⢠⣾⣳⣿⣿⣿⣿⡿⣿⣿⣄⠹⣄⣿⣿⠀⠀⠀⠈⠉⢉⣿⣿⣦⣤⣤⣿⣿⣜⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡟⠁
⠀⠸⣷⣿⣿⠿⣿⠿⠷⣸⡟⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⣭⡟
⠀⠀⠈⠉⠀⠀⠉⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⣀⠀⣀⠀⡄⢠⠀⠀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⡞⠁⢱⠀⣴⣿⣿⣿⣿⣿⣦⡀⡆⢀⢳⡄⠀⢀⣀⣻⣷⣾⣿⣿⣿⣿⡾⣿⣿⣶⣿⣏⣀⡀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠈⢷⣽⣯⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣯⡾⠃⠀⠈⠻⣿⣿⣿⣽⣿⣼⣿⣷⣿⣩⣿⣿⣿⠟⠁⠀⠘⢿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣾⣿⣿⡿⠃
⠀⠀⠀⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠟⠂⠀⠀⢀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠶⣾⣿⣿⣿⣿⣉⣿⣿⣿⣉⣿⣿⣿⣿⣷⠆⠀⢠⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⣿⡆
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠠⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠄⠀⠈⠉⠀⠈⢷⣬⣿⣿⣿⣿⣥⡾⠃⠀⠈⠁
⠀⢀⠀⡀⠀⢸⣿⡿⢿⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢿⣯⣿⣿⣿⣻
⠀⠀⠙⠒⠖⠛⡟⠙⢂⣿⡀⠋⢻⠛⠲⠖⠋⠀⠀⠀⠀⠀⠾⠿⠟⠛⠛⠛⠋⠛⠻⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⠞
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⣼⡇⠠⡄⠀⠀⠠⠆⠴⠀⠀⠀⡤⠈⣿⡄⠀⠀⠀⠀⠀⢀⣠⣴⠾⡿⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠏⠿⣿⠦
⠀⠀⢿⡇⣴⠇⠀⠰⠇⠀⠀⠣⠀⠀⢷⡄⣿⠇⠀⠀⠀⣠⠖⠋⠁⠀⠀⡧⠀⠀⠉⠛⠶⣄⠀⠀⠀⢀⣾⣛⠋⠀⠀⠀⠸⣿⡇
⠀⠀⠀⢻⡇⠀⠀⢶⣷⣤⣴⣷⠄⠀⠀⣿⠂⠀⠀⠀⠘⠁⢠⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠘⣷⣀⢀⣴⣿⣿⣿⣿⣄⢀⣼⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⣵⣾⣷⣤⣄⣀⡀
⠀⠀⠀⠀⠈⠛⢛⣿⣿⣿⣿⣿⣟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠛⢓⠶⣦⡀
⠀⠀⠀⠀⡠⠚⡡⠜⢛⠿⢟⠛⢤⡙⠢⡀⠀⠀⠀⠈⢿⣿⡿⠃⠀⠀⢠⣿⡀⠀⠀⠹⣿⣿⠟⠀⠀⠀⡼⠋⣻⠿⣿⣿⣟⠁⠀⠈⠁⠈⠃
⠀⠀⠀⠈⠀⠈⠀⠔⠁⠀⠀⠑⠄⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠰⠃⠀⠹⢧⣤⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣾⣯⣈⠁⢈⣁⣽⣿⣷⣄⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⣤⡾⠃⠀⠀⠀⠐⢉⠉⠉⠉⠉⠙⠛⢿⣿⣦⣄⠀⣀⣤⡖
⠀⠀⠀⠀⠀⣼⣿⣯⣠⡄⢠⣈⣽⣿⣿⣿⣆⠀⠀⠀⠀⠀⠙⣵⣿⣶⣶⣶⣿⣯⠃⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⣈⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⡟⠋⢻⣿⣿⣿⠀⠀⠀⠀⣤⢺⣿⣿⣿⣿⣿⣿⣿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠸⣿⣷⡄
⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣾⡯⠉⢹⠀⠀⠀⣾⣿⠸⣿⡟⣿⣿⣟⣻⣿⠧⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⢿⣷
⠀⢀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟⠀⠀⠀⠿⠋⠀⠘⣿⣿⣿⣿⡿⠁⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣿⠆
⢰⠋⠉⠁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠈⢷⣄⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⡠⠃

⠀⠠⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢄
⠀⠀⣿⣿⣷⣤⣤⣤⡤⠀⠀⠀⣠⣶⣿⣿⣿⣦⡄⠀⠀⠀⢠⠪⡆⠉⢹⡿⡆⠀⠀⠀⠀⠘⢦⣶⣶⡴⠃
⠀⠀⠈⣿⡟⠛⡟⡏⠀⠀⠀⠀⠀⠹⢾⠛⠛⢻⠃⠀⠀⠀⠈⣰⠂⣻⡃⢀⡀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠛⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠉
⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡔⣿⣿⣦⣤⡍⠀⠀⠀⠀⠀⠊⠉⢉⣿⣿⠀⠀⠀⠀⢀⣤⣀⡴⣿⣷⠤⡄⠀⠀⢀⣾⡏⠉⢿⣶⣤⡀
⠀⠀⠀⡖⢻⣿⣿⠓⠀⠀⠀⠀⠁⠀⣡⠞⠙⢯⠀⠀⠀⠀⠈⠜⠿⠿⣿⣿⡃⠁⠀⠀⠈⠙⠛⢷⣀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠐⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠀⠀⣀⣀⣀
⠀⠀⠐⠙⠻⣿⡿⠳⠀⠀⠀⠀⢹⣿⣿⣦⠀⡀⠀⠀⠀⠀⠈⠃⢀⣴⡄⠀⠋⠀⠀⠀⠐⠻⠢⠐⣿⣿⣽
⠀⠀⠀⣤⠞⠉⠷⡀⠀⠀⠀⠀⡦⢴⡿⣿⣷⣟⠉⠀⠀⠀⠀⠀⠉⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠂
⠀⠀⠚⠉⠀⠀⠀⠈⠂⠀⠀⠀⠁⠋⠂⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⡄⠀⠀⠀⠀⣄⠀⠀⠀⡀⠠⠜⠁⠀⠀⠳⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤⣀⣤⣶⣶⣦⡀
⠀⠀⢠⡿⠃⠀⠀⠀⠀⢿⡆⠀⠸⡀⠃⡤⠀⠀⠀⢤⠐⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⡟⠃⠀⠘⢻⠋⠈⡇
⠀⠀⠀⠝⣒⣤⣶⣤⣔⣚⠅⠀⠀⠈⠙⢃⡀⠀⢀⡘⠛⠉⠀⠀⡠⠀⠠⡄⢠⠄⠀⠤⡀⠀⢸⡐⢶⡾⡂⠉⢐⢿⡶⢃⡇
⠀⠀⠀⠘⠝⠛⣿⡛⠛⠘⠀⠀⠀⠀⠀⠘⠀⠀⠀⠃⠀⠀⠀⡈⠶⢆⣤⡅⢨⣤⡐⠶⢡⠀⠀⠙⠛⠃⢤⡄⡄⠈⠓⠋
⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠆⠀⠀⠠⢈⠡⠞⠉⠀⢇⡼⠀⠉⠳⠌⡉⠄⠀⠀⠀⠈⠬⠄
⠀⠀⠀⠀⠀⠒⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠠⠾⡏⣹⠷⠀⠀⠀⠈⠉⠀⠀⠐⠒⠒⠂⠂
⠀⠀⠀⠀⠀⠂⠐⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡅⠀⠀⠀⠈⣄⣀⡀
⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⠀⠀⡠⢐⡘⣖⡀⠀⠁⠀⠀⢀⣀⠀⠀⣴⣾⣿⣿⣿⣿⣶⡄⠀⠈⠉⢙⣁⣀⠀⣀⣀⡃⠀⠁
⠀⠀⠀⠀⠀⢋⠀⢙⠀⠀⠀⠀⠺⡒⠎⢩⠛⠛⠋⠋⠓⠲⣿⡋⠀⣼⣿⣟⡋⠀⠈⣛⣿⣿⡄⠀⠀⠀⣙⡉⠶⠉⣋⠁
⠀⠀⠠⠤⠴⢄⠀⢠⠦⠤⠄⠀⢶⣄⠉⠋⠠⠀⠀⠀⣏⣄⢠⣍⠀⣿⣧⡜⠛⠀⠘⠋⣤⣿⡇⠀⠀⠀⠘⠇⠀⠰⠃
⠀⠀⠀⠀⢀⡒⠂⢓⡂⠀⠀⠀⠴⠿⠃⠛⠳⠲⠞⡿⢣⢀⡩⠈⠀⠘⢿⣧⠀⠻⠁⢠⣿⡏⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠠⣄⣀⠄⠰⢀⠔⠀⣄⣀⠄⠀⠀⠀⢀⣀⢀⢀⠀⠀⠁⠀⠀⠀⠀⠈⠻⣤⣉⣡⡼⠏⠀⠀⠀⠀⠂⠀⠀⠀⠐⠒⠐⠘
⠀⠀⠀⠀⠰⠒⠛⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛
⠀⠀⠀⠀⠀⠘⠃⠃⠂⠀⠀⠀⠀⠀⠀⠀⣴⣾⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢸⡆⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⢻⡄⢀⣀⢀⣾⠁
⠀⠀⠀⡄⡗⠒⠒⠒⠒⠒⡇⠀⠀⠀⠀⠀⣠⠀⠀⣼⠇⠀⠀⠀⣘⣾⣿⣷⣾⣵⢿⡾⣿⣖⡀⠀⠀⠀⢈⣿⡧⠰⣿⣇
⠀⠀⠀⠤⡛⠿⢷⡶⠿⠟⠁⠀⠀⠀⣀⣣⠖⣩⣾⣏⣾⠆⠀⠀⠺⡿⡋⢃⡀⠁⣀⡓⢳⡽⠀⠀⠀⠀⢸⠦⠀⢂⠰⡎
⠀⠀⠀⠀⠎⠑⣀⣀⡀⠉⠀⠀⠀⢸⢸⡿⣯⣾⡧⣬⢈⠀⠀⠀⠀⠀⢉⠛⢋⣆⠛⢋⠀⠀⠀⠀⠀⠀⠀⢱⠈⠈⠐⠁
⠀⠀⠀⠀⠀⠀⢸⣹⠀⠀⠀⠀⠀⢿⠃⠉⠁⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⢿⠄⠀⢼⠃⠀⠀⠀⠀⠀⠀⠀⠈⡶⢰⡆
⠀⠀⠀⠀⠐⠒⠛⠛⠓⠂⠀⠀⠀⠀⢀⠀⠀⣉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠩⠼
⠀⠀⠀⠀⠁⠈⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠀⠂⠀⠒⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡠⢀⠢⠅⡄⢄⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠈⠻⠢⡀⣑⢊⠀⠀⡁⣎⢀⢀⠟⠀⡄⠀⠐⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣘⠀⠀⠀⠌⡢⢡⠖⡭⠀⢏⠈⠀⡡⠀⢭⠲⡌⢔⠡⠀⠄⠠⠃⠂⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠐⠮⢆⠠⠈⠁⠀⢄⠀⠗⠺⠀⡀⠀⠈⠁⠄⡰⠰⠂⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠂⠀⠘⠦⠠⣏⡁⣠⠲⠆⠈⣄⠌⠣⣠⠉⡰⠖⣄⠀⣻⡤⠴⠃⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠂⠀⠌⠁⠉⢀⡐⢤⠈⡄⢠⢁⡤⢂⣐⠉⠈⠡⠁⠰⠀⠐⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⣀⡈⠉⠛⠒⠀⡄⣈⡒⠦⣁⠀⠑⢨⡁⠊⠀⣈⠴⢒⣁⣄⠀⠒⠙⠉⢁⢀⡄⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣠⠀⢀⠄⡖⠠⢂⠤⠭⠑⠀⠀⢤⣹⣏⡄⠈⠀⠊⠡⠤⡐⠄⢲⠠⡀⠀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⣈⡴⠁⠀⡘⠍⡃⠒⢰⠆⢀⣉⡁⠀⠀⠀⠀⢀⣉⡀⠐⡆⠒⢘⠩⢇⠀⠈⢧⣁⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠩⠁⢠⠃⡌⣄⢠⠁⡀⠀⠄⢤⣠⢌⡁⠀⠀⢉⣡⣀⠤⠐⠄⢀⠈⡆⣰⢡⠘⡄⠈⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⣆⢁⠛⢠⣎⠀⠁⢦⡀⠉⠀⠁⠀⠀⠈⠀⠉⢀⡴⠈⠀⣱⡴⠋⡘⣴⡁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡹⢻⡜⠃⡀⠀⢠⠘⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⡄⠀⢀⠘⢣⡛⢇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⡴⠯⢋⠚⠀⠀⢟⠀⢻⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡟⠀⣳⠀⠀⠓⡙⠽⠦⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡈⠂⠤⠭⢀⠴⡙⠀⠺⣆⠀⠀⠶⠀⠀⠶⠀⠀⣰⠇⠀⢛⠦⡀⠭⡤⠘⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢀⠀⠡⠆⠁⠀⢠⠸⢷⡄⠀⠀⠀⠀⠠⡾⠇⡄⠀⡈⠰⠄⠀⡀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢘⣤⠀⢀⠀⠀⠀⢡⠠⠁⠀⠀⠈⠄⡎⠀⠀⠀⡀⠀⡤⡃⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢋⠀⠀⡘⠀⣀⠤⡀⢐⠀⠀⠀⠀⠊⢀⠤⣄⠀⠃⠀⠀⡙⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠛⢢⡄⡚⢤⡲⠀⠀⠀⠀⢗⡬⢓⢤⡄⠚⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠲⣏⡁⠀⠀⠀⠀⢈⣹⠶⡁⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠆⠁⠀⠈⠉⡅⢨⠉⠁⠀⢈⠰⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠆⣑⢀⢰⠀⠀⠀⠘⠂

⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠤⠀⠀⠀⢤⣤⣄
⠀⠀⢰⣿⡿⣿⣿⡿⣆⣀⣠⣿⣾⣿⢿⣿⡄⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⠿⠷⠂⠀⠀⠀⠒⠿⢿⠇
⠀⠀⢸⡏⢀⠙⠛⣿⣿⣿⣿⣿⠛⢋⠀⣹⡇⠀⠀⠀⠀⢛⣛⣾⣿⣿⣿⣷⣿⡛⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣦⣀⣠⣤⣤⣤⣄⣀⡴⠁
⠀⠀⠈⠛⠚⠁⠀⣿⣻⣿⣿⣻⠀⠙⠚⠋⠀⠀⠀⠀⠘⠛⠋⣿⣿⣿⣿⣿⡍⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠬⢿⣿⣿⣿⣿⣿⠯⡥⠚
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠁⢘⡯⢻⢿⣿⠿⣟⢭⡃⠈
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⢺⣿⣿⣿⣿⣷⣤⣤⣄⠀⠈⠀⠻⣄⠀⣠⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣿⣿⣿⣿⣿⡿⠁⠀⠀⣾⠏⡀⠀⠉⣉⡛
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣸⣿⢿⣿⢿⣿⣧⢠⣦⣤⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⢈⣁⢀⣀⡠⠖⠁
⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣞⣻⡷⠂⠀⠀⠀⠀⠹⠟⠋⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⣄⣀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠹⣿⣿⣿⡿⣿⠏⠀⠀⠀⢲⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠋⢺⠙⠁⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠘⠋⠏⠛⠿⠿⠿⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡁⠀⠀⠀⢩⠀⠀⠀⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢴⡷⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠻⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠺⠛⣳⣿⣿⣿⣿⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣿⣟⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⢶⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⠀⢀⣤⣶⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢰⢻⣿⣿⡃⠀⢤⡄⠀⢀⣿⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⣷⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠈⠈⣿⡟⠀⠀⠙⠉⠀⣾⢿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⠿⠿⠿⠛⠉⠉⠉⣹⠇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢻⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠉⠉⠛⠂⠀⠀⠀⠄⠀⠀⣤⣶⣶⣶⠖⠁⠀⠀⠀⠀⠀⠘⣶⡶⠶⠶⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⢠⣶⠀⣸⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠛⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠰⠒⠒⠒⠒⠒⠒⠒⠐⠆⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠑⣄⠱⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠶⡈⠑⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠉⠀⡷⠀⠁⠀⠀⠀⠀⠀⠀⠀⡽⠏⠀⠁⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⣀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠒⠀⠀⠈⠀⠐⠀⠠⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠞⠁⡤⠚⢀⡤⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢲⣶⣶⣾⣷⡄⠀⠈⠀⠢⠀⠀⠀⠀⢀⠴⣢⡱⣄
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠔⠊⠠⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣉⣺⣿⠋⠀⠐⠶⡔⠒⠤⡀⠀⠀⠁⠀⠀⠋⠞⡵⠁⠈⢢⡀
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣱⡏⡶⣻⠀⠀⣠⣦⣴⣶⣿⠁⠀⠀⠀⠀⠀⢀⠪⡀⠀⠀⠀⠱⡄
⠀⠀⢀⠎⠀⠀⠀⠀⢀⠐⠀⠀⠀⡰⠀⠀⡠⠄⡀⣄⠀⢀⣯⠀⢹⣿⠾⣷⣄⠀⢰⡿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠘⡄
⠀⠀⡜⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠣⠀⠐⠇⠘⠆⠉⠀⢸⣧⠀⡨⢿⣶⣽⣯⣀⡀⠻⠲⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠱
⠀⡸⠀⠑⠶⠄⠄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣦⣍⣈⣹⡛⠻⣿⣶⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⢠⠤⣄⢣
⠀⠇⠈⠈⠉⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠈⠙⠿⢿⣯⣟⣿⣍⣺⣿⣌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⡄⡦⠊⠘
⢸⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠈⢧⠀⠀⠲⢍⡉⢻⣿⣿⣿⣟⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⢸⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⡀⠀⡀⠀⠀⠀⠑⠀⢈⣻⣿⡿⢹⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠠⢝⠀⠀⠀⠀⠀⢡⡿⡿⣡⡿⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠸
⢸⡄⣀⠀⠀⠀⠘⠀⣀⣤⡶⢶⢖⠀⠀⡰⠀⠐⢄⢀⣀⣀⣀⣀⠀⠑⣄⠀⠃⠂⠻⢿⡿⠿⣣⣈⣲⣦⠀⠑⠀⠀⠀⠀⠀⡇
⠀⡞⠤⢀⣠⡤⢴⡿⠋⠀⡀⠀⠀⠀⡀⢀⣤⣤⣀⣉⡉⠉⢉⣁⣠⣶⣿⠂⢆⡉⠀⠈⢣⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠐⠀⠀⣁⡒⢦⢸
⠀⠰⡴⠋⠀⢠⣿⠎⠐⠈⠐⠀⠀⢈⡵⢿⢿⠿⠛⠛⠛⠛⠟⠉⠉⠀⠀⠙⢦⡉⠁⡀⠀⠣⡀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠁⠈⠳⡐⣮⠝⡝
⠀⠀⢣⠁⠀⠘⣿⣄⣠⣶⣶⣶⣾⡏⠀⠙⢌⡀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠈⡄⠀⠳⢦⣤⠇⠀⠀⣠⡀⠀⠀⠀⠀⡸
⠀⠀⠈⢆⠀⠀⡃⢍⢉⠻⣆⠉⠿⠁⠀⠀⠀⠙⢦⣀⡐⢈⣻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣦⣄⣠⠃
⠀⠀⠀⠈⢦⠈⠑⠛⠚⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣄⡀⠀⠈⣉⣰⣾⡻⡏⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠐⠉⠑⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢚⣻⣿⣧⣶⡝⣿⡶⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠘⢮⡉⠉⡆⠚⠀⠀⠈⠀⢢⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠐⣿⣿⣿⣿⣿⣿⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠀⢀⣀⣀⡀⠀⠤⠔⠂⠉⠈⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⢸⠁⡒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢂⠈⠁⡂⠀⠀⠀⢈⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠠⠎⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⢀⡠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠐⠒⠠⠤⠤⠤⠤⠤⠤⠤⠒⠒⠚⠉⠁
⡿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠴⠶⠖⠒⠶⠒⠶⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠲⣀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣭⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠙⠀⢷⢁⠇⠀⠀⠀⠀⠀⠀⠀⠐⣤⠊⠘⠀⠀⠀⠀⠉⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡰⠃⠄⠀⢀⣀⣒⣀⠀⠠⠄⣈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡐⠀⠀⠀⠀⠀⡀⢄⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⡀⢀⡀⠀⠀⠀⠀⠀⠀⣈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⡖⠃⢠⠼⠉⡀⡐⠅⠀⠀⠀⢸⠁⢀⢀⡄⠂⡲⠰⢲⠂⡆⢰⢒⠆⡲⠀⠀⠀⠁⠀⠄⢄⠀⠀⢀⡴⣩⠟⡮⠳⡀
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠘⠉⡠⠪⠂⠀⠀⠀⠀⠀⢈⢂⡞⠈⠃⠀⠁⠁⠘⠀⠁⠉⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠐⣡⠊⠀⠀⠘⢆
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⢀⣠⣬⣄⣀⠀⡀⣠⠂⠉⠁⠉⠉⠛⠲⣄⡠⢂⣤⣤⣄⡀⢀⡴⠒⠋⠀⠁⠈⠉⠲⣀⡄⠀⠐⠤⣄⣳⠀⠀⠀⠀⠈⢧
⠀⠀⠀⡼⠁⠀⠀⠀⡖⠉⠁⠤⠀⠀⠀⠀⢀⣉⢛⡺⢆⠀⠀⠀⠀⢸⣦⠔⣈⠌⢀⣈⣽⡁⠀⠀⠀⢀⠴⣞⣛⣁⠀⠀⠀⠀⠒⠈⠑⣦⠀⠀⠀⠈⢣
⠀⠀⢰⠁⠀⠀⠀⠀⠘⣆⠀⠤⢤⣴⡶⠛⠛⠛⠙⠂⠈⣿⣷⠲⠆⣼⠉⠑⠀⠈⠉⠀⢸⡅⠢⢲⣿⣟⠀⠚⠉⠉⠛⠛⢻⣿⠿⠁⢠⡏⠀⠀⠀⠀⠈⢇
⠀⢀⠇⠈⠢⡤⠀⠄⢀⠘⢢⡀⠀⠈⠛⢄⣀⠸⡄⠰⢾⣿⣶⢿⡿⠏⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣾⣿⡷⠀⢸⠀⢀⡴⠋⠁⠀⣰⠋⠀⠀⢠⠠⡀⢄⠘⡄
⠀⣼⠀⠂⠘⠒⡄⢀⠄⠀⠀⠙⢦⣀⠀⠀⠈⢲⡄⢶⣿⠛⠸⠈⢧⣀⣀⡀⠀⠀⠀⠀⠀⣀⡼⢿⢹⢿⣷⡶⣤⢶⡏⠁⠀⢀⡴⠁⠀⠘⡄⠈⡇⢠⠠⠃⢳
⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⢷⡄⣸⠀⠉⡏⠁⠀⠄⢦⣄⢉⣻⡿⣦⠀⡰⠿⠿⣏⡀⠞⢸⡀⠻⠎⠌⠎⣧⠀⡴⠋⠀⠀⠀⠀⢠⠀⠁⠈⠀⠀⢸
⢰⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⡇⠀⠀⡨⠊⠀⠀⡠⠞⠛⢻⡞⠉⠑⡄⠉⢶⣄⠀⠀⠀⡘⠸⢸⡿⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘
⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⡇⣠⠞⠀⠀⡼⣿⠀⠀⠀⠀⢷⠀⠀⢸⡄⢈⠻⣷⣄⢀⣿⡌⠀⡅⠀⠀⠀⠀⠀⠀⡃⠂⠀⠀⠀⠀⠐
⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⠌⢰⣧⢟⡅⠀⠀⠘⣧⣿⡀⠤⡀⠀⣸⡀⢰⣿⡿⠈⣆⠙⣿⢾⡟⡿⢶⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠
⠀⣇⡤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢻⢀⣾⠋⠉⠀⠀⠀⢸⡜⢿⣷⡄⠀⠀⠈⢳⣿⡾⠃⢀⡟⠀⠈⠇⠻⣷⣿⡇⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⢸
⠀⢹⠑⠂⣤⠴⠖⠀⠆⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⢘⢀⠙⠄⣿⠇⠀⠀⠀⠈⣿⢁⡄⠈⢠⠀⠀⠠⡄⣙⢿⠁⠀⠀⠀⠀⠀⢠⠃⡀⠦⠭⣙⣒⡾
⠀⠈⣦⡋⣀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⣾⣷⠎⠀⠀⠀⠀⡘⢎⠓⢠⠏⠀⢣⡀⠀⠃⢹⡀⠀⢠⡾⠀⠀⠀⠻⣜⣿⠃⠀⠀⠀⠀⠠⠀⠈⠙⢍⢲⠞⢳⠃
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢿⣿⡄⠀⡀⠀⡀⠳⡄⠀⣿⠠⠆⠀⢳⢄⠔⠛⣷⠀⠁⣠⠞⣠⠊⢠⡼⠃⠀⠀⠀⠀⠀⡱⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠑⡂⠀⠀⠀⠈⠻⣷⡤⠼⠶⠞⠛⠓⠚⣿⠀⠀⠀⢰⡄⠀⢀⣿⠓⠖⠚⠛⠓⢚⣾⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⣷⣶⡒⠶⠶⡒⠛⣛⣹⣇⠀⠀⠘⡇⣠⢾⣿⣿⣿⠿⢛⣵⣿⣿⣧⠀⠀⠀⡢⠂⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⢀⠖⠒⡄⠐⢠⣾⣯⣿⢭⡓⠒⠛⠀⠈⠉⣹⣦⠀⠀⣿⠁⠀⠀⠀⠀⢈⡩⣾⠟⣿⣿⡷⠠⢋⡀⢎⣱⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠣⡄⠒⠚⣄⣠⠞⠀⠀⠊⠟⢯⡓⢮⣅⣀⣀⡀⣀⠁⢈⣷⠀⢹⡄⠀⠀⢈⣡⣵⠞⠉⡼⠡⠍⠈⠀⠁⠈⢦⡀⠈⣃⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡆⠀⠀⠀⠀⠀⠀⣼⣶⣥⣎⣉⠁⠀⣇⠀⠘⣮⠇⠀⡑⠀⠀⢉⡉⠤⠢⠑⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⣼⣿⣿⣿⣯⠛⡿⠿⣶⣶⣶⣾⣶⠒⠒⠀⠉⢁⠤⠐⠒⠤⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣼⣿⣿⣿⣿⡟⣼⡟⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠀⠁⢐⢁⠄⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣯⣥⣟⡑⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⠉⢉⣉⠥⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠿⠷⠖⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉⡁⣈⠁⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠠⢌⡣⠥⢠⡌⢱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡃⡞⠀⠊⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠇⠈⠧⣀⢀⢀⡓⠀⠰⠀⡆⠢⡀⠀⣠⠆⠀⠀⠀⣰⢦⠀⢠⢶
⠀⠀⠀⠀⠀⢁⠀⠁⢖⡛⠛⡓⠊⠁⣠⠟⢡⠀⠃⠙⠀⡨⠬⠥⣼⣆⠀⠀⠙⢾⡤⡾⠋⠀⠀⣰⠉⣇⡐⣲⠀⡀
⠀⠀⠀⠀⠀⠂⡖⣀⠀⢙⡆⠀⡀⠈⣈⠕⠂⡎⡤⠒⠢⡀⠠⠤⣄⢉⡄⠀⠀⢈⠀⠇⠀⠀⢠⠤⠀⡄⠀⠙⢠⡞⢀
⠀⠀⠀⠀⠀⠸⡇⣧⠄⠚⢉⣈⠁⢰⠃⠀⠀⠿⠁⠀⠀⠑⣠⠴⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣦⠀⢡⠚⠁
⠀⠀⠀⠀⠀⠡⠙⢀⡴⠋⠉⠁⠀⠚⠂⠀⢀⡴⠋⠉⡥⠋⠁⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠰⠄⠸⡏⠀⠀⣩⠁⢈⡅⠶
⠀⠀⠀⠀⠀⠠⠀⠈⣠⠀⠘⣇⣠⠴⠒⠒⡿⠁⡔⠉⢧⠀⠀⠀⢈⠐⠲⡅⠩⠥⠀⠬⠅⠌⠀⣄⠀⣀⠴⠏⢀⣘⢢⡄
⠀⠀⠀⠀⠀⠥⠶⠄⣏⠀⡀⠈⠃⠀⠀⠀⡇⠀⠃⣠⠄⠂⠀⠀⠀⢱⡀⠘⡇⠻⣿⠟⠀⠀⢠⠏⠈⠁⠀⣤⢌⣧
⠀⠀⠀⠀⠀⠀⢀⠴⠛⢲⡧⠖⠒⠂⠀⠀⠁⢀⡞⠀⡔⠂⠀⠀⢀⠼⠁⠀⡇⠀⠘⡆⠀⢠⠞⠀⣸⠇⠀⠙
⠀⠀⠀⠀⠀⠀⠏⠀⢠⠏⠀⡔⠀⣠⣤⣄⠀⢸⠀⠀⣇⠀⢄⡀⠀⠀⠀⠀⠀⠀⢀⡇⠐⢁⠤⢰
⠀⠀⠀⠀⠐⣴⠒⠂⢾⠀⠸⡅⠈⢀⡠⠤⣄⣘⣆⠀⠘⣆⠀⠉⠛⠂⠀⠀⢀⡠⠊⠀⣰⠁
⠀⠀⠀⠀⠀⢳⠀⣀⠘⢆⠀⠈⠁⡏⠀⠀⠀⠈⠉⠀⠀⠘⡆⠀⢸⠀⣀⡤⠤⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⢀⣥⠀⠀⠒⠂⣶⠐⠀⠀⠁⢀⣘⣛⣁⣀⣀⣀⣀⣘⠀⠀⠀⢢⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉⠀⢠⡀⠀⣿⠠⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⢠⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣴⡾⠟⠛⠋⢉⡻⠛⠉⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃⠀⠀⣠⠾⢀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠀⠀⠀⠀⠉⠻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠊⠀⠀⠄⠀⠀⠀⢀⠁⠀⠠⡃⠀⢀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠁⠢⡀⠀⠀⢀⠈⠿⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠀⠈⠐⠀⠈⠛⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⠀⠀⠀⡠⢤⢄⠀⠀⠀⠈⠈⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠈⠦⣡⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣛⣿⣿⡟⠁⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠂⠀⠃⠀⠻⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⣤⣤⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠻⣿⣿⣿⣿⠋⠀⠈⠀⠄⣀⣠⣦⣄⣀⣈⣤⣶⡄⠀⠂⠀⠀⠀⠿⡿⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡀⠀⣶⣿⣿⣿⣿⣿⣭⣿⣿⠹⢿⣿⠄⢠⣷⣄⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠿⠟⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠠⠾⢛⠉⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⢸⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂⠀⠀⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠁⠀⠀⠀⠀⠻⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⠿⠿⠛⠋⢹⡇⠀⠀⢀⠀⠀⠀⣿⠉⠛⠻⠿⢷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⣀⣠⣤⢤⡆⠈⢷⡀⢀⠎⣆⠀⣰⠏⠀⣶⠤⣤⣀⡀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⠴⢛⡉⠀⠀⣾⠁⠀⠀⠙⠛⠀⠘⠛⠁⠀⠀⢹⡆⠀⠈⣙⠳⢄⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠁⣴⡿⡻⢦⠀⣿⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⢸⡇⢠⠞⡻⣷⡀⠁⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⣸⡿⢹⠙⢦⡀⢻⡀⠀⠤⠖⢻⣇⣿⠓⠦⠄⠀⣸⠃⣠⠞⢹⠹⣿⡀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠖⠀⠀⠀⢿⡇⠈⠀⠔⢁⡼⠃⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠻⣄⠑⠄⠈⠀⣿⠇⠀⠀⠐⢦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣠⣴⣾⠃⠀⠙⠀⠀⠀⠋⠀⠀⠀⠀⢤⣼⠏⢿⣤⠄⠀⠀⠀⠈⠃⠀⠀⠘⠁⠀⢳⣶⣤⡀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡇⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣴⡆⠈⠀⠀⠀⢠⣶⡅⢤⣀⣀⡀⠀⠀⢸⡇⣿⠀⠀⠀⣀⣀⣠⠀⣵⣆⠀⠀⠀⠈⠀⣶⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣼⠉⠻⣄⠀⠀⠀⢸⣧⠱⣄⠉⢯⡛⢷⡄⠀⠀⠁⠀⣴⠟⣫⠏⢁⡴⢡⣿⠀⠀⠀⣀⡼⠋⢻⡄⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠃⠀⠀⠈⠙⠒⠀⠘⣿⡄⢸⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠀⣾⡏⠀⠐⠚⠉⠀⠀⠀⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⡴⠞⠓⠢⠀⠀⠀⠀⠘⢿⣼⡀⠀⠀⠀⡼⠀⠀⠀⢸⡄⠀⠀⠀⢘⣼⠟⠁⠀⠀⠀⠀⠒⠛⢶⣄⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⢀⡆⠀⠀⠀⠀⠀⠀⠙⠆⢠⡆⠘⠁⠀⠀⠀⠀⠑⢀⣦⠀⠞⠁⠀⠀⠀⠀⠀⠀⣆⠀⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣾⣿⠀⠀⠀⠀⠀⡄⠀⢠⣿⡽⢰⣤⣤⣤⣤⣤⡴⠺⣽⣧⠀⠀⡆⠀⠀⠀⠀⢸⣿⡆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⣧⡀⠀⠀⢠⡇⠀⢸⣿⡅⠘⠷⣤⡀⣠⠴⠟⠀⣿⣿⠀⠈⣧⠀⠀⠀⣠⡿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠻⠶⠶⠋⠀⣶⡈⠻⣷⣤⣤⣄⠀⢀⣤⣤⣴⡿⠃⣰⡄⠈⠳⠶⠞⠋⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠢⢤⣤⡴⠞⠉⠳⣄⠀⠘⣯⣄⣀⣀⣼⡟⠀⢀⡴⠋⠙⠶⣤⣤⠤⠂⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠉⠉⠉⠁⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⢣⣀⠀⠲⣤⣀⡀⠐⢦⣄⡀⠀⠀⠀⠀⣀⣤⠖⠁⣀⣠⡴⠂⢀⣠⢋⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣉⠳⢦⣄⣉⠉⠛⠛⠛⠛⠛⠒⠚⠛⠛⠛⠛⠋⢉⣁⣤⠶⢋⣡⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡉⠛⠶⣤⣄⠀⠀⡀⡀⠀⢀⣤⠶⠞⠋⣁⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⡉⠻⢦⡇⣧⠾⠋⣠⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡈⠁⠉⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣷⢀⣤⡀⢠⣄⢀⡁⣆⢀⡇⣠⣀⢠⣄⢈⢡⠀⡀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠑⢤⣨⠤⣷⣏⢨⢸⡏⡏⠹⣯⠥⢸⢸⠀⢸⢸⡀⣇⡱⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠥⣈⣹⠀⠁⠁⠀⠁⠉⠉⠀⠀⠈⠀⠉⠁⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣦⣀⡁
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠛⠋⠀⠀⠀⠠⠄⣉⠉⠀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⢿⣿⣆⣀⠁⠀⠀⠀⠸⠿⣣⡾⠋⢹⠄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠟⠀⠀⠀⠀⠀⣴⠟⣁⠀⠻⢄⡉⠀⠀⠀⠀⠀⠀⠻⢿⡆
⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⣻⣿⢀⠀⠀⢹⠄⣀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠁⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⡿⠃⠀⠀⠘⠋⠸⣿⠂⣾⠟
⠀⠀⠀⠀⠀⠁⠀⢻⣇⠘⠿⡿⢇⣄⠐⠃⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⣀⠀⠙⠀⠁⢰⣿⣴⠆
⠀⠀⠀⠀⠀⠸⢿⣦⡀⠀⠀⠀⠸⠃⢴⡄⣠⣤⣴⣶⣶⣶⣶⣶⣷⠀⢸⣦⣧⣈⠀⠀⠀⠀⢈⣙⡋
⠀⠀⠀⠀⠀⠀⢠⣠⣀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⠿⠿⠟⠿⡟⢛⣣⡀⠻⣿⣿⣶⠀⠀⠀⢙⣉⠁⠴
⠀⠀⠀⠀⠀⠀⢺⣿⡯⠀⠀⠄⠀⠠⣆⠁⢸⠀⠀⠀⠀⠀⢤⡙⠈⠛⡵⣢⡙⢿⣿⣇⢀⡠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠈⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢠⡀⡀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠃⠀⠈⢻⣿⣿⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⢀⣿⡟⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⡟⢀⣿⡿⣀⠀⠀⠀⠀⣀⣀⡀⣀⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⡿⢁⣾⣿⠛⢋⣵⣶⣿⣿⣿⣷⠁⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⣀⢄⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠚⠆
⠀⠀⠀⠀⠁⢈⡆⠀⠀⠂⠀⢄⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠠⠀⠄⠀⠂⠤⠀⠀⠀⠀⠀⠀⢀⠀⡀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠆⢉⡀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⢀⡴⠀⡀⡲⠅⠀⠁⠀⠈⡁⠀⠀⠀⠀⠀⠌⠜⠀⠀⠀⠀⠀⠀⣦⠀⠁⣈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢠⣀⣠⣤⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠌
⠀⠀⠀⢔⠒⢂⢀⠀⠀⠤⠤⠌⠀⠀⠀⠀⠄⠀⠀⠠⠄⢀⠈⡀⠁⠀⡇⠀⠀⠈⠐⠀⠀⠀⠀⠀⢀⠀⣠
⠀⠀⠀⠸⠂⠄⢘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠸⣀
⠀⠀⠀⢀⡐⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠚⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⠈⠣⡼⢆⠂⠀⠂⠀⠀⠀⠀⠀⠇⠀⠀⠀⠂⠀⠀⠚⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠁⠂⠈⠄⠶⠀⡀⠘⢄⠃⣀⠂⠀⡀⠄⠀⠀⠀⠀⡀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⡠⠀⢺⠀⠉⠀⠈⠀⠀⠀⠠⠤⢀⠁⣛⡇⠂⠀⠀⢀
⠀⠀⠘⢀⠀⠀⠀⢧⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠒⢈⠤⡀⠀⠡⠐⠀⠀⢂⠀⠘⠋⢸⠆⠀⠀⠆
⠀⠀⠀⠀⠉⠣⠀⠈⠀⠀⠀⡠⡆⠀⠀⠀⠄⠀⠀⠀⠀⠄⠉⠁⠀⠨⠀⠀⢀⠀⡆⠀⠀⣸⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠛⠂⣀⠀⠀⠦⠀⠀⢀⠀⡀⣀⢀⢄⡁⢀⠔⠒⠁⠁⠀⠈⢣⣬⠂⠰⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣄⣀⠀⠀⠀⠀⢀⠴⢒⠞⣁⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⣴⡦⠤⠌⣫⡙⠦⠀⠔⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⣓
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢀⡤⠤⠤⣀⠀⠀⠀⠉⠒⠤⡀⠀⠀⢠⠊⠀⡠⠔⠀⠀⠀⠀⠀⢀⡤⠤⢄⠀⠀⠀⠀⢩⡃
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⡎⢠⠤⢄⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠃⡴⠋⠠⠒⠉⠉⠀⠀⢠⠋⢀⡀⠀⠱⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⢸⠀⠂⠀⠈⢣⠀⠡⠀⠀⠀⠀⠀⠀⠀⠌⢀⣠⠄⠤⠤⣄⡀⠀⠇⠀⡜⠀⢠⠀⡇⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⣆⠣⡀⠀⠀⡇⠀⠀⠉⠁⠈⠐⠤⠀⠔⠁⢀⠀⠀⠀⠀⡈⠐⠀⠀⠃⠀⡸⢠⠃⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠒⠌⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠀⡴⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠔⢁⠎⠀⠀⠀⠀⠀⠀⢈⣣
⠀⠀⠀⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⣠⡾⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢰⣢⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠈⠁⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⢈⣁⣄⣀⣀⣀⣤⠶⠒⠀⠠⡄⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⠎⠀⠀⠀⠀⠈⠡⡞⠁⠀⠀⠀⠀⠈⢦⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣮⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⡼⠃⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⠀⠠⢔⣲⠶⠀⠄⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠁⠤⣀⣀⣤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠐⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡔⣒⢶⠋⠙⡆⠀⠀⠰⣏⣳⡀⣠⠚⣓⡆⠀⠀⠐⠛⢦⡤⢴⠚⠙⠀⠀⠀⢀⡴⣖⠒⠲⣄
⠀⠀⠀⠳⠼⢸⠘⠒⠃⠀⠀⠀⢠⠏⠉⠙⢦⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠘⠒⠋⠀⡨⣟
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢧⣀⣠⠞⠀⠀⠀⠀⠀⣄⡼⠤⢼⣀⣤⠀⠀⠀⠀⠻⢧⣈⣧⠼

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣄⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡞⠉⠙⡆⠀⠀⠀⠉⢸⠁⢹⢉⡇⠀⠀⠀⠀⠀⢀⡔⠒⢦⠀⠀⠀⠀⠀⠘⢻⡏⢹⢹
⠀⠀⣖⠻⡇⢀⡜⠁⠀⠀⠀⠀⢸⠀⡿⢸⣤⡄⠀⠀⠀⠤⠤⠷⠀⢯⡤⠶⠀⠀⠀⠀⢸⠁⣼⢸⡀
⠀⠀⠉⠉⠁⠘⠶⠂⠀⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠀⠁⠀⠿

⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⣩⠛⡇⠀⠀⠀⠀⠚⣿⠉⢹⠀⠀⠀⠀⠀⢀⡔⠳⠞⠶⠳⠆⠀⠀⠀⠀⠀⢳⠀⢰⠃
⠀⠀⠀⢲⣴⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⣤⣄⠀⠀⠀⠀⡴⢲⠞⣶⢻⠄⠀⠀⠀⠀⠛⡟⠉⢻⠛
⠀⠀⠀⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠼⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠈⠧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢀⣀⣀⠀⣸⣿⣿⣿⡄⠀⠀⢠⣶⣶⣾⡂⠀⠀⢀⣤⡀⠀⠀⠀⠰⢾⣿⣿⣿⠆
⠀⢀⣤⡴⠶⣶⣮⡅⠀⠀⠀⣈⡉⣻⡇⠸⠿⠿⠛⠐⢦⡀⠸⣿⣿⣿⡃⠀⠀⠘⠿⠛⠓⠲⠶⠶⢬⠟⣻⣯
⠀⠸⡇⠀⣸⣿⠏⠻⠶⠞⠊⠉⠁⢻⡇⠀⠀⢸⣿⣠⡾⠃⠀⠈⠉⢿⣿⣄⠀⠀⠀⠀⠀⠻⠷⣤⣄⣸⣿⣿⡆
⠀⢰⣧⢀⣿⣟⣀⣀⣀⡀⠀⡴⠀⢸⡇⠀⢀⣼⣿⠋⠀⠀⠀⠀⣀⠀⠻⣿⣿⡛⠛⣿⠂⠀⠀⠀⠈⠉⢸⣿⣷
⠀⠀⠀⢸⣿⠉⠉⠉⢹⡇⠈⠀⠀⢸⣇⣴⠟⣻⣿⠀⠀⢀⣴⣿⣿⣿⡄⠙⣿⡟⠛⠛⠛⠶⣤⣼⠃⠀⣴⡿⢻⡆
⠀⠀⠀⢸⣿⠀⠀⠀⠘⠳⠀⢀⣴⣿⠟⠁⠀⠙⣿⠀⠀⠀⠙⠿⣿⠿⡃⠀⠘⣷⡀⠀⠀⡀⠀⠀⣠⡾⠋⠀⠘⣿
⠀⠀⠀⠸⡿⠀⠀⣤⣶⣶⣴⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⣤⣦⡄⠀⣿⣿⡀⠀⠘⣷⠀⠀⠇⠀⠀⣿⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠘⠿⠦⠀⣿⣿⣿⡟⠙⢷⣀⣀⡀⠀⣴⡷⠀⠈⠿⠿⠛⠲⢾⣿⣇⠀⠀⠐⣷⡴⠆⠀⢀⣿⠀⠀⠀⠀⠈⣿
⠀⣠⣴⣦⡀⠀⠀⠈⠙⣯⡁⠀⠀⠉⠉⢉⡧⠀⠀⢲⣀⣀⣤⣤⣤⣌⣿⣿⡀⢀⣤⣤⣄⠀⠈⠉⠉⠀⣀⣀⠈⠉⠉
⠐⣿⣿⣿⡿⠂⠀⠀⠀⣌⡗⢀⣠⣾⣿⣦⡀⠀⠀⠈⠉⠉⣁⣀⠀⠉⠉⣿⡇⢾⣿⣿⣿⠷⠀⢀⣀⣼⣿⣿⠆
⠀⠈⣛⣯⡀⠀⢀⣀⡀⠈⠀⠈⠹⣿⣿⡿⠃⢐⣧⡀⠀⣾⣿⣿⣷⡄⠀⣿⡇⠈⢻⣟⣡⠶⠟⠋⠁⠀⢉⣡⣤⣄
⠀⠀⣿⣿⣿⡄⣿⣿⡿⠶⠶⠶⠶⠶⣶⣶⠀⠀⠀⠁⠀⠙⠿⠿⠋⠀⠀⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⡇
⠀⠀⣿⠈⢿⣿⡆⣉⣤⣶⣄⠀⠀⠀⢹⣿⣧⠀⠀⢸⡟⠛⣿⣿⠀⠀⠀⣽⡅⠀⣿⣿⣠⣴⡶⣷⡀⠀⠀⠙⣿⡏
⠀⠀⣿⠀⠈⢻⣿⠋⠁⠀⠙⠷⣄⣰⠎⢿⣿⡆⠀⢸⡇⠀⣿⣿⠀⠐⠛⠛⠃⠀⢿⡟⠉⠀⠀⠈⢻⣄⣀⡀⣿⡇
⠀⠀⠋⠀⠀⠈⢿⡄⢸⡀⠀⠀⠈⢃⣴⠿⠻⣿⠀⠈⠁⠀⣹⣿⣤⣤⣄⠀⠀⠀⢸⡇⠀⠈⢦⠀⠀⠛⠉⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣠⡾⠛⠁⠀⠀⢻⡇⠀⠀⠀⣿⡏⠉⢩⣿⠀⠀⡶⢟⠁⡀⠀⠀⢀⣀⣤⣤⠶⢹⡇
⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣾⡇⢸⣿⣿⣿⡌⢻⣄⠀⠀⣿⡇⠀⠨⣿⣤⠘⠟⠋⣄⠹⠿⠘⠋⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⣠⣿⠀⠘⠿⣿⡿⠃⢀⣹⣷⡀⢿⡇⠀⠀⠀⠀⠀⢀⣾⠋⠀⣀⣤⠀⠀⠀⡇⠀⢸⡇
⠀⠀⡈⢧⣀⣼⡿⣦⣄⡀⠀⠀⢀⣶⣶⠶⣶⠀⠀⠀⠀⠀⠘⢛⣦⣀⠀⠀⢠⡾⢁⣴⠾⠋⠀⠀⠀⠀⠁⢀⣼⡇
⠀⠀⣇⣈⣉⣉⣀⣈⣙⣿⣶⣶⣿⡿⠃⠀⣿⠀⠀⢸⠀⠀⢀⣿⣿⣿⣷⢰⣿⣷⣿⣷⣶⣶⣶⣶⣶⣾⢦⣿⡇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠀⠀⠀⠀⠀⠈⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢨⡽⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡁⠀⣎⠀⠐⠀⠀⢩⡆⣠⠖
⠀⠀⠀⠀⠀⠀⡞⠁⠉⠉⠉⠉⠉⢈⣷⢿⠉⣹⣶⠻⢌⢀⡼⠛⡿⠈⣁⠉⠉⠉⠉⠉⠉⠳
⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠠⠒⢊⠟⣡⣛⣛⢿⡟⠻⠷⠄⠔⣺⣷⡾⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢀⢴⢗⠉⠉⢁⡼⠳⢾⡯⠘⣧⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣰⣷⢊⡞⠁⠀⠀⡜⠀⠀⠀⠀⠀⠙⠤⢤⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠐⠒⠛⠏⠸⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡔⣿⠀⠀⠀⠀⠀⢄⡀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣄⠀⠰⠋⠀⠈⠀⠀⠀⠀⠀⠀⢙⣄⣠⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠋⠙⣅⠀⠀⠀⠀⠀⠀⡀⠀⣠⠆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠈⠑⠀⠀⠀⠀⠀⣿⠜
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⡆⣸⣤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠓⠒⣄⠀⠀⠀⠀⠀⡜⠀⠀⢀⡼⡡⢿⠏
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢻⡄⣺⡷⢦⡞⢁⡀⣀⢴⠗⠁⢰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡾⢻⡯⠖⠐⢶⣦⣼⣷⣭⣭⢋⣴⡁⠄⠂⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢦⡀⡀⣀⣀⡀⣀⣉⣀⣾⣤⡞⠁⡑⣦⠿⣏⣀⣷⢿⡁⣀⣀⣀⣀⣀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠸⣃⠀⠀⢄⠀⡹⠀⢈⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣾⡃⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⢐⠀⠀⠀⡰⠒⠐⠒⡄⠀⠐⢀⠀⠀⣀⠀⠠⠑⠀⠀⠐⠠⠀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢰⠀⢰⠃⠀⠀⠀⢳⠀⣃
⠀⠀⠀⠀⠀⣇⠀⠀⠐⢀⡀⠀⢠⠀⡀⠀⠀⡎⠀⠀⠀⡂⠤⡀⠀⢢⠄⠂⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⢀⠕
⠀⠀⠀⠀⠀⠙⠦⠀⠈⠢⣁⠀⡡⠊⠀⠠⠜⠁⠀⠀⠀⠘⠄⡁⠀⡀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡄⠀⠀⠀⠀⢀⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⢸⠃⠀⠀⠀⢳⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠁⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠴⠳⢄⡗⠀⢠⣏⠛⠆
⠀⠀⠀⠀⠀⠀⠨⠀⠐⢁⠀⢀⠑⠀⢠⠀⠀⠀⠀⠀⢰⡠⠄⠀⠠⠄⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⡏⠀⠀⡗⠦⡍
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠂⠀⢠⠄⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠈⡌⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡂⣐⡠⠄⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠈⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⠀⠀⠀⠀⠄⠠⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⡑⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣴⠧⠊⠀⠀⡀⠀⠀⠈⠁⣀⣠⡬⠁⠀⠉⣉⠀⠖
⠀⠀⠀⠀⠀⢀⣶⡄⢀⠀⠐⠁⢁⠀⣼⠀⡦⠄⠀⠰⠄⠴⢀⠀⠀⣀⠐⠀⠴⠀⠀⠀⠀⠈⠃⠀⡀⠀⠀⣱⠀⠀⠀⠈⣉⠀⠁⠀⠀⠀⢾⠖⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠍⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠈⠐⠊⠈⠀⠀⠀⠀⠀⠰⣖⠒⠀⢀⠁⠀⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠒⠚⠛⠛⠛⠛⠛⣦⣶⠛⠛⠛⠛⠛⠓⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠂⠀⠀⢀⠤⠒⠀⠓⠀⣿⣿⠀⠘⠀⠒⠤⡀⠀⠀⠐⣬⡳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠟⠁⢠⣷⣤⡛⠂⠒⠒⠦⣄⣻⣏⣠⠴⠒⠒⠒⢛⣴⢾⡄⠈⠻⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣛⣥⡴⠂⣾⠁⠐⠋⠀⠀⣀⣀⣀⣽⣯⣀⣀⣀⠀⠀⠙⠂⠈⣷⠐⢦⣬⣛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⢻⡿⠋⠀⠀⠻⠄⢀⣤⢦⠞⠉⠁⠀⡽⢧⠀⠈⠉⠷⡴⣤⡀⠠⠟⠀⠀⠙⢿⡟⢿⣆
⠀⠀⠀⠀⠀⠀⠐⠋⣰⡟⡀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠸⡅⢨⠇⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⢀⢻⣆⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡟⡜⠀⢀⣀⠀⢠⡏⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢹⡄⠀⣀⡀⠀⢣⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣿⢱⣥⠊⠁⠀⠀⢸⡇⠀⠀⠯⡍⣿⠃⠀⠀⠀⠀⠘⣿⢩⠽⠀⠀⢸⡇⠀⠀⠈⠑⣬⡞⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠘⠁⣠⠂⠀⠀⠀⡇⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⣄⠈⠃⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣄⣾⠇⠀⠀⠀⠀⢹⣀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠸⡇⠀⠀⠀⣀⡏⠀⠀⠀⠀⢸⣷⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⢰⠀⠀⠀⠀⠈⠛⣦⢀⠘⣶⠤⠤⠤⠤⣾⠇⡀⣴⠛⠁⠀⠀⠀⠀⡇⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣧⠸⡄⣰⢺⠀⠀⠀⠈⣟⠀⠈⠑⠶⠶⠊⠁⠀⣿⠁⠀⠀⠀⡷⣆⢠⠇⣼⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢳⡇⠈⢦⠀⠀⠀⣿⠀⠀⢀⣠⣄⡀⠀⠀⣿⠀⠀⠀⡴⠁⢸⡞⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢁⡀⠀⢳⠀⠀⠈⣳⠚⠉⠀⠀⠉⠓⣚⠁⠀⠀⡞⠀⢀⡈⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠟⢳⡄⠀⠇⠀⠀⠙⣶⣀⠀⠀⣀⣶⠋⠀⠀⠸⠀⢠⡟⠻⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠙⠳⠞⠋⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣠⡌⢦⠀⠀⠀⠀⠀⠀⡴⢡⣄⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠙⣄⠳⣄⡇⢸⣠⠞⣠⠋⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⠣⠜⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢦⡄⠀⠀⠀⠀⢀⡀⠀⠀⠐⢢⡀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⣿⠀⠀⠀⣠⣞⣱⡟⢚⣷⣞⠁⠀⠀⠀⠀⣾⡷⢂⣤⣤⣐⢾⣷
⠀⠀⠀⠀⠀⠀⠀⣶⡛⠛⣽⡿⣷⠄⠀⠀⠈⠉⣿⣱⡋⠙⣿⡀⠀⠀⠀⠀⠉⢳⣿⣿⣿⣿⡟⠋⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⣀⣡⡆⠀⠀⠀⢠⠤⠿⣿⣉⡾⠛⢛⡿⠀⠀⠀⠈⠶⠛⠛⠛⠛⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠂⠐⠉⠉⠉⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⠀⠀⠀⠀⠀⠲⢶⣿⡄⠤⠠⠀⠂⠀⠰⡁⠀⠘⠄⢀⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣾⣿⣷⡄⠀⠀⠀⠀⢨⣿⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢿⣿⣿⠆⠀⠀⠨⠙⠻⠿⣿⣿⠙⠷⠀⠀⢠⡿⣿⠋⠙⣿⠉⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠃⠀⠀⠀⠘⠓⠈⠉⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⢠⣾⣿⡏⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⡀⠂⠀⠀⠀⢺⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡄⠘⢿⣏⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⠻⡇⠀⠀⠀⢠⣾⣿⣿⡟⣁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠙⠷⠂⠀⠘⣷⠛⣿⠏⣽⠏⢀⡇⠀⠀⠀⢠⡿⢹⢹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⢿⠀⠀⠀⠀⠶⠒⠒⠒⡟⠒⠀⠂⠀⠀⣠⣾⣷⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣖⠀⠀⠀⣀⣀⣀⠀⢰⣿⠀⠰⠶⠀⠠⣿⢟⣿⠯⠉⣿⡷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠉⠁⠈⠟⠀⠀⠀⠙⠛⠉⠀⣾⣿⡆⠀⠀⠀⠀⠀⠈⣡⣄⣠⡿⠷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡏⠀⠀⡀⢀⣤⣄⣀⠀⠀⣀⣤⢤⠀⡀⠀⠀⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠃⢠⢤⡄⠉⠀⠈⠉⠤⠤⠄⠃⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠲⠶⠦⢤⣤⣤⠤⠶⠶⠚⢛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡷⢎⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣉⠛⢿⣍⠉⠉⠉⠛⣶⣴⠛⠉⠉⠉⣩⡿⠛⣉⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣍⡉⠉⠉⠉⠛⠻⠿⣧⠀⣄⠀⡯⣻⠀⣠⠀⣼⡿⠟⠛⠉⠉⠉⢉⣩⣿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣴⠶⣿⣿⣦⡀⠀⠀⠀⠀⠈⣀⣂⣑⣇⣸⣊⣈⣀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠶⣦⡑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣴⠟⠁⠰⣶⣽⣻⣿⡄⣴⠾⠛⠻⠭⢭⣍⡿⢯⣩⡽⠭⠟⠛⠷⣶⢠⣿⣟⣯⣶⠆⠈⠻⣦⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⡇⠀⢤⣤⣬⣿⡟⠿⣧⣼⢃⠀⠀⠀⢈⡾⠁⠈⢷⡁⠀⠀⠀⡘⣧⣼⠿⢻⣿⣥⣤⡤⠀⢸⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⣇⠀⠀⢙⣿⡍⠛⠀⢈⠋⢈⠇⠀⠀⠺⣄⠀⠀⣠⠗⠀⠀⢸⡅⠙⡁⠀⠛⢩⣿⡋⠀⠀⣸⡟⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣿⣦⡻⢷⣴⡿⠋⠀⢐⠟⠁⠘⠁⠀⠀⠀⠘⢟⠂⠐⡻⠃⠀⠀⠀⠈⠃⠈⢻⡂⠀⠙⢿⣦⡾⢟⣴⣿⣤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠛⠻⠿⠦⠉⠀⠀⣰⢋⠄⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠠⡙⣆⠀⠀⠉⠴⠿⠟⠛⠉⢻⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢰⢣⡟⠀⢤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡤⠀⢻⡜⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⡄⠀⠀⠀⠀⠀⠀⠀⠿⣾⠁⠀⠀⠸⣯⣉⢹⡇⡄⠀⠀⠀⠀⢠⢰⡏⣉⣽⠇⠀⠀⠈⣷⠿⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿
⠀⠀⠀⠀⠀⢰⣿⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⡆
⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡰⡄⠀⠀⣠⢠⡟⠀⠀⠀⠀⠀⠀⢻⡄⣀⠀⠀⢠⢆⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡹⠄⢰⡇⢿⣀⣀⡀⠀⠀⢀⣀⣀⣿⢸⡆⠠⢏⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡆⢻⣄⣠⣤⣤⣤⣀⠀⠀⠀⠀⠸⣷⣾⡿⠥⠻⣖⣢⡀⠀⠀⢀⣔⣲⠟⠮⢿⣷⣾⠇⠀⠀⠀⠀⣀⣤⣤⣤⣄⣠⡟⢰⣿
⠀⠀⠀⠀⠀⠀⠹⣷⡈⢋⣥⡴⢦⣬⡙⢷⡄⠀⠀⠀⢻⣯⡶⠤⠤⠸⠃⠙⠤⠤⠃⠈⠇⠤⠤⢶⣽⡟⠀⠀⠀⢠⡾⢋⣥⡴⢦⣬⡙⢁⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⠃⠀⠀⠈⢻⡈⣿⡀⠀⠀⠸⣏⢳⣶⠶⠀⠀⠀⢠⡄⠀⠀⠀⠶⣶⡞⣸⠇⠀⠀⢀⣿⢡⡟⠁⠀⠀⠘⣷⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢸⢃⣿⠃⠀⠀⠀⠹⣮⡢⠤⠒⣣⡴⠋⠙⢦⣜⠒⠤⢖⣵⠏⠀⠀⠀⠈⣿⡘⡇⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⣾⠏⠀⠀⠀⠀⠀⢨⡿⡿⠿⠋⠀⠀⠀⠈⠙⠿⢿⣿⡅⠀⠀⠀⠀⠀⠹⣷⣌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠁⠀⠀⠀⠀⠀⠀⢸⡄⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢷⣰⡄⢰⡆⢀⣆⡾⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠙⠞⠛⢷⣸⣇⡼⠛⠳⠋⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠙⢷⣄⠀⠀⣠⠴⠒⠒⠀⠀⠈⠻⠟⠁⠀⠐⠒⠒⠦⣄⠀⠀⣠⡾⠋⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⠳⡾⢁⣴⠾⠿⢶⣄⠀⠀⠀⠀⣠⡶⠿⠷⣦⡈⢷⠾⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⢸⣧⠀⠀⢀⣿⠀⡇⢸⠀⣿⡀⠀⠀⣼⡇⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣆⠐⣿⠁⠀⡇⢸⠀⠈⣿⠆⣰⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⡇⢸⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡇⢸⡾⠋

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

⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠄⠀⠀⢠⠀⠀⠀⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠐⢀⣀⣀⡀⣸⣀⢀⣀⣐⠀⠀⠀⠉⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢫⡀⢀⡀⠤⠐⠂⢉⣀⡀⠀⠀⢸⠀⠀⢀⣀⡈⠑⠒⠠⠄⠀⠀⣨⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⢀⠤⠻⠁⠀⠀⠰⠿⣿⠿⠿⠆⠀⠸⠀⣼⣯⠽⣷⡾⠷⣦⠀⠀⠹⠣⢄⠀⠀⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⡠⠀⠀⠀⠀⠐⡀⠀⣾⠀⠀⣀⣀⣾⠀⢰⠀⢾⠿⠀⣿⣮⣶⡿⠀⡐⠁⢀⣀⠀⢠⡀⢀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⣠⣤⣤⣄⡄⠀⠹⣄⠘⠐⢿⡟⠛⠁⠀⢸⠀⠀⠀⠐⠛⠈⠙⢀⡔⠀⠀⣾⣉⠁⠀⠙⢢⡀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣊⠀⢀⠔⠀⠀⣿⣿⣿⣿⣇⠀⠀⠈⣄⣀⡤⠶⠒⠚⠉⢹⠁⠘⠒⠂⢤⣄⣀⡞⠀⠈⢩⣼⣿⣷⡀⢰⡀⠀⢀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⢰⠏⡈⢳⠮⣄⡀⠀⠼⢟⢿⣿⡯⠀⣀⡴⠛⢧⠀⠈⡶⠇⠀⢸⠀⠈⠸⠁⠆⢠⠟⠳⣄⡀⣿⡿⠿⣽⠛⠋⠀⣀⡬⢶⠋⠈⢷
⠀⠀⠀⠀⠀⢠⠏⠄⢁⠎⠀⠀⠙⠲⢤⡈⠉⠉⢁⡞⢁⠔⢀⠈⢧⣠⡥⠶⠶⠛⠒⠶⠦⣤⣠⠏⠀⢀⡂⠙⣆⠀⠀⠈⣡⠴⠊⠁⠀⠈⢆⠀⠈⢧
⠀⠀⠀⠀⠀⡾⠀⠀⡌⢀⣤⡶⠷⠶⣄⠉⠓⣶⣏⠐⢁⠖⢡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠠⠋⠀⢈⣷⡖⠋⠀⣠⣶⣶⣤⣄⠈⡄⢠⠸⡆
⠀⠀⠀⠀⢸⠃⠈⠀⠀⢻⣿⣷⢰⠋⢹⡆⢰⠇⠈⠙⠲⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠚⠉⠀⢷⡀⠸⣻⠻⠏⠛⠿⡄⠰⠐⡀⢛
⠀⠀⠀⠀⠸⠠⠄⠂⠀⢸⣧⠉⢀⠙⡻⠃⡟⢀⠐⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠂⠨⠈⡇⠐⢶⣾⣷⣾⡞⠀⠀⠂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⡆⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠉⠏⠉⠉⠉⠉⠁⠉⠉⠙⠉⠉⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⣉⠈⠉⠉⠉⠉⠉⠈⠈⠏⠁⢸⡇
⠀⠀⠀⠀⢷⠀⠀⡀⠀⠀⠀⣠⣤⣔⠀⠀⡆⠲⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⢀⠀⠆⡟⠀⠀⣀⣠⢤⣦⡀⠀⡁⠀⢸⠇
⠀⠀⠀⠀⢸⡄⠁⢡⠀⠉⣿⣏⣻⣿⠀⡀⢻⡀⠀⢀⣨⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢯⣀⠁⠀⣼⠁⠀⣸⣿⣿⠏⠉⠁⢠⠀⡂⣼
⠀⠀⠀⠀⠈⣧⠀⠀⠆⠀⠉⢀⡟⠛⠃⠀⣀⣷⡞⠉⠀⠆⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠈⠙⣶⣇⡀⠀⠘⠟⢿⣧⠖⠀⠇⠈⢠⡏
⠀⠀⠀⠀⠀⠸⡄⠀⠘⡄⠀⠈⢀⣠⠴⠋⠁⠀⠙⣦⠢⣀⠀⠀⡽⠳⠦⣤⣤⣀⣠⣤⡤⠶⠻⡅⠐⣆⡠⢀⡞⠁⠀⠉⠲⢤⣀⠀⠀⠀⡜⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠹⡀⠀⠐⡤⠖⠋⠀⣠⠞⢛⣿⡆⠈⠓⢦⣀⡼⠁⢠⠀⡄⠀⢸⠀⠒⠠⠀⠀⠹⣄⣠⠞⠋⣠⣶⣿⣷⢦⠀⠈⠑⠀⣔⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠻⡎⠀⠘⢆⠀⠰⣿⠟⢻⣿⡗⠃⠀⠀⠟⠓⠶⣤⣁⡀⠀⢸⠀⣈⣀⡁⠵⠖⠛⣇⠀⠀⡟⢿⣿⣫⢸⡦⠀⢀⠖⠀⠉⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⢀⠻⠄⣰⣿⡆⠀⢠⠞⢀⡀⠲⣶⣤⡄⠉⢹⠁⠉⣁⠀⣤⣄⡀⠘⢆⠀⠻⣿⣿⠀⠾⢃⡴⠋⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⠀⠀⠀⠀⢠⠋⠀⠀⠀⢠⡟⠀⠶⠄⠸⠀⢸⣿⣣⣿⢿⣿⡄⠀⢂⠀⠀⠀⠀⠀⠋⠄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠐⠤⣀⠀⠀⠀⠀⠀⣸⠁⢀⠀⠀⢰⠀⠸⣿⡅⣿⣷⣽⠃⠀⠀⢡⡠⠔⠉⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡰⠃⠀⠢⠤⣀⡰⠿⠀⠀⠀⠀⢸⠀⠀⠈⠛⠀⠈⣁⠠⠤⠒⠉⠳⣄⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠐⠀⢤⠉⠑⠒⠒⠂⢺⠒⠒⠒⠂⠉⠁⠄⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⠤⠀⠀⠸⠀⠀⠠⠤⠴⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐

⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⡰⢻⣿⣿⣶⡀⠀⠀⠀⠀⠀⣠⣤⠴⠧⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢰
⠀⠀⠀⢣⠀⣽⣿⠉⠁⠀⠀⠀⠀⢸⣿⡀⠀⠿⢯⣷⣤⠀⠀⠀⠀⠀⢹⣿⣷⣤⣤⡠⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣮⣄
⠀⠀⠀⢈⡼⠋⠘⢇⠀⠀⠀⠀⠀⠈⠻⠻⠧⠀⣈⣿⠏⠀⠀⠀⠀⠀⠈⢿⡿⠿⢿⠃⠀⠀⠀⠀⠀⠈⠹⣟⣿⡿⡿⣿⣿⠂
⠀⠀⠠⠞⠇⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠈⠻⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⠇⠜⠜⠀⠀⠀⠀⠀⠀⠀⠀⠨⠽⠄⡠⠃⠃
⠀⢰⣶⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⡤⡆⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⢲⠂⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢰⡆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢺⣶⣴
⠀⠀⠸⡯⣦⠉⢹⣿⠷⠀⠀⠀⠀⠠⣐⠋⣁⣀⡉⢙⡢⠀⠀⠀⠀⠀⠴⣿⣿⣤⣌⡁⠀⠀⠀⠀⠀⠀⠈⠟⠒⣿⣿⣿
⠀⠀⠈⢀⡅⢶⡏⠃⠆⠀⠀⠀⠀⠀⠉⠻⢿⣿⡿⠯⠁⠀⠂⠀⠀⠀⠀⣫⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠹
⠀⠀⠀⡜⠃⠞⢆⠱⡄⠀⠀⠀⠀⠀⢀⡆⠀⠁⠀⢠⠈⠀⠀⠀⠀⠀⠀⠁⠙⠛⠻⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲
⠀⠀⠠⠤⠈⠀⠈⣀⡀⠀⠀⠀⠀⢠⢤⠄⠀⢀⠀⠈⢀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢤⣤⠀⠀⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠤⠄
⠀⠀⠀⣰⣒⠒⣿⣦⠀⠀⠀⠀⠀⠀⠀⠁⢀⣄⠀⠋⠀⠀⠀⠀⠀⠀⣀⠀⣠⣿⣷⢀⣐⠀⠀⠀⠀⠈⡿⣿⣿⣆⠀⡀
⠀⠀⠠⠄⠠⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠀⠀⠀⠀⠀⠀⠀⠼⢻⣿⣷⣾⣿⣏⠸⠀⠀⠀⠀⣀⣉⣿⣿⣿⣧⣼⠖⠂
⠀⠀⠀⠀⢠⠏⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⡀⠀⠀⠀⠀⠀⠀⢐⠋⠋⠙⠻⢿⢦⠀⠀⠀⠀⠀⢨⣨⢟⡙⠛⣿⣿⡇
⠀⠀⠀⣀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢀⢠⠄⠀⠀⠀⠈⠉
⠀⠀⠘⠒⠃⠀⠈⠁⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃

⠀⠀⠀⠀⢀⣤⠶⣤⡀⠀⠀⢀⣤⠦⣄⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⠀⠀⠀⠀⠀⢀⡴⠆⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⣀⣀⣀⣤⣤⠤
⠀⠀⠀⠀⢸⡇⠀⠈⢷⡀⢠⡟⠁⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠃⠀⠀⠘⡇⣼⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠚⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠚⠶⠰⠲⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠒⠆⠛⠷⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠲⠆⠲⠐⠆⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣴⡄⢠⣴⡄
⠀⠀⠀⠀⣠⠶⠿⣏⠉⠉⠉⠉⠙⠛⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠘⣷⠏⠘⡗⣠⣄
⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠋⢻⡄
⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⢸⡇⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⢷⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⣸⠇
⠀⠀⠀⠐⠶⣤⣄⣀⣀⣀⣀⣹⣶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⡼⠁⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⡟⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣿⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⣤⣴⣿⠁
⠀⠀⠀⠀⠀⠠⠂⣠⢠⣄⠄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⢠⡀⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠋⢉⣁⣀⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡴⢻⣆⡴⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⣿⣿
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⡿⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⣠⣾⠟⠁⣿⡀
⠀⠀⠀⣀⣀⣀⣿⣦⠀⠀⢀⣼⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣾⡟⠁⠀⠀⠛⠃
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠻⣶
⠀⠀⠀⢴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⣷⠆
⠀⠀⠀⠀⠀⠀⠤⠠⠸⠔⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠶⠀⠆⠆⠶⠰⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠐⠆⠦⠶⠒⠆⠆⠲
⠀⠀⠀⠀⣀⣀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⢹⡆⢠⡞⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡀⢀⣠⣦⠀⢀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⣿⡾⠁⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠁⠘⠷⠛⠁⠘⠷⠛⠁⠈⠧⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢹⣇⡴⠖⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣿⠷⠶⢾⡷⠶⠶
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢘⣿⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⡴⢿⡀⣠⡴⢷⡀⣠⡴⢿⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠙⠓⠞⠋⠀⠀⠀⠀⠀⠐⠟⠉⠀⠈⠛⠉⠀⠈⠟⠉⠀⠈⠓⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⡀⣤⣤⡤⠄⢤⠠⣤⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⢤⡠⣤⢤⠠⡄⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠠⡤⠀⡤⢤⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⣀⣀⣠⣀⠀⠀⢀⣀⣀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢼⡁⠀⡇⠀⡇⠀⠙⠂⢸⠀⢸⠀⢠⡾⠋⠀⠀⠉⠙⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⣀⠔⡆
⠀⠀⠀⡞⠁⠀⠧⠀⡇⠀⡆⠀⣿⠠⠘⠀⠉⡀⠐⠀⠀⠀⠆⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠄⠀⠐⠒⠉⠁⠀⠀⠋⡖⠒⢲⡏⠂⡁
⠀⠀⠈⠁⠀⠀⠀⠊⠁⠀⠁⠐⠋⠐⠋⠀⠀⠐⠈⠉⠀⠀⠁⠀⠈⠓⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠄⠀⡠⠂⠐⠐⠒⠠⡀⠀⠀⣄⣧⣤⢼⡇⡄
⠀⠀⣀⠂⠀⢐⠂⠋⠘⡒⠀⠀⢰⠈⠉⡆⢀⡄⠀⠐⠖⠀⠠⠲⡂⠀⣀⠀⠀⣠⠄⢠⡤⢠⣄⢠⣶⠠⣴⠌⢰⡀⢠⠄⡀⢰⡀⠀⡄⠉⠀⠀⡀⠈⢁⡄
⠀⢸⡌⠀⣀⢠⠤⠀⠠⡄⣀⠀⠀⠓⠂⠀⠈⠁⠀⠀⠀⢐⡀⠐⠀⠀⠋⠁⠀⠙⠂⠈⠓⠉⠉⠋⠃⠀⢋⠀⠀⠌⠀⠀⠀⠐⠀⢀⡀⠀⠄⢀
⠀⠀⠇⠀⠰⢀⣁⠑⡈⡀⠇⠐⠀⠅⠈⠀⠀⠄⠀⠁⠤⠄⠀⠠⠂⠰⠄⠀⠀⡁⠀⠀⠀⠘⠈⠀⠈⠐⠉⣤⠀⠈⠳⠲⢞⠀⣠⡄⠁⠀⠸⡏⠀⢄⠀⢰
⠀⠀⠄⠀⠀⠅⣙⢒⡋⠰⠀⠀⠀⠁⠠⠀⠠⠀⢀⠀⢰⡆⠀⠿⠀⠀⢠⡠⠆⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠉⠠⠅⠈⠀⢈⡠⡲⠀⠀⠇⠀⠈⠉⠙
⠀⠀⠀⠠⢀⠀⠹⠈⠅⠤⠀⠂⠠⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠉⠀⠉⠀⠀⣀⣀⣈⣀⣂⣀⣁⡀⠀⠀⢄⠀⠠⠀⠠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠂⢴⠀⠠⠄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⣿⡁⢀⣀⡀⠀⠀⠐⠉⠉⣉⣉⣉⣉⢉⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠡⠈⠀⠀⠀⣀
⠀⠀⠀⡀⠓⠌⠀⠀⠀⠓⠒⠁⠰⠀⠈⠈⠀⠀⢰⡏⠀⠉⠁⣿⠈⠉⠻⠋⠉⢳⡄⠿⡄⢠⣭⣀⢀⣥⡿⠀⢀⣀⠀⢀⠀⠀⠈⠀⠠⠁⠈⠀⠀⢀⡄⣿
⠀⠀⠠⠄⠀⢸⠀⠈⠀⠙⠀⠀⢀⡀⠈⠀⠀⣀⠘⢳⡀⠀⠀⣿⢸⣷⠀⢠⣿⠀⡇⠀⡇⢸⣿⣿⠸⡇⠀⠐⠛⠛⠒⠛⠛⠀⠀⠀⠀⠀⠂⠀⠁⠀⣤⠄
⠀⢀⡀⠀⠆⠈⢀⠀⢀⠈⠀⠀⠐⢰⠢⠤⠄⡞⠀⠈⢪⣀⠀⣿⢸⡿⠀⠸⠟⢰⡇⣀⡇⢸⣿⣿⠠⡇⠠⢠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⢀
⠀⢨⣧⠀⡀⠃⢚⠀⢐⠒⢂⡀⢀⣼⠀⣼⢀⡇⠀⠀⠀⣽⡀⢿⡤⠴⠻⣤⡴⠎⠘⡯⠤⠤⠿⣷⣴⡇⠀⠁⢈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⢀⣈⡁⣤⡁⠀⢀⡀⡀⠁⠀⠉⠉⠉⠀⠀⠀⣀⣀⣬⣥⣄⣀⣀⣀⣋⣀⣀⣀⡀⢀⣀⢀⣀⣛⣀⣀⡀⢀⣀⣈⣀⡈⠈⠈⢀⠀⠀⠀⡀⠀⠀⠅⣀⠬
⠀⠐⠈⠡⡤⣤⡄⢹⡅⠠⡤⠀⠀⡀⠠⠂⢻⡙⣿⠏⣩⠀⣤⡀⠉⢿⣿⠈⠀⣶⡌⠹⢋⣉⠻⢡⣤⣾⡇⣿⢸⡁⠶⢾⡇⠠⠀⠀⠀⠠⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣶⡀⠁⠄⠀⠈⢷⡘⢀⡏⢸⣿⡇⠀⢸⣿⠀⠀⠛⣡⣴⡸⢿⣶⠘⢛⣿⡇⣿⢸⡇⡶⠺⡇⠀⠄⠀⠀⠄⢸⡄⠲⢖⢾⡄
⠀⠠⠰⠻⢻⠁⠄⠈⠈⠀⠀⠈⠁⠀⠀⠀⠀⢸⡇⢸⣿⠘⠿⠃⡄⢸⣿⠀⠀⣷⡌⣿⢻⡆⠙⢠⣿⣿⠀⣏⣸⡇⣷⠀⡠⢆⡤⡄⠀⠀⠀⢆⢹⣘⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⠀⢠⣤⣤⡄⠀⠀⣧⣸⡟⠶⠴⠞⠳⣤⣥⣾⣤⣿⣧⣽⣦⣤⡾⠦⠤⠼⣧⠭⠽⣧⣿⠈⢋⠉⠉⠁⢠⡎⠀⠘⢛⠉⠉
⠀⠀⠈⢶⠃⠀⠀⠀⠄⠠⠀⠀⠸⠿⠿⠇⠀⠀⠀⡉⣁⣀⡤⣠⠀⠉⠀⠈⠉⠙⠀⠀⢸⠇⠀⢴⠒⠠⣀⠀⠀⠁⠀⠀⠘⣆⡇⠀⠉⠀⠀⠀⠀⠀⢢⠶⠂
⠀⠀⢀⡀⠀⠀⠀⠈⠉⠉⠁⢀⣀⢀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠠⡀⠀⠀⠂⠀⢀⠀⢀⠀⡢⣀⡠⠋⢀⣠⣄⣀⣀⠀⠙⠁⠐⣿⢿⣿⡏⠃⠀⢸⡆
⠀⠀⠈⡀⣤⠀⠀⠄⠠⠤⠄⠠⣬⠐⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠰⠂⠁⠆⠀⢠⡈⡾⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠏⡁⠀⠀⠘⠇
⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡀⠁⠇⣆⠀⠀⠈⠀⢀⣾⠀⠀⡀⠄⠀⠀⠈⠢⠤⠁⠀⠀⣀⠀⡀⠀⢀⡀⠀⠥⠁⠀⠀⢀⠄⠀⠐⠑⠃⢈⠃⠁⠀⠀⢤⡀
⠀⠀⠀⠀⠁⠀⡶⠂⠀⠐⠻⠇⠀⡇⡴⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⢶⡄⣒⡆⠀⠀⠠⠀⠰⡆⠀⠊⠂⠍⠀⠀⡀⣀⡩⠂⠘⠈
⠀⠀⠀⠀⡀⠀⠀⠅⠀⠨⠄⠀⠀⡆⡆⠀⠈⠀⠀⠀⠀⠀⣀⠈⠀⡁⢀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠈⠀⠂⠀⡀⠀⠨⠅⠀⢶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠸⠀⠇⠀⠀⠀⠨⠐⡀⠀⡄⠃⡧⠀⠀⡡⠁⠀⠀⠂⠈⠐⠀⠄⠂⠀⠀⠀⠂⠀⡀⠂⢄⣐⠄⠀⠀⠀⠀⠀⠈⡅⠀⠉⠁⣠⣤⠀⠀⢠⣤⡀⠀⢹
⠀⠀⢰⠀⢀⡀⡀⠘⠓⠛⢀⣀⡅⠀⠃⠀⠀⡮⠀⠀⠀⠂⠀⠀⠂⠀⠀⠈⢀⢀⠀⠀⢀⠀⠀⡇⠀⣀⢀⡁⠀⣀⠀⡉⢀⣀⡈⡉⠉⠀⠀⠈⣉⣉⣀
⠀⠀⢀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠩⠐⠂⠀⠀⠃⠀⢠⢀⠈⠀⠘⠁⣀⠀⢀⡘⠀⠀⠠⠀⠀⠀⡇⢸⣏⡸⠇⠀⡷⠀⠿⠌⠉⠁⡇⠹⣶⠁⡏⢉⠉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⣤⠶⠞⠛⠉⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠞⠉⠀⠀⠴⠚⠋⠉⠉⠩⠯⢽⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠚⢛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢻⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⣀⣀⠀⠀⠀⠀⠋⠈⡁⠀⠀⠀⠀⠀⠀⢂⠀⠳⡄⠀⠀⠀⢳⠈⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠛⠛⠋⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⠀⠀⠀⠸⡀⠀⠹⣆⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⠀⠀⠀⠀⡇⠀⠀⢻⡄⠀⠀⢀⠇
⠀⠀⠀⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡀⠀⠀⠈⣷⠀⠀⢸
⠀⠀⠸⠋⠀⠀⠤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣿⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⢛⣶⣤⡀⠀⠀⠤⣀⡀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣸⠀⠀⣿⠀⠀⠀⢸⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢹⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢾⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢛⠛⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡴⠀⠀⠀⠀⣸⠏⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠂⠀⠀⡼⠀⠀⡼⠁⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⣰⠇⠀⠀⣼⠁⠀⣼⠃⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡿⠀⠀⢰⡏⠀⣸⠇⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⣿⠀⠀⡿⠀⠀⠀⠀⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⠀⠀⢰⡇⠀⢸⠇⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⠀⠀⢸⠇⠀⠸⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠘⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘


⣴⣦⣦⣦⣤⣴⣴⣦⣤⣤⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣤⣤⣴⣦⣴⣦⣤⣤⣴⣶⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠀⠀⡴⠀⠀⠠⠁
⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⢠⡆⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⢿⡆⠠⣀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠆
⠀⢀⠀⠠⠀⠔⣤⢸⣇⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠈⠃⢞⠀⠁⠀⠀⢰⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠉⠐⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠄⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠆⠀⠀⠰
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠆⠐⠀⠀⠀⠀⠂⠀⠀⢤⠤⠄⠀⠀⠘
⠀⠊⢳⡎⠁⢈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣴⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⢰⣠⡀⠀⠂⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⣸⠇⠀⠀⠀⠐⠂⠒⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠂
⠀⠀⠀⡀⠄⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠈⢺⡆
⠀⠀⠀⠀⠀⠀⣫⡔⣮⡂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠸⠃⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⢜
⠀⠀⠀⠀⠂⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠈
⠀⠀⠀⠀⠀⠀⠓⣊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠆⠠⠂
⠀⠄⠀⢈⠃⠀⠀⠡⠀⠀⢄⠀⢀⣀⡀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠖⢆⠀⠀⠀⠀⠀⣀⣄⣤⡀⢀
⠀⠀⠠⠈⠀⢀⠀⠀⠀⠀⣀⠀⠿⠽⣉⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠸⢆⠇⠀⠀⠀⠀⠀⠹⠘⢈⡁
⠀⠀⠀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠑⠚⠒⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠰⠀⠀⠀⠠⠀⠂⠰⠤⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡊⠇⠀⠀⢠⠂⠀⠀⠀⠚⡰⠄⠣⠔⠂⠜⣂⠀⢀
⡄⠀⠀⠀⠀⠀⡆⠀⠈⡀⢀⠀⠀⠀⠀⠀⠀⠒⠀⢣⠀⠀⠀⠀⠀⣐⡶⠃⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⢉
⠁⠉⠀⠀⠀⠀⠁⡀⣀⠀⠉⠀⠰⠀⠀⠁⠀⠈⠄⠀⠀⠤⢀⠀⠀⠈⠋⠰⠟⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠃⠀⠀⠀⠀⠁⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⢀⡘⠀⢀⣀⣄⣠⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⡀⠀⡀⠀⢠⡘
⠀⠀⠀⠀⢠⠛⢣⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠥⠀⠀⡅⠀⠀⡏⠈⠀⢃⡀⠀⠀⠀⠀⢀⣀⠶⠀⠀⢀⠁⠈⡁⠂⠁
⠀⠀⠀⠀⠛⠒⠚⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠀⠀⠀⠀⠀⠘⠾⠀⠀⠀⠀⠀⠚⠁⠀⠠⠤⠉⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⢄⠖⠓⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢈⠆⠒⡃⡐⠁⠀⠈⠉⠃⠀⠀⠀⠀⠰⣉⢭⣓⡃⠀⠀⠀⠀⠀⠛⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⡀⠀⠀⠀⣀⡋⠉⠂
⠀⠀⠘⣡⣄⣵⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠉⠀⠀⠀⠀⠀⠐⠀⣰⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⡞⠁⠡⠀⠀⡠⠀⠙⡆
⠀⠀⠀⠀⢈⠅⢀⠀⠀⠀⠈⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠘⡀⠀⠂⠀⠀⠀⠀⠰⠀⠀⠶⠧⠀⠀⠀⠀⠠⠶⢆⠀⠰
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠘⠂⠀⠀⠐⠀⢀⠀⠆⢀⠀⠀⠀⠀⡀⠏
⠢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⡀⠀⠀⠀⡀⢠⠀⢀⠀⠀⡀⠀⠀⠠
⠀⠃⢸⣗⡄⠘⠂⡄⠀⠀⠀⠐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⢻⠃⠀⠁
⣀⣀⣀⣀⣀⣐⣂⣉⣀⣀⣀⣐⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣄⡀
⠀⣠⠞⣡⠀⢠⣶⣄⣉⢳⡄⠀⠀⠀⢠⡞⠉⠀⢀⠀⢀⠈⠳⣄⠀⠀⠀⢀⡾⠫⣄⠀⠀⠀⣬⠙⣦⠀⠀⠀⠀⡴⠋⣡⣤⣀⣤⣌⠙⢦⡀
⢠⡏⠀⠘⡇⡾⢸⡧⠞⠀⢻⠀⠀⠀⡟⠀⠹⡴⠛⢦⠏⢳⠀⢹⡆⠀⠀⣾⠀⠀⢹⣄⠀⣸⡃⠀⠸⡇⠀⠀⣸⠁⠸⡇⠀⣿⠀⢨⠇⠈⣧
⠘⡇⠀⠀⢻⠇⢸⠇⠀⠀⣸⠀⠀⠀⣧⠀⢲⣠⠶⣄⡴⣆⠀⢸⠇⠀⠀⢿⠀⠈⢹⠏⠉⢻⡉⠀⢠⡇⠀⠀⢻⡀⠀⠀⠀⣿⠀⠀⠀⢀⡟
⠀⠹⣆⡀⠈⢀⡟⠀⢀⣴⠋⠀⠀⠀⠘⢦⡀⠉⠀⠉⠀⠈⣠⠟⠀⠀⠀⠘⢧⡐⠋⠀⠀⠈⠓⣠⠞⠀⠀⠀⠈⢷⣄⠀⠀⠛⠀⠀⣠⡾⠁
⠀⠀⠈⠙⠲⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠛⠉

⠀⠀⣠⠶⠛⠛⠛⠓⢦⣄⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⠛⠶⣄⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠲⣤⡀⠀⠀⠀⠀⢀⡴⠞⣛⣉⣛⠳⢦⡀
⠀⡾⠁⠘⢧⣤⣤⡾⠁⠘⣧⠀⠀⠀⣼⠋⠀⢲⠶⠶⣶⠃⠈⢷⠀⠀⠀⣰⠏⢠⠴⠒⢛⡶⢦⡈⢳⡀⠀⠀⢠⡟⠀⣾⠁⠀⠈⢻⠀⠹⡄
⢸⡇⠀⢰⠋⠀⠀⠹⡆⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⢠⠴⢤⠘⠦⠼⠃⠈⡇⠀⠀⣾⠀⣠⠿⡄⠀⢠⡟⠀⠀⣷
⠈⢧⠀⠘⣦⣀⣀⡼⠃⢀⡾⠀⠀⠀⢷⡀⢀⣸⣤⣤⣿⡀⠀⣼⠁⠀⠀⢻⡄⠸⢦⣼⣁⣠⡴⠃⣸⠃⠀⠀⠸⣆⠘⠛⠁⢠⡏⠀⠀⣠⠇
⠀⠈⠳⣤⣀⣉⣉⣀⡴⠞⠁⠀⠀⠀⠈⠻⢮⣁⣀⣀⣀⣥⠞⠁⠀⠀⠀⠀⠙⢦⣄⣈⣉⣀⣠⠾⠃⠀⠀⠀⠀⠙⠦⣄⣀⣈⣓⣾⡶⠋
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⢠⡞⣏⣀⣀⣄⠈⠙⢷⡀⠀⠀⠀⢀⡶⠋⠁⠀⠀⠀⠉⠳⣄⠀⠀⠀⢀⡴⢻⣁⣀⣠⡀⠉⠳⣄⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣈⡛⣦
⢠⡏⠀⡏⠈⡇⢸⡧⡄⠀⢻⠀⠀⠀⡿⠀⣀⣰⠋⠉⣧⣀⠀⢹⡄⠀⠀⣾⠁⢸⠀⣿⠀⢹⠀⠀⠸⡆⠀⠀⣸⠃⠀⠀⠀⢀⡼⠿⠁⠘⣇
⠸⡇⠀⡇⠀⡇⢸⡇⡿⠀⣸⠁⠀⠀⣇⠀⠬⠤⠤⠤⠬⠥⠀⢸⡇⠀⠀⣿⠀⢸⠀⣿⢀⡞⢀⠀⢠⡇⠀⠀⢻⠀⠀⢐⡶⣯⠀⠀⠀⠀⡟
⠀⠻⣄⠁⠀⣁⡼⢿⣁⣰⠏⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠘⣧⡈⠀⠉⢺⣀⠟⣃⡾⠁⠀⠀⠈⢧⡐⠋⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠈⠛⠶⠤⠴⠶⠛⠁⠀⠀⠀⠀⠀⠈⠛⠶⠦⠴⠶⠚⠁⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠶⠞⠋⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠒⠂⢓⣚⣓⡚⠂⠳⠶⠴⣄⡀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⡠⠔⠙⠉⠀⠀⠀⠀⠀⠸⡟⠉⠉⠛⠋⠛⠛⠶⢮⣌⡒⠆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠐⢉⡀⠀⠀⠀⠀⢰⣶⣦⡀⠐⣗⠀⠀⡀⣠⠀⠀⠀⢠⡟⠀⠀⠈⠚⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠛⠋⠙⢧⣀⠀⠀⠻⣟⠋⠀⠀⠩⠀⠘⠋⠀⠘⠀⠠⠶⠀⢀⡐⠀⠀⠀⠙⢧⡄
⠀⠀⠀⠀⠀⠀⣀⠟⣡⣿⠀⣠⢦⠀⠀⠹⢦⡀⢀⣠⣤⣴⣶⣾⣶⣦⣦⣂⣠⢄⠂⠀⠀⠋⢉⣷⠘⠀⠀⢀⡙⡄⠣
⠀⠀⠀⠀⣰⢶⠒⣤⡟⠁⢾⠁⠀⡀⠀⢀⡜⣯⠴⣿⡉⠀⠀⠀⢸⠄⠀⠀⠉⡿⠾⣷⣄⠀⠀⠀⢀⣠⡶⠟⠷⣄⢠
⠀⠀⠀⠀⣀⠇⠘⠾⢦⡄⠀⠀⠀⢀⣔⡵⠋⠀⠀⠘⣷⡀⠀⢀⣸⣄⡀⠀⡼⠁⠀⠈⠛⣿⡤⣴⠍⠉⢀⣀⠈⢻⡘⠳
⠀⠀⠀⢀⡍⠀⠀⠀⠀⠈⠳⢦⣠⣶⡾⠷⣄⡀⠀⣤⠶⣚⠩⠥⠔⠇⠭⣝⠢⣀⠀⣠⠶⠋⠱⡽⡄⣸⣿⣿⣷⠈⢿⡀⠇
⠀⠀⠀⢴⠁⠀⠆⠈⠅⠀⠀⢀⣾⡟⠀⠀⠈⢩⡾⡡⠞⠂⠀⠀⠀⡀⠀⠈⠙⢪⡣⡁⠀⠂⠀⠹⡜⡀⠉⠷⠁⠀⠀⡧⢨⡄
⠀⠀⠀⡻⠀⠀⠀⢄⠀⠄⠀⣺⣿⠁⠀⠀⢀⡟⡸⠁⠀⠀⠠⣴⣴⣆⡀⠀⠀⠈⢻⣹⣤⠤⠤⠴⣷⡃⣀⣀⣀⣠⣴⣶⠘⠠⡄
⠀⠀⠈⡇⠀⠀⠀⠉⠀⠀⠀⠾⣿⣤⠶⠶⢸⠁⡅⠀⠀⠀⣺⡏⠁⠙⣿⠤⠀⠀⠐⡃⡗⠀⠀⡄⣼⣯⠉⠉⠀⠀⠀⠀⢈⡖⠁
⠀⠀⣄⡖⢰⣶⠶⠶⠒⠂⠛⡆⣧⠀⠄⠀⢿⡆⣆⠀⠀⠀⢺⣷⣤⣾⠟⠉⠀⠀⢰⢃⡇⡈⠐⠀⣿⡯⢀⡶⠶⠀⠀⠀⢀⢂
⠀⠀⠀⢰⠀⢒⠀⠀⡀⠀⠀⢷⢹⡀⠀⠀⢘⣿⡘⣀⠀⠀⠀⠟⠏⠈⠀⠀⠀⣠⢍⡾⠀⠛⠶⣄⣿⠇⠘⡇⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠘⡄⢸⡄⠀⣋⡄⠀⠈⠄⢷⣠⡶⠋⠈⠻⣌⡓⠄⣀⢀⠀⡀⣀⡤⢚⣵⡷⠃⡀⠀⢀⡼⣱⡀⠀⠈⠃⠀⠀⢀⣧
⠀⠀⠀⠀⢹⠈⢷⠀⠀⠀⠀⠀⣀⢿⣷⣆⠀⠀⠀⠈⢛⣶⣾⣭⣵⢩⡤⠖⠓⠋⠙⣦⡀⢠⠞⡕⠀⠈⠓⠶⣄⣀⠀⡨⠒
⠀⠀⠀⠀⠀⠡⡌⢱⣀⣀⡴⠞⠋⠈⠹⢿⣷⣤⡀⣠⡿⠀⡀⠉⠉⢰⡇⠐⠀⠄⢀⣼⣿⠣⠞⣀⠏⠁⡀⠀⢀⠋⠴⠻⠆
⠀⠀⠀⠀⠀⠠⡓⣀⠋⠁⠀⠀⣴⣤⣦⠀⠘⠿⠛⡿⠧⣤⣁⣀⣀⣘⣛⣤⣴⣾⡿⠟⡋⠀⠘⠋⢀⠐⠀⣰⢋⡤
⠀⠀⠀⠀⠀⠀⠉⠸⠧⡀⠀⠀⠛⠛⠛⠀⠀⣰⠋⠉⠓⠢⠴⠨⠍⣭⠑⠛⠙⠈⠁⠀⠘⢆⠀⠀⢀⡠⠞⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡧⣄⠀⠀⠀⠀⣰⠃⠀⢠⠆⠀⡀⠀⠀⢹⠀⠠⠀⣶⡄⠀⠀⠀⠻⠶⢉⠥⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠤⣀⡸⢧⡄⢀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠊⠉⠃⠀⠀⣀⡀⢼⣤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠗⠐⠤⣍⢙⣓⡒⠶⠶⠖⠾⠀⣀⡀⠀⡰⠴⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠘⠌⠃⠽⠆⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⢏⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠖⣀⡤⠴⠚⠃⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠋⡁⠀⠀⢀⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠻⣷⣤⠴⠚⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠐⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠒⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠐⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢠⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠟⠉⠉⠙⢛⢿⣿⣿⣿⣿⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⣿⠟⠉⠙⢿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡟⠛⠛⣿⣿⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢸⣿⣿⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣿⣿⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠍⠉⠉⠉⠡⣿⣿⣿⣿⣿⣿⣿⣿⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠸⣷⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣷⣎⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢻⣿⣿⣿⣿⡁⠀⢀⣿⣿⣿⣿⣉⣿⣿⣶⣄⢠⣤⣹⣿⣿⣿⣿⡿⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠙⣿⣿⣿⣿⣶⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠛⠿⢿⣿⣿⣷⣶⣾⡿⠿⠛⠉⢀⡀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣀⣀⠉⠉⣀⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⣤⣤⣤⣄⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣩⣿⣯⣍⠙⠛⠳⢶⣤⣀⠀⠀⣀⣤⡶⠞⠛⠉⣩⣽⣿⣭⣀⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣭⣉⠉⠉⠙⠛⠷⢦⣍⡉⠁⠀⠀⠀⠀⠙⠻⠟⠉⠀⠀⠀⠀⠈⢉⣩⡴⠾⠛⠛⠉⠉⣉⣭⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠛⢦⣀⣠⠴⠚⠉⣁⣀⣤⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣤⣀⣈⠉⠓⠦⣄⣀⡴⠟⠉⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢀⡴⠛⠱⠶⣿⣿⣯⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣽⣿⣿⠶⠎⠛⢦⡀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⡴⠋⠀⢀⡴⠛⠉⠀⠈⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠉⠛⢦⡀⠀⠙⢦⡀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠿⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢸⣿⠛⠻⣶⣦⣄⡀⠀⠀⠀⠀⢻⡄⢠⡟⠀⠀⠀⠀⢀⣠⣤⣶⠟⠻⣿⡆⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⡀⠙⢿⣷⣦⣄⡀⠀⣧⡼⠀⢀⣠⣴⣾⡿⠋⢀⣿⠀⠀⢸⣿⡀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠀⣀⣤⠀⠀⢸⣿⣄⠀⠀⠘⠳⠤⠼⣿⣷⠀⠉⠓⠾⠷⠚⠉⠀⣾⣿⠧⠤⠞⠃⠀⠀⣠⣿⡇⠀⠀⣤⣀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣾⡷⠛⣽⠃⠀⠀⣿⡏⠻⣧⡀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⢀⣼⠟⢹⣿⠀⠀⠘⣯⠛⠾⣷
⠀⠀⠀⠀⠀⠀⠁⠀⣼⠃⠀⠀⠀⣿⡇⠀⠹⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⠏⠀⢸⣿⠀⠀⠀⠘⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠈⠻⣆⠀⢹⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣸⡏⠀⣰⠟⠁⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⣿⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⢀⣿⢁⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⢀⣾⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣷⡀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡿⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢿⣾⡿⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢠⡀⠀⠀⠀⠀⢸⡟⠀⠀⠙⠛⠿⣷⣶⣶⣶⣶⣾⠿⠛⠋⠀⠀⢻⡇⠀⠀⠀⠀⢀⡄⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⠻⣦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣴⠟⣇⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠈⠿⠶⠶⣾⣷⣀⠀⠀⠀⠀⣀⣼⠟⠻⣧⣀⠀⠀⠀⠀⣀⣾⣷⠶⠾⠿⠁⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⡶⠛⠋⠁⠀⠀⠈⠙⠓⢶⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣤⣤⣤⣤⣤⣤⠿⠃

⠀⠀⢀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡀⢀⢀⠝⠀⠀⠀⠀⠐⢃⣀⠀⠀⠀⠀⠀⠀⡐⠀⠁⠀⠈⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣐⠓⠀⠀⠀⠀⠐⠔⡌⠁
⠀⠀⠀⠁⠀⠀⠠⡀⠄⠀⠀⠈⠄⠅⡀⠀⠀⠣⠉⠐⠀⠀⢀⠀⠱⠜⠀⠀⠀⠀⠀⠀⢀⣀⠂⠀⢀⣐⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⠀⠀⢀⠀⠈⠊
⠀⠀⠀⡠⠰⣤⠄⠀⠠⡼⠀⠀⠀⠉⠀⠀⠀⠀⠠⣤⠀⠀⢠⣤⠄⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠦⢶⣤⠀⠈⢠⡵⡤⡇
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡳⠀⠀⠀⠀⠀⠀⠔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠘⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠔⠀⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⢁⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢀⠀⡄⠀⠂⠀⣄⡀⠈⠀⠄⠀⠐⠒⠐⠂⣁⠀⠀⢀⠁⠓⠐⠂⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠂⠂⠀⠐⠂⠄⠠⠀⠀⠀⠀⢁⠘⠠⠶⠀⠀⠰⠶⠑⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠉⢁⠠⣤⡄⠀⢠⣥⠄⡄
⠀⠀⠀⠀⠀⠛⠁⠀⠘⠋⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠶⠀⠀⠰⠖⠀⠣⠀⠀⠀⠀⠐⠊⠀⠄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢀⠀⠀⠄⠀⠀⠀⢀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡐⡀⠀⠀⠀⠀⠐⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄


⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠛⠀⠄⢄⡀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠇⠀⠀⠀⠀⠀⠀⣄⢐⣁⠀⠀⣠⣄⢀
⠀⠐⠀⣀⡠⠀⠀⠐⠀⠀⠀⠴⠄⠀⠀⠀⠀⠁⠀⠛⠂⠀⠘⠛⠀⠁⠀⠀⠀⠀⡀⠘⣀⠀⠀⢂⣀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉
⠀⠀⢁⠘⠛⠀⠀⠙⠃⠀⠁⠀⠀⠀⠀⠀⠀⠈⡄⢀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠙⠉⣀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⢶⠂⠀⠀⠀⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣈⣁⣂⣀⣉⣙⣀⣈⣘⣀⣀⣀⣀⣀⣀⣁⣃⣁⣈⣀⣁⣀⣁⣀⣀⣀⣀⣀⣁⣑⣀⣈⣈⣀⣀⣃⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣁⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢷⡀
⠀⠀⢀⠔⠋⠁⣉⠙⢦⡀⠀⠀⠀⠀⠀⠀⡀⠀⢔⣩⣴⠃⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠙⠡⣤
⠀⠀⣘⠛⠆⣜⠐⠊⢸⣷⡀⠀⠀⠀⠀⠀⠓⠏⡄⠈⠻⡲⡆⢷⣜⡆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙
⠀⠀⠿⠖⠒⠪⣐⣶⡿⠋⠁⠀⠀⠀⠀⠀⠘⠂⢈⣾⢣⠇⣧⡔⢺⡇⠀⠀⠀⠀⠀⣮⣁⣠⡴⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢴⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⣀⠋⠣⠟⠀⠀⠀⠀⠀⠀⠈⠍⠭⠢⠰⣤⣌⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⠀⠤⠤⠀⠀⠀⠀⠀⢰⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠤⠶⠶⠦⣄⣄⣀⠀⠀⠀⠀⠀⠀⢴⡚⠱⠃⠈⠆⢓⣢
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢒⠃⠰⠀⠀⠀⠈⠧⢉⠴⠀⠐⠀⠜⠛⣣⡀⠀⠀⠀⠀⠀⡴⣁⠋⠙⠋⡉⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠗⠁⠐⢨⠿⠇⠀⠀⠀⠀⠀⠙⠻⠷⠔⠣⢜⠯⠋
⠀⠀⠀⠀⠀⠀⠐⠂⠉⠑⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀
⠀⠀⡄⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠔⣂⢍⠀⠀⠀⠀⠀⠀⠀⡈⡷⠂⠀⠐⢢⢷
⠀⠀⠹⡾⠆⠀⢀⣦⠀⡀⡀⠀⠀⠀⠀⠀⠰⠻⠒⠚⡙⢧⣀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⡕⣰⣈⣧⣜⣚
⠀⠀⠀⠀⢰⠃⠨⢁⡾⡟⠮⣓⠦⠀⠀⠀⢰⡆⠒⠀⣀⠈⠁⢀⣀⡀⡘⠃⠀⠀⠀⠀⠀⢸⡿⣌⠨⡽⢿
⠀⠀⠀⢀⡈⢁⡀⡜⠐⢓⣪⠝⠓⢢⠀⠀⠀⠛⠥⠤⠜⠛⠶⣉⣩⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⢀⠄
⠀⠀⠀⠘⠓⠀⠀⠖⠈⠉⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⠀⠀⢠⠄⡄
⠀⠀⠀⠀⠀⠀⡌⡀⢀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⣠⢤⠀⢠⢬⡜⠁⠀⠀⠀⠀⠀⢰⣫⠋⡍⢳⣄
⠀⠀⠀⠀⠀⢀⣅⢼⢣⣠⡇⠀⠀⠀⠀⠀⠀⠘⢄⣹⢶⡊⣲⢾⣅⠜⠀⠀⠀⠀⠀⣸⣯⡁⢉⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠟⢻⡋⠄⠀⠀⠀⠀⠀⠀⠈⠋⠁⣐⣿⣟⡀⠉⠋⠀⠀⠀⠀⠀⣼⣿⡉⢇⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠫⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡄⣼⡗⠀⠀⠀⠀⠀⠀⠀⠈⠈⣙⠟⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⢄⣠⣂⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⢀⣴⢖⣒⣢⣄⡀⠀⠀⠀⠀⠀⠀⠪⠑⢦⣄⣐⡀⠀⠀⠀⠀⠀⠀⣠⡆⢀⡆⠀⠀⣀⠀⠀⠀⢀⣤⡶⠀⠀⠐⢦⣤⡀
⠀⠀⠀⠀⡰⣋⠱⢿⢏⡍⢻⣷⡀⠀⠀⠀⠀⠷⡀⣤⡶⢿⣯⠁⠀⠀⠀⠀⣀⡾⣛⠁⠈⠻⢶⣴⡌⢷⠀⠀⠹⡾⠆⣤⠁⡄⣼⢿⠃
⠀⠀⢠⠀⠀⣉⣤⣾⡿⣿⡆⢿⡇⠀⠀⢀⣾⠇⠀⠔⣻⢸⡿⠀⠀⠀⠀⢸⣿⡇⣁⠀⠀⠀⠙⢧⣽⢸⠀⠀⠀⣁⠀⣷⠀⣆⢹⡜
⠀⠀⠀⠑⡿⣿⣿⣍⠴⠟⣹⠸⠃⠀⠀⢸⣿⠀⡤⠼⠅⢸⡆⠀⠀⠀⠀⢸⡿⣶⣯⠃⠀⠀⠠⣾⡿⠞⠀⠀⠀⠘⣾⠛⡀⡟⣾⡇
⠀⠀⠀⠀⢰⣿⣤⣤⠴⠞⠃⠀⠀⠀⠀⠈⡿⢀⡇⠀⠀⣽⣇⡤⣄⠀⠀⠀⠻⠬⠛⠻⡦⠀⣾⠛⠁⠀⠀⠀⠀⠈⡈⢀⣿⡀⢁⠇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⡠⠀⣀⢐⣄
⠀⠀⠀⠀⠲⡤⠒⢾⠗⠒⡴⠂⠀⠀⠀⠈⣹⡇⣰⠀⠀⢤⡤⠤⣀⠀⠀⠀⠀⠐⠗⠶⠶⢸⢻⡶⠛⠁⠀⠀⠀⢸⡟⠁⡏⠘⣿
⠀⠀⠀⠀⡐⠁⠀⣼⠀⠠⠉⠄⠀⠀⢨⣟⣡⠶⠃⠀⢀⣀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⢈⠣⣴⡇⢠⢟⡀
⠀⠀⠀⢰⣤⠶⡧⣿⡤⣧⡶⢾⠄⠀⠀⠻⣥⣶⡀⡶⣿⣿⣧⡀⠀⠀⠀⠀⢀⡍⠁⣶⣦⡶⠾⢯⡻⠀⠀⠀⠀⢑⣿⣿⣏⣿⣟⡊
⠀⠀⠀⠀⠉⠉⠀⣿⢖⠈⠉⠁⠀⠀⠀⠀⣿⠋⣷⠀⠈⣧⠀⢱⠀⠀⠀⠀⠈⠁⢴⠉⢿⠀⠀⠋⠀⠀⠀⠀⠀⠉⠊⢸⣿⡃⠱⠉
⠀⠀⠀⠀⠀⠀⠛⠈⠈⠓⠀⠀⠀⠀⠀⠚⠃⠚⠃⠀⠚⠋⠐⠛⠀⠀⠀⠀⠀⠀⠘⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⢀⡀⠀⢀⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠴⠾⠓⡢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣄⠋⣩⣹⡇⠀⠀⢸⢷⣦⣘⠛⣃⣴⣿⠆
⠀⠀⠀⠀⢀⣀⣐⠛⠋⠀⢀⡀⠀⠀⠀⢀⣀⡤⠶⠷⠟⢛⣩⣵⡇⠀⠀⠀⢀⡤⠀⡀⠀⠀⢉⢠⠇⠀⠀⠀⠀⠙⠻⠿⡇⠻⠟⠋
⠀⠀⠀⠀⢸⣟⣶⣴⣤⡀⢰⢌⡆⠀⠰⠋⠉⠙⠧⣴⣾⣿⣿⡿⣍⠂⠀⠀⠀⢅⣼⣿⠶⡇⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣟⡧
⠀⠀⠀⢠⡏⠀⠙⣿⠋⠻⢿⡀⠀⠀⠀⠀⠀⠀⢀⣘⡯⠉⡇⠳⠈⠀⠀⠀⢰⠋⠹⣇⠐⠶⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠈⠈⠀⠀⠒⠃⠀⠀⠚⠁⠀⠀⠀⠀⠈⠭⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠙⠂⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣷⡀⢾⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⡯⡍⣻⣿⡟⣻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⣿⡿⠋⢠⠾⢿⣿⣽⣿⡽⠾⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⣡⠶⠞⢋⣁⡀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⢰⠁⠀⢸⣿⣿⣿⠀⠀⡎⠀⢀⠀⠀⠐⠀⠈⠶⣴⣿⣿⣿⠀⠀⠰⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⢸⣿⣿⣿⠓⢤⠷⠄⠀⠁⢰⢀⠀⠀⠀⠂⠀⢹⣿⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣿⣿⣏⡀⠀⠀⠡⠚⠈⠐⠂⠀⠀⣠⣶⣤⣾⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢿⡿⡿⠀⢄⢀⠀⠀⠀⠀⠀⠀⢬⣿⣿⠏⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣾⣃⣹⡿⠇⠀⣀⣀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠤⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⣠⣴⣾⠿⠛⠋⣁⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣠⣾⡿⠋⠀⣠⡶⠟⠛⠉⠉⠉⠙⠛⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡏⠀⢀⡼⠋⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢿⣆⠀⠘⠀⠀⡠⠒⠋⠉⠉⠉⠛⠻⣶⣄⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠙⠢⠀⠀⠀⠀⠴⠶⢶⣶⣤⣄⠀⠀⠙⣷⣄⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠤⣤⣄⡙⠻⣿⣦⠀⠈⢿⡆⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣧⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠕⠀⠀⠀⠀⠀⢀⡆⠀⠀⡀⠘⡇⠀⢻⠀⠀⣿⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢰⠇⠀⠀⠀⠈⠀⠀⣿⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠐⠂⠀⠀⡔⠀⠀⡄⢀⡟⠀⠀⢀⣴⠟⠀⣰⠀⠀⠀⠀⢸⡿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠾⠄⠀⠀⠀⠈⠀⢀⣴⡿⠋⢀⡼⠃⠀⠀⠀⢠⡿⠁⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⣠⣿⠋⢀⣴⠟⠁⠀⠀⠀⣠⡟⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣰⡿⠋⠀⠀⠀⢀⡼⠋⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣸⡿⠁⠀⠀⠠⠞⠁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠃⠀⠀⠀⢀⣠⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠀⠀⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⠀⠀⠀⣀⡄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⢀⠈⢻⠀⢠⣾⣿⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⡄⠀⣿⣿⡗⢾⠀⠀⡀⢾⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣾⡇⠀⣿⣿⡇⠀⢀⣼⡇⠘⢿⣿⡀⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠁⠈⠉⠉⠁⠀⠉⠉⠁⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣴⣢⣤⣤⣤⣄⡀⠀⠀⠀⢰⣄⣀⣀⡦⠀⠀⠀⠀⠀⠀⠀⢲⣴⣶⣴⠂⠀⠀⠀⠀⠀⠀⠀⣠⢂⣾
⠀⠀⠀⠀⡀⠀⠀⠀⣸⡿⠛⡿⣿⣿⣿⣿⣿⣆⠀⠀⢹⣿⣿⣿⡁⠀⠀⠀⠤⠀⠀⠀⠻⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠿⢻⠟
⠀⠀⠀⠀⢀⠀⠀⠀⠋⢀⡜⢁⠇⢹⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⠀⠀⠠⣴⣾⣿⣦⣴⣶⣿⣶⣄
⠀⠀⠀⠄⠀⠐⠂⠀⠀⠀⡠⡀⠀⠚⣽⠋⢩⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡁⠀⠀⢀⡠⠏⠀⠀⠀⠀⢸⡆⣿⢩⡟⣿⠀⠀⠀⠀⠀⠰⠿⢠⡏⣻⣿⣿⠿⠁⠀⠀⠠⢞⡽⢁⡤⠟⣻⡿⠉⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⠺⣇⠀⠀⠀⠀⠀⣀⣀⠀⢀⣴⡾⠻⣶⡄⠀⠀⣰⣶⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⡿⠿⠍⡂
⠀⠀⠀⠀⠛⠀⠀⠀⠀⣿⣿⣁⡈⢿⣷⣿⣿⣶⡄⠀⣸⡟⢻⡇⢹⡿⢃⡀⠸⣿⠀⢸⣿⣿⣿⣄⣀⣀⣄⣀⣤⣴⠀⠀⠘⢹⣿⣷⣄⣀⣀⣀⡤
⠀⠀⠀⠀⠒⠀⠀⠀⣰⣿⣿⡟⠹⠆⠀⣴⠞⠛⣿⠀⢸⡆⢸⡇⠀⠀⠧⣹⢀⣿⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⠉⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠠⡀⠀⠁⠀⠀⠀⢸⣦⠿⣿⠀⢰⣧⠀⠀⣴⠏⠀⠈⢿⣌⡻⠦⠤⢖⣣⡾⠃⠀⠸⣇⢸⡟⠛⠛⠻⣿⢿⡄⠀⠀⠀⢠⠞⢻⠟⠛⠻⢿⣿⠁
⠀⠰⠁⠀⢀⡀⠀⠀⠀⠀⠀⠘⠃⠀⠛⠓⠚⠉⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠿⠸⠁⠀⠀⠀⠛⠀⢸⠀⠀⠀⡏⠛
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠑⠀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠈⠂
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⡀⣀⣤⣀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠘⣇⢿⣿⠏⠀⢀⡀⠀⠀⢠⣾⡟⠁⠀⣀⣤⣤⣄⠀⠀⢠⡟⢻⠆⠀⠀⠀⣿⣿⡇⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠆⠀⠀⠀⢀⡤⠤⡌⢱⣿⣞⠛⠉⠁⠀⠀⣿⣿⣷⣄⣰⣿⣿⣿⣿⡆⠀⠸⢿⢶⣿⣿⣶⣾⣿⣿⠃⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠄⠀⠀⠀⣏⢰⣃⡼⠘⣿⣿⡷⢶⡄⠀⠀⠹⣿⣿⣿⣿⡿⢿⢻⣿⠃⠀⠀⠀⣸⣿⡿⠿⢿⡿⣏⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠏⠈
⠀⠀⠠⠀⠀⠀⠀⠀⠙⠦⢤⠤⠖⢹⣏⣀⠺⠇⠀⠀⠀⢈⠛⢛⠁⠀⠀⠈⠃⠀⠀⠀⠀⢻⡘⣧⠀⠀⢷⡘⠦⠀⠀⠀⠛⠼⠁⠀⠼⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⡤⠠⠄⠠⠀⠀⠀⠀⠀⠑⠀⠀⡐⠤⠄⠀⠀⠀⠀⠀⠀⡀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠐⠐⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣿⠃⠀⠀⠀⠀⠹⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠛⠉⠁⠀⢿⡀⠀⠀⠀⠀⢀⡿⠀⠈⠉⠛⠛⠿⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⣠⠴⠚⠛⣿⠃⠈⠳⣤⠇⠸⣤⠞⠁⠘⣿⠛⠓⠦⣄⡀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠐⢋⣴⢶⣄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣠⡶⣦⡙⠂⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⣾⢯⠶⣍⠀⣷⠀⢀⣠⣴⣇⣸⣦⣄⡀⠀⣾⠀⣩⠶⡽⣷⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠄⠀⠀⠸⣿⠘⠀⡸⢃⡸⠆⠀⠀⠀⣿⣿⠀⠀⠀⠰⢇⡘⢇⠀⠃⣿⠇⠀⠀⠤⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣠⣴⡖⠀⠙⠆⠀⠀⠋⠀⠀⠀⠀⣴⡟⢻⣦⠀⠀⠀⠀⠙⠀⠀⠰⠋⠀⢲⣦⣄⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠸⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢹⡇⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠇⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⣶⡀⠈⠀⠀⠀⣶⡅⢠⣤⣄⣀⠀⠈⡇⢸⠀⠀⣀⣠⣤⡄⢸⣶⠀⠀⠀⠁⢀⣶⡈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⡞⠀⠻⢦⡀⠀⠀⣿⠘⢦⣈⠯⠝⠣⠀⠀⠀⠀⠞⠫⠽⣁⡴⠃⣿⠀⠀⢀⡤⠟⠀⢳⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⡀⠀⠈⠀⠀⢻⣇⣼⠁⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠈⣷⣸⡟⠀⠀⠁⠀⢀⣀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠟⠉⠉⠁⠀⠀⠀⠀⠻⣷⡀⢀⠀⡰⠁⠀⠀⠈⢆⠀⡀⢀⣾⠟⠀⠀⠀⠀⠈⡉⠉⠻⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⣼⡇⠀⠀⠀⠀⡀⠈⢁⡾⣇⡀⠀⠀⠀⠀⢀⣸⢷⡈⠁⢀⠀⠀⠀⠀⢸⣧⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⡀⠀⠀⠀⡇⠀⢸⣿⠁⢿⣉⠉⠉⣩⡿⠈⣿⡇⠀⢸⠀⠀⠀⢀⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠈⠻⢦⡤⠞⠁⣄⠘⢿⣄⣀⣈⠉⠉⣁⣀⣠⡿⠃⣤⠈⠳⢤⡴⠟⠁⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠢⢤⣀⣤⠾⠛⢧⡀⠈⢿⣍⡀⢀⣹⡿⠁⢀⡼⠛⠷⣤⣀⡤⠔⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⡀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠈⠉⠉⠁⠠⠒⠉⠀⠀⠀⠀⠀⠀⢀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡘⢄⡀⠐⢤⣄⣀⣓⢦⣄⡀⠀⠀⢀⣠⡴⣊⣀⣠⠤⠂⢀⡠⢃⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣉⠳⠦⣄⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣠⠴⠞⣋⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣉⠛⠶⣤⡀⠀⠀⢀⣤⠶⠛⣉⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⠙⠇⠸⠋⣠⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠚⠁


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

⠀⢠⠴⣄⠀⣤⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠠
⠀⠀⠀⠸⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡎⠀⠀⠀⠀⠀⠀⠀⠀⢊⣀⣥⡤⢤
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠸⠥⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣢⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁⠈


⠀⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠐⠳⡖⠳⡖⢳⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠦⡄⠀⠀⠀⠀⠀⠀⠚⢶⠚⣶⠛⡆
⠀⣠⠖⠲⣄⡼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠸⡏⡷⠀⠀⠀⠀⠀⢸⡅⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⡇
⠀⠻⣄⣀⠟⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢁⠀⠀⠀⠀⠀⣉⣥⣤⣤⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠑⠑⠂


⠀⠀⠀⠈⠒⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠀⠀⠀⠀⠀⠀⠀⠴⠒⢦⣀⣠⠖⠲⠄⠀⠀⠀⠀⠀⠀⢠⡀⢠⠂
⠀⠀⠀⢤⡞⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠐⢤⠀⠀⠀⠀⠀⠀⣠⢤⣀⠉⢁⣤⢤⡀⠀⠀⠀⠀⠀⠀⢺⡷⢿⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠂

⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠉⠉⠀⠉⠉⠉⠁⠀⠀⠀⠐⠀⠁⠁⠁⠉⠉⠉⠀⠀⠀⠀⠀⠉⠋⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⣿⡀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠈⠻⠶⣦⡶⠾⠋⠀⠀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⣠⡶⠟⠛⢿⣶⠶⠶⣾⠟⠛⠶⣤⡀⠀⠀⠀⠈⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠻⣦⣾⠃⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⡾⠛⠋⠛⠻⣦⡀⠀⠀⠀⢠⡿⠀⢀⣾⠀⠀⠀⠙⠁⠀⠀⢸⡆⠀⠸⣧⠀⠀⠀⠀⣠⡾⠛⠋⠛⠻⣦⡀
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢸⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠈⠻⠶⣶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢶⡶⠾⠋
⠀⠀⣠⡶⠟⠛⢿⡶⠶⠶⣾⠟⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⢿⣶⠶⠶⣾⠟⠛⠷⣦⡀
⠀⣼⠏⠀⠀⠀⠀⢻⣦⣾⠋⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣦⡀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢻⣆⣼⠋⠀⠀⠀⠈⢿⡄
⢸⡏⠀⢀⣶⠀⠀⠀⠹⠃⠀⠀⢰⣆⠀⠈⣷⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢰⡏⠀⢀⣶⠀⠀⠀⠹⠃⠀⠀⢰⣆⠀⠈⣷
⣾⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻
⠿⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⠇⠀⠀⠈⠻⣦⣀⣀⣀⣠⡾⠋⠀⠀⠀⠿⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣌⣉⣉⣉⣉⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠈⠻⣏⠉⢉⣿⠋⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠹⣧⡾⠁⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣼⠇⠀⠀⠀⠘⠁⠀⠀⠀⢿⡄⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠠



⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⢀⡤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠄⠀⢠⡀⠠⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⣷⠀⠀⠀⢠⡆⠀⠚⢻⠹⠀⠀⠀⠀⠀⠀⣾⡿⠋⠉⣙⣿⣧
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⢶⣄⡀⠀⠀⠀⠀⠀⣠⡴⢦⡀⣿⠇⠀⠀⠀⢰⣿⣷⣶⣷⣤⢤⠀⠀⠀⠀⣿⣧⠠⡤⢁⣾⣿
⠀⠀⠀⠀⠈⠁⠙⠻⠟⠟⠀⠉⠀⠀⠀⠀⠹⣄⣿⣀⣀⡿⠏⠀⠀⠀⠀⠉⢍⠉⠉⣙⠕⠃⠀⠀⠀⠀⠘⢿⣧⣤⣼⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⣴⣿⣿⡏⠐⢠⠀⠀⠀⠀⠀⠰⢾⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠰⣾⣦⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠐⠄
⠀⠀⠀⠀⢸⠋⠛⠉⣿⣶⣶⡄⠀⠀⠀⢠⣾⠉⠁⠈⠈⠟⠓⠀⠀⠀⠀⡄⠀⣻⣿⡐⠀⠀⠀⠀⠀⠈⡚⠀⠀⣀⠀⠐⡃
⠀⠀⠀⠀⠘⢧⣤⣷⣤⠟⠋⠀⠀⠀⠀⠀⢉⠀⠐⠓⠢⠤⠆⠀⠀⠀⠀⠓⠄⠉⠿⠁⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡿⠿⠄
⠀⠀⠀⠀⠀⠈⠙⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠈⠀⠛⠺⠿⠖⠋

⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣤⣄⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⣴⣍⣁⣨⣭⠲⠀⠀⠀⠀⣸⠏⣝⠛⠉⣋⠉⢷⠀⠀⠀⠀⢸⠋⠉⡏⠉⠉⠁⠀⠀⠀⠀⣠⠿⠄⠀⠤⠧⡄
⠀⠀⠀⠀⠸⢳⣿⣗⣺⣿⡲⡁⠀⠀⠀⢹⢾⠿⡅⠈⡹⢿⣾⠀⠀⠀⠀⣀⠀⢠⣷⢀⣀⠀⠀⠀⠀⠈⢿⣇⣀⣀⡀⣸⡍
⠀⠀⠀⠀⠀⠋⠟⠹⢝⡟⠅⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠉⡉⠈⠉⠈⡉⠁⠀⠀⠀⠀⠘⢽⠋⠉⠙⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠇⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠟⠀⠐⠋⠻⠵⢢⣀⢕⣀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢑⣀⣠⢻⠀⠀⠀⠀⠘⠈⠈⠠⠤⠠⡐⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣢⠓⡽⣶⠂⡀⠀⠀⠠⡞⢉⣀⠠⠂⠀⢀⠀⣷⠀⡏⣠⡇
⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⣷⢄⣄⣢⠇⠀⢀⠁⠀⠀⠀⠀⠉⡀⠀⠀⢹⡄⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠢⠽⡿⠀⠀⠀⠈⠀⡄⣶⡶⠈⠁⠀⢸⡀⠻⢃⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠄⠀⠀⢸⡦⣄⣀⢸⢘⠀⣿⠃⡅⢠⣄⠸⣇⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡎⠀⠀⠀⠀⣧⡀⠉⠚⠚⢲⠃⠠⡄⠀⢻⣆⣫⣴⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢣⡀⠀⠐⢀⣿⣿⣦⣤⣠⣏⣀⣴⣷⠀⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣴⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⣀⣠⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⠗⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⠀⠙⠋⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠴⠆⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⠿⣿⣿⣿⣿⣿⡷⠀⠀⠠⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡤⡔⠁⠀⠀⠭⣿⡍⣥⣆⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡾⡀⢢⠀⠖⠂⢸⡆⠀⠉⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣾⡇⢻⡌⠀⠀⠀⡈⣰⡃⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠊⣧⣷⠈⠁⠀⠀⠀⠐⠊⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡚⣋⢱⢼⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡶⠔⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⣰⡯⠥⠤⠤⠰⠆⠤⠤⠬⢽⣔⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣩⡄⠂⠉⠀⠀⢀⡀⠀⢸⡇⠀⠀⢀⣀⠀⠉⠐⢠⣍⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⠖⠊⠀⠹⡄⠀⠀⠊⢸⣿⠀⠠⠀⠀⠒⠿⡧⠀⠀⢠⠏⠀⠑⠦⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢫⠜⠁⢠⠀⠀⣠⠈⠄⠀⠠⠋⠹⠀⢠⡄⠀⡠⣴⠀⠀⠠⢁⣤⡀⠀⠀⠈⢢⡙⢄
⠀⠀⠀⠀⠀⠀⡜⣠⠊⠀⠀⠀⢰⡦⠀⠀⠀⢦⠀⠀⠀⠀⣘⣃⠀⠀⠀⠀⡴⠀⠌⣏⣷⡦⠀⠀⠀⠑⣄⢣
⠀⠀⠀⠀⠀⡼⡠⠉⠓⠦⡀⠀⠀⠃⠀⠀⠀⣈⢶⠂⠩⠀⠀⠀⣀⡉⠐⠾⣁⠀⠀⠋⠙⠣⠄⢀⠴⠚⠉⠄⢡
⠀⠀⠀⠀⡰⢵⠃⠀⠀⢀⠀⠁⠠⣄⡀⢠⠊⣀⠀⠣⡈⠁⠠⠄⢈⢀⠜⢀⠀⠑⡄⢀⣠⠄⠈⠀⠀⠀⠀⠘⣆⢃
⠀⠀⠀⠀⣏⡌⠀⣰⣤⣾⣷⠀⠀⠀⠹⠓⠤⡀⠠⠂⢱⣦⣾⠷⠤⣮⠐⢄⢀⠄⠚⠇⠀⠀⢺⣥⣦⡵⠀⠀⢹⣸
⠀⠀⠀⢸⠘⡆⠀⠀⠉⠙⠿⠀⠀⠀⠃⠂⠀⠈⠑⣿⣿⣿⡏⠠⠄⠈⢳⠎⠅⠰⠒⠘⠀⠀⠀⠀⠁⠀⠀⠀⢰⠁⡇
⠀⠀⠀⢸⠰⠆⠶⠶⠀⠂⠀⠶⠶⢸⠆⠀⠀⠆⠾⣿⣿⣶⡟⠀⠀⠆⢸⡷⠰⠀⠀⠠⡇⠶⠶⠀⠀⠀⠶⠶⠰⠆⡇
⠀⠀⠀⢸⡀⡇⠀⠐⣿⣤⠀⠀⠀⠀⡄⠀⠀⢐⡰⠿⣿⣿⣧⠀⠂⢀⣼⢄⡂⠀⠀⢠⠀⠀⢜⡓⢰⣷⠀⠀⠸⠀⡇
⠀⠀⠀⠀⣷⢇⠀⠠⢼⡿⣷⡗⠀⠀⢰⡤⠒⠁⠐⢄⡻⠛⢿⡗⠺⢇⠠⠂⠈⠒⢠⡆⠀⠀⠀⠛⡾⣇⠀⠀⡸⣿
⠀⠀⠀⠀⢸⣌⡄⠈⠉⠀⠉⡁⠐⠊⠁⠘⢄⠀⠀⡔⠁⠁⠐⠂⠀⠈⢢⠐⠀⡤⠃⠈⠑⠦⣤⡀⠀⠀⠀⢠⠷⡇
⠀⠀⠀⠀⠀⢋⠑⣀⡤⠖⠁⠀⠰⣾⣆⠀⠀⢉⠾⠄⣀⠀⠀⠀⠈⣁⠠⢷⡀⠀⠐⣶⣀⣀⠈⠙⠳⢦⣠⠏⡼⠁
⠀⠀⠀⠀⠀⠀⢣⠙⢄⠀⠀⠀⣠⠿⡄⠀⠀⠞⠀⠀⠀⠀⢩⡍⠀⠀⠀⠀⠻⠀⠈⢻⠿⡿⠀⠀⠀⡠⠋⡝
⠀⠀⠀⠀⠀⠀⠀⠑⣮⠣⡀⠀⠁⠀⠈⢀⠂⣰⡖⢹⣤⣀⠘⠃⣀⣴⣷⣦⡀⠐⡀⠈⠀⠁⠀⢀⢜⣥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢎⡢⢄⠀⣰⠃⠀⠛⠿⣆⣽⠟⠀⠀⠉⢻⠛⠻⠏⠀⠘⣆⠀⡠⠖⡱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣙⠃⠄⣀⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⠠⠘⣋⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠴⢾⣱⡖⠒⠒⠰⠆⠒⠂⢒⣨⠥⠆⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠛⠛⠿⣿⣷⣦⣀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣤⡶⠶⠿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠁⠀⠀⠀⡀⠀⠈⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢿⠀⠀⠀⠈⠉⠙⠛⢶⣄⠀⠙⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠈⢷⣄⣀⠀⢀⣀⣴⡿⠁⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣉⠙⠛⠛⠋⢁⣠⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢰⣿⡇⠀⢀⣾⠏⠀⠀⣠⣴⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⣿⡇⠀⣼⠏⠀⠀⠸⠉⠀⠀⠉⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⣿⣿⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠙⢿⣿⣷⣤⣄⣀⣀⣤⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⠙⠛⠿⢿⣿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⡀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⢹⣿⢻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⢰⡿⠻⣦⢀⣴⠟⣷⡄⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⠻⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠿⠿⠷⢶⠄
⠀⠈⠛⠀⢻⣿⡇⠐⠟⠀⠀⠀⠀⠀⠀⠀⢈⣿⡶⢾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⣻⡇⢠⡾⠿⣶
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⡛⠋⠀⣘⣷⣶⡿
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠟⠛⠛⠻⠷⣦⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁


⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⡶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣶⢶⡶⣶⡶⣦
⠀⠀⠀⣾⡏⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣿⢷⡄⠀⠀⠀⠀⠀⠀⣠⣍⣉⡛⠛⣋⣉⣥⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⣿
⠀⠀⣴⢿⣧⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⣿⣸⡏⠀⠀⠀⠀⠀⠀⠈⢹⡿⠁⠀⠹⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⣿
⠀⠀⠿⣶⡿⣾⡇⣠⡤⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠁⣿⣤⣄
⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁

⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣤⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣠⡄
⠀⠀⠀⡀⠈⢉⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢿⣾⠋⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣦⡿⣦⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⢸⡟
⠀⠀⠈⢻⣶⡟⠁⠘⠗⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⡿⠛⣷⡆⠀⠀⠀⠀⠀⠉⢉⣤⣉⣤⣈⣡⣌⠁⠀⠀⠀⠀⠀⠀⠐⠾⢿⡿⢿⡿⠿⠂
⠀⠀⣴⠟⠙⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣼⠟⠿⠛⠁⠀⠀⠀⠀⠀⠶⠾⠛⠿⠋⠻⠟⠻⠆⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⡀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣰⡄
⠀⠀⠀⠀⠀⢠⡞⣲⡀⠀⠀⠀⢀⠀⠄⠀⠊⠈⠃⠁⠎⢱⠀⠀⢱⠁⠀⠁⠀⠁⠀⠀⠐⠀⠂⠀⠀⠰⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠁⠀⠋⣠
⠀⠀⠀⠀⠀⠁⠀⠀⣣⠀⡆⠐⠐⠆⠠⡆⡀⠠⠀⠄⠀⠈⠀⠠⠈⠀⠀⠠⡀⠀⠀⠀⠀⠃⠸⠄⡰⠶⠀⠘⠅⠴⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠇⢃⠈⠁⠀⠀⡄⠀⡀⢃⠀⠠⠁⠁⠌⠙⠕⠀⠠⠀⢈⠉⠀⠄⠀⠀⠀⠀⠀⠁⠰⠈⡄⠒⠂⡀⠀⠀⠀⠀⢀⠂⠀⡀⠀⠀⢌⠀⠁⠘⢁⡀
⠀⠀⠀⠀⠀⠀⣀⡂⢀⡀⠐⡄⠠⠁⡠⢄⠀⠀⢠⠴⠉⠃⠀⠀⠀⠀⠤⠀⠼⢠⠈⠀⠀⠀⠀⠀⠀⠀⡋⡶⠁⢤⠰⠆⠀⠀⠰⠀⠀⠀⠀⠀⠂⡀⠀⠠⡠
⠀⠀⠀⠀⠈⠀⢨⣞⠡⠠⠀⠭⠗⠜⠀⠀⠂⠀⠀⢠⠀⠀⠀⠂⠀⠀⠀⠳⠤⠊⠀⠐⠀⠀⠀⠒⠀⠀⣤⡇⠀⠀⡠⡠⠁⠀⠠⠀⠄⠈⠂⡀⠀⠈⢔⠆
⠀⠀⠀⠀⠀⢀⠈⠁⠀⠰⠈⠤⠀⠤⠀⢠⠤⢠⢠⡤⣄⠀⠀⡠⢀⠀⠀⠀⡔⠀⠀⢄⠀⠀⠀⠀⠠⠀⢁⡀⠀⠐⠠⠁⠀⢠⠀⠀⠀⠀⠀⡀⠐⠀⠈
⠀⠀⠀⠀⡠⠀⣍⠏⠀⠀⠀⠀⡀⠀⠀⢸⠋⠉⠉⠉⠹⠀⠐⠁⠁⠀⢀⢀⠁⠀⠀⡄⠀⠆⠀⢰⠈⡀⠈⠀⠄⢈⠄⠐⠀⠸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣀⡆⠀⠐⢆⡦⠀⠀⠙⠉⠉⢉⠀⠀⢠⠄⠈⠁⡁⢠⡀⠂⠸⠁⠀⠐⡀⢢⠀⢀⠀⡘⠀⠀⠀⠀⢀⠂⠄⠀⠀⠈⠀⠐⠀⠠⠆⣤⠖⠁
⠀⠀⠀⢀⠕⢫⠐⡄⠀⡁⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠐⠀⠛⠓⠂⠈⠚⠀⠉⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠒⠀⠀⠁⠤⠀⣻⣀⡄⠠⠄
⠀⠀⠀⠀⠀⠈⠆⠼⠂⠠⠀⠠⠀⠂⢼⠂⡄⠀⠘⠀⢀⠀⠄⠀⠠⠄⢜⠠⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢉⠄⡜⠀⡌⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢐⣛⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠀⠀⠠⢏⠄⡀⠀⠰⠅⠻⣘⠕⢮⢀⡐⠀⢐⡅⠀⠀⠀⠀⠀⠀⠀⡟⢀⠁⠀⠀⢃⡀⢐⠀⢀⢶⣷⠊⠀⠆
⠀⠀⠀⠀⠐⠚⠃⠐⠀⠂⠐⠀⠀⠀⡀⠁⣀⣀⡀⠀⠀⣴⠀⠀⠅⠀⠈⢶⡁⢣⡈⡈⠀⠦⠀⠀⠀⠀⠀⡀⠀⣋⡊⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠠⠀⢀⠄
⠀⠀⠀⠂⡀⠠⠀⠡⠁⠞⡆⠀⠀⢈⠀⠀⠁⠁⠀⠊⠀⠘⠶⠰⡆⠀⣀⠄⠱⣂⠱⡀⠐⠀⠀⢀⡠⠂⠀⠀⠀⠄⠀⠀⡀⠀⠙⠂⡁⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠂⠀⠈⠃⠐⠀⠀⠀⠀⠀⠒⣀⠄⢴⠀⠈⣹⡄⠂⡃⠀⠀⠀⠀⣀⢀⠀⠐⠃⣾⢿⡀⠅⠀⠄⠀⠠⠀⠀⠈
⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠀⢠⠀⡃⠐⠀⠠⠀⠀⠀⠈⠛⡀⣀⡼⠁⠀⠀⠀⢤⠑⡀⠀⠈⠀⡈⠉⠀⣦⡶⢹⡎⠀⠘⡀⢀⠂⠒⡆
⠀⠀⠀⠀⢀⣄⡀⢀⢀⠨⡶⠀⢀⠀⠀⠀⢲⠀⠃⡀⠀⢐⠂⡀⠀⠀⠀⠈⠀⠠⡖⢲⠀⠄⠐⡂⠁⠀⠀⡈⢠⠀⠀⠁⠀⣸⡗⠀⠀⠀⠉⠅⢀⠁⠀⢄
⠀⠀⠀⠀⠀⠀⡸⠣⡀⠀⠉⠀⠀⠠⠐⢀⢴⠀⠀⠄⠀⠠⡀⠁⠀⡄⠀⠀⢀⡀⠑⠉⠀⠀⠀⡥⢘⠀⠀⣂⣀⡇⠀⠀⠃⡞⡏⠀⠒⠂⠀⠁⠸
⠀⠀⠀⠓⠀⠀⠀⠲⡇⠀⠀⠀⠀⠐⠀⠹⠿⡦⠤⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⡀⠚⠈⠀⢀⣠⡄⠧⠄⠒⠒⠻⢇⢤⡐⣦⣀⣀⠄⡠⠆
⠀⠀⠀⠀⠈⠀⠀⠅⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠁⠀⠁⠀⢌⠀⠆⠐⠒⠐⠂⠒⠈⠒⠃⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠽⠤⢄⠈⠉⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣟⣯⡙⠛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⠟⠻⠿⣿⣿⣧⢹⠏⢫⡹⡇⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⢰⣿⣿⣿⣿⣿⣿⣎⣧⢀⡘⢙⣯⠀⠀⢸⡃⡄⠀⢀⠀⠘⠺⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⣿⣟⣿⣿⣿⠿⡿⣿⣿⣿⣿⣴⣿⣤⣦⣾⣱⣤⣀⣮⣴⠂⠀⣻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⣡⡦⢘⠻⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢟⡻⣿⣫⣩⡷⠎⠀⠀⠀⢹⣿⣟⠻⠭⠉⠙⠛⠛⠿⢿⣿⣿⣌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠈⠙⠻⠻⣿⣽⣷⢦⣄⠁⠀⠻⣟⠻⢶⣦⣀⠀⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠘⠻⠷⢶⣦⣌⠐⠈⢢⡌⢠⣿⣷⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⡂⠀⠀⣼⣿⠻⣦⣤⡝⣦⡀⠐⣻⣾⡟⠉⠦⣀⠀⠀⠀⠀⠐⠲⣤⣀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡛⠉⡘⣧⠀⠀⠙⢷⣄⣸⣿⣟⣿⡻⠊⣝⡟⠃⠀⠀⠈⠂⠠⣁⠀⠀⠀⠀⠈⠉⠠⡘⣿
⠀⠀⠀⠀⠀⠀⠀⣤⠃⢁⠀⢀⡘⢳⢠⡄⠀⠈⠉⠉⠉⠘⠁⠀⣹⣆⠀⠐⢄⠐⠀⠀⢙⢷⣄⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠸⡻⢷⡈⠶⣤⢤⡛⢶⠀⠀⣾⣿⣿⣦⣀⣾⠿⠻⣿⠀⠀⠈⢳⡀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⢀⡼⡿⠊⠙⠀⠈⠂⢳⣟⢠⣠⢸⣿⣿⣿⠗⢾⣧⡀⠘⣆⠁⠀⠀⠘⠦⠀⠀⢿⡿⣿⡿⠆⠀⠀⠞
⠀⠀⠀⠀⠀⢻⣋⡟⠀⠀⠀⣢⣤⣤⣜⠀⡁⠈⠀⢉⢀⡄⡉⠛⠿⠆⢈⡳⣀⠀⠀⠀⠀⠀⢸⣷⢹⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠙⠢⣤⣀⣸⣿⣿⣿⣿⣷⠀⠀⢀⣠⣾⣿⣿⣶⣶⣿⣿⣿⣏⢷⡀⢀⢀⡌⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⢯⡻⣿⣿⣿⢿⢿⣧⣾⣶⣾⣿⣿⣿⣿⣿⣛⡛⣿⢿⡿⣧⣹⣦⢣⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠸⠶⠊⠁⠀⢻⡆⠀⠀⠨⡍⠙⣻⣿⣯⣿⣿⣯⣿⣿⢸⡇⠸⠇⢻⢾⣿⠃⡸⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡄⣿⣷⣘⠄⣐⣠⣲⡿⠟⠉⢁⠿⣹⣿⡧⢸⠇⠀⠀⠀⣸⠏⠀⠃⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣵⣿⣿⣿⣷⢫⢟⡟⡷⠂⡀⠃⠀⠻⠛⢀⡿⣠⠀⠀⢸⡏⠀⠀⠀⣠⡾⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠈⡟⠛⠑⣿⢻⡿⣷⣯⢟⡞⣽⠃⠆⠀⠀⣰⣶⡿⠁⢣⣀⣴⣿⠁⠀⢀⣾⠋
⠀⠀⠀⠘⠇⠀⠀⠀⡰⣧⠀⠀⣿⣄⠁⠹⣷⣏⣼⣱⣿⣤⠀⢠⣿⡟⡱⢃⣾⠉⣸⠇⠀⠀⣼⠃
⠀⠀⠀⠀⠐⣤⣴⣿⣿⡿⡄⠀⣿⡈⡄⠀⠙⠛⣻⣟⣿⣿⡄⠷⢟⠔⣵⣿⡏⢀⡟⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠁⣴⣿⣸⡇⠀⠀⢸⠿⣯⣿⣿⣷⠀⡊⠀⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡣⠁⠀⠀⢸⣿⣿⡙⣿⣿⡄⠁⠸⠏⠿⢷⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢋⠁⢄⡤⠠⠌⠶⣦⡹⣾⣿⣿⡀⠀⠀⠀⠸⣿⠃⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠠⣶⠠⠌⠀⠰⠀⠐⡦⢷⢿⣿⣿⡦⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⡡⠈⠴⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠄⢒⢒⠎⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⡿⠓⢫⡾⠀⠀⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣀⣺⣾⡇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠿⠿⠿⢿⡿⢿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣴⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠃⠀⠀⠉⠙⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⠀⡀⠀⠀⠀⠄⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠦⠀⠀⠉⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣯⡝⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠛⠛⠛⠻⠛⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠄⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⣴⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⢻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠊
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠙⣴⡇⢸⠘⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣞⢑⠄⣀⠀⡎⢱⠀⣀⠠⡊⣳⠀⠀⠀⠀⣿⣿⣿⣿⣻⡯⣿⣿⣿⢹⡏⣿⣿⣿⢽⣟⣿⣿⣿
⠀⠀⠀⡤⠚⠓⠸⣄⠗⡺⢗⠺⣠⠇⠚⠓⢤⠀⠀⠀⣿⣿⣿⢿⣭⣬⣯⡿⣺⢇⡸⣗⢿⣼⣥⣯⡿⣿⣿
⠀⠀⠀⡇⡇⢰⡉⠉⢁⣳⣞⠈⣉⡉⣂⡸⢸⠀⠀⠀⣿⣿⣿⢿⣿⣯⣷⣷⡾⢜⡫⣷⠶⣾⡽⢿⡇⣿⣿
⠀⠀⠀⠈⡇⡸⡄⠀⢡⠀⢠⠊⢠⠔⢤⠈⢣⠀⠀⠀⣿⣿⣿⣷⣿⣟⢿⣿⣾⣿⡟⣵⡿⣫⡻⣷⣝⣿⣿
⠀⠀⠀⠀⠇⠇⡅⢠⢲⠀⢎⠀⢈⠓⢢⠀⢀⠇⠀⠀⣿⣿⣿⣿⣿⣼⣿⣿⣟⣿⣿⣿⣿⣯⣟⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠈⠦⣈⠳⠊⣠⠞⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢷⣯⣷⢿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣤⣤⡀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄
⠀⠀⠁⠈⢷⢸⠇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⠿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱⢀⣀⡀
⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⡏⠀⣹
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠴⠛⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠁⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠊⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠉⠉⠁


⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣠⣀⣠⣄
⠀⠀⠀⠀⠸⠉⢻⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠁⢸⡇⠘⣇⣀⡀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⢹⠁⢹
⠀⢠⠞⠉⠻⣆⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⡇⢸⠇⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢸⠀⢸⡄
⠀⠘⢦⣀⡠⠏⢧⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡋⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣇⢀
⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠄⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡫⠡⠤⠄⡄⠄⠄



⠀⠀⠀⠀⠙⠒⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠲⣤⣀⣠⡴⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⡔
⠀⠀⠀⠀⣠⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠉⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣧⣼⡧⠂
⠀⠀⠀⠰⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣸⠇⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠦⠶⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠠⠒⣀⣀⡀⡤⣀⣀⢀⠀⠀⠀⠀⠀⠀⣠⢀⠀⣀⡀⠀⣀⡀⡀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢠⠄
⠈⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠂⠈⠈⠀⠉⠁⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⢰⡦⠀⠀⣤⣴⣶⣦⠀⢠⣦⡀⠀⣴⣶⣦⡄⠀⣦⣾⣿⡄⣴⣶⣦⣄⠀⣴⣶⣄⠀⣴⡄⣴⡆⢠⣦⢠⣦⣦⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣯⣭⡅⠀⣾⣿⣇⠀⣿⣿⣿⣿⠀⣿⣭⣭⣿⣿⣏⣹⡿⣿⣿⣬⣉⠀⣿⣿⣿⡇⢸⣿⢸⣿⣉⣿⡇
⠀⠀⠀⠀⠀⢸⣧⣀⡀⣿⣏⣉⣁⢀⣿⣾⣿⠀⣿⣿⣼⡿⠀⣿⣉⣉⣿⣽⡟⢻⣷⣹⣿⣙⣿⠀⣿⣿⣿⡇⢸⣿⢸⣿⠛⠛
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠙⠛⠛⠋⠘⠋⠀⠛⠃⠛⠛⠛⠁⠀⠛⠛⠛⠃⠛⠃⠘⠛⠋⠛⠛⠋⠀⠛⠃⠛⠃⠘⠛⠈⠛







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣯⣽⣶⣶⣾⣭⣽⣛⠿⣿⣿⣿⣿⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡙⣿⣷⡔⣿⣿⣿⡿⢟⣫⣭⣿⣯⣭⣭⣛⢿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⡻⢏⣶⣿⢏⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡜⣿⡇⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⢣⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠿⡿⢇⣫⣾⡟⣵⣿⣿⣿⣿⣿⠿⣛⣫⣭⣭⣽⡻⣧⢿⠿⣛⣯⣭⣽⣛⠿⣿⣿⣿⣿⣿⣧⡛⠸⣿⡟⣼⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⡿⣋⣵⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣶⣝⢿⣿⣿⣿⡏⣿⣶⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣇⢿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣯⣭⣭⣟⡻⣷⡹⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢙⡿⢿⣿⣿⣿⣿⣿⣮⢡⣿⢫⣵⣶⣶⣶⣾⣽⡻⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⢞⣷⢹⣿⣿⣜⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣭⣛⢿⣿⣿⢸⣿⣷⣭⣛⣛⢛⢛⡛⢿⣜⣿⣿⣿⣿⣿⢳⣿⠟⣫⢛⢛⣭⣭⣶⣿⣿⣸⣿⣿⣿⣧⡙⢿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡝⣿⡏⣿⣿⣿⣿⣿⣷⣤⣤⣵⣿⢸⣿⣿⣿⣿⣾⣿⣶⣴⣶⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣷⣍⣤⣤⣤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⣱⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣌⠉⠁
⠀⠀⠀⠀⠀⠰⡆⣸⠿⠿⣛⣭⣾⣿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣧⣛⡶⣶⣶⡶⣛⣼⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣷⡹⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣼⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⡿⣿⣿⣿⣿⡟⢸⣿⣿⣿⡿⢿⣿⣿⡿⢣⣾⣿⣿⣿⠿⣛⣛⣯⣭⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣧⣙⢿⣮⣙⣛⣫⣼⣷⣽⣛⣫⣾⣿⣿⢟⣵⣿⣿⡟⣩⣶⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⣭⡻⠿⣿⣿⣿⣿⣿⣿⠿⣛⣵⣿⣿⣿⣿⢸⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣭⣭⣿⣶⣿⣿⣛⢿⣿⣿⣿⡸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠟⢫⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣷⣍⡛⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣮⡻⣿⣿⣿⣿⣿⣿⠿⣫⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⠟⠉⠁⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣮⣭⣷⡼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠠⣀⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡶⢀⠀⠀⢀⡄⠀⠀⠀⠐⠀⠀⡆⠀⠀⠀⡄⠀⣀⣀⠀⠀⠂
⠀⠀⠀⠀⢠⠄⡄⠀⠁⠀⠀⠄⠀⡂⠈⠰⠃⠂⠠⠀⡌⠁⠀⠀⠀⢺⡠⠀⠁⠀⡀⠐⠂⠈⠀⠡⠄⠁
⢠⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡇⢃⠀⢀⡀⣠⡀⠀⠀⢲⣖⣾⠀⠈⣻⠖⠀⠐⠀⠀⢁⠈⢀⠁⠁⠐
⠈⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⠀⢸⠘⠐⠀⠀⠉⠀⠀⣠⠟⠛⠛⠀⣄⠉⠀⠘⠃⠀⠀⣤⢀⡄⣤
⠀⠰⠆⠀⠘⠀⠀⠀⠀⠀⠀⢀⠸⠐⠀⢦⠐⠄⠀⡀⠀⠰⠐⡀⠀⢄⠀⡄⠂⠀⠀⠀⠁⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠉⡐⠀⢴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⣌⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠰⠂⠀⠀⡦⡆⠀⠀⠀⡰⠀⠀⠅⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠆⣀⢈⠳⡷
⠀⠒⠀⣀⠉⠁⠀⠀⠀⠁⠀⠶⠀⠀⠄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠃⠠⠐⠀⠀⠀⡀⠀⠀⠀⠀⠐⠒⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘
⠀⠀⠀⠀⠀⠀⢆⠀⠀⢈⠀⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂
⠀⠀⡀⠀⠀⠀⢀⠀⠐⠀⠠⠄⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠁⠀⡀⠄⠀⠃⢘⠀⠀⠀⠀⠐⠒⠙⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠐⠟⠐⠀⢘⠃⠘⠁⠬⠁⠀⢠⠀⠀⢠⠄⠀⠀⠀⠀⠀⠐⠀⠃⠘⠐⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠘⠀⠸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠒⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠻⡏⠉⠠⣤⣤⡀⢀⣠⣤⡀⠀⣹⠛⠛⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁⣠⠀⠀⠀⢿⠀⠂⢿⠛⣿⣿⢟⡿⠁⢠⡟⠀⢰⣆⠀⠈⠙⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢁⣀⣀⡀⣿⣤⡆⠂⠸⡇⠀⠀⠀⢹⡋⠀⠀⠀⣼⠁⠀⣨⣿⣶⣄⣈⣀⡈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠻⣦⠀⠄⠨⢉⣿⣿⠻⣧⣤⠀⢹⠀⠁⠀⠘⠁⠐⠂⢠⠇⠀⣾⣿⠉⠉⣿⡟⠛⠁⢀⡴⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⣀⢈⠳⣄⠀⠘⡀⣿⠀⠀⠀⣀⣬⣶⡶⠶⠶⠶⠶⢶⣾⣤⣀⠈⠿⣶⠾⡿⠃⢁⡴⠋⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⣰⡿⠃⢀⣹⡏⣤⣤⠈⠳⢄⠀⢉⣠⡶⠟⣫⣥⣶⣶⣿⣿⢿⣿⡶⣶⣬⣝⡻⢶⣄⡀⢀⡴⠋⠀⠐⢈⣹⣿⣧⠈⢿⣆
⠀⠀⠀⠀⣰⡿⠁⢠⣿⠉⣶⠾⠂⠀⠀⢈⣵⡿⣫⣴⣿⣯⣿⣿⡟⣿⣿⣶⣿⡿⣿⣯⣿⣿⢶⣍⠻⣯⡀⠛⢷⣾⠟⢁⣤⣿⣆⠈⢿⣆
⠀⠀⠀⢠⣿⠃⠀⠘⠇⢻⡟⠂⠐⠂⢠⣾⢋⣾⣯⣿⣿⣽⣿⣼⡿⠛⠉⠉⠉⠙⠛⠿⣥⣿⡿⢿⣷⡜⢷⡄⠀⢻⣶⠛⠉⠈⠙⠃⠈⣿⡄
⠀⠀⠀⣾⡿⢦⣤⣀⡀⠘⠃⠀⠀⢠⡿⢡⣿⣉⣿⣯⡿⠋⠉⠉⠑⠾⠋⠹⣦⠀⠀⠈⠐⠝⣷⣾⣿⣿⣎⢿⡄⠀⠘⠒⢀⣀⣤⠴⠶⢻⣧
⠀⠀⢀⣿⠁⠀⠀⠈⠉⠉⠑⠒⢤⣿⢃⣿⢻⣿⣿⡿⠷⢔⣴⠄⠀⡀⠀⠃⢘⡆⠀⠀⠀⠈⢎⣿⣷⣿⣿⡌⣿⠖⠊⠉⠉⡀⣀⣀⠀⠀⣿⡀
⠀⠀⢸⣿⠀⠄⠀⢠⡶⢦⠀⠁⢸⣿⢸⣿⠿⣿⢃⠀⠀⠀⠀⠀⣀⠃⠐⣩⡝⡧⢨⡍⡀⠀⠀⢸⣿⢿⣿⣧⢻⡇⠀⢠⣟⣸⣿⣹⣷⡄⣿⡇
⠀⠀⢸⣿⠀⣤⣶⡾⠷⠿⢷⡄⢸⣿⢸⣿⣿⣿⣦⡤⠖⠀⠀⠀⣸⠀⠀⠉⢨⣃⣀⠉⠀⠀⠀⢸⣿⡿⠿⣿⢸⡇⠀⢸⣧⣤⣍⡉⢻⡇⣿⡇
⠀⠀⢸⣿⠀⣾⠿⠟⠃⠈⠀⠀⠸⣿⢸⣦⣼⣟⣿⠀⠀⠀⠀⠀⠈⣆⠀⠀⠤⢾⡡⠄⠀⠀⠀⣸⣿⣿⣾⡏⣼⠇⠀⠈⢽⣯⣿⢇⡟⠃⣿⡇
⠀⠀⠈⣿⡄⠀⠀⠀⣀⣀⡤⠤⠚⢿⡆⢿⡟⣿⡿⣶⣤⣤⠀⠀⠀⠘⠀⠀⠀⢺⠀⠀⠀⠠⣢⣿⣉⣿⣿⢡⡿⠓⠒⠤⢤⣄⣀⡀⠀⢀⣿
⠀⠀⠀⢻⣷⠖⠛⠋⠉⠄⠀⠀⠀⠘⢿⡜⢿⣿⡷⣾⣿⡇⠈⠀⢀⣀⠢⠁⠀⠁⠀⢀⣪⣾⣿⣏⣿⣿⢣⣿⠃⠀⣀⣤⣄⢀⠉⠉⠛⣿⡟
⠀⠀⠀⠈⣿⡄⢀⣀⣰⣞⣀⣸⣆⠀⠈⢿⣮⡻⣷⣿⣿⣿⣷⣾⣏⣿⣿⣖⣶⣶⠾⢿⣿⡟⣿⣿⠟⣱⡿⢁⡄⢠⢿⣤⣿⣀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠘⣿⡄⠙⠻⣏⠉⣿⠟⠻⠀⠀⡹⢷⣮⡻⢿⣿⣭⣿⣿⠛⣿⣿⣿⣿⣷⣾⣿⡿⢟⣡⡾⠯⡀⠸⠋⠙⢷⣄⠉⢻⡤⢰⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠈⠀⠉⠀⠀⡠⢋⣀⡀⠉⠻⢷⣮⣝⣛⡿⠿⠿⠷⠿⠿⣛⣫⣥⡶⠟⠉⠀⠀⠙⢦⡈⠀⠈⠙⠿⠟⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⢀⣴⠋⠀⠈⢉⣿⠀⠐⠀⠈⠉⡟⠛⠛⠿⠟⠛⠛⠛⡋⢡⣶⡄⠀⡀⠰⣄⠀⠙⢶⣄⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⡟⠁⠀⠁⣴⠏⣡⠆⢂⠀⠀⣸⢀⣴⡆⢢⣄⣀⣠⡔⣥⠉⢷⣹⡆⢳⣄⣸⣷⣄⡀⠙⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠻⢶⣯⣠⠾⠀⢠⡇⠨⢾⣷⠏⡩⣿⣿⠆⢸⡆⠈⠉⠿⣶⠿⠛⠁⢉⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⠉⠻⡇⠀⣼⠁⠀⠸⣿⣄⣠⠿⠀⠀⠀⢿⠀⠀⠀⠀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⣴⣧⣀⣀⡀⠈⠙⠋⠀⢀⣀⣀⣸⣧⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⣰⣆⠀⠀⣀⣠⣆⠀⠀⠀⠀⠀⠀⠙⣤⣈⣀⡴⠃⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣤⣧⣤⣠⡴⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠐⡦
⠀⠀⠀⠀⣹⣿⠀⠐⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣾
⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⢹⣧⠺⠛⠒⣾⣿⣿⡷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢼⣿⠀⡟⢸⡇⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠾⠿⠀⠀⠀⠼⠇⠀⠀⠀⠀⠀⠀⠀⠸⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠙⠠⠃⠘⠇⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣤⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠐⠟
⠀⠀⠀⠀⢼⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟
⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⡿⢿⡿⠃⠀⠀⠀⠀⠀⠈⣿⡏⠀⣿⢻⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠘⢻⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⢸⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡟⠀⠀⠀⠀⠀⠀⢰⣿⣧⢸⣿⣹⡇⣸⠻⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠹⠃⠿⠁⠉⠀⠀⠀⠀⠀⠀⠸⠿⠟⠸⠿⠾⠇⠻⠛⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣬⠀⣀⠉⠀⣘⢁⣤⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⡄⠐⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣁⣰⠆
⠀⠀⠀⠞⢹⣿⣶⣿⣧⣸⣿⣯⡄⠀⠀⠀⠀⠀⠈⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢾⣻⣿⡿⡴⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣷⡟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣏⢻⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠰⡆⠦⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⢠⡞⠁⣸⠏⣹⡿⠋⠉⠙⣆⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⣿
⠀⠀⠀⠋⠀⢾⠻⠟⠉⠧⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⢼⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠠⠄⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⡌⠙⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣶⡀⠀⢰⣶⠶⠖⠂⠀⠀⠀⠀⠈⢳⣶⡞⠁⠀⠀⠀⠀⠀⠠⠄⠠⣿⣿⣿⡇⠀⠀⠙
⠀⠀⠀⠀⠀⢻⣶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⣶⣄⡀⠹⣿⣿⠓⠀⢀⣾⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣿⡇⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⠀⠀⠀⠀⢺⣿⡟⠿⠶⡻⠟⠈⠾⢻⣿⣧⠁
⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡗⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢀⡿⠺⣇⠀⠀⠀⠀⠀⠘⠛⠛⣿⡿⡇⢹⢀⣿⠛⠛⠛
⠀⠀⠀⠀⠀⣸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⣿⡇⠀⠰⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣼⣷⡄⠀⠀⠀⠀⠀⠀⣿⡇⠃⠘⢻⡿⠄
⠀⠀⠀⠀⠀⠁⠀⠈
⣿⢿⠿⠿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣿⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⣤⣄⣄⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠛⠘⠁⠀⠰⠲⢢⣄⠀⠈⠋⠚⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⡋⠢⠀⠀⠀⠀⠀⠈⠸⠁⠀⡀⠀⢠⠔⢐⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠀⠀⠀⠑⠊⢄⢀⠨⠄⠀⠒⠒⠀⠤⠅⡀⠀⠀⠣⠀⠀⠀⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠗⠁⠀⠀⠀⠄⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠠⠀⠀⢀⢈⣻⡄
⠀⠀⠀⠀⠀⠀⠠⠏⣜⠢⠉⠁⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠄⢂⣡⡘⠸⡄
⠀⠀⠀⠀⠀⢀⡛⠀⠀⠈⠀⢂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⣙⡀
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⣺⠰⢆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢄⢠⠀⣗
⠀⠀⠀⠀⠀⢿⠀⠀⠑⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠈⠁⡯
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠄⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠈⣩⠀⠀⣄⡀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡡⠀⠀⠀⠀⣽⠁
⠀⠀⠀⠀⠀⠀⠘⣇⣔⡡⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠁⠯⣵⠚⣾⢏⠂
⠀⠀⠀⠀⠀⠀⠀⠘⣬⡀⠀⠀⠀⠂⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠐⠀⠀⠀⢘⣿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠀⠀⠠⣀⠀⠈⠉⢐⠒⠀⠤⠄⠀⠒⡂⠁⠀⢠⡀⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⠑⠋⠀⠀⠈⠀⠀⢂⡘⠀⠀⠀⠀⠀⣘⠈⡊⡔⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣄⢠⡀⠀⠀⠈⠁⠀⠀⠀⣄⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠙⠛⠛⠋⠈⠁⠁