The 3-digit number 376 in the decimal numbering system is an example of numbers with the special property that its square ends with the same digits: ${376}^2 = 141376$. Let’s call a number with this property a steady square.
Steady squares can also be observed in other numbering systems. In the base 14 numbering system, the 3-digit number $c37$ is also a steady square: $c37^2 = aa0c37$, and the sum of its digits is $c+3+7=18$ in the same numbering system. The letters $a$, $b$, $c$ and $d$ are used for the 10, 11, 12 and 13 digits respectively, in a manner similar to the hexadecimal numbering system.
For $1 ≤ n ≤ 9$, the sum of the digits of all the $n$-digit steady squares in the base 14 numbering system is $2d8$ (582 decimal). Steady squares with leading 0’s are not allowed.
Find the sum of the digits of all the $n$-digit steady squares in the base 14 numbering system for $1 ≤ n ≤ 10000$ (decimal) and give your answer as a string in the base 14 system using lower case letters where necessary.