On a blackboard, a stranger writes the values of $s_7(n)^2$ for $n = 0, 1, \ldots, 7^{20} - 1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.

Let's think step by step and output the final answer within \boxed{}.
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