Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r{ }$th root of $N$. Find the value of \[f(2) + f(3) + f(4) + f(5)+ f(6).\]

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