13 LHS Students attend the LHS Math Team tryouts. The students are numbered $1, 2, \ldots, 13$. Their scores are $s_1, s_2, \ldots, s_{13}$, respectively. There are 5 problems on the tryout, each of which is given a weight, labeled $w_1, w_2, \ldots, w_5$. Each score $s_i$ is equal to the sum of the weights of all problems solved by student $i$. On the other hand, each weight $w_j$ is assigned to be $\frac{1}{\sum_{s_i}}$, where the sum is over all the scores of students who solved problem $j$. (If nobody solved a problem, the score doesn't matter). If the largest possible average score of the students can be expressed in the form $\frac{\sqrt{a}}{b}$, where $a$ is square-free, find $a+b$.

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