There are some balls, each marked with a positive integer not exceeding $14$, and the sum of these numbers is $S$. Determine the greatest possible value of $S$ such that, irrespective of the specific integers on the balls, it is possible to divide the balls into two piles where the sum of the numbers on the balls in each pile does not exceed $129$.

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