Let $S$ be the set of integer triplets $(a, b, c)$ with $1 \le a \le b \le c$ that satisfy $a + b + c = 77$ and:
\[\frac{1}{a} +\frac{1}{b}+\frac{1}{c}= \frac{1}{5}.\]
What is the value of the sum $\sum_{a,b,c \in S} a\cdot b \cdot c$?

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