Let $p_1, p_2, p_3, p_4, p_5, p_6$ be distinct primes greater than $5$. Find the minimum possible value of 
\[
p_1 + p_2 + p_3 + p_4 + p_5 + p_6 - 6\min\left(p_1, p_2, p_3, p_4, p_5, p_6\right)
\]

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