How many 6-tuples \((a_1,a_2,a_3,a_4,a_5,a_6)\) are there such that each of \(a_1,a_2,a_3,a_4,a_5,a_6\) is from the set \(\{1,2,3,4\}\) and the six expressions  
\[
a_j^2 - a_j a_{j+1} + a_{j+1}^2
\]  
for \(j = 1,2,3,4,5,6\) (where \(a_7\) is to be taken as \(a_1\)) are all equal to one another?

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