Compute the number of permutations $(a,b,c,x,y,z)$ of $(1,2,3,4,5,6)$ which satisfy the five inequalities:
\[ a < b < c, \quad x < y < z, \quad a < x, \quad b < y, \quad \text{and} \quad c < z. \]

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