Let $S$ be the set of all $10-$term arithmetic progressions that include the numbers $4$ and $10.$ For example, $(-2,1, 4,7,10,13, 16,19,22,25)$ and $(10,8\tfrac{1}{2}, 7,5\tfrac{1}{2}, 4, 2\tfrac{1}{2},1,\tfrac{1}{2},-2, -3\tfrac{1}{2})$ are both members of $S.$ Find the sum of all values of $a_{10}$ for each $(a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}) \in S,$ that is, $\sum_{a_1, a_2, a_3, ... , a_{10} \in S}  a_{10}.$

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