Robert has $4$ indistinguishable gold coins and $4$ indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the $8$ coins.

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