In a $2 \times 8$ squared board, you want to color each square red or blue such that on each $2 \times 2$ sub-board there are at least $3$ squares painted blue. In how many ways can this coloring be done?

**Note:** A $2 \times 2$ board is a square made up of $4$ squares that have a common vertex.

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