
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In a  $2 \times 8$  squared board, you want to color each square red or blue in such a way that on each  $2 \times 2$  sub-board there are at least  $3$  boxes 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.

Present the answer in LaTex format: \boxed{Your answer}
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