
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
Given a rectangular playing field of size \( 13 \times 2 \) and an unlimited number of dominoes of sizes \( 2 \times 1 \) and \( 3 \times 1 \), the playing field is to be covered completely without gaps and without overlapping, and no domino should extend beyond the playing field. Additionally, all dominoes must be oriented the same way, i.e., their long sides must be parallel to each other.

How many such coverings are possible?

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