
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
A sequence of twelve numbers comprised of 1s, 2s, or 3s is randomly generated. Each number is chosen independently at random from {1, 2, 3}. Determine the probability that the sequence does not contain two consecutive numbers that are the same or does not restart in the form 1 following 3 directly. If this probability can be expressed as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are relatively prime integers, find $p+q$.

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