
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 \( U_{1}, U_{2}, U_{3}, \ldots \) is defined as follows:
- \( U_{1} = 2 \);
- if \( U_{n} \) is prime then \( U_{n+1} \) is the smallest positive integer not yet in the sequence;
- if \( U_{n} \) is not prime then \( U_{n+1} \) is the smallest prime not yet in the sequence.
The integer \( k \) is the smallest such that \( U_{k+1} - U_{k} > 10 \).
What is the value of \( k \times U_{k} \)?

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