
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
Consider the sequence \( u_{n} \) defined as follows: \( u_{0} = 2010^{2010} \), and \( u_{n+1} = u_{n} + 7 \) if \( u_{n} \) is odd, and \( \frac{u_{n}}{2} \) if it is even. What is the smallest integer that this sequence will reach?

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