
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
Define a new tower function $T'(n)$ such that $T'(1) = 3$ and $T'(n + 1) = 3^{T'(n)}$ for $n \ge 1$. Let $A' = (T'(2009))^{T'(2010)}$ and $B' = 3^{(T'(2009))^A'}$. Determine the largest integer $k$ for which \[\underbrace{\log_3\log_3\log_3\ldots\log_3B'}_{k\text{ times}}\] is defined.
A) 2008
B) 2009
C) 2010
D) 2011
E) 2012

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