
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 modified tower function $U(n)$ as follows: $U(1) = 3$ and $U(n + 1) = 3^{U(n)}$ for $n \ge 1$. Consider $X = (U(5))^2$ and $Y = (U(5))^X$. Determine the largest integer $k$ for which \[\underbrace{\log_3\log_3\log_3\ldots\log_3 Y}_{k\text{ times}}\] is defined.
**A)** 5
**B)** 6
**C)** 7
**D)** 8
**E)** 9

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