
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 recursive function $S(n)$ such that $S(1) = 3$ and $S(n + 1) = 3^{S(n)}$ for $n \geq 1$. Let $C = (S(2023))^{S(2023)}$ and $D = (S(2023))^C$. Determine the largest integer $k$ for which \[\underbrace{\log_3\log_3\log_3\ldots\log_3D}_{k\text{ times}}\] is defined.
A) 2022
B) 2023
C) 2024
D) 2025

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