
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 a sequence $y_1, y_2, y_3, \dots$ defined by
\begin{align*}
y_1 &= \sqrt[3]{5}, \\
y_2 &= (\sqrt[3]{5})^{\sqrt[3]{5}},
\end{align*}
and in general,
\[y_n = (y_{n - 1})^{\sqrt[3]{5}}\]
for $n > 1.$ What is the smallest value of $n$ for which $y_n$ is an integer?

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