
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 $x_1,$ $x_2,$ $x_3,$ $\dots$ defined by
\begin{align*}
x_1 &= \sqrt[4]{4}, \\
x_2 &= (\sqrt[4]{4})^{\sqrt[4]{4}},
\end{align*}and in general,
\[x_n = (x_{n - 1})^{\sqrt[4]{4}}\]for $n > 1.$   What is the smallest value of $n$ for which $x_n$ is an integer?

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