
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
\[
y_1 = \sqrt[4]{4}, \quad y_2 = (\sqrt[4]{4})^{\sqrt[4]{4}},
\]
and in general,
\[
y_n = (y_{n - 1})^{\sqrt[4]{4}}
\]
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