
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
The denominator of a geometric progression \( b_{n} \) is \( q \), and for some natural \( n \geq 2 \),

$$
\log_{4} b_{2}+\log_{4} b_{3}+\ldots+\log_{4} b_{n}=4 \cdot \log_{4} b_{1}
$$

Find the smallest possible value of \( \log_{q} b_{1}^{2} \), given that it is an integer. For which \( n \) is this value achieved?

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