
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
Let \(d\) and \(f\) be positive integers and \(a_{1} = 0.9\). If \(a_{i+1} = a_{i}^{2}\) and \(\prod_{i=1}^{4} a_{i} = \frac{3^{d}}{f}\), determine the smallest possible value of \(d\).

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