
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 the following property of arithmetic sequences: If the sequence $\{a_n\}$ is an arithmetic sequence, and $b_n = \frac{a_1 + a_2 + ... + a_n}{n}$, then the sequence $\{b_n\}$ is also an arithmetic sequence. By analogy, if the sequence $\{c_n\}$ is a geometric sequence with $c_n > 0$, then the sequence $\{d_n\}$ is also a geometric sequence when $d_n = $\_\_\_\_\_\_\_\_\_\_.

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