
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 \(C_n\) be the sum of the first \(n\) terms of the geometric series
\[256 + \frac{256}{4} + \frac{256}{16} + \dotsb,\]
and let \(D_n\) be the sum of the first \(n\) terms of the geometric series
\[1024 - \frac{1024}{4} + \frac{1024}{16} - \dotsb.\]
Find the smallest value of \(n \ge 1\) for which \(C_n \neq D_n.\)

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