
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
\[ 512 + \frac{512}{4} + \frac{512}{16} + \dotsb, \]
and let $D_n$ be the sum of the first $n$ terms of the geometric series
\[ 3072 - \frac{3072}{3} + \frac{3072}{9} - \dotsb. \]
Calculate the smallest integer value of $n \ge 1$ for which $C_n = D_n$.

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