
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 \( f:[0,1) \rightarrow \mathbb{R} \) be a function that satisfies the following condition: if
\[ 
x=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{n}}=. a_{1} a_{2} a_{3} \ldots 
\]
is the decimal expansion of \( x \) and there does not exist a positive integer \( k \) such that \( a_{n}=9 \) for all \( n \geq k \), then
\[
f(x)=\sum_{n=1}^{\infty} \frac{a_{n}}{10^{2 n}} .
\]
Determine \( f^{\prime}\left(\frac{1}{3}\right) \).

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