
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
\[ f(x) = \sum_{n=1}^{\infty} \frac{a_n}{10^{2n}} . \]

Determine \( f'\left(\frac{1}{3}\right) \).

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