
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 \( f(x) \) be a function defined on \( \mathbf{R} \) such that \( f(0) = 0 \), \( f(x) + f(1-x) = 1 \), \( f\left(\frac{x}{3}\right) = \frac{1}{2} f(x) \), and for \( 0 \leqslant x_{1} < x_{2} \leqslant 1 \), we have \( f(x_{1}) \leqslant f(x_{2}) \). Find \( f\left(\frac{1}{2011}\right) \).

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