
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
Define a function \( f(x) \) for \( 0 \leq x \leq 1 \) with the following properties:

(i) \( f(0) = 0 \).
(ii) If \( 0 \leq x < y \leq 1 \), then \( f(x) \leq f(y) \).
(iii) \( f(1 - x) = \frac{3}{4} - \frac{f(x)}{2} \).
(iv) \( f \left( \frac{x}{3} \right) = \frac{f(x)}{3} \) for \( 0 \leq x \leq 1 \).

Determine the value of \( f \left( \frac{2}{9} \right) \).

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