
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 $g(x),$ for $0 \leq x \leq 1,$ with the following properties:

(i) \( g(0) = 0 \) and \( g(1) = 1 \).
(ii) If \( 0 \leq x < y \leq 1, \) then \( g(x) \leq g(y) \) (non-decreasing function).
(iii) \( g(1 - x) = 1 - g(x) \) for all \( 0 \leq x \leq 1 \).
(iv) \( g\left(\frac{x}{4}\right) = \frac{g(x)}{3} \) for \( 0 \leq x \leq 1 \).

Find \( g\left(\frac{1}{2}\right) \).

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